Handbook of mathematical analysis in mechanics of viscous fluids 9783319133447, 3319133446, 9783319133430, 9783319133454

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Yoshikazu Giga Antonín Novotný Editors

Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Yoshikazu Giga • Antonín Novotný Editors

Handbook of Mathematical Analysis in Mechanics of Viscous Fluids With 62 Figures and 3 Tables

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Editors Yoshikazu Giga Graduate School of Mathematical Sciences University of Tokyo Meguro-ku, Tokyo Japan

Antonín Novotný Université de Toulon IMATH, Toulon France

ISBN 978-3-319-13343-0 ISBN 978-3-319-13344-7 (eBook) ISBN 978-3-319-13345-4 (print and electronic bundle) https://doi.org/10.1007/978-3-319-13344-7 Library of Congress Control Number: 2017957054 © Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer International Publishing AG part of Springer Nature. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

In memoriam Mariarosaria Padula, our respected colleague and friend

Preface

Fluid mechanics has a long history since Greek philosopher Archimedes discovered his famous law of forces acting on bodies in motionless fluid. A good understanding of its principles and its mathematical formulation and properties is crucial in many branches of contemporary science and technology. Not only its modern foundation relies on mathematics but also many branches of mathematics were developed or even emerged through the research in fluid mechanics or through the mathematical formulation of problems of fluid mechanics. The examples include the theory of functions of one complex variable, topology, dynamical systems, differential equations, differential geometry, probability theory, and functional analysis, to name only a few. Mathematics has been always playing a key role in the research on fluid mechanics. Many imminent problems in various branches of mathematics have their origin or can be interpreted as problems of fluid mechanics although in many cases the community of pure mathematicians fails to notice this fact. The purpose of this handbook is to provide a synthetic review of the state of the art in the theory of viscous fluids, present fundamental notions, formulate problems of fluid mechanics representing the development of the theory during last several decades, and show the methods and mathematical tools for their resolution. Since the field of mathematical fluid mechanics is huge, it is impossible to cover all topics. In this handbook, we focus on mathematical analysis in mechanics of viscous Newtonian fluids. The first part consisting of two chapters is devoted to derivation of basic equations by physical modeling. The second part is devoted to mathematical analysis of incompressible fluids, while the third part is dealing with the mathematical theory of viscous compressible fluids. There are many topics that are not covered by the handbook. In particular, this is the case of numerical analysis of the equations which would deserve by itself an independent volume. The handbook reviews important problems and notions that marked the development of the theory. It explains the methods and techniques that may be used for their resolution. We hope that it will be useful not only to mathematicians who work on the future development of the theory but also to physicists and engineers who need to know the tools of mathematical analysis for developing applications.

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Part 1: Derivation of Basic Equations There are several ways to derive the basic equations. One of the most typical ways is to start from the balance equations for mass, momentum, and total energy incorporating to the process the second law of thermodynamics. This approach is discussed in the first chapter by J. Málek and V. Pr˚uša. The approach based on variational principles is discussed in the contribution by M.-H. Giga, A. Kirshtein, and C. Liu. Part 2: Incompressible Fluids The Navier-Stokes system is a conventional macroscopic model consisting of partial differential equations describing the motion of viscous incompressible Newtonian fluids. The modern mathematical analysis of these equations goes back to seminal works of J. Leray in 1933 and 1934. Leray introduced the notion of weak solutions for the Navier-Stokes equations and developed several key tools of functional analysis for their mathematical treatment. In his above mentioned papers, Leray formulated many imminent open problems. One of them, on the regularity of weak solutions, was named among the seven Millennium Prize Problems of the Clay Institute of Mathematics. The fundamental question is whether, in three space dimensions, the global in time (weak) solutions of the Navier-Stokes system emanating from the smooth initial data must be smooth. This problem is related to the question whether or not the system of Navier-Stokes equations is still a good model for fluid flows in regimes with large Reynolds numbers. Despite a lot of effort of excellent mathematicians, this problem is still open. Many mathematical tools and specific techniques, e.g., the theory of partial differential equations, theory of interpolation, and maximal regularity theory, have been developed or refined within the process of its investigation. Many important open problems have been formulated during its investigation, and some of them solved. The accelerated development of technology at the end of the last century gave rise to many new mathematical problems in fluid mechanics. We can name as examples free boundary problems and problems related to complex fluids. In all these problems, the Navier-Stokes equations play an important or even a crucial role. We shall include in the handbook these modern topics, as well as recent development in the problems on inviscid limits. In this part, we intend to provide key notions and tools for mathematical understanding of equations of incompressible fluids. We mainly discuss existence and uniqueness problems as well as behavior of solutions and different notions of their stability. When the flow is slow, or more precisely when the Reynolds number is small, it is convenient to consider a linearized version of Navier-Stokes equations called Stokes equations. The investigation of the Stokes system is not only extremely important for itself, but it is fundamental also for the development of the nonlinear theory. Well-posedness and regularity questions for the various initial-boundary value problems for the Stokes equations in various types of sufficiently smooth domains are discussed in the first chapter by M. Hieber and J. Saal. Similar

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problems in domains with less regular boundaries are investigated by S. Monniaux and Z. Shen. There are several long-standing problems opened by Leray for the stationary Navier-Stokes equations related to general inflow-outflow boundary conditions in the case of bounded domains and to velocity profiles at infinity in the case of unbounded domains. The recent development in the former case is reported in the contribution by M. V. Korovkov, K. Pileckas, and R. Russo, while the contribution of T. Hishida deals with the latter case in 3-D exterior domains. The contribution of G. P. Galdi and J. Neustupa is devoted to the stationary flows around a rotating body. After the short excursion to the stationary Navier-Stokes equations, the handbook continues by several chapters dealing with weak and strong solutions to the nonsteady Navier-Stokes equations. The existence of a weak solution in a smooth general domain is discussed by R. Farwig, H. Kozono, and H. Sohr. Self-similar solutions are introduced and investigated by H. Jia, V. Šverák, and T.-P. Tsai. The problem of existence of time periodic solutions is addressed in the contribution by G. P. Galdi and M. Kyed. Large time behavior of solutions is discussed by L. Brandolese and M. E. Schonbek. Since the Navier-Stokes equations have a regularizing property, it is interesting to investigate the structure of sets of “rough” initial data still allowing smooth solutions globally in time. These questions are addressed in the contribution by I. Gallager. Stability of some special solutions, e.g., of the so-called Lamb-Oseen vortex, is investigated by T. Gallay and Y. Maekawa. Some important exact solutions are constructed in the contribution by H. Okamoto. Asymptotic behavior of solutions near the boundary endowed with no-slip conditions is investigated in the chapter by Y. Maekawa and A. Mazzucato. Regularity and regularity criteria for weak solutions are discussed by G. Seregin and V. Šverák and by H. Beirão da Veiga, Y. Giga, and Z. Grujic from several different viewpoints. Behavior of solution of an ideal flow is discussed by T. Y. Hou and P. Liu. The Navier-Stokes flow coupled with other effects is increasingly important. A class of models for geophysical flows is discussed by J. Li and E. S. Titi. Equations for polymetric materials are investigated by N. Masmoudi, while nematic liquid crystal flows are discussed by M. Hieber and J. Prüss. Some problems for viscoelastic fluids are discussed by X. Hu, F.-H. Lin, and C. Liu. A problem dealing with two-phase flows is a typical free boundary problem. There are several chapters devoted to this topic by J. Prüss and S. Shimizu; by V. A. Solonnikov and I. V. Denisova; by G. Simonett and M. Willa; and by H. Abels and H. Garke. The first three chapters handle smooth solutions, while the latter chapter handles weak solutions which allow topological change of region occupied by the fluid. Finally, the classical free boundary value problem of water wave is discussed by D. Córdova and C. Fefferman. Part 3: Compressible Fluids The mathematical models that take into account the compressibility of the fluid and thermodynamical effects lead to a rich variety of systems of partial differential

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equations. Their mathematical character depends on the physical assumptions on transport phenomena (describing the viscous effects and the heat transport) and on constitutive laws (usually prescribing pressure and internal energy as functions of density and temperature), not speaking about the situations when these equations are coupled with systems describing other phenomena (e.g., in magnetohydrodynamics, multifluid models, and theory of chemically reacting mixtures). In these circumstances, the classical formulation may give rise to different nonequivalent weak formulations, which are all physically reasonable. This abundance of possibilities imposes severe restrictions on the material that can be treated in one handbook volume. In particular, as in the “incompressible part” we have restricted our subject of interest only to the so-called Newtonian fluids (when, loosely speaking, the stress tensor depends linearly on the gradient of velocity), and the heat flux is proportional to the gradient of temperature through the Fourier law. The resulting system is called Navier-Stokes-Fourier system. This system describes the most simple thermodynamically consistent model of fluid dynamics. If the pressure depends on density only, one obtains a simpler model communally called compressible Navier-Stokes equations in barotropic regime. Although simpler than the complete Navier-Stokes-Fourier system, it inherits most of its mathematical difficulties. The present handbook is exclusively treating the two above mentioned systems. The mathematical difficulties encountered in mathematical investigation of compressible Navier-Stokes equations are many fold: they are related to (1) the enormous range of scales of motion described by the system, (2) the absence of mechanisms preventing density to create a vacuum and the absence of dissipation in the mass conservation, (3) the mixed parabolic-hyperbolic (or elliptic-hyperbolic in the steady case) character of the underlying linearized equations, and (4) the nonlinear character of equations (which still allows, through a priori estimates, to prevent the field quantities of concentrations but does not defend them from oscillations). The history of mathematical investigation of the compressible Navier-Stokes equations is more recent than the history of investigation of the incompressible Navier-Stokes equations. It starts with the works of D. Graffi and J. Nash on the local in time existence of strong solutions in the 1960s and continues with global in time existence results for data close to the equilibrium in works of A. Matsumura and T. Nishida in the 1980s. The first results on the existence of weak solutions similar to those introduced for incompressible fluids by Leray in 1934 were established by P.-L. Lions only in 1998. Likewise, the stability analysis introduced for the Navier-Stokes equations by Prodi and Serrin in the 1960s was waiting for its “compressible” counterpart till 2012. Being much younger than its “incompressible” counterpart, the mathematical theory of compressible fluids undergoes still a tumultuous development, and the synthetic level of contributions in “compressible” part is therefore necessarily objectively less high than in the “incompressible” part. The handbook chapters report the state of the art of some parts of the theory at the end of 2016. It covers mostly the questions of well-posedness of solutions to

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various boundary and initial-boundary value problems to these equations (meaning the existence of strong and weak solutions, uniqueness or conditional uniqueness, stability, longtime behavior, conditional regularity, and qualitative properties of solutions) in one and several space dimensions. We have made a preference to cover in the handbook solely these topics knowing well that this is a subjective and nonexhausting choice. Various types of weak and strong solutions are introduced and discussed in the contribution of E. Feireisl. Qualitative theory of strong solutions has longer history and its solid foundations are nowadays relatively well established. These questions are treated in three chapters by R. Danchin; J. Burczak, Y. Shibata, and W. M. Zaja¸czkowski; and M. Kotschote. Related questions of blowup criteria and existence of strong solutions for small data with large oscillations are developed in two contributions by Z. Huang and Z. Xin and by J. Li and Z. Xin. The longtime behavior of strong solutions is investigated in the contribution of Y. Shibata and Y. Enomoto, while an overview of some results for free boundary value problems is given in the contribution by I. Denisova and V. A. Solonnikov. The existence, stability, and asymptotic behavior of solutions to the equations in one dimension, or of spherically and axially symmetric solutions, are treated in three chapters by A. Zlotnik, by Y. Qin, and by S. Jiang and Q. Ju. The chapter by A. Matsumura is a comprehensive introduction to waves in 1-D compressible fluids. The theory of weak solutions has a short history and is still in agitated development. The existence of different types of weak solutions, stability, longtime behavior, and weak-strong uniqueness principle are discussed by A. Novotný and by H. Petzeltová. The existence problem for weak solutions with degenerate viscosity coefficients is presented by D. Bresch and B. Desjardins. The contribution by P. I. Plotnikov and W. Weigant is devoted to the existence results in the case of critical adiabatic coefficients. A particular role in the theory of weak solutions is played by solutions belonging to the so-called intermediate regularity class. These solutions are introduced in the contribution by M. Perepelitsa. The contribution of Y. Sun and Z. Zhang on the conditional regularity of weak solutions provides a link between the theory of weak solutions on the one hand and strong solutions/blowup criteria on the other hand. Three chapters are devoted to the stationary solutions, written by S. Jiang and C. Zhou, by P. Mucha, M. Pokorný, and E. Zatorska, and by P. Mucha, M. Pokorný, and O. Kreml. Compressible Navier-Stokes equations are applicable to modeling of a large variety of fluid motions ranging from small-scale motions (as acoustic waves) to large-scale motions of planetary size. The specific regimes of some of these flows are described sufficiently by simplified models characterized by extreme values of nondimensional numbers (as Mach, Reynolds, Péclet, Strouhal, and other numbers). Some of the simplified models can be obtained rigorously from the compressible Navier-Stokes equations as singular limits of nondimensional numbers. Typically, the incompressible Navier-Stokes system is known to be a low Mach number limit of the compressible Navier-Stokes equations. Three handbook chapters by N. Jiang and N. Masmoudi, by Feireisl, and by R. Klein are devoted to the investigation of the

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singular limits involving low Mach number and related questions. These chapters can be viewed as a bridge between the “compressible” and “incompressible” part of the handbook. Finally, we have involved three chapters giving examples of the investigation of compressible Navier-Stokes equations coupled with other equations. The contributions by X. Blanc and B. Ducoment, by V. Giovangigli, and by D. Bresch, B. Desjardins, J.-M. Ghidaglia, E. Grenier, and M. Hillairet deal with and discuss compressible magnetohydrodynamics, chemically reacting mixtures, and various multifluid models, respectively. The original idea of this handbook started in early 2012 in a discussion of Yoshikazu Giga with Professor Mariarosaria Padula. The proposal was finalized at the end of September 2012. Unfortunately, Professor Padula passed away prematurely at the end of September 2012 and the plan was almost abandoned until the second editor has joined the project. We rewrote the proposal in early 2013 and invited several distinguished authors to participate at this enterprise. Most of the invited contributors, specialists, and leading experts in the field accepted our invitation. In spite of their academic duties and responsibilities, they accepted to work hard on their chapters. Without their deep engagement and generous investment, the project could never see the light of the day. Since the field of mathematical fluid mechanics is huge even after the restriction to viscous fluids, there are several important fields which are not covered in its table of contents. In spite of this fact, the volume of the handbook is already considerable. Since this work is not an encyclopedia, we believe that such selection is allowed. We hope that readers will find this handbook comprehensive and that it will help them to learn about typical problems and approaches in mathematical analysis in fluid mechanics and encourage them to go further. We believe that its review character and extended bibliography will help to improve the orientation in this vast subject especially among young scientists. We hope that the overview material and mathematical tools gathered in the handbook chapters will encourage applications in other fields of mathematics and even beyond mathematics. March 2018

Yoshikazu Giga Antonín Novotný

Acknowledgments

The idea of editing Handbook of Mathematical Analysis in Mechanics of Viscous Fluids began with a brief conversation in 2011 at the occasion of the International Congress on Industrial and Applied Mathematics in Vancouver between Hans Koelsch, one of the editors of Springer, and Yoshikazu Giga. Yoshikazu Giga and Mariarosaria Padula build up the preliminary proposal which was available in the middle of September 2012. Unfortunately, Professor Padula passed away on September 29, 2012, and the project was interrupted until Antonín Novotný joined the editorial team around the end of January 2013. With several aids from Springer team led by Achi Dosanjh, executive editor, we finally reached a concrete and explicit plan around fall of 2013 and adjusted the proposal by suggestions of several potential authors. Namely, our numerous discussions with Giovanni Paolo Galdi and Eduard Feireisl contributed a lot to establish the first global structure of the handbook. We thank them for their serious interest in our project. After Springer sent an official invitation to each of the potential contributors around September 2014, the project was gradually growing by contributions of authors and reports of referees. Our thanks go particularly to the authors for their deep, generous, and unselfish investment into the uneasy task of writing their chapters and to the referees for careful reading and critical remarks allowing to improve the quality of chapters. Last but not least, we would like to thank the production team at Springer, led by Achi Dosanjh, for the help, efficiency, and advice. We would like to thank the editorial assistant Saskia Ellis who coordinated the day-to-day interactions between the editors, the authors, and the production departments. Without the high level of commitment and devotion on her part, we would neither have had the bandwidth nor the patience to complete this work. Coeditors Yoshikazu Giga Antonín Novotný

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Contents

Volume 1 Part I Derivation of Equations for Incompressible and Compressible Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Josef Málek and Vít Pr˚uša 2 Variational Modeling and Complex Fluids . . . . . . . . . . . . . . . . . . . . . Mi-Ho Giga, Arkadz Kirshtein, and Chun Liu Part II

Incompressible Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3 73

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3 The Stokes Equation in the L -Setting: Well-Posedness and Regularity Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matthias Hieber and Jürgen Saal

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4 Stokes Problems in Irregular Domains with Various Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sylvie Monniaux and Zhongwei Shen

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5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mikhail V. Korobkov, Konstantin Pileckas, and Remigio Russo 6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Toshiaki Hishida 7 Steady-State Navier-Stokes Flow Around a Moving Body . . . . . . . . Giovanni P. Galdi and Jiˇrí Neustupa 8 Stokes Semigroups, Strong, Weak, and Very Weak Solutions for General Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reinhard Farwig, Hideo Kozono, and Hermann Sohr

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299 341

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9 Self-Similar Solutions to the Nonstationary Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hao Jia, Vladimír Šverák, and Tai-Peng Tsai

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Time-Periodic Solutions to The Navier-Stokes Equations . . . . . . . . . Giovanni P. Galdi and Mads Kyed

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11

Large Time Behavior of the Navier-Stokes Flow . . . . . . . . . . . . . . . . Lorenzo Brandolese and Maria E. Schonbek

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12

Critical Function Spaces for the Well-Posedness of the Navier-Stokes Initial Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . Isabelle Gallagher

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Existence and Stability of Viscous Vortices . . . . . . . . . . . . . . . . . . . . . Thierry Gallay and Yasunori Maekawa

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Models and Special Solutions of the Navier-Stokes Equations . . . . . Hisashi Okamoto

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The Inviscid Limit and Boundary Layers for Navier-Stokes Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yasunori Maekawa and Anna Mazzucato

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Regularity Criteria for Navier-Stokes Solutions . . . . . . . . . . . . . . . . . Gregory Seregin and Vladimir Šverák

17

Stable Self-Similar Profiles for Two 1D Models of the 3D Axisymmetric Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thomas Yizhao Hou and Pengfei Liu

869

Vorticity Direction and Regularity of Solutions to the Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hugo Beirão da Veiga, Yoshikazu Giga, and Zoran Gruji´c

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Recent Advances Concerning Certain Class of Geophysical Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jinkai Li and Edriss S. Titi

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19

20

Equations for Polymeric Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nader Masmoudi

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Volume 2 21

Modeling of Two-Phase Flows With and Without Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007 Jan W. Prüss and Senjo Shimizu

22

Equations for Viscoelastic Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045 Xianpeng Hu, Fang-Hua Lin, and Chun Liu

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23

Modeling and Analysis of the Ericksen-Leslie Equations for Nematic Liquid Crystal Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075 Matthias Hieber and Jan W. Prüss

24

Classical Well-Posedness of Free Boundary Problems in Viscous Incompressible Fluid Mechanics . . . . . . . . . . . . . . . . . . . . . . . 1135 Vsevolod Alexeevich Solonnikov and Irina Vladimirovna Denisova

25

Stability of Equilibrium Shapes in Some Free Boundary Problems Involving Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1221 Gieri Simonett and Mathias Wilke

26

Weak Solutions and Diffuse Interface Models for Incompressible Two-Phase Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1267 Helmut Abels and Harald Garcke

27

Water Waves with or Without Surface Tension . . . . . . . . . . . . . . . . . 1329 Diego Córdoba and Charles Fefferman

Part III

Compressible Viscous Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1351

28

Concepts of Solutions in the Thermodynamics of Compressible Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1353 Eduard Feireisl

29

Weak Solutions for the Compressible Navier-Stokes Equations: Existence, Stability, and Longtime Behavior . . . . . . . . . . 1381 Antonín Novotný and Hana Petzeltová

30

Weak Solutions for the Compressible Navier-Stokes Equations with Density Dependent Viscosities . . . . . . . . . . . . . . . . . . 1547 Didier Bresch and Benoît Desjardins

31

Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in Critical Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1601 P. I. Plotnikov and W. Weigant

32

Weak Solutions for the Compressible Navier-Stokes Equations in the Intermediate Regularity Class . . . . . . . . . . . . . . . . . 1673 Misha Perepelitsa

33

Symmetric Solutions to the Viscous Gas Equations . . . . . . . . . . . . . . 1711 Song Jiang and Qiangchang Ju

34

Local and Global Solutions for the Compressible Navier-Stokes Equations Near Equilibria via the Energy Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1751 Jan Burczak, Yoshihiro Shibata, and Wojciech M. Zaja¸czkowski

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Fourier Analysis Methods for the Compressible Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1843 Raphaël Danchin

36

Local and Global Existence of Strong Solutions for the Compressible Navier-Stokes Equations Near Equilibria via the Maximal Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1905 Matthias Kotschote

37

Local and Global Solvability of Free Boundary Problems for the Compressible Navier-Stokes Equations Near Equilibria . . . 1947 Irina Vladimirovna Denisova and Vsevolod Alexeevich Solonnikov

Volume 3 38

Global Existence of Regular Solutions with Large Oscillations and Vacuum for Compressible Flows . . . . . . . . . . . . . . . 2037 Jing Li and Zhou Ping Xin

39

Global Existence of Classical Solutions and Optimal Decay Rate for Compressible Flows via the Theory of Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2085 Yoshihiro Shibata and Yuko Enomoto

40

Finite Time Blow-Up of Regular Solutions for Compressible Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2183 Xiangdi Huang and Zhou Ping Xin

41

Blow-Up Criteria of Strong Solutions and Conditional Regularity of Weak Solutions for the Compressible Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2263 Yongzhong Sun and Zhifei Zhang

42

Well-Posedness and Asymptotic Behavior for Compressible Flows in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . 2325 Yuming Qin

43

Well-Posedness of the IBVPs for the 1D Viscous Gas Equations . . . 2421 Alexander Zlotnik

44

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2495 Akitaka Matsumura

45

Existence of Stationary Weak Solutions for Isentropic and Isothermal Compressible Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2549 Song Jiang and Chunhui Zhou

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46

Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2595 Piotr Bogusław Mucha, Milan Pokorný, and Ewelina Zatorska

47

Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2663 Ondˇrej Kreml, Piotr Bogusław Mucha, and Milan Pokorný

48

Low Mach Number Limits and Acoustic Waves . . . . . . . . . . . . . . . . . 2721 Ning Jiang and Nader Masmoudi

49

Singular Limits for Models of Compressible, Viscous, Heat Conducting, and/or Rotating Fluids . . . . . . . . . . . . . . . . . . . . . . 2771 Eduard Feireisl

50

Scale Analysis of Compressible Flows from an Application Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2827 Rupert Klein

51

Weak and Strong Solutions of Equations of Compressible Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2869 Xavier Blanc and Bernard Ducomet

52

Multi-Fluid Models Including Compressible Fluids . . . . . . . . . . . . . 2927 Didier Bresch, Benoît Desjardins, Jean-Michel Ghidaglia, Emmanuel Grenier, and Matthieu Hillairet

53

Solutions for Models of Chemically Reacting Compressible Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2979 Vincent Giovangigli

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3031

About the Editors

Yoshikazu Giga is professor at the Graduate School of Mathematical Sciences of the University of Tokyo, Japan. He is a fellow of the American Mathematical Society as well as of the Japan Society for Industrial and Applied Mathematics. Through his more than 200 papers and 2 monographs, he has substantially contributed to the theory of parabolic partial differential equations including geometric evolution equations, semilinear heat equations, as well as the incompressible Navier-Stokes equations. He has received several prizes including the Medal of Honor with Purple Ribbon from the Government of Japan.

Antonín Novotný is professor at the Department of Mathematics of the University of Toulon and member of the Institute of Mathematics of the University of Toulon, France. Coauthor of more than 100 papers and 2 monographs, he is one of the leading experts in the theory of compressible Navier-Stokes equations.

xxi

Contributors

Helmut Abels Fakultät für Mathematik, Universität Regensburg, Regensburg, Germany Hugo Beirão da Veiga Dipartimento di Matematica, Università di Pisa, Pisa, Italy Xavier Blanc Univ. Paris Diderot, Sorbonne Paris Cité, Laboratoire Jacques-Louis Lions, UMR 7598, UPMC, CNRS, F-75205 Paris, France Lorenzo Brandolese Institut Camille Jordan, Université Lyon 1, Villeurbanne, France Didier Bresch LAMA UMR 5127 CNRS Batiment le Chablais, Université de Savoie Mont-Blanc, Le Bourget du Lac, France Jan Burczak Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland OxPDE, Mathematical Institute, University of Oxford, Oxford, UK Diego Córdoba Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, Madrid, Spain Raphaël Danchin Université Paris-Est Créteil, LAMA UMR CNRS 8050, France Irina Vladimirovna Denisova Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, St. Petersburg, Russia BenoOıt Desjardins Fondation Mathématique Jacques Hadamard, CMLA, ENS Cachan, CNRS and Modélisation Mesures et Applications S.A., Paris, France Bernard Ducomet Département de Physique Théorique et Appliquée, CEA/DAM Ile De France, Arpajon, France Yuko Enomoto Department of Mathematical Sciences, Shibaura Institute of Technology, Saitama, Japan

xxiii

xxiv

Contributors

Reinhard Farwig Department of Mathematics, Darmstadt University of Technology, Darmstadt, Germany International Research Training Group Darmstadt-Tokyo (IRTG 1529), Darmstadt, Germany Charles Fefferman Department of Mathematics, Princeton University, Princeton, NJ, USA Eduard Feireisl Evolution Differential Equations (EDE), Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague, Czech Republic Giovanni P. Galdi Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA, USA Isabelle Gallagher Department of Mathematics, Paris-Diderot University, Paris, France Thierry Gallay UMR 5582 – Mathematics Laboratory, Institut Fourier, Université Grenoble Alpes, Gières, France Harald Garcke Fakultät für Mathematik, Universität Regensburg, Regensburg, Germany Jean-Michel Ghidaglia Centre de Mathématiques et de Leurs Applications (CMLA), Ecole Normale Supérieure Paris-Saclay, Centre National de la Recherche Scientifique, Université Paris-Saclay, Cachan, France Mi-Ho Giga Graduate School of Mathematical Sciences, University of Tokyo, Meguro-ku, Tokyo, Japan Yoshikazu Giga Graduate School of Mathematical Sciences, University of Tokyo, Meguro-ku, Tokyo, Japan Vincent Giovangigli CMAP-CNRS, Ecole Polytechnique, Palaiseau Cedex, France Emmanuel Grenier Unité de Mathématiques Pures et Appliquées, ENS Lyon, Lyon Cedex, France Zoran Gruji´c Department of Mathematics, University of Virginia, Charlottesville, VA, USA Matthias Hieber Angewandte Analysis, Fachbereich Mathematik, Technische Universität Darmstadt, Darmstadt, Germany University of Pittsburgh, Benedum Engineering Hall, Pittsburgh, PA, USA Matthieu Hillairet Institut Montpelliérain Alexander Grothendiek, UMR 5149 CNRS, Université de Montpellier, Montpellier, France Toshiaki Hishida Graduate School of Mathematics, Nagoya University, Nagoya, Japan

Contributors

xxv

Thomas Yizhao Hou Division of Engineering and Applied Science, Computing and Mathematical Sciences Department, California Institute of Technology, Pasadena, CA, USA Xianpeng Hu Department of Mathematics, City University of Hong Kong, Hong Kong, China Xiangdi Huang Institute of Mathematics, AMSS, Chinese Academy of Sciences, Beijing, China Hao Jia Institute for Advanced Studies, Princeton and Department of Mathematics, University of Minnesota, Chicago, IL, USA Song Jiang Institute of Applied Physics and Computational Mathematics, Beijing, China Ning Jiang School of Mathematics and Statistics, Wuhan University, Wuhan, China Qiangchang Ju Institute of Applied Physics and Computational Mathematics, Beijing, China Arkadz Kirshtein Department of Mathematics, Pennsylvania State University, University Park, PA, USA Rupert Klein FB Mathematik and Informatik, Institut für Mathematik, Freie Universität Berlin, Berlin, Germany Mikhail V. Korobkov Sobolev Institute of Mathematics, Novosibirsk, Russia Novosibirsk State University, Novosibirsk, Russia Matthias Kotschote Department of Mathematics and Statistics, University of Konstanz, Konstanz, Germany Hideo Kozono Department of Mathematics, Waseda University, Tokyo, Japan Japanese-German Graduate Externship Program, Japan Society of Promotion of Science, Tokyo, Japan Ondˇrej Kreml Institute of Mathematics, Czech Academy of Sciences, Praha, Czech Republic Mads Kyed Fachbereich Mathematik, Technische Universität Darmstadt, Darmstadt, Germany Jinkai Li Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong, China Jing Li Institute of Applied Mathematics, AMSS and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, China

xxvi

Contributors

Fang-Hua Lin Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, New York, NY, USA Chun Liu Department of Mathematics, Pennsylvania State University, University Park, PA, USA Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL, USA Pengfei Liu Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA Yasunori Maekawa Department of Mathematics, Graduate School of Sciences, Kyoto University, Kyoto, Japan Josef Málek Faculty of Mathematics and Physics, Charles University in Prague, Praha 8 – Karlín, Czech Republic Nader Masmoudi Department of Mathematics, New York University in Abu Dhabi, Saadiyat Island, Abu Dhabi, United Arab Emirates Courant Institute of Mathematical Sciences, New York University, New York, NY, USA Akitaka Matsumura Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka, Japan Anna Mazzucato Mathematics Department, Eberly College of Science, The Pennsylvania State University, University Park, State College, PA, USA Sylvie Monniaux Aix-Marseille Université, CNRS, Centrale Marseille, Marseille, France Piotr Bogusław Mucha Institute of Applied Mathematics and Mechanics, University of Warsaw, Warsaw, Poland Jiˇrí Neustupa Institute of Mathematics, Czech Academy of Sciences, Praha 1, Czech Republic Antonín Novotný Université de Toulon, IMATH, Toulon, France Hisashi Okamoto Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan Department of Mathematics, Gakushuin University, Toshima-ku, Tokyo, Japan Misha Perepelitsa Department of Mathematics, University of Houston, Houston, TX, USA Hana Petzeltová Department EDE, Mathematical Institute of the Academy of Sciences of the Czech Republic, Praha 1, Czech Republic

Contributors

xxvii

Konstantin Pileckas Faculty of Mathematics and Informatics, Vilnius University, Vilnius, Lithuania P. I. Plotnikov Mathematical Department, Novosibirsk State University, Novosibirsk, Russia Siberian Division of Russian Academy of Sciences, Lavryentyev Institute of Hydrodynamics, Novosibirsk, Russia Milan Pokorný Faculty of Mathematics and Physics, Charles University, Praha, Czech Republic Vít Pruša ˚ Faculty of Mathematics and Physics, Charles University in Prague, Praha 8 – Karlín, Czech Republic Jan W. Prüss Institut für Mathematik, Martin-Luther-Universität, HalleWittenberg, Halle, Germany Yuming Qin Department of Applied Mathematics, College of Science, Donghua University, Shanghai, China Remigio Russo Department of Mathematics and Physics, Second University of Naples, Caserta, Italy Jürgen Saal Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf, Düsseldorf, Germany Maria E. Schonbek Department of Mathematics, University of California Santa Cruz, Santa Cruz, CA, USA Gregory Seregin Mathematical Institute, University of Oxford, Oxford, UK Zhongwei Shen University of Kentucky, Lexington, KY, USA Yoshihiro Shibata Department of Mathematics and Research Institute for Science and Engineering, Waseda University, Shinjuku-ku, Tokyo, Japan Senjo Shimizu Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, Japan Gieri Simonett Department of Mathematics, Vanderbilt University, Nashville, TN, USA Hermann Sohr Faculty of Electrical Engineering, Informatics and Mathematics, Department of Mathematics, University of Paderborn, Paderborn, Germany Vsevolod Alexeevich Solonnikov Laboratory of Mathematical Physics, St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian Academy of Sciences, St. Petersburg, Russia Yongzhong Sun Department of Mathematics, Nanjing University, Nanjing, China Vladimir Šverák School of Mathematics, University of Minnesota, Minneapolis, MN, USA

xxviii

Contributors

Edriss S. Titi Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel Department of Mathematics, Texas A&M University, College Station, TX, USA Department of Mathematics, University of California, Mathematics, Mechanical and Aerospace Engineering, Irvine, CA, USA Tai-Peng Tsai Department of Mathematics, University of British Columbia, Vancouver, BC, Canada W. Weigant Institute für Angewandte Mathematik, Universität Bonn, Bonn, Germany Mathias Wilke Fakultät für Mathematik, Universität Regensburg, Regensburg, Germany Zhou Ping Xin The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, Hong Kong, China Wojciech M. Zaja¸czkowski Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland Institute of Mathematics and Cryptology, Cybernetics Faculty, Military University of Technology, Warsaw, Poland Ewelina Zatorska Department of Mathematics, Imperial College London, London, UK Department of Mathematics, University College London, London, UK Zhifei Zhang School of Mathematical Sciences, Peking University, Beijing, China Chunhui Zhou Department of Mathematics, Southeast University, Nanjing, China Alexander Zlotnik Faculty of Economic Sciences, Department of Mathematics, National Research University Higher School of Economics, Moscow, Russia

Part I Derivation of Equations for Incompressible and Compressible Fluids

1

Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids Josef Málek and Vít Pr˚uša

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Balance Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Balance of Mass, Momentum, and Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . 2.2 Balance of Total Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Stress Power and Its Importance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Stability of the Rest State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 General Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Compressible and Incompressible Viscous Heat-Conducting Fluids . . . . . . . . . . . . . 4.3 Compressible Korteweg Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Compressible and Incompressible Viscoelastic Heat-Conducting Fluids . . . . . . . . . . 4.5 Beyond Linear Constitutive Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Boundary Conditions for Internal Flows of Incompressible Fluids . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 5 7 9 13 18 19 22 23 26 31 42 46 60 64 67 68 68

Abstract

The chapter starts with overview of the derivation of the balance equations for mass, momentum, angular momentum, and total energy, which is followed by a detailed discussion of the concept of entropy and entropy production. While the

Dedicated to professor K. R. Rajagopal on the occasion of his 65th birthday. J. Málek () • V. Pr˚uša Faculty of Mathematics and Physics, Charles University in Prague, Praha 8 – Karlín, Czech Republic e-mail: [email protected]; [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_1

3

4

J. Málek and V. Pr˚uša

balance laws are universal for any continuous medium, the particular behavior of the material of interest must be described by an extra set of material-specific equations. These equations relating, for example, the Cauchy stress tensor and the kinematical quantities are called the constitutive relations. The core part of the chapter is devoted to the presentation of a modern thermodynamically based phenomenological theory of constitutive relations. The key feature of the theory is that the constitutive relations stem from the choice of two scalar quantities, the internal energy and the entropy production. This is tantamount to the proposition that the material behavior is fully characterized by the way it stores the energy and produces the entropy. The general theory is documented by several examples of increasing complexity. It is shown how to derive the constitutive relations for compressible and incompressible viscous heat-conducting fluids (Navier-Stokes-Fourier fluid), Korteweg fluids, and compressible and incompressible heat-conducting viscoelastic fluids (Oldroyd-B and Maxwell fluid). Keywords

Continuum mechanics  Constitutive relations  Thermodynamics

2000 Mathematics Subject Classification. 76A02, 76A05, 74A15, 74A20

1

Introduction

Continuum mechanics and thermodynamics are based on the idea of continuously distributed matter and other physical quantities. Originally, continuum mechanics was equated with hydrodynamics, aerodynamics, and elasticity. The scope of the study was the motion of water and air and the deformation of some special solid substances. However, the concept of continuous medium has been shown to be useful and extremely viable even in the modelling of the behavior of much more complex systems such as polymeric solutions, granular materials, rock and land masses, special alloys, and many others. Moreover, the range of physical processes modelled in the continuum framework nowadays goes beyond purely mechanical processes. Processes such as phase transitions or growth and remodeling of biological tissues are routinely approached in the setting of continuum thermodynamics. Finally, the systems studied in continuum thermodynamics range from the very small ones studied in microfluidics (see, e.g., Squires and Quake [89]) up to the gigantic ones studied in planetary science; see, for example, Karato and Wu [48]. It seems that the laws governing the motion of a continuous medium must be extremely complicated in order to capture such a wide range of physical systems and phenomena of interest. This is only partially true. In principle, the laws

The authors were supported by the project LL1202 in the program ERC-CZ funded by the Ministry of Education, Youth and Sports of the Czech Republic.

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

5

governing the motion of a continuous medium can be seen as reformulations and generalizations/counterparts of the classical physical laws (the laws of Newtonian physics of point particles and the laws of classical thermodynamics). Surprisingly, the continuum counterparts of the classical laws are relatively easy to derive. These laws are – in the case of a single continuum medium – the balance equations for the mass, momentum, angular momentum, and energy and the evolution equation for the entropy. The balance equations are supposed to be universally valid for any continuous medium, and their derivation is briefly discussed in Sect. 2. The critical and the most difficult part in formulating the system of governing equations for a given material is the specification of the response of the material to the given stimuli. The sought stimulus-response relation can be, for example, a relation between the deformation and the stress or a relation between the heat flux and the temperature gradient. The task of finding a description of the material response is the task of finding the so-called constitutive relations. Apparently, the need to specify the constitutive relation for a given material calls for an investigation of the microscopic structure of the material. The reader who is interested in examples of the derivation of constitutive relations from microscopic theories is referred to Bird et al. [4] or Larson [52] to name a few. However, the investigation of the microscopic structure of the material is not the only option. The constitutive relations can be specified staying entirely at the phenomenological level. Here the phenomenological level means that it is possible to deal only with phenomena directly accessible to the experience and measurement without trying to interpret the phenomena in terms of ostensibly more fundamental (microscopic) physical theories. Indeed, the possible class of constitutive relations is in fact severely restricted by physical requirements stemming, for example, from the requirement on Galilean invariance of the governing equations, perceived symmetry of the material, or the second law of thermodynamics. As it is apparent from the discussion in Sect. 4, such restrictions allow one to successfully specify constitutive relations.

2

Balance Equations

Before discussing the theory of constitutive relations, it will be convenient to briefly recall the fundamental balance equations for a continuous medium. The reader who is not yet familiar with the field of continuum mechanics and thermodynamics is referred to Truesdell and Toupin [93], Müller [62], Truesdell and Rajagopal [92], or Gurtin et al. [35] for a detailed treatment of balance equations and kinematics of continuous medium. Note that the same formalism can be applied even for several interacting continuous media which constitute the continuum approach to the theory of mixtures; see, for example, Samohýl [84] and Rajagopal and Tao [81]. The continuous body B is assumed to be a part of Euclidean space R3 . The motion of the body is described by the function  that maps the positions X of points at time t0 to their respective positions x at a later time instant t, such that x D .X ; t /. Concerning a suitable framework for the description of processes in a continuous medium, one can, in principle, choose from two alternatives.

6

J. Málek and V. Pr˚uša

Either one expresses the quantities of interest as functions of time and the initial position X or as functions of time and the chosen position x in space R3 . The former alternative is called the Lagrangian description, while the latter is referred to as the Eulerian description. The Lagrangian description is especially suitable for the description of change of form or shape, since in order to describe the change, a reference point is needed. (Change with respect to some initial state.) As such, the Lagrangian description is a popular choice for studying the motion of solids. On the other hand, the Eulerian description is not primarily focused on change with respect to some initial state, but on the rate of change. Naturally, the rate of change (the time derivative) can be captured by referring to the state of the material at the current time and in its infinitesimally small-time neighborhood. In other words, if one is interested in the instantaneous velocity field, then there is no need to know complete trajectories of the individual points from the possibly distant initial state. As such, the Eulerian approach is useful for the description of the motion of fluids, where one is primarily interested in the velocity field, and the knowledge of the motion (individual trajectories) is of secondary importance. (Note, however, that the Eulerian description can be advantageous even in the description of solids, in particular in problems that involve phenomena like fluid–structure interaction or growth; see, e.g., Frei et al. [29, 30].) In what follows a theory for fluids is of primary interest; hence the Eulerian description is used. The balance laws are derived by applying the classical laws of Newtonian physics and classical thermodynamics to a volume of the moving material V .t / Ddef .V .t0 /; t /, where V .t0 / is an arbitrarily chosen part of the continuous body B. The main mathematical tool is the Reynolds transport theorem; see, for example, Truesdell and Toupin [93]. Theorem 1 (Reynolds transport theorem). Let .x; t / be a sufficiently smooth function describing the motion of the body B, and let V .t0 / be an arbitrary part of B at the inital time t0 . Let .x; t / be a sufficiently smooth scalar Eulerian field, and let V .t / D .V .t0 /; t / be the volume transported by the motion . Then d dt

Z

Z .x; t / dv D

V .t/

V .t/



 d.x; t / C .x; t / div v.x; t / dv; dt

where d.x; t / @.x; t / Ddef C v.x; t /  r.x; t / dt @t

(1)

denotes the material ˇ time derivative, and v.x; t / is the Eulerian velocity field, ;t/ ˇ . v.x; t / Ddef @.X ˇ @t 1 X D

.x;t/

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

2.1

7

Balance of Mass, Momentum, and Angular Momentum

2.1.1 Balance of Mass Classical Newtonian physics assumes conservation of mass m, which can be written as dm D 0. The balance of mass is a generalization of this equation to the continuum dt mechanics setting. The mass mV .t/ of volume R V .t / is expressed in terms of the Eulerian density field .x; t / as mV .t/ Ddef V .t/ .x; t / dv. Using the notion of the density field, the Reynolds transport theorem, and the requirement dmV .t/ D0 dt

(2)

then immediately yield the integral form of the balance of mass Z V .t/



 d.x; t / C .x; t / div v.x; t / dv D 0; dt

(3)

where V .t / is an arbitrary volume in the sense that it is the volume obtained by tracking an arbitrarily chosen initial volume V .t0 /. Since the volume in (3) is arbitrary, and the considered physical quantities are assumed to be sufficiently smooth, the integral form of the balance of mass leads to the pointwise evolution equation d.x; t / C .x; t / div v.x; t / D 0: dt

(4)

Having (4), it follows that Reynolds transport theorem for the quantity .x; t / Ddef .x; t /w.x; t /, where w is an arbitrary Eulerian field, in fact reads d dt

Z

Z .x; t /w.x; t / dv D V .t/

.x; t / V .t/

dw.x; t / dv: dt

(5)

This simple identity will be useful in the derivation of the remaining balance equations. Note that the integral form of the balance of mass and the other balance equations as well can be reformulated in a weak form without the need to use the pointwise equation (4) as an intermediate step. See Feireisl [27] or Bulíˇcek et al. [8] for details.

2.1.2 Balance of Momentum Balance of momentum is the counterpart of Newton second law dp D F for dt point particles, where p D mv is the momentum of particle with mass m moving with velocity v. In continuum setting the momentum pV .t/ of volume V .t / is expressed in terms of the Eulerian density and the velocity field as R pV .t/ Ddef V .t/ .x; t /v.x; t / dv. The balance of momentum for V .t / then reads

8

J. Málek and V. Pr˚uša

dpV .t/ dt

D F:

(6)

The force F on the right-hand side of the continuum counterpart of the Newton second law consists of two contributions, F D F volume C F contact :

(7)

The first physical mechanism that contributes to the force acting on the volume V .t / is the specific body force b. This is the force that acts on every part of V .t /, hence the specific body force contributes to the total force F via the volume integral Z F volume Ddef

.x; t /b.x; t / dv:

(8)

V .t/

A particular example of specific body force is the electrostatic force or the gravitational force. The second contribution to the force F acting on the volume V .t / is the force F contact due to the resistance of the material surrounding the volume V .t /. Since this force contribution arises due to wading of the volume V .t / through the surrounding material, it acts on the surface of the volume V .t/. Consequently, it is referred to as the contact or surface force. The contact force contribution is a new physical mechanism that goes beyond the concept of forces between point particles, and it is the key concept in mechanics of continuous media. The total contact force F contact acting on V .t/ is assumed to take the form Z F contact Ddef

t.x; t; n.x; t // ds;

(9)

@V .t/

where t is the contact force density and n denotes the unit outward normal to the surface of volume V .t/. It is worth emphasizing that the formula for the contact force is a fundamental and nontrivial assumption concerning the nature of the forces acting in a continuous medium. Assuming that the contact force is given in terms of the contact force density t.x; t; n.x; t //, one can proceed further and prove that the contact force density is in fact given by the formula t.x; t; n.x; t // D T.x; t /n.x; t /;

(10)

where T.x; t / is a tensorial quantity that is referred to as the Cauchy stress tensor. The corresponding theorem is referred to as the Cauchy stress theorem, and the interested reader will find the proof, for example, in the classical treatise by Truesdell and Toupin [93]. Note that the proof of the Cauchy stress theorem could be rather subtle if one wants to work with functions that lack smoothness; see, for example, Šilhavý [87] and references therein for the discussion of this issue.

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

9

Using (10) and the formulae for the body force and the contact force, the equality D F volume C F contact that holds for arbitrary volume V .t / can be rewritten in the form Z Z Z d .x; t /v.x; t / dv D .x; t /b.x; t / dv C T.x; t /n.x; t / ds; dt V .t/ V .t/ @V .t/ (11) dp V .t / dt

which upon using the Reynolds transport theorem, the Stokes theorem, and the identity (4) reduces to Z Z dv.x; t / Œdiv T.x; t / C .x; t /b.x; t / dv; dv D (12) .x; t / dt V .t/ V .t/ denotes the material time derivative of the Eulerian velocity field. (The where dv.x;t/ dt divergence of the tensor field is defined as the operator that satisfies .div A/  w D div.A> w/ for an arbitrary constant vector field w.) Since the considered physical quantities are assumed to be sufficiently smooth and the volume V .t / can be chosen arbitrarily, the integral form of the balance of momentum (12) reduces to the pointwise equality .x; t /

dv.x; t / D div T.x; t / C .x; t /b.x; t /: dt

(13)

2.1.3 Balance of Angular Momentum Balance of angular momentum is for a single point particle a simple consequence of Newton laws of motion, and it is in fact redundant. In the context of a continuous medium, the balance of angular momentum however provides a nontrivial piece of information concerning the structure of the Cauchy stress tensor. In the simplest setting it reduces to the requirement on the symmetry of the Cauchy stress tensor T.x; t / D T> .x; t /;

(14)

See, for example, Truesdell and Toupin [93] for details.

2.2

Balance of Total Energy

The need to work with thermal effects inevitably calls for reformulating the laws of classical thermodynamics in continuum setting. Classical thermodynamics is essentially a theory applicable to a volume of a material wherein the physical fields are homogeneous and undergo only infinitesimal (slow in time) changes. This is clearly insufficient for the description of the behavior of a moving continuous medium. In particular, the concepts of energy and entropy as introduced in classical thermodynamics (see, e.g., Callen [15]) need to be revisited.

10

J. Málek and V. Pr˚uša

a

b h b

b

b

d

b

c

c

c

d d

a

f

a

d d

d d

d d c

e e k

k

b a

b

g

a

c

Fig. 1 Joule experiment (Original figures from Joule [47] with edits). (a) Overall sketch of Joule experimental apparatus. (b) Horizontal and vertical cross section of the vessel with a paddle wheel

A good demonstration of the (in)applicability of classical thermodynamics and its continuum counterpart is the analysis of the famous Joule experiment; see Joule [46]. In the experiment concerning the mechanical equivalence of heat, Joule studied the rise of temperature due to the motion of a paddle wheel rotating in a vessel filled with a fluid. The motion of the paddle wheel was driven by the descent of weights connected via a system of pulleys to the paddle wheel axis; see Fig. 1. The potential energy of the weights is in this experiment transformed to the kinetic energy of the paddle wheel and due to the resistance of the fluid in the vessel also to the kinetic and thermal energy of the fluid. Classical thermodynamics is restricted to the description of the initial and final state where the weights are at rest. At the beginning, the temperature is constant all over the vessel. This is the initial state. As the weights descend, the fluid moves, and its temperature rises. Once the weights are stopped, the temperature reaches, after some time, homogeneous distribution in the vessel. This is the final state. In the classical setting, it is impossible to say anything about the intermediate states since the descent of the weight induces substantial motion in the fluid and inhomogeneous distribution of the temperature in the vessel. (Recall that the energy, entropy, and temperature are in the classical setting defined only for the whole vessel.) On the other hand, the ambition of continuum thermodynamics is to describe the whole process and the time evolution of the spatial distribution of the physical quantities of interest. In order to describe the spatial distribution of the physical quantities of interest, one obviously needs to talk about specific internal energy e.x; t / and the specific entropy .x; t /. The internal energy or the entropy of the volume V .t / of the material is then obtained by volume integration of the corresponding densities. Besides the introduction of the specific internal energy e.x; t / and the specific entropy .x; t /, it is necessary to identify the energy exchange mechanisms in the continuous medium. Naturally, one part of the energy exchange is due to processes of mechanical origin.

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

11

In order to identify the energy exchange due to mechanical processes, it suffices to recall the same concept in classical Newtonian physics. The energy balance for a single particle of constant mass m is in the classical setting obtained via the multiplication of the Newton’s second law dtd .mv/ D F by the particle velocity v,   which yields dtd 12 mjvj2 D F  v. Repeating the same steps in the setting of continuum mechanics – see the pointwise equality (13) – yields Z .x; t / V .t/

dv.x; t /  v.x; t / dv D dt

Z Œdiv T.x; t / C .x; t /b.x; t /  v.x; t / dv; V .t/

(15) where v.x; t / is the Eulerian velocity field. Further, one needs to introduce a quantity describing the nonmechanical part of the energy exchange between the given volume of the material and its surroundings. It is assumed that this part of the energy exchange can be described using the energy flux j e and that the energy exchange through the volume boundary is then given by the means of a surface integral Z  @V .t/

j e .x; t /  n.x; t / ds;

(16)

where n denotes the unit outward normal to the surface of the volume. (The minus sign is due to the standard sign convention. If the energy flux vector j e points into the volume V .t / – that is, in the opposite direction to the unit outward normal – then the energy is transferred from the surrounding into the volume, and the surface integral is positive.) The energy flux j e is, in the simplest setting, tantamount to the heat flux j q .x; t /. Consequently, for the sake of clarity and specificity of the presentation, it is henceforward assumed that j e  j q . Using the heat flux j q , the nonmechanical energy exchange – the transferred heat – between the volume V .t / and its surrounding is given by the surface integral Z  @V .t/

j q .x; t /  n.x; t / ds:

(17)

Alternatively, one can also introduce volumetric heat source contribution to the energy exchange, Z .x; t /q.x; t / dv;

(18)

V .t/

but the volumetric contribution shall not be considered here for the sake of simplicity of presentation. (Similarly, volumetric contribution is not considered in the discussion on the concept of entropy and entropy production.)

12

J. Málek and V. Pr˚uša

Now one is ready to formulate the total energy balance for the volume V .t /. The total energy of the continuous medium in the volume V .t / is assumed to be given R by the volume integral V .t/ .x; t /etot .x; t / dV of the specific total energy etot . Further, the specific total energy is assumed to be given as the sum of the specific kinetic energy of the macroscopic motion 12 jvj2 and the specific internal energy e, 1 .x; t /etot .x; t / Ddef .x; t /e.x; t / C .x; t / jv.x; t /j2 : 2

(19)

(In the simplest setting, the internal energy represents the thermal energy, i.e., the energy of the microscopic motion; see Clausius [17].) The integral form of the balance law then reads Z Z d .x; t /etot .x; t / dv D .x; t /b.x; t /  v.x; t / dv dt V .t/ V .t/ Z ŒT.x; t /n.x; t /  v.x; t / ds C Z

@V .t/

 @V .t/

j q .x; t /  n.x; t / ds;

(20)

where the right-hand side contains all possible contributions to the energy exchange, namely, the mechanical ones (see the right-hand side of (15)) and the nonmechanical ones (see (17)). If the physical quantities of interest are sufficiently smooth, then (20) can be in virtue of the Reynolds theorem, the Stokes theorem, and the symmetry of the Cauchy stress tensor (14) reduced to the pointwise equation .x; t /

detot .x; t / D .x; t /b.x; t /  v.x; t / C div ŒT.x; t /v.x; t /  div j q .x; t /: dt (21)

Equation (21) can be further manipulated in order to get an evolution equation for the specific internal energy e only. Multiplying the balance of momentum (13) by v and subtracting the arising equation from the balance of total energy (21) yield, in virtue of the identity div.A> a/ D .div A/  a C A W ra, the equation .x; t /

de.x; t / D T.x; t / W D.x; t /  div j q .x; t /; dt

(22)

where the symmetry of the Cauchy stress tensor has been used.  The symbol  D stands for the symmetric part of the velocity gradient, D Ddef 21 rv C rv> , and   A W B Ddef Tr AB> . The term T.x; t / W D.x; t / in (22) is called the stress power and plays a fundamental role in thermodynamics of continuous medium. A few hints on its relevance are given in Sect. 3. Later, in the discussion concerning constitutive

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

13

relations for viscous fluids, it is shown that the stress power is one of the terms in the entropy production. Note that although equations (22) and (21) are, in virtue of the balance of momentum (13), equivalent for sufficiently smooth functions, they are different from the perspective of a weak solution; see Feireisl and Málek [28], Bulíˇcek et al. [8], and Bulíˇcek et al. [9] for details.

2.3

Entropy

Concerning the concept of entropy in the setting of continuum mechanics, two assumptions must be made. First, it is assumed, following the classical setting, that an energetic equation of state holds even for the specific internal energy and the specific entropy. Note that the particular form of the energetic equation of state depends on the given material; hence, the energetic equation of state is in fact a constitutive relation. In the simplest setting, the energetic equation of state reads

e.x; t / D e..x; t/; .x; t//;

(23)

which is clearly a straightforward generalization of the classical relation E D E.S; V /; see, for example, Callen [15]. A general energetic equation of state can however take a more complex form

e.x; t / D e..x; t /; y1 .x; t /; : : : ; ym .x; t //;

(24)

where y1 ; : : : ; ym , m 2 N, m  1 are other state variables. If the energetic equation of state is obtained as a direct analogue of a classical equilibrium energetic equation of state, then it is said that the system under consideration satisfies the assumption of local equilibrium and that the system is studied in the framework of classical irreversible thermodynamics. However, if one wants to describe the phenomena that go beyond the classical setting (see, e.g., Sect. 4), then the energetic equation of state must be more complex. In particular, the energetic equation of state can contain the spatial gradients as variables. In such a case, it is said that the system is studied in the framework of extended irreversible thermodynamics. Second, the energetic equation of state is assumed to hold at every time instant in the given class of processes of interest. This is again a departure from the classical setting, where the relations of type E D E.S; V / hold only in equilibrium or in quasistatic (infinitesimally slow) processes. In particular, in the current setting, it is

14

J. Málek and V. Pr˚uša

assumed that one can take the time derivative of the equation of state. For example, if the equation of state is (23), then ˇ ˇ @e.; / ˇˇ @e.; / ˇˇ de.x; t / d.x; t / d.x; t / D C dt @ ˇD.x;t/; D.x;t/ dt @ ˇD.x;t/; D.x;t/ dt (25) is assumed to be valid in the given class of processes of interest, no matter how rapid the time changes are or how large the spatial inhomogenities are.

2.3.1 Thermodynamic Temperature Besides the energetic equation of the state, one can, following classical equilibrium thermodynamics, specify the entropy as a function of the internal energy e and the other state variables y1 ; : : : ; ym , m 2 N, m  1, which leads to the entropic equation of state .x; t / D .e.x; t /; y1 .x; t /; : : : ; ym .x; t //:

(26)

The specific entropy  is assumed to be a differentiable function, and further it is assumed that ˇ ˇ @ .e; y1 ; : : : ; ym /ˇˇ >0 (27) @e eDe.x;t/;y1 Dy1 .x;t/;:::;ym Dym .x;t/ holds for any fixed m-tuple y1 ; : : : ; ym and point .x; t /. This allows one to invert (26) and get the energetic equation of state e.x; t / D e..x; t /; y1 .x; t /; : : : ; ym .x; t //:

(28)

Further, following classical equilibrium thermodynamics, the thermodynamic temperature  is defined as .x; t / Ddef

ˇ ˇ @e .; y1 ; : : : ; ym /ˇˇ : @ D.x;t/;y1 Dy1 .x;t/;:::;ym Dym .x;t/

(29)

2.3.2 Clausius-Duhem Inequality The last ingredient one needs in the development of thermodynamics of continuous medium is a counterpart of the classical Clausius inequality dS 

dQ ¯ ; 

(30)

where  is the thermodynamic temperature; see Clausius [18]. Recall that in the ideal situation of reversible processes, the inequality reduces to the equality

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

dS D

dQ ¯ : 

15

(31)

The generalization of the Clausius inequality is the Clausius-Duhem inequality d dt

Z

Z

j q .x; t /

.x; t /.x; t / dv   V .t/

@V .t/

.x; t /

Z  n ds C

.x; t / V .t/

q.x; t / dv; .x; t / (32)

which can be obtained from (30) by appealing to the same arguments as in the discussion of the concept of the energy. (Recall that the transferred heat is expressed in terms of the surface integral (17).) The last term in (32) accounts for entropy changes due to volumetric heat sources. As before (see the discussion in Sect. 2.2), the volumetric term shall not be henceforward considered for the sake of simplicity of presentation. In what follows, the Clausius-Duhem inequality is therefore considered in the form d dt

Z

Z

j q .x; t /

.x; t /.x; t / dv   V .t/

@V .t/

.x; t /

 n ds;

(33)

where V .t / is again an arbitrary volume in the sense that it is the volume obtained by tracking an arbitrarily chosen initial volume V .t0 /. Since the classical Clausius (in)equality (30) is the mathematical formalization of the second law of thermodynamics, the Clausius-Duhem inequality (33) can be understood as the reinterpretation of the second law of thermodynamics in the setting of continuum thermodynamics. As such the Clausius-Duhem inequality can be expected to play a crucial role in the theory which is indeed the case. It is convenient, for the sake of later reference, to rewrite the inequality in a slightly different form. Using the Stokes theorem, the Reynolds transport theorem, and the identity (5), the inequality (33) can be rewritten in the form   j q .x; t / d dv  0: .x; t / .x; t / C div dt .x; t /



Z V .t/

(34)

The first term in (34) is the change of the net entropy,  Z dSV .t/ d .x; t /.x; t / dv ; Ddef dt dt V .t/

(35)

while the other term is the entropy exchange – the entropy flux – with the surrounding of volume V .t/, Z J@V .t/ Ddef @V .t/

j q .x; t / .x; t /

 n ds:

(36)

16

J. Málek and V. Pr˚uša

The general statement of a balance law in continuum mechanics and thermodynamics is that the time change of the given quantity in the volume V .t / plus the flux J@V .t/ of the quantity through the boundary of V .t/ is equal to the production of the given quantity in the volume V .t/, dSV .t/ C J@V .t/ D „V .t/ : dt

(37)

Using this nomenclature, it follows that the left-hand side of (34) deserves to be denoted as the net entropy production in the volume V .t /. Further, it is assumed that the net entropy production can be obtained via a volume integral of the corresponding spatially distributed quantity .x; t /, Z „V .t/ Ddef

.x; t / dv:

(38)

V .t/

In such a case, the Clausius-Duhem inequality can be rewritten as „V .t/ D

dSV .t/ C J@V .t/  0: dt

(39)

Now it is clear that the Clausius-Duhem inequality in fact states that the net entropy production in the volume V .t/ is nonnegative, „V .t/  0:

(40)

Note that if the heat flux j q vanishes on the boundary of V .t /, then the flux term in (39) vanishes, and it follows that the net entropy of the volume V .t / increases in time. This is the classical statement concerning the behavior of isolated systems. Finally, using the concept of entropy production density (38), it follows that the localized version of (34) reads .x; t / D .x; t /

  j q .x; t / d  0: .x; t / C div dt .x; t /

(41)

The fact that (33) is indeed a generalization of (30) is best seen in the case of a vessel filled with a fluid with “almost” constant uniform temperature distribution .x; t / Ddef  that is subject to heat exchange with its surrounding. In such a case, time integration of (33) from t1 to t2 yields Z

Z

1 .x; t2 /.x; t2 / dv .x; t1 /.x; t1 / dv  V .t2 / V .t1 /

Z t2Z t1

@V .t/

 j q .x; t /  nds dt: (42)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

17

The left-hand side is the difference between the net entropy of the fluid in the vessel at times t2 and t1 , and the right-hand side is the heat exchanged with the surrounding in the time interval Œt1 ; t2 ; hence, one gets S .t2 /  S .t1 / 

Q ; 

(43)

which is tantamount to (30).

2.3.3 Entropy Production In the setting of continuum mechanics, it is assumed that the energetic equation of state is known, which means that a relation of the type (25) holds for the time derivative of the entropy. The knowledge of the energetic equation of state for the given material allows one to identify the entropy production mechanisms in the material, that is, the quantity .x; t / in (41). In other words, one can rewrite the lefthand side of (41) in terms of more convenient quantities than the entropy. Indeed, inspecting carefully (25) it is easy to see that (25) is in fact a formula for the material time derivative of the specific entropy d.x;t/ , dt  d.x; t / 1 de.x; t / pth .x; t / d.x; t / D  ; dt .x; t / dt Œ.x; t /2 dt

(44)

where the thermodynamic temperature .x; t / and the thermodynamic pressure pth .x; t / have been defined by formulae .x; t / Ddef pth .x; t / Ddef

ˇ @e.; / ˇˇ ; @ ˇD.x;t/;D.x;t/ ˇ ˇ 2 @e.; / ˇ Œ.x; t / : @ ˇD.x;t/;D.x;t/

(45a) (45b)

(These relations are counterparts of the well-known classical relations; see, e.g., Callen [15].) The time derivatives de.x;t/ and d.x;t/ are known from the balance laws (22) dt dt and (4), and using the balance laws in (44) yields  d.x; t / 1

D T.x; t / W D.x; t /  div j q .x; t / C pth .x; t / div v.x; t / : dt .x; t / (46) Finally, using (46) in (41) leads one, after some manipulation, to .x; t /

1 1 ŒT.x; t / W D.x; t / C pth .x; t / div v.x; t /  j q .x; t /  r.x; t / .x; t / Œ.x; t /2   j q .x; t / d  0: (47) D .x; t / .x; t / C div dt .x; t /

18

J. Málek and V. Pr˚uša

The expression on the left-hand side of (47), that is, .x; t / Ddef 

1 ŒT.x; t / W D.x; t / C pth .x; t / div v.x; t / .x; t / 1 Œ.x; t /2

j q .x; t /  r.x; t /

(48)

determines the entropy production in the given material; see the local version of the Clausius-Duhem inequality (41). In other words, in the continuum setting, one is able to explicitly evaluate, for the given material, the “uncompensated transformation” N Ddef dS 

dQ ¯ 

(49)

introduced by Clausius [18]. The Clausius-Duhem inequality (41) requires .x; t / to be nonnegative; hence (47) can be expected to impose some restrictions on the form of the constitutive relations. This is indeed the case, and this observation is the key concept exploited in the theory of constitutive relations discussed in Sect. 4. Furthermore, relations of the type (47) provide one an explicit criterion for the validity of the Clausius equality (31). In particular, a process in a material with energetic equation of state of the type (23) is reversible, that is, (31) holds, if the entropy production (48) vanishes. From (48) it follows that the entropy production vanishes if there are no temperature and velocity gradients in the material. This means that the process in the material is very close to an ideal reversible process if the gradients r and D are during the whole process kept extremely small. The other possibility is that the entropy production identically vanishes because of the particular choice of the constitutive relation for T and j q ; see Sect. 4.2.2 for details.

3

Stress Power and Its Importance

The second law of thermodynamics requires the entropy production to be a nonnegative quantity. However, such statement concerning the entropy production does not seem to be very intuitive. In what follows it is shown that the nonnegativity of the entropy production is in fact closely related to some fundamental direct observations concerning the qualitative behavior of real materials. Let us, for example, consider the entropy production for an incompressible homogeneous viscous heat nonconducting fluid. The incompressibility means that div v D 0 is required to hold in the class of considered processes, while the fact that a heat nonconducting fluid is considered translates into the requirement j q D 0. In such a case, the entropy production (48) reduces to

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

.x; t / D

1 T.x; t / W D.x; t /: .x; t /

19

(50)

(Details concerning the derivation of the entropy production in an incompressible fluid can be found in Sect. 4. Note that for an incompressible fluid, one gets T W D D Tı W Dı ; see the discussion in Sect. 4.2.3, in particular formula (119).) Since the thermodynamic temperature in (50) is a positive quantity, it follows that the requirement on the nonnegativity of the entropy production simplifies to T W D  0:

(51)

This means that the second law is satisfied provided that the stress power is a pointwise nonnegative quantity. The importance of the requirement concerning the nonnegativity of the stress power term T W D – and consequently the nonnegativity of the entropy production – is documented below by means of two simple mechanical examples. These two examples show that the seemingly esoteric requirement on the nonnegativity of entropy production leads to very natural consequences. First, it is shown that the pointwise positivity of the stress power is sufficient for establishing the fact that the drag force acting on a rigid body moving through an incompressible fluid decelerates the moving body; see Sect. 3.1. Second, it is shown that the pointwise positivity of the stress power is sufficient to establish the stability of the rest state of an incompressible fluid; see Sect. 3.2.

3.1

Drag

Let us first consider the setting shown in Fig. 2. A rigid body B moves with constant velocity U in an infinite domain, and it is assumed that no body force is acting on the surrounding fluid. Further, it is assumed that the velocity field in the fluid that is generated by the motion of the body decays sufficiently fast at infinity and that the fluid adheres to the surface of the body, that is, vj@B D U . The drag force is the projection of the force acting on the body to the direction parallel to the velocity U , Fig. 2 Body moving in a fluid

Flift

U B

Fdrag

20

J. Málek and V. Pr˚uša

U ˝U

F drag Ddef

Z

 Tn ds :

jU j2

(52)

@B

The key observation is that if the fluid is assumed to be a homogeneous incompressible fluid with symmetric Cauchy stress tensor, then the formula for the drag force (52) can be rewritten in the form Z F drag D 

 U T W D dv : jU j2 exterior of B

(53)

The derivation of (53) proceeds as follows. The balance of momentum (13) and the balance of mass (5) for the motion of the fluid outside the body B are 

dv D div T; dt

(54a)

div v D 0;

(54b)

and the boundary conditions read vj@B D U ;

(54c)

v.x; t / ! 0 as jxj ! C1:

(54d)

Note that the velocity field is considered to be a steady velocity field; hence, the D Œrv v. material time derivative is dv dt ˚ Let D x 2 R3 ; jxj < R n B denote the ball centered at origin with radius R (large number) with excluded body B and its boundary @B. Governing equations (54) hold in this domain. Now one can multiply (54a) by v and integrate over , Z

Z .div T/  v dv:

.Œrv v/  v dv D

(55)



Application of standard identities yields Z

  1 v  rjvj2 dv D 2

Z

  div T> v dv 

Z T W D dv:

(56)



The first term on the right-hand side can be in virtue of the Stokes theorem rewritten as a sum of integrals over the boundary of the ball and the boundary of the body

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

Z

  div T> v dv D

Z

 >  T v  n ds 

fx2R3 ; jxjDRg Z



D

Z

21



 T> v  n ds

@B

 >  T v  n ds  U 

Z

fx2R3 ; jxjDRg

Tn ds;

(57)

@B

where the boundary condition (54c) has been used. (Note that the last integral is taken over the surface of the body; hence, one needs to add the minus sign in order to compensate the opposite orientation of the surface of the body and the surface of .) Concerning the left-hand side of (56), integration by parts implies Z

  1 v  rjvj2 dv D 2

Z @

Z

1 jvj2 v  n ds  2

D f

x2R3 ;

Z

1 jvj2 div v dv 2

1 jvj2 v  n ds; 2 jxjDRg

(58)

where the boundary condition (54c) and the incompressibility, div v D 0, have been used. Summing up all partial results, it follows that (56) reduces to Z

Z  >  1 T v  n ds jvj2 v  n ds D fx2R3 ; jxjDRg 2 fx2R3 ; jxjDRg Z Z U  Tn ds  T W D dv: @B

(59)



Now one can take the limit R ! C1 and use the fact that the velocity vanishes sufficiently fast for R ! C1. This means that the surface integrals in (59) vanish for R ! C1 and that (59) reduces to Z

Z Tn ds D 

U @B

T W D dv;

(60)



which yields the proposition. The implication of formula (53) is, that if the stress power T W D is pointwise nonnegative, then the drag direction is opposite to the direction of the motion. (The drag force is acting against the motion of the body.) This is definitely a desirable outcome from the point of everyday experience, and, as it has been shown, this outcome is closely related to the second law of thermodynamics.

22

3.2

J. Málek and V. Pr˚uša

Stability of the Rest State

Let denote a vessel occupied at time t0 by a homogeneous incompressible viscous fluid, and let no external body force act on the fluid. The Cauchy stress tensor is again assumed to be symmetric. Further, it is assumed that the fluid adheres to the surface of the vessel, vj@ D 0:

(61)

The everyday experience is that whatever has been the initial state of the fluid, the fluid comes, after some time, to the state of rest, that is, v D 0 in . The question is whether it is possible to recover this fact for a general homogeneous incompressible viscous fluid. A good measure of the deviation from the rest state is the net kinetic energy of the fluid Z 1 Ekin Ddef .x; t /jv.x; t /j2 dv: (62) 2 In order to assess the stability of the rest state, one needs to find an evolution equation for a measure of the deviation from the rest state. This is an easy task if the measure of the deviation from the rest state is the net kinetic energy introduced in (62). The evolution equations for the motion of the fluid are 

dv D div T; dt

(63a)

div v D 0;

(63b)

and the multiplication of (63a) with v followed by integration over the domain yields Z

d dt



1  vv 2



Z .div T/  v dv:

dv D

(64)



(See identity (5).) The left-hand side is equal to the time derivative of the net kinetic energy (62). Concerning the right-hand side, one can use the identity div.A> a/ D .div A/  aCA W ra, the Stokes theorem, and the fact that v vanishes on the boundary to get Z

Z .div T/  v dv D 

T W D dv:

This implies that the evolution equation for the net kinetic energy reads

(65)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

dEkin D dt

23

Z T W D dv:

(66)



It follows that if the stress power T W D is pointwise positive, then the righthand side of (66) is negative, and the net kinetic energy decays in time. If one was interested in the rate of decay, and in the proof that the net kinetic energy decays to zero, then one would need further information on the relation between the Cauchy stress tensor and the symmetric part of the velocity gradient. (The details are not elaborated here; interested reader is referred to Serrin [86] for the detailed treatment of the Navier-Stokes fluid, T D pI C 2 D. Note that the net kinetic energy does not completely vanish in finite time.) The decay of the kinetic energy of the fluid contained in a closed vessel is definitely a desirable outcome from the point of view of everyday experience, and, as it has been shown, the decay is closely related to the second law of thermodynamics.

4

Constitutive Relations

The motion of a single continuous medium is governed by the following system of partial differential equations d.x; t / C .x; t / div v.x; t / D 0; dt dv.x; t / .x; t / D div T.x; t / C .x; t /b.x; t /; dt T.x; t / D T> .x; t /; .x; t /

de.x; t / D T.x; t / W D.x; t /  div j q .x; t / dt

(67a) (67b) (67c) (67d)

that are mathematical expressions of the balance law for the mass, linear momentum, angular momentum, and total energy. (See equations (4), (13), (14), and (22).) The unknown quantities are the density , the velocity field v, and the specific internal energy e. (For the first reading, the internal energy can be thought of as a proxy for the temperature field .x; t /.) These equations are insufficient for the description of the evolution of the quantities of interest. The reason is that the Cauchy stress tensor T and the heat flux j q act in (67) as additional unknowns a priori unrelated to the density , the velocity field v, and the specific internal energy e. Therefore, in order to get a closed system of governing equations, system (67) must be supplemented by a set of equations relating the stress and the heat flux to the other quantities. These relations are called the constitutive relations, and they describe the response of the material to the considered stimuli.

24

J. Málek and V. Pr˚uša

It is worth emphasizing that the constitutive relations are specific for the given material. For example, the Cauchy stress tensor is a quantity that describes how the given volume of the material is affected by its surrounding; see Sect. 2.1.2. Clearly, the interaction between the given volume of the material and its surrounding depends on the particular type of the material. Similarly, the heat flux describes the thermal energy transfer capabilities of the given material; hence, it is material specific. The fact that the constitutive relations are material specific means that they indeed provide an extra piece of information supplementing the balance laws. Consequently, the constitutive relations can be derived only if one appeals to physical concepts that go beyond the balance laws. Moreover, the constitutive relations are inevitably simplified and reduced descriptions of the physical reality, and as such they are designed to describe only the behavior of the given material in a certain class of processes. For example, the same material, say steel, can be processed by hot forging, and it can deform as a part of a truss bridge or move as a projectile. The mathematical models for the response of the material in these processes are however different despite the fact that the material remains the same. The specification of the constitutive relations therefore depends on two things – the material and the considered processes. The most convenient form of the constitutive relations would be a set of equations of the form T D T.; v; e/;

(68a)

j q D j q .; v; e/:

(68b)

This would allow one to substitute for T and j q into (67) and get a system of evolution equations for the unknown fields , v, and e. An example of a system of constitutive relations of the type (68) are the well-known constitutive relations for the compressible viscous heat-conducting Navier-Stokes fluid, T D pth .;  /I C  .div v/ I C 2 D; j q D  r ;

(69a) (69b)

where pth .;  / denotes the thermodynamic pressure determined by an equation of state. (See Sect. 4.2 for details.) Substituting (69) into (67) then leads to the wellknown Navier-Stokes-Fourier system of partial differential equations. Besides the constitutive relations, the system of balance laws must be supplemented with initial and boundary conditions. The boundary conditions can be seen as a special case of constitutive relations at the interface between two materials. As such the boundary conditions can be very complex. In fact, the processes at the interface can have, for example, their own dynamics, meaning that they can be governed by an extra set of partial differential equations at the interface. Unfortunately, discussion of complex boundary conditions goes beyond the scope of the present contribution, and the issue is only briefly touched in Sect. 4.6. Consequently, overall the discussion of boundary conditions is mainly restricted

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

25

to the claim that the considered boundary conditions are the standard ones such as the no-slip boundary condition for the velocity field. The reader should be however aware of the fact that the specification of the boundary conditions is far from being trivial and that it again requires additional physical insight into the problem. The early efforts concerning the theory of constitutive relations for fluids (see, e.g., Rivlin and Ericksen [83], Oldroyd [65], and Noll [64] or the historical essay by Tanner and Walters [90]) were mainly focused on the specification of the Cauchy stress tensor in terms of the kinematical variables. These early approaches were based almost exclusively on mechanical considerations, such as the symmetry of the material and the requirement of the invariance of the constitutive relations with respect to the change of the observer. Thermodynamic considerations were rarely the leading theme and were mainly reduced to the a posteriori verification of the nonnegativity of the stress power. Since then thermodynamics started to play an increasingly important role in the theory of constitutive relations; see, for example, Coleman [19] and Truesdell and Noll [91]. This paradigm shift went hand in hand with the development of the field of nonequilibrium thermodynamics; see, for example, de Groot and Mazur [34], Glansdorff and Prigogine [32], Ziegler [97], and Müller [62]. In what follows a modern thermodynamics based approach to the phenomenological theory of constitutive relations is presented. The presented approach is essentially based on the work of Rajagopal and Srinivasa [78] (see also Rajagopal and Srinivasa [80]), who partially took inspiration from various earlier achievements in the field, most notably from the work of Ziegler [97] and Ziegler and Wehrli [98]. The reader who is interested into some comments concerning the precursors of the current approach is referred to Rajagopal and Srinivasa [78]. The advantage of the presented approach is that the second law of thermodynamics plays a key role in the theory of constitutive relations. Unlike in some other approaches, the second law is not used a posteriori in checking whether the derived constitutive relations conform to the second law. In fact, the opposite is true; the second law is the starting point of the presented approach. The constitutive relations for the Cauchy stress tensor, the heat flux, and the other quantities of interest are in the presented approach derived as consequences of the choice of the specific form for the internal energy e and the entropy production . If the entropy production  is chosen to be nonnegative, then the second law is satisfied. Since the relations of the type (68) are in the presented approach the consequences of the choice of e and , then it is clear that the arising constitutive equations cannot violate the second law. The second law is automatically built in the arising constitutive relations of the type (68). Further, the specification of two scalar quantities e and  is apparently much easier than the direct specification of the constitutive relations between the vectorial and tensorial quantities as in (68). This brings simplicity into the theory of constitutive relations. Two scalar quantities determine everything. Finally, the advocated approach is robust enough to handle complex materials. Nonlinear constitutive relations and constitutive relations for constrained materials as well as constitutive relations for materials reacting to the stimuli of various

26

J. Málek and V. Pr˚uša

origins (thermal, mechanical, electromagnetic, chemical) are relatively easy to work with in the presented approach. A general outline of the advocated approach is given in Sect. 4.1, and then it is shown how the general approach can be used in various settings. First, the approach is applied in a very simple setting, namely, the well-known constitutive relations for the standard compressible Navier-Stokes-Fourier fluid are derived in Sect. 4.2. This example should allow the reader to get familiar with the basic concepts. Then the discussion proceeds to more complex settings, namely, that of Korteweg fluid; see Sect. 4.3. Finally (see Sect. 4.4), the discussion of the approach is concluded by the derivation of constitutive equations for viscoelastic fluids.

4.1

General Framework

The main idea behind the presented approach is that the behavior of the material in the processes of interest is determined by two factors, namely, its ability to store energy and produce entropy. The energy storage mechanisms are specified by the choice of the energetic equation of state, that is, by expressing the internal energy e as a function of the state variables. (Other thermodynamic potentials such as the Helmholtz free energy, Gibbs potential, or the enthalpy can be used as well. The discussion below is however focused exclusively on the internal energy.) The entropy production mechanisms are specified by the choice of the formula for the entropy production . The derivation of the constitutive relations (68) from the knowledge of e and  then proceeds in the following steps: STEP 1: Specify the energy storage mechanisms by fixing the constitutive relation for the specific internal energy e in the form of the energetic equation of state e D e.; y1 ; : : : ; ym /

(70)

or in the form of the entropic equation of state (26). At this level it is sufficient to determine the state variables yi that enter (70). STEP 2: Find an expression for the material time derivative of the specific entropy . This can be achieved by the application of the material time derivative to (70), and by the multiplication of the result by .x; t /, which yields de.x; t / D dt ˇ d.x; t / @e.; y1 ; : : : ; ym / ˇˇ .x; t / ˇ @ dt D.x;t/;y1 Dy1 .x;t/;:::;ym Dym .x;t/

.x; t /

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

C

m X

.x; t /

iD1

27

ˇ dyi .x; t / @e.; y1 ; : : : ; ym / ˇˇ : ˇ @yi dt D.x;t/;y1 Dy1 .x;t/;:::;ym Dym .x;t/ (71)

The definition of the thermodynamic temperature (see (29)) and (71) then yield

d.x; t / de.x; t / D .x; t / dt dt ˇ m X @e.; y1 ; : : : ; ym / ˇˇ dyi .x; t / .x; t / ;  ˇ @yi dt yj Dyj .x;t/; D.x;t/ iD1

.x; t /.x; t /

(72)

. which is the sought formula for d dt STEP 3: Identify the entropy production. Use the balance equations and kinematics to write down (derive) the formulae for the material time derivative of the state variables dyidt.x;t/ , i D 1; : : : ; m, and substitute these formulae into (72). Rewrite in the form the equation for .x; t / d.x;t/ dt

.x; t /

m 1 X d.x; t / j .x; t /  a˛ .x; t /; C div j .x; t / D dt .x; t / ˛D1 ˛

(73)

where j ˛ .x; t /  a˛ .x; t / denotes the scalar product of vector or tensor quantities, respectively. The right-hand side of (73) is the entropy production, where each summand is supposed to represent an independent entropy-producing mechanism. The quantities j ˛ .x; t / are called the thermodynamic fluxes, and the quantities a˛ .x; t / are called the thermodynamic affinities. The affinities are usually the spatial gradients of the involved quantities, for example, r or D, while the fluxes are, for example, the heat flux j q or the Cauchy stress tensor T. (As a rule of thumb, the fluxes are the quantities that appear under the divergence operator in the balance laws (67); see also Rajagopal and Srinivasa [78] for further comments.) The quantity j .x; t / is called the entropy flux, and in standard j .x;t/

q . cases, it is tantamount to .x;t/ STEP 4* (Linear nonequilibrium thermodynamics): The second law of thermodynamics states that the entropy production .x; t / Ddef .x; t / d.x;t/ C dt div j .x; t / is nonnegative; see Sect. 2.3.3. Referring to (73), it follows that

.x; t / D

m 1 X j .x; t /  a˛ .x; t / .x; t / ˛D1 ˛

(74)

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J. Málek and V. Pr˚uša

must be nonnegative. A simple way to fulfill this requirement is to consider linear relations between each pair of j ˛ .x; t / and a˛ .x; t /, that is, j ˛ .x; t / D ˛ a˛ .x; t /;

(75)

where ˛ , ˛ D 1; : : : ; m, are nonnegative constants. Alternatively, one can consider cross effects by assuming that the relations between j ˛ .x; t / and a˛ .x; t / take the form j ˛ .x; t / D

m X

˛ˇ aˇ .x; t /;

(76)

ˇD1

where ˛ˇ is a symmetric positive definite matrix. This is essentially the approach of linear nonequilibrium thermodynamics; see de Groot and Mazur [34] for details. STEP 4 (Nonlinear nonequilibrium thermodynamics): Since the linear relationships between the fluxes j ˛ .x; t / and affinities a˛ .x; t / can be insufficient for a proper description of the behavior of complex materials, an alternative procedure is needed. In particular, the procedure should allow one to derive nonlinear constitutive relations of the type j i D j i .a1 ; : : : ; am / or vice versa. Here it comes to the core of the approach suggested by Rajagopal and Srinivasa [78]. As argued by Rajagopal and Srinivasa [78], one first specifies function  in one of the following forms  D a1 ;:::;am .j 1 ; : : : ; j m /;

(77a)

 D j1 ;:::;jm .a1 ; : : : ; am /:

(77b)

or

Note that the state variables can enter the constitutive relations (77) as well, but they are not, for the sake of compactness of the notation, written explicitly in (77). Formula (77a) or (77b) is a constitutive relation that determines the entropy production ,  Ddef  . Since the constitutive function  is – up to the positive factor  – tantamount to the entropy production, it must be nonnegative, which guarantees the fulfillment of the second law of thermodynamics. Further,  should vanish if the fluxes vanish. Other restrictions concerning the formula for the entropy production can come from classical requirements such as the material symmetry and the invariance with respect to the change of the observer. Moreover, the assumed form of the entropy production must be compatible with the already derived form of the entropy production (74). Consequently, the following equation must hold

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

j1 ;:::;jm .a1 ; : : : ; am / 

m 1 X j .x; t /  a˛ .x; t / D 0; .x; t / ˛D1 ˛

29

(78)

and similarly for a1 ;:::;am .j 1 ; : : : ; j m /. In terms of the constitutive function , this reduces to the requirement a1 ;:::;am .j 1 ; : : : ; j m / 

m X

j ˛ .x; t /  a˛ .x; t / D 0;

(79a)

j ˛ .x; t /  a˛ .x; t / D 0;

(79b)

˛D1

or to j1 ;:::;jm .a1 ; : : : ; am / 

m X ˛D1

respectively, depending whether one starts with (77a) or (77b). Relations (79) can be rewritten in the form A .J /  J  A D 0;

(80a)

J .A/  J  A D 0;

(80b)

> >

0 and J D j 1 : : : j m are vectors in RM , where A Ddef a1 : : : am M 0 > m that contain all the affinities and fluxes, respectively. Here the tensorial quantities such as the Cauchy stress tensor T are understood as column vectors >

T11 T12 : : : T33 . Having fixed a formula for the constitutive function  via (77a) or (77b), respectively, the task is to determine the fluxes j 1 ; : : : ; j m or affinities a1 ; : : : ; am that are compatible with the single constraint (80a) or (80b), respectively. If the fluxes/affinities are vectorial or tensorial quantities, then the single constraint is not sufficient to fully determine the sought relation between the fluxes and the affinities. (This means that in general there exist many constitutive relations of the type j i D j i .a1 ; : : : ; am / or vice versa such that (80a) or (80b) holds.) Rajagopal and Srinivasa [78] argued that the choice between the multiple constitutive relations that fulfill the constraint (80a) or (80b) can be based on the assumption of maximization of the entropy production. The assumption simply requires that the sought constitutive relation is the constitutive relation that leads to the maximal entropy production in the material and that is compatible with other available information concerning the behavior of the material. In more operational terms, the maximization of the entropy production leads to the following problem. Given the function A .J / and the values of the state variables and the values of the affinities A, the corresponding values of the fluxes J are those which maximize A .J / subject to the constraint (80a),

30

J. Málek and V. Pr˚uša

and alternatively to other possible constraints. (Note that since  is a positive quantity, then the task to maximize  with respect to the fluxes is indeed tantamount to the task of maximizing the entropy production  with respect to the fluxes.) A similar reformulation can be made for J .A/; it suffices to switch the role of the fluxes and affinities. The constrained maximization problems arising from the assumption on the maximization of the entropy production read max A .J /

(81)

J 2J

o n 0 where J Ddef J 2 RM ; A .J /  J  A D 0 , and max J .A/

(82)

A2A

o n 0 where A Ddef A 2 RM ; j .A/  J  A D 0 . Assuming that A or J are smooth and strictly convex with respect to their variables, then the corresponding values of J and A, respectively, are uniquely determined and can be found easily by employing the Lagrange multipliers; see Rajagopal and Srinivasa [78]. For example, if the starting point is the constitutive function A .J /, then the auxiliary function for the constrained maximization reads ˆ.J / Ddef A .J / C ` ŒA .J /  J  A, where ` is a Lagrange multiplier. The condition for the value / of the flux J that corresponds to maximal entropy production is @.J D 0 @J which leads to equation AD

1 C ` @A .J / ; ` @J

(83)

where 1C` D `

A .J / @A .J / J @J

:

(84)

Relation (83) is the sought relation between J and A. Consequently, if one accepts the assumption on the maximization of the entropy production, then the relations between the fluxes and affinities are due to the strict convexity of A .J / indeed uniquely determined by the choice of the constitutive function A .J /. Moreover, if the constitutive function A .J / is quadratic in J , then one recovers the result known from linear nonequilibrium thermodynamics, namely, that the fluxes are linear functions of the affinities; see (76). Apparently, a similar procedure can be followed if the starting point is J .A/ instead of A .J /. STEP 5: Relations (75) or (76) or (83) are the sought constitutive relations.

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

31

The rest of the present contribution is focused mainly on linear relations between the fluxes and affinities; hence, a quadratic ansatz for the constitutive function  is predominantly used in the rest of the text. This means that STEP 4 is rarely referred to in its full complexity and that one basically stays in the framework of STEP 4*. The reason is that the following text is mainly focused on the most difficult step of the advocated approach which is the correct specification of the functions e and . This is a nontrivial task in the development of the mathematical model, since the correct specification of these functions requires physical insight into the problem. However, the key physical arguments concerning the choice of the functions e and  can be easily documented in the simplest possible case of a quadratic ansatz; generalization of the quadratic ansatz to a more complex one is then a relatively straightforward procedure. Also note that once the functions e and  are specified, the derivation of the constitutive relations is only a technical problem not worth of lengthy discussion.

4.2

Compressible and Incompressible Viscous Heat-Conducting Fluids

The application of the outlined procedure is first documented in a very simple case of the derivation of constitutive relations for the compressible Navier-Stokes-Fourier fluid (see Sect. 4.2.1), which is a simple model for a compressible viscous heatconducting fluid. The next section (see Sect. 4.2.2) is devoted to reduced models that are variants of the compressible Navier-Stokes-Fourier fluid model. Finally, the incompressible counterparts of the former models are discussed in Sect. 4.2.3.

4.2.1 Compressible Navier-Stokes-Fourier Fluid STEP 1: The specific internal energy e is assumed to be a function of the specific entropy  and the density , e D e.; /:

(85)

This is a straightforward generalization of the classical energetic equation of state E D E.S; V / known for the equilibrium thermodynamics; see, for example, Callen [15]. (For the sake of clarity of the notation, the spatial and temporal variable is omitted in (85). If written in full, the energetic equation of state should read e.x; t / D e..x; t /; .x; t //. The same approach is applied in the rest of this section.) STEP 2: Taking the material derivative of (85) yields 

@e d de @e d D  : @ dt dt @ dt

(86)

32

J. Málek and V. Pr˚uša

Recalling the definition of the temperature (see (29)) and introducing the thermodynamic pressure pth through  Ddef

@e ; @

pth Ddef 2

@e ; @

(87a) (87b)

which are again the counterparts of the classical relations known from equilibrium thermodynamics, one can rewrite (86) as 

d de pth d D  : dt dt  dt

(88)

STEP 3: Using the evolution equation for the internal energy (67d) and the balance and d which yields of mass (67a), one can in (88) substitute for de dt dt 

d D T W D  div j q C pth div v: dt

(89)

The right-hand side can be further rewritten as T W D  div j q C pth div v D Tı W Dı C .m C pth / div v  div j q ;

(90)

where m Ddef

1 Tr T 3

(91)

denotes the mean normal stress, and 1 .Tr T/ I; 3 1 D  .Tr D/ I 3

Tı Ddef T 

(92)

Dı Ddef

(93)

denote the traceless part of T and D, respectively. Since the thermodynamic temperature is positive, one can, after some manipulation, rewrite (89) as    jq 1 r d D Tı W Dı C .m C pth / div v  j q  C div  dt    which is the evolution equation for  in the desired form (73).

(94)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

33

The rationale for using the traceless parts of T and D in (94) is the following. The quantities D and div v that appear in the first and the last term on the righthand side of (89) are not independent quantities. (Recall that Tr D D div v.) However, the fluxes one would like to identify when writing (89) in the form (73) should be independent quantities. Therefore, it is necessary to split D to mutually independent quantities Dı and div v, which requires one to split the Cauchy stress tensor in a similar manner. A closer inspection of (94) indicates that there exist three “independent” entropy-producing mechanisms in the material. The first one, j q  r , is  entropy production due to heat transfer; the second one, .m C pth / div v, is entropy production due to volume changes; and the last entropy production mechanism, Tı W Dı , is due to isochoric processes such as shearing. STEP 4: The entropy production  – that is, the right-hand side of (94) – takes the form  3 1X 1 r Tı W Dı C .m C pth / div v  j q  : D j  a˛ D  ˛D1 ˛  

(95)

Using the flux/affinity nomenclature, the fluxes are the traceless part of the Cauchy stress tensor Tı , the mean normal stress plus the thermodynamic pressure m C pth , and the heat flux j q . The affinities are Dı , div v, and r . . However, the positive (Strictly speaking the last affinity is not r but  r  1 factor  is of no importance.) The entropy production is a positive quantity if the constitutive relations are chosen as follows, Tı D 2 Dı ;

(96a)

m C pth D Q div v;

(96b)

j q D  r ;

(96c)

where > 0, Q > 0, and > 0 are given positive functions of the state variables  and . This choice of constitutive relations follows the template specified in (75). (Note that since Dı is a symmetric tensor, then Tı is also symmetric; hence, the balance of angular momentum (67c) is satisfied.) The Q and are called the shear viscosity, the bulk viscosity, and the coefficients , , heat conductivity. Traditionally, the bulk viscosity is written in the form 3 C 2 Q D ; 3 where  is a function of the state variables such that Q remains positive.

(97)

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J. Málek and V. Pr˚uša

STEP 5: It follows from (96) that the constitutive relation for the full Cauchy stress tensor T D Tı C mI is T D 2 Dı C Q .div v/ I  pth I

(98)

which can be in virtue of (97) rewritten as T D pth I C 2 D C .div v/I:

(99a)

This is the standard formula for the Cauchy stress tensor in the so-called compressible Navier-Stokes fluid. The constitutive relation for the heat flux reads j q D  r ;

(99b)

which is the standard Fourier law of thermal conduction. The coefficients in the constitutive relations can be functions of the state variables  and . This choice of variables is however inconvenient in practice. Solving the equation (87a) for the entropy, one can see that the entropy could be written as a function of the temperature and the density  D f .; /:

(100)

If this is possible, then the coefficients can be rewritten as functions of the temperature and the density, pth ? . ; / D pth .f .; /; / ;

(101a)

? . ; / D .f .; /; / ;

(101b)

?

 . ; / D  .f .; /; / ;

(101c)

? . ; / D .f .; /; / ;

(101d)

?

e . ; / D e .f .; /; / :

(101e)

Consequently, the final set of the governing differential equations arising from the balance laws (67) and the constitutive relations (99) reads

 

d D  div v; dt

(102a)

dv D div T C b; dt

(102b)

de ? . ; / D T W D  div j q ; dt

(102c)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

T D pth ? . ; /I C 2 ? . ; /D C ? . ; /.div v/I; j q D  ? . ; /r :

35

(102d) (102e)

These are the standard governing equations for the so-called compressible NavierStokes-Fourier fluids. If necessary, the evolution equation for the internal total energy (102c) can be reformulated as a balance of total energy, that is, 

d dt



1 e ? . ; / C jvj2 2

 D b  v C div .Tv/  div j q :

(103)

Note that if the internal energy as a function of the temperature and the density takes the simple form e ? D cV  , where cV is a constant referred to as the specific heat capacity, then the evolution equation for the internal energy (102c) takes the well-known form d D T W D  div j q : cV (104) dt

4.2.2 Other Linear Models for Compressible Fluids Referring to (95) one can notice that the entropy production  can be written in the form 

 r D J  A: (105)  D Tı ; m C pth ; j q  Dı ; div v;  If the constitutive relation for any of the quantities Tı , m C pth , j q is trivial, that is, if Tı D 0, m C pth D 0 or j q D 0, then the corresponding term in the entropy production vanishes. Such models are referred to as the reduced models. For example, Tı D 0 corresponds to an inviscid compressible heat-conducting fluid (Euler-Fourier fluid). On the other hand, if any of the quantities Dı , div v, r vanishes, then the corresponding entropy production also vanishes regardless of the particular constitutive equation for Tı , mCpth , or j q , respectively. The processes in which Dı , div v, or r vanishes are the motion with a velocity field in the form v.x; t / Ddef a.t /  x C b.t /, where a and b are arbitrary time-dependent vectors, the isochoric motion, that is, volume-preserving motion, and isothermal process, that is, a process with no temperature variations, respectively. Going back to the constitutive relations, the entropy production completely vanishes if the constitutive relations take the form Tı D 0;

m D pth ;

and

j q D 0:

(106)

This means that T D pth ? . ; /I;

and

j q D 0;

(107)

36

J. Málek and V. Pr˚uša

which are the constitutive equations for a compressible Euler fluid. The corresponding system of partial differential equations reads d D  div v; dt dv  D rpth ? . ; / C b; dt   1 d  e ? . ; / C jvj2 D  div .pth ? . ; /v/ C b  v: dt 2

(108a) (108b) (108c)

There is a hierarchy of models between the compressible Euler fluid (108) with no entropy production and the compressible Navier-Stokes-Fourier fluid (102) in which all entropy production mechanisms are active. These models are, for example, the Euler-Fourier fluid where the constitutive relations read Tı D 0;

m D pth ;

and

j q D  ? . ; /r ;

(109)

the fluid where thermodynamic pressure coincides (up to the sign) with the mean normal stress, that is, the fluid with constitutive relations m D pth ;

j q D  ? . ; /r.x; t /;

Tı D 2 ? . ; /Dı ;

(110)

which leads to T D pth ? . ; /I C 2 ? . ; /Dı , or the fluid dissipating due to volume changes but not due to shearing, that is, the fluid with constitutive relations Tı D 0;

m D pth ? . ; / C ? . ; / div v;

j q D  ? . ; /r :

(111)

Finally, one can observe that if one deals with the ideal gas, that is, with the fluid given by constitutive relations (102) and e ? . ; / D cV  , where cV is a constant, then one can in some cases consider isothermal processes. (The temperature is a constant .x; t / D  ? .) Indeed, if the initial temperature distribution is uniform, and if the fluid undergoes only small volume changes, and if the entropy production mechanisms in the fluid are weak, then the terms T W D D pth div v C Q .div v/2 C 2 Dı W Dı ;

(112)

and div j q do not significantly contribute to the right-hand side of (104). In such a case, and under favorable boundary conditions for the temperature, the evolution equation (104) for the internal energy/temperature is of no interest, and one can focus only on the mechanical part of the system, which leads to the compressible Navier-Stokes equations

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

d D  div v; dt   dv Q div v C b; Q C r ./  D r p f th ./ C div .2 ./D/ dt

37

(113a) (113b)

? ? Q with ./ Q > 0 and 2 ./ Q C 3./ > 0. Here p f Q Ddef th ./ Ddef pth . ; /, ./ ? ? ? ? Q . ; /, and ./ Ddef  . ; /. Note that genuine isothermal processes are in fact impossible in a fluid with nonvanishing entropy production due to mechanical effects. Once there is entropy production due to the term T W D, kinetic energy is lost in favor of thermal energy, and the temperature must change; see (102c) or (104). The isothermal process is in this case only an approximation. Other simplified systems that arise from (113) are the compressible Euler equations

d D  div v; dt dv  D r p f th ./ C b; dt

(114a) (114b)

the compressible “bulk viscosity” equations d D  div v; dt 

(115a) !

Q dv 2 ./ Q C 3./ D r p f div v C b: th ./ C dt 3

(115b)

and the compressible “shear viscosity” equations d D  div v; dt dv Q C b:  D r p f th ./  div .2 ./D/ dt

(116a) (116b)

4.2.3 Incompressible Navier-Stokes-Fourier Fluid Incompressibility means that the density of any given material point X is constant in time. In terms of the Eulerian density field .x; t /, this requirement translates into the requirement of vanishing material time derivative of .x; t /, d D 0: dt

(117)

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J. Málek and V. Pr˚uša

(Note that the density can still vary in space, i.e., the density assigned to different material points can be different. In such a case, the material is referred to as inhomogeneous material.) Using the balance of mass (67a), it follows from (117) that div v D 0;

(118)

which means that the material can undergo only isochoric – that is, volume preserving – motions. In this sense, the incompressible material can be understood as a constrained material, meaning that the class of motions possible in the given material is constrained by the requirement div v D 0. If one takes into account the constraint div v D 0, then the term T W D D Tı W Dı C m div v

(119)

in the evolution equation for the internal energy (67d) reduces to Tı W Dı . (Recall that m denotes the mean normal stress, m Ddef 31 Tr T, and that Tr D D div v.) This implies that the mean normal stress cannot be directly present in the entropy production in the form of a flux/affinity pair, because the corresponding flux div v identically vanishes. This observation is easy to document by inspecting the entropy production formula (95) for compressible heat-conducting fluids, where the constraint div v D 0 would imply that .m C pth / div v D 0. However, the specification of the constitutive relations is based on the presence of the quantities of interest in the formula for the entropy production in the form of a flux/affinity product. In this context one could say that the mean normal stress cannot be specified constitutively. (This means that the mean normal stress m is not – in contrast to the traceless part Tı of the Cauchy stress tensor – given by a constitutive relation in terms of other variables such as the kinematical quantities.) If thought of carefully, this is a natural consequence of the concept of incompressibility. The mean normal stress in an incompressible fluid must incorporate the force mechanism that prevents the fluid from changing its volume. (In mathematical terms this means that it must depend on the Lagrange multiplier enforcing the incompressibility constraint (118).) As such it cannot be specified without the knowledge of boundary conditions. Think, for example, of a vessel completely filled with an incompressible fluid at rest. The value of the mean normal stress in the vessel cannot be determined without the knowledge of the forces acting on the vessel walls. This makes the normal stress in an incompressible fluid a totally different quantity than the mean normal stress in a compressible fluid. In a compressible fluid (see, e.g., (96b)), the mean normal stress is a function of the thermodynamic pressure pth and the velocity field, and the thermodynamic pressure is a function of the density and the temperature. Therefore, the mean normal stress in a compressible fluid is fully specified in terms of the local values of the state variables and the velocity field. On the other hand, the mean normal stress in an incompressible fluid is not a function of the local values of the state variables and the velocity field; it constitutes another unknown in the governing equations. Unfortunately, the simplest model for an incompressible viscous fluid is the Navier-Stokes fluid where the Cauchy stress tensor is given by the formula

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

T D pI C 2 D;

39

(120)

where p is called the “pressure.” This could be misleading since the quantity called “pressure” also appears in the compressible Navier-Stokes fluid, albeit the pressure in the compressible Navier-Stokes fluid is the thermodynamic pressure, which is a totally different quantity than the pressure p in (120). Further, the mean normal stress m in the incompressible Navier-Stokes fluid model (120) is m D p; hence it differs from the “pressure” p by a mere sign. Consequently, the notions of the mean normal stress and the “pressure” are often used interchangeably, which is a bad practice since these notions do not coincide if one deals with incompressible fluid models that go beyond (120). Concerning the compressible Navier-Stokes fluid model (102d), the mean normal stress is given by the formula m D pth C 3C2 div v; hence the mean normal stress 3 differs from the thermodynamic pressure by more than a sign. On the other hand, the mean normal stress coincides with the thermodynamic pressure if one adopts so-called Stokes assumption 3 C 2 D 0. In such a case, one is allowed to use interchangeably terms thermodynamic pressure and mean normal stress, which is otherwise not correct. The subtleties concerning the notion of the pressure, Lagrange multiplier enforcing the incompressibility constraint, thermodynamic pressure, and the mean normal stress can be of importance if one, for example, wants to talk about viscoelastic materials with “pressure”-dependent coefficients. The different contexts in which the term pressure is used are discussed in detail by Rajagopal [76] (see also comments by Huilgol [43] and Dealy [22]), and this issue is not further commented in the present treatise. The reader should be however aware of the fact that the notion of “pressure” becomes complicated in particular if one deals with non-Newtonian fluids. Concerning the possible approaches to the derivation of constitutive relations for incompressible materials, one can either opt for the strategy outlined in Sect. 4.1 and enforce the incompressibility constraint via an additional Lagrange multiplier. Examples of such approaches can be found in Málek and Rajagopal [54] (incompressible Navier-Stokes fluid) and Málek et al. [58] (incompressible viscoelastic fluids). This approach allows one to find a relation between the Lagrange multiplier enforcing the incompressibility constraint and the mean normal stress. However, this relation is usually of no interest in the final constitutive relations. Therefore, a different and more pragmatic approach is adopted in the rest of the text. The constitutive relations are derived in a standard manner with the constraint (118) in mind, which leads to a constitutive relation for the traceless part of the Cauchy stress tensor Tı . As shown above, the incompressibility constraint makes the search for a constitutive relation for the mean normal stress pointless; hence one can leave the final formula for the full Cauchy stress tensor in the from

T D mI C Tı ;

(121)

40

J. Málek and V. Pr˚uša

where m is understood as a new unknown quantity to be solved for. (In (121) one can further exploit the fact that the incompressibility constraint (118) allows one to write D D Dı .) In particular, if the outlined approach is applied in the case of the constitutive relations derived in Sect. 4.2.1, one gets T D mI C Tı D mI C 2 ? . ; /Dı ; hence T D mI C 2 ? . ; /D;

(122a)

where the relation D D Dı has been exploited. The constitutive relation for the heat flux remains formally the same, j q D  ? . ; /r :

(122b)

These are the constitutive relations for the incompressible Navier-Stokes-Fourier fluid. Note that unlike in the majority of the literature, the constitutive relation for the Cauchy stress tensor is written as (122a) and not as T D pI C 2 ? . ; /D:

(123)

This notation emphasizes the different nature the thermodynamic pressure pth in compressible fluids, which is frequently denoted as p, and the “pressure” p Ddef m in incompressible fluids. The governing equations for the incompressible Navier-Stokes-Fourier fluid then take the form d D 0; dt

(124a)

div v D 0;

(124b)

dv D rm C div .2 ? . ; /D/ C b; (124c) dt   1 d e ? . ; / C jvj2 D div .mv C 2 ? . ; /Dv C ? . ; /r / C b  v:  dt 2 (124d) 

The first equation in (124), the characteristics,

@ @t

C v  r D 0, implies that  is transported along

  .x; t / D 0 .X / D 0 1 .x; t / ;

(125)

where 0 is the density in the initial (reference) configuration. Consequently the density .x; t / is constant in space and time if the initial (reference) density 0 is uniform, that is, if 0 .X 1 / D 0 .X 2 /

(126)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

41

holds for all X 1 , X 2 2 B. If (126) does not hold, then the fluid at its initial (reference) state is inhomogeneous. This is why the system (124) is sometimes called inhomogeneous incompressible Navier-Stokes-Fourier equations. Obviously the incompressible Navier-Stokes-Fourier fluid model can be further simplified. Setting Tı D 0, one gets a model for a fluid without shear viscosity. Such a model can be referred to as inhomogeneous incompressible Euler-Fourier fluid model, and the governing equations reduce to d D 0; dt

(127a)

div v D 0;

(127b)

dv D rm C b; dt   1 d e ? . ; / C jvj2 D div .mv C ? . ; /r / C b  v:  dt 2 

(127c) (127d)

Note that equations (127a)–(127c) are not coupled with the temperature and can provide a solution for  and v independently of the energy equation. The fluid described by (127a)–(127c) is called the inhomogeneous incompressible Euler fluid. If the viscosity in (124) does not depend on the temperature, ? . ; / D ./, Q then energy equation again decouples from the system, and one can solve only the mechanical part of (124). This leads to inhomogeneous incompressible NavierStokes equations d D 0; dt

(128a)

div v D 0;

(128b)



dv D rm C div .2 ./D/ Q C b: dt

(128c)

If the density at initial (reference) configuration is uniform, then as a consequence of (126) and (125), one finds that .x; t / Ddef ? 2 .0; C1/

(129)

for all t > 0, x 2 B. System (124) then simplifies to the so-called homogeneous incompressible Navier-Stokes-Fourier equations div v D 0; dv D rm C div .2 ./D/ O C b; dt   1 2 d O C . O /r / C b  v; eO . / C jvj D div .mv C 2 ./Dv  dt 2 

(130a) (130b) (130c)

42

J. Málek and V. Pr˚uša

where ./ O Ddef ? . ; ? /, eO . / Ddef e ? . ; ? /, and . O / Ddef ? . ; ? /. Following the same template as above, one can introduce homogeneous incompressible Euler-Fourier fluids. If the viscosity does not depend on the temperature, ? Ddef . O ? / D ? .? ; ? /, then (130c) decouples from the rest of the system (130), and one can solve only the mechanical part of (130) for the mechanical variables m and v. This leads to the classical incompressible Navier-Stokes equations div v D 0; 

(131a)

dv D rm C div .2 ? D/ C b; dt

(131b)

where the right-hand side of (131b) can be rewritten as rm C div .2 ? D/ C b D rm C ? v C b. Finally, if Tı D 0, then one ends up with the incompressible Euler equations div v D 0; 

4.3

(132a)

dv D rm C b: dt

(132b)

Compressible Korteweg Fluids

Another popular model describing the behavior of fluids is the model introduced by Korteweg [51]. The model has been developed with the aim to describe phase transition phenomena, but it is also used for the modelling of the behavior of some granular materials. The key feature of the model is that the Cauchy stress tensor includes terms of the type r ˝ r, which is natural given the fact that the model is designed to take into account steep density changes that can occur, for example, at a fluid/vapour interface. In what follows it is shown how to derive the model using the thermodynamical procedure outlined in Sect. 4.1. The reader interested in various applications of the model and the details concerning the derivation of the model is referred to Málek and Rajagopal [55] and Heida and Málek [38]. Since the density gradient is the main object of interest, for later use, it is convenient to observe that the balance of mass (67a) implies d .r/ D r . div v/  Œrv> r: dt

(133)

STEP 1: Consider the constitutive equation for the specific internal energy – or the specific entropy  – in the form  D .e; Q ; r/

or

e D eQ .; ; r/:

(134)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

43

The presence of the density gradient r in the formula for the internal energy/entropy means that in this specific example, one goes, for the first time in this text, beyond classical equilibrium thermodynamics. Moreover, the energetic equation of the state is considered in a special form e Ddef eQ .; ; r/ D e.; / C

 jrj2 ; 2

(135)

where  is a positive constant. This choice is motivated by the fact that one expects the density gradient r to contribute to the energy storage mechanisms and the fact that the norm of the gradient r is the simplest possible choice if one wants the internal energy to be a scalar isotropic function of r. STEP 2 and 3: Applying the material time derivative to (135), multiplying the result by  and using the evolution equation for the internal energy (22), the balance of mass (4) and identity (133) yield @e d @Qe DTWDdiv j q C2 div v C .r/r . div v/C .rv/W.r ˝ r/: @ dt @ (136) Introducing the thermodynamic pressure and temperature via 

 Ddef

@Qe ; @

K Ddef 2 pth

@Qe ; @

(137a) (137b)

one ends up with a formula for the material time derivative of the entropy 

  d  K D .Tı C  .r ˝ r/ı / W Dı C m C pth C jrj2 div v  div j q dt 3 C div . .div v/ r/   div v;

(138)

where the relevant tensors have been split to their traceless part and the rest; see Sect. 4.2 for the rationale of this step. Notice however that this thermodynamic NSE K pressure pth is different from the thermodynamic pressure pth Ddef 2 @e for @ the Navier-Stokes fluid as they are related through K NSE D pth  pth

 jrj2 : 2

(139)

Equation (138) can be further rewritten as 

  d  K C jrj2   div v D .Tı C  .r ˝ r/ı / W Dı C m C pth dt 3   (140)  div j q  .r/ div v ;

44

J. Málek and V. Pr˚uša

and using the notation j qQ Ddef j q  r;

(141)

and applying the standard manipulation finally yields 

   j qQ 1 d D .Tı C  .r ˝ r/ı / W Dı C div dt      r r 2 K : C m C pth C jrj   div v  j q  C  .div v/ .r/  3   (142)

The affinities are in the present case Dı , div v, and r . Apparently, the last term on the right-hand side of (142) can be interpreted either as 

r .r/  div v 

(143)

  K C 3 jrj2   div v, or it can and grouped with the second term m C pth be read as Π.div v/ .r/ 

r 

(144)

. The question is left unresolved, and a and grouped with the third term j q  r  parameter ı 2 Œ0; 1 that splits the last term as  r r r div vCıŒ .div v/ .r/   .div v/ .r/  D.1ı/ .r/  ;    (145) is introduced. The arising terms are then grouped with the corresponding terms in (142). This yields    j d 1 .Tı C  .r ˝ r/ı / W Dı  D C div dt    C m C pth

K

 r C jrj2   C .1  ı/.r/  3 

 r  :  j q  ı .div v/ r  

 div v

(146)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

45

STEP 4: The entropy production  – that is, the right-hand side of (146) – takes the form D

3 1X j  a˛ ;  ˛D1 ˛

(147)

where the affinities are Dı , div v, and r . Requiring the linear relations between the fluxes and affinities in each term contributing to the rate of entropy production leads to Tı C  .r ˝ r/ı D 2 Dı ; K m C pth C

(148a)

2 C 3  r D div v; jrj2   C .1  ı/.r/  3  3 (148b) j q  ı .div v/ r D  r

(148c)

where > 0, 2 C3 > 0, and > 0 are positive constants. This choice of 3 constitutive relations follows the template specified in (75). STEP 5: It follows from (148) that the constitutive relation for the full Cauchy stress tensor T D Tı C mI is r K T D pth IC2 DC .div v/ I .r˝r/ C ./ I.1  ı/.r/  I  (149) that can be upon introducing the notation NSE   Ddef pth

 r jrj2   C .1  ı/.r/  2 

(150a)

rewritten as T D I C 2 D C  .div v/ I   .r ˝ r/ :

(150b)

Concerning the heat/energy flux, one gets j q D  r C ı .div v/ r:

(150c)

Note that if only isothermal processes are considered, then (149) reduces to ! jrj2 TD C  ./ I; C 2 D C  .div v/ I C  .r ˝ r/ C  2 (151) which is the constitutive relation obtained by Korteweg [51]. NSE pth I

46

4.4

J. Málek and V. Pr˚uša

Compressible and Incompressible Viscoelastic Heat-Conducting Fluids

Viscoelastic materials are materials that exhibit simultaneously two fundamental modes of behavior. They can store the energy in the form of strain energy; hence they deserve to be called elastic, and they dissipate the energy; hence they deserve to be called viscous. Since the two modes are coupled, the nomenclature viscoelastic is obvious. A rough visual representation of materials that exhibit viscoelastic behavior is provided by systems composed of springs and dashpots. The springs represent the elastic behavior, while the dashpots represent the dissipation. Such a visual representation is frequently used in the discussion of viscoelastic properties of materials; see, for example, Burgers [14] or more recently Wineman and Rajagopal [95]. The use of the visual representation as a motivation for the subsequent study of the constitutive relations for viscoelastic fluids is also followed in the subsequent discussion. A simple example of a spring–dashpot system is the Maxwell element consisting of a viscous dashpot and an elastic spring connected in series; see Fig. 3a. The response of the element to step loading is the following. (The step loading is a loading that is constant and is applied only over time interval Œt0 ; t1 ; otherwise the loading is zero.) Initially, the element is unloaded, and it is in an initial state; see Fig. 3a. Once the loading is applied at time t0 , the spring extends, and the dashpot starts to move as well; see Fig. 3b. When the loading is suddenly removed at some time t1 (see Fig. 3c), the spring, that is, the elastic element, instantaneously shrinks to its initial equilibrium length. On the other hand, the sudden unloading does not change the length of the dashpot at all. (The term “length of the dashpot” denotes the distance between the piston and the dashpot left wall.)

a

b

ls

ls

l0

l0

c

ls

l0

Fig. 3 Maxwell element. (a) Initial state. (b) Loaded state. (c) Unloaded state

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

47

This means that the instantaneous response to the sudden load removal is purely elastic, while the configuration (length) to which the material relaxes after the sudden load removal is determined by the length of the dashpot. Therefore, the unloaded configuration is determined by the behavior of the purely dissipative element.

4.4.1 Concept of Natural Configuration and Associated Kinematics Now the question is whether one can use this observation in the development of the constitutive relations for viscoelastic materials. It turns out that this is possible. One can associate these three states (initial/loaded/unloaded) of the Maxwell element with the three states of a continuous body: 1. the initial (reference) configuration 0 .B/, 2. the current configuration t .B/ that the body takes at time t > 0 during the deformation process due to the external load, 3. the natural configuration p.t/ .B/ that would be taken by the considered body at time t > 0 upon sudden load removal. The evolution from the initial configuration to the current configuration is then virtually decomposed to the evolution of the natural configuration (dissipative process) and instantaneous elastic response (non-dissipative process) from the natural configuration to the current configuration. Figure 4 depicts the situation. Referring back to the spring–dashpot analogue, the evolution of the natural configuration plays the role of the dashpot, while the instantaneous elastic response from the current configuration to the natural configuration plays the role of the spring.

current configuration

κt (B)

κ0 (B) elastic response

κp(t) (B) reference configuration dissipative response natural configuration

0

t

time

Fig. 4 Initial (reference), current and natural configurations associated with a material body

48

J. Málek and V. Pr˚uša

Derivation of constitutive relations for viscoelastic materials by appealing to the procedure outlined in Sect. 4.1 can then proceed as follows. Since the energy storage mechanisms in a viscoelastic fluid are determined by the elastic part – the spring – the plan is to enrich the constitutive relation for the internal energy (see (70)) by a state variable that measures the deformation of the elastic part (deformation from the natural to the current configuration). The next step of the thermodynamic procedure requires one to take material time derivative of the internal energy; hence a formula for the material time derivative of the chosen measure of the deformation of the elastic part is needed. This task requires a careful analysis of the kinematics of continuous media with multiple configurations. Recall that the key quantities in the kinematics of continuous media are the deformation gradient @ 0 .X ; t / ; @X

(152)

ˇ @ 0 .X ; t / ˇ ˇ : ˇ @t X D1

.x;t/

(153)

F.X ; t / Ddef

and the spatial velocity field

v.x; t / Ddef

0

(The subscript 0 recalls that the deformation gradient is taken with respect to the initial configuration.) Other quantities of interest are the left Cauchy-Green tensor and the right Cauchy-Green tensor B Ddef FF> , C Ddef F> F, the (spatial) velocity @v gradient L Ddef @x , and its symmetric part D. The material time derivative of F is then given by the formula dF D LF: dt

(154)

Within the framework consisting of three configurations 0 .B/, t .B/, and

p.t/ .B/, the standard kinematical setup is extended by the deformation gradient F p.t / describing the deformation between the natural and the current configuration and the deformation gradient G describing the deformation between the initial (reference) configuration and the natural configuration. Obviously (see Fig. 4), the following relation holds

F D F p.t / G:

(155)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

49

For later use, one also introduces notation B p.t / Ddef F p.t / F>

p.t / ;

(156a)

C p.t / Ddef F>

p.t / F p.t / ;

(156b)

for the left and right Cauchy-Green tensors for the deformation from the natural to the current configuration. The left Cauchy-Green tensor B p.t / is the sought measure of the deformation associated with the instantaneous elastic part of the deformation. The reason is the following. In the standard finite elasticity theory, one uses the left Cauchy-Green tensor B Ddef FF> in order to characterize the finite deformation of a body. In the current setting, the elastic response is the response from the natural to the current configuration; hence the left Cauchy-Green B p.t / tensor built using the relative deformation gradient F p.t / instead of the full deformation gradient F is used. Now one needs to find an expression for the material time derivative of B p.t / . (The knowledge of an expression for the time derivative is needed in the STEP 2 of the procedure discussed in Sect. 4.1.) The aim is to express B p.t / as a function of F p.t / and a rate quantity associated with the dissipative part of the response and a rate quantity associated with the total response. Noticing that (154) implies that L D dF F1 , a new tensorial quantity L p.t / is dt introduced through dG 1 G ; dt

(157)

 1 L p.t / C L>

p.t / : 2

(158)

L p.t / Ddef and its symmetric part is denoted as D p.t / , D p.t / Ddef

Relations (155), (154), and (157) together with the formula dG d  1  G D G1 G1 dt dt

(159)

then yield dG dF 1 dG1 G CF D LFG1  FG1 G1 D LF p.t /  F p.t / L p.t / ; dt dt dt dt (160) which implies dF p.t /

D

dB p.t / dt

D LB p.t / C B p.t / L>  2F p.t / D p.t / F>

p.t / :

(161)

50

J. Málek and V. Pr˚uša

This is the sought expression for the time derivative of the chosen measure of the deformation. As required, the time derivative is a function of F p.t / and the rate quantities D p.t / and L that are associated to the dissipative part of the response and the total response, respectively. Introducing the so-called upper convected time derivative through the formula O

A Ddef

dA  LA  AL> ; dt

(162)

where A is a second order tensor, it follows from (161) that O

B p.t / D 2F p.t / D p.t / F>

p.t / :

(163)

For later reference it is worth noticing that (162) implies O

I D 2D;

(164)

and that (161) implies

Since

d dt

d Tr B p.t / D 2B p.t / W D  2C p.t / W D p.t / : dt   det A D .det A/ Tr dA A1 , one can also observe that dt

   dB p.t / d 

ln det B p.t / D Tr B p.t / 1 D 2I W D  2I W D p.t / : dt dt

(165)

(166)

Finally, the upper convected derivative of the traceless part of B p.t / reads O



B p.t /

 ı

D

D

  d B p.t / ı dt

dB p.t / dt O

D B p.t / 



     L B p.t / ı  B p.t / ı L>

 1 dTr B p.t / 2 I  LB p.t /  B p.t / L> C Tr B p.t / D 3 dt 3

   2 2 2 B p.t / W D I C C p.t / W D p.t / I C Tr B p.t / D 3 3 3

D 2F p.t / D p.t / F p.t / > 

4.4.2

   2 2 2 B p.t / W D I C C p.t / W D p.t / I C Tr B p.t / D: 3 3 3 (167)

Application of the Thermodynamic Procedure in the Context of Natural Configuration The application of the thermodynamic procedure discussed in Sect. 4.1 in the setting of a material with an evolving natural configuration goes as follows:

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

51

STEP 1: The fact that the energy storage mechanisms are required to be related to the elastic part of the deformation (see Fig. 4) suggests that the internal energy e should be enriched by a term measuring the elastic part of the deformation. As it has been already noted, a suitable measure of the deformation is the left Cauchy-Green tensor B p.t / ; hence the entropic/energetic equation of state is assumed to take the form      D Q e; ; B p.t / or e D eQ ; ; B p.t / : (168) For the sake of simplicity of the presentation, the energetic equation of state is further assumed to take the form   

 e D eO ; ; Tr B p.t / ; det B p.t / Ddef e .; / C Tr B p.t /  3  ln det B p.t / ; 2 (169) where is a constant. The motivation for this particular choice comes from the theory of constitutive relations for isotropic compressible elastic materials; see for example Horgan and Saccomandi [41] and Horgan and Murphy [40] for a list of some frequently used constitutive relations in the theory of compressible elastic materials. STEP 2: Applying the material time derivative to (169), multiplying the result by  and using the balance of mass (67a) and the evolution equation for the internal energy (67d) together with the identities (165) and (166), one gets 

d M D T W D  div j q C pth div v  B p.t / W D C C p.t / W D p.t / dt C D W I  D p.t / W I

(170)

where M pth Ddef 2

 @Oe @e  D 2  Tr B p.t /  3  ln det B p.t / @ @ 2  

NSE D pth Tr B p.t /  3  ln det B p.t /  2

(171)

NSE Ddef 2 @e . denotes the “pressure”, and pth @ STEP 3: Splitting the tensors on the right-hand side of (170) into the traceless part and the rest yields



   d D Tı  B p.t / ı W Dı dt      

M C m C pth  Tr B p.t / C div v C C p.t / ı W D p.t / ı 3   Tr B p.t / C

 1 Tr D p.t /  div j q ; 3

(172)

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J. Málek and V. Pr˚uša

where the notation m Ddef 31 Tr T for the mean normal stress has been used. (See Sect. 4.2.1 for the rationale of the splitting.) The standard manipulation finally leads to   jq d  C div dt      

1 M  Tr B p.t / C div v D Tı  B p.t / ı W Dı C m C pth  3       Tr B p.t / r (173) C C p.t / ı W D p.t / ı C

 1 Tr D p.t /  j q  3  which is the sought formula that allows one to identify the entropy production.  The flux/affinity pairs are Tı  B p.t / ı versus Dı and so on. STEP 4: The entropy production – the right-hand side of (173) – is positive provided that the linear relations between the fluxes and affinities are   (174a) Tı  B p.t / ı D 2 Dı ; M  m C pth

2 C 3 Tr B p.t / C D div v; 3 3    

C p.t / ı D 2 1 D p.t / ı ; 



Tr B p.t / 3

 1 D

2 1 C 31 Tr D p.t / ; 3

j q D  r ;

(174b) (174c) (174d) (174e)

1 where > 0, 2 C3 > 0, 1 > 0, 2 1 C3 > 0, and > 0 are constants. This 3 3 choice of constitutive relations follows the template specified in (75). STEP 5: The first two equations in (174) allow one to identify the constitutive relation for the full Cauchy stress tensor T D mI C Tı ,

   2 C 3

 M .div v/ I  pth Tr B p.t / I  I; T D 2 Dı C B p.t / ı C IC 3 3 (175) which can be further rewritten as   M T D pth I C 2 D C  .div v/ I C B p.t /  I ; (176) or, in virtue of (171), as 

 NSE IC Tr B p.t /  3  ln det B p.t / I C 2 D C  .div v/ I T D pth 2   (177) C B p.t /  I :

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

53

The last step leading to the complete description of the material behavior is the elimination of D p.t / from the constitutive relations. Note that the Cauchy stress is given in terms of D, that is, the velocity field v, and the left Cauchy-Green tensor B p.t / . The evolution equation for B p.t / is (163), that is, O

B p.t / D 2F p.t / D p.t / F>

p.t / :

(178)

The equation expresses the time derivative of B p.t / as a function of L, F p.t / , and D p.t / . If the right-hand side of (163) can be rewritten as a function of B p.t / and L, then the time evolution of B p.t / would be given in terms of the other quantities directly present in the governing equations, and the process of specification of the constitutive relation would be finished. This can be achieved as follows. Summing (174c) and (174d) multiplied by the identity tensor yields    

C p.t /  I D 2 1 D p.t / C 1 Tr D p.t / I

(179)

and Tr D p.t / D

3

2 1 C 31



Tr C p.t / 3

 1 ;

(180)

which leads to  

C p.t /  I D 2 1 D p.t / C

3 1 2 1 C 31



Tr C p.t / 3

  1 I:

(181)

This is an explicit relation between D p.t / and C p.t / . Recalling the definition of C p.t / and B p.t / (see (156)) and noticing that Tr C p.t / D Tr B p.t / , it follows that the multiplication of (181) from the left by F p.t / and from the right by F>

p.t / yields



B2 p.t /



 B p.t / D

2 1 F p.t / D p.t / F>

p.t /

3 1 C 2 1 C 31



Tr C p.t / 3

  1 B p.t / : (182)

Substituting the formula just derived for F p.t / D p.t / F>

p.t / into (178) then gives   O 1 B p.t / C B2 p.t /  B p.t / D

3 1 2 1 C 31



Tr B p.t / 3

  1 B p.t / ;

(183)

which is the sought evolution equation for B p.t / that contains only B p.t / and L. The set of equations (67) supplemented with the constitutive equations for the heat flux (see (174e)), the Cauchy stress tensor (see (177)), and the evolution

54

J. Málek and V. Pr˚uša

equation for the left Cauchy-Green tensor B p.t / (see (183)) forms a closed system of equations for the density , the velocity field v, the specific internal energy e (or the temperature ), and the left Cauchy-Green tensor B p.t / . The constitutive relation for the Cauchy stress tensor (see (177)) can be further manipulated as follows. Introducing the notation NSE  Ddef pth C



 Tr B p.t /  3  ln det B p.t / ; 2

(184a)

  S Ddef B p.t /  I ;

(184b)

and

the equations (177) and (183) take the form T D I C 2 D C  .div v/ I C S;

(185)

and O

1 S C S2 C S D 2 1 D C

1 .Tr S/ .S C I/ : 2 1 C 31

(186)

Note that if 1 D 0, then (186) simplifies to O

1 S C S2 C S D 2 1 D;

(187)

hence the derived model could be seen as a compressible variant of the classical model for a viscoelastic incompressible fluid developed by Giesekus [31]. If the material is constrained in such a way that Tr D D Tr D p.t / D 0;

(188)

which means that the material is assumed to be an incompressible material and that the response from the initial to the natural configuration is assumed to be isochoric, it is clear that neither Tr B p.t / nor m can be specified constitutively. (These quantities correspond to the forces that make the material resistant to the volumetric changes; see Sect. 4.2 for a discussion.) However, formula (174a) still holds, and the only difference is that D D Dı ; hence   T D mI C Tı D mI C 2 D C B p.t / ı :

(189)

Similarly, (174c) is also valid; hence      1

C p.t /  Tr C p.t / I D 2 1 D p.t / ı ; 3

(190)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

55

  where, in virtue of (188), D p.t / D D p.t / ı . Multiplication of (190) by F>

p.t / from the right and F p.t / from the left and the evolution equation for the left Cauchy-Green tensor (163) then yield   O  1 Tr B p.t / B p.t / D 0 1 B p.t / C B2 p.t /  3

(191)

as the evolution equation for B p.t / .

4.4.3

Application of the Maximization of the Entropy Production: Constitutive Relations for Oldroyd-B and Maxwell Viscoelastic Fluids The last example is devoted to discussion of the derivation of incompressible Oldroyd-B and Maxwell models for viscoelastic fluids and their compressible counterparts. Unlike in the previous example, the full thermodynamic procedure based on the assumption of the maximization of the entropy production is now applied. The starting point is the formula for the entropy production derived in the previous section (see (173)) that can be rewritten in the form D

     1 

M  Tr B p.t / C div v Tı  B p.t / ı W Dı C m C pth  3   r : (192) C C p.t /  I W D p.t /  j q  

Using this formula one can continue with STEP 4 of the full thermodynamic procedure discussed in Sect. 4.1. STEP 4: At this point the constitutive function ,  D  , is chosen as   2 C 3  Ddef Q Dı ; div v; D p.t / ; r D 2 jDı j2 C .div v/2 3 C 2 1 D p.t / W C p.t / D p.t / C

jr j2 ; 

(193)

where the right Cauchy-Green tensor C p.t / is understood as a state variable and , 2 C3 , 1 , and are positive constants. Using the definition of the right 3 Cauchy–Green tensor (see (156b)) and the properties of the trace, it is easy to ˇ2 ˇ see that D p.t / W C p.t / D p.t / D ˇF p.t / D p.t / ˇ  0; hence the chosen constitutive function is nonnegative as required by the second law of thermodynamics. Further it is a strictly convex smooth function.

56

J. Málek and V. Pr˚uša

Introducing the auxiliary function ˆ for constrained maximization problem max

Dı ;div v;D p.t / ;r

  Q Dı ; div v; D p.t / ; r

(194)

subject to the constraint (192) as      ˆ Ddef Q Dı ; div v; D p.t / ; r C ` Q Dı ; div v; D p.t / ; r     

M  Tı  B p.t / ı W Dı  m C pth  Tr B p.t / C div v 3   r  C p.t /  I W D p.t / C j q  ; (195)  where ` is the Lagrange multiplier, the conditions for the extrema are   1 C ` @Q D Tı  B p.t / ı ; ` @Dı

1 C ` @Q M  Tr B p.t / C ; D m C pth ` @div v 3 Q   1 C ` @ D C p.t /  I ; ` @D p.t / jq 1 C ` @Q D : ` @r 

(196a) (196b) (196c)

(196d)

On the other hand, direct differentiation of (193) yields @Q D 4 Dı ; @Dı 2 @Q D .2 C 3/ div v; @div v 3   @Q D 2 1 C p.t / D p.t / C D p.t / C p.t / ; @D p.t / @Q r D 2 : @r 

(197a) (197b) (197c)

(197d)

Now it is necessary to find a formula for the Lagrange multiplier `. This can be done as follows. First, each equation in (196) is multiplied by the corresponding affinity, that is, by Dı , div v, D p.t / , and r, respectively, and then the sum of all equations is taken. This manipulation yields

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

57

"

# @Q @Q @Q @Q W Dı C W D p.t / C div v C  r @Dı @div v @D p.t / @r     

M  Tr B p.t / C div v D Tı  B p.t / ı W Dı C m C pth 3   r C C p.t /  I W D p.t /  j q  : (198) 

1C` `

The right-hand side is identical to Q D   (see (192)), while the term on the leftQ Consequently 1C` D Q , hand side reduces, in virtue of (193) and (197), to 2. ` 2Q which fixes the value of the Lagrange multiplier 1C` 1 D : ` 2

(199)

Inserting (197) and (199) into (196), one finally concludes that   Tı  B p.t / ı D 2 Dı ; M  m C pth

2 C 3 Tr B p.t / C D div v; 3 3    

C p.t /  I D 1 C p.t / D p.t / C D p.t / C p.t / ; j q D  r :

(200a) (200b) (200c) (200d)

Further, if (200c) holds, then it can be shown that the symmetric positive definite matrix C p.t / and the symmetric matrix D p.t / commute. (In order to prove that C p.t / and D p.t / commute, it suffices to show that the eigenvectors of C p.t / and D p.t / coincide, which is in virtue of (200c) an easy task. See the Appendix in Rajagopal and Srinivasa [77] for details.) If the matrices commute, then (200c) in fact reads  

C p.t /  I D 2 1 C p.t / D p.t / :

(201)

STEP 5: The first two equations in (200) allow one to identify the constitutive relation for the full Cauchy stress tensor T D mI C Tı ,   M I C 2 D C B p.t /  I C  .div v/ I: T D pth

(202)

Further, (201) can be rewritten as  

C p.t /  I D 2 1 F>

p.t / F p.t / D p.t / :

(203)

58

J. Málek and V. Pr˚uša > which upon multiplication by F>

p.t / from the right and by F p.t / from the left

yields a formula for F p.t / D p.t / F>

p.t / . This formula can be substituted into (163) that yields the evolution equation for the left Cauchy-Green tensor, O   1 B p.t / C B p.t /  I D 0:

(204)

The set of equations (67) supplemented with the constitutive equations for the heat flux (see (200d)), the Cauchy stress tensor (see (202)), and the evolution equation for the left Cauchy-Green tensor B p.t / (see (204)) forms a closed system of equations for the density , the velocity field v, the specific internal energy e (or the temperature ), and the left Cauchy-Green tensor B p.t / .   Introducing the extra stress tensor S by S Ddef B p.t /  I , equations (202) and (204) can be rewritten as M I C 2 D C S C  .div v/ I; T D pth

1 O S C S D 2 1 D:

(205a) (205b)

Further manipulation based on the redefinition of the extra stress tensor SQ Ddef S C 2 D allows one to rewrite system (205) as M I C SQ C  .div v/ I; T D pth

1 OQ 2 1 O D: S C SQ D 2 . 1 C / D C



(206a) (206b)

The expressions (206) suggest that the derived model can be seen as a compressible variant of the classical Oldroyd-B model for viscoelastic incompressible fluids developed by Oldroyd [65]. On the other hand, if the constitutive relation for the traceless part of the Cauchy stress tensor reads   Tı  B p.t / ı D 0;

(207)

which corresponds to D 0 in (200a), and the constitutive relation for the mean normal stress is M  m C pth

which corresponds to the choice Cauchy stress tensor reads

Tr B p.t / C D 0; 3

2 C3 3

(208)

D 0 in (200b), then the formula for the full

  M T D pth I C B p.t /  I :

(209)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

59

  Introducing the extra stress tensor S Ddef B p.t /  I , it follows that the system (205) reduces to M I C S; T D pth

1 O S C S D 2 1 D:

(210a) (210b)

This model can be denoted as a compressible variant of the classical Maxwell model for viscoelastic incompressible fluids; see, for example, Tanner and Walters [90]. Finally, if the fluid is assumed to be incompressible, then it is subject to constraint Tr D D 0:

(211)

However, then the same procedure as above can be still applied. One starts from (193) to (194) with the only modification that the terms with div v disappear, and one ends up with the same system of equations as those given in (200) (with the only exception that the second equation in (200b) is missing). The derived constitutive relations read



C p.t /

  Tı D 2 Dı C B p.t / ı ;   I D 2 1 C p.t / D p.t / ;

(212a) (212b)

j q D  r ;

(212c)

  T D mI C 2 Dı C B p.t / ı

(213)

which translates to

where the mean normal stress m is a quantity that cannot be specified via a constitutive relation. (See Sect. 4.2 for discussion.) Concerning the alternative approach based on the enforcement of the incompressibility constraint by an addition of an extra Lagrange multiplier to the maximization procedure, the interested reader is referred to Málek et al. [58]. Introducing the notation  Ddef m 

Tr B p.t / C ; 3

(214)

it follows that (213) can be rewritten as   T D I C 2 D C B p.t /  I :

(215a)

(Recall that in virtue of (211), one has D D Dı .) Further, the evolution equation for the left Cauchy-Green tensor B p.t / reads

60

J. Málek and V. Pr˚uša O   1 B p.t / C B p.t /  I D 0:

(215b)

  Defining the extra stress tensor as S Ddef B p.t /  I or as SQ Ddef 2 D C S, it follows that (215) can be converted into the equivalent form T D I C 2 D C S; 1 O S C S D 2 1 D;

(216a) (216b)

or into the following equivalent form Q T D I C S; 1 OQ 2 1 O D; S C SQ D 2 . 1 C / D C



(217a) (217b)

which are the frequently used forms of constitutive relations for the Oldroyd-B fluid, which is a popular model for viscoelastic fluids derived by Oldroyd [65]. (Oldroyd [65] has used formulae (217).) Further, if one formally sets D 0, then one gets the standard incompressible Maxwell fluid. Note however that the classical derivation by Oldroyd [65] is based only on mechanical considerations and that the compatibility of the model with the second law of thermodynamics is not discussed at all. Second, the present approach naturally gives one an evolution equation for the internal energy that automatically takes into account the storage mechanisms related to the “elastic” part of the deformation. If the internal energy is expressed as a function of the temperature and other variables, then the evolution equation for the internal energy leads, upon the application of the chain rule, directly to the evolution equation for the temperature. (See Sect. 4.2.1 for the same in the context of a compressible Navier-Stokes-Fourier fluid.) Moreover, the inclusion of the storage mechanisms in the internal energy is compatible with the specification of the constitutive relation for the Cauchy stress tensor. Again, such issues have not been discussed in the seminal contribution by Oldroyd [65] or for that matter by many following works on viscoelasticity. Obviously, more complex viscoelastic models can be designed by appealing to the analogy with more involved spring–dashpot systems; see, for example, Karra and Rajagopal [49, 50], Hron et al. [42], Pr˚uša and Rajagopal [72], or Málek et al. [59].

4.5

Beyond Linear Constitutive Theory

In the previous parts, the development of the constitutive equations has been based either on linear relationships between the affinities and fluxes or on postulating quadratic constitutive equations for the entropy production. In general, this

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

61

limitation to linear constitutive relations or to a quadratic ansatz for the entropy production is not necessary. In fact a plethora of nonlinear models can be developed in a straightforward manner following the method based on the maximization of entropy production. Their relevance with respect to real material behavior should be however carefully justified. A list of some popular simple nonlinear models for incompressible fluids is given below. Clearly, the list is not complete, and other models can be found in the literature as well. In particular, the list does not include nonlinear viscoelastic models at all. The first class of nonlinear models is the class of models where the Cauchy stress tensor is decomposed as T D mI C S;

(218)

where m is the mean normal stress, and the relation between the traceless extra stress tensor S and the symmetric part of the velocity gradient D takes the form of an algebraic relation f.S; D/ D 0;

(219)

where f is a tensorial function. This implicit relation is a generalization of the standard constitutive relation for the incompressible Navier-Stokes fluid (see Sect. 4.2.3) where S  2 ? D D 0:

(220)

Note that the standard way of writing algebraic constitutive relations for a nonNewtonian fluid is S D f.D/. However, it has been argued by Rajagopal [73,74] that the standard setting is too restrictive and that (219) should be preferred to S D f.D/. (See, e.g., Málek et al. [57], Pr˚uša and Rajagopal [71], Le Roux and Rajagopal [53], Perlácová and Pr˚uša [69], and Janeˇcka and Pr˚uša [45] for further developments of the idea. Mathematical issues concerning some of the models that belong to the class (219) have been discussed, e.g., by Bulíˇcek et al. [12] and Bulíˇcek et al. [13].) The implicit relation (219) opens the possibility of describing – in terms of the same quantities that appear in (220) – various non-Newtonian phenomena such as stress thickening or stress thinning, shear thickening or shear thinning, yield stress, and even normal stress differences. None of these important phenomena can be captured by the standard Navier-Stokes model (220). (The reader is referred to Málek and Rajagopal [54] or any textbook on non-Newtonian fluid mechanics for the discussion of these phenomena and their importance.) The mechanics of complex fluids is indeed an unfailing source of qualitative phenomena that go beyond the reach of the Navier-Stokes model. For interesting recent observations concerning the behavior of complex fluids, see, for example, the references in the reviews by Olmsted [66] and Divoux et al. [23].

62

J. Málek and V. Pr˚uša

Particular models that fall into class (219) are the models in the form S D 2 .D/D;

(221)

where the generalized viscosity .D/ is given by one of the formulae listed below. Model .D/ D 0 jDjn1 ;

(222)

where 0 is a positive constant and n is a real constant, is called Ostwald–de Waele power law model; see Ostwald [67] and de Waele [94]. Models 0  1

.D/ D 1 C

n ; .1 C ˛jDj2 / 2  n1  .D/ D 1 C . 0  1 / 1 C ˛jDja a ;

(223) (224)

are called Carreau model (see Carreau [16]) and Carreau–Yasuda model (see Yasuda [96]), respectively. Here 0 and 1 are positive real constants, and n and a are real constants. Other models with nonconstant viscosity are the Eyring models (see Eyring [26] and Ree et al. [82]), where the generalized viscosity takes the form .D/ D 1 C . 0  1 / .D/ D 0 C 1

arcsinh .˛jDj/ ; ˛jDj

arcsinh .˛1 jDj/ arcsinh .˛2 jDj/ C 2 ; ˛1 jDj ˛2 jDj

(225) (226)

and 0 , 1 , 2 , 1 , ˛1 , and ˛2 are positive real constants. Finally, the model named after Cross [21] takes the viscosity in the form 0  1 1 C ˛jDjn

(227)

.D/ D 1 C ˛jDjn1 ;

(228)

.D/ D 1 C and the model named after Sisko [88] reads

where 0 , 1 , and ˛ are positive real constants and n is a real constant. Another subset of general models of the type (219) are models where the generalized viscosity depends on the traceless part of the Cauchy stress tensor Tı D S, that is, models, where S D .S/D:

(229)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

63

Examples are the Ellis model (see, e.g., Matsuhisa and Bird [61]) 0

.S/ D

1 C ˛jSjn1

;

(230)

where 0 and ˛ are positive real constants and n is a real constant or the model proposed by Glen [33], .S/ D ˛jSjn1 ;

(231)

where ˛ is a positive real constant and n is a real constant, the model by Seely [85]  jSj

.S/ D 1 C . 0  1 / e

0

;

(232)

or the models used for the description of the flow of the ice, A

.S/ D 

jSj2 C 02

;  n1 2

(233)

See, for example, Pettit and Waddington [70] and Blatter [5]. Here 0 , 1 , 0 , and A are positive real constants, and n is a real constant. Particular parameter values for models (222), (223), (224), (225), (226), (227), (228) and (230), (231), (232), (233) can be found in the referred works and/or in the follow-up works. Other popular models are the models with activation criterion introduced by Bingham [3] and Herschel and Bulkley [39]. These models are usually (see, e.g., Duvaut and Lions [25]) written down in the form of a dichotomy relation jSj   , D D 0

and

jSj >  , S D

 D C 2 .jDj/D; jDj

(234)

where is a positive function and  is a positive constant. (Constant  is called the yield stress.) It is worth emphasizing that these models can be rewritten as DD

1 .jSj   /C S; 2 .jDj/ jSj

(235)

where x C D maxfx; 0g, which shows that the yield stress models also fall into the class (219). This observation can be exploited in the discussion of the physical and mathematical properties of the models; see Rajagopal and Srinivasa [79] and Bulíˇcek et al. [13]. Note that if (234) holds, then S cannot be considered as a function of D, while (235) gives a functional (continuous) dependence of D on S. Since the function at the right-hand side of (235) is tacitly supposed to be zero for S D 0, the alternative viewpoint given by (219) with a continuous tensorial function f defined on the

64

J. Málek and V. Pr˚uša

Cartesian product of S and D allows one to avoid description of the material via a multivalued or discontinuous function. See Bulíˇcek et al. [13] and Bulíˇcek et al. [10] for exploiting this possibility as well as the symmetric roles of S and D in (219) in the analysis of the corresponding initial and boundary value problems. Another advantage of the formulation (235) is that it allows one to replace, in a straightforward way, the constant yield stress  by a yield function that can depend on the invariants of both S and D,  D  .S; D/. Finally, the class of implicit relations (219) can be viewed as a subclass of models where the full Cauchy stress tensor T and the symmetric part of the velocity gradient D are related implicitly, f.T; D/ D 0:

(236)

This apparently subtle difference has significant consequences. Constitutive relations in the form (236) provide, contrary to (219), a solid theoretical background to incompressible fluid models where the viscosity depends on the pressure (mean normal stress); see Rajagopal [74] and Málek and Rajagopal [56] for an in-depth discussion. Such models have been proposed a long time ago by Barus [2], and the practical relevance of such models has been demonstrated by Bridgman [6] and Bridgman [7]. In the simplest settings, the model for a fluid with pressure dependent viscosity can take the form T D pI C 2 ref eˇ.ppref / D;

(237)

where pref – the reference pressure – and ref – the reference viscosity – are positive constants. If necessary, this model can be combined with power law type models. This is a popular choice in lubrication theory; see, for example, Málek and Rajagopal [56].

4.6

Boundary Conditions for Internal Flows of Incompressible Fluids

As it has been already noted, the specification of the boundary conditions is a nontrivial task. Since the boundary conditions can be seen as a special case of constitutive relations at the interface between two materials, thermodynamical considerations could be of use even in the discussion of the boundary conditions. A simple example concerning the role of thermodynamics in the specification of the boundary conditions for incompressible viscous heat nonconducting fluids is given below. The problem of internal flow in a fixed vessel represented by the domain has been discussed in Sect. 3.2, and it is revisited here in a slightly different setting. In Sect. 3.2 the boundary condition on the vessel wall has been the no-slip boundary condition

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

vj@ D 0:

65

(238)

In the present case, one enforces only the no-penetration boundary condition v  nj@ D 0;

(239)

where n denotes the unit outward normal to , and the question concerning the value of the velocity vector in the tangential direction to the vessel wall is for the moment left open. Introducing the notation u Ddef u  .u  n/ n

(240)

for the projection of any vector u to the tangent plane to the boundary, it follows that the product .Tv/  n can be rewritten as .Tv/  n D mv  n C S W .v ˝ n/ D S W .n ˝ v/ D .Sn/  v D .Sn/  v :

(241)

This observation is a consequence of the symmetry of the Cauchy stress tensor and the decomposition of the Cauchy stress tensor to the mean normal stress and the traceless part; see (218). The multiplication of the balance of momentum (63a) by v followed by integration over the domain yields Z

d dt



1  vv 2



Z .div T/  v dv:

dv D

(242)



The right-hand side can be, following Sect. 3.2, manipulated as Z

Z

Z

.div T/  v dv D

Z

.div Tv/ dv 

Z

T W D dv D 

T W D dv C

Tv  n ds: @

(243)

Consequently, the evolution equation for the net kinetic energy Ekin reads dEkin D dt

Z

Z T W D dv C

Tv  n ds;

(244)

@

which can be in virtue of (241) and the incompressibility condition div v D 0 further rewritten as dEkin D dt

Z

Z S W D dv 

s  v ds; @

(245)

66

J. Málek and V. Pr˚uša

where the notation s Ddef  .Sn/

(246)

has been used. Unlike in Sect. 3.2, one now gets a boundary term contributing to the evolution equation for the net kinetic energy. The boundary term is the key to the thermodynamically based discussion of the appropriate boundary conditions. The desired decay of the net kinetic energy is in the considered case guaranteed if one enforces pointwise positivity of the product T W D in and the pointwise positivity of the product s  v at @ . This restriction can be used to narrow down the class of possible relations between the value of the velocity in the tangential direction v and the projection of the stress tensor .Sn/ , where the relation between v and .Sn/ is the sought boundary condition. R Note that in the full thermodynamic setting, both the volumetric term T W D dv R and the boundary term @ s  v ds would appear in the entropy production for the whole system. Both the boundary term and the volumetric term have the flux/affinity structure. In linear nonequilibrium thermodynamics, the fluxes and affinities in the volumetric term are connected linearly via a positive (nonnegative) coefficient of proportionality in order to guarantee the validity of the second law of thermodynamics. Application of the same approach to the boundary term leads to a linear relation between s and v s D  v ;

(247)

where  is a positive constant. This is the Navier slip boundary condition; see Navier [63]. Moreover, one can identify two cases where the boundary term vanishes. If s D 0, then one gets the perfect slip boundary condition, and if v D 0, then one gets the standard no-slip boundary condition. Further, a general relation between s and v can take, following (219), the form of an implicit constitutive relation f .s; v / D 0;

(248)

which considerably expands the number of possible boundary conditions. In particular, the threshold-slip (or stick-slip) boundary condition that is usually described by the dichotomy relation jsj   , v D 0

and

jsj >  , s D

 v C  v  ; jv j

(249)

where  and  are positive constants, can be seen as a special case of (248). Indeed, (249) can be rewritten as v D

1 .jsj   /C s:  jsj

(250)

1 Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids

67

(Note the similarity between the threshold-slip boundary condition (249) and the Bingham/Herschel-Bulkley model for fluids with the yield stress behavior (235). The formulation of the threshold-slip boundary condition in the form (250) can be again exploited in the mathematical analysis of the corresponding governing equations; see Bulíˇcek and Málek [11].) An example of more involved thermodynamical treatment of boundary conditions can be found in the study by Heida [37]. Concerning a recent review of the nonstandard boundary conditions used by practitioners in polymer science, the reader is referred, for example, to Hatzikiriakos [36].

5

Conclusion

Although the classical (in)compressible Navier-Stokes-Fourier fluid models have been successfully used in the mathematical modelling of the behavior of various substances, they are worthless from the perspective of modern applications such as polymer processing. Navier-Stokes-Fourier models are simply incapable of capturing many phenomena observed in complex fluids; see, for example, the list of non-Newtonian phenomena in Málek and Rajagopal [54], the historical essay by Tanner and Walters [90], or the classical experimentally oriented treatises by Coleman et al. [20], Barnes et al. [1], or Malkin and Isayev [60] to name a few. The need to develop mathematical models for the behavior of complex materials leads to the birth of the theory of constitutive relations. In the early days of the theory, constitutive relations have been designed by appealing to purely mechanical principles. This turns out to be insufficient as the complexity of the models increases. Nowadays, the mathematical models aim at the description of an interplay between various mechanisms such as heat conduction, mechanical stress, chemical reactions, electromagnetic field, or the interaction of several continuous media in mixtures; see, for example, Humphrey and Rajagopal [44], Rajagopal [75], Dorfmann and Ogden [24], and Pekaˇr and Samohýl [68]. In such cases the correct specification of the energy transfers is clearly crucial. Consequently, one can hardly hope that an ad hoc specification of the complex nonlinear constitutive relations for the quantities that facilitate the energy transfers is the way to go. A theory of constitutive relations that focuses on energy transfers and that guarantees the compatibility of the arising constitutive relations with the second law of thermodynamics is needed. The modern theory of constitutive relations outlined above can handle the challenge. The theory from the very beginning heavily relies on the concepts from nonequilibrium thermodynamics, and the restrictions arising from the laws of thermodynamics are automatically built into the derived constitutive relations. Naturally, the models designed to describe the behavior of complex materials in farfrom-equilibrium processes are rather complicated. In particular, the arising systems of partial differential equations are large and nonlinear.

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Fortunately, the available numerical methods for solving nonlinear partial differential equations as well as the available computational power are now at the level that makes the numerical solution of such systems feasible. This is a substantial difference from the early days of the mechanics and thermodynamics of continuous media; see, for example, Truesdell and Noll [91]. Since the quantitative predictions based on complex models are nowadays within the reach of the scientific and engineering community, they can actually serve as a basis for answering important questions in the applied sciences and technology. This is a favorable situation for a mathematical modeller equipped with a convenient theory of constitutive relations. The design of suitable mathematical models for complex materials undergoing farfrom-equilibrium processes does matter – from the practical point of view – more than ever.

6

Cross-References

 Concepts of Solutions in the Thermodynamics of Compressible Fluids  Equations for Viscoelastic Fluids  Multi-Fluid Models Including Compressible Fluids  Solutions for Models of Chemically Reacting Compressible Mixtures

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2

Variational Modeling and Complex Fluids Mi-Ho Giga, Arkadz Kirshtein, and Chun Liu

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Nonequilibrium Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Energetic Variational Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Hookean Spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Gradient Flow (Dynamics of Fastest Descent) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Basic Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Flow Map and Deformation Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Newtonian Fluids and Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Elasticity and Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Other Approaches to Elastic Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Generalized Diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Complex Fluid Mixtures: Diffusive Interface Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Surface Tension and the Sharp Interface Formulation . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Diffusive Interface Approximations (Phase Field Methods) . . . . . . . . . . . . . . . . . . . 4.3 Boundary Conditions in the Diffusive Interface Models . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74 76 76 79 80 82 82 84 86 89 93 97 97 99 105 106 108 108

M.-H. Giga Graduate School of Mathematical Sciences, University of Tokyo, Meguro-ku, Tokyo, Japan e-mail: [email protected] A. Kirshtein () Department of Mathematics, Pennsylvania State University, University Park, PA, USA e-mail: [email protected] C. Liu Department of Mathematics, Pennsylvania State University, University Park, PA, USA Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL, USA e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_2

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Abstract

In this chapter, a general energetic variational framework for modeling the dynamics of complex fluids is introduced. The approach reveals and focuses on the couplings and competitions between different mechanisms involved for specific materials, including energetic contributions vs. kinematic transport relations, conservative parts vs. dissipative parts and kinetic parts vs. free energy parts of the systems, macroscopic deformation or flows vs. microscopic deformations, bulk effects vs. boundary conditions, etc. One has to notice that these variational approaches are motivated by the seminal works of Rayleigh (Proc Lond Math Soc 1(1):357–368, 1871) and Onsager (Phys Rev 37(4):405, 1931; Phys Rev 38(12):2265, 1931). In this chapter, the underlying physical principles and background, as well as the limitations of these approaches, are demonstrated. Besides the classical models for ideal fluids and elastic solids, these approaches are employed for models of viscoelastic fluids, diffusion, and mixtures.

1

Introduction

The focus of this chapter is on the mathematical modeling of anisotropic complex fluids whose motion is complicated by the existence of mesoscales or subdomain structures and interactions. These complex fluids are ubiquitous in daily life, including wide varieties of mixtures, polymeric solutions, colloidal dispersions, biofluids, electro-rheological fluids, ionic fluids, liquid crystals, and liquid-crystalline polymers. On the other hand, such materials often have great practical utility since the microstructure can be manipulated by external field or forces in order to produce useful mechanical, optical, or thermal properties. An important way of utilizing complex fluids is through composites of different materials. By blending two or more different components together, one may derive novel or enhanced properties from the composite. The properties of composites may be tuned to suit a particular application by varying the composition, concentration, and, in many situations, the phase morphology. One such composite is polymer blends [121]. Under optimal processing conditions, the dispersed phase is stretched into a fibrillar morphology. Upon solidification, the long fibers act as reinforcement and impart great strength to the composite. The effect is particularly strong if the fibrillar phase is a liquidcrystalline polymer [99]. Another example is polymer-dispersed liquid crystals, with liquid crystal droplets embedded in a polymer matrix, which have shown great potential in electro-optical applications [127]. Unlike solids and simple liquids, the model equations for complex fluids continue to evolve as new experimental evidences and applications become available [97]. The complicated phenomena and properties exhibited by these materials reflect the coupling and competition between the microscopic interactions and the macroscopic dynamics. New mathematical theories are needed to resolve the ensemble of microelements, their intermolecular and distortional elastic interactions, their coupling to hydrodynamics, and the applied electric or magnetic fields. The most

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common origin and manifestation of anomalous phenomena in complex fluids are different “elastic” effects [77]. They can be attributed to the elasticity of deformable particles; elastic repulsion between charged liquid crystals, polarized colloids, and multicomponent phases; elasticity due to microstructures; or bulk elasticity endowed by polymer molecules in viscoelastic complex fluids. These elastic effects can be represented in terms of certain internal variables, for example, the orientational order parameter in liquid crystals (related to their microstructures), the distribution density function in the dumbbell model for polymeric materials, the magnetic field in magnetohydrodynamic fluids, the volume fraction in mixture of different materials, etc. The different rheological and hydrodynamic properties will be attributed to the special coupling (interaction) between the transport (macroscopic fluid motions) of the internal variable and the induced (microscopic) elastic stress [115,116]. This coupling gives not only the complicated rheological phenomena but also formidable challenges in analysis and numerical simulations of the materials. The common feature of the systems described in this chapter is the underlying energetic variational structure. For an isothermal closed system, the combination of the first and second laws of thermodynamics yields the following energy dissipation law [6, 11, 39, 56]: d total E D ; dt

(1)

where E total is the sum of kinetic energy and the total Helmholtz free energy and  is the entropy production (here the rate of energy dissipation). The choices of the total energy functional and the dissipation functional, together with the kinematic (transport) relations of the variables employed in the system, determine all the physical and mechanical considerations and assumptions for the problem. The energetic variational approaches are motivated by the seminal work of Rayleigh [106] and Onsager [100, 101]. The framework, including Least Action Principle and Maximum Dissipation Principle, provides a unique, well-defined, way to derive the coupled dynamical systems from the total energy functionals and dissipation functions in the dissipation law (1) [67]. Instead of using the empirical constitutive equations, the force balance equations are derived. Specifically, the Least Action Principle (LAP) determines the Hamiltonian part of the system [2, 5, 50], and the Maximum Dissipation Principle (MDP) accounts for the dissipative part [11, 101]. Formally, LAP represents the fact that force multiplies distance is equal to the work, i.e., ıE D force  ıx; where x is the location and ı the variation/derivative. This procedure gives the Hamiltonian part of the system and the conservative forces [2, 5]. The MDP, by Onsager and Rayleigh [67, 100, 101, 106], produces the dissipative forces of the system, ı 12  D force  ı x: P The factor 12 is consistent with the choice of quadratic form for the “rates” that describe the linear response theory for longtime near-equilibrium dynamics [74]. The final system is the result of the balance of all these forces (Newton’s Second Law). Both total energy and energy dissipation may contain terms related to microstructure and those describing macroscopic flow. Competition between different parts

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of energy, as well as energy dissipation, defines the dynamics of the system. The main goal of this chapter is on describing the role of microstructures in the special coupling between the kinematic transport and the induced “elastic” stresses.

2

Nonequilibrium Thermodynamics

In this section, some basic thermodynamic principles and general relations between energy laws and differential equations are described. We first clarify notation of variations of the functionals [50, 54]. Let E D E. / be a functional depending on a function in a space H which is equipped with an inner product h ; iH . The variation ıE D ı E of a function E is defined as ı E. / D lim ŒE. h!0

C hı /  E. / =h;

where ı is a function so that C hı is a variation of . The quantity ı E is often called a directional derivative in the direction of ı at . It is formally the Gâteau derivative of E at in the direction of ı . If ı E can be written as ı E. / D hf; ı iH ; with some f for a big class of ı , we often write f by H_

ıE ı

or simply

ıE : ı

This quantity corresponds to the total derivative or the Fréchet derivative if the latter is well defined [55]. It is simply called the variational derivative . In this notation, denominator points to the function with respect to which the variation of the functional in the numerator is taken.

2.1

Energetic Variational Approaches

The first law of thermodynamics [56] states that the rate of change of the sum of kinetic energy K and the internal energy U can be attributed to either the work WP P done by the external environment or the heat Q: d P .K C U / D WP C Q: dt In other words, the first law of thermodynamics is really the law of conservation of energy. It is noticed the internal energy describes all the interactions in the system. In order to analyze heat, one needs to introduce the entropy S [56], which naturally leads to the second law of thermodynamics [39, 56]:

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77

dS P C DQ dt

where T is the temperature and S is the entropy.  is the entropy production which is always nonnegative and gives the rate of energy dissipation in irreversible systems. Subtracting the two laws, in the isothermal case when T is constant, one arrives at: d .K C U  T S/ D WP  ; dt where F D U  T S is called the Helmholtz free energy. We denote E total D K C F to be total energy of the system. If the system is closed, when work done by the environment WP D 0, the energy dissipation law of the system can be written as dE total D 2D: dt

(2)

The quantity D D 12  is sometimes called the energy dissipation. The dissipative law allows one to distinguish two types of systems: conservative (or Hamiltonian) and dissipative. The choices of the total energy components and the energy dissipation take into consideration all the physics of the system and determine the dynamics through the two distinct variational processes: Least Action Principle (LAP) and Maximum Dissipation Principle (MDP). To derive the differential equation describing the conservative system ( D 0), one employs the Least Action Principle (LAP) [2, 5], which says that the dynamics is determined as a critical point of the action functional (Remark 1 below). We RT RT give its equivalent form. We consider functionals 0 Kdt and 0 Fdt defined for a function x (the trajectory in Lagrangian coordinates, if applicable) depending on space time variables. The inertial and conservative force from the kinetic and free energies are, respectively, defined as RT ı 0 Kdt forceinertial D H _ ; ıx RT ı 0 Fdt : forceconservative D H _ ıx The space H is typically taken as the space time L2 space, L2x;t , i.e., L2x;t D L2 .0; T I L2x /, where L2x is the L2 space in the spatial variables. (These are called variational forces.) In other words, for all ıx, Z T Z T Kdt D hforceinertial ; ıxiL2x;t D hforceinertial ; ıxiL2x dt ı Z ı 0

0

T

Fdt D hforceconservative ; ıxiL2x;t D

Z

0

T 0

hforceconservative ; ıxiL2x dt:

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The LAP can be written as Z

T

ı 0

Kdt D ı

Z

T

Fdt

0

for all ıx. This gives the natural variational form (the weak form) of the forces with suitable test functions ıx. The strong form of the system (Euler-Lagrange equation) can be also written as a force balance (without dissipation). forceinertial D forceconservative :

(3)

The inertial force corresponds to the inertial term ma in Newton’s Second Law, where a is the acceleration and m is the mass. Note that if the variation is performed on a bounded domain and involves integration by parts, one has to assume specific boundary conditions to cancel the boundary terms, so that no boundary effects are involved. Remark 1. The standard approach [5] dictates to define the Lagrangian functional RT L D K  F and the action functional as A .x/ D 0 Ldt . The Euler-Lagrange D 0. equation in this case is H _ ıA ıx For a dissipative system . D 2D > 0/, according to Onsager [100, 101], the dissipation D is taken to be proportional to a “rate” xt raised to a second power. The Maximum Dissipation Principle (MDP) [67] implies that the dissipative force may be obtained by minimization of the dissipation functional D with respect to the above mentioned “rate.” Hence, through MDP, the dissipative force (linear with respect to the same rate function) can be derived as follows: ıD D hforcedissipative ; ıxt iHQ : In other words, Q _ıD=ıxt D forcedissipative : H Note that the test function in MDP is different from that in LAP before. Remark 2. It is important to note that although the limitation for the dissipation D to be quadratic in “rate” is rather restrictive, strong nonlinearities can be introduced through coefficients independent of the “rate.” When all forces are derived, according to the force balance (Newton’s Second Law, where inertial force plays role of ma): forceinertial D forceconservative C forcedissipative :

(4)

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Notation. For shorter notation, one can write Eq. (4) as H _ H_

ı

RT

0 F dt ıx

ı

RT 0

Kdt ıx

D

Q _ ıD with H D L2 .0; T I H Q /. CH ıxt

It is important to notice that Eq. (4) uses the strong form of the variational result. This might bring ambiguity in the original variational weak form, since the test functions may be in different spaces.

2.2

Hookean Spring

As a start, a simple ordinary differential equations (ODE) example of a dissipative system is considered here, which had been originally proposed by Lord Rayleigh [106]: the Hookean spring of which one end is attached to the wall and the other end to a mass m (see Fig. 1). Let x .t / be a displacement of the mass from the equilibrium. Consider that the spring has friction-based damping which is proportional to the velocity (relative friction to the resting media). Under these assumptions, KD

mxt2 ; 2

FD

kx 2 ; and 2

DD

 xt2 ; 2

where k is spring strength material parameter and  is damping coefficient. The energy dissipation law is clearly as follows: d dt



mxt2 kx 2 C 2 2



D  xt2 :

The corresponding action functional of the spring [50] in terms of the position x.t /: AD

Z 0

T



mxt2 kx 2  2 2

 dt:

Then the Least Action Principle, i.e., variation with respect to the trajectory x.t /, yields [50]:

Fig. 1 Spring attached to a wall on one end, with mass m on the other end

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Z ıA D 0

Z

T

Œmxt .ıx/t  kxıx dt D  D

T

.mxt t  kx/ ıx dt 0

ıA L2t _ ; ıx ıx



Z

T



D L2 .0;T /

0

ıA L2t _ ; ıx ıx

 dt: R

Here the space H with inner product is L2t D L2 .0; T / because here L2x is just R. On the other hand, the principle of maximum dissipation gives R_

ıD D  xt : ıxt

Indeed, looking at forces involved and formulating Newton’s Second Law (F D ma) for this system, one can get mxt t D kx   xt ; or equivalently mxt t C  xt C kx D 0;

(5)

D which is equivalent to the variational force balance (corresponding to (4)) L2t _ ıA ıx ıD R_ ıx for this example. t Looking at the explicit solution of (5), it is clear that the Hamiltonian part describes the transient dynamics, the oscillation near initial data, while the dissipative part gives the decaying longtime behavior near equilibrium.

2.3

Gradient Flow (Dynamics of Fastest Descent)

The energetic variational approaches have many different forms in practices and applications. Next look at the familiar example of gradient flow (dynamics of fastest descent): ıF.'/ (6)  C 't D 0; ı' where F is a general energy functional in terms of '. Here ' is a function of spatial variables with parameter time t, and the constant  > 0 is the dissipation rate which determines the evolution approaching the equilibrium. Such a system has been used in many applications both in physics and in mathematics; in particular, it is commonly used in both analysis and numerics to achieve the minimum of a given energy functional. It is clear that with natural boundary conditions (Dirichlet or Neumann), the solution of (6) satisfies the following energy dissipation law (by chain rule and integration by parts, if needed): Z d 1 F D j't j2 d x; dt   where  is a domain in a Euclidean space.

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On the other hand, one can put this in the general framework of energetic variational approaches. Notice that there is no kinetic energy in this system, indicating the nature of being the longtime near-equilibrium dynamics. Notation. When working on a bounded domain, one should consider a variation up to the boundary. Generally, for a functional E depending on a function defined in , the variation ı E is often of the form ı E. / D

Z

Z f ı dx C 

gı dS: @

Then, we denote f D

ıE ; ı

gD

ıE : ı@

Here f gives variational force inside the domain, while g is a kind of a boundary force. So if the boundary is taken into account, boundary forces should be also balanced. Unless mentioned otherwise, in this chapter, specific boundary conditions are taken to cancel the boundary effects (i.e., make boundary integralR equal to zero). R In the case of F D  W .'; rx '/ d x and D D 21  j't j2 d x, the variation leads to the following two variational derivatives: L2x _

ıF @W @W D r  C ; ı ' @r' @'

L2x _

ıD 1 D 't ; ı  't 

which after substitution in (4) yield equation (6). In this case, the boundary effects would be canceled out in case of homogeneous Dirichlet or Neumann boundary conditions. Remark 3. To derive implicit Euler’s time discretization scheme [8], one may consider minimization of the following functional: Z ( min ' nC1 given ' n



) ˇ ˇ2  nC1  1 ˇ' nC1  ' n ˇ C W ' ; r' nC1 d x:  2t

By introducing time discretization, one can avoid the two different variations and only take the variation with respect to ' nC1 . However, the scheme often fails in the case of  dependent on ', since it is unclear whether to take it explicit or implicit: explicit may cause stability issues and implicit will lead to a highly nonlinear system.

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3

Basic Mechanics

Before moving on to more complicated and realistic applications, it is important first to introduce some basic terminologies and concepts in continuum mechanics [36,59]. In particular, in this section, the relation between Eulerian (space reference) and Lagrangian (material reference) coordinates [119] is explored, and the variational techniques in terms of deformable medium are described. In this section, the boundary conditions are not in the focus of attention. However they may and should also be derived through the variational procedure with various specific boundary energy terms and dissipative terms.

3.1

Flow Map and Deformation Gradient

For a given velocity field u .x; t/, one can define the corresponding flow map (trajectory) x .X; t/ as xt D u;

x .X; 0/ D X:

(7)

In other words, x .X; t/ describes the position of a particle moving with velocity u and initial position X. Here x are the Eulerian coordinates, and X – the Lagrangian coordinates or initial configuration (see Fig. 2). Since the flow map should satisfy (7), its recovery is possible only if u .x; t/ has certain regularity properties, for instance, being Lipschitz in x [36]. In order to describe the evolution of structures or patterns (configurations), it is clear that one needs to consider the matrix of partial derivatives, the Jacobian matrix, the deformation gradient (or deformation tensor) [61]: F .X; t/ D

@x .X; t/ : @X

If one writes F by components .Fij /, our convention is Fij D

Fig. 2 A schematic illustration of a flow map x .X; t/. For t fixed x maps 0X to tx . For X fixed x .X; t/ is the trajectory of X

@xi : @Xj

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Then by chain rule: @Fij @ D @t @t



@xi @Xj

 D

@ @Xj



@xi @t

 D

X @ui @xk @ ui .x .X; t/ ; t / D ; @Xj @xk @Xj k

which in Eulerian coordinates will take the form as: @ @F Q D u .x .X; t/ ; t / D .rx u/ F: FQ t C .u  rx / FQ D @t @X Here FQ .x .X; t/ ; t / D F .X; t/ and rx denote the gradient. In Eulerian coordinates, FQ satisfies the following important identity: Q FQ t C .u  rx / FQ D .rx u/ F:

(8)

Remark 4. The form of (8) is directly related to the equation of vorticity w D curl u in inviscid incompressible fluids [94]: in two-dimensional cases, the solution of wt C .u  r/ w D 0 is expressed along the trajectory as w .x .X; t/ ; t / D w0 .X/; in three-dimensional case wt C .u  r/ w  .w  r/ u D 0, the solution becomes w .x .X; t/ ; t / D F w0 .X/. It is clear that the stretch term .w  r/u is the direct consequence of the deformation F , although F itself is absent from the original fluid equations. Remark 5. Incompressibility condition is actually a restriction on deformation det F D 1. By using Jacobi’s formula, 

0 D @t det F D det F  tr F

1





@X @u @t F D 1  tr @x @X

 D tr .rx u/ D rx  u;

which yields the usual incompressibility condition in conventional descriptions of fluids. Notice that the nonlinear constraint in Lagrangian coordinates becomes a linear one in Eulerian coordinates. (Here rx  u denotes the divergence of u.) Remark 6. F also determines the kinematic relations of transport of various physical quantities. The following formulations of kinematic relations describing transport of scalar quantities are expressed in Eulerian and Lagrangian coordinates as: 't C .u  rx / ' D 0 is equivalent to ' .x .X; t/ ; t / D ' .X; 0/ ; 't C rx  .'u/ D 0 is equivalent to ' .x .X; t/ ; t / D

' .X; 0/ : det F

(9) (10)

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3.2

Newtonian Fluids and Navier-Stokes Equations

Next the classic Newtonian fluids [36] are examined, and the Navier-Stokes equations are derived from the energetic variational approaches. Consider fluid with density  and velocity field u. Here the local mass conservation law is postulated, i.e., t C rx  .u/ D 0:

(11)

For fluids, one should consider the free energy depending only on the density  (the single most important characterization of the material being a fluid), which implies the following energy dissipation law: d dt

Z " 

3 ˇ ˇ  ˇ T ˇ2  2  juj ˇ ru C .ru/ ˇ 2 C ! ./ d x D  42 ˇ ˇ C    jr  uj 5d x; ˇ ˇ 2 2 3  2

#

Z

2

(12) ˇ ˇ P .ui;j Cuj;i /2 T ˇ2 ˇ where ˇ ruC.ru/ , ui;j D @ui =@xj . In general for matrix ˇ D i;j 2 2 p M , we write jM j D trMM T which is often called the Hilbert-Schmidt norm R R 2 or the Frobenius norm. Then K D  juj d x; F D  ! ./ d x; D D 2  ˇ ˇ  R ˇ ruC.ru/T ˇ2  1 2  ˇ ˇ C 2   13  jr  uj d x. The last being the viscosity contribu 2 tion [76], the relative friction between particles of the fluids. The constants  and  are called coefficients of viscosity ( is second viscosity coefficient), and !./ is free energy density. Since the rate in the dissipation is u D xt , one will have to take the variation with respect to the flow map x in the Lagrangian coordinates X. Since d x D .det F /X, and since (11) and (10) imply  .x .X; t/ ; t/ D 0 .X/ = det F .X; t/ with 0 .X/ D  .X; 0/, we observe that ı

ı

RT 0

RT 0

R T R 0 .X/ RTR .X; t/j2 det FdXdt D ı 0 12 0 .X/ jxt .X; t/j2 d Xdt Kdt D ı 0 12 det F jxt RT R RT R D 0 0 xt  ıxt d Xdt D  0 0 xt t  ıxd Xdt  RT R

d  dt u .x .X; t/ ; t /  ıx d xdt D 0 RT R D 0 Œ .ut C .u  r/ u/  ıx d xdt D h .ut C .u  rx /u/ ; ıxiL2x;t ;  RT R  Fdt D ı 0 ! det0F det F d Xdt       RT R

d Xdt ! det0F det0F C ! det0F det F tr F 1 @ıx D 0 @X  RT R

D 0 ! ./  C ! ./ .rx  ıx/d xdt  ˛  ˝

RT R D 0 rx ! ./   ! ./  ıx d xdt D r ! ./  !./ ; ıx L2 : x;t

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The normal component of the variation ıx  is assumed to be zero at the boundary @, which follows from no penetration boundary condition. This gives the following force terms expressed in the strong PDE form: L2x;t _

ı

RT

Kdt D  .ut C .u  r/ u/ ; ı x

0

L2x;t _

ı

RT

 Fdt D r ! ./   ! ./ ; ı x

0

The second (conservative) force term is exactly the gradient of the thermodynamic pressure. In the absence of the dissipation, from the force balance (3) with LAP, one obtains the compressible Euler equations [118]: 8 ˆ ˆ 0 is the Hookean constant and constant 3 is subtracted to null the energy of the nondeformed material. Here and hereafter  denotes the Laplace operator, i.e.,  D r  r. This model is closely related to neo-Hookean materials, where free energy may also depend on det F [33]. In [59] free energy density for linear

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elastodynamics depends on E, symmetric part   of displacement gradient, related to deformation gradient F by E D 12 F C F T  I . Remark 8. In case of incompressible elasticity , in order to enforce the nonlinear constraint det F D 1, we are tempted to consider the variation of action integral " under volume preserving diffeomorphism x" , i.e., det ddxX D 1. However, it is more convenient to introduce Lagrange multiplier 'ˇand consider variation of I.x/ under " no constraint of variation x" , i.e., x0 D x; ddx" ˇ"D0 D ıx: Then the variation of I .x/ D

Z

T



KF 

Z

0



 @x ' .X; t/ det  1 d X dt @X

yields the following incompressible elasticity equation: 8 2 C p1 , B can be continuously extended to a bounded operator s;p

sC1;p

from H0 ./ to H0

./n satisfying

krBgkH s;p  C . 0

s;p

diam ./ n diam ./ / .1 C /kgkH s;p ; 0 R R 0

for some C > 0, where H0 ./ D .H s;p .//0 for s < 0. The proof of Proposition 4 relies on the observations that @k .Bf /i consists of weakly singular and Calderón-Zygmund type operators. The assertion for s > 0 then follows from Calderón-Zygmund theory, whereas the cases for s < 0 follow from interpolation and a study of the adjoint kernels. Observe that bounded Lipschitz domains   Rn can be written as a finite union of star-shaped domains. More precisely, there exists m 2 N and fG1 ; : : : ; Gm g   such that S  \ Gi is star-shaped with respect to some ball Bj for j D 1; : : : ; m and D m iD1 . \ Gi /. Applying this procedure to the divergence problem yields the following result. Theorem 4. RLet 1 < p < 1,   Rn be a bounded Lipschitz domain and f 2 Lp ./ with  f D 0. Then there exists B W Cc1 ./ ! Cc1 ./n with div Bf D f: s;p

sC1;p

Moreover, B can be extended continuously from H0 ./ to H0 s > 2 C p1 .

./n provided

A complete proof of the above theorem in the case s  0 is due to Galdi [74]; for a detailed proof, see [76]. The general case s > 2 C p1 is due to Geissert, Heck, and Hieber; see [82]. Remarks 4. a) Note that Bf is defined for all f 2 Lp ./, whereas Bogovskii [25] and Galdi [76] constructed solutions to (14). Hence, B may be regarded as an extension to a solution operator to (14). However, Bf is a solution to (14) R only if  f D 0.

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b) Mitrea, Mitrea, and Monniaux [151] studied generalizations of Bogovskii-type integral operators on large classes of function spaces and various boundary conditions. c) Costabel and McIntosh followed a different approach and proved that Bogovskiitype integral operators are classical pseudo-differential operators of order 1 1 with symbols in the Hörmander class S1;0 .Rn /. This implies that the associated operators act as bounded operators in a wide range of function spaces, including Hölder, Hardy, Sobolev, Besov, and Triebel-Lizorkin spaces. For details, see [45]. s d) The setting of mixed estimates of type L1 .Bp;q / is investigated in [49]. e) Note that the above Theorem 4 does not hold for p D 1 or p D 1. For a detailed discussion of these questions, we refer here to the work of Bourgain and Brezis [39]. In addition, a different approach to (14) is presented there.

2.4

The Stationary Stokes Equation and Elliptic Estimates

In this section, the stationary Stokes equation 8 < u C rp D f in ; div u D 0 in ; : u D 0 on @;

(15)

is considered, where f 2 Lp ./, 1 < p < 1 and   Rn ; n  2, is a domain. It is the aim to establish estimates of the form kr 2 ukLp ./ C krpkLp ./  C kf kLp ./ ; which one refers to as elliptic estimates. To this end, for m 2 N, let D m;p ./ WD fu 2 L1loc ./ W D ˛ u 2 Lp ./ for all j˛j D mg, and let the homogeneous Sobolev b m;p ./ be defined as space W b m;p ./ WD D m;p ./=Pm1 : W

(16)

Here Pm1 denotes the space of all polynomials of order at most .m  1/. Note b m;p ./ becomes a Banach space, when equipped with kuk m;p that W WD b W ./ 1=p P p ˛ . j˛jDm kD ukLp ./ n If  D RC , one may use Fourier transform in tangential direction to obtain the following result. For a proof, see, e.g., [76], Sect. 4.3. Lemma 1. Let 1 < p < 1, m 2 N0 and f 2 W m;p .RnC /n . Then Eq. (15) admits a solution .u; p/ 2 D mC2;p .RnC /  D mC1;p .RnC /, and for all k 2 Œ0; m, there is a constant C > 0 such that

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133

kr 2 ukW k;p .RnC / C krpkW k;p .RnC /  C kf kW k;p .RnC / : < 1 for Moreover, if .v; q/ is another solution to (15) satisfying kvkb W kC2;p .RnC / D kp  qkb D 0. some k 2 Œ0; m, then ku  vkb W kC2;p W kC1;p Applying a localization procedure yields the following result for bounded domains. Proposition 5 ([76]). Let 1 < p < 1, m 2 N0 , f 2 W m;p ./ and   Rn be a bounded domain of class C mC2 . Then there exists a solution .u; p/ 2 W mC2;p ./  W mC1;p ./ of (15). Moreover, u is unique and p is unique up to an additive constant. Furthermore, for k 2 Œ0; m, there is a constant C > 0 such that kukW kC2;p ./ C krpkW k;p ./  C kf kW k;p ./ : Remark 1. Note that the constant C in Proposition 5 depends on . For certain applications it is important to know whether C is independent of the size of . In this context, the reader is referred to a result by Heywood [107] saying that for p D 2; m D 0, and n D 3, the above constant C does not depend on the size of @ or . Indeed, if   R3 is bounded and of class C 3 and .u; p/ 2 H 2 ./n  H 1 ./ is a solution of (15) with f 2 L2 ./n , then kr 2 ukL2 ./  C .kP;2 f kL2 ./ C krukL2 ./ /; where C depends on the C 3 -regularity of @, only. Here P;2 denotes the Helmholtz projection for L2 ./. For exterior domains   Rn , the situation reads as follows: Proposition 6 ([76]). Let 1 < p < 1; m 2 N0 and   Rn be an exterior domain of class C mC2 and f 2 W m;p ./n . Let .u; p/ 2 D 2;p ./n  D 1;p ./ be a solution of (15). Then, for all k 2 Œ0; m, there exists C > 0 such that kr 2 ukW k;p ./ C krpkW k;p ./  C .kf kW k;p ./ C kukLp .D/ /;

(17)

where D WD  \ BR for some sufficiently large R > 0. In particular, kr 2 ukW k;p ./ C krpkW k;p ./  C .kf kW k;p ./ C krukLp ./ /: For a proof of the above estimates, see [76], Chap. 5, as well as [42] and [146]. One finally notes that if p < n2 , then estimate (17) can be improved to kr 2 ukW k;p ./ C krpkW k;p ./  C kf kW k;p ./ for all k 2 Œ0; m.

(18)

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The Stokes Equation and Operator in L2 ./

In this section, the L2 -theory for the Stokes operator and the Stokes semigroup is presented by making use of form methods. To this end, let H and V be Hilbert spaces with V ,! H and denote by h; i the scalar product on H . Consider a sesquilinear form a W V  V ! C, which is continuous, i.e., ja.u; v/j  M kukV kvkV ;

u; v 2 V;

for some M > 0 and coercive, i.e., Re a.u; u/ C ıkuk2H  ˛kuk2V ;

u 2 V;

for some ı 2 R and some ˛ > 0. The space V is called the domain of the form a. Given such a form, one associates with a an operator A on H by setting D.A/ WD fu 2 V W there exists v 2 H such that a.u; '/ D hv; 'i for all ' 2 V g; Au WD v: The A is called the operator induced by the form a W V  V ! C. In the following, let   Rn be an open set, n 2 N, and assume that  ¤ ;. One then sets 1 ./ WD fu 2 Cc1 ./n W div u D 0g; Cc; 1 ./ L2 ./ WD Cc;

kkL2 ./

1 1 ./ H0; ./ WD Cc;

kkH 1 ./

;

;

1 1 and defines the form a W H0; ./  H0; ./ ! C by setting

a.u; v/ WD

n Z X j D1 

ruj  rvj dx;

1 u; v 2 H0; ./:

(19)

Then a is continuous, coercive, and symmetric, i.e., a.u; v/ D a.v; u/ for all u; v 2 1 ./. Based on this form, the Stokes operator on L2 ./ is defined as follows. H0; Definition 2. Let   Rn be open, n 2 N,  ¤ ;, and a be defined as in (19). Then the Stokes operator with Dirichlet boundary conditions A;2 in L2 ./ is defined as the operator induced by a. The following properties of the Stokes operator A;2 are consequences of the definition via the form.

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Proposition 7. Let   Rn ; n 2 N, be an open set. Then the Stokes operator A;2 has the following properties. a) A;2 is injective, self-adjoint, and densely defined. b) The numerical range W .A;2 / is contained in Œ0; 1/. c) A;2 generates a bounded analytic semigroup T;2 on L2 ./ of angle =2 satisfying kT;2 .z/kL.L2 .//  1;

z 2 †=2 ; and

tkA;2 T;2 .t /kL.L2 .//  C;

t >0

for some C > 0. The semigroup T;2 is called the Stokes semigroup on L2 ./. It is remarkable that the definition of the Stokes semigroup on L2 ./ does not need any assumptions on the open set   Rn . However, in general, no characterization of D.A;2 / in terms of suitable function spaces seems to be available in this setting. Note that if   Rn , n  2, is either Rn ; RnC or a domain with compact boundary of class C 2 , then it will be shown in Sect. 2.8.1 (see also [185]) that D.A;2 / D H 2 ./n \ H01 ./n \ L2 ./: For bounded domains, Poincaré’s inequality and Rellich’s theorem on compact embeddings imply the following result. Proposition 8. Let   Rn be a bounded domain. Then the following assertions hold true. a) The Stokes semigroup T;2 is exponentially stable, i.e., there exists ı > 0 such that kT;2 .t /kL.L2 .//  e ıt ;

t > 0:

2 2 b) 0 2 %.A;2 / and A1 ;2 W L ./ ! L ./ is compact. 2 c) The space L ./ has an orthonormal basis . j /  D.A;2 / consisting of eigenfunctions of A;2 corresponding to a sequence of eigenvalues .j / such that

0 < 1  2     ! 1: Consider now the case where   Rn is an exterior domain of class C 2 . Then, Proposition 6 implies the following global mapping properties of T;2 . Proposition 9. For an exterior domain   Rn of class C 2 , there exists a constant C > 0 such that for all f 2 L2 ./

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kT;2 .t /f kL2 ./  kf kL2 ./ ; t > 0, 1 krT;2 .t /f kL2 ./  C t  2 kf kL2 ./ ; t > 0, kA;2 T;2 .t /f kL2 ./  C t 1 kf kL2 ./ ; t > 0, 1 kr 2 T;2 .t /f kL2 ./  C t  2 kf kL2 ./ ; t > 1.

Note that the proof of the last assertion d) makes use of the elliptic estimate for the Stokes system given in Proposition 6. Observe also that for 0 < t  1 one has kr 2 T;2 .t /f kL2 ./  C t 1 kf kL2 ./ .

2.6

The Stokes Equation in the Half-Space: The Case 1 < p < 1

In this section, the Stokes equation is being considered in the half-space RnC WD fx 2 Rn W xn > 0g. In this case, several rather explicit representation formulas for the solution of the Stokes equation or the associated resolvent problem may be derived. The representation formulas due to Solonnikov [187], Ukai [202] and Desch, Hieber, and Prüss [56] will be discussed in the following in some detail. The reader is also referred to the representation formula given by McCracken [139] and to the monograph [76] by Galdi. Some words about notation are in order. Given x 2 Rn , the components of x will be written as x D .x 0 ; xn /, where x 0 2 Rn1 . Using this notation, one writes u D .u0 ; un / for functions and R D .R0 ; Rn / for operators. Similarly, F 0 denotes then the Fourier transformation with respect to the variable x 0 2 Rn1 . Consider the Stokes problem in the half-space RnC , which is given by 8 @t u  u C rp D f ˆ ˆ < div u D 0 ˆ uD0 ˆ : u.0/ D u0

in RC  RnC ; in RC  RnC ; on RC  @RnC ; in RnC :

(20)

Denote by Rj , j D 1; : : : ; n, and by Sj , j D 1; : : : ; n  1, the Riesz operators corresponding to the symbols i j =j j and i j =j 0 j, respectively. Furthermore, one puts R WD .R1 ; : : : ; Rn /;

S WD .S1 ; : : : ; Sn1 /:

(21)

In the following, solution formulas either for the instationary system (20) or for the corresponding resolvent problem 8 < u  u C rp D f in RnC ; div u D 0 in RnC ; : u D 0 on @RnC :

(22)

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137

will be discussed. Note that (22) is obtained by applying Laplace transformation to (20). Most of the explicit formulas below strongly rely on the orthogonality of x 0 and xn and therefore do not generalize to curved boundaries, at least not in such an explicit form.

2.6.1 The Green Tensor In the following, in order to simplify the presentation, assume that n D 3. Based on the fundamental solution G of the heat equation given by G.t; x/ WD

1 exp.jxj2 =4t/; .4t/3=2

x 2 R3 ; t > 0;

which is the Gaussian heat kernel, and on the fundamental solution E of the Laplace equation given by E.x/ WD

1 ; 4jxj

x 2 R3 n f0g;

(23)

the fundamental solution for (20) was constructed first by Solonnikov in [187]. In terms of the Green tensor, the solution .u; p/ of (20) with right-hand side f satisfying div f D 0 and f3 jx3 D0 D 0 has the following explicit representation Z tZ



 G.t  ; x  y/  G.t  ; x  y  / f . ; y/d dy

u.t; x/ D R3C

0

C4

2 X

Z @xj

p.t; x/ D 4

Z tZ rE.x  y/

0

j D1 2 X

x3Z R2

0

"Z @xj

j D1

R2

@x3 E.x y/

G.t  ; y z/fj . ; z/ d d zdy1 dy2 jy3 D0

0

0

R3C

#

Z tZ E.x  y/

R2

G.t  ; y  z /fj . ; z/ d d zdy;

Z tZ

Z C

R3C

R3C

@z3 G.t  ; y  z/fj . ; z/ d d zdy1 dy2 jy3 D0 ; (24)

where y  D .y 0 ; yn /. Formula (24) is probably the most classical tool in order to examine well-posedness and asymptotic properties of the instationary Stokes equation in R3C . Until today the formula (24) serves as a powerful tool in order to derive decay and regularity properties for the solution .u; p/ of (20). It is very remarkable that Solonnikov already proved maximal regularity estimates for (20) in [187] in 1977. This pioneering result implies in particular that p the associated Stokes operator generates an analytic semigroup on L .R3C / for 1 < p < 1. For a comprehensive study of the fundamental solution and Green functions for Stokes systems, the reader is also referred to Galdi’s monograph [76].

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There exists also a version of Green’s formula for the resolvent problem (22), which was constructed first by McCracken in [139]. Based on this representation of the resolvent, she proved resolvent estimates for (22), and hence the Stokes operator p is the generator of a bounded analytic C0 -semigroup on L .R3C / for 1 < p < 1.

2.6.2 Ukai’s Formula The representation formula derived by Ukai [202] in 1987 reduces the desired representation for the solution of the Stokes equation to the one of the heat equation combined with Riesz operators. In a first step of its derivation, one formally applies div to (20) with f D 0 and obtains that p D 0. Taking tangential Fourier transform, one obtains O xn / D 0; .@2n  j 0 j2 /p. ;

xn > 0; 0 2 Rn1 :

Hence, the integrable solution of the above equation must be of the form 0

p. O 0 ; xn / D e j jxn pO 0 . /;

(25)

with a certain, at this point still unknown, trace p0 . The key point in Ukai’s approach is given by the fact that in view of (25), the pressure p also solves the equation .@n C j 0 j/p. O 0 ; xn / D 0;

xn > 0; 0 2 Rn1 :

Setting z WD .@n Cj 0 j/Oun and applying @n Cj 0 j to the n-th line of (20), a calculation shows that z solves the equation 8 t > 0; xn > 0; < @t z  z D 0; t > 0; zjxn D0 D 0; : z.0/ D jr 0 jV1 u0 ; xn > 0;

(26)

V1 f WD S  f 0 C fn ;

(27)

where

and where jr 0 j denotes the pseudo-differential operator with symbol j 0 j. This yields z D jr 0 je tD V1 u0 ; where D denotes the Dirichlet Laplacian. Since uO n jxn D0 D 0, one may recover uO n as uO n .t; 0 ; xn / D

Z

xn 0

k. 0 ; xn  s/.F 0 e tD V1 u0 /.t; 0 ; s/ ds;

3 The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties

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0

where k. 0 ; y/ WD j 0 je j jy for y > 0. For a function f W RnC ! R, denote its trivial extension to Rn by ef , and for a function g W Rn ! R, denote its restriction to RnC by rg. Then un D Ue

tD

0 1

V1 u0 WD r.F /

Z R

.ek/.x 0 ; xn  s/e.F 0 e tD V1 u0 /.t; x 0 ; s/ ds:

A direct calculation yields Uf D rR0  S .R0  S C Rn /ef:

(28)

In order to derive a corresponding representation for u0 , one sets V2 f WD f 0 C Sfn :

(29)

Then, similarly as above, one sees that also V2 u satisfies a homogeneous heat equation with initial data V2 u0 . Consequently, u0 D e tD V2 u0  S un : Plugging the representation for un into the n-th line of (20), one may derive the corresponding representation formula for the pressure term. Summarizing, one obtains thus the following result. Theorem 5 (Ukai’s formula [202]). Let u0 2 Lp .RnC / and f D 0. Then the solution of (20) can be represented as un D Ue tD V1 u0 ;

u0 D e tD V2 u0  S un ;

0

pD

e jr jxn .@n e tD V1 u0 /jxn D0 ; jr 0 j

where V1 , U , and V2 are given as in (27), (28), (29). Since V1 , U , and V2 essentially consist of Riesz operators, many Lp -properties for the Stokes equation are reduced by Theorem 5 to the ones for the heat equation. In particular, the following corollary holds true. Corollary 2. The solution operator u0 7! u to Eq. (20) for t  0 with f D 0, where u is given as in Theorem 5, defines a bounded analytic C0 -semigroup T on p L .RnC / for 1 < p < 1.

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The semigroup T is called the Stokes semigroup. It is not difficult to show that the generator of T coincides with the Stokes operator ARnC ;2 introduced in Definition 2 provided p D 2.

2.6.3 The Formula of Desch, Hieber, and Prüss The following representation formula for the solution of the Stokes resolvent problem (22) due to Desch, Hieber, and Prüss [56] will be discussed in the sequel. This formula also relies on the representation (25) for the pressure term p. For u, the following Ansatz is made: 0

Z

0

1

uO . ; xn / D

k . 0 ; xn ; s/.fO 0 . 0 ; s/  i 0 p. O 0 ; s// ds;

(30)

kC . 0 ; xn ; s/.fO n . 0 ; s/  @n p. O 0 ; s// ds;

(31)

0 n

0

Z

1

uO . ; xn / D 0

where k˙ . 0 ; xn ; s; / WD

1 0 0 .e !.j j/jxn sj ˙ e !.j j/.xn Cs/ /; 2!.j 0 j/

and !.j 0 j/ WD

p  C j 0 j2 ;

 2 C n .1; 0:

Here k and kC represent the tangential Fourier transform of the kernel corresponding to the resolvent of the Laplacian with Dirichlet and Neumann boundary conditions, respectively. By construction, div u D 0. While u0 satisfies Dirichlet boundary conditions, un jxn D0 has to be enforced. This is the key step, since due to (25) and by taking the trace in (31), the trace of the pressure is given by pO 0 . ; xn / D 

!.j 0 j/ C j 0 j j 0 j

Z

1

0 e !.j j/s fOn . 0 ; s/ ds:

(32)

0

Plugging this term into (31) and (30) leads to the following representation formula for the Stokes resolvent problem. Theorem 6 (Desch, Hieber, Prüss [56]). Let  2 C n .1; 0, 0 2 Rn1 and xn > 0. Then the solution of (22) can be represented as un D .  D /1 fn C Kfn ; 0

F 0 p.; 0 ; xn / D e j jxn pO 0 ;

u0 D .  D /1 f 0  SKfn ;

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141

where pO 0 is given by (32), S is defined as in (21), and K is defined by its Fourier transform by

F 0 Kfn .; 0 ; xn / WD

e !.z/xn  e zxn !.z/  z

Z

1

0

e !.j j/s fOn . 0 ; s/ ds:

0

Remarks 5. a) Theorem 6 says that the resolvent of the Stokes problem may be represented as the resolvent of the Dirichlet-Laplacian plus an additive remainder term. b) Based on this formula, one may easily show that the Stokes operator generates a bounded analytic semigroup on Lp .RnC / for 1 < p < 1. c) A slightly different method in order to estimate the terms in Theorem 6, based on the H 1 -calculus for the operator jr 0 j, is performed in [167]. This approach generalizes also to partial slip boundary conditions. d) Considering the spaces L1 .RnC / and L1 .RnC /, the above formula of Desch, Hieber, and Prüß exploits an advantage compared to the other formulas. In fact, the above representation formula yields resolvent estimates for the Stokes resolvent problem in L1 .RnC /. Hence, one may deduce from this formula that the Stokes operator generates a bounded analytic semigroup on solenoidal subspaces of L1 .RnC /. More detailed information in this direction will be given in Sect. 4. Since the Riesz operators are unbounded in L1 , it seems not to be possible to obtain such a result by Ukai’s formula. Note, however, that Solonnikov proved in [194] also estimates for the solution u of the Stokes equation in spaces of bounded functions. His approach is based on the Green tensor (24). e) It was shown by Maekawa and Miura [141] that in the case of the half-space RnC , p there exists an isomorphism V W Lp .RnC / ! L .RnC / such the Stokes semigroup p n tA tA e on L .RC / can be represented as e D Ve tD V 1 , where D denotes the Dirichlet Laplacian on Lp .RnC /. Since the negative Dirichlet-Laplacian D admits an R-bounded H 1 -calculus on Lp .RnC / for 1 < p < 1, a kernel estimate of the remainder term yields the following result. Corollary 3 ([56]). The negative Stokes operator ARnC ;p admits an R-bounded p R1 D 0. H 1 -calculus on L .RnC / for 1 < p < 1 with RH1 -angle A Rn ;p C

Remark 2. Note that the existence of a bounded H 1 -calculus for ARnC ;p on p L .RnC / can be deduced also from Ukai’s formula. In fact, applying Laplace transformation to the formula for u given in Theorem 5 and taking into account that the negative Dirichlet-Laplacian D admits a bounded H 1 -calculus on Lp .RnC / for 1 < p < 1, the latter property can be transferred in this way to the Stokes operator.

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The Stokes Equation in Layers: The Case 1 < p < 1

Techniques similar to those described above for the half-space RnC work also for infinite layer domains of the form  WD Rn1  .R; R/; where R > 0. In fact, applying tangential Fourier transformation, explicit solution formulas can be derived in this case, too. These formulas will not be recalled here in detail, but note that their structure is similar to the one of the half-space. The interested reader may find more information on layer-like domains, e.g., in [158, 159]. An approach to asymptotically flat layers is provided in [9], and the boundedness of the H 1 -calculus for the Stokes operator in this situation was proved in [10]. p Sectoriality of the Stokes operator on L ./ was proved by Abe and Shibata in [6, 7] and by Abels and Wiegner in [15]. The boundedness of the imaginary powers for the negative of the Stokes operator in a layer within the Lp -framework was obtained by Abels in [8]. Only one characteristic result is formulated explicity in the following. It follows, for instance, from the results obtained in [10]. Defining the Stokes operator on a layer as A;p u WD P u;

1;p

u 2 D.A;p / WD W 2;p ./ \ W0 ./ \ Lp ./;

the following result holds true. Theorem 7. Let n  2, 1 < p < 1 and  D Rn1  .1; 1/. Then A;p p R;1 admits an R-bounded H 1 -calculus on L ./ with RH 1 -angle A D 0. ;p Furthermore, 0 2 .A;p /. For interesting results with respect to the spaces L1 ./ and L1 ./, where  denotes a layer, see Sect. 4.1.

2.8

The Stokes Equation in Lp ./ for 1 < p < 1 and for Domains with Compact Boundaries

2.8.1 The Stokes Equation with Dirichlet Boundary Conditions In the following, the Stokes equation on domains   Rn with smooth boundaries is investigated by means of a localization procedure. This setting is the most classical situation, and its analysis started with the pioneering results by Sobolevskii [184] and Ladyzhenskaja [135]. The procedure will be explained for the resolvent problem

3 The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties

8 < u  u C rp D f in ; div u D 0 in ; : u D 0 on @;

143

(33)

where  is assumed to lie in a suitable sector of the complex plane. For simplicity, in this section only, domains with compact boundaries will be investigated. By employing finite coverings, one then may include domains of the following type: given n  2, a domain   Rn is called a standard domain, if  coincides with Rn , RnC , a bounded domain, an exterior domain, or a perturbed half-space. The first step in the localization procedure consists of choosing a finite covering m n 2 .Uj /m j D1 of   R with boundary of class C and a partition of unity .'j /j D1 m 2 subordinate to .Uj /m j D1 . In other words, .Uj ; 'j /j D1 is an atlas for the C -manifold @. Multiplying (33) with 'j leads to a localized perturbed version in Uj . Since the Stokes equations are invariant under rotations and translations, and by choosing Uj sufficiently small, one may assume that the localized version of Eq. (33) is either an equation on Rn or on a bent half-space Hj WD fx 2 Rn W xn > hj .x 0 /g;

(34)

with a certain bending function hj W Rn1 ! R. One further transforms the localized system on Hj by v.x 0 ; xn / WD .u ı /.x 0 ; xn / WD u.x 0 ; xn C hj .x 0 //;

.x 0 ; xn / 2 RnC :

(35)

The resulting system for v then is an equation on RnC . Summarizing, by this procedure the Stokes resolvent problem on a domain is reduced to finitely many equations on RnC or Rn . A fundamental problem arising at this stage is the fact that the condition div u D 0 is not preserved, neither under multiplication with a cutoff function nor by transformation (35). This might be the reason why the first proofs of analyticity of the Stokes semigroup or other properties like bounded imaginary powers for the Stokes operator on domains rely on different methods. In order to investigate the Eq. (33) via a localization procedure and to overcome this difficulty, the following three strategies were established: Strategy 1. One may replace the transformation (35) by   v D T u WD u ı  0; : : : ; 0; .r 0 hj ; 0/  .u ı / :

(36)

Geometrically speaking, this transformation leaves the outer normal at the boundary invariant. This implies div v D 0, i.e., the above transformation is volume preserving. The price one has to pay for this is a lift of the boundary smoothness from C 2 to C 3 . This is due to the fact that r 0 hj appears in the transformation, and since one deals with a second-order system, one requires

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existence of third-order derivatives of the bending functions hj . The transformation (36) is utilized, e.g., in [24, 84, 160, 187]. Strategy 2. Suppose the divergence problem div w D g, wjxn D0 D 0 admits a solution given by a Bogovskii operator B W g 7! w, as investigated in Sect. 2.3. Then one may correct the lacking solenoidality by the term Qv WD v  B div v:

(37)

Note that Q is a projection onto solenoidal functions that keeps Dirichlet boundary values, which is not the case for the Helmholtz projection. Besides the appearance of a couple of additional perturbation terms in the transformed equations, the price to pay here is that one has to prove independently the existence of a Bogovskii type operator including sufficient regularity properties. This approach to the Stokes system is performed, e.g., in [84] and [83]. Strategy 3. It is also possible to work with an inhomogeneous divergence condition right from the beginning. In this case, the inhomogeneous Stokes systems in Rn and in RnC , where div u D 0 replaced by div u D g, have to be solved. Localizing the inhomogeneous equations produces then perturbation terms not only in the first n lines of (33) but also in the divergence condition. In this context, it is not possible to work on an operator theoretical level, since p the ground space of the Stokes operator is L ./. Instead, the full system including div u D g and the pressure term rp has to be handled. Another inconvenience is that one has to find a “good” conditions for the function space for the right-hand side g in the divergence condition. Considering, e.g., exterior domains in the strong 1;p Lp ./-setting, i.e., u 2 W 2;p ./ \ W0 ./, a sufficient regularity condition for g is that b 1;p ./: g D div u 2 W 1;p ./ \ W Estimating the additional perturbation terms leads to further technicalities (see, e.g., [68]). Of course, each one of the above strategies (1), (2), and (3) has its advantages and its disadvantages. If one is interested in results, where the regularity of the boundary is not so important, then a combination of strategies (1) and (2) might be an elegant and a convenient way. This was already demonstrated by Solonnikov in [187] (see also [84] and [83]), and this approach will be outlined in the following for the resolvent problem. For a characterization of the existence of a strong Lp -solution of the inhomogeneous Stokes equation by its data, see Theorem 11 below. In order to explain the strategy outlined above in more detail, one first considers the Stokes resolvent problem on  D H , where H is a bent half-space with h being a suitable bending function defined in (34). Since  is a C 3 domain, one may assume h 2 BUC 3 .Rn1 /. This implies that the transformation T defined in (36) is an isomorphism between the Sobolev spaces involved up to order two. In particular,

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145

it can be shown that T 2 Lis .Lp .H /; Lp .RnC // \ Lis .D.AH;p /; D.ARnC ;p //; where D.A;p / denotes the domain of the Stokes operator A;p on . Hence, one may define AT WD TAH;p T 1 ; p

which is an operator in L .RnC / with domain D.AT / D D.ARnC ;p /. Utilizing the smallness of r 0 h, one proves that B WD AT  ARnC ;p is a relatively bounded perturbation of the Stokes operator ARnC ;p . Depending on the property one wishes to verify, one then needs to have this property valid for AT as well as a perturbation result for this property. Concerning the generation of a bounded holomorphic semigroup or the property of maximal Lq -regularity, a standard perturbation argument for sectorial operators or maximal regularity may be applied (see [54, 133]). For the existence of a bounded H 1 calculus, one needs a more refined perturbation result as the one which is provided in [52]. Note that these properties are invariant under conjugation with isomorphisms. Consequently, these properties remains true for AH;p . In the next step, one sets u WD

m X

'j uj ;

j D1

p WD

m X

'j pj ;

j D1

where .uj ; pj / is the restricted bent half-space solution to data fj D j f that corresponds to Uj . To be precise, modulo rotation and translation, one has Uj \ Hj D Uj \ ;

Uj \ @Hj D Uj \ @:

Then u solves the perturbed Stokes resolvent problem 8 P ˆ f C m < u  u C rp D P j D1 .uj 'j  ruj r'j C pj r'j / in ; u in ; div u D m j D1 j r'j ˆ : uD0 on @: (38) It remains to prove that the remainder terms are of lower order so that they may be absorbed into the terms on the left-hand side. The perturbation term in the divergence condition can be handled by employing properties of the Bogovskii operator as explained in strategy 2. Whereas the terms uj 'j and ruj r'j are standard, a further difficulty is represented by the terms pj r'j . This relates to the fact that, a priori, only an estimate on rp of the form

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krpkp  C kf kp uniformly in the resolvent parameter  is available, but no suitable estimate for the pressure itself. Resolvent or H 1 -estimates for u, however, require a suitable decay estimate for p in . This is provided by the following pressure lemma (see [160, Lemma 13]). Lemma 2 ([160]). Let  2 .0; =2/, 1 < p < 1, and .u; p/ 2 D.A;p /  b 1;p ./ the unique solution of the Stokes resolvent problem (33) on Lp ./. Then W for each ˛ 2 .0; 1=2p 0 / and every bounded C 1;1 -domain G  , we have kpG kp  C jj˛ kf kp ;

 2 † ; jj  1;

(39)

with C > 0 independent of  and f and where pG D p 

1 jGj

Z

p

G

p dx 2 L0 .G/ D fg 2 Lp .G/I

Z g dx D 0g: G p

A sketch of the proof of the above lemma is as follows. Note that L0 .G/0 D p0 1;p 0 2 W0 .G/ of L0 .G/. Due to (4), for every  2 L0 .G/, there is a solution div D  such that k kW 1;p0 .G/  C kkp0 . Utilizing rpG D rp and the fact that p0

rpG .x/ D .I  P /u.x/

x 2 ;

where P denotes the Helmholtz projector on , one obtains Z

Z

Z

pG  dx D  

rpG  

Z

dx D

.D u/.I  P / dx 

Œ.D /1˛ u.D /˛ .I  P / dx:

D 

It is now straight forward to derive the estimate (39) from here. Taking into account 0 the fact that D has bounded imaginary powers on Lp ./ (see, e.g., [52]), it 0 follows by (8) that D..D /˛ / D H 2˛;p ./. The crucial point here is that for 0 2˛;p 0 ./, so that one may shift the part ˛ 2 .0; 1=2p 0 /, one has H 2˛;p ./ D H0 .D /˛ of D onto .I  P / without getting boundary terms. Collecting all the estimates for the perturbation terms, it is then straightforward to derive the desired resolvent or H 1 -estimates for A;p . Summarizing, one obtains the following result. Theorem 8. Let n  2, 1 < p < 1,  2 .0; / and assume that   Rn is a standard domain with boundary of class C 3 . Then there exists a unique solution .u; p/ of (33) satisfying

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147

1

jjkukLp ./Cjj 2 krukLp ./Ckr 2 ukLp ./CkrpkLp ./  C'0 kf kLp ./ ;  2 † ; where C > 0 is independent of f; u, and p. Furthermore, setting R./f WD u./, the set  n o   1 R./;  2 rR./; r 2 R./ I  2 †  L Lp ./; Lp .; Rn /

(40)

is R-bounded. Remarks 6. a) Note that the case n D 2 and small  for exterior domains and perturbed half-spaces requires special methods and techniques, which have been developed by Abels in [9]. b) The first assertion of Theorem 8 still holds true for standard domains of class C 1;1 . Theorem 8 implies the following important consequences for the Stokes operator and the Stokes equation. Theorem 9. Let n  2, 1 < p; q < 1, J D .0; T / for some T > 0 and assume that   Rn is a standard domain of class C 3 . Then the Stokes operator defined by A;p u WD P u;

1;p

D.A;p / WD W 2;p ./ \ W0 ./ \ Lp ./

(41)

p

admits maximal Lq -regularity on L ./. In particular, the solution u to the Cauchy problem u0 .t /  Ap u.t / D f .t/; t > 0;

u.0/ D u0 ;

satisfies the estimate ku0 kLq .J ILp .// C kAp ukLq .J ILp .//  C .kf kLq .J ILp .// C ku0 kX /; p

for some C > 0 independent of f 2 Lq .J I L .// and u0 2 X WD p .L ./; D.Ap //11=q;q . p Moreover, Ap generates a bounded analytic C0 -semigroup on L ./ and a)  .A;p / D .1; 0 if  is Rn , RnC or an exterior domain, b)  .A;p / D .1;  for some  D ./ > 0 provided  is bounded, 22=q p c) u0 2 X if and only if u0 2 Bp;q ./ \ L ./ and u D 0 on @. Setting r D .Id  P /.  Ap /1 , one obtains the following results for the Stokes equation (1) and its corresponding resolvent Eq. (33).

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Corollary 4. Given the assumptions of Theorem 9, the Stokes equation (1) admits a 1;p p unique solution .u; / 2 W 1;q .J I Lp .//\Lq .J I W 2;p ./\W0 ./\L .// q b 1;p L .W .//, and there exists a constant C > 0 such that kut kLq .J ILp .// C kukLq .J ILp .// C kr 2 ukLq .J ILp .// C krkLq .J ILp .//  C .kf kLq .J ILp .// C ku0 kX /: Corollary 5. Let 1 < p < 1,   Rn as in Theorem 9,  2 .0; /, and f 2 p 1;p p L ./. Then there exists a unique .u; / 2 W 2;p ./ \ W0 ./ \ L .//  b 1;p ./ satisfying (33) and a constant C > 0 such that W jjkukLp ./ C kr 2 ukLp ./ C krkLp ./  C kf kLp ./ ;

 2 † ; f 2 Lp ./:

The above Theorems 8 and 9 go back to several authors, who proved various assertions of the above theorems by different methods and approaches during the last decades. Some comments on key contributions are in order at this point. First results on maximal Lp -regularity estimates for the instationary Stokes system (1) go back to the pioneering work of Solonnikov; see [186] for  D Rn ; RnC and [187] for domains with compact boundaries. For a modern approach to his results, then also in the mixed Lq  Lp -context, based on the characterization of maximal Lp -regularity by the R-boundedness property of the resolvent, see the work of Geissert, Heck, Hieber, Schwarz and Stavrakidis [84], and [83]. For the half-space RnC , results on the existence and analyticity of the Stokes semigroup go back to [32, 56, 68, 139, 167, 202]. Giga and Sohr [97] proved for the first time global-in-time mixed Lq  Lp maximal regularity estimates for smooth exterior domains by combining a result on the boundedness of the imaginary powers of the Stokes operator with the DoreVenni theorem [62]. A different approach to maximal Lp -regularity of the Stokes equation (1) based on pseudo-differential methods was developed by Grubb and Solonnikov [101]. It will be discussed in some detail in Sect. 3.1. Concerning the existence and analyticity of the Stokes semigroup, a different proof for the case of bounded domains with smooth boundary is due to Giga [89]. His approach is based on Seeley’s theory on pseudo-differential operators rather than on localization methods. Borchers and Sohr [34] and Borchers and Varnhorn [35] were the first to prove that the Stokes semigroup on exterior domains is uniformly bounded. The fact that the Stokes semigroup on exterior domains is a bounded analytic semigroup is due to Giga and Sohr [78]. The maximal regularity results in the case n D 2 go back to Abels [9], who was in particular able to treat the case of small resolvent parameters. In [68] Farwig and Sohr also developed an Lp -approach to the Stokes equation on C 1;1 standard domains yielding the analyticity of the Stokes semigroup.

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The characterization of the space X  in terms of function spaces given in Theorem 9 c) is due to Amann (see [18], Thm. 3.4). The above theorem allows in particular to consider also the fractional powers of the Stokes operators. In fact, by the sectoriality of the negative Stokes operator, fractional powers .A;p /z x of A;p are well-defined for x 2 D.Ak;p /\R.Ak;p / and z 2 C such that jRezj < k (see, e.g., [17, 54, 133]). It follows from the abstract theory of sectorial operators that .A;p /z is linear, closed, injective, and densely defined and has dense range. An important tool in the investigation of nonlinear problems is the representation of the domain of the fractional powers of a sectorial operator in terms of suitable function spaces. In this context, the property of bounded imaginary powers introduced in Sect. 2.1 plays an important role. In fact, as described in Sect. 2.1, assuming this property for a sectorial operator on a Banach space X , one has ŒX; D.A/˛ D D.A˛ /;

˛ 2 .0; 1/:

(42)

The following result deals with the H 1 -calculus for the Stokes operator on

p L ./.

Theorem 10 ([160]). Let n  2, 1 < p < 1, and assume that   Rn is a standard domain of class C 3 . Then A;p admits an R-bounded H 1 -calculus p on L ./. In particular, A;p has bounded imaginary powers, and relation (42) p holds for A D A;p and X D L ./. The first proof of the boundedness of the imaginary powers of the Stokes operator on bounded domains with smooth boundaries goes back to Giga [91]. His proof is again based on Seeley’s theorem. The case of an exterior domain, due to Giga and Sohr, was treated in [98]. A first proof of the fact that ARnC ;p admits an R-bounded H 1 -calculus on p L .RnC / was given by Desch, Hieber, and Prüss in [56]. Moreover, the proof of the p existence of an R-bounded H 1 -calculus on standard domains in L ./ is due to Noll and Saal [160]. For the case n D 2, see the work of Abels [9]. p 1;p Theorem 10 implies that ŒL ./; D.A;p /1=2 D W0 ./ holds for standard domains. Thus, in particular, the norms k  kW 1;p ./

and

k  kLp ./ C k.A;p /1=2  kLp ./

(43)

are equivalent on D..A;p /1=2 /. In the theory of the equations of Navier-Stokes, it is often important to know that the norms kr  kLp ./

and

k.A;p /1=2  kLp ./

(44)

are equivalent on D..A;p /1=2 /. In the case of a bounded domain, this property follows immediately from the equivalence of the norms in (43), thanks to Poincaré’s

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inequality. In scaling invariant domains such as Rn , RnC , this property may be derived from (43) by a scaling argument (see [167]). On exterior domains and perturbed half-spaces, the equivalence of the norms in (44) cannot be obtained on the full scale for p. Here one needs to restrict the value of p to p 2 .1; n=2/, similarly to (18). Summarizing, one obtains the following proposition, which was proved first by Borchers and Miyakawa in [33], however, by different methods. Proposition 10. Let n  2 and assume that   Rn is a standard domain of class C 3 . Then the norms in (44) are equivalent on D..A;p /1=2 / provided a)  is bounded, Rn , or RnC and 1 < p < 1, b)  is an exterior domain or a perturbed half-space and 1 < p < n=2. Finally, consider the Stokes equation with inhomogeneous data of the form 8 .@t C !/u  u C r D f ˆ ˆ < div u D g ˆ uDh ˆ : u.0/ D u0

in RC  ; in RC  ; on RC  @; in :

(45)

where   Rn is a domain with compact boundary of class 3 and ! 2 R. In the following, the maximal Lq  Lp -regularity estimates for the solution .u; / of (45) are characterized by the following conditions on the data .f; g; h; u0 /. One may view this characterization for the solution of the Stokes equation as the counterpart of the characterization of the solution of the inhomogeneous parabolic boundary value problems subject to general boundary conditions in terms of the data given; see [55]. To this end, the set of conditions (D) is introduced: Condition (D): 22=q

a) f 2 Lq .RC I Lp .//; u0 2 Bp;q ./, P 1;p .// \ Lq .RC I H 1;p .//, div u0 D g.0/, b) g 2 H 1;q .RC I H 11=2p 21=p .RC I Lp . @// \ Lq .RC I Bp;p .@// and h.0/ D u0 on @ if c) h 2 Fq;p q > 3=2, P 1;p .// and h .0/ D .ju0 / on @. d) .gjh / 2 H 1;q .RC I H Then the following theorem holds: Theorem 11 ([165] Chapter 7). Let   Rn be a domain with compact boundary @ of class 3, let 1 < p; q < 1 and q ¤ 3; 3=2. Then there exists !0 2 R such that for each ! > !0 , there exists a unique solution .u; / of Eq. (45) within the class P 1;p .// u 2 H 1;q .RC I Lp .// \ Lq .RC ; H 2;p .// and  2 Lq .RC I H if and only if the data .u0 ; f; g; h/ satisfy the above condition (D).

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The proof of Theorem 11 is rather involved; see Sect. 7 of [165] for a very thorough study of the Stokes equation with inhomogeneous data. In addition, other types of boundary conditions as pure slip, outflow and free boundary conditions are studied there. For previous results in this direction, see also [19].

2.8.2 Energy Preserving Boundary Conditions In this section, various types of boundary conditions for the Stokes equation as well as inhomogeneous right-hand sides are considered. Boundary conditions different from the Dirichlet condition arise in many applications and in related quasilinear problems. For instance, considering a free boundary value problem, e.g., a free ocean surface, then there is no stress at the surface of the fluid. Hence, it is natural to impose the condition T .u; p/ D 0

(46)

at the free boundary. Here T .u; p/ D 2 D.u/  Ip denotes the stress tensor, D.u/ D 12 .ru C .ru/T / the deformation tensor, and  the outer normal. On the other hand, in many engineering applications, such as in the design of hydrophobic surfaces, the partial slip condition ˛u C .D.u// D 0;

  u D 0;

(47)

also called Navier condition, plays a fundamental role as a macroscopic model. Here ˛ 2 R is a parameter (that relates to the slip length) and v D v  .  v/ denotes the tangential part of a vector field v. For this reason, results related for the Stokes system 8 @t u  div T .u; p/ D f ˆ ˆ < div u D g ˆ B.u; p/ D h ˆ : u.0/ D u0

in RC  ; in RC  ; on RC  @; in ;

(48)

subject to a given boundary operator B.u; p/ will be discussed in the following. The conditions (46) and (47) with ˛ D 0 belong to the large class of so-called energy preserving boundary conditions. This notion arises from the fact that the kinetic energy balance related to (48) with homogeneous right-hand sides is given by 1d 2 dt

Z

juj2 dx C 2



Z

jD.u/j2 dx D 

Z Œu  T .u; p/ d  : @

Whenever B.u; p/ is chosen in such a way that the right-hand side vanishes, B.u; p/ D 0 is called an energy preserving boundary condition. Note that the constraint div u D 0 allows for a second variant. In fact, if one defines the rate

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of the rotation tensor by R.u/ D 12 .ru  .ru/T /, one still has div R.u/ D div D.u/ D

1 u: 2

Setting V .u; p/ WD 2 R.u/  Ip, one obtains the alternative form 1d 2 dt

Z

juj2 dx C 2



Z

jR.u/j2 dx D 

Z Œu  V .u; p/ d  @

of the kinetic energy balance. This yields another set of energy preserving boundary conditions in terms of the antisymmetric counterpart V of the stress tensor T . Examples of energy preserving boundary conditions for the case D 1 include B.u; p/ WD T .u; p/; free boundary (Neumann) condition, B.u; p/ WD .D.u// C .  u/; full slip boundary condition, B.u; p/ WD u; no slip boundary condition, B.u; p/ WD u C p; tangential velocity and pressure condition, B.u; p/ WD .R.u// C p; vorticity and pressure condition, B.u; p/ WD u C 2.@ u  /  p; outflow condition, B.u; p/ WD .R.u// C .u  /; adjoint full slip condition. To keep the notational cost moderate, a corresponding maximal regularity result for the solution of (48) is formulated here only for the first boundary operator B.u; p/ D T .u; p/, which is, however, nevertheless one of the most representative energy preserving boundary conditions. To this end, one introduces the following classes of data and solution spaces. For the class of data spaces, for 1 < p < 1, one sets Ff WD Lp ..0; T /; Lp .//; Fg WD H 1=2;p ..0; T /; Lp .// \ Lp ..0; T /; W 1;p .//; Fh WD Wp1=21=2p ..0; T /; Lp . @// \ Lp ..0; T /; Wp11=p .@//; Fu0 WD Wp22=p ./; and defines F as the class of all .f; g; h; u0 / 2 Ff  Fg  Fh  Fu0 such that div u0 D g.0/; if p  2, .D.u0 // D h .0/; if p > 3 and such that there exists  2 Wp11=2p ..0; T /; Lp . @// \ Lp ..0; T /; Wp11=p .@// with u0   D .0/ if p > 3=2, and b 1;p .//; .g; / 2 W 1;p ..0; T /; 0 W

3 The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties

153

where .g; / acts as a functional via Z

Z

.g; /. / WD

 d  @

g dx;

0

2 W 1;p ./:



For the corresponding classes of solution spaces, one puts Eu WD W 1;p ..0; T /; Lp .// \ Lp ..0; T /; W 2;p .//; n Ep WD p 2 Lp ..0; T /; W 1;p .// W pj@ 2 Wp1=21=2p ..0; T /; Lp . @// o \ Lp ..0; T /; Wp11=p .@// ; and E WD Eu  Ep : Then one obtains the following result. Theorem 12 ([36]). Let p 2 .1; 1/nf3=2; 3g, n  2, > 0, and assume that   Rn is a bounded domain with smooth boundary. Then (48) with B.u; p/ D T .u; p/ admits a unique solution .u; p/ 2 E if and only if the data satisfy .f; g; h; u0 / 2 F. In particular, the solution operator S W .f; g; h; u0 / 7! .u; p/ is an isomorphism between the corresponding classes of function spaces. Remark 3. The corresponding results remain valid for each of the other boundary operators B.u; p/ listed above. Even a much larger class is admissible, including, for instance, also partial slip-type conditions. The reader is referred to [36] for a comprehensive study of this topic. First results on other boundary conditions than Dirichlet conditions were derived by Miyakawa in [154] for classical Neumann conditions and by Giga in [90] for another first-order type condition. An approach to maximal regularity subject to partial slip-type boundary conditions for  D RnC was developed in [166, 167]. Its extension to standard domains can be found in [174, 180]. A first approach to the Neumann conditions related to the free boundary condition B.u; p/ D T .u; p/ was given by Solonnikov in [189–191]. Comprehensive approaches to this topic were also developed by Shibata and Shimizu in [177, 178], by Boyer and Fabrie in [40] and by Prüss and Simonett in [164,165]. Maximal regularity results for a wide class of boundary conditions, based on pseudo-differential operator methods, were derived in [101]. This approach will be discussed in more detail in Sect. 3.1. As already mentioned in Remark 3, an

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extensive approach to the class of energy preserving boundary conditions, as well as to further relevant classes, was derived in [36, 128]. This approach includes all boundary conditions discussed in the previously cited papers. Note that the characterization of the space for initial data u0 in Theorem 12 relies on Proposition 1 and that the general theory of the trace operator implies the the conditions on the boundary trace spaces given above are necessary. Observe finally that results corresponding to Theorem 12 are available on certain classes of domains with noncompact boundaries, too. For details, see the Sects. 2.7, 2.9 and 2.11.

2.9

The Stokes Equation in Lp ./ for 1 < p < 1 and for Domains with Noncompact Boundaries

In this section, one considers possibly unbounded domains, which are uniformly of class C k for some k 2 N. A key problem in the investigation of the Stokes operator in general unbounded domains is the fact that the Helmholtz decomposition for p Lp ./ into L ./ ˚ Gp ./ does not exist, in general. The reader is referred to the results by Maslennikova and Bogosvkii [147] described in Remark 1 b), where an example of an unbounded domain  with smooth boundary is constructed for which the Helmholtz decomposition exists only for certain values of p. In the following, one assumes that   Rn is a domain with uniformly C 3 -boundary and that the Helmholtz projection P exists for Lp ./. Then it is shown that the Stokes operator p Ap , defined as in (50) below, generates an analytic semigroup on L ./ and that the solution of the Stokes equation ut  u C r D f;

in   .0; T /;

div u D 0;

in   .0; T /;

u D 0; u.0/ D u0 ;

on @  .0; T /;

(49)

in 

satisfies the maximal Lq -Lp -regularity estimate. Theorem 13 ([83]). Let n  2, p; q 2 .1; 1/ and J D .0; T / for some T 2 .0; 1/. Assume that   Rn is a domain with uniform C 3 -boundary and p that the Helmholtz projection P exists for Lp ./. Let f 2 Lq .J I L .//. Then 1;q equation (49) with u0 D 0 admits a unique solution .u; / 2 W .J I Lp .// \ 1;p p b 1;p .//, and there exists a constant Lq .J I W 2;p ./\W0 ./\L .//Lq .J I W C > 0 such that kut kLq .J ILp .// C kukLq .J ILp .// C kr 2 ukLq .J ILp .// CkrkLq .J ILp .//  C kf kLq .J ILp .// :

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155

Assuming as in the above theorem that the Helmholtz projection P exists for p Lp ./, one may define the Stokes operator Ap in L ./ as 1;p

D.Ap / WD W 2;p ./ \ W0 ./ \ Lp ./;

(50)

Ap u WD P u for u 2 D.Ap /: p

Then maximal Lq -regularity for the Stokes operator A;p in L ./ u0 .t /  Ap u.t / D f .t/;

t > 0; (51)

u.0/ D u0 ;

holds true. Some comments about the strategy of the proof of Theorem 13 are in order. a) The above assumptions imply that one may choose balls Bj WD Br .xj / with centers xj 2  and C 3 -functions hj , (j D 1; 2; : : : ; N ) if  is bounded and j 2 N if  is unbounded, such that   [1 j D1 Bj ;

Bj  U .xj / if xj 2 @;

Bj   if xj 2 ;

where U .xj / are suitable neighborhhods of xj . Given the covering .Bj /, there exists a partition of unity 'j 2 Cc1 .Rn / satisfying supp 'j  Bj and 0  'j  1. Consider now

uQ WD

N X

'j uj ;

j D1

Q D

N X

'j  j ;

j D1

where .uj ; j / is the push forward of the solution .Ouj ; O j / to the resolvent problem on the half-space with a suitable right-hand side fOj . Note that uQ is not P solenoidal in general since div uQ D N j D1 .r'j /uj ¤ 0. In order to circumvent these difficulties, one therefore uses the modified ansatz u WD

1 X 

 'j uj  rvj ;

j D1

where vj is a weak solution to the Neumann problem v D div f

in ;

@v Df  @

on @;

(52)

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with f D 'j uj . Note that the existence and uniqueness of vj is guaranteed by the existence of the Helmholtz projection and by Theorem 2. By construction, one then obtains div u D

1 X

.div.'j uj /  vj / D 0I

j D1

however, the tangential component of u does not vanish at the boundary anymore, which leads to additional correction terms. b) The above strategy needs higher-order estimates for weak solutions of the above Neumann problem. In this context, the following lemma is important (see [83]). Lemma 3. Let p and  as above. Then, for k D 2; 3, there exists C > 0 such that b 1;p ./ of (52) for f 2 W k1;p ./ with   f D 0 on @, the weak solution v 2 W satisfies k X

  kr j vkp  C kf kp C kr  f kW k2;p :

(53)

j D1

For different approaches to the Stokes and Navier-Stokes equation on special domains with noncompact boundaries, e.g., domains with strip-like or cylindrical outlets at infinity or parabolically growing layers, the reader is referred to the works of Heywood [106], Solonnikov [188], and Pileckas [161–163]. In [14], Abels and Terasawa considered the reduced Stokes operator in unbounded domains with the additional assumption on  that the associated space for the pressure can be decomposed suitably. For related results, see also [13] and [173]. The following result, due to Geissert and Kunstmann, shows that under the present assumptions, the Stokes operator even admits a bounded H 1 -calculus on p L ./ for suitable values of p. Proposition 11 ([86]). Let   Rn be a domain with uniform C 3 -boundary and assume that the Helmholtz decomposition exists for Lp ./ for some p 2 .1; 1/. Then there exists ! 2 R such that Ap C ! admits a bounded H 1 -calculus on q L ./ for all q 2 .minfp; p 0 g; maxfp; p 0 g/. The proof relies on the maximal regularity Theorem 13 and an abstract result for H 1 -calculus in complemented subspaces. This section is being finished with a solution of a long outstanding question, whether or not the existence of the Helmholtz projection for Lp ./ is necessary for the Stokes operator being the generator of an analytic semigroup on Lp ./? A very recent result due to Bolkart, Giga, Miura, Suzuki, and Tsutsui [29] says that the existence of the Helmholtz projection for Lp ./ is not necessary for the

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157

Stokes operator being the generator of an analytic semigroup on Lp ./. More precisely, the following holds true. Theorem 14 ([29]). Let p 2 Œ2; 1/. Then there exists a sector-like domain   R2 with C 3 -boundary such that the solution operator T .t/ W u0 7! u.t; / of the p Stokes equation (49) in  with f D 0 defines an analytic semigroup on L ./, p while the Helmholtz decomposition for L ./ fails to exist. As written in the introduction, a characterization of those domains, even with smooth boundaries, for which the solution of the Stokes equation is governed by an p analytic semigroup on L ./ remains a challenging open problem until today.

2.10

The Stokes Equation in Noncylindrical Space-Time Domains

This subsection is motivated by applications of fluid flows in spatial regions with a moving boundary including moving obstacles. One hence considers the Stokes system 8 vt  v C rp ˆ ˆ < div v ˆ v ˆ : v.0/

D f in QT ; D 0 in QT ; S D 0 on t2.0;T / @.t /  ft g; D v0 in .0/ DW 0 ;

(54)

S on noncylindrical space-time domains of the form QT WD t2.0;T / .t/  ft g  RnC1 . Assume that the moving boundary, i.e., the evolution of the domain .t/, is determined by a level-preserving diffeomorphism W 0  .0; T / ! QT ;

. ; t / 7! .x; t / D

. ; t / WD . . ; t /; t /;

such that for each t 2 Œ0; T /, .; t / maps 0 onto .t/. More precisely, one assumes the following conditions on and , respectively: Let T 2 .0; 1, 0  Rn be a standard domain of class C 3 . The domains .t/, t 2 Œ0; T , shall be of the same type as 0 , i.e., f.t/gt2Œ0;T  is either a family of bounded domains, of exterior domains, or of perturbed half-spaces. Furthermore, it is assumed that (a) for each t 2 Œ0; T , .; t / W 0 ! .t/ is a C 3 -diffeomorphism with inverse 1 .; t /. (b) if Q0 WD 0  .0; T / and BC .Q0 / denotes the space of all bounded and continuous functions on Q0 , then 2 BC 3;1 .Q0 / WD ff 2 C .Q0 / W @kt Dx˛ f 2 BC .Q0 /; 1  2k C j˛j  3; k 2 N0 ; ˛ 2 Nn0 g. (c) det r . ; t/ 1, . ; t / 2 Q0 , (volume preserving). (d) if T D 1, then @kt .; t / ! @kt .; 1/ in BC 32k .0 /, k D 0; 1, for t ! 1.

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For t 2 Œ0; T / and 1 < p < 1, the Stokes operator in the time-dependent domain .t/ is given by

A.t/ WD P.t/ ; 1;p

D.A.t/ / WD W 2;p ..t // \ W0 ..t // \ Lp ..t //:

p

Here P.t/ W Lp ..t // ! L ..t // denotes the Helmholtz projection associated to p the Helmholtz decomposition Lp ..t // D L ..t // ˚ Gp ..t //. The following maximal regularity result for the Stokes equation on noncylindrical regions was obtained in [168]. Theorem 15 ([168]). Let n  2, 1 < p; q < 1, and T 2 .0; 1. Let the evolution of .t/, t 2 Œ0; T , be determined by a function satisfying the above assumptions. Then problem (54) has a unique solution t 7! .v.t /; p.t // 2 b 1;q ..t //, and t 2 Œ0; T . Furthermore, for T < 1, this solution D.A.t/ /  W satisfies the maximal regularity estimate Z

T 0

q

q

q

Œkvt .t /kLp ..t// C kv.t /kW 2;p ..t// C krp.t/kLp ..t//  dt  q  C .T / kv0 kI q .A / C 0

Z 0

T

q

kf .t/kLp ..t// dt



p

for all v0 2 I q .A0 / WD .L ..0//; D.A.0/ //11=q;q , and all f 2 Lq ..0; T /I Lp ..t ///. If 0 is bounded, then the above inequality is also valid for T D 1. Special cases of Theorem 15 were already considered earlier. First investigations of the solvability of Eq. (54) and the corresponding Navier-Stokes equations can be found in [170]. The L2 ..t //-situation for a family of bounded domains f.t/gt2Œ0;T  was considered by Inoue and Wakimoto in [120] (see also [24]). The periodic case for the Stokes equation, i.e., if t 7! .t/ is periodic, is covered by results given in [206]. Note that the assumptions on the evolution and regularity of .t/ differ in the existing literature. A rough sketch of the proof of Theorem 15 is as follows: first one transforms (54) to a nonautonomous Cauchy problem on the time-independent domain .0/ by utilizing properties of the diffeomorphism . The resulting system then can be handled by a result on nonautonomous Cauchy problems, which was derived in [168].

3 The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties

2.11

159

An Approach for General Domains

By the results described in Sect. 2.2, the Helmholtz decomposition for Lp ./ does not exist for general unbounded domains, unless p D 2. Farwig, Kozono, and Sohr presented in [65, 66] an approach to the Stokes equation for a large class of unbounded domains, more precisely for uniform C 1 -domains of type .˛; ˇ; K/ (see [65] for the definition). Their main idea is to replace the space Lp ./ by the space Q p ./, where L Q p ./ WD Lp ./ \ L2 ./ for 2  p < 1 and L Q p ./ WD Lp ./ C L2 ./ for 1 < p < 2: L kk Q p

p

Q  ./ WD fu 2 Cc1 ./ W div u D 0g L ./ . Moreover, let L Given a uniform C 1 -domain   Rn , n  2, of type .˛; ˇ; K/ and 1 < p < 1, Q p ./ can de decomposed uniquely as u D u0 C rp they showed that each u 2 L Q p ./ and satisfying with u0 2 L ku0 kLQ p ./ C krpkLQ p ./  C kukLQ p ./ : p

Q p ./ with range L Q  ./. In particular, PQ u WD u0 defines a bounded projection on L Setting Q p u WD PQ p u; A with ( Q p/ D D.A

D.Ap / \ D.A2 /;

2  p < 1;

D.Ap / C D.A2 /;

1 < p < 2;

1;p

p

where D.Ap / D W 2;p ./ \ W0 ./ \ L ./, one obtains the following result. Theorem 16 ([65, 66]). Let 1 < p < 1,  as above and J D .0; T / for some Q p generates an analytic semigroup on L Q p ./. Moreover, for all T > 0. Then A p Q  .//, there exists a unique u solving the inhomogenous Stokes f 2 Lp .J; L equation (49) with u0 D 0 and satisfying Q p uk p Q p ku0 kLp .J ILQ p .// C kA Q p .// L .J IL .//  C kf kLp .J IL for some C > 0.

160

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The Stokes Equation on Domains with Edges and Vertices

Whereas the Lp -theory for classical elliptic and parabolic problems on singular domains is well developed, corresponding results for the Stokes equation are very rare, in particular for the instationary case. For the stationary Stokes equation, there are the classical regularity results, which go back to Kondrat’ev [129], Kellogg and Osborn [125], Dauge [51], as well as to Maz’ya and Rossmann [150] and Grisvard [99]. For results concerning domains with conical boundary points, see the work of Deuring [57]. He also proved a negative result concerning the generation of an analytic semigroup in three dimensions in [58]. More recently, an approach to analytic regularity was presented by Guo and Schwab in [102]. Most of the approaches in the articles listed above rely on a thorough analysis of the Green tensor (see, e.g., [150]) or on the Kondrat’ev technique, that is, on the transformation onto a layer by introducing polar coordinates. This leads in a natural way to a treatment in weighted Sobolev or Hölder spaces, the so-called Kondrat’ev spaces. Dauge [51] presented a detailed analysis for the stationary Stokes system for a large class of two- and three-dimensional singular domains. Kellogg and Osborn [125] provided H 2 -regularity in two-dimensional convex domains. In [99], an approach to the stationary Stokes equation in two dimensions on polygonal domains is developed. The approach relies on the fact that in two space dimensions the stationary Stokes equation is equivalent to the biharmonic equation. In fact, since div u D 0, there is a vector potential such that D curl u D @1 u2  @1 u2 . Applying curl to the Stokes equation, one obtains the scalar equation 2 D 0. Note that this method does not generalize to three space dimensions or to the instationary counterpart. Hence, it is very restrictive. For a recent local well-posedness result of the contact line problem for twodimensional Stokes flow, see [208]. It seems that a general approach to the instationary Stokes equation on singular domains does not exist in the literature, even for domains having a simple structure such as wedges. Just in the case of reflecting boundary conditions or for their lower-order perturbations, well-posedness is assured. For example, partial slip (also called Navier)-type boundary conditions given by ˛u C .D.u// D 0;

uD0

on @

(55)

belong to this class. Here ˛ 2 R is a parameter related to the slip length. For many types of domains, the condition (55) can be reformulated (modulo lower-order perturbation terms) as ˛u    curl u D 0;

uD0

on @:

(56)

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161

This reformulation is a key step in this investigation, since the Stokes equation subject to (56) reduces then to a heat equation. This idea is outlined in the following for perfect slip conditions, that is, for the case ˛ D 0. The analysis of this special situation is justified by the fact that in many situations the term ˛u can be treated also by perturbation arguments. For the perfect slip condition, it is well known that the Helmholtz projection P and the Laplacian  commute. In fact, one has curl P u D curl .u  rp/ D curl u; which implies P .D.ps //  D.ps /. Here, ps denotes the Laplacian subject to perfect slip conditions. Relying on Gauß’ theorem, one shows that   curl 2 uj@ D 0 provided   curl j@ u D 0. This yields P u D P curl 2 u D curl 2 u D curl 2 .P u C rp/ D P u:

(57)

Thus, in the case of perfect slip boundary conditions, one has A;p D ps jLp ;

(58) p

which means that the Stokes operator is the part of the Laplacian in L ./. In other words, in order to prove well-posedness results for the Stokes equation subject to perfect slip boundary condtions, it suffices to consider the following resolvent problem for the Laplacian: 8 < u  u D f in ;   curl u D 0 on @; :   u D 0 on @:

(59)

Thus, once sectoriality (or maximal Lp regularity, or a bounded H 1 -calculus, respectively) is proved for ps , then this property immediately transfers to the Stokes operator subject to perfect slip boundary conditions. As mentioned above, employing suitable perturbation arguments, in a number of cases, this remains true even for the Stokes operator subject to partial slip-type boundary conditions (55). The above approach was used in [143] for three-dimensional wedges, i.e., on domains of the form G D fx D .x1 ; x2 ; x3 / 2 R3 W 0  x2  sx1 g; where s > 0. Theorem 17 ([143]). Let 1 < p < 1 and AG;p be the Stokes operator subject to perfect slip conditions defined as in (58). Then AG;p admits an R-bounded H 1 p R1 calculus on L .G/ with A < =2. G;p

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Remarks 7. a) The above approach relies heavily on the property (57) and hence does not extend to other relevant types of boundary conditions. In fact, the question whether the negative of the Stokes operator with Dirichlet conditions or p other than partial slip-type boundary conditions is a sectorial operator on L ./ for wegde-type domains  remains an open question until today. This fact is very unsatisfactory, in particular in regard to applications. Indeed, solvability of the Stokes equation on wedge-type domains, for instance, is a crucial ingredient for the well-posedness of the moving contact line problem, which has been an open problem for decades. Only a few results for very restrictive values of contact angles c are known (see, e.g., [191] for c D 0, [205] for c D =2). For very recent results in this direction in the twodimensional setting, see [208]. b) Results similar to the one described in Theorem 17 are available also on socalled weakly singular domains for the class of reflecting boundary conditions. For instance, cylindrical domains G D fx D .x1 ; x2 ; x3 / 2 R3 W x12 C x22  r; 0  x3  hg for r; h > 0 belong to this class. Imposing, e.g.,  ˇ .D.u// C .  u/ ˇx3 2f0;hg D 0; ˇ uˇfx 2 Cx 2 Drg D 0; 1

(60)

2

then a version of Theorem 12 remains valid for  D G. The reader is referred, e.g., to [128] for the precise definition of weakly singular domains and for more results in this direction.

3

Other Approaches to the Stokes Equation for 1 < p < 1

In this section, three further approaches to the Stokes equation are presented. They differ from the approaches described above in Sect. 2. The first one concerns the so-called reduced Stokes equation, whereas the second one is based on the theory of layer potentials. The last one relies on Muckenhoupt weights and on an extrapolation theorem due to Rubio de Francia.

3.1

The Reduced Stokes Equation and the Pseudo-differential Operator Approach

Consider the Stokes equation (1) with f D 0 on a domain   Rn . Applying div to the first equation yields the following Neumann problem for p: p D 0 in ;

@ p D   curl 2 u on @:

3 The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties

163

Let G W u 7! rp denote the operator mapping the velocity u to rp via the above Neumann problem for p. Then, replacing rp in the Stokes equation by Gu, the velocity u solves the system 8 < .@t   C G/u D u D : u.0/ D

0 in RC  ; 0 on RC  @; u0 in :

(61)

The set of Eq. (61) is called reduced Stokes system. It was first introduced by Grubb and Solonnikov in [100, 101] in order to solve the Stokes equation on domains with smooth boundaries by a pseudo-differential operator approach. The advantage of system (61) in comparison to the usual Stokes equation is given by the fact that there is no condition on the divergence of u anymore. Hence, an approach in standard Lp -spaces is possible, in principle. However, the price to pay is that G is a nonlocal operator G. On the other hand, G is a singular Green operator, and thus the structure of (61) fits into the Boutet de Monvel calculus of pseudodifferential operators [101]. Thus formally, system (61) is equivalent to the Stokes equation. To see this, it remains to ensure that the condition div u D 0 can be recovered from a solution u of (61). In fact, one may apply div to (61), which yields .@t  /div u D 0; due to the properties of G. Next, applying the normal  to (61) and employing the relation  D curl 2  rdiv , one obtains 0 D   curl 2 u C   Gu D @ div u; again thanks to the properties of G. Thus div u solves a homogeneous Neumann problem for the heat equation and hence is constant. This constant, however, must be zero in view of   uj@ D 0 and the divergence theorem. This formal argumentation can be made rigorous in many concrete situations. In [100] and [101], Grubb and Solonnikov presented a comprehensive approach to the Stokes equations on bounded domains for inhomogeneous boundary conditions Bk .u; p/ D h of the type B1 .u; p/ WD u;

B2 .u; p/ WD T .u; p/;

B4 .u; p/ WD @ u  p;

B3 .u; p/ WD ŒD.u/ C .  u/;

B5 .u; p/ WD @ u C .u  /:

As before, the tangential part of u is denoted u D u  .  u/ of u; moreover D.u/ D 1=2.ruCruT / denotes the deformation tensor and T .u; p/ D 2D.u/Ip the stress tensor. Maximal and higher-order regularity for the inhomogeneous Stokes equation subject to either one of the above inhomogeneous boundary conditions, i.e.,

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8 < .@t   C G/u D Bk .u; p/ D : u.0/ D

f in RC  ; 0 on RC  @; u0 in ;

(62)

can be obtained following this approach. In order to avoid the introduction of trace classes and higher-order compatibility conditions, the main result on the Stokes equation is formulated in the following only for homogeneous boundary conditions and right-hand sides in mild regularity classes. For the general result, the reader is referred to [101, Theorem 7.5]. Setting Hk WD



L2 ./; k D 1; 3; 5; L2div ./; k D 2; 4;

where L2div ./ D fv 2 Lp ./I div v D 0g, the following result holds true. Theorem 18 ([100,101]). Let T 2 .0; 1/, J D .0; T /, k 2 f1; : : : ; 5g and assume that  is a bounded domain of class C 1 . Then for each pair ˚ .f; u0 / 2 L2 .J; Hk /  H 1 ./ \ Hk satisfying the compatibility condition .u0 / D 0 on @ if k D 1, there exists a unique solution ˚ b 1 .// .u; p/ 2 H 1 .J; Hk / \ L2 .J; H 2 .//  L2 .J; H of (62). In the case where k D 2; 4, one has pj@ 2 H 1=2 .J; L2 . @// \ L2 .J; H 1 . @//: Abels employed the reduced Stokes system approach in order to develop an Lp theory to the above system and also for the Stokes operator on asymptotically flat layers (see [9–11] and also Theorem 7). In [157], the Ansatz via the reduced Stokes system is used in order to prove maximal Lp -regularity for the Stokes equation on Lipschitz domains subject to partial slip-type boundary conditions (see also Theorem 22).

3.2

Lp -theory for Lipschitz Domains via Double Layer Potentials

Consider a bounded Lipschitz domain   R3 . Then, due to the roughness of the boundary, an approach to the Stokes equation on  via localization as presented in the sections above is no longer possible. For this reason, various other approaches

3 The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties

165

have been developed in order to deal with the situation of Lipschitz domains, see, e.g., [60, 79, 153, 171, 200]. A common approach in this situation is the approach by layer potentials. To this end, let G.x; / be the fundamental solution to the Helmholtz equation in R3 ;  > 0:

.  /u D 0

An explicit representation for G may be given in terms of Hankel functions (cf. [172]). In three dimensions, G is given by p

e  jxj ; G.x; / D 4jxj

x 2 R3 ;  > 0:

Next, introducing the functions 1 @j @k .G.x; /  G.x; 0// and 

j k .x; / WD G.x; /ıj k  ˆk .x/ WD @k G.x; 0/;

j; k D 1; 2; 3;

a calculation shows that the pair .j k ; ˆk / is a fundamental tensor for the Stokes resolvent problem (33) on R3 nf0g. For a suitable h defined on @, say h 2 Lp . @/, the double layer potential for the Stokes system then is defined as Z vj .x; / WD .D h/j .x/ WD

3 3 X X

@ kD1

@i j k .y  x; /i .y/

iD1

  ˆj .y  x/k .y/ hk .y/ d  .y/; p.x/ WD

3 X

j D 1; 2; 3;

Z @i @k

i;kD1

C

G.y  x; 0/i .y/hk .y/ d  .y/ @

3 Z X kD1

G.y  x; 0/k .y/hk .y/ d  .y/: @

Then the pair .v; p/ solves the first two lines in Eq. (64) in R3 n @. Note, however, that v does not satisfy the boundary condition vj@ D h in Eq. (64), but it can be shown that 1 v./ D . I C K /h 2

on @;

(63)

for  > 0 and with an operator K 2 L.Lp . @//. Note that the trace of v in (63) is understood as a nontangential limit taken inside , cf. [123]. A crucial point now is

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M. Hieber and J. Saal

to show that  12 I C K is invertible on Lp . @/, since then the pair .w; p/, where  1 h; is a solution of w./ WD D  12 I C K 8 < w  w C rp D 0 in ; div w D 0 in ; : w D h on @:

(64)

This strategy was already used by Giga in [89] in the case of smooth boundaries. In this case, K is a compact operator, and K can be viewed as a lower-order perturbation for large . In the case of Lipschitz domains, one sets Z X n

hj ./ WD .

j k .x  y; /fk .y/ dy/j@ ;

j D 1; : : : ; n;

 kD1

and sees that a solution of (33) is given by .u; p/, where uj ./ D

Z X n

j k .x  y; /fk .y/ dy  wj ./;

j D 1; : : : ; n:

 kD1

This was proved first rigorously by Shen in [171] for the case  D 0. Theorem 19 ([171]). Let   R3 be a bounded graph Lipschitz domain and 3=2  p  3. Then, for each f 2 W 1;p ./, there is a unique solution 1;p p .u; p/ 2 W0 ./  L0 ./ of the stationary Stokes system (15) satisfying kukW 1;p C kpkp  C kf kW 1;p ; where C > 0 is independent of f , u, and p. The corresponding assertion for fixed  2 † and  2 .0; =2/ can be a carried out in an analogous way. On the other hand, it is not obvious how to derive uniform resolvent estimates for u, i.e., estimates of the form ku./kp  C kf kp ;

 2 † ; f 2 Lp ./:

This was known as Taylor’s conjecture (see [198]). An affirmative answer to Taylor’s conjecture was given by Shen in [172]. His result reads as follows. Theorem 20 ([172]). Let   R3 be a bounded graph Lipschitz domain. Then there exists " > 0 such that for all 3=2  " < p < 3 C ", the Stokes operator subject p to Dirichlet boundary conditions generates an analytic semigroup on L ./.

3 The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties

167

The strategy of Shen’s proof is to apply a Calderón-Zygmund argument in order to extrapolate sectoriality from L2 to Lp . For doing this, the main task is to derive a weak reverse Hölder estimate of type

1 jB.x0 ; r/ \ j C

Z

jujp dx

1=p

B.x0 ;r/\

1 jB.x0 ; 2r/ \ j

Z

2

juj dx

1=2 ;

B.x0 ;2r/\

for local solutions u of the Stokes resolvent problem, which is uniform in . Shen established this estimate in [172] based on an argument, which shows that  1 1  2 I C K is bounded on L2 by a constant independent of  2 † . The potential theoretical approach is standard in the theory of partial differential equations. However, there is a significant difference between smooth and nonsmooth domains. For smooth domains with compact boundary of class C 1C˛ , say, the operator K usually is compact. Then, invertibility follows easily by a Fredholm argument. However, compactness of K fails to be true in nonsmooth domains. Thus, invertibility of  12 I C K is much harder to prove in this setting and requires more subtle tools from Harmonic Analysis. The reader is referred to the classical papers [64, 123] for a potential theoretical approach to elliptic and parabolic equations on graph Lipschitz domains. p Recently, it was shown by Tolksdorf that the Stokes operator on L ./ for q bounded Lipschitz domains  admits maximal L -regularity provided the following condition on p is satisfied. Theorem 21 ([200]). Let   Rn be a bounded Lipschitz domain and n  3. Then there exists " > 0 such that for all 2n 2n "

0 such that jj kvkLp ./ CkvkW 2;p ./ CkkW 1;p .G/  C kf kLp ./ ;  2 †" [f0g; f 2 Lp ./: (84) As in the case of the classical Helmholtz projection, the existence of the hydrostatic Helmholtz projection is closely related to the unique solvability of the Poisson problem in the weak sense. In the given situation, the equation H  D divH f in G, subject to periodic boundary conditions, plays an essential role.

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Lemma 6 ([109]). Let p 2 .1; 1/ and f 2 Lp .G/. Then there exists a unique 1;p p  2 Wper .G/ \ L0 .G/ satisfying hrH ; rH iLp0 .G/ D hf; rH iLp0 .G/ ;

0

p0

1;p 2 Wper .G/ \ L0 .G/:

(85)

Furthermore, there exists a constant C > 0 such that kkW 1;p .G/  C kf kLp .G/ ;

f 2 Lp .G/:

(86)

The above Lemma 6 allows to define the hydrostatic Helmholtz projection Pp W 1;p p Lp ./ ! Lp ./ as follows: given v 2 Lp ./, let  2 Wper .G/ \ L0 .G/ be the unique solution of Eq. (85) with f D v. N One then sets Pp v WD v  rH ;

(87)

and calls Pp the hydrostatic Helmholtz projection. It follows from Lemma 6 that Pp2 D Pp and that thus Pp is indeed a projection. In the following one defines the closed subspace Xp of Lp ./ as Xp WD RgPp . This space plays the analogous role in the investigations of the primitive equations p as the solenoidal space L ./ plays in the theory of the Navier-Stokes equation. The hydrostatic Helmholtz projection Pp defined as in (87) allows then to define the hydrostatic Stokes operator as follows. In fact, let 1 < p < 1 and Xp be defined as above. Then the hydrostatic Stokes operator Ap on Xp is defined as (

Ap v WD Pp v; 2;p ./2 W divH vN D 0 in G; @z v D 0 on u ; v D 0 on b g: D.Ap / WD fv 2 Wper (88)

The resolvent estimates for Eqs. (82) and (83) given in Proposition 22 yield that Ap generates a bounded analytic semigroup on Xp . More precisely, one has the following result. Theorem 42 ([109]). Let 1 < p < 1. Then the hydrostatic Stokes operator Ap generates a bounded analytic C0 -semigroup Tp on Xp . Moreover, there exist constants C; ˇ > 0 such that kTp .t /f kXp  C e ˇt kf kXp ;

t > 0:

Recently, it was shown by Giga, Gries, Hieber, Hussein, and Kashiwabara [93] that Ap even admits an R-bounded H 1 -calculus on Xp of angle 0, which implies in particular maximal Lq  Lp -estimates for the solution of the hydrostatic Stokes equation and allows further to characterize the domains D.Ap / of the fractional powers Ap for 0 <  < 1 in terms of Sobolev spaces subject to the boundary conditions given. More precisely, one has the following results.

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195

Theorem 43 ([93]). Let p 2 .1; 1/. Then the operator Ap admits a bounded RH 1 -calculus on Xp with AR;1 D 0. p Combining this result with a characterization of the complex interpolation spaces ŒXp ; D.Ap / proved in [108] allows then to characterize the domains D.Ap / of the fractional powers Ap for 0 <  < 1 as follows. Corollary 7 ([93]). Let 1 < p < 1 and  2 Œ0; 1 with  … f1=2p; 1=2 C 1=2pg. Then 8 ˇ ˇ 2;p ˇ ˇ ˆ ˆ 0 such that

ke

tAp

kre

tAp

ke

6.6

tAp

Pp f kLq ./  C t Pp f kLq ./  C t

Pp div f kLq ./  C t



3 2



3 2



3 2







1 1  p q

for f 2 Lp ./; t > 0;

kf kLp ./ ;

1 1 1   p q 2

1 1 1   p q 2

kf kLp ./ ;

for f 2 Lp ./; t > 0;

kf kLp ./ ;

for f 2 Lp ./; t > 0:

The Stokes Operator in the Rotating Setting

Consider the linearization of Navier-Stokes equation with Coriolis force on all of R3 , i.e., consider the equation ut  u C e3  u C rp D 0;

in R3  .0; 1/;

div u D 0;

in R3  .0; 1/;

u.0; x/ D u0 .x/;

(89)

x 2 R3 ;

where denotes the viscosity coefficient of the fluid,  the speed of rotation, and e3 the unit vector in x3 -direction. This equation gained quite some attention due to its importance in geophysical flows. Taking Fourier transforms in (89) and

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denoting this equation by F(89), an explicit solution formula for F(89) was derived by different methods in [94] and [116]. Lemma 7. There exists a unique solution .Ou; p/ O of equation F(89), where uO for

D 1 is given by

uO .t; / D cos.

3 3 2 2 u0 . /; t /e j j t I ub0 . / C sin. t /e j j t R. /b j j j j

t  0; 2 R3 ;

and 0

0 B 3 R D R. / D @  j j 2 j j

3 j j

2  j j

0 1  j j

1 j j

1 C A:

0 p

One may deduce from Lemma 7 that the solution of Eq. (89) in L .R3 / for D 1 p is governed by a C0 -semigroup Tp on L .R3 /, which is explicitly given by

 3 3 2 2 Tp .t /f WD F 1 cos. t /e j j t I fO . / C sin. t /e j j t R. /fO . / ; j j j j t  0; f 2 Lp .R3 /: This semigroup is called the Stokes-Coriolis semigroup. Mikhlin’s theorem implies then the following result. Theorem 45. Let 1 < p < 1 and Tp be the Stokes-Coriolis semigroup defined as p above. Then Tp is a C0 -semigroup on L .R3 /, which may be represented as Q 3 t /I C sin.R Q 3 t /Re t f; Tp .t /f D Œcos.R

t  0; f 2 Lp .R3 /:

Q 3 denotes the operator associated to the symbol 3 . Here R j j Further information on the Stokes operator in the rotating setting can be found, e.g., in [43, 94, 95, 116]. The dispersive effect of the Coriolis force and its consequences concerning global well-posedness for the Navier-Stokes equations in the rotational framework was investigated in detail by Mahalov and Nicolaenko [142], by Chemin, Desjardin, Gallagher, and Grenier [43] as well as by Koh, Lee, and Takada in [127], by Iwabuchi and Takada in [121], and by Kozono, Mashiko, and Takada in [131].

3 The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties

7

197

Conclusion

This survey article is concerned with well-posedness questions and regularity properties of the Stokes equation in various classes of domains   Rn within the Lp -setting for 1  p  1. Taking the point of view of evolution equations, classical as well as modern approaches for obtaining well-posedness results for the Stokes equation in the strong sense are presented. Topics being discussed include the Helmholtz decomposition, the Stokes operator, the Stokes semigroup, and maximal Lp -regularity results for 1 < p < 1 via the theory of R-sectorial operators. Of concern are domains having compact or noncompact, smooth, or nonsmooth boundaries. In addition, the endpoints of the Lp -scale, i.e., p D 1 and p D 1, are considered, and recent well-posedness results for the case p D 1 are described. Results on Lp  Lq -smoothing properties of the associated Stokes semigroups and on various variants of the Stokes equation complete this article.

8

Cross-References

 Critical Function Spaces for the Well-Posedness of the Navier-Stokes Initial Value

Problem  Derivation of Equations for Continuum Mechanics and Thermodynamics of

Fluids  Large Time Behavior of the Navier-Stokes Flow  Local and Global Existence of Strong Solutions for the Compressible Navier-S-

tokes Equations Near Equilibria via the Maximal Regularity  Modeling and Analysis of the Ericksen-Leslie Equations for Nematic Liquid

Crystal Flows  Models and Special Solutions of the Navier-Stokes Equations  Recent Advances Concerning Certain Class of Geophysical Flows  Regularity Criteria for Navier-Stokes Solutions  Self-Similar Solutions to the Nonstationary Navier-Stokes Equations  Stokes Problems in Irregular Domains with Various Boundary Conditions  Stokes Semigroups, Strong, Weak, and Very Weak Solutions for General Domains  The Inviscid Limit and Boundary Layers for Navier-Stokes Flows  Time-Periodic Solutions to the Navier-Stokes Equations  Vorticity Direction and Regularity of Solutions to the Navier-Stokes Equations

References 1. K. Abe, Y. Giga, Analyticity of the Stokes semigroup in spaces of bounded functions. Acta Math. 211, 1–46 (2013) 2. K. Abe, Y. Giga, The L1 -Stokes semigroup in exterior domains. J. Evol. Equ. 14, 1–28 (2014)

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4

Stokes Problems in Irregular Domains with Various Boundary Conditions Sylvie Monniaux and Zhongwei Shen

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Dirichlet Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Linear Dirichlet-Stokes Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Nonlinear Dirichlet-Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Neumann Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Linear Neumann-Stokes Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Nonlinear Neumann-Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Hodge Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Hodge-Laplacian and the Hodge-Stokes Operators . . . . . . . . . . . . . . . . . . . . . . 4.2 The Nonlinear Hodge-Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Robin Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Robin-Hodge-Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Robin-Hodge-Stokes Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Nonlinear Robin-Hodge-Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

208 210 210 216 223 223 225 226 226 233 234 235 240 244 245 246 246

Abstract

Different boundary conditions for the Navier-Stokes equations in bounded Lipschitz domains in R3 , such as Dirichlet, Neumann, Hodge, or Robin boundary conditions, are presented here. The situation is a little different from the case of smooth domains. The analysis of the problem involves a good comprehension

S. Monniaux () Aix-Marseille Université, CNRS, Centrale Marseille, Marseille, France e-mail: [email protected] Z. Shen University of Kentucky, Lexington, KY, USA e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_4

207

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of the behavior near the boundary. The linear Stokes operator associated to the various boundary conditions is first studied. Then a classical fixed-point theorem is used to show how the properties of the operator lead to local solutions or global solutions for small initial data.

1

Introduction

The aim of this chapter is to describe how to find solutions of the Navier-Stokes equations 8 ˆ @ u  u C r C .u  r/u D 0 in .0; T /  ; ˆ < t div u D 0 in .0; T /  ; ˆ ˆ : u.0/ D u0 in ;

(NS)

in a bounded Lipschitz domain   R3 and a time interval .0; T / (T  1), for initial data u0 in a critical space, with one of the following boundary conditions on @: 1. Dirichlet boundary conditions: u D 0;

(Dbc)

also called “no-slip” boundary conditions, which can be also decomposed as a nonpenetration condition   u D 0 and a tangential part   u D 0 which model the fact that the fluid does not slip at the boundary; this is commonly used for a boundary between a fluid and a rigid surface; 2. Neumann boundary conditions: Œ.ru/ C .ru/>    D 0;

 2 .1; 1;

(Nbc)

which can be rewritten as T .u; / D 0 where T .u; / WD .ru/ C .ru/>   Id; if  D 0, (Nbc) becomes @ u D ; if  D 1, T1 .u; / is the Cauchy’s stress tensor so that (Nbc) can be viewed, for instance, as an absence of stress on the interface separating two media in the case of a free boundary; (Nbc) can be decomposed into its normal and tangential parts and can be rewritten in the following form:    .1 C /   @ u D ; .ru/ C .ru/>  tan D 0I (1) 3. Hodge boundary conditions:   u D 0;

  curl u D 0;

(Hbc)

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also called “absolute” boundary conditions (see [53, Section 9] or “perfect wall” condition (see [1]); they have been studied in, e.g., [4] and [24]; they are related to the more traditionally used “Navier’s slip” boundary condition:   u D 0;



  ru/> C .ru  tan D 0:

(2)

See discussion below (see also a detailed discussion in [35, Section 2]). 4. Robin boundary conditions:   u D 0;

  curl u D ˛ u;

˛ > 0I

(Rbc)

since   u D 0, u is a tangential vector field at the boundary, so it makes sense to compare it to the tangential part of the vorticity: it describes the fact that the fluid slips with a friction proportional to the vorticity. Remark that (Hbc) is recovered if ˛ D 0 and (Dbc) if ˛ D 1. In the boundary conditions above, .x/ denotes the unit exterior normal vector at a point x 2 @ (defined almost everywhere when @ is a Lipschitz boundary). As explained in [35, Section 2 and Section 6], the Hodge boundary conditions (Hbc) are close to the Navier’s slip boundary conditions (2). Indeed, if  is assumed to be smooth enough, say of class C2 , under the condition   u D 0, the following holds: 

  ru/> C .ru  tan D   curl u C 2 Wu

where W is the Weingarten map (also called the shape operator, see [45, Chapter 5]) on @ acting on tangential fields (see also [17, Section 3]). In particular, the term Wu is a zero-order term, depending linearly on the velocity field u and is equal to 0 on flat portions of the boundary. The strategy in this chapter to solve the Navier-Stokes equations with one of the boundary conditions described above is to find a functional setting in which the Fujita-Kato scheme applies, such as in their fundamental paper [20]. In all situations, the idea is to study the linear problem to prove enough regularizing properties of the Stokes semigroup so that the nonlinear problem can be treated via a fixed-point method. For the last two types of boundary conditions (Hbc) and (Rbc), the Navier-Stokes system is rewritten as follows: 8 ˆ @ u  u C r  u  curl u D 0 in .0; T /  ; ˆ < t div u D 0 in .0; T /  ; ˆ ˆ : u.0/ D u0 in :

(NS’)

This is motivated by the form of the boundary conditions and the fact that, for a smooth enough vector field u,

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.u  r/u D 12 rjuj2  u  curl u; so that (NS) becomes (NS’) with the pressure  replaced by the so-called dynamical pressure  C 12 juj2 (see, e.g., [24] or [4]). In this chapter,   R3 is a bounded, simply connected, Lipschitz domain. The chapter is organized as follows. In Sect. 2, the Dirichlet-Stokes operator is defined in 2 p the ˚ L setting and then in the L theory. Existence of a local 2solution of the system (NS); (Dbc) for initial values in a critical space in the L -Stokes scale is then shown. In Sect. 3, the previous proofs are adapted in the case of Neumann boundary ˚  ˚ conditions, i.e., for the system (NS); (Nbc) . In Sect. 4, the˚system (NS’); (Hbc) 3 3 is studied for initial conditions  in the critical space u 2˚ L .I R /Idiv u D 0 in ;   u D 0 on @ , whereas in Sect. 5, the system (NS’); (Rbc) is considered in a C1 domain.

2

Dirichlet Boundary Conditions

For a more complete exposition of the results in this section, as well as an extension to more general domains, the reader can refer to [34, 41] and [51]. The case where  is smooth was solved by Fujita and Kato in [20]. In [15], the case of bounded Lipschitz domains  was studied for initial data not in a critical space.

2.1

The Linear Dirichlet-Stokes Operator

2.1.1 The L2 Theory The following remarks about L2 vector fields on  will be used throughout this chapter. Remark 1. For   R3 a bounded Lipschitz domain, let u 2 L2 .I R3 / such that div u 2 L2 .I R/. Then   u can be defined on @ in the following weak sense in 1 H  2 .@I R/: for  2 H 1 .I R/, hu; ri C hdiv u; i D h  u; 'i@ ;

(3)

where ' D Trj@ , the right-hand side of (3) depends only on ' on @ and not on the choice of , its extension to . The notation h; iE stands for the L2 -scalar product on E. The following Hodge decomposition holds on vector fields: L2 .I R3 / is equal ?

to the orthogonal direct sum HD ˚ G, where ˚  HD D u 2 L2 .I R3 /I div u D 0 in ;   u D 0 on @

(4)

4 Stokes Problems in Irregular Domains With Various Boundary Conditions

211

and G D rH 1 .I R/. This follows from the following theorem due to Georges de Rham [12, Chap. IV §22, Theorem 17’]; see also [55, Chap.I §1.4, Proposition 1.1]. 3 0 Theorem 1 (de Rham). Let T be a distribution in C1 c .I R / such that hT; i D 1 3 0 for all  2 Cc .I R / with div  D 0 in . Then there exists a distribution 1 0 0 S 2 C1 c .I R/ such that T D rS . Conversely, if T D rS with S 2 Cc .I R/ , 1 3 then hT; i D 0 for all  2 Cc .I R / with div  D 0 in .

Remark 2. In the case of a bounded Lipschitz domain   R3 , the space HD 3 coincides with the closure in L2 .I R3 / of the space of vector fields u 2 C1 c .I R / with div u D 0 in . Denote by J W HD ,! L2 .I R3 / the canonical embedding and P W L2 .I R3 / ! HD the orthogonal projection, called either Leray or Helmholtz projection. It is clear that PJ D IdHD. Define now the space VD D H01 .I R3 / \ HD : it is a closed subspace of H01 .I R3 /. The embedding J restricted to VD maps VD to H01 .I R3 /: denote it by J0 W VD ,! H01 .I R3 /. Its adjoint J00 D P1 W H 1 .I R3 / ! VD0 is then an extension of the orthogonal projection P. The space HD is endowed with the norm u 7! kuk2 and VD with the norm u 7! kruk2 . The definition of the Dirichlet-Stokes operator then follows. Definition 1. The Dirichlet-Stokes operator is defined as being the associated operator of the bilinear form: a W VD  VD ! R;

a.u; v/ D

3 X h@i J0 u; @i J0 vi: iD1

Proposition 1. The Dirichlet-Stokes operator AD is the part in HD of the bounded operator A0;D W VD ! VD0 defined by A0;D u W VD ! R, .A0;D u/.v/ D a.u; v/, and satisfies ˚  D.AD / D u 2 VD I P1 . D /J0 u 2 HD ; AD u D P1 . D /J0 u u 2 D.AD /; 2 3 where  D denotes the weak vector-valued Dirichlet-Laplacian in L .I R /. The operator AD is self-adjoint, invertible, AD generates an analytic semigroup of 1

contractions on HD , D.AD2 / D VD , and for all u 2 D.AD /, there exists  2 L2 .I R/ such that JAD u D J0 u C r and D.AD / admits the following description: ˚  D.AD / D u 2 VD I 9  2 L2 .I R/ W J0 u C r 2 HD :

(5)

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Proof. By definition, for u 2 D.AD / and for all v 2 VD , hAD u; vi D a.u; v/ D

n X

h@j J0 u; @j J0 vi

j D1

D

n X

2 H 1 h@j J0 u; J0 viH01

D

H 1 h./J0 u; J0 viH01

j D1

D VD0 hP1 ./J0 u; viVD : The third equality comes from thePdefinition of weak derivatives in L2 ; the fourth equality comes from the fact that nj D1 @2j D . The last equality is due to the fact that J00 D P1 . Therefore, AD u and P1 ./J0 u are two linear forms which coincide on VD ; they are then equal, which proves that A0;D D P1 ./J0 W VD ! VD0 . Moreover, the fact that u 2 D.AD / implies that AD u is a linear form on HD , so that the linear form P1 ./J0 u, originally defined on VD , extends to a linear form on HD (since VD is dense in HD by de Rham’s theorem). The fact that AD is selfadjoint and AD generates an analytic semigroup of contractions comes from the properties of the form a: a is bilinear, symmetric, sectorial of angle 0, and coercive 1

on VD VD . The property that D.AD2 / D VD is due to the fact that AD is self-adjoint, applying a result by J.L. Lions [29, Théorème 5.3]. To prove the last assertions of this proposition, let u 2 D.AD /. Then AD u 2 HD and P1 J .AD u/ D PJ .AD u/ D u. Moreover, if u 2 D.AD /, u belongs, in particular, to VD . Therefore, J0 u 2 H01 .I R3 / and ./J0 u 2 H 1 .I R3 /. The following identities take place in VD0 :   P1 J .AD u/  ./J0 u D P1 J .AD u/  P1 ./J0 u D AD u  AD u D 0: 0 By de Rham’s theorem, this implies that there exists p 2 C1 c .I R/ such that 1 3 Q J .AD u/  ./J u D rp: rp 2 H .I R /, which implies that p 2 L2 .I R/. t u

The relations between the spaces and the operators described above are summarized in the following commutative diagram:

VD

J0

d A0,D

HD

d J P=J

d

VD

H01 L2 d

P1 =J0

H −1

(−ΔΩ D)

4 Stokes Problems in Irregular Domains With Various Boundary Conditions

213

In the case of a bounded Lipschitz domain   R3 , the following property of 3

D.AD4 / also holds; see [34, Corollary 5.5]. 3

Proposition 2. The domain of AD4 is continuously embedded into W01;3 .I R3 /. It has been proved by R. Brown and Z. Shen [7] that the domain of AD 3 1;p is embedded into W0 .I R3 / \ W 2 ;2 .; R3 / for some p > 3. The proof Proposition 2 uses the well-posedness result for the Poisson problem of the Stokes system [16, Theorem 5.6], similar to the corresponding result proved in [26] for the Laplacian.

2.1.2 The Lp Theory P. Deuring provided in [14] an example of a domain with one conical singularity such that the Dirichlet-Stokes semigroup does not extend to an analytic semigroup in Lp for p large, away from 2. M.E. Taylor in[54], however, conjectured that this should be true for p in an interval containing 32 ; 3 , which was indeed proved 12 years later by the second author in [51]. 1 3 Let C1 c; ./ denote the space of vector fields u 2 Cc .I R / with div u D 0 in  and p 3 Lp ./ D the closure of C1 c; ./ in L .I R /:

(6)

Note that if  is Lipschitz and p D 2, L2 ./ D HD . In view of Proposition 1, the Dirichlet-Stokes operator in the Lp setting for 1 < p < 1 is defined by AD;p D u C r;

(7)

with the domain n 1;p D.AD;p / D u 2 W0 .I R3 /I div u D 0 in  and  u C r 2

Lp ./

o for some  2 L ./ :

(8)

p

p

1 ./  D.AD;p /, the operator AD;p is densely defined in L ./ and Since Cc; 1 AD;p .u/ D P./u for u 2 Cc; ./. If p D 2, AD;p agrees with the DirichletStokes operator AD defined in the previous subsection. The following theorem was proved in [51].

Theorem 2. Let  be a bounded Lipschitz domain in R3 . Then there exists " > 0, depending only on the Lipschitz character of , such that AD;p generates p a bounded analytic semigroup in L ./ for .3=2/  " < p < 3 C ".

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It was in fact proved in [51] that if  is a bounded Lipschitz domain in Rd , p d  3, then AD;p generates a bounded analytic semigroup in L ./ for 2d 2d "

0 depends only on d and the Lipschitz character of . This was done by establishing the following resolvent estimate in Lp : k.AD;p C /1 f kLp .ICd /  Cp jj1 kf kLp .ICd /

(10)

d for any f 2 C1 c .I C / with div f D 0 in , where p satisfies (9),

˚   2 † WD z 2 C W  ¤ 0 and j arg.z/j <   ; and 2 .0; =2/. The constant Cp in (10) depends only on d , , p, and . It has long been known that if  is a bounded C2 domain in Rd , the resolvent estimate (10) holds for  2 † and 1 < p < 1 (see [21]). Consequently, the operator AD;p generates a bounded analytic semigroup in Lp for any 1 < p < 1, if  is C2 . The case of nonsmooth domains is much more delicate. As mentioned earlier, P. Deuring constructed a three-dimensional Lipschitz domain for which the Lp resolvent estimate (10) fails for p sufficiently large. This was somewhat unexpected. Indeed it was proved in [48] that the Lp resolvent estimate holds for 1 < p < 1 in bounded Lipschitz domains in R3 for any second-order elliptic systems with constant coefficients satisfying the Legendre-Hadamard conditions (the range is d2d  " < p < d2d C " for d  4). It is worth mentioning that C3 3 it is not known whether the range of p in Theorem 2 is sharp. The approach used in [51] to the proof of (10) is described below. Consider the operator T on L2 .I Cd /, defined by T .f / D u, where  2 † and u 2 H01 .I Cd / are the unique solution to the Stokes system: 8 u C r C u D f ˆ ˆ < div u D 0 ˆ ˆ : uD0

in ; in ;

(11)

on @:

Note that T is bounded on L2 .I Cd / and kT kL2 !L2  C . To show that T is bounded on Lp .I Cd / and kT kLp !Lp  C for 2 < p < d2d C ", a real 1 variable argument is used, which may be regarded as a refined (and dual) version of the celebrated Calderón-Zygmund lemma. According to this argument, which originated from [8] and was further developed in [49,50], one only needs to establish the weak reverse Hölder estimate:

4 Stokes Problems in Irregular Domains With Various Boundary Conditions



jujpd

1=pd

C



juj2

B.x0 ;r/\

for pd D

2d , d 1

1=2

215

(12)

B.x0 ;2r/\

whenever u 2 H01 .I Cd / is a (local) solution of the Stokes system: (

u C r C u D 0;

(13)

div u D 0

in B.x0 ; 3r/ \  for some x0 2  and 0 < r < c diam./. The extra " in the range of p is due to the self-improvement property of the weak reverse Hölder inequalities (see, e.g., [25]). To prove the estimate (12), the Dirichlet problem for the Stokes system (13) is considered in a bounded LipschitzR domain  in Rd , with boundary data u D f on @, where f 2 L2 .@I Cd / and @ f   D 0. The goal is to show that k.u/ kL2 .@/  C kf kL2 .@/ ;

(14)

where .u/ denotes the nontangential maximal function of u and is defined by n o .u/ .Q/ WD sup ju.x/j W x 2  and jx  Qj < C0 dist.x; @/ for any Q 2 @ (C0 > 1 is a large fixed constant depending on d and ). This, together with the inequality Z

juj

pd

1=pd

C

Z



j.u/ j2

1=2

;

@

which holds for any continuous function u in , leads to Z 

jujpd

1=pd

C

Z

juj2

1=2

:

(15)

@

The desired estimate (12) follows by applying (15) in the domain B.x0 ; t r/ \  for t 2 .1; 2/ and then integrating the resulting inequality with respect to t over .1; 2/. Finally, the nontangential maximal function estimate (14) is established by the method of layer potentials. The case  D 0 was studied in [11, 18], where the L2 Dirichlet problem as well as the Neumann type boundary value problems with boundary data in L2 for the system u C r D 0 and div u D 0 in a Lipschitz domain  was solved by the method of layer potentials, using the Rellich-type estimates: @u  krtan ukL2 .@/ : 2 @ L .@/

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Here @u is a conormal derivative and rtan u denotes the tangential derivative of @ u on @. The reader is referred to the book [27] by C. Kenig for references on related work on Lp boundary value problems for elliptic and parabolic equations in nonsmooth domains. In an effort to solve the L2 initial boundary value problems for the nonstationary Stokes equations @t u  u C r D 0 and div u D 0 in a Lipschitz cylinder .0; T /  , the Stokes system (13) for  D i with 2 R was considered by the second author in [47]. One of the key observations in [47] is that if  D i and 2 R is large, the Rellich estimates for the system (13) involve two extra terms j j1=2 kukL2 .@/ and j jkukH 1 .@/ , where H 1 .@/ denotes the dual of H 1 .@/. While the first term j j1=2 kukL2 .@/ was expected in view of the Rellich estimates for the Helmholtz equation  C i in [6], the second term j jku  kH 1 .@/ was not. Let @u @u D  : @ @ By following the general approach in [47], it was proved in [51] that if .u; / is a suitable solution of (13) in , then

k

@u kL2 .@/  krtan ukL2 .@/ C jj1=2 kukL2 .@/ C jjku  kH 1 .@/ @

(16)

holds uniformly in  for  2 † with jj  c > 0. As in the case of Laplace’s equation [56], the estimate (14) follows from (16) by the method of layer potentials. The reader is referred to [51] for the details.

2.2

The Nonlinear Dirichlet-Navier-Stokes Equations

˚  The system (NS); (Dbc) is invariant under the scaling u .t; x/ D  u.2 t; x/, ˚ 2 . t; x/ 2 .0; T /   ( > 0): if u is a solution of (NS); (Dbc) in .0; T /   for  ˚   the initial value u0 , then u is a solution of (NS); (Dbc) in 0; T2  1  for the initial value x 7! u0 .x/. ˚  The goal here is to find the so-called mild solutions of the system (NS); (Dbc) for initial values u0 in a critical space, in the same spirit as in [20]. 1

Lemma 1. The space D.AD4 / is a critical space for the Navier-Stokes equations. 1

Proof. The space D.AD4 / is invariant under the scaling u .x/ D u0 .x/ for 1 x 2 1 ,  > 0. Indeed, it suffices to check that ku k2 D  2 kuk2 and 1

1

kru k2 D  2 kruk2 and apply the fact that D.AD4 / is the interpolation space

4 Stokes Problems in Irregular Domains With Various Boundary Conditions

217 1

(with coefficient 12 ) between HD , closed subspace of L2 .I R3 /, and VD D D.AD2 /, t u closed subspace of H01 .I R3 /. For T > 0, define the space ET by n 1 3 1 ET D u 2 Cb .Œ0; T I D.AD4 //I u.t / 2 D.AD4 /; u0 .t / 2 D.AD4 / for all t 2 .0; T  1

3

o

1

and sup kt 2 AD4 u.t /k2 C sup ktAD4 u0 .t /k2 < 1 t2.0;T /

t2.0;T /

endowed with the norm 1

1

3

1

kukET D sup kAD4 u.t /k2 C sup kt 2 AD4 u.t /k2 C sup ktAD4 u0 .t /k2 : t2.0;T /

t2.0;T /

t2.0;T /

The fact that ET is a Banach space is straightforward. Assumenow that u 2 ET ˚ and that .J0 u; p/ (with p 2 L2 .I R/) satisfy (NS); (Dbc) in H 1 .I R3 /: indeed, every term rp, @t J0 u, J0 u, and .J0 u  r/J0 u independently belongs to H 1 .I R3 /. Apply P1 to the equations and obtain   u0 .t / C AD u.t / D P1 .J0 u  r/J0 u ˚  since P1 rp D 0 and P1 ./J0 u D A0;D u. The problem (NS); (Dbc) is then reduced to the abstract Cauchy problem:   u0 .t / C A0;D u.t / D P1 .J0 u  r/J0 u u.0/ D u0 ;

u 2 ET ;

(17)

for which a mild solution is given by the Duhamel formula: u D ˛ C .u; u/;

(18)

where ˛.t/ D e tAD u0 and Z

t

.u; v/.t / D 0

   e .ts/AD  12 P1 .J0 u.s/  r/J0 v.s/ C .J0 v.s/  r/J0 u.s/ ds:

(19) The strategy to find u 2 ET satisfying u D ˛ C .u; u/ is to apply a fixed-point theorem. For that, ET needs to be a “good” space for the problem, i.e., ˛ 2 ET and .u; u/ 2 ET . The fact that ˛ 2 ET follows directly from the properties of the Stokes operator AD and the semigroup .e tAD /t0 . Proposition 3. The mapping  W ET  ET ! ET is bilinear, continuous, and symmetric.

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Proof. The fact that  is bilinear and symmetric is immediate, once it is proved that it is well-defined. For u; v 2 ET , let   f .t/ D  12 P1 .J0 u.t /  r/J0 v.t / C .J0 v.t /  r/J0 u.t / ;

t 2 .0; T /:

(20)

By the definition of ET and Sobolev embeddings, it is easy to see that .J0 u.t /  r/J0 v.t / C .J0 v.t /  r/J0 u.t / 2 L2 .I R3 / and .J0 u.t /  r/J0 v.t / C .J0 v.t /  r/J0 u.t /  C t  34 kukE kvkE T T 2 where C is a constant independent from t , which gives the following estimate: f .t/  C t  34 kukE kvkE T T 2

(21)

Therefore, Z

1 4

kAD .u; v/.t /k2  C

t 0

1

Z

t 0

and since obtained:

Rt

0 .t

1

3

 s/ 4 s  4 ds D

3

kAD4 e .ts/AD kL.HD / C s  4 kukET kvkET ds  1 3 .t  s/ 4 s  4 ds kukET kvkET ;

R1 0

1

3

.1  s/ 4 s  4 ds, the following estimate is finally

1

kAD4 .u; v/.t /k2  C kukET kvkET :

(22)

1

The proof of the continuity of t 7! AD4 .u; v/.t / on HD is straightforward once the estimate (22) is established. The proof of the fact that p 3 k t AD4 .u; v/.t /k2  C kukET kvkET 1

(23)

3

is proved the same way, replacing AD4 by AD4 and using the fact that 3

3

kAD4 e .ts/AD kL.HD /  C .t  s/ 4 and Z 0

t

3

3

1

.t  s/ 4 s  4 ds D t  2

Z 0

1

3

3

.1  s/ 4 s  4 ds:

4 Stokes Problems in Irregular Domains With Various Boundary Conditions

219

It remains to prove the estimate on the derivative with respect to t of .u; v/. Rewrite f as defined in (20) as follows:   f .s/ D  12 P1 r  J0 u.s/ ˝ J0 v.s/ C J0 v.s/ ˝ J0 u.s/

(24)

where u ˝ v denotes the matrix .ui vj /1i;j 3 and the differential operator r acts on matrices M D .mi;j /1i;j 3 the following way: r M D

3 X

@i mi;j

iD1

 1j 3

:

For u; v 2 ET and s 2 .0; T /,  f 0 .s/ D 12 P1 r  J u0 .s/ ˝ J0 v.s/ C J0 u.s/ ˝ J v 0 .s/  C J v 0 .s/ ˝ J0 u.s/ C J0 v.s/ ˝ J u0 .s/ For all s 2 .0; T /, 5

1

s 4 kJ u0 .s/ ˝ J0 v.s/k2  ksJ u0 .s/k3 ks 4 J0 v.s/k6 1

1

1

 ksAD4 u0 .s/k2 ks 4 AD2 v.s/k2  kukET kvkET ; where the first inequality comes from the fact that L3  L6 ,! L2 , the second 1

inequality comes from the Sobolev embeddings D.AD4 / ,! L3 .I R3 / and 1

D.AD2 / ,! L6 .I R3 /, and the third inequality follows directly from the definition of the space ET . Of course the same occurs for the other three terms J0 u.s/˝J v 0 .s/, 1

J v 0 .s/ ˝ J0 u.s/, and J0 v.s/ ˝ J u0 .s/. Therefore, since AD 2 maps Vd0 to HD , 5

1

sup ks 4 AD 2 f 0 .s/k2  c kukET kvkET :

(25)

0 0 independent of such that k.u; v/kLp .0; ID.A1=4 //  Cp kukLp .0; ID.A1=4 // kvkL1 .0; ID.A1=4 // : D

D

(26)

D

If v 2 L1 .0; I VD /, the following improved estimate holds k.u; v/k

1

1 Lp .0; ID.AD4 //

 Kp 4 kukLp .0; ID.A1=4 // kvkL1 .0; IVD / ;

(27)

D

where Kp > 0 is a constant independent of . Proof. First, let M be the maximal regularity operator on HD : for all ' 2 Lp .0; I HD /, M' is defined by M'.t/ WD

Z

t

AD e .ts/AD '.s/ ds;

t 2 .0; /:

0

Since HD is a Hilbert space and AD generates an analytic semigroup in HD , the operator M is bounded on Lp .0; I HD / for all p 2 .1; 1/ and all > 0; see, e.g., [13]. Moreover, kMkL.Lp .0; IHD // is independent of . Then 1  3  AD4 .u; v/ D M AD 4 f

1

1

where f is defined by (24). For u 2 Lp .0; I D.AD4 / and v 2 L1 .0; I D.AD4 /, by Sobolev embeddings, J u ˝ J v C J v ˝ J u 2 Lp .0; I L3=2 .I R3 //, with the estimate kJ u ˝ J v C J v ˝ J ukLp .0; IL3=2 .IR3 //  C kukLp .0; ID.A1=4 // kvkL1 .0; ID.A1=4 // ; D

D

1

where the constant C depends only on the constant of the embedding D.AD4 / ,! 3   L3 .I R3 /. This implies that f 2 Lp 0; I P1 .W 1;3=2 / . Since D.AD4 / ,! 3     0 W01;3 .I R3 / (see Proposition 2), the embedding P1 W 1;3=2 .I R3 / ,! D.AD4 / 3

holds and therefore AD 4 f 2 Lp .0; I HD / with 3

kAD 4 f kLp .0; IHD /  C kukLp .0; ID.A1=4 // kvkL1 .0; ID.A1=4 // : D

Using the Lp maximal regularity result in HD gives (26).

D

222

S. Monniaux and Z. Shen 1

To prove (27), let u 2 Lp .0; I D.AD4 // and v 2 L1 .0; I VD /. Using the 1

embeddings D.AD4 / ,! L3 .I R3 / and VD ,! L6 .I R3 /, kJ u ˝ J v C J v ˝ J ukLp .0; IL2 .;R3 //  C kukLp .0; ID.A1=4 // kvkL1 .0; IVD / : D

As before, this implies that f 2 Lp .0; I VD0 /, and therefore Z

1 4

AD .u; v/.t / D

t 0

3  1  AD4 e .ts/AD AD 2 f .s/ ds;

t 2 .0; /:

Using the analyticity of the semigroup .e tAD /t0 in HD and Young’s inequality, 1

3

kAD4 .u; v/kLp .0; IHD /  C kt 7! t  4 kL1 .0; / kukLp .0; ID.A1=4 // kvkL1 .0; IVD / : D

t u Proof of Theorem 4. The proof is inspired by the method described in [39] (see also [2, Section 8]). Let p 2 .1; 1/, " > 0 to be chosen later and w WD u  v 2 1

1

Cb .0; T I D.AD4 //  Lp .0; T I D.AD4 //: w satisfies w D .u; w/ C .w; v/ D .w; u C v  2˛/ C 2.w; ˛/ D .w; u C v  2˛/ C 2.w; ˛  ˛" / C 2.w; ˛" / where ˛" .t / D e tAD u0;" , with u0;" 2 VD satisfying the estimate ku0;"  u0 kD.A1=4 / D

1

 ". Using Lemma 2, w is estimated in Lp .0; I D.A 4 // as follows: kwkLp .0; ID.A1=4 //

  1  kwkLp .0; ID.A1=4 // Cp .ku C v  2˛kL1 .0; ID.A1=4 // C "/ C Kp 4 ku0;" kVD D    p " C g" . / kwkLp .0; ID.A1=4 // ; 1

where g" . / D ku C v  2˛kL1 .0; ID.A1=4 // C 4 ku0;" kVD ! 0. This shows that D

t!0

choosing " > 0 small enough, there exists > 0 such that kwkLp .0; ID.A1=4 //  1 kwkLp .0; ID.A1=4 // ; in other terms, w D 0 on Œ0; / (recall that w is continuous 2 on Œ0; T /). If D T , then it was proved that u D v on Œ0; T /. If < T , by continuity, w. / D 0 also holds. The previous reasoning can be iterated on intervals of the form Œk ; .k C 1/ / to prove ultimately that w D 0 on Œ0; T / (remark again that all constants Cp ; Kp ; p appearing in the estimates above are independent of ). t u

4 Stokes Problems in Irregular Domains With Various Boundary Conditions

3

223

Neumann Boundary Conditions

˚  In this section, the system (NS); (Nbc) is studied. The results proved in [37] will be only surveyed, the method to prove existence of solutions being similar to what has been done in Sect. 2.

3.1

The Linear Neumann-Stokes Operator

Before defining the Neumann-Stokes operator, the following integration by parts formula will be useful. Lemma 3. Let  2 R, u; w W  ! R3 , ;  W  ! R be sufficiently nice functions defined on the Lipschitz domain   R3 . Let L u D u C r.div u/ and define the conormal derivative:   @ .u; / D ru C .ru/>   

on @:

(28)

Then the following integration by parts formula holds: Z

Z





Z

I .ru; rw/   div w dx C

.L u  r/  w dx D  



@

@ .u; /  w d (29)

Z

Z

D

.L w  r/  u dx C 



  div w   div u dx



Z



(30)

.i;j i;j C i;j j;i /;

for  D .i;j /1i;j 3 and  D .i;j /1i;j 3 :

C @

 @ .u; /  w  @ .w; /  u d ;

where I .; / D

3 X

i;j D1

Recall that ru D .@i uj /1i;j 3 . The space L2 .I R3 / admits the following Hodge decomposition, dual to the one ˚  ? shown in Sect. 2: HN ˚ G0 , where G0 WD rI  2 H01 .I R/ and ˚  HN WD u 2 L2 .I R3 /I div u D 0 :

(31)

Following the steps of the previous section, define VN D H 1 .I R3 / \ HN and JN W HN ,! L2 .I R3 / the canonical embedding, PN D JN0 W L2 .I R3 / ! HN

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S. Monniaux and Z. Shen

the orthogonal projection, and JQ N W VN ,! H 1 .I R3 / the restriction of JN on VN and JQ N0 D PQ N W .H 1 .I R3 //0 ! VN0 , extension of PN to .H 1 .I R3 //0 . The Neumann-Stokes operator is defined as follows. Definition 2. Let  2 R. The Neumann-Stokes operator A is defined as being the associated operator of the bilinear form: a W VN  VN ! R;

Z

I .r JQ N u; r JQ N v/ dx

a .u; v/ D 

In the case where  2 .1; 1, the bilinear form a is continuous, symmetric, coercive, and sectorial. So its associated operator is self-adjoint, invertible and the negative generator of an analytic semigroup of contractions on HN . The following proposition is a consequence of the integration by parts formula (29), [37, Theorem 6.8] and [29, Théorème 5.3]. Proposition 4. Let  2 .1; 1. The Neumann-Stokes operator A is the part in HN of the bounded operator A0; W VN ! VN0 defined by .A0; u/.v/ D a .u; v/. The operator A is self-adjoint, invertible, A generates an analytic semigroup 1

of contractions on HN , D.A2 / D VN and for all u 2 D.A /, there exists  2 L2 .I R/ such that JN A u D JQ N u C r

(32)

and D.A / admits the following description: ˚  D.A /D u 2 VN I 9  2 L2 .I R/ W f DJQ N uCr 2 HN and @ .u; /f D 0 ; where @ .u; /f is defined in a weak sense for all f 2 .H 1 .I R3 //0 by h@ .u; /f

Z ; i@ D

.H 1 /0 hf; ‰iH 1

C

I .r JQ n u; r‰/ dx 

L2 h; div ‰iL2



for ‰ 2 H 1 ./ and

D Tr@ ‰.

Remark 3. If f 2 .H 1 .I R3 //0 , the quantity @ .u; /f exists on @ in the Besov 1 space B2;21 .@I R3 / D H  2 .@; R3 / according to [37, Proposition 3.6]. 2

Thanks to [37, Sections 9 & 10], a good description of the domain of fractional powers of the Neumann-Stokes operator A can be given. In particular, in [37, Corollary 10.6], it was established that 3

D.A4 / is continuously embedded into W 1;3 .I R3 /:

(33)

4 Stokes Problems in Irregular Domains With Various Boundary Conditions

3.2

225

The Nonlinear Neumann-Navier-Stokes Equations

The results in Sect. 3.1 allow to prove a result similar ˚to Theorem 3 for the Navier-Stokes system with Neumann boundary conditions (NS); (Nbc) . As in the 1

previous section, it is not difficult to see that D.A4 / ,! L3 .I R3 / is a critical space for the system. For T 2 .0; 1, following the definition of ET in Sect. 2, define n 1 3 1 FT D u 2 Cb .Œ0; T I D.A4 //I u.t / 2 D.A4 /; u0 .t / 2 D.A4 / for all t 2 .0; T  3

1

1

o

and sup kt 2 A4 u.t /k2 C sup ktA4 u0 .t /k2 < 1 t2.0;T /

t2.0;T /

endowed with the norm 1

1

3

1

kukFT D sup kA4 u.t /k2 C sup kt 2 A4 u.t /k2 C sup ktA4 u0 .t /k2 : t2.0;T /

t2.0;T /

t2.0;T /

The same tools as in 2.2 apply, so the following result can be proved (see [37, Theorem 11.3]). 1

Theorem 5. Let   R3 be a bounded Lipschitz domain and let u0 2 D.A4 /. Let ˇ and be defined by ˇ.t / D e tA u0 ;

t  0;

and for u; v 2 FT and t 2 .0; T /, Z

t

.u; v/.t / D 0

  e .ts/A . 12 PN / .JN u.s/  r/JQ N v.s/ C JN v.s/  r/JQ N u.s/ ds W

1

(i) If kA4 u0 k2 is small enough, then there exists a unique u 2 F1 solution of u D ˇ C .u; u/. 1

(ii) For all u0 2 D.A4 /, there exists T > 0 and a unique u 2 FT solution of u D ˇ C .u; u/. A comment here may be necessary to link ˚  the solution u obtained in Theorem 5 and a solution of the system (NS); (Nbc) . If u 2 FT , then u0 2 HN and .JN u  r/JQ N u 2 L2 .I Rn /. Moreover, if u satisfies the equation u D ˇ C .u; u/, then u is a mild solution of   A u D u0  PN .JN u  r/JQ N u 2 HN :

226

S. Monniaux and Z. Shen

Going further,   JN PN .JN u  r/JQ N u D .JN u  r/JQ N u  rq where q 2 H01 .I R/ satisfies q D div .JN u  r/JQ N u/ 2 H 1 .I Rn /: Therefore, by definition of A , there exists  2 L2 .; R/ such that JQ n u C r D JN .A u/ D JN u0  .JN u  r/JQ N u C rq and at the boundary, .u; / satisfies (Nbc) in the weak sense as in Proposition 4. Since q 2 H01 .I R/, .u;˚  q/ satisfies  also (Nbc). This proves that .u;   q/ is a solution of the system (NS); (Nbc) . 1 The uniqueness is true in a larger space than FT : for each u0 2 D.A 4 /, there is ˚  1 at most one u 2 Cb .Œ0; T /I D.A 4 //, mild solution of the system (NS); (Nbc) . For a more precise statement, see [37, Theorem 11.8].

4

Hodge Boundary Conditions

Most of the results presented here are proved thoroughly in [36] for the linear theory and [35] for the nonlinear system. The linear Hodge-Laplacian on Lp -spaces is first studied and then the Hodge-Stokes operator before applying the properties of this operator to prove the existence of mild solutions of the Hodge-Navier-Stokes system in L3 . Some recent developments/improvements can be found in [30].

4.1

The Hodge-Laplacian and the Hodge-Stokes Operators

We denote by H the space L2 .I R3 /. Let

and

˚  WT WD u 2 H I curl u 2 H; div u 2 L2 .I R/ and   u D 0 on @ ; ˚  WN WD u 2 H I curl u 2 H; div u 2 L2 .I R/ and   u D 0 on @ ;

(subscript T is for “tangential” and N for “normal”) both endowed with the scalar product hhu; viiW WD hcurl u; curl vi C hdiv u; div vi C hu; vi ; where h; iE denotes the L2 .E/-pairing.

4 Stokes Problems in Irregular Domains With Various Boundary Conditions

227

Remark 4. As in Remark 1 for a bounded Lipschitz domain  and a vector field w 2 H satisfying curl w 2 H , define   w on @ in the following weak sense in 1 H  2 .@I R3 /: for  2 H 1 .I R3 /, hcurl w; i  hw; curli D h  w; 'i@

(34)

where ' D Trj@  , the right-hand side of (34) depends only on ' on @ and not on the choice of , its extension to . Remark 5. In the case of smooth bounded domains, i.e., with a C1;1 boundary or convex, the spaces WT and WN are contained in H 1 .I R3 / (see, e.g., [3, Theorems 2.9, 2.12, and 2.17]). This is not the case if  is only Lipschitz. The Sobolev embedding associated to 1 the spaces WT;N is as follows: WT;N ,! H 2 .I R3 / with the estimate   kukH 1=2  C kuk2 C kcurl uk2 C kdiv uk2 ;

u 2 WT;N I

(35)

see, for instance, [9] or [32, Theorem 11.2] where it was proved moreover that if u 2 WT;N ; then u has an L2 trace at the boundary @ W uj@ D .  u/ C .  u/   2 L2 .@I R3 /;   and kuj@ kL2 .@IR3 /  C kuk2 C kcurl uk2 C kdiv uk2 :

(36) (37)

Remark 6. If  is of class C1 , the previous result applies also if u 2 Lp .I R3 / with curl u 2 Lp .I R3 /, div u 2 Lp .I R/, and   u D 0 on @ (or   u D 0 on @) if p 2 .1; 1/ (see [32, Theorem 11.2], where it was proved that if  is only Lipschitz, it is also true for p in a range around 2). Remark 7. The Helmholtz projection P˚ W L2 .I R3 / ! HD defined in Sect. 2 (after Remark 2) maps also WT to the space u 2 WT I div u D 0 DW VT . 3 in Sect. 3 (before Definition 2) The projection PN W L2 .I ˚ R / ! HN defined  maps also WN to the space u 2 WN I div u D 0 DW VN . On WT  WT , we define the following form: bT W WT  WT ! R;

bT .u; v/ D hcurl u; curl vi C hdiv u; div vi;

where h; i denotes either the scalar or the vector-valued L2 -pairing. Similarly, we define bN W WN  WN ! R;

bN .u; v/ D hcurl u; curl vi C hdiv u; div vi:

228

S. Monniaux and Z. Shen

Proposition 5. The Hodge-Laplacian operators BT and BN , defined as the associated operators in H of the forms bT and bN , satisfy ˇ n o ˇ   curl u D 0 on @ D.BT;N / D u 2 WT;N I rdiv u 2 H; curl curl u 2 H and ˇˇ .div u/ BT;N u D  u;

u 2 D.BT;N /:

(38)

Proof. Let u 2 WT;N and v 2 H01 .I R3 /  WT;N . Then bT;N .u; v/ D H 1 hrdiv u C curl curl u; viH 1 D H 1 hu; viH 1 0

0

so that BT;N u D u in H 1 .I R3 /. The proof of Proposition 5 is described now in the case of bT defined on WT WT . The case of bN defined on WN  WN can be proved with the same arguments (using PN instead of P in what follows). Let D be the space  ˚ D WD u 2 WT I rdiv u 2 H; curl curl u 2 H and   curl u D 0 on @ : If u 2 D, then BT u D u 2 H and therefore u 2 D.BT /. Conversely, assume that u 2 D.BT /. Then .Id  P/BT u 2 H satisfies for all v 2 WT h.Id  P/BT u; vi D hBT u; .Id  P/vi D bT .u; v/  bT .u; Pv/ D hdiv u; div vi D WT0 hrdiv u; viWT ; so that rdiv u D .Id  P/BT u 2 H . Then curl curl u D BT u C rdiv u 2 H . It remains to prove that   curl u D 0 on @. Remark that it makes sense to consider the tangential part of w WD curl u on the boundary @ since it was just proved 1 that curl w 2 H , and, therefore, thanks to (34),   w 2 H  2 .@I R3 /. For all 1 ' 2 H 2 .@I R3 / \ L2tan .@I R3 /, there exists  2 H 1 .I R3 / such that j@ D '. In that case,  2 WT , and therefore hrdiv u C curl curl u; i D hBT u; i D bT .u; / D hdiv u; div i C hcurl u; curl i D hrdiv u C curl curl u; i  H 1=2 .@/ h  curl u; 'iH 1=2 .@/ : 1

It proves that H 1=2 .@/ h  curl u; 'iH 1=2 .@/ D 0 for all ' 2 H 2 .@I R3 / \ L2tan .@I R3 /, and then   curl u D 0 on @. t u Since the forms bT;N are continuous, bilinear, symmetric, coercive, and sectorial, the operators BT;N generate analytic semigroups of contractions on H ; BT;N is

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229

1=2

self-adjoint and D.BT;N / D WT;N . The following property will be useful in the next section; it links BT and BN , as shown in [43, Proposition 2.2]. Lemma 4. For u 2 H such that curl u 2 H , the following commutator property occurs for all " > 0: curl .1 C "BT /1 u D .1 C "BN /1 curl u:

(39)

Proof. Let u 2 H such that curl u 2 H . Let u" D .1 C "BT /1 u and w" D .1 C "BN /1 curl u. Step 1: curl u" 2 D.BN /. By (38), it holds curl u" 2 H , curl curl u" 2 H , div .curl u" / D 0 2 H 1 ./,   curl u" D 0 on @, and div .curl u" / D 0 on @. To prove that curl u" 2 D.BT /, it remains to show, thanks to (38), that curl curl .curl u" / 2 H . This is due to the fact that curl curl .curl u" / D curl .u" /

in H 1 .; R3 /:

Since u" D BT .1 C "BT /1 u D

 1 u  u" "

and curl u" ; curl u 2 H , the claim follows. Step 2: curl u" D w" . By Step 1, curl u" 2 D.BN /. Moreover, in the sense of distributions,   .1 C "BN /.curl u" / D curl u"  "curl u" D curl u"  "u" D curl u since u"  "u" D .1 C "BT /.1 C "BT /1 u D u. Therefore, curl u" D .1 C "BN /1 curl u D w" which proves the claim.

t u

To prove that the operators BT;N extend to Lp -spaces, it suffices to prove that their resolvents admit L2  L2 off-diagonal estimates. This was proved in, e.g., [36, Section 6] (see also [30]). Proposition 6. There exist two constants C; c > 0 such that for any open sets E; F  R3 such that dist .E; F / > 0 and for all t > 0, f 2 H and u D .Id C t 2 BT;N /1 .1lF f /;

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it holds k1lE uk2 C tk1lE div uk2 C t k1lE curl uk2  C e c

dist .E;F / t

k1lF f k2 :

(40)

Proof. Start by choosing a smooth cutoff function  W R3 ! R satisfying  D 1 on k E,  D 0 on F , and krk1  dist .E;F . Then define  D e ı where ı > 0 is to be / chosen later. Next, take the scalar product of the equation: u  t 2 u D 1lF f;

u 2 D.BT;N /

with the function v D 2 u. Since  D 1 on F and kuk2  k1lF f k2 , it is easy to check then that k uk22 C t 2 k div uk22 C t 2 k curl uk22    k1lF f k22 C 2˛krk1 t 2 k uk2 k div uk2 C k curl uk2 and therefore, using the estimate on krk1 and choosing ı D

dist .E;F / , 4kt

k uk22 C t 2 k div uk22 C t 2 k curl uk22  2k1lF f k22 : Using now the fact that  D e ı on E, p dist .E;F / k1lE uk2 C tk1lE div uk2 C t k1lE curl uk2  2e  4k t k1lF f k2 ; which gives (40) with C D

p 2 and c D

1 . 4k

t u

With a slight modification of the proof, it can be shown that for all 2 .0; /, 3 there exist two constants C; c > 0 such that ˚ for any open sets E; F  R such that dist .E; F / > 0, and for all z 2 † D ! 2 C n f0gI j arg zj <   , f 2 H and u D .zId C BT;N /1 .1lF f /; it holds 1

1

1

jzjk1lE uk2 C jzj 2 k1lE div uk2 C jzj 2 k1lE curl uk2  C e c dist.E;F /jzj 2 k1lF f k2 : (41) Following [31] and [10] (see also [30]), there exist Bogovski˘ı-type operators Ri , Ti , i D 1; 2; 3, and K1;2 ; L1;2 such that for all p 2 .1; 1/, R1 W Lp .I R3 / ! W 1;p .I R/; R2 W Lp .I R3 / ! W 1;p .I R3 /;

1;p

T1 W Lp .I R3 / ! W0 .I R/; 1;p

T2 W Lp .I R3 / ! W0 .I R3 /;

4 Stokes Problems in Irregular Domains With Various Boundary Conditions

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1;p

R3 W Lp .I R/ ! W 1;p .I R3 /;

T3 W Lp .I R/ ! W0 .I R3 /;

K1;2 W Lp .I R3 / ! W 1;p .I R3 /;

1;p

and L1;2 W Lp .I R3 / ! W0 .I R3 /

satisfying R2 curl u C rR1 u D u  K1 u

8 u 2 Lp .I R3 / with curl u 2 Lp .I R/ and curl K1 u D 0 if curl u D 0;

R3 div u C curl R2 u D u  K2 u;

8 u 2 Lp .I R3 / with div u 2 Lp .I R/ and div K2 u D 0 if div u D 0;

T2 curl u C rT1 u D u  L1 u;

(42)

(43)

8 u 2 Lp .I R3 / with curl u 2 Lp .I R/;

  u D 0 on @ and curl L1 u D 0 if curl u D 0; (44) T3 div u C curl T2 u D u  L2 u;

8 u 2 Lp .I R3 / with div u 2 Lp .I R/;

  u D 0 on @ and div L2 u D 0 if div u D 0:

(45)

With these potential operators (at this point, only the relations (43) and (45) are needed) and (41), it is easy to prove that (see, e.g., [30])  ; 2 uniformly in z 2 †

(46) ˚  p where HD WD u 2 Lp .I R3 / s.t. div u D 0 and   u D 0 on @ and Gp WD rW 1;p .I R/ are defined for p 2 .1; 1/; if p D 2, then HD2 D HD and G2 D G defined in Sect. 2. With the same reasoning, one can prove that p

z.zId C BT /1 is bounded in HD and in Gp for p 2

6 5

 ; 2 uniformly in z 2 †

(47) ˚  p 1;p where HN WD u 2 Lp .I R3 / s.t. div u D 0 and Gp;0 WD rW0 .I R/ are defined for p 2 .1; 1/; if p D 2, then HN2 D HN and G2;0 D G0 defined in Sect. 3. p

z.zId C BN /1 is bounded in HN and in Gp;0 for p 2

6 5

˚  1 Proposition 7. The resolvents z.zIdCB † are T;N / ; z 2 ˚    uniformly bounded in Lp .I R3 / for all p 2 q00 ; q0 , where q0 WD min 6; 3 C " (" > 0 depends on @). Proof. By [19, Theorems 11.1 and 11.2], the projections defined in Sect. 2 and Sect. 3   P and PN extend to bounded projections on Lp .I R3 / for p 2 .3 C "/0 ; 3 C " ; (48)

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where " > 0 depends on @ (and .3 C "/0 D 3C" < 32 ); if  is of class C1 , then 2C" p p " D 1. This means in particular that HD coincides with thespace L ./ defined  0 0 ˚in (6) for all p12 .3 C "/ ; 3 C " . Therefore for all pp 2 q0 ;3 2 , the resolvents z.zId C BT;N  / ; z 2 † are uniformly bounded in L .I R /. The same result for all p 2 2; q0 is obtained by duality. t u tBT;N /t0 extend to bounded analytic semigroups Corollary 1. The semigroups  0  .e p 3 on L .I R / for p 2 q0 ; q0 and satisfy

p p t div .e tBT;N f /  Cp kf kp t curl .e tBT;N f /  C 0 kf kp p p p t rdiv .e tBT;N f /  Kp kf kp t curl curl .e tBT;N f /  K 0 kf kp p p p

(49) (50)

for all f 2 Lp .I R3 /. Proof. The (49) and (50) in the corollary above come from the fact that  estimates  p for p 2 q00 ; q0 , the negative generators BT;N of the semigroups .e tBT;N /t0 satisfy ˚ p D.BT;N / D u 2 Lp .I R3 /I div u 2 W 1;p .I R3 /; curl u 2 Lp .I R3 /;  curl curl u 2 Lp .I R3 /;   u D 0 and   curl u D 0 on @ p

BT;N u D  u;

(51)

p

u 2 D.BT;N /:

This can be proved the same way we proved Proposition 5, (case p D 2) using the fact that P and PN are bounded in Lp .I R/. t u Remark 8. Let w 2 L2 .I R3 / such that curl w 2 L2 .I R3 / and   w D 0 on @. 1 Then   curl w D 0 in H  2 .@/. If the operator BT is restriced on HD and the operator BN on HN , the following Hodge-Stokes operators AT and AN defined by n o D.AT / D u 2 HD \ WT I curl curl u 2 L2 .I R3 / and   curl u D 0 on @ AT u D curl curl u

for u 2 D.AT /

and n o D.AN / D u 2 HN \ WN I curl curl u 2 L2 .I R3 / ; AN u D curl curl u for u 2 D.AN /

4 Stokes Problems in Irregular Domains With Various Boundary Conditions

233

are obtained. The construction of the Hodge-Stokes operators is strongly related to the particular Hodge boundary conditions. As seen in Sect. 2 and 3, this doesn’t hold in general. The reason behind this is that the Helmholtz projection P commutes with BT and PN commutes with BN . Remark 8 ensures that if u 2 D.AT / as defined above,   curl curl u D 0 on @, so that curl curl u 2 HD . The properties (46) and (47), together with a duality argument and the fact   that the projections P and PN are bounded on Lp .I R3 / for p 2 .3 C "/0 ; 3 C " , prove p that .e tAT /t0 extends to an analytic semigroup on HD (its generator is denoted p tAN by AT;p ) and .e /t0 extends  analytic semigroup on HN (its generator  6 to an is denoted by AN;p ) for all p 2 5 ; q0 . Moreover, the estimates (49) and (50) are   valid if BT;N is replaced by AT;N for all p 2 65 ; q0 .   p Lemma 5. If u 2 HD3 and curl u 2 L3 .I R3 /, then u 2 HD for all p 2 3; q0 . Proof. Thanks to the relation (42),   u D Pu D P R2 curl u C K1 u since PrR1 u D 0. The mapping properties of R2 and K1 show that R2 curl u C K1 u 2 L3 .; R3 / \ L6 .; R3 /, which proves the claim of the lemma. This has been done in, e.g., [35, Sections 3 and 4]. t u Remark 9. One can actually prove that the operator AT;p generates an analytic p p semigroup in HD for all p 2 .1; 3 C "/. The same holds for AN;p on HN . See [30] for more details. Remark 10. In [54], M.E. Taylor conjectured that the Dirichlet-Stokes operator  p generates an analytic semigroup in HD for p 2 .3 C "/0 ; 3 C " , which was proved in [51]. The question of optimality of this range is still open; the counterexample provided by P. Deuring in [14] is for p > 6. We see here that, for the Hodge-Stokes operator, one can allow all p 2 .1; 3 C "/.

4.2

The Nonlinear Hodge-Navier-Stokes Equations

  The nonlinear Hodge-Navier-Stokes system (NS’); (Hbc) 8 @t u  u C r  u  curl u D 0 in .0; T /  ; ˆ ˆ ˆ ˆ < div u D 0 in .0; T /  ; ˆ   u D 0;   curl u D 0 on .0; T /  @; ˆ ˆ ˆ : u.0/ D u0 in ;

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is considered for initial data u0 in the critical space HD3 in the abstract form:   u0 .t / C AT;p u.t /  P u.t /  curl u.t / D 0;

u0 2 HD3 :

(52)

The idea to solve (52) is to apply the same method as in Sect. 2 and 3. With the properties of the Hodge-Stokes semigroup listed in the previous subsection (and more particularly Lemma 5), the following existence result for (52) is almost immediate. For T 2 .0; 1, define the space GT by n 3.1Cı/ /I curl u 2 C..0; T /I L3 .; R3 // GT D u 2 Cb .Œ0; T /I HD3 / \ C..0; T /I HD o   p ı with sup ks 2.1Cı/ u.s/k3.1Cı/ C k s curl u.s/k3 < 1 0 0 such that (55) holds follows from the closed graph theorem since fu 2 H I curl u 2 H g is complete for the norm kuk2 C kcurl uk2 . 2. Assume now that g 2 L2tan .@I R3 /. Let w 2 H such that curl w 2 H and (54) holds. Since   g 2 L2 .@I R3 /, we can approach it in L2 .@I R3 / by a sequence .'n /n2N of vector fields 'n 2 H 1=2 .@; R3 /. In particular, 'n   ! .  g/   D g

in L2 .@I R3 / as n ! 1:

By assertion 1, for each n 2 N, there exists wn 2 H such that curl wn 2 H satisfying h'n  ; i@ D hcurl wn ; i  hwn ; curl i

for all  2 WT :

Thanks to the estimate (55), it is immediate that wn ! w n!1

and

curl wn ! curl w n!1

in H:

Let now  2 H 1 .I R3 /. For " > 0, let " D .1 C "BT /1 . Then " 2 WT , and thanks to Lemma 4, " !  "!0

and

curl " D .1 C "BN /1 curl  ! curl  "!0

in H:

This implies also that   " !    "!0

in H 1=2 .@I R3 /:

Therefore, for all " > 0 and n 2 N h  " ; 'n i@ D h'n  ; " i@ D hcurl wn ; " i  hwn ; curl " i : First take the limit as " goes to 0 and obtain (recall that 'n 2 H 1=2 .@I R3 /) H 1=2 h

 ; 'n iH 1=2 D hcurl wn ; i  hwn ; curl i :

Since  2 H 1 .; R3 /, the first term of the latter equation is also equal to h'n  ; i@ . Taking the limit as n goes to 1 yields hg; i@ D hcurl w; i  hw; curl i which proves the claim made in 2.

t u

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Remark 12. If  is of class C1 , one can prove that Lemma 6 is also valid in Lp 0 instead of L2 for all p 2 .1; 1/, identifying the dual of Lp with Lp (noting that q0 defined in Proposition 7 is equal to 1). Proof of (53). For the time being, denote by D˛ the set on the right-hand side of (53). Let u 2 D˛ : u D curl curl u C rdiv u 2 H , and for all v 2 WT \ H 1 .I R3 /, hu; vi D hcurl curl u; vi  hrdiv u; vi D hcurl u; curl vi C h  curl u; vi@ C hdiv u; div vi D hcurl u; curl vi C hdiv u; div vi C ˛hu; vi@ D b˛ .u; v/: The second equality comes from the integration by parts formula. In the third equality, the characterization of elements in D˛ has been used. Thanks to the density of WT \ H 1 .I R3 / in WT , this proves the inclusion D˛ D.B˛ / and that B˛ u D u for u 2 D˛ . Conversely, let u 2 D.B˛ /. Let  D B˛ u 2 H , g D ˛ u. Since uj@ 2 L2tan .@I R3 /, Lemma 6 shows the existence of w 2 H with curl w 2 H such that ˛ u D   w on @. Therefore, the boundary value g D ˛ u satisfies the conditions of [33, Theorem 1.2] with p D 2. Then there exists a unique uQ satisfying 8 < uQ 2 WT ; curl curl uQ 2 H; div uQ 2 H 1 ./; Qu D  2 H; :   curl uQ D g 2 H 1=2 .@I R3 /;

(58)

For all v 2 WT , integrating by parts, hcurl uQ ; curl vi C hdiv uQ ; div vi D hQu; vi  h  curl uQ ; vi@ D h; vi  hg; vi@ D hB˛ u; vi  h˛ u; vi@ D b˛ .u; v/  ˛hu; vi@ D hcurl u; curl vi C hdiv u; div vi : The second equality comes from the fact that uQ is the solution of (58). The third equality is a simple reformulation of the previous line using the notations introduced before. The fourth equality uses the fact that B˛ is the operator associated with the form b˛ . Finally, the last equality comes directly from the definition of b˛ . Therefore, we proved that v D u  uQ 2 WT and satisfies curl v D 0 and div v D 0. Since  is simply connected, this proves that v D 0, or equivalently u D uQ , and then that u 2 D˛ from which follows the inclusion D.B˛ / D˛ . Ultimately, it has been proved that D.B˛ / D D˛ . t u

4 Stokes Problems in Irregular Domains With Various Boundary Conditions

239

As in the case of Proposition 6, Gaffney-type estimates hold. Proposition 8. There exist two constants C; c > 0 such that for any open sets E; F  R3 such that dist .E; F / > 0 and for all t > 0, f 2 H and u D .Id C t 2 B˛ /1 .1lF f /; it holds p dist .E;F / k1lE uk2 Ctk1lE div uk2 Ct k1lE curl uk2 Ct ˛ k1lE ukL2 .@IR3 /  C e c t k1lFf k2 : (59) Proof. The proof goes as in the case ˛ D 0 (Proposition 6 for BT ). Choose a smooth cutoff function  W R3 ! R satisfying  D 1 on E,  D 0 on F , and krk1  k . Then define  D e ı where ı > 0 is to be chosen later. Next, take the dist .E;F / scalar product of the equation: u  t 2 u D 1lF f;

u 2 D.B˛ /

with the function v D 2 u. Since  D 1 on F and kuk2  k1lF f k2 , it is easy to check then that k uk22 C t 2 k div uk22 C t 2 k curl uk22 C t 2 ˛k uk2L2 .@IR3 /    k1lF f k22 C 2˛krk1 t 2 k uk2 k div uk2 C k curl uk2 and therefore, using the estimate on krk1 and choosing ı D

dist .E;F / , 4kt

k uk22 C t 2 k div uk22 C t 2 k curl uk22 C t 2 ˛k uk2L2 .@IR3 /  2k1lF f k22 : Using now the fact that  D e ı on E, p p dist .E;F / k1lE uk2 Ctk1lE div uk2 Ct k1lE curl uk2 Ct ˛ k1lE ukL2 .@IR3 /  2e  4k t k1lFf k2 ; which gives (59) with C D

p 2 and c D

1 . 4k

t u

As before, with a slight modification of the proof, it can be shown that for all 3

2 .0; / there exist two constants C; c > 0 such that ˚ for any open sets E; F  R such that dist .E; F / > 0 and for all z 2 † D ! 2 C n f0gI j arg zj <   , f 2 H and u D .zId C B˛ /1 .1lF f /;

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it holds 1

1

jzjk1lE uk2 C jzj 2 k1lE div uk2 C jzj 2 k1lE curl uk2 1

C jzj 2

p

1

˛ k1lE ukL2 .@IR3 /  C e c dist.E;F /jzj 2 k1lF f k2 :

(60)

With the same arguments as for the Hodge-Laplacian, the analogue of 7   Proposition and Corollary 1 can be obtained, as well as (51) for B˛ : for all p 2 q00 ; q0 :  ˚ z.zId C B˛ /1 ; z 2 † is uniformly bounded in Lp .I R3 /I

(61)

.e tB˛ /t0 extends to a bounded analytic semigroup on Lp .I R3 /I

(62)

p t div .e tB˛ f /  Cp kf kp ; p

p t curl .e tB˛ f /  C 0 kf kp I p p

(63)

t rdiv .e tB˛ f /  Kp kf kp ; p

t curl curl .e tB˛ f /  K 0 kf kp : p p

(64)

Moreover, if  is of class C1 , the following description of B˛;p , the negative generator of .e tB˛ /t0 in Lp .I R3 / holds: ˚ D.B˛;p / D u 2 Lp .I R3 /I div u 2 W 1;p .I R3 /; curl u 2 Lp .I R3 /;  curl curl u 2 Lp .I R3 /;   u D 0 and   curl u D ˛ u on @ B˛;p u D u;

(65)

u 2 D.B˛;p /;

To prove that, the result in Remark 6 has been used, as well as the solvability of (58)  in Lp for p in the interval .3 C "/0 ; 3 C " D .1; 1/ in that case ([33, Theorem 1.2] is also valid in this range of p).

5.2

The Robin-Hodge-Stokes Operator

From now on, assume that  is of class C1 . Let p 2 .1; 1/. Let g 2 Lp .I R3 /, with div g D 0. By Remark 1 (also valid for p 2 .1; 1/ with the obvious 1=p changes), it holds   g 2 Bp;p .@/ and also   g satisfies the condition h  g; 1liB 1=p .@/ D 0. By [19, Corollary 9.3], the problem 1=p B .@/ p;p

p 0 ;p 0

q 2 W 1;p ./;

q D 0 in ;

@ q D   g on @

(66)

has a unique (modulo constants) solution satisfying moreover krqkp . k  gkB 1=p .@/ : p;p

(67)

4 Stokes Problems in Irregular Domains With Various Boundary Conditions

241

Consider the operator p W D.B˛;p / ! W 1;p ./;

u 7! q

where q is the solution of (66) with g D curl curl u. Lemma 7. For p 2 .1; 1/, u 2 D.B˛;p /, the following estimate holds   krp ukp . ˛ kcurl ukp C kdiv ukp :

(68) 0

1=p

Proof. Let p 2 .1; 1/ and u 2 D.B˛;p /. Let ' 2 Bp0 ;p0 .@/. Let ˆ 2 W 1;p ./, so that ˆj@ D ' (recall that p1 D 1  p10 ). Thanks to the description of D.B˛;p / given by (65) and the formula (34) (also valid in Lp ), there holds 1=p

Bp;p .@/

h  curl curl u; 'iB 1=p

p 0 ;p 0

.@/

D hcurl curl u; rˆi D h  curl u; rˆi@ D ˛ hu; rˆi@ D ˛ hcurl w; rˆi ;

where w 2 Lp .I R3 / with curl w 2 Lp .I R3 / is determined by Lemma 6, 2 (for g D u; see Remark 6). Therefore by Remark 11, k  curl curl ukB 1=p .@/  C kcurl wkp  C kukLp .@IR3 / p;p    C kukp C kcurl ukp C kdiv ukp : Since  is bounded, kukp can be estimated in terms of kcurl ukp and kdiv ukp , which gives (68). t u Next result links the operator p and B˛;p with the Robin-Hodge-Stokes resolvent problem for z 2 † : 8
0 if p  2. Proof. Let z 2 † . By Proposition 9,   .zId C A˛;p / D Id  rp .zId C B˛;p /1 .zId C B˛;p /: Lemma 7 and (63) imply that for all f 2 Lp .I R3 /,  krp .zId C B˛;p /1 f kp . ˛ kcurl .zId C B˛;p /1 f kp  C kdiv .zId C B˛;p /1 f kp  C p˛jzj kf kp : p

This proves that, for jzj large enough (jzj  4C 2 ˛ 2 ), zId C A˛;p W D.A˛;p / ! HD is invertible with  1 .zId C A˛;p /1 D .zId C B˛;p /1 Id  rp .zId C B˛;p /1 and z.zId C A˛;p /1

p

L.HD /

 2 z.zId C B˛;p /1 L.Lp .IR3 // . 1:

Moreover, the same reasoning gives p jzj curl .zId C A˛;p /1

p

L.HD ILp .IR3 //

p  2 jzj curl .zId C B˛;p /1 L.Lp .IR3 // . 1

(72)

and curl curl .zId C A˛;p /1

p

L.HD ILp .IR3 //

2 curl curl .zId C B˛;p /1 L.Lp .IR3 // .1 (73) p

To prove that zId C A˛;p W D.A˛;p / ! HD is invertible if z 2 † with jzj  4C 2 ˛ 2 , proceed by induction. The assertion is proved for p  2 (the range is obtained A˛;2 is self-adjoint in HD ). Assume first that  1 < p  2 by duality since p p 2 2; 92 , so that D.A˛;2 / ,! HD by Lemma 8. Let z 2 † with jzj  4C 2 ˛ 2 and let ! D z C 8C 2 ˛ 2 . There holds ! 2 † and j!j  8C 2 ˛ 2  jzj  4C 2 ˛ 2 . p Therefore, for f 2 HD ,! HD , .zId C A˛;2 /1 f D .!Id C A˛;p /1 f C 8C 2 ˛ 2 .!Id C A˛;p /1 .zId C A˛;2 /1 f;

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which gives .zId C A˛;2 /1 f  C˛ kf kp ; p p

and this proves that zId C A˛;p W D.A˛;p / ! HD is invertible with the norm of its inverse controlled by a constant depending on ˛. For any p  2, the previous procedure can be iterated using again Lemma 8 valid for all p  2. Estimates of the form (72) and (73) are straightforward. Eventually, the result claimed in Theorem 7 is obtained for p  2. As mentioned earlier, the case 1 < p  2 is obtained by duality. t u

5.3

The Nonlinear Robin-Hodge-Navier-Stokes Equations

  The nonlinear Robin-Hodge-Navier-Stokes system (NS’); (Rbc) 8 ˆ ˆ @t u  u C r  u  curl u D 0 in ˆ ˆ < div u D 0 in ˆ   u D 0;   curl u D ˛ u on ˆ ˆ ˆ : u.0/ D u0 in

.0; T /  ; .0; T /  ; .0; T /  @; ;

for initial data u0 is considered in the critical space HD3 in the abstract form:   u0 .t / C A˛;p u.t /  P u.t /  curl u.t / D 0;

u0 2 HD3 :

(74)

Recall that C1 domains  are considered here. The idea to solve (74) is to apply the same method as in previous sections. With the properties of the Robin-Hodge-Stokes semigroup listed in particular in Theorem 7, the following existence result for (74) is almost immediate. For T 2 .0; 1, define the space HT by n HT D u 2 Cb .Œ0; T /I HD3 /I curl u 2 C..0; T /I L3 .; R3 // o p with sup k s curl u.s/k3 < 1 0 0. Hence, k are separated from zero.

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

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Now take in (12)  D Jk2 , where  is an arbitrary vector field from H ./. l Passing again to the limit as kl ! 1 yields the integral identity Z



 v  r v   dx D 0

8 2 H ./:

(15)



Hence, v 2 H ./ is a weak solution of the Euler equation 8  v  r v C rp D 0; ˆ ˆ < div v D 0; ˆ ˆ : v D 0;

x 2 ; (16)

x 2 ; x 2 @:

The function p in (16) belongs to the space W 1;s ./, where s 2 Œ1; 2/ for n D 2 and s 2 Œ1; 3=2 for n D 3. Since v D 0 on @, it can be proved, using the equations (16), that the pressure p is equal to some constants pO j on the connected components j of the boundary @. More precisely, it was proved in [30, Lemma 4] and independently in [2, Theorem 2.2] that the following equalities p.x/ji D pO i ;

pO i 2 R; j D 0; 1; : : : ; N:

(17)

hold. Multiply the Euler system (16) by A and integrate the obtained equality over . Integrating by parts and using (17), we obtain Z 



 vr vA dx D 

Z p An dS D 

N X iD0

@

b pi

Z An dS D  i

N X

pO i Fi :

(18)

i D0

If N D 0 or Fi D 0; i D 0; 1; : : : ; N (the condition (4) is satisfied), then (18) gives Z



 v  r v  A dx D 0:

(19)



The last relation contradicts (14). Therefore, the assumption is wrong and the norms of all possible solutions w./ to the operator equation (8) are uniformly bounded with respect to  2 Œ0; 1. Thus, by the Leray-Schauder theorem, equation (7) has at least one solution. An analogous conclusion is obtained when all constants b p j are equal: pO 0 D pO 1 D : : : D pO N :

(20)

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Indeed, in virtue of (3), N X iD0

pO i Fi D b p0

N X

Fi D 0;

iD0

and from (18) again follows (19). However, in the general case, one cannot claim that all constants pO i are equal. Amick [2] exhibited a solution to problem (16), for which equalities (20) are not valid. Let  D fx 2 R2 W 1 < jxj < 2g be annulus on the plane, 2 C 1 .Œ1; 2/; 0 .1/ D 0 .2/ D 0, and 00 2 L2 ..1; 2//. A solution of the Euler problem (16) is defined by  v.x/ D

x2 jxj

0

x1 .jxj/;  jxj

0

 .jxj/ 2 H ./;

Zjxj p.x/ D

j

0

.s/j2 ds: s

(21)

1

It is easy to see that p.x/jjxjD1 D 0, and p.x/jjxjD2 D

R2 j 1

3

0

.s/j2 ds > 0. s

An Existence Theorem in the General Planar Case

In this section the problem (2) is studied in the general case. For the two-dimensional domains, the result reads as follows. Theorem 1. Assume that   R2 is a bounded domain with C 2 -smooth boundary @. If f 2 W 1;2 ./ and a 2 W 3=2;2 .@/ satisfies condition (3), then problem (2) admits at least one weak solution u. Remark 1. It is well known (see [47]) that under the hypotheses of Theorem 1, 3;2 every weak solution u of problem (2) is more regular, i.e., u 2 W 2;2 ./ \ Wloc ./. Generally speaking, the solution is as regular as the data allow; in particular, u is C 1 -smooth when f, a, and @ are C 1 -smooth. Similar result holds for the 3D axially symmetric case (see Theorem 6). Moreover, for the axially symmetric case, also the existence theorem for an exterior domain could be proved (see Theorem 7). Below (Sect. 3.4) the main ideas of the proof of Theorem 1 are shown. In order to make it easer, consider the case when @ has only two connected components of the boundary and assume that f D 0. Some needed auxiliary results are formulated in the subsections below.

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

3.1

257

Properties of Sobolev Functions and an Analog of the Morse-Sard Theorem for Functions from W 2,1 .R2 /

Recall some classical differentiability properties of Sobolev functions. Working with such functions, we always assume that the “best representatives” are chosen. If w 2 L1loc ./, then the best representative w is defined by ( 

w .x/ D

R lim Br .x/ w.z/d z; if the finite limit existsI

r!0

0

otherwise;

R R 1 w.z/d z, Br .x/ D fy W jy  xj < rg is a ball where Br .x/ w.z/d z D meas.B r .x// Br .x/ of radius r centered at x. Lemma 1 (see Proposition 1 in [12]). Let 2 W 2;1 .R2 /. Then the function is continuous, and there exists a set A such that H1 .A / D 0 and the function is differentiable (in the classical sense) at each x 2 R2 n A . Furthermore, the R classical derivative at such points x coincides with r .x/ D lim Br .x/ r .z/d z, r!0 R and lim Br .x/ jr .z/  r .x/j2 d z D 0. r!0

Hausdorff measure, i.e., Here and henceforth, denote by H1 the one-dimensional 1 1 P S diamFi W diamFi  t; F  Fi . H1 .F / D lim H1t .F /, where H1t .F / D inf t!0C

iD1 1;q

i D1

It is well known that for functions w 2 Wloc ./,   R2 , H1 -almost all points x 2  are the Lebesgue points, i.e., the above limit exists H1 -almost everywhere in . The next theorem has been proved recently by J. Bourgain, M. Korobkov, and J. Kristensen [6] (see also [7, 35] for multidimensional case). The statement (i) of this theorem is the analog for Sobolev functions of the classical Morse-Sard Theorem. Theorem 2. Let   R2 be a bounded domain with Lipschitz boundary and W 2;1 ./. Then

2

(i) H1 .f .x/ W x 2  n A & r .x/ D 0g/ D 0; (ii) for every " > 0, there exists ı > 0 such that H1 . .U // < " for any set U   with H11 .U / < ı; in particular, H1 . .A // D 0; (iii) for every " > 0, there exists an open set V  R with H1 .V / < " and a function g 2 C 1 .R2 / such that for each x 2  if .x/ … V , then x … A and .x/ D g.x/, r .x/ D rg.x/ ¤ 0; (iv) for H1 –almost all y 2 ./  R, the preimage 1 .y/ is a finite disjoint family of C 1 -curves Sj , j D 1; 2; : : : ; N .y/. Each Sj is either a cycle in  .i.e., Sj   is homeomorphic to the unit circle S1 / or a simple arc with endpoints on @ .in this case Sj is transversal to @/.

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Some Facts from Topology

Below some topological definitions and results will be needed. By continuum we mean a compact connected set. The connectedness is understood in the sense of general topology. A subset of a topological space is called an arc if it is homeomorphic to the unit interval Œ0; 1. A locally connected continuum T is called a topological tree, if it does not contain a curve homeomorphic to a circle or, equivalently, if any two different points of T can be joined by a unique arc. This definition implies that T has topological dimension 1. A point C 2 T is an endpoint of T (resp., a branching point of T ), if the set T nfC g is connected (resp., if T nfC g has more than two connected components). Let us shortly present some results from the classical paper of A.S. Kronrod [45] concerning level sets of continuous functions. Let Q D Œ0; 1  Œ0; 1 be a square in R2 , and let f be a continuous function on Q. Denote by Et a level set of the function f , i.e., Et D fx 2 Q W f .x/ D tg. A connected component K of the level set Et containing a point x0 is a maximal connected subset of Et containing x0 . By Tf denote a family of all connected components of level sets of f . It was established in [45]  that Tf equipped by a natural topology is a one-dimensional topological tree. The convergence in Tf is defined by the following rule: Tf 3 Ci ! C iff sup dist.x; C / ! 0.) Endpoints of this tree are the components C 2 Tf which do x2Ci

not separate Q, i.e., Q n C is a connected set. Branching points of the tree are the components C 2 Tf such that Q n C has more than two connected components (see [45, Theorem 5]). By results of [45, Lemma 1], see also [51] and [62], the set of all branching points of Tf is at most countable. The main property of a tree is that any two points could be joined by a unique arc. Therefore, the same is true for Tf . Lemma 2 (see Lemma 13 in [45]). If f 2 C .Q/, then for any two different points A 2 Tf and B 2 Tf , there exists a unique arc J D J .A; B/  Tf joining A to B. Moreover, for every inner point C of this arc, the points A; B lie in different connected components of the set Tf n fC g. Remark 2. The assertion of Lemma 2 remains valid for level sets of continuous functions f W  ! R, where  is a multi-connected bounded domain of type (1), provided f  j D const on each inner boundary component j with j D 1; : : : ; N . Indeed, f can be extended to the whole 0 by putting f .x/ D j for x 2 j , j D 1; : : : ; N . The extended function f will be continuous on the set 0 which is homeomorphic to the unit square Q D Œ0; 12 .

3.3

Euler Equation

Most of the results of this section are obtained under the following assumptions. (E) Let   R2 be a bounded domain of type (1) with Lipschitz boundary. Assume that v 2 W 1;2 ./ and p 2 W 1;s ./, s 2 Œ1; 2/, satisfy the

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

259

Euler equations (

 v  r v C rp D 0; div v D 0;

for almost all x 2 , and let Z v  n ds D 0; i D 0; 1; : : : ; N;

(22)

(23)

i

where i are connected components of the boundary @. If instead of (23) the solution v satisfies the homogeneous boundary conditions vji D 0; i D 0; 1; : : : ; N;

(24)

it will be said that v satisfies the condition Eı . Under the conditions (E), it is easy to see that there exists a stream function 2 W 2;2 ./ such that r D .v2 ; v1 / (note that by the Sobolev embedding jvj2 theorem, is continuous in ). Denote by ˆ D p C the total head pressure 2 1;s corresponding to the solution .v; p/. Obviously, ˆ 2 W ./ for all s 2 Œ1; 2/. By direct calculations one easily gets the identity    @v2 @v1  rˆ   v2 ; v1 D !r in ; (25) @x1 @x2 where ! denotes the corresponding vorticity: ! D @2 v 1  @1 v 2 D  . Since the stream lines in our case coincide with the level sets of , from (25), in the case of smooth functions ; ˆ, the classical Bernoulli law follows immediately: The total head pressure ˆ is constant along any stream line. But the Sobolev case is more delicate: now the stream function 2 W 2;1 ./ 1 is not C -smooth, and the total head pressure ˆ belongs to the spaces W 1;q ./ with q < 2, but functions of this space need not to be continuous and they are well defined everywhere except for some “bad” set of H1 -measure zero (see, e.g., Theorem 1 of §4.8 and Theorem 2 of §4.9.2 in [13]). So the formulation of the Bernoulli law for solutions in Sobolev spaces has to be modulo negligible “bad” set Av of one dimensional Hausdorff measure zero. Such version of Bernoulli’s law was obtained in [34, Theorem 1] (see also [36, Theorem 3.2] for a more detailed proof). Theorem 3 (The Bernoulli law). Assume the conditions (E). Then there exists a set Av with H1 .Av / D 0 such that any point x 2  n Av is a Lebesgue point for v; ˆ, and for every compact connected set K  , the following property holds: if ˇ ˇ D const; (26) K

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then ˆ.x1 / D ˆ.x2 /

for all x1 ; x2 2 K n Av :

(27)

(Here, in order to define a Lebesgue point at x 2 @, the usual Sobolev extension of v; ˆ to the whole R2 is considered.) Remark 3. In particular, if v D 0 on @ .in the sense of trace/, then by the MorseSard Theorem 2, there exist constants 0 ; : : : ; N 2 R such that .x/  j on each component j , j D 0; : : : ; N (Indeed, if v D 0 on @, then r D 0 on @, and by Theorem 2 (i)–(ii), the image .@/ has zero H1 -measure. This implies, by continuity of , that  const on each connected subset of @.). Therefore, by the above Bernoulli law, the pressure p.x/ ˇ is constant on @. Note that p.x/ could take different constant values b p j D p.x/ˇj on different connected components j of the boundary @. This fact was already mentioned in Sect. 2 (see example (21)). Using the assertion of Remark 3, one could prove the following regularity result for the pressure. Theorem 4. Let the conditions .Eı / be satisfied. Then p 2 C ./ \ W 2;1 ./:

(28)

The proof of this theorem is based on the div–curl lemma with two cancelations (e.g., [10, Theorem II.1]) and classical results concerning the Poisson equation (see, e.g., [48, Chapter II]). Under .Eı /-conditions by Remarks 2 and 3, one can apply Kronrod’s results to the stream function . Define the total head pressure on the Kronrod tree T (see Sect. 3.2) as follows. Let K 2 T with diam K > 0. Take any x 2 K n Av and put ˆ.K/ D ˆ.x/. This definition is valid by Bernoulli’s law (see Theorem 3). Lemma 3. Assume that the conditions .Eı / are satisfied. Let A; B 2 T , diam A > 0; diam B > 0. Consider the corresponding arc ŒA; B  T joining A to B (see Lemma 2). Then the restriction ˆjŒA;B is a continuous function. Remark 4. The continuity of ˆjŒA;B was proved in [41, Lemma 3.5]. The proof relies on the fact that each Sobolev function is continuous (in classical sense) on almost all straight line. Note that the total head pressure ˆ.x/ itself is not necessary continuous function in the whole  since about the velocity field v, it is only known that v 2 W 1;2 ./. For x 2  denote by Kx the connected component of the level set fz 2  W .z/ D .x/g containing the point x. Under .Eı /-conditions by Remark 3, Kx \

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

261

@ D ; for every y 2 ./ n f 0 ; : : : ; N g and for every x 2 1 .y/. Thus, Theorem 2 (ii), (iv) implies that for almost all y 2 ./ and for every x 2 1 .y/, the equality Kx \ Av D ; holds, and the component Kx   is a C 1 – curve homeomorphic to the circle. Such Kx is called an admissible cycle. The next lemma was obtained in [36, Lemma 3.3]. Lemma 4. Let the conditions .Eı / be satisfied. Assume that there exists a sequence 1;q 1;q of functions fˆ g such that ˆ 2 Wloc ./ and ˆ * ˆ in the space Wloc ./ for all q 2 Œ1; 2/. Then there exists a subsequence ˆkl such that ˆkl jS converges to ˆjS uniformly ˆkl jS  ˆjS on almost all admissible cycles S .here, “almost all cycles” means cycles in preimages 1 .y/ for almost all values y 2 .//. In connection with Lemma 4, note that in [2] Amick proved the uniform convergence ˆk  ˆ on almost all circles. However, his method can be easily modified to prove the uniform convergence on almost all level lines of every C 1 smooth function with nonzero gradient. Such modification was done in the proof of Lemma 3.3 of [36]. Below assume (without loss of generality) that the subsequence ˆkl of Lemma 4 coincides with ˆk . Admissible cycles S satisfying the statement of Lemma 4 will be called regular cycles. Let  be a bounded domain with Lipschitz boundary. The function f 2 W 1;s ./ is said to satisfy a one-side maximum principle locally in  if ess sup f .x/  ess sup f .x/ x20

(29)

x2@0

holds for any strictly interior subdomain 0 ( 0  / with the boundary @0 not containing singleton connected components. (In (29) negligible sets are the sets of two-dimensional Lebesgue measure zero in the left esssup and the sets of onedimensional Hausdorff measure zero in the right esssup.) If (29) holds for any 0   (not necessary strictly interior) with the boundary @0 not containing singleton connected components, then f 2 W 1;s ./ satisfies a one-side maximum principle in  (in particular, we can take 0 D  in (29)). Using Lemma 4, it could be proved that the one-side maximum principle is inherited by the limiting solutions. Theorem 5. Let the conditions .E/ be satisfied. Assume that there exists a sequence 1;q 1;q of functions fˆ g such that ˆ 2 Wloc ./ and ˆ * ˆ in the space Wloc ./ for all q 2 Œ1; 2/. If all ˆ satisfy the one-side maximum principle locally in , then ˆ satisfies the one-side maximum principle in . Theorem 5 was obtained in [34, Theorem 2] (see also [36, Theorem 3.4] for the detailed proof).

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Note that some version of a local weak one-side maximum principle was proved by Ch. Amick [2] (see Theorem 3.2 and Remark thereafter in [2]).

3.4

Arriving at a Contradiction

To prove the solvability of problem (2), we follow the arguments described in Sect. 2. First repeating Leray’s argument of getting an a priori estimate by a contradiction, we arrive to the following assertion: Lemma 5. Assume that   R2 is a bounded domain with C 2 -smooth boundary @, f 2 W 1;2 ./, and the boundary value a 2 W 3=2;2 .@/ satisfies the necessary condition (3). Then, if problem (2) admits no weak solutions, then there exists a sequence of functions uk 2 W 1;2 ./, pk 2 W 1;q ./ and numbers k ! 0C, k ! 0 > 0 with the following properties: (E-NS) the norms kuk kW 1;2 ./ , kpk kW 1;q ./ are uniformly bounded for every q 2 Œ1; 2/, and the pairs .uk ; pk / satisfy the system of equations 8   ˆ ˆ k uk C uk  r uk C rpk D fk ; x 2 ; < div uk D 0; x 2 ; ˆ ˆ : uk D ak ; x 2 @;

with fk D

(30)

k k2 k k f, ak D a, and 2 

kruk kL2 ./ ! 1;

uk * v in W 1;2 ./;

pk * p in W 1;q ./

8 q 2 Œ1; 2/;

where the pair of functions v 2 W 1;2 ./, p 2 W 1;q ./ is a solution to the Euler system (16). (In this lemma uk D wk C Jk1 A, k D .k Jk /1 ; fk D k k2  2 f, where the objects Jk ; wk were defined in Sect. 2.) Assume, in what follows, that the conditions (E-NS) are satisfied. As it is shown in Sect. 2, if all the fluxes Fi are zero (see (4)), then the conditions (E-NS) lead to a contradiction, thereby proving that (2) is solvable. In this section the goal is to demonstrate that these conditions also lead to a contradiction in the general case when the boundary data satisfy only the necessary condition (3). This will justify the existence of Theorem 1. Assume for simplicity that @ consists of two connected components 0 and 1 . Moreover, suppose that f D 0. The pressure p is equal to constants on 0 and 1 : pj0 D pO 0 ;

pj1 D pO 1

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

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(see (17)). If pO 0 D pO 1 , then, as it is shown in Sect. 2, a contradiction arises with the equality (14), and the required a priori estimate follows. Assume that b p 0 ¤ pO 1 . Normalizing the pressure (and changing the numeration of the components i , if necessary), one can assume without loss of generality that pO 0 D 0;

pO 1 < 0:

(31)

Introduce the main idea of the proof in a heuristic way. It is well known that total head pressures ˆk D pk C 12 u2k (under above assumptions f D 0) satisfy the linear elliptic equation ˆk D !k2 C

1 div .ˆk uk /; k

(32)

where !k D @2 u1k  @1 u2k is the corresponding vorticity. By Hopf’s maximum principle, in a subdomain 0 b  with C 2 – smooth boundary @0 , the maximum of ˆk is attained at the boundary @0 , and if x 2 @0 is a maximum point, then the normal derivative of ˆk at x is strictly positive. It is not sufficient to apply this property directly. Instead we will use some “integral analogs” that lead to a contradiction by using the Coarea formula. Namely, we construct a set Ei   consisting of level lines of ˆk such that ˆk jEi ! 0 as i ! 1 and Ei separates the boundary component 0 (where ˆ D 0) from the boundary component 1 (where ˆ < 0). On the one hand, the length of each of these level lines is bounded from below by a positive constant (since they separate the boundary components), and R by the Coarea formula, this implies the estimate from below for Ei jrˆk j. On the other Rhand, elliptic equation (32) for ˆk and boundary conditions allow us to estimate Ei jrˆk j2 from above, and this asymptotically contradicts the previous one. Describe this heuristic idea in more details. From (32) and the mentioned Hopf theorem, one concludes that all ˆk satisfy the strong maximum principle globally in . Then by conditions (E-NS) and Theorem 5, the limiting total head pressure ˆ satisfies the weak maximum principle globally in , i.e., max b p j D ess sup ˆ.x/ D 0:

j D1;2

(33)

x2

Using the results of Kronrod (see Sect. 3.2), one can construct a decreasing sequence of domains with the following properties. Let T be a family of all connected components of level sets of . Take B0 ; B1 2 T , B0  0 , and B1  1 , and set ˛D

min

C 2ŒB1 ;B0 

ˆ.C / < 0:

(this minimum exists by Lemma 3). Let ti 2 .0; ˛/; ti C1 D 12 ti and ti is such that ˆ.C / D ti ) C 2 .B1 ; B0 / is a regular cycle:

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t

k

0

k

Ai0

n

t

k

k

t

hk

1

Wik (t )

Ai0 1 Sik (t)

Fig. 1 The case of annulus-type domain (here, Ai is denoted as A0i )

(See the definition of the regular cycles in the commentary to Lemma 4.) Note that the existence of such a sequence ti follows from the fact that   H1 fˆ.C / W C 2 ŒB1 ; B0  and C is not a regular cycleg D 0I see [41, Corollary 3.2]. The proof of this equality is based on the Coarea formula (see [41]). Denote by Ai an element from the set fC 2 ŒB1 ; B0  W ˆ.C / D ti g which is closest to 0 . Let Vi be a connected component of the set nAi such that 0  @Vi , i.e., @Vi D Ai [ 0 . Obviously, Vi  ViC1 (since ti C1 D 12 ti ). Note that Ai are regular cycles and, therefore, ˆk jAi  ˆjAi D ti . Take t 2 Œ 58 ti ; 78 ti . Let Wik .t / be the connected component of the set fx 2 Vi n V iC1 W ˆk .x/ > tg such that @Wik .t /  Ai C1 (see Fig. 1). Put Sik .t / D .@Wik .t // \ Vi n V iC1 . Then ˆk jSi k .t/ D t, @Wik .t / D Sik .t / [ Ai C1 . Since 2 ˆk 2 W2;loc ./, by the Morse-Sard theorem for almost all t 2 Œ 58 ti ; 78 ti , the level set Sik .t / consists of a finite number of C 1 -cycles; moreover, ˆk is differentiable at every point x 2 Sik .t / and rˆk .x/ ¤ 0. Such values t are called .k; i /-regular. By construction Z

Z rˆk  ndS D 

Si k .t/

Si k .t/

jrˆk jdS < 0;

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

265

where n outward with respect to Wik .t / normal to @Wik .t / (see Fig. 1). Indeed, Sik .t / is a subset of the level set fx 2  W ˆ.x/ D t g, and by construction the nonzero gradient rˆ.x/ is directed inside the domain Wik .t / for x 2 Sik .t /, i.e., rˆk .x/ D n. jrˆk .x/j The key step in the proof is the following estimate Lemma 6. For any i 2 N, there exists k.i/ 2 N such that the inequality Z jrˆk .x/jdS  C t Si k .t/

holds for every k k.i/ and for almost all t 2 Π58 ti ; 78 ti . The constant C is independent of t; k, and i . The proof of Lemma 6 is based on the integration of the equality (32) over the suitable subdomain k .t / with @k .t / D Sik .t /[hk , where the cycle hk D fx 2  W dist.x; 0 / D hk g lies near the boundary component 0 and the parameter hk is taken in such a way that Z

ˆ2k

2

ds <  ;

hk

ˇZ ˇ ˇZ ˇ ˇ ˇ ˇ ˇ ˇ rˆk  n ds ˇ D ˇ !k u?  n ds ˇ < "; k ˇ ˇ ˇ ˇ hk

(34)

hk

Z

juk j2 ds < C" k2 ;

(35)

hk

where  and " are some fixed sufficiently small numbers and C" does not depend on k and  . For sufficiently large k k.i/ such hk can be found, using the weak convergences ˆk * ˆ, uk ! v from the assumptions (E-NS) and the boundary conditions kuk kL2 .@/ k , v  0, ˆ  0 on @ (see (303 ) and (163 )) When Lemma 6 is proved, the required contradiction can be obtained using the Coarea formula. For i 2 N and k k.i/, put [

Ei D

Sik .t /:

t2Π58 ti ; 78 ti 

By the Coarea formula (see, e.g., [50]), for any integrable function g W Ei ! R, the equality 7

Z8 ti Z

Z gjrˆk j dx D Ei

5 8 ti

Si k .t/

g.x/ d H1 .x/ dt

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holds. In particular, taking g D jrˆk j and using Lemma 6 yield Z

7

7

Z8 ti Z

jrˆk j2 dx D

Ei

5 8 ti

jrˆk j d H1 .x/dt 

Z8 ti

C t dt D C ti2 :

5 8 ti

Si k .t/

Now, taking g D 1 in the Coarea formula and using the Hölder Inequality, we get 7

Z8 ti

  H1 Sik .t / dt D

5 8 ti

Z jrˆk j dx Ei

0

Z

@

1 12 jrˆk j2 dx A



 12

meas.Ei /



p

 1 C ti meas.Ei / 2 :

Ei

By construction, for almost all t 2 Π58 ti ; 78 ti , the set Sik .t / is a smooth cycle and each set Sik .t / separates 0 from 1 . In Sik .t / separates Ai from AiC1 . Thus,  particular, H1 .Sik .t // C D min diam 0 ; diam 1 /. Hence, 7

Z8 ti

  1 H1 Sik .t / dt C ti : 4

5 8 ti

So, it holds p  1 1 C ti  C ti meas.Ei / 2 ; 4 or p  1 1 C  C meas.Ei / 2 : 4

(36)

  By construction meas.Ei /  meas Vi n ViC1  . But since  Vi  Vi C1 is a decreasing sequence of bounded sets, we have meas Vi n ViC1 ! 0 as i ! 1; therefore, inequality (36) gives the contradiction. Thus, our assumption is wrong; the norms of all possible solutions w./ to the operator equation (8) are uniformly bounded with respect to  2 Œ0; 1, and by the Leray-Schauder theorem, the equation (7) and equivalently the problem (2) has at least one solution. Hence, in the case when f D 0, Theorem 1 is proved. If f ¤ 0, then the maximum principle is not valid, and one has to consider two cases

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(a) Maximum of ˆ is attained on the boundary @: p 1 g D ess sup ˆ.x/: maxfpO 0 ; b x2

(b) Maximum of ˆ is not attained on @: p 1 g < ess sup ˆ.x/ maxfpO 0 ; b x2

(the case ess sup ˆ.x/ D C1 is also possible). x2

In the case (a) the proof is literally the same as above, while in the case (b) it can be proved that there exists a regular cycle F 2 T such that diam F > 0, F \ @ D ;, and ˆ.F / > ˇ, where ˇ D maxfpO 0 ; pO 1 g. For such F we consider the behavior of ˆ on the Kronrod arcs ŒBj ; F , j D 0; 1. The remaining part of the proof is similar to that of the proof for the case (a) with the following difference: F plays now the role which was played before by B0 , and the calculations become easier since F lies strictly inside . The main idea of the proof for a general multiply connected domain is the same as in the case of annulus-like domains (when @ D 1 [ 0 ). The proof has an analytical nature and unessential differences concern only well-known geometrical properties of level sets of continuous functions of two variables.

4

3D Axially Symmetric Case

First, specify some notations. Let Ox1 ; Ox2 ; Ox3 be the coordinate axis in R3 and D arctg.x2 =x1 /, r D .x12 C x22 /1=2 , z D x3 be the cylindrical coordinates. Denote by v ; vr ; vz the projections of the vector v on the axes ; r; z. A function f is said to be axially symmetric if it does not depend on . A vectorvalued function h D .hr ; h ; hz / is called axially symmetric if hr , h , and hz do not depend on . A vector-valued function h0 D .hr ; h ; hz / is called axially symmetric with no swirl if h D 0, while hr and hz do not depend on .

4.1

Bounded 3D Axially Symmetric Domains

The main result of this section is as follows. Theorem 6 ([37,41]). Assume that   R3 is a bounded axially symmetric domain of type (1) with C 2 -smooth boundary @ (Fig. 2). If f 2 W 1;2 ./, a 2 W 3=2;2 .@/ are axially symmetric and a satisfies condition (3), then (2) admits at least one weak axially symmetric solution. Moreover, if f and a are axially symmetric with no swirl, then (2) admits at least one weak axially symmetric solution with no swirl.

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x3

Fig. 2 Axially symmetric domain (N D 3)

0 1

3

2

The proof of Theorem 6 follows the same ideas as for the two-dimensional case, so it is not discussed here. (Some specific details for axially symmetric case could be found in the next section where the more complicated case of exterior domains is discussed.)

4.2

Exterior 3D Axially Symmetric Domains

This section is based on results of the paper [39]. Consider the Navier-Stokes problem   8 u C u  r u C rp D f ˆ ˆ ˆ ˆ ˆ < div u D 0 ˆ ˆ ˆ ˆ ˆ :

uDa lim

jxj!C1

in ; in ; on @;

(37)

u.x/ D u0

in the exterior domain of R3  D R3 n

N  S j D1

 Nj ; 

(38)

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

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Nj \ where i are bounded domains with connected C 2 -smooth boundaries i ,  N i D ; for i ¤ j , and u0 is a constant assigned vector.  Let Z (39) Fi D a  n dS; i D 1; : : : N; i

where n is the unit outward normal to @. Under suitable regularity hypotheses on  and a and assuming that Fi D 0;

i D 1; : : : N;

(40)

in the celebrated paper [49] of 1933, J. Leray showed that (37) has a solution u with finite Dirichlet integral: Z jruj2 dx < C1; (41) 

and u satisfies (374 ) in a suitable sense for general u0 and uniformly for u0 D 0. In the 1950s, the problem was reconsidered by R. Finn [16] and O.A. Ladyzhenskaya [46, 47]. They showed that the solution satisfies the condition at infinity uniformly. Moreover, condition (40) and the regularity of a have been relaxed by requiring N P jFi j to be sufficiently small [16] and a 2 W 1=2;2 .@/ [47]. i D1

In 1973 K.I. Babenko [4] proved that if .u; p/ is a solution to (37), (41) with u0 ¤ 0, then .u  u0 ; p/ behaves at infinity as the solutions to the linear Oseen system. In particular, u.x/  u0 D O.r 1 /;

p.x/ D O.r 2 /:

(42)

(See also [21]. Here the symbol f .x/ D O.g.r// means that there is a positive constant c such that jf .x/j  cg.r/ for large r.) However, nothing is known, in general, on the rate of convergence at infinity for u0 D 0. (For small kakL1 .@/ existence of a solution .u; p/ to (37) such that u D O.r 1 / is a simple consequence of Banach contractions theorem [69]. Moreover, one can show that p D O.r 2 /, and the derivatives of order k of u and p behave at infinity as r k1 , r k2 , respectively [75]; see also [21, 58].) One of the most important problems in the theory of the steady-state NavierStokes equations concerns the possibility to prove the existence of a solution to (37) without any assumptions on the fluxes Fi (see, e.g., [21]). To the best of our knowledge, the most general assumptions assuring the existence is expressed by N X iD1

max i

jFi j < 8  jx  xi j

(43)

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(see [67]), where Fi is defined by (39) and xi is a fixed point of i (see also [5] for analogous conditions in bounded domains). In the recent paper [39], the above question was solved for the axially symmetric case. Note that for axially symmetric solutions u of (37), the vector u0 has to be parallel to the symmetry axis. The main result is as follows. Theorem 7 ([39]). Assume that   R3 is an exterior axially symmetric domain (38) with C 2 -smooth boundary @, u0 2 R3 is a constant vector parallel to the symmetry axis, and f 2 W 1;2 ./ \ L6=5 ./, a 2 W 3=2;2 .@/ are axially symmetric. Then (37) admits at least one weak axially symmetric solution u satisfying (41). Moreover, if a and f are axially symmetric with no swirl, then (37) admits at least one weak axially symmetric solution with no swirl satisfying (41). Remark 5. It is well known (see, e.g., [47]) that under hypothesis of Theorem 7, 2;2 every weak solution u of the problem (37) is more regular, i.e., u 2 Wloc ./ \ 3;2 Wloc ./. Emphasize that Theorem 7 furnishes the first existence result without any assumption on the fluxes for the stationary Navier-Stokes problem in exterior threedimensional domains.

4.2.1 Extension of the Boundary Values The next lemma concerns the existence of a solenoidal extensions of boundary values. Lemma 7 (see, e.g., [39]). Let   R3 be an exterior axially symmetric domain (38). If a 2 W 3=2;2 .@/, then there exists a solenoidal extension A 2 W 2;2 ./ of a such that A.x/ D  .x/ for sufficiently large jxj, where  .x/ D 

N x X Fi 4 jxj3 iD1

(44)

and Fi are defined by (39). Moreover, the following estimate kAkW 2;2 ./  ckakW 3=2;2 .@/

(45)

holds. Furthermore, if a is axially symmetric (axially symmetric with no swirl), then A is axially symmetric (axially symmetric with no swirl) too.

4.2.2 Leray’s Argument “Invading Domains” Consider the Navier-Stokes problem (37) with f 2 W 1;2 ./ \ L6=5 ./ in the C 2 smooth axially symmetric exterior domain   R3 defined by (38). Without loss of generality, assume that f D curl b 2 W 1;2 ./ \ L6=5 ./. (By the HelmholtzWeyl decomposition, f can be represented as the sum f D curl b C r' with curl b 2

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

271

W 1;2 ./ \ L6=5 ./, and the gradient part is included then into the pressure term; see, e.g., [21, 47].) Below the proof of Theorem 7 will be discussed in the case u0 D 0: The proof for u0 ¤ 0 follows the same steps with minor standard modification. A function u is called a weak solution of problem (37) if w  A 2 H ./ and the integral identity  R R R  rw  r dx D  rA  r dx  A  r A   dx 





R

  R A  r w   dx  w  r w   dx







(46)



R

 R w  r A   dx C f   dx





holds for any  2 J01 ./. Here A is the extension of the boundary data constructed in Lemma 7. We shall look for the axially symmetric (axially symmetric with no swirl) weak solution of problem (37) and find this solution as a limit of weak solution to the Navier-Stokes problem in a sequence of bounded domain k that in the limit exhausts the unbounded domains  (this is the main idea of the “invading domain” method). Namely, consider the sequence of the boundary value problems 8 uk C .b uk  r/b uk C r pO k D f in k ; ˆ < b (47) div b uk D 0 in k ; ˆ : b uk D A on @k ; jxj < kg, 12 Bk0 

where k D Bk \  for k k0 , Bk D fx W

N S

N i . By 

i D1

Theorem 6, each problem (47) has an axially symmetric solution b uk D A C b wk with b wk 2 H .k /. To prove the assertion of Theorem 7, it is sufficient to establish the uniform estimate Z jrb wk j2  c: (48) 

Estimate (48) will be proved following a classical reductio ad absurdum argument of J. Leray and O.A. Ladyzhenskaia (see [47, 49]). Indeed, if (48) is not true, then there exists a sequence fb wk gk2N such that Z wk j2 : lim Jk2 D C1; Jk2 D jrb k!C1

k

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The sequence wk D b wk =Jk is bounded in H ./. Extracting a subsequence (if necessary), one can assume that wk converges weakly in H ./ and strongly in q Lloc ./ .q < 6/ to a vector field v 2 H ./ with Z

jrvj2  1:

(49)



It is easy to check that v 2 H ./ is a weak solution to the Euler equations, and for some p 2 D 1;3=2 ./ the pair .v; p/ satisfies the Euler equations almost everywhere: 8  ˆ ˆ v  r v C rp D 0 < div v D 0 ˆ ˆ : vD0

in ; (50)

in ; on @:

Adding some constants to p (if necessary) by virtue of the Sobolev inequality (see, e.g., [21] II.6), it may be assumed without loss of generality that kpkL3 ./  const:

(51) 2

Put k D .Jk /1 . Multiplying equations (47) by J12 D k2 , one sees that the pair k   wk C J1k A; pk D J12 pO k satisfies the following system: uk D J1k b k

8   ˆ k uk C uk  r uk C rpk D fk ˆ < div uk D 0 ˆ ˆ : uk D ak

in k ; in k ;

(52)

on @k ;

2

3;2 2;2 where fk D k2 f, ak D k A, uk 2 Wloc ./, pk 2 Wloc ./ (the interior regularity of the solution depends on the regularity of f 2 W 1;2 ./, but not on the regularity of the boundary value a; see [47]). Using the classical local estimates for ADN-elliptic problems (see [1, 73]), one could prove the following uniform estimate:

kuk kL6 .k / C krpk kL3=2 .k /  C;

(53)

where C does not depend on k. By construction, there holds the weak convergences 1;3=2 1;2 uk * v in Wloc ./; pk * p in Wloc ./ (the weak convergence in 1;2 1;2 Wloc ./ means the weak convergence in W .0 / for every bounded subdomain 0  ). As in the two-dimensional case, in conclusion, one can prove the following lemma.

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

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Lemma 8. Assume that   R3 is an exterior axially symmetric domain of type (38) with C 2 -smooth boundary @, and a 2 W 3=2;2 .@/, f D curl b 2 W 1;2 ./ \ L6=5 ./ are axially symmetric. If the assertion of Theorem 7 is false, then there exist v; p with the following properties. (E-AX) The axially symmetric functions v 2 H ./, p 2 D 1;3=2 ./ satisfy the Euler system (50) and kpkL3 ./ < 1. (E-NS-AX) Condition (E-AX) is satisfied, and there exist sequences of axially symmetric functions uk 2 W 1;2 .k /, pk 2 W 1;3=2 .k /, k D  \ BRk , Rk ! 1 as k ! 1, and numbers k ! 0C, such that estimate (53) holds, 2 the pair .uk ; pk / satisfies (52) with fk D k2 f, ak D k A (here A is solenoidal extension of a from Lemma 7), and kruk kL2 .k / ! 1;

1;3=2

1;2 uk * v in Wloc ./; pk * p in Wloc Z  D .v  r/v  A dx

./;

(54) (55)

 3;2 2;2 Moreover, uk 2 Wloc ./ and pk 2 Wloc ./.

4.2.3 Euler Equation in 3D Axially Symmetric Case (Exterior Domains) Suppose that the assumptions (E-AX) (from Lemma 8) are satisfied, and, for definiteness, assume that (SO)  is the domain (38) symmetric with respect to the axis Ox3 and j \ Ox3 ¤ ;; j \ Ox3 D ;;

j D 1; : : : ; M 0 ; j D M 0 C 1; : : : ; N:

(The cases M 0 D N or M 0 D 0, i.e., when all components (resp., no components) of the boundary intersect the axis of symmetry, are also allowed.) Denote PC D f.0; x2 ; x3 / W x2 > 0; x3 2 Rg, D D  \ PC , Dj D j \ PC . Of course, on PC the coordinates x2 ; x3 coincide with coordinates r; z. Then v and p satisfy the following system in the plane domain D: 8 @vz @vz @p ˆ C vr C vz D 0; ˆ ˆ ˆ @z @r @z ˆ ˆ 2 ˆ @p .v / @vr @vr ˆ ˆ <  C vr C vz D 0; @r r @r @z @v @v v vr ˆ ˆ ˆ C vr C vz D 0; ˆ ˆ r @r @z ˆ ˆ ˆ @.rvz / @.rvr / ˆ : C D0 @r @z (these equations are fulfilled for almost all x 2 D).

(56)

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1;2 There hold the following integral estimates: v 2 Wloc .D/, Z Z rjrv.r; z/j2 drd z C rjv.r; z/j6 drd z < 1: D

(57)

D

Also, the inclusions rp 2 L3=2 ./, p 2 L3 ./ can be rewritten in the following two-dimensional form: Z Z 3=2 rjrp.r; z/j drd z C rjp.r; z/j3 drd z < 1: (58) D

D

The next statement was proved in [30, Lemma 4] and in [2, Theorem 2.2]. Theorem 8. Let the conditions (E-AX) be fulfilled. Then 8j 2 f1; : : : ; N g 9 pO j 2 R W

p.x/  pO j

for H2  almost all x 2 j :

(59)

In particular, by axial symmetry, p.x/  pO j

for H1  almost all x 2 M j :

(60)

Here and below the following convenient agreement is used: for a set A  R3 M WD A \ PC , and for B  PC denote by BQ the set in R3 obtained by rotation put A of B around Oz -axis. The following result gives more precise information about the constants from the previous theorem. Corollary 1 ([39]). Assume that the conditions (E-AX) are satisfied. Then ˆjj  0 whenever j \ Oz ¤ ;, i.e., pO 1 D    D pO M 0 D 0; where pO j are the constants from Theorem 8. This phenomenon is connected with the fact that the symmetry axis can be approximated by stream lines (see Theorem 10 below), where the total head pressure is constant according to the Bernoulli law (see Theorem 9 below). To formulate the last result, some preparation is needed. Below without loss of generality, assume that the functions v; p are extended to the whole half-plane PC as follows: v.x/ WD 0;

x 2 PC n D;

p.x/ WD pO j ; x 2 PC \ DN j ; j D 1; : : : ; N:

(61) (62)

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275

Obviously, the extended functions inherit the properties of the previous ones. 1;3=2 1;2 .PC /, p 2 Wloc .PC /, and the Euler equations (56) are fulfilled Namely, v 2 Wloc almost everywhere in PC . The last equality in (56) (which is fulfilled, after the above extension agreement, in the whole half-plane PC ) implies the existence of a continuous stream function 2;2 2 Wloc .PC / such that @ D rvz ; @r Denote by ˆ D p C .v; p/. From (57) we get Z

@ D rvr : @z

(63)

jvj2 the total head pressure corresponding to the solution 2

rjˆ.r; z/j3 drd z C

PC

Z

rjrˆ.r; z/j3=2 drd z < 1:

(64)

PC

By direct calculations one easily gets the identity vr ˆr C vz ˆz D 0

(65)

for almost all x 2 PC . The identities (61)–(62) mean that ˆ.x/  pO j

8x 2 PC \ DN j ; j D 1; : : : ; N:

(66)

Theorem 9 (Bernoulli law for Sobolev solutions [39]). Let the conditions (E-AX) be valid. Then there exists a set Av  PC with H1 .Av / D 0, such that every point x 2 PC n Av is a Lebesgue point for v; ˆ, and for any compact connected set K  PC , the following property holds : if ˇ ˇ D const; K

(67)

then ˆ.x1 / D ˆ.x2 / for all x1 ; x2 2 K n Av :

(68)

Theorem 9 is a space version of the above Theorem 3 (for the plane case). For the axially symmetric bounded domains, the result was proved in [37, Theorem 3.3]. The proof for exterior axially symmetric domains is similar: one has to overcome two obstacles. First difficulty is the lack of the classical regularity, and here the results of [6] have a decisive role (according to these results, almost all level sets of plane W 2;1 -functions are C 1 -curves; see Sect. 3.1). The second obstacle is the set where r .x/ D 0 ¤ rˆ.x/, i.e., where vr .x/ D vz .x/ D 0, but v .x/ ¤ 0. Namely, without assuming the boundary conditions (503 ), in general,

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the equality (67) even for smooth functions does not imply (68). For example, if vr D vz D 0 in the whole domain, v D r, then  const on the whole domain, while ˆ D r 2 ¤ const. Without the boundary assumptions, one can prove only that ˆ.r; z/ D f .r/ along every level set K of the stream function for some absolutely continuous function f .r/ (see [39, Lemma 4.5]). But the last equality together with the boundary conditions (503 ), (66) easily implies Theorem 9. For " > 0 and R > 0 denote by S";R the set S";R D f.r; z/ 2 PC W r "; r 2 C z2 D R2 g. From the assumptions (64) one gets Lemma 9. For any " > 0, there exists a sequence j > 0, j ! C1, such that S";j \ Av D ; and sup jˆ.x/j ! 0 as j ! 1:

(69)

x2S";j

One of the main results of this section is the following. Theorem 10. Assume that the conditions (E-AX) are satisfied. Let Kj be a sequence of continuums with the following properties: Kj  DN n Oz , jKj D const, and lim inf r D 0, lim sup r > 0. Then ˆ.Kj / ! 0 as j ! 1. Here we j !1 .r;z/2Kj

j !1 .r;z/2Kj

denote by ˆ.Kj / the corresponding constant cj 2 R such that ˆ.x/ D cj for all x 2 Kj n Av (see Theorem 9).

4.2.4 Obtaining a Contradiction From now on assume that the assumptions (E-NS-AX) (see Lemma 8) are satisfied. The goal is to prove that they lead to a contradiction. This implies the validity of Theorem 7. For simplicity assume that f D 0, N D 2 and M 0 D 1, i.e., the boundary @ splits into the two components @ D 1 [ 2 , where 1 \ Oz ¤ ;;

2 \ Oz D ;:

(70)

The main idea of the proof is similar to that for the two-dimensional case. Since every ˆk D pk C 12 juk j2 satisfies the linear elliptic equation ˆk D !k2 C

1 div .ˆk uk /; k

(71)

where !k D curl uk and !k .x/ D j!k .x/j, a contradiction is obtained by using some “integral analog” of Hopf’s maximum principle and the Coarea formula. Consider the constants pO j from Theorem 8 (see also Theorem 1). From the equality (55), the Euler equations (501 ), and the regularity assumptions (64), the identity follows

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

 D

N X

pO j Fj D b p 2 F2

277

(72)

j DM 0 C1

(since pO 1 D 0 by Theorem 1). Therefore, pO 2 ¤ 0. Further consider separately three possible cases. (a) The maximum of ˆ is attained at infinity, i.e., it is zero: 0 D ess sup ˆ.x/:

(73)

x2

(b) The maximum of ˆ is attained on a boundary component which does not intersect the symmetry axis: 0 < pO 2 D

max

j DM 0 C1;:::;N

b p j D ess sup ˆ.x/;

(74)

x2

(c) The maximum of ˆ is not zero and it is not attained on @: max

j DM 0 C1;:::;N

b p j < ess sup ˆ.x/ > 0

(75)

x2

(the case ess sup ˆ.x/ D C1 is not excluded). x2

4.2.5

The Case ess sup ˆ.x/ D 0. x2

Let us consider the case (73). By Theorem 1, pO 1 D ess sup ˆ.x/ D 0:

(76)

x2

Then pO 2 < 0. The arguments below are similar to the plane situation (see Sect. 3.4). Take the positive constant ıp D pO 2 . The first goal is to separate the boundary components where ˆ < 0 from infinity and from the singularity axis Oz by level sets of ˆ compactly supported in D. More precisely, for any t 2 .0; ıp / we construct a continuum A.t/ b PC with the following properties: M WD A \ PC ) lies in (i) The set M j (recall that for a set A  R3 by definition A a bounded connected component of the open set PC n A.t/; (ii) jA.t/  const, ˆ.A.t // D t; (iii) (monotonicity) If 0 < t1 < t2 < ıp , then the set A.t1 / [ M 1 lies in the unbounded connected component of the set PC n A.t2 / (in other words, A.t2 / [ M 2 lies in the bounded connected component of the set PC n A.t1 /; see Fig. 3).

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Fig. 3 The surface Sk .t1 ; t2 ; t / ( here A.t/ is denoted as A2 .t/)

For this construction, the results of Sect. 3.2 are used for the restrictions of the stream function on suitable compact subdomains of PC (see details in [39]). We have also one additional property (cf. with Lemma 4 for the plane case):   (iv) There exists a set T  .0; ıp / of full measure (i.e., meas .0; ıp / n T D 0) such that for all t 2 T the set A.t/ is a regular cycle, i.e., it is a C 1 -curve homeomorphic to the circle and ˆk .x/  ˆ.x/  t

on A.t/:

(77)

issue is to construct Let t1 ; t2 2 T and t1 < t 0 < t 00 < t2 . The very important  for sufficiently large k k0 and for almost all t 2 t 0 ; t 00 a C 1 -circle Sk .t / which separates A.t1 / from A.t2 / and satisfies ˆk jSk .t/  t . Moreover, the gradient of ˆk is directed toward 1 . For this purpose, for t 2 Œt 0 ; t 00  denote by Wk .t1 ; t2 I t / the bounded connected component of the open set fx 2 PC n A.t1 / W ˆk .x/ > tg such that @Wk .t1 ; t2 I t /  A.t1 / (see Fig. 3). This definition is valid since for sufficiently large k by the convergence (77), the estimates ˆk jA.t1 / > t;

ˆk jA.t2 / < t

8t 2 Œt 0 ; t 00 

hold. Now put Sk .t1 ; t2 I t / D .@Wk .t1 ; t2 I t // n A.t1 /:

(78)

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279

Clearly, ˆk  t on Sk .t1 ; t2 I t /. Moreover, Sk .t1 ; t2 I t / separates A.t1 / from A.t2 / because of the monotonicity property (iii) and (78) (see Fig. 3). Q denotes the set in R3 obtained by rotation of A Recall, that for a set A  PC , A around Oz -axis. By construction, for every regular value t 2 Œt 0 ; t 00  b .t1 ; t2 /, the set Sk .t1 ; t2 I t / is a C 1 -circle; consequently, SQ k .t1 ; t2 I t / is a torus, and Z

Z rˆk  n dS D 

SQ k .t1 ;t2 It/

jrˆk j dS < 0;

(79)

SQ k .t1 ;t2 It/

where n is the unit outward normal vector to @WQ k .t1 ; t2 I t /. Now formulate the key estimate. Lemma 10. For any t1 ; t2 ; t 0 ; t 00 2 T with t1 < t 0 < t 00 < t2 , there exists k D k .t1 ; t2 ; t 0 ; t 00 / such that for every k k and for almost all t 2 Œt 0 ; t 00 , the inequality Z

jrˆk j dS < Ft;

(80)

SQ k .t1 ;t2 It/

holds with the constant F independent of t; t1 ; t2 ; t 0 ; t 00 and k. Proof. Fix t1 ; t2 ; t 0 ; t 00 2 T with t1 < t 0 < t 00 < t2 . Below always assume that k is sufficiently large; in particular, the set SQ k .t1 ; t2 I t / is well defined for all t 2 Œt 0 ; t 00 . The main idea of the proof of (80) is quite simple: to integrate the equation ˆk D !k2 C

1 div .ˆk uk / k

(81)

over the suitable domain k .t / with @k .t /  SQ k .t1 ; t2 I t / such that the corresponding boundary integrals ˇ ˇ ˇ ˇ 

Z 

ˇ ˇ rˆk  n dS ˇˇ

(82)

@k .t/ nSQ k .t1 ;t2 It/

ˇ 1 ˇˇ k ˇ

Z 

ˇ ˇ ˆk uk  n dS ˇˇ

(83)

@k .t/ nSQ k .t1 ;t2 It/

are negligible. Such domain k .t / can be constructed because of the weak convergences ˆk * ˆ, uk ! v from the assumption (E-NS-AX) and the boundary conditions kuk kL2 .@/ k , v  0, ˆ  0 on @ (see (523 ) and (503 )).

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The following technical fact from the one-dimensional real analysis is needed. Lemma 11. Let f W S ! R be a positive decreasing function defined on a measurable set S  .0; ı/ with measŒ.0; ı/ n S  D 0. Then 4

Œf .t2 / 3 .t2  t1 / D 1: t1 ;t2 2S .t2 C t1 /.f .t1 /  f .t2 // sup

(84)

The proof of this fact is elementary (see, e.g., [39, Appendix]). For t 2 T denote by U .t/ the bounded connected component (the torus) of the Q By construction, U .t2 / b U .t1 / for t1 < t2 . set R3 n A.t/. From estimate (80), the isoperimetric inequality and from the Coarea formula, one can easily deduce Lemma 12. For any t1 ; t2 2 T with t1 < t2 , the estimate

4

meas U .t2 / 3  C

 t2 C t1

meas U .t1 /  meas U .t2 / t2  t1

(85)

holds with the constant C independent of t1 ; t2 . The R proof of this lemma is based on the same idea (Coarea formula for and jrˆk j2 ) as in Lemma 6 discussed for the plane case. The last estimate leads us to the main result of this subsection.

R

jrˆk j

Lemma 13. Assume that   R3 is an exterior axially symmetric domain of type (38) with C 2 -smooth boundary @ and f 2 W 1;2 ./ \ L6=5 ./, a 2 W 3=2;2 .@/ are axially symmetric. Then the assumptions (E-NS-AX) and (73) lead to a contradiction. Proof. By construction, U .t1 /  U .t2 / for t1 ; t2 2 T, t1 < t2 . Thus the just obtained estimate (85) contradicts Lemma 11. This contradiction finishes the proof of Lemma 13. t u

4.2.6

The Case ess sup ˆ.x/ > 0. x2

The cases (b) and (c), where ess sup ˆ.x/ > 0 (see (74) and (75)), are easily reduced x2

to the plane case, because now one can separate, by the level sets of ˆ, the region where ˆ is close to maximum both from infinity and from the singularity axis Oz and carry out all arguments in the constructed bounded plane domain.

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

4.3

281

A Simple Proof of the Existence Theorem in the Case Without Swirl

Here we discuss an alternative (and much more simple) approach to the existence result for the 3D exterior domain in the axially symmetric case without swirl. The proof is based on the idea of [36] of using some divergence identities for solutions to the Euler equations (see also [42]).

4.3.1 Some Identities for Solutions to the Euler System Let the conditions (E-AX) from Lemma 8 be fulfilled, i.e., the axially symmetric functions .v; p/ satisfy the Euler equations (50) and v 2 L6 .R3 /; rv 2 L2 .R3 /;

p 2 L3 .R3 /;

rp 2 L3=2 .R3 /;

r 2 p 2 L1 .R3 /

(these properties were discussed in Sect. 4.2.3; without loss of generality, assume that v is extended by zero into R3 n  and put p.x/ D pO j for x 2 j , j D 1; : : : ; N ). Assume also that (73) is valid, i.e., ˆ.x/  0:

(86)

First of all, discuss the integrability properties of these functions on half-plane PC . For any axially symmetric vector function g D .g ; gr ; gz /, the following equality jrgj2 D

jg j2 jgr j2 C C j@r gr j2 C j@z gr j2 C j@r g j2 r2 r2 2

2

Cj@z g j C j@r gz j C j@z gz j

(87)

2

holds. Thus, jgrr j  jrgj. Applying this formula to g D rp D .@r p; 0; @z p/ we get, by virtue of r 2 p 2 L1 .R3 /, @r p 2 L1 .R3 /: r

(88)

Hence @r p 2 L1 .PC /. Since p.r; z/ ! 0 for each z 2 R as r ! 1, the inclusion p.r; / 2 L1 .R/

(89)

is valid for each r > 0; moreover, Z1 Z

Z p.t; z/ d z D  R

@r p.r; z/ d z dr ! 0 t

R

as t ! 1:

(90)

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From the last formula and the inequality ˆ  0, it follows that jvj2 .r; / 2 L1 .R/

(91)

for each r > 0. Further, (87) and rv 2 L2 .R3 / imply jv j2 jvr j2 C 2 L1 .PC /: r r

(92)

From the Euler system (50), it follows by direct calculation that for any smooth function g, the following identity

 div Œp g C .v  g/v D p div g C .v  r/g  v

(93)

holds. Apply this formula two times for g D rer and g D 1r er , where er is the unit vector parallel to the r-axis. 2 2 (I) g D rer , div Œp g C .v  g/v this identity  D 2p C v C vr . Integrating over the q 2 2 3 3D infinite cylinder Ct D .x1 ; x2 ; x3 / 2 R W r D x1 C x2 < t yields

t2

Z



 p.t; z/ C vr2 .t; z/ d z D

R

“



r 2p C v 2 C vr2 d z dr;

(94)

Pt

where Pt D f.r; z/ 2 PC W r < t g. (II) g D 1r er , div Œp g C .v  g/v D r12 .v 2  vr2 /. Since there is an essential singularity at r D 0, one needs to integrate this identity over Ct0 t D  q .x1 ; x2 ; x3 / 2 R3 W r D x12 C x22 2 .t0 ; t / to obtain R

R

D

  R

p.t; z/ C vr2 .t; z/ d z  R p.t0 ; z/ C vr2 .t0 ; z/ d z

’ 1 Pt0 t

r

 .v 2  vr2 / d z dr;

where Pt0 t D f.r; z/ 2 PC W r 2 .t0 ; t /g. Since

R

R

 p.t; z/ C vr2 .t; z/ d z ! 0 as t ! C1 and “ PC

ˇ1  2 ˇ ˇ v  v 2 ˇ d z dr < 1; r r

(95)

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

it follows from the above formulas that

“ Z

  1 2 vr  v 2 d z dr; p.t; z/ C vr2 .t; z/ d z D r R

283

(96)

Pt1

where Pt1 D f.r; z/ 2 PC W r 2 .t; C1/g.

4.3.2 Proof of the Existence Theorem As in the beginning of Sect. 4.2.4, assume that the assumptions (E-NS-AX) (see Lemma 8) are satisfied, but now suppose, additionally, that all functions a; f; uk ; v have no swirl. Our goal is to receive a contradiction. This implies the validity of Theorem 7 for the case with no swirl. It turns out that a contradiction for this case could be obtained extremely easy. Consider the limit solution .v; p/ to the Euler equations (50) from Lemma 8 (E-AX). It is necessary to discuss only the case (73), since for other two cases (74)–(75), the arguments are carried outR for

bounded plane domains (see Sect. 4.2.6). From (73), (94), it follows that R p.t; z/R C vr2 .t; z/ d z  0 for all t > 0. But in the case v  0, the equality (96) implies R p.t; z/Cvr2 .t; z/ d z > 0 a contradiction. Remark 6. The similar idea was used in [36] to obtain the existence theorem for annulus-type plane domain under inflow conditions (the flux through the external boundary component is nonpositive) and in [42] to prove the Liouville theorem in R3 for the D-solution without swirl of the stationary Navier-Stokes system. Note that this result of [42] could be easily derived from the Liouville-type theorem for ancient solutions of nonsteady Navier-Stokes system in [33].

5

2D Axially Symmetric Case: Exterior Domain

5.1

Formulation of the Problem and Historical Review

Let  be an exterior domain of R2 :  D R2 n

N  S

 Nj ; 

(97)

j D1

where j  R2 , j D 1; : : : ; N; are bounded, simply connected domains with Nj \  N i D ; for i ¤ j . Look for a solution of the Lipschitz boundaries and  steady-state Navier-Stokes problem   8 u C u  r u C rp D f in ; ˆ ˆ < div u D 0 in ; (98) ˆ ˆ : uDh on @;

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satisfying the additional condition at infinity lim

jxj!C1

u.x/ D e1 ;

(99)

where for simplicity it is assumed that f vanishes outside a disk. The two-dimensional problem in an exterior domain is much harder than the above three-dimensional case (see Sect. 4.2). The main difficulty is to find a solution satisfying the condition at infinity (99). In 1933 J. Leray [49] proved that if the boundary data are sufficiently regular, f D 0, and the fluxes through every @i vanish Z h  n dS D 0; (100) Fi D @i

then problem (98) has a weak solution .u; p/ with finite Dirichlet integral Z

jruj2 dx < C1:

(101)



To show this, Leray introduced an elegant argument, known as invading domains method, which consists in proving first that the Navier-Stokes problem   8 C u  r uk C rpk u k k ˆ ˆ ˆ ˆ < div uk ˆ uk ˆ ˆ ˆ : uk

D0

in k ;

D0

in k ;

Dh

on @;

D e1

on @Bk

(102)

has a weak solution uk for every bounded domain k D  \ Bk , Bk D fx 2 R2 W jxj < kg, Bk c {, and shows that the following estimate holds Z

jruk j2 dx  c;

(103)

k

for some positive constant c independent of k. While (103) is sufficient to assure existence of a subsequence ukl which converges weakly to a solution u of (98) satisfying (101), it does not give any information about the behavior at infinity of the velocity u (Indeed, the unbounded function log˛ jxj .˛ 2 .0; 1=2/) satisfies (101).), i.e., we do not know whether this limiting solution u satisfies the condition at infinity (99). That means that the limiting solution u does not “remember” about the boundary value e1 despite the fact that this boundary value was used in the construction of ukl * u (cf. with Sect. 4.2.2 for 3D case).

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

285

In 1961 H. Fujita recovered, by means of a different method, Leray’s result. Nevertheless, due to the lack of a uniqueness theorem, the solutions constructed by Leray and Fujita are not comparable, even for very small . The solution to (98) constructed by the invading domains method is called Leray’s solution, while any solution satisfying (101) is called D-solution. Only 40 years after Leray’s paper, D. Gilbarg and H.F. Weinberger [25] were able to show that the velocity u in Leray’s solution is bounded, p converges uniformly to a constant at infinity, and there is a constant vector uN such that Z2 lim

r!C1

N 2d D 0 ju.r; /  uj

(104)

0

(here .r; / denote polar coordinates with center at O). Moreover, they proved that if uN D 0, then the convergence is uniform and ru decays at infinity as r 3=4 log r. In the subsequent paper [26], the same authors proved that a bounded D-solution meets the same asymptotic properties as the Leray solution (see also [2]). One of N To the most difficult and unanswered questions is the relation between e1 and u. point out the difficulties of the problem, recall that even the linear Stokes problem 8 ˆ ˆ u C rp ˆ ˆ ˆ < div u ˆ ˆ ˆ ˆ ˆ :

u lim

jxj!C1

D

0

in ;

D

0

in ;

D

h

on @;

(105)

u.x/ D e1 ;

does not have, in general, a solution. Indeed, the solutions of the homogeneous problem 8 v C rQ D 0 ˆ ˆ ˆ ˆ ˆ ˆ div v D 0 < ˆ ˆ ˆ ˆ ˆ ˆ :

v D 0 v.x/ D 0; lim jxj!C1 jxj

in ; in ; on @;

spans a two-dimensional linear space C, and (105) is solvable if and only if the data satisfy the following compatibility condition (Stokes’ paradox) Z @

.h  e1 /  Œ.rv C rv> /  n  QndS D 0;

8v 2 C

(106)

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(see [8, 23]). Let us observe, by the way, that this is not surprising. Indeed, the natural solution to (105)1;2;3 should behave at infinity as the fundamental solution to (105) .u D O.log r//, and the addition of (105)4 makes (105) overdetermined. Therefore, (106) appears to be quite natural. Unexpectedly, in 1967 R. Finn and D.R. Smith [17] discovered the existence of a solution to (98), (99) without any compatibility relation between h and ¤ 0, for  sufficiently large. They also showed that .u  e1 ; p/ behaves at infinity as the fundamental solution of the linear Oseen system (see also [22]). In particular, taking also into account the results in [11, 72], one obtains the following behavior: u1  D O.r 1=2 /; ru D O.r 1 log2 r/;

u2 D O.r 1 log r/; p D O.r 1 log r/;

(107)

and outside a parabolic “wake region” around axis e1 , the decay is more rapid; in particular, ! D @1 u2  @2 u1 behaves according to !.x/ D O.e c.x1 jxj/ /

(108)

for some absolute constant c. R. Finn and D.R. Smith called a solution .u; p/ to (98), (99) physically reasonable provided u  e1 D O.r 1=4" / for some positive ". D.R. Smith [72] proved that a physically reasonable solution satisfies (107) and D.C. Clark [11] that (107) implies (108). More recently, V. I. Sazonov [71] showed that a D-solution such that u  e1 D o.1/, with ¤ 0, is physically reasonable (see also [21, 24]). Notice that nothing is currently known about the asymptotic behavior, in general, for D 0 or for arbitrary . Later, in 1988, problem (98), (99) was taken up by C.J. Amick [3] under the assumption f D 0. He proved that if h D 0, then any D-solution is bounded and converges to uN in the sense of (104). Moreover, he considered a particular but physically interesting class of solutions u D .u1 ; u2 / such that u1 is an even function of x2 and u2 is an odd function of x2 : u1 .x1 ; x2 / D u1 .x1 ; x2 /;

u2 .x1 ; x2 / D u2 .x1 ; x2 /

(109)

in the symmetric domain .x1 ; x2 / 2  , .x1 ; x2 / 2 :

(110)

Using Leray’s argument C.J. Amick showed that for symmetric solutions the convergence of u at infinity is uniform; moreover, if @ is regular enough and h D 0, then u is nontrivial, i.e., u ¤ 0 whenever ¤ 0. (Amick assumes  to be of class C 3 . Recently, this result has been extended to Lipschitz domains [68].) These last results rule out the Stokes paradox for the nonlinear case for symmetric domains and homogeneous boundary data. A clear exposition of Amick’s results, as well as the results outlined above, can be found in [22]. For an exterior domain

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

287

condition (100) has been replaced in [66] by the weaker assumption that the sum P jFi j is sufficiently small. An interesting approach to the existence of solutions i

to (98)–(99) with D 0 and small data has been recently proposed by M. Hillairet and P. Wittwer [28]. Finally, in the recent paper [61] mentioned by the authors, the problem (98), (99) with D 0 was considered in exterior plane domains symmetric with respect to both coordinate axes and a solution was found in the class of vector fields v 2 C0 satisfying the following symmetry conditions: v1 .x1 ; x2 / D v1 .x1 ; x2 / D v1 .x1 ; x2 /; v2 .x1 ; x2 / D v2 .x1 ; x2 / D v2 .x1 ; x2 /:

(111)

It is proved in [61] that if data h, f 2 C0 satisfy only natural regularity assumptions, then (98) has a D-solution in C0 which converges uniformly to zero at infinity. The flux of the boundary value h over @ in this case is arbitrary. All abovementioned results (except [61]) were proved either under the condition that all fluxes Fi are equal to zero (see (100)) or assuming that fluxes Fi are “small.” N another relevant problem in the theory of the Besides the relation between e1 and u, stationary Navier-Stokes equations is to ascertain whether a solution to problem (98) exists without any restriction on the fluxes Fi . For exterior plane domains, this problem, in general, is unsolved until now (solutions of the problem for bounded plane and 3D axially symmetric domains as well as for 3D axially symmetric exterior domains were discussed above in Sects. 3 and 4). In the last paper [40] it is proved for arbitrary fluxes Fi the existence of a Dsolution to problem (98) for exterior plane domains in the case when only Amick’s symmetric conditions (109)–(110) are satisfied and every i intersects the x1 – axis, i.e., \ i (112) Ox1 ¤ ; for all i D 1; : : : ; N:

5.2

Formulation of the Main Result

Theorem 11 ([40]). Let   R2 be a symmetric exterior domain (97), (110), (110), (112) with multiply connected Lipschitz boundary @ consisting of N disjoint components j , j D 1; : : : ; N . Assume that f is a symmetric .in the sense ofR (109)/ distribution such that the corresponding linear functional H ./ 3  7! f   

is continuous .with respect to the norm k  kH ./ / and h is a symmetric field in W 1=2;2 .@/. Then problem (98) admits at least one symmetric weak solution u. The following estimate   kruk2L2 ./  c khk2W 1=2;2 .@/ C khk4W 1=2;2 .@/ C kfk2 is valid.

(113)

288

M.V. Korobkov et al.

Here the total flux FD

Z h.x/  n.x/ dS D

N X

Fi

(114)

i D1

@

is not required to be zero or small. By what was said before, if f has a compact support, then the solution converges uniformly at infinity to a constant vector ˛e1 ; moreover, for ˛ ¤ 0, it behaves at large distance according to (107), (108). However, it is not known whether this solution satisfies the condition at infinity (99). The proof of Theorem 11 is based on Leray’s method of invading domains. The needed a priori estimate is obtained using the special extension of the boundary value satisfying the Leray-Hopf inequality (cf. with (9)) which is obtained by applying a new inequality of Poincaré type (see Lemma 16) that could be useful also in other contexts.

5.3

Some Estimates for Plane Functions with Finite Dirichlet Integral

Lemma 14. Let  be the exterior domain (97), v 2 D./. Then the following inequality Z 

jv.x/j2 dx  c jxj2 log2 jxj

Z

jrv.x/j2 dx

(115)



holds. Inequality (115) is well known (e.g., [47]). As it follows from (115), functions v 2 D./ do not have to vanish at infinity. The next assertion gives some information about the behavior of a function of D./ as jxj ! 1. Lemma 15. Let  be the exterior domain (97), v 2 D./. Then 1 lim sup r!1 log r

Z2 0

jv.r; /j2 d  2

Z

jrv.x/j2 dx:

(116)



Inequality (116) is proved in [26] (see Lemma 2.1). Lemma 16 ([40]). Let  be the exterior domain (97), v 2 D./,  > 0, ˛ 2 .0; 1/, R R0 > 1. Then the following inequality

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow ˛ jx Z 1j

Z Rn.R ; R /

jv.x1 ; x2 /j2 dx1 dx2  c jxj2

0

Z

jrv.x/j2 dx

289

(117)



holds. The constant c in (117) depends only on R0 ; , and ˛.

5.4

Construction of the Extension of the Boundary Value

Below only the construction of the extension of the boundary value is given. Other details of the proof can be found in [40]. Let 2 C 1 .R/ be a nonnegative function such that 0  .t /  1,  .t / D

1; t 1; 0; t  0;

and  2 C 1 .R/ be a monotone function on RC with .t/ 0 > 0,  .t/ D

jt j˛ ; jt j 3R0 ; 1; jt j  2R0 ;

where ˛ 2 .0; 1/. Let C D fx 2  W x2 > 0g and  D fx 2  W x2 < 0g. Set   C .x/ D x2 .x1 / C .1  .x1 //ı.x/ ;

x 2 C ;

where  2 C 1 .R/ is a monotone function with 8 < 1; jt j 2R0 ; .t / D 3 : 0; jt j  R0 ; 2 and ı.x/ is the regularized distance from the point x 2  to @ D

N S

@j . Notice

j D1

that ı.x/ is infinitely differentiable function in R2 n@ and the following inequalities a1 d .x/  ı.x/  a2 d .x/; jD ˛ ı.x/j  a3 d 1j˛j .x/ hold. Here d .x/ D dist.x; @/ is the Euclidean distance from x to @ (see [74]). Let " 2 .0; 1/ be an arbitrary number. In the domain C , define the cutoff function    ".x1 / C .x; "/ D " ln : C .x/

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M.V. Korobkov et al.

Obviously, ( C .x; "/ D

0; ".x1 / < C .x/; 1 1; C .x/ < "e  " .x1 /:

Define b.x/ D

1 1 r ln jxj D 2 2



x 1 x2 ; jxj2 jxj2

 :

The vector field b.x/ satisfies the symmetry conditions (109). Moreover, it is well known that the flux of b.x/ over a closed curve  is equal to 1, Z b.x/  n.x/ d  D 1; 

if and only if the domain bounded by  contains the point x D 0. Here n is unit vector of outward (with respect to the domain bounded by  ) normal to  . Otherwise the flux is equal  to zero.  .j / .j / Let x D x1 ; 0 2 j ; j D 1; : : : ; N . Put   b.j / .x/ D Fj b x  x .j / : Then Z

b.j / .x/  n.x/ dS D Fj ;

j

Z

b.j / .x/  n.x/ dS D 0; i ¤ j:

i

In the domain C , the functions b.j / .x/ could be represented in the form b.j / .x/ D

Fj ? .j / r 'C .x/; 2

.j /

.j /

'C .x/ D arctg

x 1  x1 ; x 2 C ; j D 1; : : : ; N; x2

 @ @ .j / . Notice that j'C .x/j  =2 for x 2 C and j D ; where r D @x2 @x1 1; : : : ; N . Define 

?

.j /

 Fj ?  .j / r C .x; "/'C .x/ 2  Fj  .j / .j / 'C .x/r ? C .x; "/ C C .x; "/r ? 'C .x/ : D 2

BC .x; "/ D

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

291

.j /

Then div BC .x; "/ D 0, and, since C .x; "/ D 1 in the neighborhood of @C , it follows that ˇ ˇ .j / BC .x; "/ˇ

@C

Fj ? .j / ˇˇ r 'C .x/ˇ : @C 2

D

Lemma 17. Let j D 1; : : : ; N . Then for every ı > 0, there exists " D ".ı/ such that the following inequality Z ˇZ  ˇ  ˇ ˇ .j / jru.x/j2 dx u.x/  r u.x/  BC .x; "/ dx ˇ  ı ˇ C

8 u 2 HS ./

(118)

C

holds. HS ./ is the subspace of functions from H ./ satisfying the symmetry conditions (109). The proof of the Leray-Hopf inequality (118) is based on Lemmas 14–16 and is true only for functions u in HS ./, i.e., satisfying the symmetry conditions (109). For an arbitrary function u 2 H ./, this inequality can be wrong. Define 8   .j / .j / ˆ BC;1 .x1 ; x2 ; "/; BC;2 .x1 ; x2 ; "/ ; x 2 C;" ; < B.j / .x; "/D ˆ : B .j / .x ; x ; "/; B .j / .x ; x ; "/; x 2  ; C;1

1

2

C;2

1

2

;"

and B.x; "/ D

N P

B.j / .x; "/:

j D1

The vector field B is symmetric, Z divB D 0;

B  ndS D Fj ; j D 1; : : : ; N:

j

ˇ Let h1 .x/ D h.x/  B.x; "/ˇ@ . Then Z h1 .x/  n.x/dS D 0; j

j D 1; : : : ; N:

(119)

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M.V. Korobkov et al.

If h 2 W 1=2;2 .@/, then obviously h1 2 W 1=2;2 .@/ and   ˇ kh1 kW 1=2;2 .@/  c khkW 1=2;2 .@/ C kBˇ@ kW 1=2;2 .@/ h

 c khkW 1=2;2 .@/ C

N X

Fj2

1=2 i

 ckhkW 1=2;2 .@/ :

j D1

Because of condition (119), there exists a function H 2 H ./ such that supp H.x; "/ is contained in a small neighborhood of the boundary @, divH D 0;

H.x; "/j@ D h1 .x/;

H 2 L4 ./; rH 2 L2 ./;

kHkL4 ./ C krHkL2 ./  ckh1 kW 1=2;2 .@/  ckhkW 1=2;2 .@/ : Moreover, H.x; "/ satisfies Leray-Hopf’s inequality, i.e., for every ı > 0, there exists " D ".ı/ such that Z ˇZ  ˇ  ˇ ˇ u.x/  r u.x/  H.x; "/ dx ˇ  ı ju.x/j2 dx ˇ 

8 u 2 H ./



holds (see [47]). The function H.x; "/ is not necessarily symmetric. However, its boundary value is symmetric and, therefore, H.x; "/ can be symmetrized defining the function e "/ as follows: H.x;

 Q 1 .x; "/ D 1 H1 .x1 ; x2 ; "/ C H1 .x1 ; x2 ; "/ ; H 2

 1 Q 2 .x; "/ D H2 .x1 ; x2 ; "/  H2 .x1 ; x2 ; "/ : H 2 Put e "/: A.x; "/ D B.x; "/ C H.x; Lemma 18.

(i) The vector field A.x; "/ is symmetric, div A.x; "/ D 0;

ˇ A.x; "/ˇ@ D h.x/;

(ii) A 2 L4 ./, rA 2 L2 ./, kAkL4 ./ C krAkL2 ./  ckhkW 1=2;2 .@/ :

(120)

5 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

293

(iii) For every ı > 0, there exists " D ".ı/ such that the inequality Z ˇ ˇZ   ˇ ˇ u  r u  A dx ˇ  ı jruj2 dx ˇ 

8 u 2 HS ./



holds. The constant c in (120) depends on " and tends to infinity as " ! 0. This inequality is used with sufficiently small but fixed ". Remark 7. If the domain  and the data are symmetric with respect to both coordinate axes, the existence of a weak solution which also satisfies symmetry conditions (111) can be proved. In this case the solution satisfies the condition at infinity (99) with D 0: lim v.jxj; / D 0

jxj!1

uniformly in , i.e., lim

.x1 ;x2 /!1

v.x1 ; x2 / D 0

(see [61]).

6

Conclusion

The first paper devoted to the existence of solutions to the stationary NavierStokes problem without smallness assumptions on data was that of J. Leray [49], under the sole hypothesis that the fluxes through any connected component of the boundary vanish. The question whether this condition could be removed was by then a fundamental open problem in the mathematical theory of incompressible fluid dynamics and was the object of researches of several outstanding mathematicians. A comprehensive account of attempts devoted to give an answer to this question is contained in the book of G.P. Galdi [21]. Recently, the problem has been solved for (i) two-dimensional bounded domains [36, 41]; (ii) two-dimensional exterior axially symmetric domains and symmetric data [40, 61]; and (iii) three-dimensional bounded and exterior axially symmetric domains and symmetric data [37, 39, 41]. However, it remains much to do in order to get a complete picture of the flow of an incompressible fluid under adherence boundary conditions. Among the still open problems of particular interest are the following: (i0 ) to remove the symmetry assumptions required in [37, 39, 41]; (ii0 ) to determine the behavior at infinity of the solutions found in [39, 40], also under symmetry assumptions; and (iii0 ) to prove or

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disprove the Liouville theorem in the class of D-solutions vanishing at infinity in the three-dimensional case [21, 42] (see also [33]).

7

Cross-References

 Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows  Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions  Steady-State Navier-Stokes Flow Around a Moving Body Acknowledgements The research of K. Pileckas leading to these results has received funding from the Lithuanian-Swiss Cooperation Programme to reduce economic and social disparities within the enlarged European Union under project agreement No. CH-3-SMM-01/01. The research of M. V. Korobkov was partially supported by the Russian Foundation for Basic Research (Grant No. 14-01-00768-a), by the Grant of the Russian Federation for the State Support of Researches (Agreement No. 14.B25.31.0029), and by the Dynasty Foundation.

References 1. S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Commun. Pure Appl. Math. 17, 35–92 (1964) 2. Ch.J. Amick, Existence of solutions to the nonhomogeneous steady Navier–Stokes equations. Indiana Univ. Math. J. 33, 817–830 (1984) 3. C.J. Amick, On Leray’s problem of steady Navier–Stokes flow past a body in the plane. Acta Math. 161, 71–130 (1988) 4. K.I. Babenko, On stationary solutions of the problem of flow past a body. Mat. Sb. 91, 3–27 (1973). English translation: math. SSSR Sbornik. 20, 1–25 (1973) 5. W. Borchers, K. Pileckas, Note on the flux problem for stationary Navier–Stokes equations in domains with multiply connected boundary. Acta Appl. Math. 37, 21–30 (1994) 6. J. Bourgain, M.V. Korobkov, J. Kristensen, On the Morse–Sard property and level sets of Sobolev and BV functions. Rev. Mat. Iberoam. 29(1), 1–23 (2013) 7. J. Bourgain, M.V. Korobkov, J. Kristensen, On the Morse–Sard property and level sets of W n;1 Sobolev functions on Rn . Journal fur die reine und angewandte Mathematik (Crelles Journal) 2015(700), 93–112 (2015). https://doi.org/10.1515/crelle-2013-0002 8. I.-D. Chang, R. Finn, On the solutions of a class of equations occurring in continuum mechanics with applications to the Stokes paradox. Arch. Ration. Mech. Anal. 7, 388–401 (1961) 9. M. Chipot, K. Kaukalyt˙e, K. Pileckas, W. Xue, On nonhomogeneous boundary value problems for the stationary Navier–Stokes equations in 2D symmetric semi-infinite outlets. Anal. Appl. (2015, to appear). https://doi.org/10.1142/S0219530515500268 10. R.R. Coifman, J.L. Lions, Y. Meier, S. Semmes, Compensated compactness and Hardy spaces. J. Math. Pures App. IX Sér. 72, 247–286 (1993) 11. D.C. Clark, The vorticity at infinity for solutions of the stationary Navier–Stokes equations in exterior domains. Indiana Univ. Math. J. 20, 633–654 (1971) 12. J.R. Dorronsoro, Differentiability properties of functions with bounded variation. Indiana U. Math. J. 38(4), 1027–1045 (1989) 13. L.C. Evans, R.F. Gariepy, in Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics (CRC Press, Boca Raton, 1992)

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14. R. Farwig, H. Morimoto, Leray’s inequality for fluid flow in symmetric multi-connected twodimensional domains. Tokyo J. Math. 35, 63–70 (2012) 15. R. Farwig, H. Kozono, T. Yanagisawa, Leray’s inequality in general multi-connected domains in Rn . Math. Ann. 354, 137–145 (2012) 16. R. Finn, On the steady-state solutions of the Navier–Stokes equations. III. Acta Math. 105, 197–244 (1961) 17. R. Finn, D.R. Smith, On the stationary solution of the Navier–Stokes equations in two dimensions. Arch. Ration. Mech. Anal. 25, 26–39 (1967) 18. H. Fujita, On the existence and regularity of the steady-state solutions of the Navier-Stokes theorem. J. Fac. Sci. Univ. Tokyo Sect. I. 9, 59–102 (1961) 19. H. Fujita, On stationary solutions to Navier–Stokes equation in symmetric plane domain under general outflow condition. Pitman research notes in mathematics, in Proceedings of International Conference on Navier–Stokes Equations, Varenna. Theory and Numerical Methods, vol. 388 (1997), pp. 16–30 20. G.P. Galdi, On the existence of steady motions of a viscous flow with non–homogeneous conditions. Le Matematiche 66, 503–524 (1991) 21. G.P. Galdi, in An Introduction to the Mathematical Theory of the Navier–Stokes Equations, ed. by C. Truesdell, vol I, II revised edition. Springer Tracts in Natural Philosophy, vol. 38, 39 (Springer, New York, 1998) 22. G.P. Galdi, Stationary Navier–Stokes problem in a two-dimensional exterior domain, in Handbook of Differential Equations, Stationary Partial Differential Equations, ed. by M. Chipot, P. Quittner, vol. 1 (Elsevier, 2003) 23. G.P. Galdi, C.G. Simader, Existence, uniqueness and Lq estimates for the Stokes problem in an exterior domain. Arch. Ration. Mech. Anal. 112, 291–318 (1990) 24. G.P. Galdi, H. Sohr, On the asymptotic structure of plane steady flow of a viscous fluid in exterior domains. Arch. Ration. Mach. Anal. 131, 101–119 (1995) 25. D. Gilbarg, H.F. Weinberger, Asymptotic properties of Leray’s solution of the stationary twodimensional Navier–Stokes equations. Russ. Math. Surv. 29, 109–123 (1974) 26. D. Gilbarg, H.F. Weinberger, Asymptotic properties of steady plane solutions of the Navier– Stokes equations with bounded Dirichlet integral. Ann. Scuola Norm. Pisa (4) 5, 381–404 (1978) 27. J.G. Heywood, On the impossibility, in some cases, of the Leray-Hopf condition for energy estimates. J. Math. Fluid. Mech. 13, 449–457 (2011) 28. M. Hillairet, P. Wittawer, On the existence of solutions to the planar Navier–Stokes system. J. Differ. Equ 255(10), 2996–3019 (2013) 29. E. Hopf, Ein allgemeiner Endlichkeitssatz der Hydrodynamik. Math. Ann. 117, 764–775 (1941) 30. L.V. Kapitanskii, K. Pileckas, On spaces of solenoidal vector fields and boundary value problems for the Navier–Stokes equations in domains with noncompact boundaries. Trudy Mat. Inst. Steklov. 159, 5–36 (1983). English Transl.: Proc. Math. Inst. Steklov. 159, 3–34 (1984) 31. K. Kaulakyt˙e, On nonhomogeneous boundary value problem for the steady Navier–Stokes system in domain with paraboloidal and layer type outlets to infinity. Topol. Methods Nonlinear Anal. 46(2), 835–865 (2015) 32. K. Kaulakyt˙e, K. Pileckas, On the nonhomogeneous boundary value problem for the Navier– Stokes system in a class of unbounded domains. J. Math. Fluid Mech. 14(4), 693–716 (2012) 33. G. Koch, N. Nadirashvili, G. Seregin, V. Sverak, Liouville theorems for the Navier–Stokes equations and applications. Acta Math. 203(1), 83–105 (2009) 34. M.V. Korobkov, Bernoulli law under minimal smoothness assumptions. Dokl. Math. 83, 107– 110 (2011) 35. M.V. Korobkov, J. Kristensen, On the Morse-Sard theorem for the sharp case of Sobolev mappings. Indiana Univ. Math. J. 63(6), 1703–1724 (2014). https://doi.org/10.1512/iumj.2014. 63.5431

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36. M.V. Korobkov, K. Pileckas, R. Russo, On the flux problem in the theory of steady Navier– Stokes equations with nonhomogeneous boundary conditions. Arch. Ration. Mech. Anal. 207(1), 185–213 (2013). https://doi.org/10.1007/s00205-012-0563-y 37. M.V. Korobkov, K. Pileckas, R. Russo, Steady Navier–Stokes system with nonhomogeneous boundary conditions in the axially symmetric case. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 14(1), 233–262 (2015). https://doi.org/10.2422/2036-2145.201204_003 38. M.V. Korobkov, K. Pileckas, R. Russo, Steady Navier–Stokes system with nonhomogeneous boundary conditions in the axially symmetric case. Comptes rendus – Mécanique 340, 115–119 (2012) 39. M.V. Korobkov, K. Pileckas, R. Russo, The existence theorem for the steady Navier–Stokes problem in exterior axially symmetric 3D domains (2014). arXiv: 1403.6921, http://arxiv.org/ abs/1403.6921 40. M.V. Korobkov, K. Pileckas, R. Russo, The existence of a solution with finite Dirichlet integral for the steady Navier–Stokes equations in a plane exterior symmetric domain. J. Math. Pures Appl. 101(3), 257–274 (2014). https://doi.org/10.1016/j.matpur.2013.06.002 41. M.V. Korobkov, K. Pileckas, R. Russo, Solution of Leray’s problem for stationary NavierStokes equations in plane and axially symmetric spatial domains. Ann. Math. 181(2), 769–807 (2015). https://doi.org/10.4007/annals.2015.181.2.7 42. M.V. Korobkov, K. Pileckas, R. Russo, The Loiuville theorem for the steady-state Navier– Stokes problem for axially symmetric 3D solutions in absence os swirl. J. Math. Fluid Mech. 17(2), 287–293 (2015) 43. M.V. Korobkov, K. Pileckas, V.V. Pukhnachev, R. Russo, The flux problem for the Navier– Stokes equations. Russ. Math. Surv. 69(6), 1065–1122 (2014). https://doi.org/10.1070/ RM2014v069n06ABEH004928 44. H. Kozono, T. Janagisawa, Leray’s problem on the Navier–Stokes equations with nonhomogeneous boundary data. Math. Zeitschrift. 262, 27–39 (2009) 45. A.S. Kronrod, On functions of two variables. Uspechi Matem. Nauk (N.S.) 5, 24–134 (1950, in Russian) 46. O.A. Ladyzhenskaya, Investigation of the Navier–Stokes equations in the case of stationary motion of an incompressible fluid. Uspekhi Mat. Nauk. 3, 75–97 (1959, in Russian) 47. O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible flow (Gordon and Breach, New York, 1969) 48. E.M. Landis, Second Order Equations of Elliptic and Parabolic Type (Nauka, Moscow, 1971, in Russian) 49. J. Leray, Étude de diverses équations intégrales non linéaire et de quelques problèmes que pose l’hydrodynamique. J. Math. Pures Appl. 12, 1–82 (1933) 50. J. Malý, D. Swanson, W.P. Ziemer, The Coarea formula for Sobolev mappings. Trans. AMS 355(2), 477–492 (2002) 51. R.L. Moore, Concerning triods in the plane and the junction points of plane continua. Proc. Nat. Acad. Sci. U.S.A. 14(1), 85–88 (1928) 52. H. Morimoto, A remark on the existence of 2D steady Navier–Stokes flow in bounded symmetric domain under general outflow condition. J. Math. Fluid Mech. 9(3), 411–418 (2007) 53. H. Morimoto, H. Fujita, A remark on the existence of steady Navier–Stokes flows in 2D semiinfinite channel involving the general outflow condition. Mathematica Bohemica 126(2), 457– 468 (2001) 54. H. Morimoto, H. Fujita, A remark on the existence of steady Navier–Stokes flows in a certain two-dimensional infinite channel. Tokyo J. Math. 25(2), 307–321 (2002) 55. H. Morimoto, H. Fujita, Stationary Navier–Stokes flow in 2-dimensional Y-shape channel under general outflow condition, in The Navier–Stokes Equations: Theory and Numerical Methods. Lecture Note in Pure and Applied Mathematics (Morimoto Hiroko, Other), vol. 223 (Marcel Decker, New York, 2002), pp. 65–72 56. H. Morimoto, Stationary Navier–Stokes flow in 2-D channels involving the general outflow condition, in Handbook of Differential Equations: Stationary Partial Differential Equations, ed. by M. Chipot, vol. 4, Ch. 5 (Elsevier, Amsterdam/London, 2007), pp. 299–353

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57. S.A. Nazarov, K. Pileckas, On the solvability of the Stokes and Navier–Stokes problems in domains that are layer-like at infinity. J. Math. Fluid Mech. 1(1), 78–116 (1999) 58. S.A. Nazarov, K. Pileckas, On steady Stokes and Navier–Stokes problems with zero velocity at infinity in a three-dimensional exterior domains. J. Math. Kyoto Univ. 40, 475–492 (2000) 59. J. Neustupa, On the steady Navier–Stokes boundary value problem in an unbounded 2D domain with arbitrary fluxes through the components of the boundary. Ann. Univ. Ferrara. 55(2), 353–365 (2009) 60. J. Neustupa, A new approach to the existence of weak solutions of the steady Navier-Stokes system with inhomoheneous boundary data in domains with noncompact boundaries. Arch. Ration. Mech. Anal. 198(1), 331–348 (2010) 61. K. Pileckas, R. Russo, On the existence of vanishing at infinity symmetric solutions to the plane stationary exterior Navier–Stokes problem. Math. Ann. 343, 643–658 (2012) 62. C.R. Pittman, An elementary proof of the triod theorem. Proc. Am. Math. Soc. 25(4), 919 (1970) 63. V.V. Pukhnachev, Viscous flows in domains with a multiply connected boundary, in New Directions in Mathematical Fluid Mechanics. The Alexander V. Kazhikhov Memorial Volume, ed. by A.V. Fursikov, G.P. Galdi, V.V. Pukhnachev (Birkhauser, Basel/Boston/Berlin, 2009), pp. 333–348 64. V.V. Pukhnachev, The Leray problem and the Yudovich hypothesis. Izv. vuzov. Sev.-Kavk. region. Natural Sciences. The Special Issue “Actual Problems of Mathematical Hydrodynamics” (2009, in Russian), pp. 185–194 65. R. Russo, On the existence of solutions to the stationary Navier–Stokes equations. Ricerche Mat. 52, 285–348 (2003) 66. A. Russo, A note on the two-dimensional steady-state Navier–Stokes problem. J. Math. Fluid Mech. 52, 407–414 (2009) 67. R. Russo, On Stokes’ problem, in Advances in Mathematica Fluid Mechanics, ed. by R. Rannacher, A. Sequeira (Springer, Berlin/Heidelberg, 2010), pp. 473–511 68. A. Russo, On symmetric Leray solutions to the stationary Navier–Stokes equations. Ricerche Mat. 60, 151–176 (2011) 69. A. Russo, G. Starita, On the existence of solutions to the stationary Navier–Stokes equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 7, 171–180 (2008) 70. L.I. Sazonov, On the existence of a stationary symmetric solution of the two-dimensional fluid flow problem. Mat. Zametki. 54(6), 138–141 (1993, in Russian). English Transl.: Math. Notes. 54(6), 1280–1283 (1993) 71. V.I. Sazonov, Asymptotic behavior of the solution to the two-dimensional stationary problem of a flow past a body far from it. Math. Zametki. 65, 202–253 (1999, in Russian). English transl.: Math. Notes. 65, 202–207 (1999) 72. D.R. Smith, Estimates at infinity for stationary solutions of the Navier–Stokes equations in two dimensions. Arch. Ration. Mech. Anal. 20, 341–372 (1965) 73. V.A. Solonnikov, General boundary value problems for Douglis-Nirenberg elliptic systems, I. Izv. Akad. Nauk SSSR, Ser. Mat. 28, 665–706 (1964); II, Trudy Mat. Inst. Steklov. 92, 233–297 (1966). English Transl.: I, Amer. Math. Soc. Transl. 56(2), 192–232 (1966); II, Proc. Steklov Inst. Math. 92, 269–333 (1966) 74. E.M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton University Press, Princeton, 1970) 75. V. Šverák, T-P. Tsai, On the spatial decay of 3-D steady-state Navier–Stokes flows. Commun. Partial Differ. Equ. 25, 2107–2117 (2000) 76. A. Takeshita, A remark on Leray’s inequality. Pac. J. Math. 157, 151–158 (1993) 77. I.I. Vorovich, V.I. Yudovich, Stationary flows of a viscous incompres-sible fluid. Mat. Sbornik. 53, 393–428 (1961, in Russian)

6

Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions Toshiaki Hishida

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Asymptotic Structure of the Stokes Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Existence of Flows in L3,1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Asymptotic Structure of the Navier-Stokes Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

300 304 309 319 324 334 336 336

Abstract

Consider the stationary Navier-Stokes flow in 3D exterior domains with zero velocity at infinity. What is of particular interest is the spatial behavior of the flow at infinity, especially optimal decay (summability) observed in general and the asymptotic structure. When the obstacle is translating, the answer is found in some classic literature by Finn; in fact, the optimal summability is Lq with q > 2 and the leading profile is the Oseen fundamental solution. This presentation is devoted to the other cases developed in the last decade, mainly the case where the obstacle is at rest, together with several remarks even on the challenging case where the obstacle is rotating. The optimal summability for those cases is L3;1 (weak-L3 ) and the leading term of small solutions being in this class is the homogeneous Navier-Stokes flow of degree .1/, which is called the Landau solution. In any case, the total net force is closely related to the asymptotic structure of the flow. An insight into the homogeneous Navier-Stokes flow of

T. Hishida () Graduate School of Mathematics, Nagoya University, Nagoya, Japan e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_6

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degree .1/, due to Šverák, plays an important role. It would be also worthwhile finding a class of the external force, as large as possible, which ensures the asymptotic expansion of the flow at infinity.

1

Introduction

Let  be an exterior domain in R3 occupied by a viscous incompressible fluid, where a compact set R3 n  is identified with an obstacle (rigid body) and the boundary @ of  is assumed to be sufficiently smooth. Given external force f D .f1 .x/; f2 .x/; f3 .x//> , the stationary motion of the fluid is described by the velocity u D .u1 .x/; u2 .x/; u3 .x//> and pressure p D p.x/ which obey the Navier-Stokes system  u C rp C .u    !  x/  ru C !  u D f;

div u D 0

in 

(1)

subject to the boundary conditions uDC!x u!0

on @;

(2)

as jxj ! 1;

(3)

where (2) is the usual no-slip condition in which the flow attains the rigid motion in the sense of trace, while (3) is understood from the class of solutions, mostly either pointwise or summability. All vectors are throughout column ones and ./> denotes the transpose of vectors or matrices. To understand the Eq. (1) of momentum, one should start with nonstationary Navier-Stokes system in a time-dependent region exterior to a moving body and make a suitable change of variables to reduce the problem to an equivalent one in the reference frame attached to the moving body (see Galdi [26] for the details). The translational velocity  and angular velocity ! are then in general time dependent; however, both are assumed to be constant vectors, and the flow is assumed to be stationary in the reference frame. The main issue one would like to address here is the spatial behavior of solutions at infinity. It turns out that the rate of decay (3) is controlled by the total net force (momentum flux) M D M .; !; f / Z Z D ŒT .u; p/  u ˝ .u    !  y/  .!  y/ ˝ u  d y C f .y/ dy @



(4) associated with (1), which is written as the divergence form  div ŒT .u; p/  u ˝ .u    !  x/  .!  x/ ˝ u D f;

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301

where T .u; p/ D ru C .ru/>  pI

(5)

is the Cauchy stress tensor, I is the 3  3 identity matrix, and  stands for the outward unit normal to @. One has nonzero force M ¤ 0 in general; however, if in particular u, p, and ru decay sufficiently fast at infinity, then integrating the equation above over fx 2 I jxj < g and letting  ! 1 yield M D 0. This simple observation suggests that M is related to the decay structure of the flow. One may also refer to [47, section 6] in the context of the self-propelled motion of a rigid body. In his celebrated paper [61], Leray showed the existence of at least one solution in the class of finite Dirichlet integral ru 2 L2 ./ to (1), (2) and (3) (when  D ! D 0) without any smallness condition on the data. The argument relies on compactness together with a priori estimate arising from structure of the NavierStokes system. Note that this structure is kept for the case  C !  x ¤ 0 as well. His theorem thus provides even large solutions, most of which would be unstable. From the viewpoint of stability, solutions of the Leray class do not give us enough information about the asymptotic behavior at infinity. In fact, the only thing one knows is u 2 L6 ./; however, mathematical analysis developed so far requires better decay property such as ju.x/j  C jxj1 or u 2 L3;1 ./ (as well as smallness) of the stationary flow u to show its stability, where L3;1 denotes the weak-L3 space (see [6, 7, 31, 38–40, 44, 46, 49, 55, 63, 65, 70], and the references therein). When  D 0, the summability L3;1 of stationary flows observed in general is actually optimal unless assuming any specific condition such as symmetry. As compared with this case, better summability of stationary flows for the case  2 R3 n f0g mentioned in the next paragraph is helpful in the proof of stability of such flows (under smallness conditions); indeed, there is no need to analyze the full linearized operator; in other words, analysis of the Oseen semigroup is enough, while that is not the case when  D 0 (unless using an interpolation technique due to [76]). The interest is focused on optimal decay/summability at infinity of the flow together with its asymptotic structure. This was addressed in a series of papers by Finn [22–24], in which the rigid body was assumed to be translating with velocity  2 R3 n f0g; however, ! D 0. In this case the essential step is to analyze the asymptotic behavior of solutions at infinity to the Oseen system  u C rp    ru D f;

div u D 0;

in :

(6)

The Oseen fundamental solution possesses anisotropic decay structure with paraboloidal wake region behind the body. In fact, the flow decays faster outside wake than inside, and consequently the summability near infinity is better like u 2 Lq with q > 2 than the case  D 0. Finn [23], Farwig [15], and Shibata [70] proved the existence of small Navier-Stokes flow which exhibits the same decay structure with wake as mentioned above; actually, the leading profile of the flow is the Oseen fundamental solution, and its coefficient is given by the force M .; 0; f /

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(see (4)), provided f is of bounded support. Such a flow was called physically reasonable solution by Finn. Furthermore, Babenko [1], Galdi [25], and Farwig and Sohr [21] showed that any solution of the Leray class without restriction on the magnitude becomes a physically reasonable solution (see Galdi [28, Theorem X.8.1]). This is a contrast to the case  D 0; indeed, when the translation of the body is absent, one has no result on the asymptotic behavior of large solutions of the Leray class except for Choe and Jin [13], in which some pointwise decay rates of axisymmetric solutions of that class were deduced. Later on, Galdi and Silvestre [33], Galdi and Kyed [29], and Kyed [57–59] generalized the results mentioned above for purely translational regime to the case   ! ¤ 0. In fact, the presence of translation still implies fine decay/summability at infinity of the flow past a rotating obstacle except the case where ! 2 R3 n f0g is orthogonal to , and, as a consequence, the leading profile of the Navier-Stokes flow is described in terms of the linear part, in which a remarkable role of axis of rotation can be also observed. Such a role was discovered first by Farwig and Hishida [18, 19] for the flow around a purely rotating obstacle, and it will be explained later. This presentation studies the other case where the translation of the body is absent ( D 0/. In this case the existence of solutions, which decay like ju.x/j  C jxj1 ;

jru.x/j  C jxj2

.jxj ! 1/

or u 2 L3;1 ./;

ru 2 L3=2;1 ./;

for small external forces was proved by [7, 56, 69] (! D 0) and by [17, 27] (! ¤ 0). For the case ! D 0, Deuring and Galdi [14] clarified that the leading profile was no longer the Stokes fundamental solution. Indeed, the rate jxj1 of decay yields the balance between the linear part and nonlinearity since u  u  ru  jxj3 (formally). This observation would suggest that a sort of nonlinear effect is involved in the leading term of the flow. Nazarov and Pileckas [68] first derived asymptotic expansion under smallness conditions, but the leading term was less explicit. It was explicitly found much later by Korolev and Šverák [50] (case ! D 0). When the nonlinearity is balanced with the linear part, it is reasonable to expect that the self-similar solution would be a candidate of the leading term. Since the stationary Navier-Stokes system ( D ! D 0) is invariant under the scale transformation u .x/ D u. x/;

p .x/ D 2 p. x/;

> 0;

(7)

in R3 n f0g

(8)

a smooth solution fu; pg to  u C rp C u  ru D 0;

div u D 0

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

303

is called (stationary) self-similar solution if u .x/ D u.x/;

8 > 0 8x 2 R3 n f0g

p .x/ D p.x/;

or, equivalently,   1 x ; u.x/ D u jxj jxj

1 p.x/ D p jxj2



 x ; jxj

8x 2 R3 n f0g;

(9)

that is, u and p are homogeneous of degree .1/ and .2/, respectively. Landau [60] derived its exact form under the assumption of axisymmetry (see (78)), in order to describe jets from a thin pipe (see also Tian and Xin [75] and Cannone and Karch [9]). Finally, Šverák [74] has characterized completely the set S of all self-similar solutions as follows: S is parameterized by vectorial parameter as n o S D fUb ; Pb gI b 2 R3

(10)

whose member fUb ; Pb g is symmetric about the axis Rb and satisfies  Ub C rPb C Ub  rUb D bı;

div Ub D 0

in D0 .R3 /

(11)

across the origin (see also [2, 9]), where ı denotes the Dirac measure. In other words, every self-similar solution must have its own axis of symmetry, and the set S eventually agrees with the family of solutions computed by Landau. This is the reason why the self-similar solution is often called the Landau solution. The proof of Šverák [74] is closely related to geometric properties of S2 (unit sphere). Based on his profound insight, he and Korolev [50] proved that the leading term of asymptotic expansion of solutions to (1) with .; !; f / D .0; 0; 0/ (without assuming (2)), which decay like jxj1 at infinity, is given by the specific Landau solution UM with label M D M .0; 0; 0/ (see (4)), provided lim supjxj!1 jxjju.x/j is small enough, where the error term satisfies the pointwise estimate like jxj2C" for " > 0 arbitrarily small (but the smallness of u depends on "). The result was extended to small time-periodic solutions (case  D ! D 0) with period T > 0 by Kang, Miura, and Tsai [48], where the leading term is the Landau solution RT Ub with label b D T1 0 M , that is, the time average of the force (4). Note that the Landau solution must be useful to describe the local behavior related to singularity/regularity as well; indeed, it was proved by Miura and Tsai [64] that the leading term of point singularity like jxj1 at x D 0 of the Navier-Stokes flow is also given by a Landau solution provided it is small enough. Later on, Farwig and Hishida [19] studied the case where the body is purely rotating with angular velocity ! 2 R3 n f0g and proved that the leading term of solutions to (1) (with  D 0, f D 0), which are small in L3;1 ./, is another ! ! Landau solution Ub with label b D . j!j  M / j!j , whose axis of symmetry is

304

T. Hishida

parallel to the axis of rotation along which the flow is largely concentrated, where M D M .0; !; 0/ (see (4)). The solution enjoys better summability if and only if !  M D 0, while so does the solution for the case where the body is at rest if and only if the full force M vanishes. Thus, one is able to find out the effect of rotation, which was not clear until the study of [18, 19]. They considered solutions in L3;1 ./ rather than pointwise decay properties, and the error term was estimated in terms of summability. Their result was then refined by Farwig, Galdi, and Kyed [16] in the sense that the asymptotic expansion with error term satisfying a pointwise estimate (as in [50]) was deduced even for solutions of the Leray class with the energy inequality under appropriate smallness of !. Most part of this presentation is devoted to the case ! D 0, but key points for the purely rotating case ! 2 R3 nf0g and the remarkable difference between those cases are also explained. One specifies a class of the external force f , which ensures the asymptotic expansion of the Navier-Stokes flow u 2 L3;1 ./ as long as it is small enough. Since the class of f is rather large, it seems difficult to deduce pointwise estimates of the error term; instead, it is estimated in terms of summability as in [19]. Before the analysis of the Navier-Stokes flow, it is also worthwhile showing the asymptotic expansion of the Stokes flow with general external force as above (see Theorem 1). This presentation consists of six sections. After some preliminaries in the next section, asymptotic structure for the linearized system is studied in sect. 3. At the end of sect. 3, a few crucial facts which interpret why the axis of rotation is preferred for the case ! 2 R3 n f0g are mentioned. In sect. 4 the existence theorem (Theorem 2) for small solutions in L3;1 ./ is provided. It is based on the linear theory (see Theorem 3). The result is due to Kozono and Yamazaki [56], but one carries out linear analysis in a different way, which can be applied to the other cases  C !  x ¤ 0 (see [17, 71]). The asymptotic expansion of the Navier-Stokes flow obtained in Theorem 2 is studied in sect. 5 (see Theorem 4), in which one needs a bit more decay property of the external force than assumed in Theorem 2. The final section summarizes what is done and raises several open questions about the related issues. Although this is a survey article, the complete proof of Theorems 1, 3, and 4 will be presented.

2

Preliminaries

In this section some function spaces are introduced and notation is fixed. Let B be the open ball in R3 centered at the origin with radius  > 0. For sufficiently large  > 0, we set  D  \ B , where  is the exterior domain under consideration. Let D be a smooth domain in R3 , such as the exterior domain , whole space 3 R , or a bounded domain. By C01 .D/ one denotes the class of smooth functions with compact support in D. For 1  q  1 the usual Lebesgue spaces are denoted by Lq .D/ with norm k  kq;D . To introduce the Lorentz space (for details, see Bergh and Löfström [3]), given measurable function f on D, set

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

mf .t / WD jfx 2 DI jf .x/j > t gj ;

305

t > 0;

where j  j stands for the Lebesgue measure. Then mf ./ is monotonically nonincreasing, right continuous, and measurable. It is well known that f 2 Lq .D/, 1  q < 1, if and only if Z

1

˚

t mf .t /1=q

0

q dt < 1: t

With this in mind, one denotes by Lq;r .D/ the vector space consisting of all measurable functions f on D which satisfy Z

1

˚

t mf .t /1=q

0

r dt t

1=r 0

Note that Lq;r0 .D/  Lq;r1 .D/ if r0  r1 I

Lq;q .D/ D Lq .D/;

the latter of which is obvious as mentioned above. Each of finite quantities (12) is a quasi-norm; however, by the use of the average function, it is possible to introduce a norm k  kq;r;D , which is equivalent to that, unless q D 1 (see [3]). Then Lq;r .D/ equipped with k  kq;r;D (1 < q < 1; 1  r  1) is a Banach space, called the Lorentz space; in particular, Lq;1 .D/ is well known as the weak-Lq space, in which C01 .D/ is not dense. As a typical function in this space, recall that jxj˛ 2 L3=˛;1 .R3 / as long as 0 < ˛  3. One also has the weak Hölder inequality ([7, Lemma 2.1]): let 1 < p  1; 1 < q < 1 and 1 < r < 1 satisfy 1=r D 1=p C 1=q, and let f 2 Lp;1 .D/, g 2 Lq;1 .D/, then fg 2 Lr;1 .D/ with kfgkr;1;D  kf kp;1;D kgkq;1;D

(13)

where L1;1 .D/ D L1 .D/. Let 1 < q < 1 and 1  r  1. The Lorentz spaces can be also constructed via real interpolation   Lq;r .D/ D L1 .D/; L1 .D/ 11=q;r :

(14)

This together with the reiteration theorem in the interpolation theory ([3, 5.3.1]) implies that   Lq;r .D/ D Lq0 ;r0 .D/; Lq1 ;r1 .D/ ;r

(15)

306

T. Hishida

provided 1 < q0 < q < q1 < 1;

1=q D .1  /=q0 C =q1 ;

1  r0 ; r1 ; r  1:

Then one knows

kf kq;r;D  C kf k1

q0 ;r0 ;D kf kq1 ;r1 ;D

(16)

for all f 2 Lq0 ;r0 .D/ \ Lq1 ;r1 .D/  Lq;r .D/. For fixed f 2 Lp;1 .D/, the map g 7! fg is bounded from Lq;1 .D/ to Lr;1 .D/ by (13), where 1 < p  1, 1 < q < 1, and 1 < r < 1 satisfy 1=r D 1=pC1=q. Hence, the interpolation (15) leads to kfgkr;s;D  kf kp;1;D kgkq;s;D

(17)

for f 2 Lp;1 .D/ and g 2 Lq;s .D/, where p; q; r are the same as above and 1  s  1. For 1 < q < 1;

1 < r  1;

1=q 0 C 1=q D 1;

1=r 0 C 1=r D 1;

(18)

the duality relation 0

0

Lq;r .D/ D Lq ;r .D/ 0

holds, in particular, Lq;1 .D/ D Lq ;1 .D/ . In what follows, the same symbols for vector and scalar function spaces are adopted as long as there is no confusion. The abbreviations k  kq D k  kq; and k  kq;r D k  kq;r; are used for the exterior domain  under consideration. P q1 .D/ be the One needs the homogeneous Sobolev space. For 1 < q < 1, let H completion of C01 .D/ with respect to the norm kr./kq;D . For D D R3 one has P q1 .R3 / D fu 2 Lq .R3 /I ru 2 Lq .R3 /g=R: H loc When 1 < q < 3, one may take the canonical representative elements to adopt P q1 .R3 / D fu 2 Lq .R3 /I ru 2 Lq .R3 /g; H where 1=q D 1=q  1=3, together with the embedding estimate kukq ;R3  C krukq;R3 : Let 1 < q0 < q < q1 < 1;

1=q D .1  /=q0 C =q1 ;

1  r  1;

(19)

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

307

and define   1 P q1 .D/; H P q1 .D/ P q;r .D/ D H H 0 1

;r

;

which is independent of the choice of fq0 ; q1 g, with norm kr./kq;r;D . Note that 1 P q;r C01 .D/ is dense in H .D/ unless r D 1. For D D R3 the embedding relations ([56]) 1 P q;r .R3 / ,! Lq ;r .R3 /; H

kukq ;r;R3  C krukq;r;R3 ;

1 P 3;1 .R3 / ,! L1 .R3 / \ C .R3 /; H

(20)

kuk1;R3  C kruk3;1;R3 ;

hold provided 1 < q < 3, 1=q D 1=q  1=3 and 1  r  1. Let 1 < q < 3, 1=q D 1=q  1=3 and 1  r  1. Let   R3 be the exterior domain. For every u 2 L1loc ./ satisfying ru 2 Lq;r ./, there is a constant k D k.u/ such that u C k 2 Lq ;r ./ with ku C kkq ;r  C krukq;r where C > 0 independent of u (see [7, Theorem 5.9]). By taking the canonical 1 P q;r ./, one has the characterization ([34, 52, 56]) representative element of u 2 H 1 P q;r ./ D fu 2 Lq ;r ./I ru 2 Lq;r ./; uj@ D 0g; H

(21)

kukq ;r  C krukq;r :

(22)

together with

P 1 ./ ,! One can also take the canonical representative element of u 2 H 3;1 1 L ./ \ C ./, which goes to zero for jxj ! 1 and satisfies uj@ D 0 as well as kuk1  C kruk3;1 :

(23)

1 P q;r For fq; rg satisfying (18), the space H .D/ is defined as the dual space of 1 1 P q .D/ D H P q;q .D/. The duality theorem for interpolation and set H spaces ([3, 3.7.1]) implies that

P 10 0 .D/, H q ;r

  1 P q1 .D/; H P q1 .D/ P q;r .D/ D H H 0 1

(24)

;r

for q; q0 ; q1 ; r satisfying (19) with 1 < r  1. For r D 1, one also defines P 1 .D/, 1 < q < 1, as the dual space of the completion of C 1 .D/ with H q;1  0  O 10 .D/ ¤ H P 10 .D/ . respect to the norm kr./kq 0 ;1;D , which is denoted by H q ;1

q ;1

Then (24) holds for r D 1 as well (see [3, p.55]). Let 1 < q < 1 and 1  r  1,

308

T. Hishida

1 P q;r then there exists a constant C > 0 such that for every f 2 H .D/, one can take q;r F 2 L .D/ satisfying

div F D f;

kF kq;r;D  C kf kHP q;r 1 .D/ I

see [51, Lemma 2.2] and [56, Lemma 2.2]. Let D  R3 be a bounded domain. Then Lp;r .D/  Lq;s .D/ for all 1 < q < p < 1; r; s 2 Œ1; 1: Both embeddings 1 1 P q;r P q;r .D/ ,! Lq;r .D/ ,! H .D/ H

(25)

are compact, where 1 < q < 1 and 1  r  1 (see [3, 3.14.8]). For the same fq; rg, one also has 1 P q;r .D/ D fu 2 Lq;r .D/I ru 2 Lq;r .D/; uj@D D 0g; H

together with the Poincaré inequality kukq;r;D  C krukq;r;D :

(26)

Finally, consider the boundary value problem for the equation of continuity div w D f

in D;

wD0

on @D;

where D is a bounded domain in R3 with Lipschitz boundary @D. Given R f being in a suitable class, say f 2 Lq .D/, with compatibility condition D f D 0, there are a lot of solutions, some of which were found by many authors (see Galdi [28, Notes for Chapter III]). Among them a particular solution discovered by Bogovskii [4] is useful to recover the solenoidal condition in a cutoff procedure on account of some fine properties of his solution. By the following lemma, the operator f 7! his solution w (called the Bogovskii operator) is well defined, and its properties are summarized. For the proof, see Borchers and Sohr [8], Galdi [28], as well as Bogovskii [4]. Lemma 1. Let D  R3 be a bounded domain with Lipschitz boundary. Then there is a linear operator B W C01 .D/ ! C01 .D/3 such that, for 1 < q < 1 and k  0 integer, kr kC1 Bf kq;D  C kr k f kq;D with some C D C .D; q; k/ > 0 and that Z div Bf D f

f .x/dx D 0;

if D

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

309

where the constant C is invariant with respect to dilation of the domain D. P qk .D/ to By continuity, B is extended uniquely to a bounded operator from H P qk .D/ is the completion of C 1 .D/ with respect to the norm P qkC1 .D/3 , where H H 0 k kr ./kq;D . Furthermore, by real interpolation, it is extended uniquely to a bounded k kC1 P q;r P q;r .D/ to H .D/3 , where 1  r  1 and operator from H   k P qk .D/; H P qk .D/ P q;r .D/ D H H 0 1

;r

with q0 ; q1 and satisfying (19).

3

Asymptotic Structure of the Stokes Flow

Let us start with asymptotic structure of the simplest case, that is, the exterior Stokes flow without external force  u C rp D 0;

div u D 0;

in ;

(27)

where nothing is imposed at the boundary @. Note that the result below does not depend on the boundary condition on @. Since (27) admits polynomial solutions, it is reasonable to impose a growth condition, for instance, u.x/ D o.jxj/

as jxj ! 1; q;r

or

u=.1 C jxj/ 2 L ./

or

ru 2 Lq;r ./

for some q 2 .1; 1/; r 2 Œ1; 1;

(28)

for some q 2 .1; 1/; r 2 Œ1; 1;

to exclude polynomials except constants. Note that the growth condition on the pressure is not needed here since it is controlled through the Eq. (27) by the velocity (but that is not the case in Theorem 1 below; see Remark 1). Then the asymptotic structure is described in terms of the Stokes fundamental solution E.x/ D

1 8



1 x˝x IC jxj jxj3

 ;

Q.x/ D

x ; 4jxj3

(29)

to be precise (Chang and Finn [11]), for every solution to (27) subject to (28), there are constants u1 2 R3 and p1 2 R such that Z

T .u; p/ d  C O.jxj2 /;

u.x/ D u1 C E.x/ @

Z

p.x/ D p1 C Q.x/  @

(30) T .u; p/ d  C O.jxj3 /;

310

T. Hishida

as jxj ! 1. For the proof, there are two methods. One is to employ a potential representation formula (as in, for instance, [18]), and the other is a cutoff technique for reduction to the whole space problem. This section takes the latter way since it works for the Navier-Stokes system as well (see sect. 5). It will be also clarified which condition on the external force f ensures that the Stokes fundamental solution is still the leading profile at infinity for  u C rp D f;

div u D 0

in :

(31)

For simplicity, let us consider smooth solution to (31) for smooth external force. If f .x/ D O.jxj3 / or f D div F with F .x/ D O.jxj2 /, it is formally balanced with the second derivative of E.x/ and thus the situation would be delicate (the equality .jxj1 log jxj/ D jxj3 suggests that one could not expect even the rate jxj1 of decay for the former case). In order to make this point clear, one needs the following lemma on asymptotic structure of the volume potentials E f and Q f . Lemma 2. Let f 2 C 1 .R3 / and assume that there are constants ˛ > 3 and C > 0 such that jf .x/j 

C .1 C jxj/˛

8x 2 R3 :

Then 8 3 < ˛ < 4; < O.jxj˛C2j /; j j f .y/ dy C O.jxj2j log jxj/; ˛ D 4; r .E f /.x/ D r E.x/ : R3 O.jxj2j /; ˛ > 4; 8 Z 3 < ˛ < 4; < O.jxj˛C1 /; 3 .Q f /.x/ D Q.x/  f .y/ dy C O.jxj log jxj/; ˛ D 4; (32) : R3 O.jxj3 /; ˛ > 4; Z

as jxj ! 1, where j D 0; 1. Proof. One can split the error term as Z

f.r j E/.x  y/  r j E.x/gf .y/ dy R3

Z

f.r j E/.x  y/  r j E.x/gf .y/ dy  r j E.x/

D jyj2jxj

Z f .y/ dy jyjjxj=2

.r j E/.x  y/f .y/ dy

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

311

It is easy to see that jI2 j C jI4 j  C jxj˛C2j and that jI3 j  C jxj˛  C jxj˛

Z Z

jyj2jxj

dy jx  yj1Cj dy D C jxj˛C2j : jx  yj1Cj

jyxj3jxj

One also finds Z

Z

1

jI1 j  C jyj 0 such that jf0 .x/j 

C .1 C jxj/˛

8x 2 :

(36)

For every smooth solution of class u; p; ru 2 Lsloc ./ to (31) subject to (28), there is a constant u1 2 R3 such that u.x/ D u1 C E.x/M0 C v0 .x/ C v1 .x/;

(37)

for jxj  3R with Z



 T .u; p/ C F  d  C

M0 D @

Z f0 .y/ dy; 

8 3 < ˛ < 4; < O.jxj˛C2j /; j r v0 .x/ D O.jxj2j log jxj/; ˛ D 4; : O.jxj2j /; ˛ > 4;

(38)

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

313

(j D 0; 1) as jxj ! 1 and v1 2 Ls . n B3R /;

rv1 2 Ls . n B3R /;

where s 2 .3=2; 3 is defined by 1=s D 1=s  1=3. Here, R > 0 is taken such that R3 n   BR . If in particular F D 0, then v1 is absent from (37). Proof. First of all, note that the boundary integral of M0 can be understood as 1=s h.T .u; p/ C F /; 1i@ in the sense of normal trace .T .u; p/ C F / 2 Hs .@/ s 1 since T .u; p/ C F 2 Lloc ./ and div .T .u; p/ C F / D f0 2 L ./  Lsloc ./. Let us reduce the problem to the one with vanishing flux at the boundary @. Without loss one may assume 0 2 int .R3 n /. We introduce the flux carrier Z

1 ; z.x/ D ˇ r 4jxj

ˇD

  u d;

(39)

@

for given solution fu; pg. Since Z   z d  D ˇ;

div z D 0;

in R3 n f0g;

z D 0

(40)

@

the pair fQu; pg with uQ D u  z fulfills also (31) subject to Z   uQ d  D 0:

(41)

@

By using a cutoff function 2 C01 .B3R I Œ0; 1/;

.x/ D 1

.jxj  2R/;

kr k1 

C R

(42)

and the Bogovskii operator B (Lemma 1) in the domain AR D fx 2 R3 I R < jxj < 3Rg;

(43)

one sets v D .1 

/Qu C BŒQu  r ;

D .1 

/p;

(44)

3 where the Bogovskii term R is understood as its zero extension to the whole space R . It should be noted that AR uQ  r dx D 0 follows from (41). Then the pair fv; g obeys

 v C r D g C .1  /f0 C div ..1 

/F /;

div v D 0

in R3

(45)

314

T. Hishida

for some function g 2 C01 .AR /. Here, one does not need any exact form of g and what is important is structure of the Eq. (45). When either u.x/ D o.jxj/ or u=.1 C jxj/ 2 Lq;r ./, it is obvious that v 2 S 0 .R3 /. Under the alternative assumption ru 2 Lq;r ./ in (28), one has rv 2 S 0 .R3 /, which implies v 2 S 0 .R3 / ([12, Proposition 1.2.1]). Going back to the Eq. (45) yields r 2 S 0 .R3 / and, therefore, 2 S 0 .R3 / by the same reasoning. Let P s1 .R3 / ,! Ls .R3 /; v1 2 H

1 2 Ls .R3 /;

P s1 .R3 / obtained be the solution to (33) with the external force div ..1  /F / 2 H in Lemma 3. We then find

v.x/ D E fg C .1  /f0 g .x/ C v1 .x/ C Pv .x/; (46)

.x/ D Q fg C .1  /f0 g .x/ C 1 .x/ C P .x/; with some polynomials Pv and P ; however, from (28) it follows that Pv must be a constant vector, which is denoted by u1 . Thus, one concludes from Lemma 2 that u.x/ can be represented as u.x/ D uQ .x/ C z.x/ D E.x/M0 C v0 .x/ C v1 .x/ C u1 with

.jxj  3R/

Z M0 D

fg C .1 

/f0 g.y/ dy;

R3

where r j v0 .x/ behaves like the remainder of (32) since r j z.x/ D O.jxj2j / (see (39)). Let  > 3R. From Z



rz C .rz/>

jyjD

one can deduce Z fg C .1 

y 

d D

ˇ 4

Z y d D 0

(47)

jyjD

Z /f0 g.y/ dy D 

B

div fT .v; / C .1  Z

/F g dy

B

y d  jyjD Z Z .T .u; p/ C F / d  C f0 .y/ dy: D

D

.T .u; p/ C F /

@



Letting  ! 1 leads to Z

Z

M0 D

.T .u; p/ C F / d  C @

which concludes (37). This completes the proof.

f0 .y/ dy; 

t u

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

315

Remark 1. Because of less information about div F , one cannot say anything about the polynomial P in (46) unless assuming the behavior of p at infinity. If in particular F D 0 so that 1 D 0, then P must be a constant. In fact, one knows that both .E h/ and r.Q h/ belong to Lr .R3 / for every r 2 .1; 1/ because so does h WD g C .1  /f0 . By going back to (31), one finds that P is a constant, which we denote by p1 . As a consequence, 8 3 < ˛ < 4; < O.jxj˛C1 /; p.x/ D p1 C Q.x/  M0 C O.jxj3 log jxj/; ˛ D 4; : O.jxj3 /; ˛ > 4;

(48)

as jxj ! 1. Theorem 1 immediately implies the following corollary. Corollary 1. In addition to the assumptions of Theorem 1 with s D 3=2, suppose either u 2 L3 ./ or ru 2 L3=2 ./. Then M0 D 0.

Remark 2. For the Stokes system in n-dimensional exterior domains, either u 2 Ln=.n2/ ./ or ru 2 Ln=.n1/ ./ yields M0 D 0 (under suitable assumptions on the external force). Consider (31) subject to uj@ D 0. Corollary 1 then tells us that the condition P1 u 2 H 3=2;1 ./ is an optimal class observed in general even if f D div F with F 2 C01 ./. The following corollary claims the uniqueness of solutions in this class. 3=2;1 P1 Corollary 2. Let fu; pg 2 H ./ be a solution to (27) subject to 3=2;1 ./  L uj@ D 0. Then fu; pg D f0; 0g.

Proof. Theorem 1 with f D 0 can be applied. Since u 2 L3;1 ./, one has (37) with u1 D 0 as well as v1 D 0. One also knows (48) with p1 D 0 (or even more directly, the same procedure as in the proof of Theorem 1 with use of p 2 L3=2;1 ./ leads to the same expansion). As a consequence, u.x/ D O.jxj1 /;

fru.x/; p.x/g D O.jxj2 /

(49)

as jxj ! 1. Let 2 C 1 .Œ0; 1/I Œ0; 1/ satisfy .t/ D 1 for 0  t  1 and

.t/ D 0 for t  2, and set  .x/ D .jxj=/ for  > 0 large enough and x 2 R3 . Since the local regularity theory for the Stokes boundary value problem together with a bootstrap argument yields fu; pg 2 H 2 .2 /  H 1 .2 /, one can multiply (27) by  u to obtain

316

T. Hishida

Z

jruj2  dx C 

Z

Z .ru  r  /  u dx   2jyj:

(57)

Proof. A brief sketch will be presented here. Let us take the Taylor formula (with respect to y) G.O.at /x  y; t / D G.x; t / C G.x; t /

.O.at /x/  y C .remainder/ 2t

and consider each term multiplied by O.at /> . To get (57), the point is rapid decay due to oscillation ˇZ ˇ ˇ ˇ

0

1



cos at sin at



ˇ ˇ C G.x; t / dt ˇˇ  jajjxj3

and a part of non-oscillating terms comes to the leading profile ˆ.x/. The term H .O.at /x  y; t / can be discussed similarly although the computation is more complicated. The details are found in [18, section 4]. The others (54), (55) and (56) are much easier (see [18, (2.11)] and [47, (6.23)]). t u By the same splitting of the volume potential as in the proof of Lemma 2, one can make use of Lemma 4 to conclude the following asymptotic expansion. In the proof, the dominant term is I1 for the region jyj < jxj=2, in which (57) is employed.

318

T. Hishida

Lemma 5. Let ! D ae3 with a 2 R n f0g. Suppose f satisfies the same condition (with ˛ > 3) as in Lemma 2. Then Z u.x/ D

.x; y/f .y/ dy R3

enjoys 8 3 < ˛ < 4; < O.jxj˛C2 /; 2 f .y/ dy E.x/e3 C O.jxj log jxj/; ˛ D 4; u.x/ D e3  : R3 ˛ > 4; O.jxj2 /; 



Z

(58)

as jxj ! 1, where E.x/ is the Stokes fundamental solution (29). By means of harmonic analytic method developed by [20], it is possible to deduce exactly the same well-posedness for (53) as in Lemma 3, where the constants in (34) and (35) are independent of a 2 R n f0g. It was done by [43, Theorem 2.1], [17, Proposition 3.2]. By using this together with Lemma 5, one can follow the proof of Theorem 1 (with a bit more care of the flux carrier, see [47, section 6]) under the same assumptions on the external force to obtain the asymptotic expansion of solutions to (51), in which the leading term is given by  V .x/ WD

 ! !  M0 E.x/ ; j!j j!j

! D e3 ; j!j

(59)

where Z M0 D

Z ŒT .u; p/ C u ˝ .!  y/  .!  y/ ˝ u C F  d y C

@

f0 .y/ dy: 

Since e3  .e3  y/ D 0, the third term of the boundary integral does not contribute to the coefficient e3  M0 . This was proved under the no-slip condition uj@ D !  x by Farwig and Hishida [18], who deduced not only the leading term but also the second one although they restricted their consideration to the simple case where f D div F with F 2 C01 ./. The leading term (59) satisfies V C r… D .e3  M0 /e3 ı;

div V D 0

in D0 .R3 /, where ….x/ D .e3  M0 /x3 =.4jxj3 /. But the pair fV; …g enjoys  V C r…  .!  x/  rV C !  V D .e3  M0 /e3 ı;

div V D 0

(60)

as well, since .e3  x/  rV  e3  V D 0:

(61)

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

319

Note that (61) holds for all vector fields which are symmetric about Re3 (x3 -axis). In fact, because such vector fields must be of the form V D .W .r; x3 / cos ; W .r; x3 / sin ; V3 .r; x3 //> in cylindrical coordinates r; ; x3 , one finds .e3  x/  rV D @ V D e3  V . As long as V .x/  1=jxj near the origin, (61) holds also in D0 .R3 /, that is, hV ˝ .e3  x/  .e3  x/ ˝ V; r i D 0 for all 2 C01 .R3 /. Hence, the leading term (59) satisfies (60) in D0 .R3 /.

4

Existence of Flows in L3,1

Consider the problem (1) and (2) with  D ! D 0, that is,

 u C rp C u  ru D f; uD0

div u D 0

in ;

on @:

(62) (63)

Before proceeding to study of asymptotic structure of the Navier-Stokes flow, one should establish the existence of (small) solutions having optimal asymptotic behavior at infinity. The existence of solutions decaying like jxj1 was proved by Finn [23], Galdi and Simader [35], Novotny and Padula [69], and Borchers and Miyakawa [7]. Indeed, such a pointwise estimate is fine, but one may take another way by the use of function spaces, so that the proof becomes easier. Among some function spaces which are able to catch homogeneous functions of degree .1/, the weak-L3 space is probably the simplest one. Actually, Kozono and Yamazaki [56] succeeded in showing the existence of a unique Navier-Stokes flow in L3;1 ./ whenever the external force is small in a sense. Since the next section studies the asymptotic structure of solutions in L3;1 ./, one will provide the existence theorem due to [56]. It was known ([7, 53, 54]) that either u 2 L3 ./ or ru 2 L3=2 ./ necessarily yields M .0; 0; f / D 0 (see (4)), although those Lebesgue spaces are invariant under the scale transformation (7). Hence, one has no chance to find the NavierStokes flow belonging to L3 ./ in generic situation. This is a nonlinear counter part of Corollary 1 (for the Stokes flow) and can be also interpreted by asymptotic expansion in the next section. For the Stokes boundary value problem (31) subject to uj@ D 0, one has the P q1 ./ if and only if well-posedness in the class u 2 H  3  n D < q < 3 .D n: space dimension/; n1 2

(64)

320

T. Hishida

see Borchers and Miyakawa [5], Galdi and Simader [34], and Kozono and Sohr [51, 52]. To be precise, the condition q > 3=2 is necessary for solvability and it is consistent with Corollary 1, while for uniqueness one needs q < 3; in fact, the proof P 1 ./ because the constant u1 cannot of Corollary 2 does not work when u 2 H 3 be excluded in (37). Kozono and Yamazaki [56] clarified that all things for both Stokes and Navier-Stokes systems work well if replacing L3 ./ (resp. L3=2 ./) by L3;1 ./ (resp. L3=2;1 ./) for u (resp. ru). When n  4, the Lq -theory is enough to construct (small) Navier-Stokes flow u 2 Ln ./ with ru 2 Ln=2 ./ because n < n2 < n in this case. n1 The existence theorem due to [56] now reads as follows. P 1 ./ with Theorem 2. There is a constant  > 0 such that for every f 2 H 3=2;1 kf kHP 1 ./ < , problem (62) and (63) admits a unique solution 3=2;1

1 P 3=2;1 ./ ,! L3;1 ./; u2H

p 2 L3=2;1 ./;

kfru; pgk3=2;1 C kuk3;1  C kf kHP 1

; 3=2;1 ./

(65)

in the sense that hru; r'i  hp; div 'i  hu ˝ u; r'i D hf; 'i P 1 ./), where h; i stands for holds for all ' 2 C01 ./ (and, therefore, all ' 2 H 3;1 duality pairings. Remark 4. It is an open question whether the small solution fu; pg constructed 3=2;1 P1 in Theorem 2 is unique in the class H ./ without assuming 3=2;1 ./  L smallness, in other words, whether fu; pg coincides with other large solutions fv; qg in this class. The difficulty seems to stem from unremovable singularity like 1=jx  x0 j of the velocity being in L3;1 ./. If such a singular behavior is ruled out for large solutions v by assuming additionally v 2 L3 ./ C L1 ./, then the answer is affirmative (provided kuk3;1 is small enough that is accomplished by (65) when kf kHP 1 ./ is still smaller). This interesting uniqueness criterion was proved 3=2;1 by Nakatsuka [67] (see also [66]). Once the following linear theory is established, it is straightforward by using a simple contraction argument with the aid of (13) to show Theorem 2 (whose proof may be omitted). P 1 ./, problem (31) subject to (63) admits a Theorem 3. For every f 2 H 3=2;1 unique solution 1 P 3=2;1 u2H ./ ,! L3;1 ./;

p 2 L3=2;1 ./;

kfru; pgk3=2;1 C kuk3;1  C kf kHP 1

3=2;1 ./

;

(66)

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

321

in the sense that hru; r'i  hp; div 'i D hf; 'i P 1 ./), where h; i stands for holds for all ' 2 C01 ./ (and, therefore, all ' 2 H 3;1 duality pairings. 1 1 P q;r P q;r The well-posedness in the class H ./  Lq;r ./ for every f 2 H ./ was established first by Konozo and Yamazaki [56] when fq; rg satisfies

fq; rg D f3=2; 1gI

fq; rg 2 .3=2; 3/  Œ1; 1I

fq; rg D f3; 1g;

which is a generalization of (64) (case q D r). Indeed Theorem 3 is just one of those cases, but it is the most important case to solve the nonlinear problem. Later on, Shibata and Yamazaki [71] proved the well-posedness not only in the class above but in the sum of function spaces 1 1 P q;r P 3=2;1 u2H ./ C H ./;

p 2 Lq;r ./ C L3=2;1 ./

even for the other cases fq; rg 2 .1; 3=2/  Œ1; 1I

q D 3=2 and r 2 Œ1; 1/:

This result suggests that ru and p do not decay faster than jxj2 in general. In [71] they discussed the Oseen system (6) as well as the Stokes system to study the relation between solutions to (1) with  ¤ 0 and  D 0 (i.e., the behavior for the limit  ! 0). The well-posedness in the class above for (51) was proved by Farwig and Hishida [17] when the obstacle is purely rotating. It was generalized by Heck, Kim, and Kozono [37] when taking both translation and rotation of the obstacle into account. As a result, one has Theorem 2 for the Navier-Stokes boundary value problem (1) and (2) even if ! ¤ 0 provided the data ( C !  x in (2) as well as f ) are small enough; in fact, the case   ! D 0 is reduced to [17], while the other case   ! ¤ 0 is reduced to [37] by using the Mozzi-Chasles transform ([32], [28, Chapter VIII]). The pointwise estimate like ju.x/j  C jxj1 for (1) and (2) with ! ¤ 0 was successfully deduced by Galdi [27] and Galdi and Silvestre [32]. For the proof of Theorem 3, a sort of duality argument was employed in [56], but this way is not taken here; instead, a parametrix is constructed as in [71]. The latter method was also adopted in [17] and [37] since the argument of [56] does not seem to work because of lack of homogeneity of the equation with  C !  x ¤ 0. Also, one cannot use any continuity argument since C01 ./ is not dense in L3=2;1 ./. P 1 ./, one intends to construct directly a solution with the use Given f 2 H 3=2;1 of solutions in the whole space (Lemma 3) and in a bounded domain (Lemma 6 below). Let D be a bounded domain in R3 with smooth boundary @D, and consider the boundary value problem

322

T. Hishida

 u C rp D f;

div u D 0

uD0

in D;

on @D:

(67) (68)

The following lemma is due to Cattabriga [10], Solonnikov [73], Kozono and Sohr [51], and Kozono and Yamazaki [56]. In (69) below, the Poincaré inequality (26) is involved. 1 P q;r .D/, problem (67) Lemma 6. Let 1 < q < 1 and 1  r  1. For every f 2 H and (68) admits a solution 1 P q;r .D/; u2H

p 2 Lq;r .D/; (69)

kfru; u; p  pgkq;r;D  C kf kHP 1 ; q;r.D/

with p WD

1 jDj

R D

p dx, in the sense that hru; r'i  hp; div 'i D hf; 'i

holds for all ' 2 C01 .D/, where h; i stands for duality pairings. The solution is unique up to an additive constant for p. One is in a position to show Theorem 3. Proof of Theorem 3. Since the uniqueness is already known by Corollary 2, one will show the existence part. Fix R > 0 so large that R3 n   BR5 . Take functions

; 1 2 C 1 .R3 I Œ0; 1/ satisfying 

.x/ D

1; jxj  R  3; 0; jxj  R  2;



1 .x/ D

0; jxj  R  5; 1; jxj  R  4;

P 1 ./, it is easily and set A D fx 2 R3 I R  4 < jxj < R  1g. Given f 2 H 3=2;1 seen that 1 P 3=2;1 f 2H .R /; 1 P 3=2;1

1 f 2 H .R3 /;

kf kHP 1

3=2;1 .R /

k 1 f kHP 1

 kf kHP 1

3 3=2;1 .R /

3=2;1 ./

; (70)

 C kf kHP 1

; 3=2;1 ./

for the latter of which (20)2 is used. Consider (67) and (68) with f in the bounded domain D D R , and let fu0 ; p0 g be the solution obtained in Lemma 6 subject to R p dx D 0. Consider also (33) with f replaced by 1 f , and let fu1 ; p1 g be the 0 R solution obtained in Lemma 3. Set ‰f WD .1  /u1 C u0 C BŒ.u1  u0 /  r ; …f WD .1  /p1 C p0 ;

(71)

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

323

where B is the Bogovskii operator (see Lemma 1) in the bounded domain A. Since R .u u 1 0 /r dx D 0, one has div ‰f D 0. Then it follows from (34), (35), (69), A and (21), Lemma 1, and (70) that 1 P 3=2;1 .‰f; …f / 2 H ./  L3=2;1 ./;

(72)

kfr‰f; …f gk3=2;1 C k‰f k3;1  C kf kHP 1

3=2;1 ./

and that .‰f; …f / is a solution to  ‰f C r…f D f C Rf;

div ‰f D 0;

‰f j@ D 0

(73)

where Rf D 2r  r.u1  u0 / C . /.u1  u0 /  BŒ.u1  u0 /  r   .r /.p1  p0 /; P 1 .R / and satisfies which is in L3=2;1 .R / ,! H 3=2;1 kRf k3=2;1;R  C kf kHP 1

3=2;1 ./

For every

:

(74)

2 C01 ./, one finds jhRf; ij  kRf k3=2;1;R k k3;1;R

which combined with k k3;1;R  C k k1  C kr k3;1 (see (23)) implies that P 1 ./ with Rf 2 H 3=2;1 kRf kHP 1

3=2;1 ./

 C kRf k3=2;1;R :

Actually, one has even kRf kHP 1

3=2;1 ./

 C kRf kHP 1

3=2;1 .R /

:

(75)

In fact, with the use of a fixed function ' 2 C01 .R / with '.x/ D 1 .x 2 A/, one observes jhRf; ij D jh'Rf; ij  kRf kHP 1

3=2;1 .R /

 C kRf kHP 1

3=2;1 .R /

k' kHP 1

3;1 .R /

kr k3;1

for every 2 C01 ./, owing to (23) as well as Rf D 0 outside A. This implies (75).

324

T. Hishida

P 1 ./ ! H P 1 ./ is a compact operator. Now it turns out that R W H 3=2;1 3=2;1 P 1 ./, and then by (74) In fact, suppose ffj g is a bounded sequence in H 3=2;1 the sequence fRfj g is bounded in L3=2;1 .R / and, therefore, converges in P 1 .R / along a subsequence on account of the compact embedding (25). Then H 3=2;1 P 1 ./ by virtue of (75). it is also convergent in H 3=2;1 P 1 ./ fulfills .1 C One next shows that 1 C R is injective. Suppose f 2 H 3=2;1 P 1 ./. Since f D Rf 2 L3=2;1 .R / which vanishes outside R/f D 0 in H 3=2;1 A, one has f D 0 in  n A. It thus suffices to show that f D 0 in A. In view of (72) and (73), it follows from Corollary 2 that f‰f; …f g D f0; 0g. Hence, by (71) one observes fu1 ; p1 g D f0; 0g;

jxj  R  1I

fu0 ; p0 g D f0; 0g;

jxj  R  4

which shows that both fu1 ; p1 g and fu0 ; p0 g can be regarded as solutions to v C r D f;

div v D 0;

in BR I

vj@BR D 0

3=2;1 P1 and belong to H .BR /. It follows from uniqueness assertion of 3=2;1 .BR /  L Lemma 6 that u1 D u0 and that p1 D p0 C c for some constant c. One goes back to R (71) to see that 0 D …f D .1  /.p0 C c/ C p0 ; however, the side condition R p0 dx D 0 yields c D 0, so that p1 D p0 . After all, one finds fu1 ; p1 g D f0; 0g, yielding f D 0 in A. By the Fredholm alternative, 1 CR is bijective, and, therefore, the pair

u D ‰.1 C R/1 f;

p D ….1 C R/1 f

provides the desired solution, which enjoys (66) on account of (72). The proof is complete. u t

5

Asymptotic Structure of the Navier-Stokes Flow

This section is devoted to a precise look at the profile of solutions for jxj ! 1 obtained in Theorem 2. In order to make essential points clear, it would be better to avoid things caused by less local regularity of solutions. In what follows, let us consider smooth solutions which are of class fu; pg 2 L3;1 ./  L3=2;1 ./. If the solution enjoyed slightly faster decay property than described in Theorem 2, such as uru D O.jxj˛ / with ˛ > 3 or u˝u 2 Ls ./ with s  3=2, then one could regard the nonlinear term as the external force and use Theorem 1 to see that the leading profile would be still the Stokes fundamental solution. But that is not the case here. As mentioned in sect. 1, the balance between the linear part and nonlinearity implies that the leading term of asymptotic expansion would be a self-similar solution.

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

325

Given b 2 R3 n f0g, Landau [60] (see also [9, 75]) found a nontrivial exact solution to (8), which satisfies axial symmetry about Rb as well as homogeneity (9). Set x D jxj ;  D .1 ; 2 ; 3 /T 2 S2 (unit sphere). When b is parallel to e3 , the Landau solution is of the form U .x/ D



c3  1 1 2  C e 3 ; jxj .c  3 /2 c  3

(76)

4 .c3  1/ P .x/ D jxj2 .c  3 /2 with parameter c 2 .1; 1/ [ .1; 1/, and it satisfies U C rP C U  rU D ke3 ı;

div U D 0

in D0 .R3 /

(see (11)), where k is given by k D k.c/ D

  8c cC1 2 2 2 C 6c :  3c.c  1/ log 3.c 2  1/ c1

(77)

This calculation was done by Cannone and Karch [9, Proposition 2.1] (see also Batchelor [2, p.209]). The function k./ is monotonically decreasing on each of intervals .1; 1/ and .1; 1/ and fulfills k.c/ ! 0

.jcj ! 1/I

k.c/ ! 1

.c ! 1/I

k.c/ ! 1

.c ! 1/:

Hence, for every b 2 R3 nf0g parallel to e3 , there is a unique c 2 .1; 1/[.1; 1/ such that k.c/e3 D b. Since the Navier-Stokes system (8) is rotationally invariant, the Landau solution fUb ; Pb g for general b 2 R3 n f0g is given by rotation of (76). b D e3 . Then one finds Let O 2 R33 be an orthogonal matrix that fulfills O jbj

2 1 b c.O /3  1 C Ub .x/ D ; jxj fc  .O /3 g2 c  .O /3 jbj 4 fc.O /3  1g Pb .x/ D jxj2 fc  .O /3 g2

(78)

for x D jxj ;  2 S2 . Since kUb k1;S2 C kPb k1;S2 D O.jcj1 / for jcj ! 1 or, equivalently, jbj ! 0, one observes kUb k3;1;R3 C kPb k3=2;1;R3 ! 0

.b ! 0/:

(79)

326

T. Hishida

When b D 0, one may understand fU0 ; P0 g D f0; 0g. As proved by Šverák [74], for each b 2 R3 the Landau solution (78) is the only solution to (11) which is smooth in R3 n f0g and possesses the homogeneity (9) (however, without assuming axisymmetry), and the family of Landau solutions covers all of possible self-similar solutions to (8). This fact as well as the Eq. (11) itself is essential in the proof of Theorem 4 below, while the exact form (78) is not really needed except for (79). Given Navier-Stokes flow u 2 L3;1 ./, the aim is to clarify how a specific Landau solution is singled out from the set S (see (10)). One also takes care of the external force, to which less attention has been paid in previous literature except [48]. Concerning that, the situation is the same as in Theorem 1 for the Stokes P 1 ./ yields the balance between the Landau flow, that is, the class f 2 H 3=2;1 solution and the error term. Thus, one needs slightly more decay property of f . In this presentation the class of the external force is a bit larger than the one in [48], for pointwise decay of F is not assumed below. The main result reads Theorem 4. Let f D f0 C div F with F 2 Ls;1 ./ \ L3=2;1 ./ \ C 1 ./

(80)

for some s 2 .1; 3=2/. Suppose f0 2 C 1 ./ satisfies (36) for some ˛ > 3. Let fu; pg be a smooth solution of class u 2 L3;1 ./;

p 2 L3=2;1 ./;

3=2;1

ru 2 Lloc

./

(81)

to (62). Set Z

Z ŒT .u; p/  u ˝ u C F   d  C

M D @

f0 .y/ dy

(82)



and q D max f3=.˛  1/; sg 2 .1; 3=2/;

q D max f3=.˛  2/; s g 2 .3=2; 3/;

where s 2 .3=2; 3/ is defined by 1=s D 1=s  1=3 (note that 1=q D 1=q  1=3). There is a constant  > 0 such that if kuk3;1 C jM j <  ;

(83)

then uUM 2 Lr ./;

frurUM ; p PM g 2 Lr ./;

8r 2 .q; 3=2/;

(84)

where fUM ; PM g 2 S denotes the Landau solution with label given by (82), see (10), and r 2 .q ; 3/ is defined by 1=r D 1=r  1=3.

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

327

Even if F satisfies F 2 L3=2 ./ \ C 1 ./

(85)

in place of (80), the conclusion above still holds true, in which (84) is replaced by u  UM 2 L3;3=2 . n B3R /;

fru  rUM ; p  PM g 2 L3=2 . n B3R /;

(86)

where R > 0 is taken large enough. Proof. Since T .u; p/  u ˝ u  F 2 Ltloc ./ for every t 2 .1; 3=2/, the boundary integral of (82) makes sense by the same reasoning as in Theorem 1. Set ˇ D R   u d . One assumes 0 2 int .R3 n / without loss and uses the flux carrier @ 2

z.x/ given by (39), which fulfills not only (40) but also z  rz D r jzj2 . So the pair uQ D u  z;

pQ D p 

jzj2 2

belongs to the class (81) and obeys Qu C r pQ C uQ  r uQ D f  u  rz  z  ru;

div uQ D 0

in 

as well as vanishing flux condition (41). Fix R0 > 0 such that R3 n   BR0 . Let R 2 ŒR0 ; 1/ be the parameter to be determined later. One takes v D .1 

/Qu C BŒQu  r ;

D .1 

/p; Q

(87)

by using the cutoff function (42) together with the Bogovskii operator B (Lemma 1) in the domain AR (see (43)). One then finds  v C r C v  rv D h;

div v D 0

in R3

(88)

with h WD g C .1 

/f0 C div f.1 

/.F  z ˝ u  u ˝ z/g;

g 2 C01 .AR /;

where the exact form of g is not needed as in the proof of Theorem 1. One is going to show that Z fg C .1  R3

/f0 g.y/ dy D M;

(89)

328

T. Hishida

(see (82)) and that kvk3;1;R3  C kuk3;1 C

C jˇj : R

(90)

Let  > 3R. By taking 

Z jyjD

 Z y ˇ 2 jzj2 z˝z d D y d D 0 2  32 2 5 jyjD

as well as (47) into account, one finds Z fg C .1 

/f0 g.y/ dy

B

Z D

div fT .v; /  v ˝ v C .1 

/.F  z ˝ u  u ˝ z/g dy

B

Z ŒT .Qu; p/ Q  uQ ˝ uQ C F  z ˝ u  u ˝ z

D jyjD

y d 

Z

y d  jyjD Z Z ŒT .u; p/  u ˝ u C F   d  C f0 .y/ dy: D

D

ŒT .u; p/  u ˝ u C F 

@



Letting  ! 1 leads to (89). To show (90), set vu WD .1  /u C BŒu  r  and vz WD .1  /z C BŒz  r . Since the map u 7! vu is bounded from Lq ./ to Lq .R3 / for every q 2 .1; 1/, the real interpolation implies that kvu k3;1;R3  C kuk3;1 :

(91)

Fix 2 .3; 1/ arbitrarily. It then follows from (42), the Gagliardo-Nirenberg and Poincaré inequalities together with dilation invariance of the estimate of the Bogovskii operator (due to Borchers and Sohr [8], see Lemma 1) that 13=

3=

kBŒz  r k1;AR  C kBŒz  r k ;R3 krBŒz  r k ;R3  CR13= krBŒz  r k ;AR  CR13= kz  r k ;AR  C kzk1;AR D

C jˇj : R2

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

329

Thus, jxjjvz .x/j  C jˇj=R for all x 2 R3 and thereby kvz k3;1;R3 

C jˇj ; R

which together with (91) concludes (90). Let fU; P g D fUM ; PM g be the Landau solution whose label is given by (82). To regularize fU; P g around x D 0, one may follow the same cutoff procedure as in (87): V D .1  One then observes

R AR

/U C BŒU  r ;

U r

Z

Z div . U / dx D

jxjDR

AR

‚ D .1 

/P:

dx D 0 because

x  U d D R

Z jxjD"

x  U d  D O."/ "

." ! 0/:

The same reasoning as in (91) implies that kV k3;1;R3  C kU k3;1;R3 :

(92)

The pair fV; ‚g obeys  V C r‚ C V  rV D H;

div V D 0

in R3

(93)

for some function H 2 C01 .AR / with Z H .y/ dy D M:

(94)

R3

In fact, by using a test function 2 C 1 .R3 / satisfying .x/ D 1 .jxj  3R/ and

.x/ D 0 .jxj  4R/, one sees from (11) with b D M that Z

Z H .y/ dy D 

AR

Z

.T .U; P /  U ˝ U / jyjD3R

y d 3R

.T .U; P /  U ˝ U /.r / dy

D 3R 0, where if ˛ 2 .3; 4/; t D 3=.˛  1/; t D 3=.˛  2/ t 2 .1; 3=2/ is arbitrary; 1=t D 1=t  1=3 if ˛  4: Since w1 ; rw1 ; #1 2 L .BL / for every 2 .1; 1/, which follows from g C .1  /f0 C H 2 L .R3 / for such , we obtain w1 2 Lt ;1 .R3 / \ L3;1 .R3 /;

frw1 ; #1 g 2 Lt;1 .R3 / \ L3=2;1 .R3 /:

(102)

p 1  jzj 2 L3=2;1 .R3 / \ L1 .R3 /, which combined with p One observes 1  juj 2 L3;1 .R3 / and (13) imply that .1 

/.z ˝ u C u ˝ z/ 2 L ;1 .R3 /;

8 2 .1; 3:

(103)

It thus follows from (80) that ˚ div .1 

 1 1 P 3=2;1 P s;1 .R3 / \ H .R3 /: /.F  z ˝ u  u ˝ z/ 2 H

(104)

Let 1 1 P s;1 P 3=2;1 .R3 / \ H .R3 / ,! Ls ;1 .R3 / \ L3;1 .R3 /; w2 2 H

#2 2 Ls;1 .R3 / \ L3=2;1 .R3 /;

(105)

be the solution to (33) with the external force (104) obtained in Lemma 3. Then one finds w0 WD w1 C w2 2 Lq ;1 .R3 / \ L3;1 .R3 /; rw0 2 Lq;1 .R3 / \ L3=2;1 .R3 /;

(106)

#0 WD #1 C #2 2 Lq;1 .R3 / \ L3=2;1 .R3 /; 3 ; sg, where 1=q D 1=q  1=3. with q D maxf ˛1 Given w 2 Lq ;1 .R3 / \ L3;1 .R3 / (see (99)), the velocity part of the unique solution to (33) with the external force 1 1 P 3=2;1 P q;1 .R3 / \ H .R3 /  div .v ˝ w C w ˝ V / 2 H

332

T. Hishida

obtained in Lemma 3 is denoted by T w. Then problem (95) is rewritten as w D w0 C T w

(107)

and the right-hand side returns to the class (99) on account of (106). By (35) together with (98) (and the similar one in terms of kwkq ;1;R3 ), the map w 7! w0 C T w is contractive from Lq ;1 .R3 / \ L3;1 .R3 / into itself provided that kvk3;1;R3 C kV k3;1;R3 < 2

(108)

with a suitable small constant 2 D 2 .q/ > 0. One thus gets a fixed point w 2 Lq ;1 .R3 /\L3;1 .R3 / with rw 2 Lq;1 .R3 /\L3=2;1 .R3 /. Since the pressure associated with T w belongs to Lq;1 .R3 / \ L3=2;1 .R3 /, so does the pressure associated with the fixed point because of (106). In view of (90), (92) with U D UM , and (79), the parameter R 2 ŒR0 ; 1/ is first fixed so that jˇj=R is small enough, and then it is possible to take a suitable constant  > 0 such that both (97) and (108) are accomplished under the condition (83). Finally, consider the case when F satisfies (85) in place of (80), then ˚ div .1 

 1 P 3=2 /.F  z ˝ u  u ˝ z/ 2 H .R3 /

on account of (103). Hence, one has 1 P 3=2 .R3 / ,! L3;3=2 .R3 /; w2 2 H

#2 2 L3=2 .R3 /

(see (20)), instead of (105). On the other hand, (101) yields w1 2 L3;3=2 .R3 n BL /;

frw1 ; #1 g 2 L3=2 .R3 n BL /

for some L > 0, which implies w1 2 L3;3=2 .R3 /;

frw1 ; #1 g 2 L3=2 .R3 /

(109)

by the same reasoning as in (102). One thus obtains w0 2 L3;3=2 .R3 /;

frw0 ; #0 g 2 L3=2 .R3 /

(110)

instead of (106). For the proof of (86), it suffices to find a solution w 2 L3;3=2 .R3 /;

frw; #g 2 L3=2 .R3 /

to (95). Given w 2 L3;3=2 .R3 /, one observes 1 P 3=2 .R3 /  div .v ˝ w C w ˝ V / 2 H

(111)

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

333

with kdiv .v ˝ w C w ˝ V /kHP 1 .R3 /  C .kvk3;1;R3 C kV k3;1;R3 /kwk3;3=2;R3 3=2

P 1 .R3 / ,! L3;3=2 .R3 /. By by (17). Therefore, the term T w in (107) belongs to H 3=2 virtue of (110), the rest of the proof of existence of a solution of class (111) is the same as above. The proof is complete. t u t 1 Remark 5. R div F 2 L R./ \ Lloc ./ for some t > 1, then the R If, in addition, equality  f .y/ dy D  f0 .y/ dy C @ F  d  is justified, so that (82) is equal to M .0; 0; f / given by (4).

Remark 6. If F is absent (so that f D f0 ) and if ˛  4, the exponent q is chosen arbitrarily in the interval .1; 3=2/ (as close to 1 as one wishes) and the small constant  depends on the choice of q. Finally, let us consider the Navier-Stokes system around a rotating obstacle  u C rp C u  ru  .!  x/  ru C !  u D f;

div u D 0

in ;

(112)

where ! D ae3 with a 2 Rnf0g. As mentioned in [74, section 3], a scaling argument with (7) works for the case ! D 0 to see that, if solutions are asymptotically homogeneous of degree .1/, then their leading terms are the Landau solutions. But (112) is no longer invariant under the transformation (7) unless ! D 0. One thus needs another heuristic observation, which is based on knowledge of the linearized system (51). The point is the asymptotic expansion (58), which yields the leading term (59) of the linearized flow. Let u 2 L3;1 ./ be the solution to the NavierStokes system (112). In view of features of (59), it is reasonable to expect that the leading term, denoted by U , still keeps symmetry about the axis of rotation (i.e., Re3 ) as well as homogeneity of degree .1/ and that the quantity e3  M controls the rate of decay; here, M D M .0; !; f / is given by (4), or Z

Z ŒT .u; p/  u ˝ .u  !  y/  .!  y/ ˝ u C F   d y C

M D @

f0 .y/ dy 

(113) when the external force is of the form f D f0 C div F satisfying the same assumptions as in Theorem 4. One can also expect, as in (60), that the leading term U together with some scalar field P solves  U C rP C U  rU  .!  x/  rU C !  U D .e3  M / e3 ı;

div U D 0 (114)

in D0 .R3 /; however, this is reduced to  U C rP C U  rU D .e3  M / e3 ı;

div U D 0

(115)

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because U satisfies (61) under the symmetry about Re3 . Hence, U is a self-similar solution to (8), and it should be a Landau solution U.e3 M /e3 . This observation can be justified along the same way as in the proof of Theorem 4, in which (94) and (96)2 should be replaced by Z Z H .y/ dy D .e3  M /e3 ; e3  fg C .1  /f0  H g.y/ dy D 0: (116) R3

R3

Then one can use Lemma 5 to obtain (101)1 for w1 , which combined with the result of [20] implies (102)1 =(109)1 for w1 . This was done by Farwig and Hishida [19] for (112) with no external force under the no-slip boundary condition uj@ D !  x. But their result can be extended to the case where the external force satisfies the same conditions as in Theorem 4 without assuming any boundary condition on @ (see [47, section 6] for the flux carrier). Note that the corresponding Landau pressure P.e3 M /e3 is not solely the leading term of the associated pressure unlike Theorem 4 for ! D 0. This is because (116)2 is not sufficient to get faster decay of the pressure, so that (101)2 for #1 is not available. In addition to P.e3 M /e3 , however, one can use (32)2 to take the leading term of #1 given by (100) so that one gets Z P.e3 M /e3 C Q.x/ 

fg C .1  R3

/f0  H g.y/ dy

D P.e3 M /e3 C Q.x/  fM  .e3  M /e3 g; which is the leading term of the pressure, where Q.x/ is the fundamental solution (29). This was found by Farwig, Galdi, and Kyed [16]. In [16] the authors deduced the asymptotic expansion of solutions of the Leray class satisfying the energy inequality, which eventually decay like jxj1 (see [30]), as long as they are small enough.

6

Conclusion

The asymptotic structure at infinity as well as existence of 3D exterior stationary Navier-Stokes flows being in L3;1 (weak-L3 ) is discussed when the obstacle is at rest. The class L3;1 is critical from both of the following points of view (one is essential, while the other would be technical). On the one hand, it is optimal summability of generic flows in the sense that better summability than L3;1 necessarily implies the vanishing total net force, M .0; 0; f / D 0 (see (4)). On the other hand, it is difficult to conclude the stability of flows with worse summability than L3;1 (even if they are small enough) as long as one adopts the mathematical analysis developed until now. The Stokes fundamental solution, which is the leading profile of the Stokes flow, is no longer the leading profile of the Navier-Stokes flow being in L3;1 on account of the balance between the linear part and nonlinearity. The correct leading term

6 Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions

335

is the Landau solution whose label is the net force M .0; 0; f / of given NavierStokes flow provided it is small enough. The reason comes essentially from Šverák’s observation on structure of the set that consists of all homogeneous Navier-Stokes flows of degree .1/. One finds a contrast with the case where the obstacle is translating, in which the leading profile is described in terms of the linear part, that is, the Oseen fundamental solution. When the obstacle is rotating with constant angular velocity !, the leading term is still a Landau solution; however, its label ! ! is given by . j!j  M / j!j with M D M .0; !; f /, which follows from a decay structure of the associated fundamental solution. One is able to specify a condition on the external force (which is not necessarily of bounded support) such that the conclusions above hold true. Several open questions about the related issues are in order. The results above (except the case where the obstacle is translating) require smallness of the NavierStokes flow in L3;1 because a perturbation argument is adopted. Asymptotic structure of large solutions of this class is much more involved and remains open. When ! D 0, as pointed out by Šverák [74, section 3], once one establishes the asymptotic expansion with homogeneous leading term of degree .1/ without any smallness, it turns out by a scaling argument that the leading term must be a Landau solution. If the external force f has less decay property (i.e., ˛ is close to 3 and s is close to 3=2, or even F 2 L3=2 ./, in Theorem 4), it is then hopeless to find out the second term after the leading one (a Landau solution) in the asymptotic expansion. For the simple case f D 0, however, one can ask what the second term is. It is probably homogeneous of degree .2/. As compared with the 3D problem, there are many open problems concerning exterior stationary Navier-Stokes flows in 2D (see Galdi [28, Chapter XII] for the details). The most difficult case is that the obstacle is at rest (unless assuming any symmetry), where the linearization method can no longer work because of the Stokes paradox. No one knows the asymptotic structure of the Navier-Stokes flow even if it is small enough; however, a remarkable conjecture based on numerical verification has been recently proposed by Guillod and Wittwer [36]. When the obstacle is translating, the problem is less difficult on account of decay structure of the 2D Oseen fundamental solution, which is the leading profile of the NavierStokes flows without restriction on the magnitude as in 3D (see Smith [72] and Galdi [28, Theorem XII.8.1]). But the stability/instability of such flows is far from clear. The case when the obstacle is rotating in 2D has been much less studied. As for the linearized problem, it was found by Hishida [45] that the oscillation caused by rotation of the obstacle leads to the resolution of the Stokes paradox and that the leading term of the flow at infinity involves the profile x ? =jxj2 whose coefficient is (not the net force but) the torque, where x ? D .x2 ; x1 /> . Very recently, Higaki, Maekawa, and Nakahara [41] showed that asymptotic structure of small NavierStokes flow around a slowly rotating obstacle exhibits the same profile as above. Such a structure has still remained open unless imposing smallness on the angular velocity. Note that the pair

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u.x/ D

cx ? ; jxj2

p.x/ D

c 2 2jxj2

.c 2 R/;

(117)

is a self-similar solution to the Navier-Stokes system (8) in R2 n f0g and that it also satisfies (112) with f D 0 in R2 n f0g since the last two terms in the left-hand side vanish, that is, ax ?  ru C au? D 0 (by following the standard notation in R 2D). By Theorem 2 of Šverák [74, section 5], under the zero flux condition S1  u d  D 0, the homogeneous Navier-Stokes flow of degree .1/ in 2D must be either the circular flow (117) or a particular Jeffery-Hamel flow (whose component tangent to the circle S1 vanishes). When the fluid region is in particular the exterior of a rotating disk, one refers to interesting papers by Hillairet and Wittwer [42] and by Maekawa [62]: the former finds the Navier-Stokes flow which is close to the solution (117) with large jcj and whose leading profile is also given by x ? =jxj2 , and the latter successfully proves the L2 stability of the solution (117) provided jcj is sufficiently small. One can expect that this latter result would hold for small Navier-Stokes flow constructed in [41].

7

Cross-References

 Self-Similar Solutions to the Nonstationary Navier-Stokes Equations  Steady-State Navier-Stokes Flow Around a Moving Body  Time-Periodic Solutions to the Navier-Stokes Equations

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7

Steady-State Navier-Stokes Flow Around a Moving Body Giovanni P. Galdi and Jiˇrí Neustupa

Contents 1 2 3 4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Early Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 A Function-Analytic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Finn’s Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Babenko’s Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 A General Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Geometric and Functional Properties for Large Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Steady Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Time-Periodic Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Stability and Longtime Behavior of Unsteady Perturbations . . . . . . . . . . . . . . . . . . . . . . 10.1 Spectrum of Operator A,T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 A Semigroup, Generated by the Operator A,T . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Existence and Uniqueness of Solutions of the Initial–Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Attractivity and Asymptotic Stability with Smallness Assumptions on v0 . . . . . . 10.5 Spectral Stability and Related Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

342 344 346 349 351 353 363 365 366 367 367 368 372 375 376 381 386 387 392 395 398 406

G.P. Galdi () Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA, USA e-mail: [email protected] J. Neustupa Institute of Mathematics, Czech Academy of Sciences, Praha 1, Czech Republic e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_7

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11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

412 412 413

Abstract

In this chapter we present an updated account of the fundamental mathematical results pertaining the steady-state flow of a Navier-Stokes liquid past a rigid body which is allowed to rotate. Precisely, we shall address questions of existence, uniqueness, regularity, asymptotic structure, generic properties, and (steady and unsteady) bifurcation. Moreover, we will perform a rather complete analysis of the longtime behavior of dynamical perturbation to the above flow, thus inferring, in particular, sufficient conditions for their stability and asymptotic stability.

1

Introduction

The motion of a rigid body in a viscous liquid represents one of the most classical and most studied chapters of applied and theoretical fluid mechanics. Actually, the study of this problem, at different scales, is at the foundation of many branches of applied sciences such as biology, medicine, and car, airplane, and ship manufacturing, to name a few. The dynamics of the liquid associated to these problems is, of course, of the utmost relevance and, already in very elementary cases, can be quite intricate or even, at times, far from being obvious. For example, consider a rigid sphere of radius R, moving by constant translatory motion with speed v0 and entirely immersed in a surrounding liquid, of kinematic viscosity . Then, it is experimentally observed (see [67]) that if Re WD v0 R= . 200, the flow is steady, stable, and axisymmetric. However, if 200 . Re . 270, this flow loses its stability, and another stable, steady, but no longer axisymmetric flow sets in, as evidenced by the loss of rotational symmetry of the wake. It is worth emphasizing the loss of symmetry of the flow, in spite of the symmetry of the data. Moreover, if 270 . Re . 300, the steady flow is unstable, and the liquid regime becomes oscillatory, as shown by the highly organized time-periodic motion of the wake behind the sphere. The remarkable feature of this phenomenon is that the unsteadiness of the flow arises spontaneously, even though the imposed conditions are time independent (constant speed of the body). Another significant example is furnished if now the sphere, instead of moving by a translatory motion, rotates with constant angular velocity, !0 , along one of its diameters. Here, again in view of the symmetry of the data, one would guess that, at least for “small” values of j!0 j (more precisely, of the dimensionless number j!0 j R2 =), the flow of the liquid is steady with streamlines being circles perpendicular to and centered around the axis of rotation. Actually, this is not the case, unless the inertia of the liquid is entirely disregarded. In fact, though the flow is steady, due to inertia, the sphere behaves like a “centrifugal fan,” receiving the liquid near the poles and throwing it away at the equator; see [12, 85]. Already from these brief considerations, one can fairly deduce that a rigorous mathematical study of the motion of a viscous liquid around an obstacle presents a

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plethora of intriguing problems of considerable difficulties, beginning with the very existence of steady-state solutions under general conditions on the data and their uniqueness going through more complicated issues such as analysis of steady and time-periodic bifurcation and longtime behavior of time-dependent perturbations. It is the objective of this chapter to address some of these fundamental problems, as well as point out certain outstanding questions that still await for an answer. In real experiments the liquid occupies, of course, a finite (though “sufficiently large”) spatial region. However, “wall effects” are irrelevant for the occurrence of the basic phenomena of the type described above. Therefore, in order not to spoil their underlying causes, it is customary to formulate the mathematical theory of the motion of a body in a viscous liquid as an exterior problem. This corresponds to the assumption that the liquid fills the entire three-dimensional space outside the body. It should be remarked that this assumption, though simplifying on one hand, on the other hand adds more complication to the mathematical analysis, in that classical and powerful tools valid for bounded flow are no longer available in this case. As it turns out, most of the questions that we shall analyze require, for their answers, a somewhat detailed analysis of the solutions at large distance from the body. From a historical viewpoint, the mathematical analysis of the steady flow of a viscous liquid past a rigid body may be traced back to the pioneering contributions of Stokes [105], Kirchhoff [71], and Thomson (Lord Kelvin) and Tait [106] in the mid- and late 1880s. However, it was only in the 1930s that, thanks to the farreaching and genuinely new ideas introduced by Jean Leray [81], the investigation of the problem received a substantial impulse. Leray’s results, mostly devoted to the existence problem, were further deepened, extended, and completed over the years by a number of fundamental researches due, mostly, to O.A. Ladyzhenskaya, H. Fujita, R. Finn, K.I. Babenko, and J.G. Heywood. It is important to observe that the efforts of all these authors were directed to the study of cases where the body is not allowed to spin. The more general and more complicated situation of a rotating body became the object of a systematic study only at the beginning of the third millennium, with the basic contributions, among others, of R. Farwig, T. Hishida, M. Hieber, Y. Shibata, the authors of the present paper, and their associates. The main goal of this review chapter is to furnish an up-to-date state of the art of the fundamental mathematical properties of steady-state flow of a NavierStokes liquid past a rigid body, which is also allowed to rotate. Thus, existence, uniqueness, regularity, asymptotic structure, generic properties, and (steady and unsteady) bifurcation issues will be addressed. In addition, a rather complete analysis of the longtime behavior of dynamical perturbation to the above solutions will be performed to deduce, in particular, sufficient conditions for their stability and asymptotic stability as well. With the exception of part of the last section, Sect. 10, this study will be focused on the case when the translational velocity, v0 , of the body is not zero and its angular velocity is either zero or else has a nonvanishing component in the direction of v0 . The reason for such a choice lies in the fact that under these assumptions, the mathematical questions listed above have a rather complete answer. On the

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other side, if one relaxes these assumptions, the picture becomes much less clear. The interested reader is referred to [45, §§ X.9 and XI.7] for all main properties known in this case. Furthermore, for the same reason of incompleteness of results, only three-dimensional flow will be considered. An update source of information regarding plane motions can be found, for example, in [45, Chapter XII], [36], and [56]. Finally, other significant investigations are left out of this chapter, such as the motion of the coupled system body–liquid (i.e., when the motion of the body is no longer prescribed, but becomes part of the problem), as well as the very important case when the body is deformable, for which the reader is referred to [38] and [5,44], respectively. The plan of the chapter is as follows. After collecting in Sect. 2 the main notation used throughout, in Sect. 3 it is provided the mathematical formulation of the problem. Section 4 is dedicated to existence questions. There, one begins to recall classical approaches and corresponding results due to Leray, Ladyzhenskaya, and Fujita. Successively, improved findings obtained by the function-analytic method introduced by Galdi are presented, based on the degree for proper Fredholm maps of index 0. Regularity and uniqueness questions of solutions are next addressed in Sects. 5 and 6, respectively. Section 7 is dedicated to the (spatial) asymptotic behavior, beginning by recalling the original results of Finn and Babenko when the body is not spinning to the more recent general contributions of Galdi and Kyed and Deuring and their associates, valid also in the case of a rotating body. Successively, in Sect. 8, one investigates the geometric structure of the solution manifold for data of arbitrary “size.” In particular, it is shown that, generically, the number of solutions corresponding to a given (nonzero) translational velocity and angular velocity is finite and odd. Sect. 9 is devoted to steady and time-periodic (Hopf) bifurcation of steady-state solutions. There, it is provided necessary and sufficient conditions for this type of bifurcation to occur. In the final section, Sect. 10, one analyzes the longtime behavior of time-dependent perturbations to a given steady state, providing, as a special case, sufficient conditions for attractivity and asymptotic stability. These results can be, roughly speaking, grouped in two different categories. The first one is where one assumes that the unperturbed steady state is “small in size.” In the second one, instead, one makes suitable hypothesis on the location in the complex plane of eigenvalues of the relevant linearized operator (spectral stability). In conclusion to this introductory section, it is worth emphasizing that throughout this chapter, there have been highlighted a number of intriguing unsettled questions that still need an answer and represent as many avenues open to the interested mathematician.

2

Notation

The symbols N, Z and R, C stand for the sets of positive and relative integers and the fields of real and complex numbers, respectively. We also put NC WD N\.0; 1/, RC WD R \ .0; 1/.

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Vectors in R3 will be indicated by bold-faced letters. A base in R3 is denoted by fe 1 ; e 2 ; e 3 g  fe i g and the components of a vector v in the given base by v1 , v2 , and v3 . Unless stated otherwise, the Greek letter ˝ will denote a fixed exterior domain of R3 , namely, the complement of the closure, B, of a bounded, open, and simply connected set of R3 . It will be assumed ˝ of class C 2 , and the origin O of the coordinate system fO; e i g is taken in the interior of B. Also, d is the diameter of 1 B, so that, setting BR WD fx 2 R3 W .x12 Cx22 Cx32 / 2 < Rg, R > 0, one has B  Bd . For R  d , the following notation will be adopted ˝R D ˝ \ BR ; ˝ R D ˝  ˝R ; where the bar denotes closure. One puts ut WD @u=@t, @1 u WD @u=@x1 , and, for ˛ a multi-index, one denotes by D ˛ the usual differential operator of order j˛j. For j˛j D 2 one shall simply write D 2 . q Given an open and connected set A  R3 ; Lq .A/, Lloc .A/, 1  q  1; m;q 0;q W m;q .A/; W0 .A/ .W 0;q  W0  Lq ), W m1=q;q .@A/, m 2 NC [ f0g, stand for the usual Lebesgue, Sobolev, and trace space classes, respectively, of real or complex functions. (The same font style will be used to denote scalar, vector, and tensor function spaces.) Norms in Lq .A/, W m;q .A/, and W m1=q;q .@A/ are indicated by k:kq;A , k:km;q;A , and k:km1=q;q.@A/ . The scalar product of functions u; v 2 L2 .A/ will be denoted by .u; v/A . In the above notation, the subscript A will be omitted, unless confusion arises. As customary, for q 2 Œ1; 1 one lets q 0 D q=.q  1/ be its Hölder conjugate. By D m;q .˝/, 1 < q < 1, m 2 NC , one denotes the space of (equivalence classes of) functions u such that jujm;q WD

X Z j˛jDm

jD ˛ ujq

 q1

< 1;

˝

m;q

and by D0 .˝/ the completion of C01 .˝/ in the norm j  jm;q . Moreover, setting D.˝/ WD fu 2 C01 .˝/ W div u D 0g; D01;2 .˝/ is the completion of D.˝/ in the norm j  j1;2 . By D01;2 .˝/ [D01;2 .˝/], one denotes the normed dual space of D01;2 .˝/ [D01;2 .˝/] and by h; i [Œ; ] the associated duality pairing. By Hq .˝/ it is indicated the completion of D.˝/ in the norm Lq .˝/, and one simply writes H .˝/ for q D 2. Further, P is the (Helmholtz-Weyl) projection from Lq .˝/ onto Hq .˝/. Notice that, since ˝ is a sufficiently smooth exterior domain, P is independent of q. If M is a map between two spaces, by D .M /, N .M /, and R .M /, one denotes its domain, null space, and range, respectively, and by Sp .M / its spectrum.

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In the following, B is a real Banach space with associated norm k  kB . The complexification of B is denoted by BC WD B C i B. Likewise, the complexification of a map M between two Banach spaces will be indicated by MC . For q 2 Œ1; 1, Lq .a; bI B/ is the space of functions u W .a; b/ 2 R ! B such that ! q1 Z b

a

q

ku.t /kB dt

< 1; if q 2 Œ1; 1/ I ess supku.t /kB < 1; if q D 1: t2.a;b/

Given a function u 2 L1 .; I B/, u is its average over Œ; , namely, Z  1 u WD u.t / dt: 2  Furthermore, one says that u is 2-periodic, if u.t C 2/ D u.t /, for a.a. t 2 R. Set n W22;0 .˝/ WD u 2 L2 .; I W 2;2 .˝/ \ Do01;2 .˝// and ut 2 L2 .; I H .˝// W u is 2 -periodic with u D 0 with associated norm Z kukW2

2;0



WD 

kut .t /k22 dt

1=2

Z



C 

ku.t /k22;2 dt

1=2 :

One also defines n o H2;0 .˝/ WD u 2 L2 .; I H .˝// W u is 2-periodic with u D 0 : Finally, C , C0 , C1 , etc., denote positive constants, whose particular value is unessential to the context. When one wishes to emphasize the dependence of C on some parameter , it will be written C ./.

3

Formulation of the Problem

Suppose one has a rigid body, B, moving by prescribed motion in an otherwise quiescent viscous liquid, L, filling the entire space outside B. Mathematically, B will be taken as the closure of a simply connected bounded domain of class C 2 . For the sake of generality, a given velocity distribution is allowed on @B, due, for example, to a tangential motion of the boundary wall or to an outflow/inflow mechanism, as well as it is assumed that on L is acting a given body force. (The presence in the model of a body force other than gravity – whose contribution can be always incorporated in the pressure term – could be questionable on physical grounds. However, from the mathematical point of view, it might be useful in consideration of extending the results to more general liquid models, where now the “body force” would represent the contribution to the linear momentum equation

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of other appropriate fields.) In order to study the motion of L under these circumstances, it is appropriate to write its governing equations in a body-fixed frame, S, so that the region occupied by L becomes time independent. One thus gets vt C .v  V /  rv C ! v D v  rp C f

)

div v D 0

in ˝ .0; 1/:

(1)

(For the derivation of these equations, we refer to [38, Section 1, eq. (1.15)].) In these equations, v; p are absolute velocity and pressure fields of L, respectively,  and  its (constant) density and kinematic viscosity, and f is the body force acting on L. Moreover, V WD  C ! x; with  and !, in the order, velocity of the center of mass and angular velocity of B in S. Finally, ˝ WD R3 nB is the time-independent region occupied by L that will be assumed of class C 2 . (Several peripheral results continue to hold with less or no regularity at all. This will be emphasized in the assumptions occasionally.) The system (1) is endowed with the following boundary condition v D v C V at @˝ .0; 1/;

(2)

with v a prescribed field, expressing the adherence of the liquid at the boundary walls of the body, and asymptotic conditions lim v.x; t / D 0; t 2 .0; 1/;

jxj!1

(3)

representative of the property that the liquid is quiescent at large spatial distances from the body. Throughout this paper it shall be assumed that the vectors  and ! do not depend on time. This assumption imposes certain limitations on the type of motion that B can execute with respect to a fixed inertial frame. Precisely [45], the center of mass of B must move with constant speed along a circular helix whose axis is parallel to !. The helix will degenerate into a circle when   ! D 0, in which case the motion of the body reduces to a constant rotation. Without loss of generality, we set ! D ! e 1 , !  0, and  D v0 e with e a unit vector. As indicated in the introductory section, one is only interested in the case when the motion of the body does not reduce to a uniform rotation. For this reason, unless otherwise stated, it will be assumed v0 ¤ 0 and e  e 1 ¤ 0:

(4)

By shifting the origin of the coordinate system S suitably (Mozzi-Chasles transformation) and scaling velocity and length by v0 e  e 1 and d , respectively,

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one can then show that (1) can be put in the following form in the shifted frame S 0 (see [45, pp. 496–497]): vt C v C  @1 v C T .e 1 x rv  e 1 v/ D v  rv C rp C f div v D 0 v D v C V at @˝ .0; 1/ I

9 > = > ;

in ˝ .0; 1/

lim v.x; t / D 0; all t 2 .0; 1/;

jxj!1

(5) ! d2 where T WD ,  8 v d ˆ < 0 e  e 1 ; if ! ¤ 0;   WD ˆ : v0 d ; if ! D 0 .e  e 1 /; 

(6)

and V WD e 1 C

T e 1 x: 

(7)

Of course, all fields entering the equations in (5) are regarded as nondimensional. Observe also that, in the rescaled length variables, the diameter of B becomes 1. In order to simplify the presentation, the origin of the coordinate system S 0 will be supposed to lie in the interior of B. Finally, we notice that, in view of (4), it follows  ¤ 0. Since all results presented in this chapter are independent of whether  ? 0, it will be assumed throughout  > 0. Of particular relevance to this chapter are time-independent solutions (steadystate flow) of problems (5), (6), and (7), which may occur only when f and v are also time independent. From (5), one thus infers that these solutions must satisfy the following boundary value problem: v C  @1 v C T .e 1 x  rv  e 1 v/ D v  rv C rp C f div v D 0

) in ˝ (8)

v D v C V at @˝ I

lim v.x/ D 0:

jxj!1

The primary objective of this chapter is to provide an updated review of some fundamental properties of solutions to (6), (7), and (8). The latter include existence, uniqueness, regularity, asymptotic structure, generic properties, and steady and unsteady bifurcation issues. Moreover, a rather complete analysis of the longtime behavior of dynamical perturbation to these solutions will be performed that will lead, in particular, to a number of stability and asymptotic stability results, under various assumptions.

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349

Existence

The starting point is the following general definition of weak (or generalized) solution for problems (6), (7), and (8) [79]. Definition 1. Let f 2 D01;2 .˝/ and v 2 W 1=2;2 .@˝/. A vector field v W ˝ ! R3 is a weak solution to problems (8)–(7) if the following conditions hold: (a) v 2 D 1;2 .˝/ with div v D 0. (b) v satisfies the equation .rv; r'/C  .@1 v; '/ C T .e 1 x  rv  e 1 v; '/ C  .v  r'; v/ D hf ; 'i; for all ' 2 D.˝/.

(9)

(c) v D v CZV at @˝ in the trace sense.

(d) lim R2 R!1

jvj D 0. @BR

(Formally, (9) is obtained by taking the scalar product of both sides of (8)1 by ' and integrating by parts over ˝. Since D 1;2 .˝/  W 1;2 .˝R /, R > 1, condition (c) is meaningful.) Remark 1. If f 2 W01;2 .˝ 0 /, for all bounded ˝ 0 with ˝ 0  ˝, then to every weak solution, one can associate a suitable corresponding pressure field. More precisely, there exists p 2 L2loc .˝/ such that .rv; r /C  .@1 v; / C T .e 1 x  rv  e 1 v; / C  .v  r ; v/ D .p; div / C Œf ; ; for all 2 C01 .˝/, where Œ;  stands for the duality pairing D01;2 $ D01;2 . Notice that this equation is formally obtained by dot-multiplying both sides of (8)1 by and integrating by parts over ˝. The proof of this property, based on the representation of elements of D01;2 vanishing on D01;2 , is given in [45, Lemma XI.1.1]. The next step is the construction of a suitable extension, U , of the boundary data. The crucial property of such extension is condition (10) given below. In fact, as will become clear later on, this allows one to obtain the fundamental a priori estimate for the existence result. Now, the validity of (10) is related to the magnitude of the flux through the boundary @˝, ˚ , of the field v : Z ˚ WD

v  n; @˝

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with n unit outer normal to @˝. For simplicity, it will be assumed ˚ D 0, even though all main results continue to hold also when j˚ j is sufficiently “small.” The reader is referred to Open Problem 4.2 for further considerations about this issue. The existence of the appropriate extension of the boundary data is provided by the following result whose proof can be found in [45, Lemma X.4.1] Lemma 1. Let v 2 W

1=2;2

Z .@˝/;

v  n D 0: @˝

Then, for any > 0, there exists U D U . ; v ; V; ˝/ W ˝ ! R3 with bounded support such that: (i) U 2 W 1;2 .˝/. ii) U D v C V at @˝. (iii) div U D 0 in ˝. Furthermore, for all u 2 D01;2 .˝/, it holds that j.u  rU ; u/j  juj21;2 :

(10)

Finally, if kv k1=2;2.@˝/  M; for some M > 0; then kU k1;2  C1 kv C Vk1=2;2.@˝/

(11)

where C1 D C1 . ; M; ˝/. Remark 2. In view of the above result, it easily follows that the existence of a weak solution is secured if there is u 2 D01;2 .˝/ satisfying  .ru; r'/ C  .@1 u; '/ C T .e 1 x  ru  e 1 u; '/ C  .u  r'; u/ C Œ.U  r'; u/  .u  rU ; '/  .rU ; r'/ C  .@1 U  U  rU ; '/ C T .e 1 x  rU  e 1 U ; '/ D hf ; 'i; for all ' 2 D.˝/. (12) In fact, setting v WD u C U ; one gets at once that conditions (a)–(c) of Definition 1 are met. Moreover, since, by Sobolev theorem D01;2 .˝/  L6 .˝/ (e.g., [45, Theorem II.7.5]), from [45, Lemma II.6.3], it follows lim

R!1

1 3

R2

Z @BR

juj D 0; for all u 2 D01;2 .˝/;

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so that also requirement (d) is met, even with a better order of decay. In view of all the above, we may equally refer to both v and u as “weak solution.”

4.1

Early Contributions

Classical approaches and results to the existence of weak solutions due, basically, to Jean Leray [81], Olga A. Ladyzhenskaya [79], and Hiroshi Fujita [29] will be now presented and summarized. Besides their historical relevance and intrinsic interest, these results will also provide a further motivation for the entirely distinct approach – recently introduced in [41, 47]–that will be described in Sect. 5.

4.1.1 Leray’s Contribution In his famous pioneering work on the steady-state Navier-Stokes equations [81, Chapitres II & III], Leray shows that for any sufficiently regular f and v ; with ˚ D 0; there is at least one corresponding solution .v; p/ to (8)1;2;3 –(7), which, in addition, satisfies v 2 D 1;2 .˝/. (As a matter of fact, Leray requires f  0; [81, §3 at p. 32]. However, for his method to go through, the weaker assumption of a “smooth” f would suffice.) It is just in this weak sense that Leray interprets the condition at infinity (8)4 . (As noticed earlier on, this condition can be expressed in a sharper, though still weak, way; see Remark 2.) Leray’s construction, basically, consists in solving the original problem (8)1;2;3 –(7) on each elements of an increasing sequence of bounded domains f˝k gk>1 with ˝ D [1 kD1 ˝k ; under the further condition v D 0 on the “fictitious” boundary @Bk (“invading domains” technique). In turn, on every ˝k ; a sufficiently smooth solution, vk ; to the system (8) is determined by combining Leray-Schauder degree theory with a uniform bound on the Dirichlet integral jvk j21;2 . (It should be observed that, even though the demonstration provided by Leray is presented in the language of LeraySchauder fixed-point theorem, such a result was not yet available at that time; see [83, 84].) The latter is crucial, in that it allows Leray to select a subsequence that, uniformly on compact sets, converges to a solution of the original problem, meant in a suitable integral sense. It must be emphasized that in order to obtain the above bound, the property (10) of the extension is crucial. (Notice that a bound on vk can also be obtained by an alternative method, based on a contradiction argument; see [81, Chapitre II, §III]. Even though the latter is more general than the one based on the existence of an extension satisfying (10) (see [48, Introduction]), however, it does not necessarily provide a uniform bound independent of k and, therefore, is of no use in the context of the “invading domains” technique.) An important feature of the solution constructed by Leray is that it could be shown to satisfy the so-called generalized energy inequality juj21;2  .u  rU ; u/ .rU ; ru/ C  .@1 U  U  rU ; u/ CT .e 1 x  rU  e 1 U ; u/  hf ; ui  0;

(13)

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formally obtained by setting '  u in (12) and replacing “D” with “.” A more familiar form of (13) can be obtained if f and v have some more regularity. For example, if in addition f 2 L2 .˝/ and v 2 W 3=2;2 .@˝/; then it can be shown that (13) is equivalent to the following one (see [45, Theorem XI.3.1(i)]):  2kD.v/k22 C

Z

˚ @˝

  .v C V/  T .v; p/  .v C V/2 v  n  hf ; vi  0; 2

(14)

where T .v; p/ D rv C .rv/>  p I; I the identity matrix, is the Cauchy stress tensor. It is worth emphasizing that (14) would represent the energy balance for the motion .v; p/; provided one could replace “” with “D.” The inequality sign in the above formulas is, again, a consequence of the little information that this solution carries at large spatial distances. For the same reason, the uniqueness question is left out.

4.1.2 Ladyzhenskaya’s Contribution Ladyzhenskaya was the first to introduce the definition and the use of the term “generalized (or weak) solution” as currently used, for steady-state Navier-Stokes problems [79, p. 78]. Her construction still employs the “invading domains” technique utilized by Leray, but the way in which she proves the existence of the solution on each bounded domain ˝k of the sequence is somewhat simpler and more direct. More precisely, Ladyzenskaya considers (12) with T D 0 and v  0 and shows that it can be equivalently rewritten as a nonlinear equation in the Hilbert space D01;2 .˝k /: M.u/ WD u C  A.u/ D F

(15)

where F is prescribed D01;2 .˝k / and A is a (nonlinear) compact operator. (The extension to the case T ¤ 0 would be straightforward.) Therefore, the operator M; defined on the whole of D01;2 .˝k /; is a compact perturbation of a homeomorphism. Moreover, using arguments similar to Leray’s, one can show that every solution to (15) is uniformly bounded in D01;2 .˝k /; for all  2 Œ0; 0 ; arbitrary fixed 0 > 0. Then, by the Leray-Schauder degree theory, it follows that (15) has a weak solution, uk 2 D01;2 .˝k /; for the given F . Since juk j1;2 is uniformly bounded in k; Ladyzhenskaya shows that a subsequence can be selected converging to a weak solution in the sense of Definition 1; see Remark 2. It is worth emphasizing that, if ˝ is an exterior domain, the operator A is not compact (see [47, Proposition 80]) so that the “invading domains technique” is indeed necessary for the argument to work. Moreover, if u is merely in D01;2 .˝/; with ˝ exterior domain, the very equation (22) would not be meaningful in such a case. Finally, it is important to observe that Ladyzhenskaya’s solution, as Leray’s, satisfies only the generalized energy inequality (13), and, again, the uniqueness question is left open because of the little asymptotic information carried by functions from D01;2 .˝/.

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4.1.3 Fujita’s Contribution Fujita’s approach to the existence of a weak solution [29] is entirely different from those previously mentioned. In fact, it consists in adapting to the time-independent case the method introduced by Eberhard Hopf for the initial value problem [64]. The method referred to above is the by now classical Faedo-Galerkin method. (Also, strictly speaking, Fujita considers the case T D 0; even though the extension of his method to the more general case presents no conceptual difficulty.) As is well known, the idea is to look first for an “approximate solution” to (12), uN ; in the manifold M.N / spanned by the first N elements of a basis of D01;2 .˝/. This is a finite-dimensional problem whose solution, at the N -th step, is found by solving a suitable nonlinear equation. Fujita solves the latter by means of Brouwer fixedpoint theorem [29, Lemma 3.1], provided j˚j is “small enough,” and then shows that juN j1;2 is uniformly bounded in N . With this information in hand, one can then select a subsequence fuN 0 g that in the limit N 0 ! 1 converges (in a suitable sense) to a vector u 2 D01;2 .˝/ satisfying (12); see also [45, Theorem X.4.1]. The advantage of Fujita’s approach, besides being more elementary, resides also in the fact that the solution is constructed directly in the whole domain ˝. However, also in this case, solutions satisfy only the generalized energy inequality, and their uniqueness is also left out.

4.2

A Function-Analytic Approach

The most significant aspect of solutions constructed by the above authors is that their existence is ensured for data of arbitrary “size,” provided only the mass flux through the boundary is not too large. (Notice that, of course, kv k1=2;2;@˝ arbitrarily “large” and j˚j “small” are not, in general, at odds.) However, as emphasized already a few times, these solutions possess no further asymptotic information at large distances other than that deriving from the fact that v 2 L6 .˝/; consequence of the of the property v 2 D 1;2 .˝/ and Sobolev inequality; see Remark 2. With such a little information, it is, basically, hopeless to show fundamental properties of the solution that are yet expected on physical grounds, such as (i) balance of energy equation, namely, (14) with the equality sign, and (ii) uniqueness for “small” data. The main objective of this subsection is to show that, in fact, this undesired feature can be removed by using another and completely different approach. The approach, introduced in [41, 47], consists in formulating the original problem as a nonlinear equation in a suitable Banach space and then using the mod 2 degree for proper Fredholm maps of index 0 to show just under the conditions on the data stated in Definition 1 the existence of a corresponding weak solution possessing “better” properties at “large” distances. As a consequence, one proves that these weak solutions satisfy, in addition, the requirements (i) and (ii) above. Moreover, this abstract setting shows itself appropriate for the study of other important properties of solutions, including generic properties, and steady and time-periodic bifurcation; see Sects. 8 and 9.

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In order to give a precise statement of the main results, it is appropriate to introduce the necessary functional setting. To this end, let R.u/ WD e 1 x  ru  e 1 u and set  ˚ X .˝/ D u 2 D01;2 .˝/ W @1 u; R.u/ 2 D01;2 .˝/ ;

(16)

where @1 u 2 D01;2 .˝/ means that there is C > 0 such that j.@1 u; '/j  C j'j1;2 ; for all ' 2 D.˝/; and, therefore, by the Hahn-Banach theorem, @1 u can be uniquely extended to an element of D01;2 .˝/ that will still be denoted by @1 u. Analogous considerations hold for R.u/. It can be shown [47, Proposition 65] that when endowed with its “natural” norm kukX WD juj1;2 C j@1 uj1;2 C jR.u/j1;2 ; X .˝/ becomes a reflexive, separable Banach space. Obviously, X .˝/ is a strict subspace of D01;2 .˝/. The primary objective is to prove existence of weak solution in the space X .˝/. In this respect, one observes that all classical approaches mentioned earlier on furnish weak solutions in D01;2 .˝/ which embeds only in L6 .˝/; see Remark 2. The fundamental property of X .˝/; expressed in the following lemma, is that it embeds in a much “better” space. Lemma 2. Let ˝  R3 be an exterior domain and assume u 2 D01;2 .˝/ with @1 u 2 D01;2 .˝/. Then, u 2 L4 .˝/; and there is C1 D C1 .˝/ > 0 such that 1

3

4 4 juj1;2 : kuk4  C1 j@1 uj1;2

(17)

Thus, in particular, X .˝/  L4 .˝/. Proof. Obviously, if u  0; there is nothing to prove, so one shall assume u 6 0. The proof for an arbitrary exterior domain is somewhat complicated by several technical issues; see [41, Proposition 1.1 with proof on pp. 8–13]. However, if ˝  R3 ; it becomes simpler and will be sketched here. (The inequality proved in [41, Proposition 1.1] is, in fact, weaker than (17). However, one can apply to Eq. (1.31) of [41] almost verbatim the argument given in the current proof after (23) and show the stronger form (17).) For a given g 2 C01 .R3 /; consider the following problem '  @1 ' D g C rp; div ' D 0; in R3 ;

(18)

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where > 0. By [45, Theorem VII.4.1], problem (18) has at least one solution such that ' 2 Ls1 .R3 / \ D 1;s2 .R3 / \ D 2;s3 .R3 /; @1 ' 2 Ls3 .R3 / p 2 Ls4 .R3 / \ D 1;s3 .R3 /

(19)

for all s1 > 2; s2 > 4=3; s3 > 1; s4 > 3=2; which satisfies the estimate

1=4 j'j1;2  C kgk4=3 ;

(20)

with C D C .s1 ; : : : ; s4 /. Using (18)–(19) and recalling that by the Sobolev inequality D01;2 .R3 /  L6 .R3 /; one shows after integration by parts .u; g/ D .u; '  @1 '  rp/ D .ru; r'/  .u; @1 '/:

(21)

The following identity is valid for all u 2 D01;2 .R3 / with @1 u 2 D01;2 .R3 / and 6 2 D01;2 .R3 / with @1 2 L 5 .R3 / and can be shown by the arguments from [41, pp. 12–13] .u; @1 / D h@1 u; i: In view of (19), we may use the latter in (21) to get .u; g/ D .ru; r'/ C h@1 u; 'i; which implies j.u; g/j  .juj1;2 C j@1 uj1;2 / j'j1;2 :

(22)

Replacing (20) into this latter inequality, one finds  3  1 j.u; g/j  C 4 j@1 uj1;2 C  4 juj1;2 kgk 4 : 3

Since g is arbitrary in C01 .R3 /; it follows that u 2 L4 .R3 / and, furthermore,  3  1 kuk4  C 4 j@1 uj1;2 C  4 juj1;2 ; for all > 0.

(23)

By a simple calculation we show that the right-hand side of (23) as a function of

attains its minimum at

D juj1;2 =.3 j@1 uj1;2 /;

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which once replaced in (23) proves (17), provided j@1 uj1;2 ¤ 0. To show that this is indeed the case, suppose the contrary. Then .@1 u; '/ D 0 for all ' 2 D.R3 /; so that there is p 2 D 1;2 .R3 / such that @1 u D rp in R3 ; see, e.g., [45, Lemma III.3.1]. From div u D 0; we deduce p D 0 in R3 in the sense of distributions, which, by the property of p; in turn furnishes @1 u D rp  0; and this contradicts the fact that  u 2 L6 .R3 /. The proof is thus completed. One is now in a position to state the following general existence result. Theorem 1. For any  ¤ 0; T  0; f 2 D01;2 .˝/; and v 2 W 1=2;2 .@˝/ with ˚ D 0; there exists at least one weak solution, v; to (8)–(7) that in addition satisfies v  U 2 X .˝/; with U given in Lemma 1. Moreover, v obeys the estimate     kv  U kX  C1 jf j1;2 C jf j31;2 C C2 kv C Vk1=2;2;@˝/ C kv C Vk31=2;2.@˝/ (24) where C1 D C1 .; T ; ˝/ and C2 D C2 .; T ; ˝; M /; whenever kv k1=2;2.@˝/  M . A full proof of Theorem 1 is given in [47, Theorem 86(i)]. Here it shall be reproduced the main ideas leading to the result, referring the reader to the cited reference for all missing details. The first step is to write (12) as a nonlinear equation in the space D01;2 .˝/. To reach this goal, for fixed ; T ; one defines the generalized Oseen operator O W u 2 X .˝/ 7! O.u/ 2 D01;2 .˝/

(25)

where hO.u/; 'i WD .ru; r'/ C h@1 u; 'i C T hR.u/; 'i; ' 2 D01;2 .˝/:

(26)

Likewise, one introduces the operators N and K from X .˝/ to D01;2 .˝/ as follows: hN.u/; 'i WD .u  r'; u/ hK.u/; 'i WD Œ.U  ru; '/ C .u  rU ; '/

' 2 D01;2 .˝/:

(27)

(The dependence of the relevant operators on the parameters  and T will be emphasized only when needed; see Sects. 8 and 9.) Finally, let F denote the uniquely determined element of D01;2 .˝/ such that, for all ' 2 D01;2 .˝/; hF ; 'i WD .rU ; r'/  .@1 U  U  rU ; '/  T .R.U /; '/ C hf ; 'i:

(28)

In view of Lemma 1 and Lemma 2 and with the help of Hölder Inequality, it is easy to show that the operators O; N; and K and the element F are well defined.

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Setting L WD O C K;

(29)

the objective is to solve the following problem: For any F 2 D01;2 .˝/; find u 2 X .˝/ such that L .u/ C N.u/ D F :

(30)

It is plain that, if this problem is solvable, then v D u C U is a weak solution satisfying the statement of Theorem 1. The strategy to solve the above problem consists in showing that the map M WD L C N W X .˝/ 7! D01;2 .˝/ is surjective. To reach this goal, one may use a very general result furnished in [47], based on the mod 2 degree of proper C 2 Fredholm maps of index 0 due to Smale [103]. More precisely, from [47, Theorem 59(a)], it follows, in particular, the following. Proposition 1. Let Z; Y be Banach spaces with Z reflexive. Let L W Z 7! Y and N W Z 7! Y and set M D L C N . Suppose: (i) M is weakly sequentially continuous (i.e., if zn ! z weakly in Z; then M .zn / ! M .z/ weakly in Y ). (ii) N is quadratic (i.e., there is a bilinear bounded operator B W Z Z 7! Y such that N .z/ D B.z; z/ for all z 2 Z). (iii) L maps homeomorphically Z onto Y . (iv) The Fréchet derivative of N is compact at every z 2 Z. (v) There is  W RC 7! RC mapping bounded set into bounded set, with .s/ ! 0 as s ! 0; such that kzkZ   .kM .z/kY /: Then M is surjective. This proposition will be applied with Z  X .˝/; Y  D01;2 .˝/; L  L; and N  N. With this in mind, one begins to show the following. Lemma 3. The operator N is quadratic, and M WD L CN is weakly sequentially continuous. Proof. The first property is obvious, since N .u/ D  B.u; u/ where, for wi 2 X .˝/; i D 1; 2;

(31)

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hB.w1 ; w2 /; 'i WD .w1  r'; w2 /; all ' 2 D01;2 .˝/.

(32)

Suppose next uk ! u weakly in X .˝/; one has to show that M.uk / ! M.u/ weakly in D01;2 .˝/. This amounts to prove that lim hM.uk /; 'i D hM.u/; 'i; for all ' 2 D.˝/.

k!1

(33)

In fact, on one hand, being D01;2 .˝/ reflexive [45, Exercise II.6.2], the generic linear functional acting on W 2 D01;2 .˝/ is of the form hW ; 'i; for some ' 2 D01;2 .˝/. On the other hand, it is jM.uk /j1;2  C0 ; C0 > 0 independent of k; as is at once established from (26)–(27) and the uniform boundedness of kuk kX . Now, to show (33), it is observed that the latter implies that there is M1 > 0 independent of k; such that juk j1;2  C: Thus, along a subsequence fuk 0 g; lim .ruk 0 ; r'/ D .ru; r'/ I

k 0 !1

lim .R.uk 0 /; '/ D .R.u/; '/;

k 0 !1

lim .@1 uk 0 ; '/ D .@1 u; '/ I

k 0 !1

for all ' 2 D.˝/.

(34)

Moreover, by the embedding D01;2 .˝/  W 1;2 .˝R /; R > 1; Rellich compactness theorem, and Cantor diagonalization method, one can also show lim kuk 0  uk4;˝R D 0 I for all R > 1;

k 0 !1

(35)

see [45, Proposition 66] for details. The desired property (33) is then a simple consequence of (26), (27), (34), (35), and Hölder inequality.  The following result also holds. Lemma 4. Let u 2 X .˝/. Then, h@1 u; ui D hR.u/; ui D 0: Proof. If u 2 D.˝/; the proof is trivial, being a consequence of simple integration by parts. However, if u is just in X .˝/; the claim is not obvious since it is not known whether D.˝/ is dense in X .˝/. As a consequence, one has to argue in a different and more complicated way, especially to show the property for R. The proof becomes then lengthy, technical, and tricky. For this reason it will be

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omitted, and the reader is referred to [41, pp. 12–13] for the first property and to [47, Proposition 70] for the second one.  The above lemma is crucial for the next result – a particular case of that shown in [47, Proposition 78] – ensuring the validity of condition (iii) in Proposition 1. Lemma 5. The operator L WD O C K is a linear homeomorphism of X .˝/ onto D01;2 .˝/. Moreover, there is a constant C D C .; T ; ˝/ such that kukX  C jL .u/j1;2 :

(36)

Proof. Referring to the cited reference for a full proof, here only the leading ideas will be sketched. As shown in [76, Theorem 2.1] and [47], the generalized Oseen operator O is a homeomorphism of X .˝/ onto D01;2 .˝/; and, moreover, juj1;2 C j@1 uj1;2 C jR.u/j1;2  C jO.u/j1;2 : Therefore, by classical results on Fredholm operators, it is enough to show that (i) K is compact and (ii) N.L / D f0g. Let fuk g  X .˝/ be a bounded sequence and let ˝R contain the support of U . Observing that X .˝/  W 1;2 .˝R /; the Rellich compactness theorem implies that there is a subsequence of fuk g that is Cauchy in L4 .˝R /. Since by (27)1 and Hölder inequality, for all ' 2 D01;2 .˝/ jhK .uk 0 /; 'i  hK .uk 00 /; 'ij  2 kU k4 kuk 0  uk 00 k4;˝R j'j1;2 ; from Lemma 1(i), one infers (along a subsequence) lim 00

k 0 ;k !1

jK .uk 0 /  K .uk 00 /j1;2 D 0;

which proves (i). To show (ii), it must be shown that hO.u/ C K .u/; 'i D 0 for all ' 2 D01;2 .˝/ H) u D 0:

(37)

Since U 2 W 1;2 .˝/ is of bounded support with div U D 0 and u 2 D01;2 .˝/; by an easily justified integration by parts, we show .U  ru; u/ D 0. So that by replacing u for ' in (37) and using this property along with (26), (27)2 ; and Lemma 4, one deduces juj21;2  .u  rU ; u/ D 0: As a result, (37) is a consequence of the latter and of (10) in Lemma 1.



The following lemma guarantees condition (iv) in Proposition 1; see [47, Propositions 79].

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Lemma 6. The Fréchet derivative, N 0 .u/; of N is compact at each u 2 X .˝/. Proof. From (27)1 ; it follows that 1 N 0 .u/w D B.u; w/ C B.w; u/; with B defined in (32). Let fvk g  X .˝/ be such that kvk kX  C; with C independent of k 2 N; and so, by Lemma 2, one gets, in particular, kvk k4 C jvk j1;2  C1 ;

(38)

with C1 D C1 .˝/ > 0. Since X .˝/ is reflexive, there exist v 2 X .˝/ and a subsequence fvk 0 g  X .˝/ converging weakly in X .˝/ to v. As in the proof of Lemma 3, it can also be shown from (38) that (possibly, along another subsequence) lim kvk  vk4;˝R D 0; for all sufficiently large R ; 0 k

(39)

see also [47, Proposition 66]. From (32) and Hölder inequality, one finds jhB.u; vk 0 /  B.u; v/; 'ij D jhB.u; vk 0  v/; 'ij    kuk4;˝R kv  vk 0 k4;˝R C kuk4;˝ R kv  vk 0 k4;˝ R j'j1;2 ; for all sufficiently large R. Using (38) and (39) into this relation gives lim jB.u; vk 0 /  B.u; v/j1;2  C1 kuk4;˝ R ;

k 0 !1

where C1 > 0 is independent of k 0 . However, R is arbitrarily large, and so, by the absolute continuity of the Lebesgue integral, it may be concluded that lim jB.u; vk 0 /  B.u; v/j1;2 D 0:

k 0 !1

(40)

In a completely analogous way, one shows lim jB.vk 0 ; u/  B.v; u/j1;2 D 0:

k 0 !1

(41)

From (40) and (41), it then follows that the operator B.u; /; and hence N 0 .u/; is compact at each u 2 X .˝/. 

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In order to apply Proposition 1 to the operator M; it remains to show condition (v), which amounts, basically, to find “good” a priori estimates for the equation M.u/ D F . Lemma 7. There is a constant C > 0 such that all solutions u 2 X .˝/ to (30) satisfy kukX  C .jF j1;2 C jF j31;2 /:

(42)

Proof. Also using (26), (27)2 ; (10), and Lemma 4, one deduces hL .u/; ui D juj21;2  .u  rU ; u/ 

1 2

juj21;2 ; hF ; ui  jF j1;2 juj1;2 :

(43)

Moreover, it is easily checked that for all u 2 D.˝/; .u  grad u; u/ D 0:

(44)

Now, by Lemma 4, X .˝/  L4 .˝/; and so, by [45, Theorem III.6.2], one can find a sequence fuk g  D.˝/ converging to u in D01;2 .˝/ \ L4 .˝/. Since, by Hölder inequality, the trilinear form .u  grad w; v/ is continuous in L4 .˝/ D 1;2 .˝/ L4 .˝/; one may conclude that (44) continues to hold for all u 2 X .˝/; which gives hN .u/; ui D 0:

(45)

Thus, from this and (43), one obtains juj1;2  2jF j1;2 :

(46)

Since L .u/ D F  N .u/; from Lemma 5, it follows that kukX  C .jF j1;2 C jN .u/j1;2 /:

(47)

Moreover, by Lemma 2 1

3

2 2 1 jhN .u/; 'ij D j.u  r'; u/j  kuk24 j j'j1;2  C1 j@1 uj1;2 juj1;2 j'j1;2 ;

so that, by virtue of (46) and (47), one finds 3 1   2 kukX2 : kukX  C2 jF j1;2 C jF j1;2

Using Young’s inequality in the latter allows one to deduce the validity of (42), and the proof of the lemma is completed. 

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Proof of Theorem 1. The proof of the first statement follows from Proposition 1, and Lemma 3, and Lemmas 5–7. Furthermore, by property (11) of U and (28), one finds jF j1;2  jf j1;2 C C1 kv C Vk1=2;2.@˝/ ;

(48)

where C1 D C1 .; ; ˝; M / whenever kv k1=2;2.@˝/  M . Estimate (24) is then a consequence of Lemma 7 and (48).  Open Problem. Property (10) of the extension U is fundamental for the estimate (46). As mentioned earlier on, (10) is only known if the flux ˚ is of “small” magnitude. While by a procedure similar to [23, 48, 61] it is probably possible to show that such a condition on ˚ is also necessary for the existence of an extension with the above property, one may nevertheless wonder if a small j˚ j would indeed be necessary if the existence problem is approached by other methods. In this respect, by combining a contradiction argument of Leray with properties of the Bernoulli’s function in spaces of low regularity, in their deep work [72], Korobkov, Pileckas, and Russo have shown existence without restrictions on j˚ j; at least for flow and data that are axisymmetric along the direction of . Whether such a result is true in general remains open.

The following result shows an important property of weak solutions in the class X .˝/ and so, in particular, applies to those constructed in Theorem 1. Theorem 2. Let f 2 D01;2 .˝/ and v 2 W 1=2;2 .@˝/; and let v be a corresponding weak solution with v U 2 X .˝/. Then, v satisfies the energy equality, namely, (13) with the equality sign. If, in addition, f 2 L2 .˝/ and v 2 W 3=2;2 .@˝/; then the latter takes the form of the classical equation of energy balance: Z

  .v C V/2 v  n D hf ; vi 2 @˝ Proof. From (30), with F given in (28), one deduces  2kD.v/k22 C

˚

.v C V/  T .v; p/ 

(49)

L .u/; ui C hN .u/; ui D hF ; ui: Employing in this equation (43) and (45), one obtains juj21;2   .u  rU ; u/  hF ; ui D 0; which, recalling the definition of F in (28), shows that u obeys (13) with the equality sign. The second part of the theorem is shown exactly like in [45, pp. 770–771] and will be omitted. 

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Open Problem. The natural question arises whether any weak solution, v; corresponding to data satisfying merely the “natural” minimal conditions of Theorem 1, is such that v  U 2 X .˝/; and, in particular, obeys the equation of energy balance. In a remarkable work, [57] Heck, Kim, and Kozono have shown that this is indeed the case, at least when T D 0 (the body is not spinning), v  0; and f is assumed slightly more regular, namely, f 2 D01;2 .˝/. Whether this result continues to hold for T ¤ 0 is not known.

5

Regularity

It is expected that if the data f ; v and the boundary @˝ are sufficiently smooth, then the corresponding weak solution is smooth as well. In this respect, one has the following very general result about interior and boundary regularity. Theorem 3. Let v be a weak solution to (8)–(7). Then, if m;q

f 2 Wloc .˝/; m  0; where q 2 .1; 1/ if m D 0; while q 2 Œ3=2; 1/ if m > 0; it follows that mC2;q

v 2 Wloc

mC1;q

.˝/; p 2 Wloc

.˝/;

where p is the pressure associated to v in Remark 1. Thus, in particular, if f 2 C 1 .˝/;

(50)

v; p 2 C 1 .˝/:

(51)

then

Assume, further, ˝ of class C mC2 and v 2 W mC21=q;q .@˝/; f 2 W m;q .˝R /; for some R > 1 and with the values of m and q specified earlier. Then, v 2 W mC2;q .˝R /; p 2 W mC1;q .˝R /: Therefore, in particular, if ˝ is of class C 1 and v 2 C 1 .@˝/; f 2 C 1 .˝ R /;

(52)

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it follows that v; p 2 C 1 .˝ R /:

(53)

The proof of this result is rather complicated, and the interested reader is referred to [45, Theorems X.1.1 and XI.1.2]. However, if one assumes (50) [(52) and ˝ of class C 1 ], then the proof of (51) [(53)] can be obtained by classical results for the Stokes problem in conjunction with a simple bootstrap argument and will be reproduced here. To show this, one needs the following classical regularity results for weak solutions to the Stokes problem, a particular case of those furnished in [45, Theorems IV.4.1 and IV.5.1], to which the reader is also referred for their proofs. 1;q

q

Lemma 8. Let .w; / 2 Wloc .˝/ Lloc .˝/; 1 < q < 1; with div w D 0 in ˝; satisfy  .rw; r / D ŒF;   . ; div /; for all m;q

2 C01 .˝/. mC2;q

(54) mC1;q

Then, if F 2 Wloc .˝/; m  0; necessarily .w; / 2 Wloc .˝/ Wloc .˝/. Moreover, assume w 2 W 1;q .˝R / for some R > 1; and w D w at @˝. Then, if F 2 W m;q .˝R /; w 2 W mC21=q;q .@˝/; necessarily .w; / 2 W mC2;q .˝r / mC1;q Wloc .˝r /; for any r 2 .1; R/. With this result in hand, it can be proved that (50) implies (51). From Remark 1, the weak solution v and the associated pressure field p satisfy (54) with F WD  @1 v  T .e 1 x  rv  e 1 v/ C  v  rv C f : Then, by assumption, the embedding 1;2 D01;2 .˝/  Wloc .˝/  L6loc .˝/; 3=2

and the Hölder inequality one has that F 2 Lloc .˝/. From the first statement in 1;3=2 Lemma 8, it can then be deduced v 2 W 2;3=2 .˝/loc ; p 2 Wloc .˝/; and, moreover, 2;3=2 (v; p) satisfy (8)1 a.e. in ˝. Next, because of the embedding Wloc .˝/  1;3 Wloc .˝/  Lrloc .˝/; arbitrary r 2 Œ1; 1/; one obtains the improved regularity 1;s .˝/; for all s 2 Œ1; 3=2/. Using once again Lemma 8, one infers property F 2 Wloc 3;s 2;s v 2 Wloc .˝/ and p 2 Wloc .˝/ which, in particular, gives further regularity for F. By induction, one then proves the desired property v; p 2 C 1 .˝/. The proof of the boundary regularity is performed by an entirely similar argument and, therefore, will be omitted.

7 Steady-State Navier-Stokes Flow Around a Moving Body

6

365

Uniqueness

This section is dedicated to the investigation of the uniqueness property of weak solutions. Basically, the main known results depend on the summability and regularity assumptions made at the outset on the data f and v . The following theorem shows, in particular, that every solution in Theorem 1 is unique in its own class of existence, provided the size of the data is sufficiently restricted. Theorem 4. Assume vi ; i D 1; 2; are weak solutions with vi  U 2 X .˝/; corresponding to the same f 2 D01;2 .˝/; v 2 W 1=2;2 .@˝/. Then, there is C D C .; T ; ˝/ such that if jf j1;2 C kv C Vk1=2;2.@˝/ < C;

(55)

necessarily v1  v2 . Proof. Setting ui D vi  U ; i D 1; 2; with U given in Lemma 1, from (25)–(28) and the assumption, one finds L .ui / C N .ui / D F ; i D 1; 2:

(56)

Therefore, from Lemma 2, Lemma 7, and (48), one infers in particular     kui k4  C1 jf j1;2 Cjf j31;2 CC2 kv CVk1=2;2.@˝/ Ckv CVk31=2;2.@˝/ ;

(57)

where C1 D C1 .; T ; ˝/ and C2 D C2 .; T ; ˝; M /; whenever kv k1=2;2.@˝/  M . Arbitrarily fix the number M once and for all. Setting u WD u1  u2 ; from (56), one obtains L .u/ D B.u; u1 /  B.u2 ; u/;

(58)

where B is defined in (32). Thus, observing that   jB.u; u1 / C B.u2 ; u/j1;2  kuk4 ku1 k4 C ku2 k4 from (58), Lemma 2, and Lemma 6, one has, in particular,  

kuk4 1  C3 ku1 k4 C ku2 k4  0; with C3 D C3 .; T ; ˝/. The result then follows from this inequality and (57).



The natural question arises of whether the solutions constructed in Theorem 1 are unique in the class of weak solutions, that is, obeying just the requirements stated in Definition 1. The answer to this question is positive if f is assumed to possess

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“good” summability properties at large distances and v is sufficiently regular. Actually, this property is a particular consequence of the following result for whose proof the reader is referred to [45, Theorem XI.5.3], once one takes into account 6 that, by Sobolev inequality, L 5 .˝/  D01;2 .˝/ and that D01;2 .˝/  D01;2 .˝/. Theorem 5. Let f 2 L6=5 .˝/ \ L4=3 .˝/; v 2 W 5=4;4=3 .@˝/: Then, there exists C D C .˝; ; T / such that, if kf k6=5 C kv C Vk7=6;6=5.@˝/ < C;

(59)

v is the only weak solution corresponding to the above data.

Open Problem. In general, it is not known whether solutions of Theorem 1 are unique in the class of weak solutions, when f and v merely satisfy the assumptions of that theorem (and are sufficiently small).

In connection with this problem, it is worth remarking that in the special case T D 0 and v  0; and with f slightly more regular (namely, f 2 D01;2 .˝/), the result is shown in [57, Theorem 2.3].

7

Asymptotic Behavior

As shown in previous sections, some fundamental attributes of weak solution expected on physical grounds, such as verifying the energy balance and being unique for small data, can be established if one has enough information on their summability properties in a neighborhood of infinity, like the one provided by Theorem 1. However, there are other significant aspects that require a sharp pointwise knowledge of the solution at large distances, which in principle is not necessarily guaranteed just by the mild asymptotic information furnished in that theorem. These aspects include, for instance, the presence of a stationary, unbounded wake region “behind” the body and a “fast” decay of the vorticity outside the wake region, in support of boundary layer theory. Proving (or disproving) these properties constituted one of the most challenging questions in mathematical fluid dynamics since the pioneering chapter of Leray. The case T D 0 was eventually settled in the mid-1970s (about 40 years after Leray’s work), thanks to the effort of Robert Finn, Konstantin I. Babenko, and their collaborators. Their contributions will be briefly summarized in the following two subsections. The case T ¤ 0 presents much more difficulties and will be treated

7 Steady-State Navier-Stokes Flow Around a Moving Body

367

successively in Sect. 7.3, by means of a different approach, originally introduced in [39], that allows for a rather complete description of the pointwise asymptotic flow behavior also in that more general situation.

7.1

Finn’s Contribution

In the late 1950s/mid-1960s, in a series of remarkable papers [24–28], Robert Finn proved the following fundamental results. Let .v; p/ be any (sufficiently smooth) solution to (8)–(7) with T D 0 and with f of bounded support, such that jv.x/j  C jxj˛ ; some ˛ > 1=2 and all “large” jxj. Then the pointwise asymptotic structure of .v; p/ can be sharply evaluated. In particular, combining the integral representation of the solution, obtained via the (time-independent) Oseen fundamental tensor, E; along with a careful estimate of the latter, Finn showed that these solutions exhibit a paraboloidal “wake region,” R; with the property that the velocity field, v; inside R decays pointwise slower than it does outside R. More precisely, he proved that v [rv] admits an asymptotic expansion with E [rE] being the leading term. (Finn left open the question of the asymptotic behavior of the second derivatives of v [24], a problem that was finally solved another 40 years later by Deuring [13].) Finn called such solutions “physically reasonable” (PR) [27, Definition 5.1] and demonstrated their existence on a condition that the magnitude of the data is sufficiently restricted [27, Theorem 4.1]. Later on, one of his students, David Clark, showed that the vorticity field of any PR solution decays exponentially fast outside R and far from the body [10]. Thanks to its sharp asymptotic (and local regularity) properties, it is easy to show that any PR solution (regardless of the size of the data) is also weak, namely, v 2 D 1;2 .˝/. However, given that the latter is the only information that weak solutions carry in a neighborhood of infinity, the converse property is by no means obvious, to the point that some author even questioned its validity [59, p. 12]. All this seemed to cast profound doubts about the physical relevance of Leray’s weak solutions.

7.2

Babenko’s Contribution

The relation between weak and PR solutions was eventually addressed by Babenko [2]. Combining Lizorkin’s multipliers theory with anisotropic Sobolev-like inequalities and the representation formula for the solution employed by Finn, he was able to show that, if the body force f is of bounded support, every weak solution is, in fact, physically reasonable in the sense of Finn. Babenko’s paper can be divided into two main parts. In the first one, he shows, by a very elegant and straightforward argument, that any weak solution, v; corresponding to the given data must be in L4 .˝/; with corresponding pressure field in L2 .˝/. The second part of the paper is aimed to show that, actually, v 2 Lq .˝/; for any q 2 .2; 4. Once this property is established, then it is relatively simple to prove that the weak solution decays like jxj˛ for some ˛ > 1=2 and therefore is also PR. It must be noted that Babenko’s proof of these further summability properties has aspects that are not

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fully transparent. Also for this reason, a distinct and more direct proof of Babenko’s result was given later on by Galdi [35] and, successively and independently, by Farwig and Sohr [16].

7.3

A General Approach

It must be emphasized one more time that the results reported in the previous two subsections refer to the case T D 0; that is, the body is not spinning. If one allows T ¤ 0; then the detailed study of the asymptotic properties of a weak solution becomes even more complicated, for several reasons. In the first place, the linear momentum equation (8)1 contains a term that grows linearly fast at large spatial distances. As a consequence, the fundamental tensor of the linearized equations, T; is no longer the classical Oseen tensor E mentioned above, let alone a “perturbation” of it, but, rather, a much more complicated one; see [21, Section 2]. Thus, the representation of the solution that in both contributions of Finn and Babenko plays a fundamental role in the determination of the pointwise behavior becomes much more involved and, actually, useless for that matter. In fact, as shown in [21, Proposition 2.1], unlike E; the tensor T does not satisfy uniform estimates at large spatial distances. In view of these issues, in [39] Galdi introduced a completely different approach to the study of the asymptotic structure of a weak solution, that was further generalized and improved in [40, 42, 43, 46]. In this approach, the weak solution v is viewed as limit as t ! 1 along sequences of the (unique) solution, w.x; t /; to a suitable initial value problem. It can be shown that, in turn, w admits a somewhat simple space-time representation in terms of the Oseen fundamental solution to the time-dependent Oseen equation. This fact allows one to obtain a number of sharp spatial estimates for w uniformly in time, which are thus preserved in the limit t ! 1; and therefore continue to hold for the weak solution v. Referring to [45, §§X.6, X.8, XI.4, XI.6] for a full account of the (technically complicated and lengthy) proofs of all the above results, here it will only be provided an outline of the main steps of the procedure used in establishing them in the case T ¤ 0. The first step consists in determining sharp summability properties of a weak solution in a neighborhood of infinity, under appropriate hypothesis on the data. To this end, one can show the following result [45, Theorem XI.6.4]. Lemma 9. Assume, for some q0 > 3 and all q 2 .1; q0 ; that f 2 Lq .˝/ \ L3=2 .˝/; v 2 W 21=q0 ;q0 .@˝/ \ W 4=3;3=2 .@˝/: Then, every weak solution v to problems (8)–(7) corresponding to f ; v ; and the associated pressure field p (possibly modified by the addition of a constant; see also Remark 1) satisfies the following summability properties:

7 Steady-State Navier-Stokes Flow Around a Moving Body

v 2 Lr .˝/ \ D 1;s .˝/;

369

@v 2 Lt .˝/; p 2 L .˝/; @x1

for all r 2 .2; 1; s 2 .4=3; 1; t 2 .1; 1; and 2 .3=2; 1. If, in addition, f 2 W 1;q0 .˝/; v 2 W 31=q0 ;q0 .@˝/; then we have also v 2 D 2; .˝/; p 2 D 1; .˝/; for all 2 .1; 1. The next objective is to “translate” the above global asymptotic information into a pointwise one. For simplicity, it shall be assumed that f is of bounded support, which also implies, with the help of Theorem 3, that .v; p/ 2 C 1 .˝  / for sufficiently large . Thus, in the second step, one uses a standard “cutoff” procedure to rewrite (suitably) (8) in the whole space R3 . More specifically, let be a smooth function that is 0 in the neighborhood of @˝ that contains the support of f and 1 sufficiently far from it. Moreover, let Z 2 C01 .˝/ such that div Z D r  v in ˝. (Such a field Z exists, as shown in [45, Theorem III.3.3].) From (8) one can deduce that u WD v  Z and pQ WD p obey the following problem: u C  @1 u C T .e 1 x  ru e 1 u/ D  div . v ˝ div u D 0

v/ C r pQ C F c

9 = ;

in R3 (60)

where F c is smooth and of bounded support. At this point, the “classical” procedure would be to write the solution u in terms of the fundamental tensor solutions, T; associated with problem (60). However, as remarked earlier on, this would not lead anywhere due to the poor properties of T. Therefore, we argue differently. In the third step one performs a time-dependent change of coordinates which transforms (60) into a suitable initial value problem. To this end, for t  0; let 3 1 0 0 Q.t / D 4 0 cos.T t /  sin.T t / 5 ; 0 sin.T t / cos.T t / 2

and define y WD Q.t /  x; w.y; t / WD Q.t /  u.Q> .t /  y/; .y; t / WD p.Q Q > .t /  y/; > V .y; t / WD Q.t /  Π v.Q .t /  y/; H .y; t / WD Q.t /  F c .Q> .t /  y/:

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From (60) and Lemma 9, it then follows that 9 @w @w D w C   r  div ŒV ˝ V   H = in R3 .0; 1/; @t @y1 ; r wD0 lim kw.; t /  ukr D 0; all r 2 .2; 1/.

(61)

t!0C

Notice that equation (61)1 does not contain the linearly growing term. The solution to the Cauchy problem (61) has the following representation: Z

2

ejyzC t e 1 j =4t u.z/ d z w.y; t / D .4t/ Z t Z 3 R3    .y  z; t  /  Rr  ŒV ˝ V .z; / C H .z; / d z d ;  3=2

0

R

(62)

where  .; s/; .; s/ 2 R3 .0; 1/ is the well-known Oseen fundamental solution to the time-dependent Stokes system [45, §VIII.3]. In the final step one utilizes into (62) the summability properties for u and V obtained from Lemma 9 along with the classical pointwise estimates of  to produce a pointwise estimate for w.x; t /; [rw.x; t /] uniformly in t. As a result, by letting t ! 1 along sequences, the latter can be shown to provide analogous bounds for u.x/ [respectively, ru.x/], which means for the weak solution v.x/ [respectively, rv.x/] for all “large” jxj. Once the necessary asymptotic information on v is obtained, analogous estimates on the pressure field can be proved observing that from (8) it follows, for sufficiently large ; that p D r  G in ˝  ; @p D g on @˝  ; @n where G WD  v  rv; g WD Œv C .@1 v  v  rv/ C T .e 1 x  rv  e 1 v/  n j@˝  : The procedure just outlined is at the basis of the following result whose full proof is found in [45, Theorems XI.6.1–XI.6.3]. Theorem 6. Let v be a weak solution, corresponding to f of bounded support, and let p be the corresponding pressure field associated to v by Remark 1. Then, for any ı; > 0 and all sufficiently large jxj;

7 Steady-State Navier-Stokes Flow Around a Moving Body

371

  1 1 3=2Cı ; v.x/ D O jxj .1 C  s.x// C jxj   3=2 3=2 2C ; rv.x/ D O jxj .1 C  s.x// C jxj

(63)

p.x/ D p0 C O.jxj2 ln jxj/; for some p0 2 R; where s.x/ WD jxj C x1 . Remark 3. This theorem suggests, in particular, that outside any semi-infinite cone, C; whose axis coincides with the negative x1 axis, the decay is faster than inside C. This is the mathematical explanation of the existence of the wake “behind” the body, once one takes into account that the velocity of the center of mass of the body .v0 e) is directed along the positive axis x1 ( > 0). Remark 4. The fundamental tensor solution E.x; y/  fEij .x; y/g of the Oseen system (which is obtained by setting T D 0 and disregarding the nonlinear term v  rv in (8)1 ) is defined through the relations  Eij .x; y/ D ıij  

@2 @yi @yj



1 ˚.x  y/; ˚./ WD 4

Z

 2 .jjC1 /

0

1  e d :

Now, the first terms on the right-hand side of (63)1 and (63)2 are just (sufficiently sharp) bounds for E and rE at large jxj; respectively; see [45, Section VIII.3]. This is suggestive of the property that a  E and a  rE; for some suitable vector a; could be the leading terms in corresponding asymptotic expansions. Actually, if T D 0; this property is true, and one can show that, in such a case, the following formula holds for all sufficiently large jxj [45, Theorem X.8.1]: v.x/ D m  E.x/ C V.x/

(64)

where m is a constant vector coinciding with the total force, F; exerted by the liquid on the body and 

V.x/ D O jxj

3=2Cı

 ; arbitrary ı > 0:

Analogous estimate can be proven for rv.x/ [45, Theorem X.8.2]. If T ¤ 0; in [78, Theorem 1.1], Kyed has shown an asymptotic formula similar to (64) (and an analogous one for rv) where now m D .F  e 1 /e 1 and the quantity V is a “higher-order term” in the sense of Lebesgue integrability at large distances. A pointwise estimate (probably not optimal) for V is shown in [77, Theorem 5.3.1]. (An even more detailed asymptotic structure was first shown in [21] for solutions to the linearized (Stokes) problem and in the absence of translational motion.)

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This section ends with some important considerations concerning the asymptotic behavior of the vorticity field, $ WD curl v; of a weak solution, v. In this regard, one can prove the following theorem, due to Deuring and Galdi [14], that ensures that $ decays exponentially fast outside the wake region and sufficiently far from the body. Theorem 7. Under the same assumptions of Theorem 6, there are constants C; R > 0 such that j$.x/j  C jxj3=2 e.=4/ .jxjCx1 /=.1CR/

for all x 2 ˝ R :

It is worth stressing the importance of this estimate that agrees with the necessary condition supporting the boundary layer assumption, namely, that sufficiently far from the body and the wake, the flow is “basically potential.” As a matter of fact, in the case T D 0; one can prove a sharper result that provides a more accurate description of the asymptotic structure of the vorticity field. Precisely, in that case, one has, for all sufficiently large jxj;    $.x/ D r˚ m C O jxj2 e 2 .jxjCx1 /

(65)

where ˚.x/ D 

  e 2 .jxjCx1 / 4 jxj

and m is a constant vector denoting the total force exerted by the liquid on the body [3, 10]. Open Problem. In the case T ¤ 0; it is not known whether the vorticity admits an asymptotic expansion of the type (65), with an appropriate choice of the leading term.

8

Geometric and Functional Properties for Large Data

Theorem 1 shows that, for any set of data D WD .; T ; v ; f /; in the specified spaces, there exists at least one corresponding weak solution v with the further property that u WD vU 2 X .˝/; for a suitable extension field U . Also, Theorem 4 shows that this is, in fact, the only weak solution in that class, provided the data are suitably restricted, according to (55). Objective of this section is to analyze the geometric and functional properties of the solution manifold in the space X .˝/; corresponding to data of arbitrary magnitude in the class specified in Theorem 1. In order to make the presentation simpler, throughout this section, it is set v  0.

7 Steady-State Navier-Stokes Flow Around a Moving Body

373

To reach this goal, one begins to rewrite equation (30) in an equivalent way that emphasizes the dependence of the operator involved on the parameter p WD .; T /. One thus writes L .p; u/ for L .u/ and N .p; u/ for N .u/ with L and N defined in (25)–(27) and (29). Moreover, let H D H.p/ denote the uniquely determined member of D01;2 .˝/ such that hH; 'i WD .rU ; r'/.@1 U U rU ; '/T .R.U /; '/; ' 2 D01;2 .˝/:

(66)

Thus, for a given f 2 D01;2 .˝/; (30) can be written as M.p; u/ D f in D01;2 .˝/

(67)

where M W .p; u/ 2 R2C X .˝/ 7! L .p; u/ C N .p; u/ C H.p/ 2 D01;2 .˝/:

(68)

The solution manifold associated to (68) is defined next: ˚  M.f / D .p; u/ 2 R2C X .˝/ satisfying (67)–(68) for a given f 2 D01;2 .˝/ The main goal is then to address the following questions: (a) Geometric structure of the manifold M D M.f / (b) Topological properties of the associated level set S.p0 ; f / WD f.p; u/ 2 M.f /; p D p0 g; obtained by fixing also Reynolds and Taylor numbers. Clearly, “points” in S.p0 ; f / are solutions to equation (67) with a prescribed p D p0 or, equivalently, to equation (30). The following theorem collects the principal properties of the set S.p0 ; f /. Theorem 8. The following properties hold. (i) S.p0 ; f / is not empty. (ii) For any .p0 ; f / 2 R2C D01;2 .˝/; S.p0 ; f / is compact. Moreover, there is N D N .p0 ; f / 2 N such that S.p0 ; f / is homeomorphic to a compact set of RN . (iii) For any p0 2 R2C ; there is an open residual set O D O.p0 /  D01;2 .˝/ such that, for every f 2 O; S.p0 ; f / is constituted by a number of points,  D .p0 f /; that is finite and odd. (iv) The number  is constant on every connected component of O.

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Proof. As usual, only a sketch of some of the proofs of the above statements will be given while referring to the appropriate reference for whatever is missing. The statement (i) is a consequence of Theorem 1. The proofs of the other statements are based on two fundamental properties of the operator M WD L .p0 ; / C N .p0 ; /; namely, being (1) proper and (2) Fredholm of index 0. Now, Lemma 5 and Lemma 6 guarantee that the Fréchet derivative of M at every u 2 X .˝/ is a compact perturbation of a homeomorphism, which proves the Fredholm property. Properness means that if F ranges in a compact set, K; of D01;2 .˝/; all possible corresponding solutions u to M.u/ D F belong to a compact set, K  ; of X .˝/. To show that this is indeed the case, one observes that in view of the continuity of M; K  is closed, so that it is enough to show that from any sequence fun g  K  ; there is a subsequence (still denoted by fun g) and u 2 X .˝/ such that un ! u in X .˝/. Let F n D M.un /. Since fF n g  K; one deduces (along a subsequence) F n ! F in D01;2 .˝/; for some F 2 K.

(69)

Moreover, being fF n g bounded, by Lemma 7, it follows that fun g is bounded, and so there exists u 2 X such that un ! u weakly in X .˝/. By Lemma 3 and (69), the latter implies M.u/ D F; M.un / ! M.u/ in D01;2 .˝/:

(70)

One next observes that M.un /  M.u/ D L .un  u/ C N .un /  N .u/

(71)

and also, since N is quadratic (Lemma 3), N .un /  N .u/  B.un ; un /  B.u; u/ D B.un  u; u/ C B.u; un  u/ C B.un  u; un  u/  ŒN 0 .u/.un  u/ C N .un  u/: Replacing this identity in (71), one finds M.un /  M.u/  ŒN 0 .u/.un  u/ D M.un  u/:

(72)

In view of (70), the first term on the left-hand side of (72) tends to 0 as n ! 1. Likewise, since N 0 .u/ is compact, and hence completely continuous, also the second term on the left-hand side of (72) tends to 0 as n ! 1; so that properness follows from this and Lemma 7. The first property in (ii) is then a corollary of what has just been proven. As for the second one, we refer to [47, Theorem 93] for a proof. We next come to show statements (iii) and (iv). In this regard, since M is proper and Fredholm of index 0, by the mod 2 degree of Smale [103], it is enough to show that there exists F 0 2 D01;2 .˝/ with the following properties: (a) the equation

7 Steady-State Navier-Stokes Flow Around a Moving Body

375

M.u/ D F 0 has one and only one solution, u0 ; and (b) N.M0 .u0 // D f0g; see [41, Lemma 6.1]. Now, set F 0 D 0. From Lemma 7, it follows that the only solution to M.u/ D 0 is u0 D 0. Moreover, by Lemma 3, one finds N 0 .0/  0; so that  M0 .0/ D L and condition (b) is a consequence of Lemma 5. Remark 5. Taking into account that the set O in Theorem 8 is dense in D01;2 .˝/; from Theorem 8(iii), one deduces the following interesting property of weak solutions. Let  ¤ 0 and T  0 be arbitrarily fixed. Given f 2 D01;2 .˝/ and " > 0; there is g 2 D01;2 .˝/ with jf  gj1;2 < " such that the number of weak solutions given in Theorem 1 corresponding to the body force g (and v  0) is finite and odd. The next result furnishes a complete generic characterization of the manifold M.f /. Its proof, based on an infinite-dimensional version of the so-called parameterized Sard theorem [107, Theorem 4.L], is technically involved and lengthy. The interested reader is referred to [47, Theorem 88]. Theorem 9. The following properties hold. (i) There exists a dense, residual set Z  D01;2 .˝/ such that, for any f 2 Z; the solution manifold M.f / is a two-dimensional (not necessarily connected) manifold of class C 1 . (ii) For any f 2 Z; there exists an open, dense set P D P.f /  R2C such that, for each p 2 P; equation (67) has a finite number of solutions, n D n.p; f /. (iii) The integer n D n.p; f / is independent of p on every interval contained in P.

Open Problem. It is not known whether, in the physically significant case of vanishing body force f and boundary velocity v ; the number of corresponding steady-state solutions is generically finite.

9

Bifurcation

As pointed out in the introductory section, if the speed of the center of mass of the body, v0 ; reaches a critical value, it is experimentally observed, already in absence of rotation, that the characteristic features of the original steady-state flow of the liquid may change dramatically. The outcome could be either the onset of an entirely different steady-state flow or even of a time-periodic regime. The objective of this section is to provide necessary conditions and sufficient conditions for the occurrence of this phenomenon. More precisely, Sect. 9.1 will be concerned with

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time-independent problems, while Sect. 9.2 will deal with the time-periodic case. For the sake of simplicity, it will be assumed throughout v  0.

9.1

Steady Bifurcation

One is mainly interested in situations where bifurcation is generated by the “combined” action of translation and rotation of the body (provided the latter is not zero). To this end, it is convenient to use a different nondimensionalization for the equations (8)–(7), in order to introduce an appropriate bifurcation parameter. Precisely rescaling velocity with v0 and length with v0 =!; equations (8)–(7) become    v C  @1 v C e 1 x  rv  e 1 v  v  rv D rp C f in ˝ div v D 0 v D e 1 C e 1 x at @˝ I lim v.x/ D 0;

(73)

jxj!1

where now  WD v02 .e  e 1 /=.!/. Remark 6. Of course, the above nondimensionalization requires ! ¤ 0. However, for future reference, it is important to emphasize that all main results presented in this section continue to hold in exactly the same form also when ! D 0. With the notation introduced in the previous section (see (67) and (68) with p  ), the original equation (30) is equivalent to the following nonlinear equation M.; u/ WD L .; u/ C N .; u/ C H./ D f in D01;2 .˝/; u 2 X .˝/

(74)

Definition 2. Let u0 2 X .˝/ be a solution to (74) with  D 0 . The pair .0 ; u0 / .1/ is called a steady bifurcation point for (74), if there are two sequences fk ; uk g and .2/ fk ; uk g with the following properties: .i/

(i) fk ; uk g; i D 1; 2 solve (74) for all k 2 N. .i/ (ii) fk ; uk g ! .0 ; u0 / in R X .˝/ as k ! 1; i D 1; 2. .1/ .2/ (iii) uk  6 uk ; for all k 2 N.

One of the main achievements of this section is the proof that, under certain conditions that may be satisfied in problems of physical interest, bifurcation is reduced to the study of a suitable linear eigenvalue problem, formally analogous to that occurring in the study of bifurcation for flow in a bounded domain; see Theorem 11 and Remark 8.

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377

A necessary condition in order for .0 ; u0 / to be a bifurcation point is obtained as a corollary to the following result. Lemma 10. Let u0 2 X .˝/ be a solution to (74) with  D 0 and fixed f 2 D01;2 .˝/. If N.M 0 .0 ; u0 // D f0g;

(75)

namely, the (linear) equation

L .0 ; w/ C 0 B.u0 ; w/ C B.w; u0 / D 0 in D01;2 .˝/

(76)

has only the solution w D 0 in X .˝/; then there exists a neighborhood U .0 /; such that for each  2 U .0 / there is one and only one u./ solution to (74). Moreover, the map  2 U ! u./ 2 X .˝/ is analytic at  D 0 . (The prime means Fréchet differentiation with respect to u.) Proof. Consider the map F W .; u/ 2 U .0 / X .˝/ 7! M.; u/  f : Also using the fact that N .; / is quadratic (see (31)–(32)), it easily follows that F is analytic (polynomial, in fact) at each .; u/. Moreover, by assumption, F .0 ; u0 / D 0. Thus, the claimed property will follow from the analytic version of the implicit function theorem provided one shows that F 0 .0 ; u0 / is a bijection. Now, from (31)–(32),

F 0 .0 ; u0 / D M 0 .0 ; u0 /  L .0 ; / C 0 B.u0 ; / C B.; u0 / ; so that by Lemma 5 and Lemma 6, we infer that F 0 .0 ; u0 / is Fredholm of index 0 and the bijectivity property follows from the assumption (75).  From this result the following one follows at once. Corollary 1. A necessary condition for .0 ; u0 / to be a bifurcation point is that dim N.M 0 .0 ; u0 // > 0;

(77)

namely, the (linear) equation

L .0 ; w/ C 0 B.u0 ; w/ C B.w; u0 / D 0 in D01;2 .˝/ has a nonzero solution w 2 X .˝/.

(78)

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Remark 7. One can show that (77) is equivalent to the requirement that the linearization of (73) around .0 ; v0  u0 C U / corresponding to homogeneous data, namely, )   w C 0 @1 w C e 1 x  rw  e 1 w  v0  rw  w  rv0 D rp div w D 0 w D 0 at @˝ I

in ˝

lim w.x/ D 0;

jxj!1

(79) has a nontrivial solution .w; p/ 2 ŒD 2;2 .˝/ \ X .˝/ D 1;2 .˝/. In fact, saying that (78) has a nonzero solution w 2 X .˝/ means that there exists w 2 X .˝/  f0g such that (see (25)–(30) with  D T ))

 .rw; r'/ C 0 h@1 w C R.w/; 'i C .w  rv0 C v0  rw; '/ D 0;

(80)

for all ' 2 D01;2 .˝/. However, by the properties of v0 and w combined with the Hölder inequality, one shows G WD .w  rv0 C v0  rw/ 2 L4=3 .˝/; which, in turn, by classical results on the generalized Oseen equation [45, Theorem VIII.8.1] furnishes, in particular, w 2 D 2;4=3 .˝/. By embedding, the latter implies w 2 D 1;12=5 .˝/ \ L12 .˝/; so that G 2 L12=7 .˝/ which, again by [45, Theorem VIII.8.1], delivers w 2 D 2;12=7 .˝/ \ D 1;4 .˝/ \ L1 .˝/. Thus, G 2 L2 .˝/ and the property follows by another application of [45, Theorem VIII.8.1]. Notice that the asymptotic condition in (79) is achieved uniformly pointwise. The next objective is to provide sufficient conditions for .0 ; u0 / to be a bifurcation point. To this end, it will be assumed that, in the neighborhood of .0 ; u0 /; there exists a sufficiently smooth solution curve, that is, there is a map  2 U .0 / 7! u./ 2 X .˝/ of class C 2 (say), with u.0 / D u0 and satisfying (74) for the given f . Setting w WD u  u; one thus gets that w satisfies the equation F.; w/ WD L .; w/ C ŒB.w; u.// C B.u./; w/ C N .; w/ D 0:

(81)

Clearly, .0 ; u0 / is a bifurcation point for (74) if and only if .0 ; 0/ is a bifurcation point for (81). Since, as showed earlier on, F 0 .0 ; 0/  L .0 ; / C 0 ŒB.; u0 .// C B.u0 ; / is Fredholm of index 0, a classical result [107, Theorem 8.A] ensures that .0 ; 0/ is a bifurcation point provided the following conditions hold: (i) dim N .F 0 .0 ; 0// D 1; (ii) ŒF w .0 ; 0/.w1 / 62 R .F 0 .0 ; 0//; w1 2 N .F 0 .0 ; 0//; where the double subscript denotes differentiation with respect to the indicated variable. Condition (i) specifies in which sense the requirement of Corollary 1 must be met. In order to give a more explicit form to condition (ii), it is convenient to introduce the Stokes operator:

7 Steady-State Navier-Stokes Flow Around a Moving Body

379

Q 2 D1;2 .˝/; Q W u 2 D01;2 .˝/ 7! u 0

(82)

Q 'i D .ru; r'/; ' 2 D1;2 .˝/: hu; 0

(83)

with

As is well known, Q is a homeomorphism [45, Theorem V.2.1]. By a straightforward computation, one then shows that ŒF w .0 ; 0/.w1 / D 



1 Q P 0 // C B.u. P 0 /; w1 / ; w1 C 0 B.w1 ; u. 0

(with “” denoting differentiation with respect to ) and therefore condition (ii) is equivalent to the request that the equation

L.0 ; w/ C 0 B.u0 ; w/ C B.w; u0 /

1 Q P P D  w 1 C 0 B.w1 ; u.0 // C B.u.0 /; w1 / 0

(84)

has no solution. All the above is summarized in the following. Theorem 10. Suppose the solution set of the equation

L .0 ; w/ C 0 B.u0 ; w/ C B.w; u0 / D 0

(85)

is a one-dimensional subspace of X .˝/ and let w1 be a corresponding normalized element. If, in addition, equation (84) has no solution w 2 X .˝/; then .0 ; u0 / is a bifurcation point for (74). The assumptions of the result just proven admit a noteworthy conceptual P 0 / D 0. This happens, in particular, if u./ interpretation in the case when u. is constant in a neighborhood of 0 ; a circumstance that may occur by a suitable nondimensionalization of the original equation [34, Section VI]. To show the above, consider the operator L W w 2 X .˝/  D01;2 .˝/ 7! L.w/ D Q 1 Œ@1 w C R.w/ C B.v0 ; w/ C B.w; v0 / 2 D01;2 .˝/; where, as before, v0 WD u0 C U . The following lemma shows the fundamental properties of L. The proof is quite involved, and, for it, the reader is referred to [47, Lemma 111]. Lemma 11. Assume u0 2 L3 .˝/ \ L4loc .˝/. Then, the operator L is (graph) closed. Moreover, Sp.LC /\.0; 1/ consists, at most, of a finite or countable number

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of eigenvalues, each of which is isolated and of finite algebraic and geometric multiplicities, that can only accumulate at 0. (Of course, the assumption u0 2 L4loc .˝/ is redundant if u0 2 X .˝/. Also u0 2 L3 .˝/ is assured by Lemma 9 if f is suitably summable at large distances.) Combining Corollary 1, Lemma 11, and Theorem 10, one can then show the following. P 0 / D 0; with u0 2 L3 .˝/ \ L4 .˝/. Then, a necessary Theorem 11. Assume u. loc condition for .0 ; u0 / to be a bifurcation point for (74) is that 0 WD 1=0 is an eigenvalue for the operator LC . This condition is also sufficient if 0 is simple. Proof. With the help of (25), (26), and (27) and (30), one sees that condition (77) is equivalent to assuming that the following equation has a nonzero solution w1 2 X .˝/ Q 1 C 0 Œ@1 w1 C R.w1 / C B.v0 ; w1 / C B.w1 ; v0 / D 0: w Operating with Q 1 on both sides of the latter, one concludes that 0 must be an eigenvalue of LC ; which provides the first statement. Performing the same procedure on (85), it can be next shown that the first assumption in Theorem 10 is satisfied if and only if there is a unique (normalized) w1 2 X .˝/ such that L.w1 / D 0 w1 ; that is, 0 is an eigenvalue of LC of geometric multiplicity 1. Furthermore, operating P 0 / D 0; one gets again with Q 1 on both sides of (84) with u.

0 w  L.w/ D  20 w1 which, by the second assumption in Theorem 10, should have no solution, which means that the algebraic multiplicity of 0 must be 1 as well and the proof of the claimed property is completed.  Another interesting and immediate consequence of Lemma 11 and Theorem 11 is the following one. Corollary 2. Let u0 be a solution branch to (74) independent of  2 J; where J is a bounded interval with J  .0; 1/. Suppose, further, that u0 satisfies the assumption of the preceding theorem. Then, there is at most a finite number, m; of bifurcation points to (74) .k ; u0 /; k 2 J; k D 1;    ; m. Remark 8. It is significant to observe that the statements of Theorem 11 and Corollary 2 formally coincide with those of analogous theorems for steady bifurcation

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381

from steady solution to the Navier-Stokes equation in a bounded domain; see, e.g., [6, Section 4.3C]. However, in the latter case, L is a compact operator defined on the whole of D01;2 .˝/; whereas in the present case, L is a densely defined unbounded operator.

9.2

Time-Periodic Bifurcation

In spite of its great relevance and frequent occurrence in experimental fluid mechanics, time-periodic bifurcation in a flow past an obstacle has represented a long-standing and intriguing problem from a rigorous mathematical viewpoint. This situation should be contrasted with flow in a bounded domain where, thanks to the pioneering and fundamental contributions of Iudovich [66], Joseph and Sattinger [68], and Iooss [65], complicated time-periodic bifurcation phenomena, like those occurring in the classical Taylor-Couette experiment, could be framed in a rigorous mathematical setting. In order to understand the reason for this uneven situation and also provide a motivation for the approach presented here, it is appropriate to briefly describe what constitutes a rigorous treatment of the phenomenon of time-periodic bifurcation. Suppose, as will be in fact shown later on, that the relevant time-dependent problem can be formally written in the form ut C L.u/ D N .u; /;

(86)

where L is a linear differential operator (with appropriate homogeneous boundary conditions) and N is a nonlinear operator depending on the parameter 2 R; such that N .0; / D 0 for all admissible values of . Then, roughly speaking, timeperiodic bifurcation for (86) amounts to show the existence a family of nontrivial time-periodic solutions u D u. I t / of (unknown) period T D T . / (T -periodic solutions) in a neighborhood of D 0 and such that u. I / ! 0 as ! 0. Setting WD 2 t =T  ! t; (86) becomes ! u C L.u/ D N .u; /;

(87)

and the problem reduces to find a family of 2-periodic solutions to (87) with the above properties. If one now writes u D u C .u  u/ WD v C w; one gets that (87) is formally equivalent to the following two equations L.v/ D N .v C w; / WD N1 .v; w; /; ! w C L.w/ D N .v C w; /  N .v C w; / WD N2 .v; w; /:

(88)

At this point, the crucial issue to realize is that while in the case of a bounded flow, both “steady-state” component, v; and “oscillatory” component, w; may be taken in the same (Hilbert) function space [65, 66, 68]; in the case of an exterior flow,

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v belongs to a space with quite less “regularity” (in the sense of behavior at large spatial distances) than w does; see also [51]. For this basic reason, as emphasized for the first time only recently in [49, 50], in the case of an exterior flow, it is not appropriate (or even “natural”) to investigate the bifurcation problem for (87) in just one functional setting, as done, for example, in [99]; it is instead much more spontaneous to study the two equations in (88) in two different function classes. As a consequence, even though formally being the same as differential operators, the operator L in (88)1 acts on and ranges into spaces different than those the operator L in (88)2 does. With this in mind, (88) becomes L1 .v/ D N1 .v; w; / I ! w C L2 .w/ D N2 .v; w; /: The above ideas will be next applied to provide sufficient conditions for timeperiodic bifurcation in a viscous flow past a body. It will be assumed throughout T D 0, leaving the case T ¤ 0 as an open question. Set L1 W v 2 X .˝/ 7! L .0 ; v/ 2 D01;2 .˝/

(89)

with L .0 ; v/ defined in (76). From Lemma 10 and Remark 6, it follows that under the assumption N .L1 / D f0g;

(H1)

there exists a unique weak solution analytic branch vs ./ WD u./ C U to (8)–(7) in a neighborhood U .0 /; with vs .0 / D u0 C U . Thus, writing v D v.x; tI / C vs .xI /; from (5), one finds that v formally satisfies the (nondimensional) problem

 vt C .v  e 1 /  rv C vs ./  rv C v  rvs ./ D v  rp in ˝ R div v D 0 v D 0 at @˝ R; lim v.x; t / D 0; t 2 R: jxj!1

(90) The bifurcation problem consists then in finding sufficient conditions for the existence of a nontrivial family of suitably defined time-periodic weak solutions to (90), v.tI /;  2 U .0 /; of period T D T ./ (unknown as well), such that v.tI / ! 0 as  ! 0 . Following the general approach mentioned before, one thus introduces the scaled time WD ! t; split v and as the sum of its time average, v; over the time interval Œ; ; and its “purely periodic” component w WD v  v; and set WD   0 . In this way, problem (90) can be equivalently rewritten as the following coupled nonlinear elliptic-parabolic problem

7 Steady-State Navier-Stokes Flow Around a Moving Body

   v C 0 @1 v  v0  rv  v0  rv D rp C N 1 .v; w; / in ˝ div v D 0 v D 0 at @˝; lim v.x/ D 0

383

(91)

jxj!1

and   9 ! w  w  0 @1 w v0  rw  w  rv0 = in ˝2 D r' C N 2 .v; w; / ; div w D 0 w D 0 at @˝ .; /; lim w.x; t / D 0;

(92)

jxj!1

where N 1 WD  Œ@ 1 v  vs . C 0 /  rv  v  rvs . C 0 /

C 0 .vs . hC 0 /  v0 /  rviC v  r.vs . C 0 /  v0 /

(93)

N 2 WD Œ@1hw  vs . C 0 /  rw  w  rvs . C 0 / i  0 .vs . C 0 /  v0 /  rw C w  r.vs . C 0 /  v0 / h i C . C 0 / w  rv C v  rw C w  rw  w  rw ;

(94)

C . C 0 / v  rv C w  rw

and

with v0  vs .0 /. The next step is to rewrite (91), (92), (93), and (94) in the proper functional setting and to reformulate the bifurcation problem accordingly. To this end, one begins to introduce the operator

L2 W w 2 D.L2 /  H .˝/ 7!  P w C 0 .@1 w  v0  rw  w  rv0 / 2 H .˝/; D.L2 / WD W 2;2 .˝/ \ D01;2 .˝/: (95) The following result can be proved by the same arguments (slightly modified in the detail) employed in [50, Proposition 4.2]. ˚  Lemma 12. Let u0 WD v0  U 2 X .˝/. Then Sp.L2C / \ i R  f0g consists, at most, of a finite or countable number of eigenvalues, each of which is isolated and of finite (algebraic) multiplicity, that can only accumulate at 0. Consider, next, the time-dependent operator

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Q W w 2 W 22;0 .˝/ 7! !0 wt C L2 .w/ 2 H2;0 .˝/:

(96)

Again, by a slight modification of the argument used in the proof of [50, Proposition 4.3], one can show the following. Lemma 13. Let v0 be as in Lemma 12. Then, the operator Q is Fredholm of index 0, for any !0 > 0. Finally, one needs the functional properties of the quantities N i ; i D 1; 2; defined in (93)–(94), reported in the following lemma. The proof is, one more time, a slight modification of that given in [50, Lemma 4.5, and the paragraph after it] and will be omitted. Lemma 14. There is a neighborhood V.0; 0; 0/  R X .˝/ W that maps

2 2;0 .˝/

such

N1 W . ; v1 ; v2 / 2 V.0; 0; 0/ 7! P N 1 . ; v1 ; v2 / 2 D01;2 .˝/ N2 W . ; v1 ; v2 / 2 V.0; 0; 0/ 7! P N 2 . ; v1 ; v2 / 2 H2;0 .˝/ are analytic. Also in view of Lemmas 12–14, one then deduces that (91)–(94) can be put in the following abstract form L1 .v/ D N1 . ; v; w/ in D01;2 .˝/ I ! w C L2 .w/ D N2 . ; v; w/ in H2;0 : (97) Notice that the spatial asymptotic conditions on v in (91)4 are interpreted in the sense of Remark 2, while the one in (91)4 for w holds uniformly pointwise for a.a. t 2 R; see [50, Remark 3.2]. One is now in a position to give a precise definition of a time-periodic bifurcation point. Definition 3. The triple . D 0; v D 0; w D 0/ is called time-periodic bifurcation point for (97) if there is a sequence f. k ; !k ; vk ; wk /g  R RC D01;2 .˝/ W 22;0 with the following properties: (i) f. k ; !k ; vk ; wk /g solves (97) for all k 2 N. (ii) f. k ; vk ; wk /g ! .0; 0; 0/ as k ! 1. (iii) wk 6 0; for all k 2 N. Moreover, the bifurcation is called supercritical [resp. subcritical] if the above sequence of solutions exists only for k > 0 [resp. k < 0].

7 Steady-State Navier-Stokes Flow Around a Moving Body

385

The goal is to give sufficient conditions for the occurrence of time-periodic bifurcation in the sense specified above. This will be achieved by means of the general result proved in [50, Theorem 4.1]. With this in mind, one has to show that the assumptions of that theorem are indeed satisfied. In this regard, supported by Lemma 12, one supposes 0 WD i !0 is an eigenvalue of multiplicity 1 of L2C ; k 0 ; k 2 N  f0; 1g is not an eigenvalue of L2C :

(H2)

Next, consider the operator L2 . / WD L2  S; with  

S W w 2 Z 2;2 .˝/ 7! P @1 wv0 rwwrv0 0 vP s .0 /rwCwr vP s .0 / 2 H .˝/; where, as before, “” means differentiation with respect to . By [108, Proposition 79.15 and Corollary 79.16], one knows that for in a neighborhood of 0, there is a smooth map 7! . /; with . / simple eigenvalue of L2C . / and such that 0 D .0/. The following condition will be further assumed: < Œ.0/ P ¤ 0;

(H3)

which basically means that the eigenvalue . / must cross the imaginary axis with “nonzero speed” when  ! 0 . The general result proved in [50, Theorem 3.1] can be now applied to show the following time-periodic bifurcation result. Theorem 12. Suppose (H1)–(H3) hold. Then, the following properties are valid. (a) Existence. There are analytic families 

 v."/; w."/; !."/; ."/ 2 X .˝/ W 22;0 .˝/ RC R

(98)

satisfying (97), for all " in a neighborhood I.0/ and such that   v."/; w."/  " v1 ; !."/; ."/ ! .0; 0; !0 ; 0/ as " ! 0: (a) Uniqueness. There is a neighborhood U .0; 0; !0 ; 0/  X .˝/ W 22;0 .˝/ RC R such that every (nontrivial) 2-periodic solution to (97), .z; s/; lying in U must coincide, up to a phase shift, with a member of the family (98).

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G.P. Galdi and J. Neustupa

(a) Parity. The functions !."/ and ."/ are even: !."/ D !."/; ."/ D ."/; for all " 2 I.0/. Consequently, the bifurcation due to these solutions is either subcritical or supercritical, a two-sided bifurcation being excluded (unless  0).

Open Problem. Sufficient conditions for the occurrence of time-periodic bifurcation in the case when the body is also spinning (T ¤ 0) are not known.

10

Stability and Longtime Behavior of Unsteady Perturbations

In this section, v0 will denote the velocity field of a steady-state solution to (8). As usual,  is assumed to be positive. However, since the theories that will be described in this section have often been equally developed for both cases  D 0 and  6D 0; a number of cited results also concern the case when  D 0. The function v0 is supposed to satisfy v0 2 L3 .˝/;

@j v0 2 L3 .˝/ \ L3=2 .˝/

(for j D 1; 2; 3):

(99)

It follows from Lemma 9 that, if  ¤ 0; such a v0 exists for a large class of body forces f and boundary data. An associated pressure field is denoted by p0 . One is interested in the behavior of unsteady perturbations, .v0 ; p 0 / to the solution .v0 ; p0 /. Thus, writing v D v0 C v0 ; p D p0 C p 0 ; it follows from (5) that the functions v0 ; p 0 satisfy the equations v0t C v0 C  @1 v0 C T .e 1 x  rv0  e 1 v0 / D v0  rv0 C v0  rv0 C v0  rv0 C rp 0 0

div v D 0

9 > = > ;

in ˝ .0; 1/

(100)

and the conditions v0 D 0

at @˝ .0; 1/ I

lim v0 .x; t / D 0; all t 2 .0; 1/:

jxj!1

(101)

For simplicity, from now on the primes are omitted in the notation above. Thus, the formal application of the Helmholtz-Weyl projection P to the first equation in (100), as formulated in Lq .˝/ (1 < q < 1), yields the operator equation

7 Steady-State Navier-Stokes Flow Around a Moving Body

dv D Lv C Nv dt

387

(102)

in the space Hq .˝/. By suitably defining the domains of the operators L and N; it can be easily seen that (102) is, in fact, equivalent to (100) and (101). To this end, let Av WD P v; B1 v WD P @1 v 1;q

for v 2 D.A/ WD W 2;q .˝/ \ D0 .˝/; B2 v

WD P .e 1 x  rv  e 1 v/;

A;T v WD Av C B1 v C T B2 v for ( v 2 D.A;T / WD

1;q

W 2;q .˝/ \ D0 .˝/ if T D 0;  ˚ 1;q 2;q q v 2 W .˝/ \ D0 .˝/I e 1 x  rv 2 L .˝/ if T 6D 0:

Note that A  A0;0 and A;0  A C B1 are the classical Stokes and Oseen operators, respectively. Furthermore, let B3 v WD P .v0  rv C v  rv0 /; Lv WD A;T v C B3 v; Nv WD  P .v  rv/ for v 2 D.L/ WD D.A;T /. Obviously, the study of the stability of the solution .v0 ; p0 / is equivalent to that of the zero solution of problem (100), (101) or equation (102). The properties of the linear operator L and, especially, those of its “leading part” A;T play a fundamental role. Thus, the next two subsections will be concerned with a detailed analysis of these properties.

10.1

Spectrum of Operator A,T

The following notions and definitions from the spectral theory of linear operators will be relevant later on. Let X be a Banach space with norm k : k; X  be its dual, and T be a closed linear operator in X with a domain D.T / dense in X . (This guarantees that the adjoint operator T  exists.)

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– The symbols nul .T / and def .T / denote the nullity and the deficiency of T; respectively. If R.T / is closed, then nul .T / D def .T  / and def .T / D nul .T  / (see, e.g., Kato [69, p. 234]). – The approximate nullity of T; denoted by nul 0 .T /; is the maximum integer m (m D 1 being permitted) with the property that to each  > 0; there exists an m-dimensional linear manifold M in D.T / such that kT vk <  for all v 2 M ; kvk D 1. The approximate deficiency of T is denoted by def 0 .T / and defined as def 0 .T / WD nul 0 .T  /. Note that nul .T /  nul0 .T / and def .T /  def0 .T /; the equalities holding if the range R.T / is closed. ETC. On the other hand, if R.T / is not closed, then nul 0 .T / D def 0 .T / D 1. The identity nul 0 .T / D 1 is equivalent to the existence of a non-compact sequence fun g on the unit sphere in X such that T un ! 0 for n ! 1 (see [69, p. 233]). – T is called a Fredholm operator if both the numbers nul .T / and def .T / are finite. This implies, in particular, that R.T / is closed in X [108, Proposition 8.14(ii)]. Operator T is semi-Fredholm if the range R.T / is closed in X and at least one of the numbers nul .T / and def .T / is finite. Consequently, T is semiFredholm if and only if at least one of the numbers nul 0 .T / and def 0 .T / is finite. – The resolvent set Res.T / is the set of all  2 C such that R.T  I / D X and the operator T  I has a bounded inverse in X . Consequently, nul .T  I / D nul 0 .T  I / D def .T  I / D def 0 .T  I / D 0 for  2 Res.T /. Note that Res.T / is an open subset of C. – The point spectrum Spp .T / is the set of all  2 C such that nul .T  I / > 0. – The continuous spectrum Spc .T / is the set of all  2 C such that nul .T  I / D 0; R.T  I / is dense in X; but R.T  I / 6D X . (In this case, R.T  I / is not closed in X; which implies that def .T  I / D def 0 .T  I / D nul 0 .T  I / D 1.) – The residual spectrum Spr .T / is the set of all  2 C such that nul .T  I / D 0 and the range R.T  I / is not dense in X . The sets Spp .T /; Spc .T /; and Spr .T / are mutually disjoint and Spp .T / [ Spc .T / [ Spr .T / D Sp.T / D C X Res.T / (the spectrum of T ). – The essential spectrum Spess .T / is the set of all  2 C such that T  I is not semi-Fredholm. Both Sp.T / and Spess .T / are closed in C and Spess .T /  Sp.T /. Obviously, Spc .T /  Spess .T /. Any point on the boundary of Sp.T / belongs to Spess .T / unless it is an isolated point of Sp.T / (see [69, p. 244]). From [70] (if T D 0) and [102] (if T 6D 0), it follows that the operator A;T is closed in Hq .˝/ (1 < q < 1), and all  2 C with a sufficiently large real part belong to Res.A;T /. The effective shapes and types of spectra of the operator A;T ; for various values of  and T ; are described in [18] (in H .˝/; the case  D 0), [19] (in H .˝/; the general case  2 R), [22] (in Hq .˝/;  D 0), and [20] (in Hq .˝/;  2 R). The spectrum of A;0 ; as an operator in H .˝/; was studied by K.I. Babenko [4]. Babenko’s result says that Sp.A;0 / D Spc .A;0 / D ;0 ; where ;0 D f D ˛ C iˇ 2 CI ˛; ˇ 2 R; ˛  ˇ 2 =2 g

(103)

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389

for  6D 0. The set ;0 represents a parabolic region in C; symmetric about the real axis, which shrinks to the nonnegative part of the real axis if  ! 0. In fact, Sp.A/ ( Sp.A0;0 /) coincides with Spc .A/ and coincides with the interval .1; 0 in R; as mentioned, e.g., by O.A. Ladyzhenskaya in [80]. The spectrum of A;T for general T is studied in [20]. Notice that the case T 6D 0 is qualitatively different from the case T D 0; because the magnitude of the coefficient of the “new” term T e 1 x  rv becomes unbounded as jxj ! 1. Consequently, the operator T e 1 x  r cannot be treated as a lower-order perturbation of Stokes or Oseen operator. The main results in [20] read as follows. Theorem 13. Let 1 < q < 1;  6D 0 and ˝ D R3 . Then the spectrum of A;T ; as an operator in Hq .R3 /; satisfies the identities Sp.A;T / D Spc .A;T / D Spess .A;T / D ;T ; where ;T WD f D ˛ C iˇ C ikT 2 CI ˛; ˇ 2 R; k 2 Z; ˛  ˇ 2 =2 g: Note that ;T is a union of a family of overlapping solid parabolas, whose axes form an equidistant system of half-lines f 2 CI  D ˛ C kT i; ˛  0; k 2 Zg. All the parabolas lie in the half-plane Re   0; and their vertices are on the imaginary axis. Theorem 14. Let 1 < q < 1;  6D 0 and ˝  R3 be an exterior domain with the boundary of class C 1;1 . Then the spectrum of A;T lies in the left complex halfplane f 2 CI Re   0g and consists of the essential spectrum Spess .A;T / D ;T and possibly a set  of isolated eigenvalues  2 C X ;T with Re  < 0 and finite algebraic multiplicity, which can cluster only at points of Spess .A;T /. The set  of such isolated eigenvalues is independent of q 2 .1; 1/. Sketch of the proof of Theorem 13 (see [20] for the details). The proof develops along the following steps (a)–(f). (a) Using the definition of the adjoint operator, it can be verified that the adjoint operator A;T to A;T coincides with the operator A;T in Hq 0 .˝/; where 1=q C 1=q 0 D 1. (b) From [17, Theorem 1.1], one can deduce that there exist constants C4 > 0 and C5 > 0 such that if u 2 D.A;T / and f 2 Hq .˝/ satisfy the equation A;T u D f ; then kuk2;q C k.! x/  rukq  C4 kf kq C C5 kukq :

(104)

(c) If  2 CX;T ; then each solution of the resolvent equation .A;T I /u D f ; for f 2 Hq .˝/; satisfies the estimate kukq  C6 kf kq ;

(105)

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where C6 D C6 .; q/. This estimate is derived by means of the Fourier transform, and a subtle and rather technical application of the Mikhlin-Lizorkin multiplier theorem. Using inequalities (104) and (105), one can prove that R.A;T  I / is closed and the operator A;T  I is injective. The same statement also holds on the adjoint operator A;T in Hq 0 .˝/. As A;T  I is injective, the range R.A;T  I / is the whole space Hq .˝/. Consequently,  2 Res.A;T /. This shows that C X ;T  Res.A;T /. (d) We show that Spp .A;T / D ;. Assume that  2 ;T and u 2 D.A;T / satisfies the equation .A;T I /u D 0. Applying the Fourier transform F; this equation yields T .e 1   rb u/  .  i1 C jj2 /b u  T e1 b v D 0;

(106)

where b u D F.u/ and  D .1 ; 2 ; 3 / denotes the Fourier variable. The case 0 1 < q  2 is simpler, because b u is a function from Lq .˝/: if 1 ; r; and  denote the cylindrical coordinates in the space of Fourier variables, then one can calculate that e 1   rb u D @b u. Substituting this to (106), one obtains the equation T @b u  .  i1 C jj2 /b u  T e1 b v D 0: If O./ denotes the matrix of rotation about the 1 -axis by angle  and b w.1 ; r; / WD O./b u.1 ; r; /; then one arrives at the ordinary differential equation w  .  i1 C r 2 C 12 / b w D 0: T @ b This equation can be solved explicitly. The solution satisfies: b w.1 ; r;  C 2 .i1 Cr 2 C12 / . As b w is 2-periodic in variable  and 2/ D b w.1 ; r; / e Re  C r 2 C 12 D Re  C jj2 6D 0 for a.a.  2 R3 ; b w is equal to zero a.e. in R3 . It means that u is the zero element of Hq .˝/; which implies that it cannot be an eigenfunction and  therefore cannot be an eigenvalue. The case 2 < q < 1 is rather more complicated becauseb u is only a tempered distribution. Nevertheless, one can also arrive at the same conclusion, i.e., that any  2 ;T cannot be an eigenvalue of A;T . (e) The identity Spr .A;T / D ; can be proven by means of the duality argument:  2 Spr .A;T / would imply that  2 Spp .A;T /. However, the same considerations as in step (d), applied to the adjoint operator A;T ; show that Spp .A;T / D ;. (f) The identities Sp.A;T / D Spc .A;T / D Spess .A;T / follow from the facts that Spp .A;T / and Spr .A;T / are empty and Spc .A;T /  Spess .A;T /. The inclusion Sp.A;T /  ;T follows from item (c). The inclusion ;T  Spess .A;T / is proven in [20] so that  is assumed to be in ı;T (the interior of ;T ), and a concrete sequence fun g; such that k.A;T  I /un kq ! 0

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for n ! 1; is constructed on the unit sphere in Hq .˝/. The construction is quite technical, so the readers are referred to [20] for the details. Thus,  2 Spess .A;T /; which implies that ı;T  Spess .A;T /. The inclusion ;T  Spess .A;T / now follows from the fact that Spess .A;T / is closed. t u Sketch of the proof of Theorem 14 (see [20] for the details.) The proof is a consequence of the following steps (g)–(j). (g) One can show the same inequality as (104), applying the cutoff function technique and splitting the equation A;T u D f into an equation for the unknown u1 in a bounded domain ˝ (where  > 0 is sufficiently large) and an equation for the unknown u2 in the whole space R3 . Due to [17, Theorem 1.1], the function u2 satisfies (104), while u1 satisfies (104) because B1 u and T B2 u can be brought into the right-hand side and then one can apply the estimates of solutions of the Stokes problem in a bounded domain. The Lq -norms of B1 u and T B2 u over ˝ can be interpolated between kukq and ku1 k2;q ; and the norm ku1 k2;q can be absorbed by the left-hand side. Finally, the sum of the estimates of u1 (over ˝ ) and u2 (over R3 ) leads to (104). (h) The inclusion ;T  Spess .A;T /: assume that  2 ı;T . By analogy with (f), one can construct a sequence fun g on the unit sphere in Hq .˝/; such that k.A;T  I /un kq ! 0 for n ! 1. Since Spp .A;T / is not known to be empty, and it is necessary to show that  2 Spess .A;T /; it is important that the sequence fun g is non-compact in Hq .˝/. The details can be found in [20], where the functions un are defined so that they have compact supports Sn in ˝; and the intersection \1 nD1 Skn (where fSkn g is any subsequence of fSn g) is empty. (i) The opposite inclusion Spess .A;T /  ;T : if  2 Spess .A;T /; then, by definition, nul 0 .A;T I / D 1 or def 0 .A;T I / D 1. The latter means that nul 0 .A;T  I / D 1. Thus, nul 0 .A;T  I / may be assumed to be infinity; otherwise one can deal with the operator A;T instead of A;T . The identity nul 0 .A;T  I / D 1 enables one to construct, by mathematical induction, kun kq D 1; k.A;T  I /un kq ! 0 a sequence fun g in D.A  n ;T / satisfying  as n ! 1 and dist u I Ln1 D 1 for all n 2 N; where Ln1 denotes the linear hull of the functions u1 ; : : : ; un1 . Using a cutoff function technique, the functions un can be modified so that they are all supported for jxj >  (for sufficiently large ), and the modified functions (let us denote them e un ) are on the unit sphere in Hq .˝/ and satisfy k.A;T  I /e un kq ! 0 for n ! 1 as un can be considered to be a function from well. However, as suppe un  ˝; e D..A;T /R3 /; where .A;T /R3 denotes the operator A;T in Hq .R3 /. This yields the equality nul 0 ..A;T /R3  I / D 1; which implies, due to item (c) that  2 ;T . (j) The domain  2 C X ;T consists of points in Res.A;T / and possibly also of isolated eigenvalues of A;T with finite algebraic multiplicities, which may possibly cluster only at points of @;T . (See [69, pp. 243, 244].) Assume that

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 2 C X ;T is an eigenvalue of A;T with an eigenfunction u. Applying again an appropriate cutoff function technique and treating the equation .A;T  I /u D 0 separately in a bounded domain ˝ (for a sufficiently large ) and in the whole space R3 ; one can show that u is in W 2;s .˝/ for any 1 < s < 1. (This follows from estimates valid in a bounded domain and the result from item (c), implying that  in the resolvent set of .A;T /R3 .) Finally, multiplying the equation .A;T  I /u D 0 by u and integrating in ˝; one can show that Re  < 0. t u When q D 2; in [19] it is shown that if B is axially symmetric about the x1 axis, then Sp.A;T / D ;T . It means that the set of eigenvalues of A;T ; lying outside ;T ; is empty. The same statement for operator A;T in Hq .˝/ for general q 2 .1; 1/ follows from Theorem 14. The proof in [19] comes from the fact that an eigenfunction u; corresponding to a hypothetic eigenvalue ; is 2-periodic in the cylindrical variable, which is the angle ' measured about the x1 -axis. Then the proof uses the Fourier expansion of u in ' and splitting of the equation .A;T  I /u D 0 to individual Fourier modes. Open Problem. In the general case when body B is not axially symmetric, it is not known whether the set of eigenvalues of A;T in C X ;T is empty.

Finally, note that Sp.A0;T / can be formally obtained, by letting  ! 0 in ;T . Then ;T shrinks to a system of infinitely many equidistant half-lines. The spectrum of operator A0;T is studied in detail in [22].

10.2

A Semigroup, Generated by the Operator A,T

10.2.1 The Case T D 0 It is well known that the Stokes operator A generates a bounded analytic semigroup, eAt ; in Hq .˝/ [55]. The fact that the Oseen operator A;0  ACB1 also generates an analytic semigroup in Hq .˝/ was proved by T. Miyakawa [89]. The main tool is the inequality kB1 ukq   kAukq C C ./ kukq

(107)

for all u 2 D.A/ and  > 0; which implies that B1 is relatively bounded with respect to A with the relative bound equal to zero. Then the existence and analyticity of the semigroup e.ACB1 /t follow, e.g., from [69, Theorem IX.2.4]. The so-called Lr –Lq estimates of the semigroup eA;0 t play an important role in the analysis of stability of steady flow. They were first derived by T. Kobayashi and Y. Shibata in [70], whose main result is given next.

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Theorem 15. If 1 < r  q < 1; then there exists C D C .; q; r/ > 0 such that keA;0 t akq  C t

 32





1 1 r q

kakr

(108)

for all a 2 Hr .˝/ and t > 0. Moreover, if 1 < r  q  3; then jeA;0 t aj1;q  C t

 32



1 1 r q



 12

kakr

(109)

for all a 2 Hr .˝/ and t > 0. Sketch of the proof (see [70] for the details.) Following [70], the starting point is the following representation formula of the semigroup eA;0 t a D

1 2it

Z

!Ci1

@ .A;0  I /1 a d ; ! > 0: @

et !i1

In order to estimate .A;0  I /1 a; the Oseen resolvent problem .A;0  /a D f is split into the problem in the bounded domain ˝ (for sufficiently large ) and in the whole space R3 . The estimates in ˝ follow from the fact that the Oseen operator in ˝ has a compact resolvent and the spectrum (which coincides with the point spectrum) is in the left half-plane in C; with a positive distance from the imaginary axis. The estimates in R3 are obtained by means of the Fourier transform and the Mikhlin-Lizorkin multiplier theorem. The next step is the construction of a parametrix, which enables the authors to combine the estimates in ˝ and in R3 and obtain the estimates of j.A;0  I /1 aj2;r and jj k.A;0  I /1 akr in terms of C kakr in the exterior domain ˝. Then the limit procedure for ! ! 0C is considered. However, due to subtle technical reasons, the limit procedure works only in a norm over a bounded domain and one only gets the inequality 2 A;0 t k@m akrI ˝  C t 3=2 kakr t r e

(110)

for t  1n and a 2 Hr .˝/ with the support in ˝ ; where C D C .m; r; ; /. On the other hand, using the formula  u.x; t / D

1 4t

3=2 Z

2 =4t

ejxtyj

a.y/ d y:

R3

for solution of the unsteady Oseen equations 9 ut C u C  @1 u D 0 = div u D 0

;

in R3 .0; 1/

(111)

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with the initial condition u.x; 0/ D a.x/ (for x 2 R3 ), one can derive the estimate j

k@t r k u.t /kqI R3  C t

 32



1 1 r q



 k2

kakr

(112)

for all a 2 Hr .R3 / and t  1. The constant C on the right-hand side depends only on j; k; q; r; and . Finally, combining appropriately (110) with (112), one can obtain (108) and (109). 

10.2.2 The Case T 6D 0 The operator A;T  A C B1 C T B2 is the Oseen operator with the effect of rotation. T. Hishida [62] considered the case  D 0 and proved that A0;T  ACT B2 generates a C0 -semigroup eA0;T t in H .˝/. M. Geissert, H. Heck, and M. Hieber [53] also considered  D 0 and proved that A0;T generates a C0 -semigroup eA0;T t in Hq .˝/ for 1 < q < 1. The case  6D 0 was studied by Y. Shibata in [102], whose main finding is given next. Theorem 16. Let 1 < r  q < 1. The operator A;T generates a C0 -semigroup eA;T t in Hq .˝/. Moreover, it satisfies the same inequalities (108) and (109) as the semigroup eA;0 t . Sketch of the proof (see [102] for the details.) One begins to study the linear Cauchy problem, defined by the equations 9 ut C u C  @1 u C T .e 1 x  ru  e 1 u/ D rp = div u D 0

;

in R3 .0; 1/ (113)

and the initial condition u.x; 0/ D a.x/. The solution u.x; t / is expressed by means of the Fourier transform, and the solution of the corresponding resolvent problem with the resolvent parameter  (denoted by AR3 ;T ; ./) is expressed by the combined Laplace-Fourier transform. Then the estimates of kAR3 ;T ; ./kqI R3 and jAR3 ;T ; ./jm;qI R3 in terms of powers of jj and kakq are derived. The solution AR3 ;T ; ./ is then split into the part A1 ./; which “neglects” the term T .e 1 x  ru  e 1 u/ in (113), and A2 ./; which is a correction due to this term. While the estimates of A1 ./ are shown in a similar way and for the same values of  as in the proof of Theorem 15, the estimates of A2 ./ impose sharper restrictions on  and hold only for  2 CC WD f 2 CI Re  > 0g. However, remarkably enough, in [102], subtle estimates of A2 . C is/ (for  > 0 and s 2 R) are derived independent of  (for 0 <  < 0 ), provided a has a support in BR .0/ for R > 0. The essential role in the expression of the solution of (113) is played by the integral of A2 ./ on the line f D  C isI s 2 Rg; parallel to the imaginary axis. The estimates independent of  enable one to pass to the limit for  ! 0. Then the appropriate

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cutoff function procedure and the limit process for R ! 1 lead to an expression that confirms that u. : ; t / depends on the initial datum a through a C0 -semigroup. One can immediately observe from the shape of the spectrum of the operator A;T (see Theorems 13 and 14) that A;T is not a sectorial operator in Hq .˝/. Thus, unlike the case T D 0; the semigroup generated by A;T is only a C0 -semigroup and not an analytic semigroup. The idea used to derive estimates analogous to (108) and (109) is similar to that employed in the proof of Theorem 15. However, in contrast to the case T D 0 (when, expressing the solution by the line integral on a line parallel to the imaginary axis, one especially needs to control the behavior of the resolvent for the values of the resolvent parameter  near 0), the case T 6D 0 requires the control of the resolvent “uniformly” on the whole line. This is caused by the fact that as the line approaches the imaginary axis in the considered limit procedure, it approaches the spectrum of operator A;T not only in the neighborhood of 0 but in the neighborhood of the infinitely many points ik T ; k 2 Z. 

10.3

Existence and Uniqueness of Solutions of the Initial–Boundary Value Problem

This subsection presents a brief survey of results on the existence and uniqueness of weak and strong solutions to the initial–boundary value problem, consisting of equation (5) and the initial condition v.x; 0/ D a.x/

for x 2 ˝:

(114)

Referring to other chapters in this handbook for a detailed analysis, here only those results are recalled that are relevant to our study. The definition of the weak solution, in the unsteady case, is analogous to that provided for the steady state problems (7), (8). More precisely, v is called a weak solution to problem (5), (114) if: (i) v 2 L1 .0; T I H .˝// \ L2 .0; T I D 1;2 .˝//; for all T > 0: (ii) lim kv.t /  ak2 D 0. t!0C

(iii) v satisfies (5) in the sense of distributions.

10.3.1 The Case T D 0 In the absence of rotation, existence results can be found in many works. They mostly concern the Navier-Stokes equations, but their extension to the more general problem (5), (114) with T D 0 is rather straightforward. The first results on the global in time existence of weak solutions, v; assuming the initial velocity a 2 H .˝/; are due to J. Leray [82] (for ˝ D R3 ) and E. Hopf [64] (for arbitrary open set ˝  R3 ). A more recent and detailed presentation of these classical results can be found, e.g., in the book [104] or in the survey paper [37]. In particular, one shows the existence of a weak solution for any a 2 H .˝/

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and f 2 L2 .0; T I D01;2 .˝// (and v  0 at @˝). However, the uniqueness of such solutions in the same class of existence remains an open problem. The weak solution is known to be unique if, in addition, it is in Lr .0; T I Ls .˝//; where 2  r  1; 3  s  1; and 2=r C3=s  1. More precisely, if v1 and v2 are two weak solutions, with v1 in the class Lr .0; T I Ls .˝// above and v2 satisfying the so- called energy inequality (see (119) with v0  0 and s D 0), then v1 D v2 . As the solutions in the class Lr .0; T I Ls .˝// satisfy the energy inequality automatically, one can speak of “uniqueness in the class Lr .0; T I Ls .˝//.” Following [104], a weak solution v in the class Lr .0; T ; Ls .˝// with r; and s as above, is called a strong solution. In addition to be unique, strong solutions are also known to be “smooth” (= regular), provided that the body force f is either a potential vector field (and can be therefore absorbed by the pressure term) or “sufficiently smooth.” (See, e.g., [37] for more details.) In particular, if @˝ is of class C 2 and f 2 L2 .0; T I L2 .˝//; then the strong solution v belongs to C ..; T /I H .˝// \ L2 .; T I W 2;2 .˝// for any  2 .0; T /. (It depends on the regularity of the initial velocity a whether  D 0 can also be considered.) For initial velocity a and body force f in appropriate function spaces and of “arbitrary size,” strong solutions are known to exist in some time interval .0; T0 /; but it is not known whether one can take T0 D 1; in general. If, however, the size of the data is sufficiently restricted, then one can show T0 D 1. There exists a vast literature on the subject dealing with various types of domains and different choices of functions spaces for a and f ; starting from the pioneering and fundamental papers of A. A. Kiselev and O. A. Ladyzhenskaya, G. Prodi, and H. Fujita and T. Kato in the early 1960s and continuing with J.G. Heywood (1980), T. Miyakawa (1982), H. Amann (2000), and R. Farwig, H. Sohr, and W. Varnhorn (2009). Among these papers, especially [60] (by Heywood), [89] (by Miyakawa), and [1] (by Amann) deal with the Navier-Stokes problems in exterior domains. An important result concerning the length of the time interval .0; T0 /; where a strong solution exists without restriction on the “size” of the data, states (e.g., [1] or [59]) that if f   is, e.g., in L2 0; 1I L2 .˝/ ; then either T0 D 1 or else jv. : ; t /j1;2 ! 1 for t ! T0  .

10.3.2 The Case T 6D 0 The existence of a weak solution to the problem (5), (114) with T 6D 0 has been proven by W. Borchers [7]. It also follows from a more general result proven in [94] on the existence of weak solutions in domains with moving boundaries. Recall that the weak solution is a function in the same class as for the case T D 0; i.e., belongs to L1 .0; T I H .˝// \ L2 .0; T I W01;2 .˝//. Regarding the local in time existence of a strong solution to the problem (5), (114) with T 6D 0; only a few results are available. Below the contributions of T. Hishida [62], G. P. Galdi and A. L. Silvestre [40], and P. Cumsille and M. Tucsnak [11] are explained. They all concern the case when the motion of body B in the fluid reduces to the rotation and the translational velocity is zero. It means that the term  @1 v in the momentum equation (5)1 vanishes.

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T. Hishida [62] assumes that the body force f is zero and the initial velocity is in D.A1=4 / and proves the existence of a solution in the class C .Œ0; T0 ; D.A1=4 // \ C ..0; T0 I D.A// for certain T0 > 0. Recall that A denotes the Stokes operator. Hishida’s proof is based on a nontrivial generalization of the semigroup method, formerly used by Fujita and Kato [30]. G. P. Galdi and A. L. Silvestre [40] also deal with the case of the zero body force f . They assume that the initial velocity a in W 2;2 .˝/ satisfies div a D 0 and e 1 x  ra 2 L2 .˝/; and they obtain a solution in C .Œ0; T0 I W 1;2 .˝// \ C ..0; T0 /I W 2;2 .˝// for some T0 > 0. The proof is based on the construction of classical Faedo-Galerkin approximations in ˝R ; getting a solution in ˝R and letting R ! 1. The procedure is, however, not standard because of the “troublesome” term T e 1 x  rv whose influence has to be controlled. P. Cumsille and M. Tucsnak [11] consider the equations of motion of the viscous incompressible fluid around body B in a frame in which the velocity of the fluid vanishes in infinity and the body is rotating with a constant angular velocity about one of the coordinate axes. Thus, the domain filled in by the fluid is time dependent and it is denoted by ˝.t /. The authors consider a body force f locally square integrable from .0; 1/ to W 1;1 .R3 / and the no-slip boundary condition for the velocity on @˝.t/. The main theorem from [11] says that if the initial velocity a is in W01;2 .˝.0// and it is then there 0 > 0 and  divergence-free,   exists T1;2  a unique 2 2;2 strong solution u 2 L 0; T I W .˝.t // \ C Œ0; T I W .˝.t // such that 0 0   ut 2 L2 0; T0 I L2 .˝.t // . Moreover, either T0 can be extended up to infinity or the norm of u in W 1;2 .˝.t // tends to infinity for t ! T0 . In order to obtain a problem in a fixed exterior domain, the authors use a change of variables which coincides with the rotation in the neighborhood of body B; but it equals the identity far from the body. Then they solve the problem in the fixed exterior domain ˝. Using the relations between the solutions of the equations in the frame considered in [11] on the one hand and the body fixed frame on the other hand, the result of Cumsille and Tucsnak from [11] can be reformulated in terms of solution to the problem (5), (114), as follows: given a 2 H .˝/ \ W01;2 .˝/; there exists T0 > 0 and a unique solution v of the problem (5), (114), such that )     v 2 L2 0; T0 I W 2;2 .˝/ \ C Œ0; T0 I W 1;2 .˝/ ;   vt  T .e 1 x  rv  e 1 v/ 2 L2 0; T0 I L2 .˝/ :

(115)

Cumsille and Tucsnak’s result is applied in Sect. 10.5. Since it is also used in the case  6D 0; we note that following the proof in [11] and using the fact that the translation-related term  @1 v in the first equation in (5) can be considered to be a subordinate perturbation of v; the above formulated result can be extended to the case when it also includes the translation of B in the direction parallel to the axis of rotation. Consequently, the result also holds for the equations in (5) with the term  @1 v and the second inclusion in (115) can be modified:   vt   @1 v  T .e 1 x  rv  e 1 v/ 2 L2 .0; T0 I L2 .˝/ :

(116)

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10.4

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Attractivity and Asymptotic Stability with Smallness Assumptions on v0

Recall that v0 denotes (the velocity field of) a solution to problem (8) (i.e., a steadystate solution to problem (5)) and that its associated “perturbation” v satisfies (100) and (101). Since in studying longtime behavior the dependence of v on time is more relevant than the one on spatial variables, in the following considerations, v.x; t / is often abbreviated to v.t /. Thus, for example, the initial condition (114) may be written in the form v.0/ D a:

(117)

10.4.1 The Case T D 0 A number of results concern the longtime behavior of the unsteady perturbations v in the class of weak solutions. The first relevant contribution in this direction is due to K. Masuda [88], who assumes that v0 is continuously differentiable, rv0 2 L3 .˝/ along with the smallness condition which, according to our notation, yields sup  jxj jv0 .x/j < x2˝

1 : 2

(118)

The perturbed unsteady solution is supposed to satisfy the momentum equation in (5) with a perturbed body force. Thus, the corresponding perturbation v satisfies (100), (101), with an additional right-hand side f 0 in (100), representing the perturbation to the steady body force f. The Helmholtz-Weyl projected function Pf 0 is assumed to be in C 1 .Œ0; 1/I H .˝// \ L1 .0; 1I H .˝// and such that R tC1 R1 supt>0 t k.d =ds/Pf 0 .s/k22 ds C 0 s 1=2 k.d =ds/Pf 0 .s/k2 ds < 1. As for the class of perturbations, the author assumes that v is a weak solution to the problem (100) (with a nonzero f 0 on its right-hand side), (101), and (117), with initial data a 2 H .˝/; and satisfies the so-called strong energy inequality, namely, 1 kv.t /k22 2

 12 kv.s/k22 

Z s

t



.v. /  rv0 ; v. // C jv. /j21;2 C .f 0 . /; v. // d ; (119)

for a.a. s > 0 (including s D 0) and all t 2 Œs; T ; arbitrary T > 0. Notice that the latter is formally obtained by multiplying equation (100) by v and integrating over ˝ .s; t / and relaxing the equality sign to the inequality one. Under the above conditions, Masuda shows that there exists T > 0 such that v.t / becomes regular for t > T and decays at the following rate jv.t /j1;2  C t 1=4 ; kv.t /k1  C t 1=8 ; for all t > T :

(120)

The proof uses (119) and the assumptions on the integrability of f 0 to deduce, first, that v.t / is “small” for large t . Then, combining this with the estimates of A1=2 v;

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one shows that v.t / is regular and tends to zero for t ! 1 in the norm j : j1;2 . The rate of decay is calculated from the energy-type inequality, satisfied by vt . The author also generalizes these results to the case when the unperturbed solution v0 is time dependent. It should be noted that in [88], no assumption on the size of the initial perturbation a is used: it can be arbitrarily large. However, nothing can be said about the behavior of v.t / for t 2 .0; T /. If v0  0; the decay rates (120) are sharpened by J.G. Heywood in [60]. The above results have been further elaborated on by P. Maremonti in [86]. Maremonti studied the attractivity of steady as well as unsteady solutions v0 to problem (5) in the same class of weak solutions considered by Masuda with f 0  0. In particular, for the case v0 steady, he shows the following decay rates kvt .t /k2  C t 1 ; jv.t /j1;2  C t 1=2 ; kv.t /k1  C t 1=2 ; thus improving and extending the analogous finding of [88] and [60]. Instead of condition (118), the author assumes that the maximum of certain variational problem involving v0 is not “too large.” The latter condition is certainly satisfied if v0 is sufficiently regular and obeys (118). The somehow more complicated question of asymptotic stability of v in the L2 norm was first addressed by P. Maremonti in [87]. In particular, he shows that all v in the class of weak solutions, with a 2 H .˝/ and satisfying the strong energy inequality (119) with f 0  0; must decay to 0 in the L2 -norm, provided that the magnitude of v0 is restricted in the same way as specified in [86] and discussed earlier on. An important contribution to the studies of the asymptotic stability of the steady solution v0 was also made by T. Miyakawa and H. Sohr in [90]. The authors show that if the basic steady solution v0 of (5) is such that v0 2 L1 .˝/; rv0 2 L3 .˝/; and the smallness condition (118) is satisfied, and if, in addition, the perturbation f 0 to the body force f is in L2 .Œ0; T /I H .˝// for all T > 0 and in L1 .Œ0; 1/I H .˝// ; then the L2 -norm of each weak solution v to problem (100), (101) satisfying the energy inequality (119) tends to 0 for t ! 1. In [90] it is also shown that the class of such weak solutions is not empty, thus solving a problem left open in [87] and partially solved in [33]. Further results concerning the L2 -decay of the perturbation v.t / (as a weak solution to (100), (101)) for t ! 1 are provided in the paper [8] by W. Borchers and T. Miyakawa: the authors assume that the steady solution v0 of (5) is in L3 .˝/; rv0 2 L3 .˝/; the smallness condition (118) is satisfied, and the perturbation f 0 to the body   force f is 1;2 2 1 1 in Lloc .Œ0; 1/I H .˝// \ L .0; 1I H .˝// \ L 0; 1I D0 .˝/ . They show that then the L2 -norm of each weak solution v to problem (100), (101) obeying (119) tends to 0 for t ! 1. Moreover, if keLt ak2 D O.t ˛ / for some ˛ > 0; then kv.t /k2 D O .ln t /1=2 for any  > 0. (Here, eLt denotes the semigroup generated by operator L; see Sect. 10.5.1.) The results of [8] are generalized by the same authors to the case of n-space dimensions (n  3) in [9].

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Even sharper rates of decay of the norms kv.t /kr (2  r  1) and krv.t /kr (2  r  3) were obtained by H. Kozono f 0 to  in [75], provided  the perturbation 1  1 2 2 the body force is in L 0; 1I L .˝/ \C .0; 1/I L .˝/ and decays like t for t ! 1. Kozono does not use any condition of smallness of the basic flow v0 or its initial perturbation a; but needs v0 in Serrin’s class Lr .0; 1I Ls .˝// (2=r C3=s D 1; 3 < s  1). This implies that v0 is in fact a strong Solution, and it is in a suitable sense small for large t. Obviously, the only time-independent solution in the considered Serrin class is v0 D 0. There exists a series of results on stability of solution v0 in the class of strong unsteady perturbations, which, unlike the cited papers [8, 9, 60, 86, 88], and [75], provide an information on the size of the perturbations at all times t > 0 and not just for “large” t . However, on the other hand, the initial value of the perturbation is always required to be “small” as well as v0 is also supposed to be “sufficiently small” in appropriate norms. The first results of this kind come from the early 1970s of the twentieth century, and new results on this topic still appear. The next paragraphs contain the sketch of the main steps to obtain a result of the above type. Assume that v is a strong solution to problem (100), (101) in the time interval .0; T0 /; for some T0 > 0. Multiplying the first equation in (100) (where v0 D v) by v and integrating by parts over ˝; one obtains 1 d kvk22 C jvj21;2 D  .v  rv0 ; v/   jv0 j1;3=2 kvk26 2 dt  C72  jv0 j1;3=2 jvj21;2 :

(121)

(The norm kvk6 has been estimated by Sobolev’s inequality: kvk6  C7 jvj1;2 ; see, e.g., [45, p. 54].) Multiplying the first equation in (100) by Av  Pv and integrating over ˝; one obtains 1 d jvj21;2 C kAvk22 D  ..@1 v C v0  rv C v  rv0 C v  rv/ ; Av/ d x 2 dt   1  kAvk22 C 42 k@1 vk22 C kv0  rvk20 C kv  rv0 k22 C kv  rvk22 : (122) 4 The first term on the right-hand side can be absorbed by the left- hand side. The other terms on the right-hand side can be estimated by means of the inequalities 1=2 1=2   jvj1;6  C krvk1;2 D C jvj21;2 C jvj22;2  C kAvk22 C jvj21;2 ; where the first one follows from the continuous imbedding W 1;2 .˝/ ,! L6 .˝/ and the second one follows, e.g., from [45, pp. 322–323]. Thus, if YŒv is defined by the formula YŒv WD kAvk22 C jvj21;2 ;

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then k@1 vk22  22 jvj21;2 ; kv0  rvk22  kv0 k26 jvj1;2 jvj1;6  C jv0 j21;2 jvj1;2 YŒv1=2     kAvk22 C C ./ jv0 j21;2 C jv0 j41;2 jvj21;2 ; kv  rv0 k22  jv0 j21;3 kvk26  C72 jv0 j21;3 jvj21;2 ; kv  rvk22  kvk26 jvj21;3  C72 jvj31;2 jvj1;6  C jvj31;2 YŒv1=2  C jvj21;2 YŒv: Employing these inequalities into (122) and choosing, e.g.,  D 14 ; one gets   d jvj21;2 C kAvk22  C8 2 1 C jv0 j21;2 C jv0 j41;2 jvj21;2 C C9 2 jvj21;2 YŒv: dt (123) Adding the inequalities (121) (multiplied by 2) and (123) (multiplied by ˛ > 0) and passing everything to the left-hand side, one obtains  d  kvk22 C ˛ jvj21;2 dt  

C jvj21;2 2  2C72  jv0 j1;3=2  C8 2 ˛ 1 C jv0 j21;2 C jv0 j41;2  C9 2 ˛ jvj21;2

C kAvk22 ˛  C9 2 ˛ jvj21;2  0: This implies that    d  kvk22 C ˛ jvj21;2 C jvj21;2 2  2C72  jv0 j1;3=2  C8 2 ˛ 1 C jv0 j21;2 C jv0 j41;2 dt  

 

C9 2 kvk22 C ˛ jvj21;2 C kAvk22 ˛  C9 2 kvk22 C ˛ jvj21;2  0: (124) This inequality shows that if   2C72  jv0 j1;3=2 C C8 2 ˛ 1 C jv0 j21;2 C jv0 j41;2 < 2

(125)

and kvk22 C ˛ jvj21;2 is initially so small that   C9 2 kak22 C ˛ jaj21;2 ˚    < min 2  2C72  jv0 j1;3=2  C8 2 ˛ 1 C jv0 j21;2 C jv0 j41;2 I ˛

(126)

(recall that v.0/ D a), then kv.t /k22 C˛ jv.t /j21;2 is nondecreasing for t in some right neighborhood of 0. This consideration can be simply extended, by the bootstrapping

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argument, to the whole interval of existence of the strong solution v (let it be .0; T0 /) so that one obtains: kv.t /k22 C ˛ jv.t /j21;2 < kak22 C ˛ jaj21;2 for all t 2 .0; T0 /. This shows, among other things, that the norm kv.t /k1;2 cannot blow up when t ! T0 . Consequently, T0 D 1 and the inequality kv.t /k22 C ˛ jv.t /j21;2 < kak22 C ˛ jaj21;2 holds for all t 2 .0; 1/. Note that if C72  jv0 j1;3=2 < 1;

(127)

then one can choose ˛ > 0 so small that (125) holds. (˛ is further supposed to be chosen in this way.) Integrating inequality (124) with respect to t; one can also derive an information on the integrability of YŒv.t / and on the asymptotic decay of jv.t /j1;2 . Thus, also including the information on the uniqueness of strong solutions and using the result of [87], one obtains the following Lyapunov-type asymptotic stability of v0 in the W 1;2 -norm. Theorem 17. Suppose the steady solution v0 to the problem (8) satisfies conditions (99) and (127) and ˛ > 0 is chosen so that (125) holds. Then, if a 2 H .˝/ \ W01;2 .˝/ satisfies (126), problem (100), (101) with the initial condition v.0/ D a has a unique strong solution v on the time interval .0; 1/. Furthermore, there exists C10 > 0 such that this solution satisfies Z t   jv.s/j21;2 C ˛ kAv.s/k22 ds  kak22 C ˛ jaj21;2 kv.t /k22 C ˛ jv.t /j21;2 C C10 0

(128) for all t > 0 and lim kv.t /k1;2 D 0:

t!1

(129)

The noteworthy assumption in the above theorem is condition (127) of “sufficient smallness” of the solution v0 . The ideas of proof described previously and similar energy-type considerations have been applied to many other studies of stability or instability of steady-state solutions to Navier-Stokes and related equations. Concerning flows in exterior domains, the readers are referred, e.g., to [31, 32, 34, 58, 59]. A different approach, based on a representation of a solution by means of semigroups generated by the operators A;0 or L and on estimates of the semigroups, has been employed by H. Kozono and T. Ogawa [73], H. Kozono and M. Yamazaki [74], and Y. Shibata [100]. In particular, Kozono and Yamazaki [74] study the flow in an exterior “smooth” domain ˝ in Rn (n  3), under the assumption that the translational velocity of the moving body is zero, which in our notation means  D 0 in the first equation in (100). The steady-state solution v0 is supposed to belong to Ln;1 .˝/\L1 .˝/; and its gradient is supposed to be in Lr .˝/ for some

r;q r 2 .n; 1/. (The Lorentz-type space L .˝/ for 1 < r < 1 and 1  q  1

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  is defined by means of the real interpolation to be Hr0 .˝/; Hr1 .˝/ ;q ; where 1 < r0 < r < r1 < 1 and 1 <  < 1 satisfy 1=r D .1   /=r0 C =r1 ; see r;q [74]. It is shown in [9] that L .˝/ coincides with the space of all u 2 Lr;q .˝/ such that div u D 0 in ˝ in the sense of distributions and u  n D 0 on @˝ in the sense of traces.) Equations (100) are treated in the equivalent form (102). The operators L and N are defined in the introductory part of this section for n D 3; but the definition in the general case n 2 N; n  3 is analogous. In the considered case, L has the concrete form L D A C B3 . The operator L generates a quasi-bounded analytic semigroup in Hq .˝/ – this is shown in [74] by means of appropriate resolvent estimates which imply that operator L is sectorial. The strong solution is identified with the mild solution, which satisfies the integral equation v.t / D e

Lt

Z

1

aC

eL.ts/ Nv.s/ ds:

(130)

0

The solution of this equation is constructed as a limit of a sequence of approximations, which are defined by the equations v0 WD eLt a and j

Z

0

t

v .t / WD v .t / C

eL.ts/ Nvj 1 .s/ ds

.j D 1; 2; 3; : : : /:

0 n



1



1



The authors define Kj WD sup0 0 be fixed. The operator A C .1 C /B3s is self-adjoint in H .˝/. The spectrum of AC.1C/B3s consists of Spess .A C .1 C /B3s / D .1; 0 and at most a finite set of positive eigenvalues, each of whose has a finite multiplicity. Let the positive eigenvalues be 1  2      N; each of them being counted as many times as is its multiplicity. Let 1 ; : : : ; N be the associated eigenfunctions. They can be chosen in a way that they constitute an orthonormal system in H .˝/. – Denote by H .˝/0 the linear hull of 1 ; : : : ; N and by P 0 the orthogonal projection of H .˝/ onto H .˝/0 . Furthermore, denote by H .˝/00 the orthogonal complement to H .˝/0 in H .˝/ and by P 00 the orthogonal projection of H .˝/ onto H .˝/00 . Then H .˝/ admits the orthogonal decomposition H .˝/ D H .˝/0 ˚ H .˝/00 ; and the operator A C .1 C /B3s is reduced on each

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of the subspaces H .˝/0 and H .˝/00 . Moreover, it is positive on H .˝/0 and nonpositive on H .˝/00 .   – Since A C .1 C /B3s ;   0 for all  2 H .˝/00 \ D.A/; operator L satisfies    .A; / .L; / D .A C B3s /;  2 D 1C  1   .A; / D C12 jj21;2 ; .A C B3s C B3s /;   C 1C 1C for all  2 H .˝/00 \ D.A/; where C12 D =.1 C /. This inequality expresses the so-called essential dissipativity of L in space H .˝/00 . – All functions 1 ; : : : ; N belong to D01;2 .˝/ (the dual to D01;2 .˝/). The main result of the paper [93] is the following. Theorem 21. Suppose that the steady solution v0 to the problem (8) satisfies conditions (99), and let  > 0 be so large that jv0 j1;3=2;˝   18 . Moreover, assume (A) there exists a function ' 2 L1 .0; 1/\L2 .0; 1/ such that keLt i k2I ˝  '.t/ for all i D 1; : : : ; N and t > 0. Then there are positive constants ı; C13 ; C14 such that if a 2 H .˝/ \ W01;2 .˝/ and kak1;2  ı; the equation (102) with the initial condition v.0/ D a has a unique solution v on the time interval .0; 1/. The solution satisfies kv.t /k21;2

Z C C13 0

t



 jv.s/j21;2 C kAv.s/k22 ds  C14 kak21;2

(135)

(for all t > 0) and lim jv.t /j1;2 D 0:

t!1

(136)

The proof is based on splitting the equation (102) into an equation in H .˝/00 ; where L is essentially dissipative, and a complementary equation, where one uses the decay of the semigroup eLt following from assumption (A). Theorem 21 tells us that the question of stability of the steady solution v0 reduces to the L1 - and L2 -integrability of a finite family of certain functions in the interval .0; 1/; i.e., condition (A). In the paper [15], the authors consider the case ˝ D R3 and show that condition (A) is indeed satisfied under some assumptions on the spectrum of L. The latter amounts to assume that all eigenvalues of L have negative real parts, without any request on the essential spectrum of L. The important tool used in [15] is the fundamental solution of the Oseen equation in R3 and the estimates for the corresponding resolvent problem.

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Sufficient conditions for the stability of the null solution to (100), in terms of eigenvalues of L and when ˝ 6 R3 ; have been recently formulated in the paper [95]. Here, the author shows at first that condition (A) in Theorem 21 can be replaced by the following one (B) Given  > 0 there exist functions ' 2 L1 .0; 1/ \L2 .0; 1/ and H .˝/ \ D01;2 .˝/ such that kj  j k1;2   ˇ Lt ˇ ˇ e i ; j ˇ  '.t/

1; : : : ;

for j D 1; : : : ; N; for t > 0 and i; j D 1; : : : ; N:

N

2

(137) (138)

ˇ ˇ (Condition (B) reduces to the requirement that ˇ eLt i ; j ˇ  '.t/ if one chooses j D j for j D 1; : : : ; N .) In [95] it is further assumed that the steady solution v0 is in Lr .˝/ for all r 2 .2; 1 and @j v0 2 Ls .˝/; for j D 1; 2; 3 and for all s 2 . 43 ; 1. This assumption is fulfilled if the acting body force f is in Lq .˝/ for all q 2 .1; q0 ; where q0 > 3; see Lemma 9. Moreover, if f has a compact support, then v0 D E.x/  m C v00 .x/; where m is a certain constant vector, E is the Oseen fundamental tensor, and v00 .x/ is a perturbation which decays faster than E.x/ for jxj ! 1; see Theorem 6. This form of v0 is used in [95], where the main result states the following. Theorem 22. Let the conditions (C1 ) there exists ı > 0 and a0 > 0 such that all eigenvalues  of operator L satisfy Re  < maxfıI a0 .Im /2 g, (C2 ) 0 is not an eigenvalue of the operator Lext be fulfilled. Then the conclusions of Theorem 21 hold. Here, Lext denotes the operator L with the domain extended to D 2;2 .˝/\D01;2 .˝/. Condition (C1 ) implies that L has no eigenvalues with nonnegative real parts. Sketch of the proof of Theorem 22 (see [95] for the details.) The proof is based on showing that Œ(C1 ) ^ (C2 ) H) (B). The function eLt i is expressed by the formula Lt

e

i D .2i/

1

Z 

et .I  L/1 i d ;

(139)

where   is a curve in C X Sp.L /; which depends on a parameter  > 0. Recall that Sp.L / consists of the essential spectrum in the half-plane f 2 CI Re   0g; see (103), and at most a countable number of isolated eigenvalues. Curve   has three parts 1 ; 2 ; and 3 ; where 1 and 3 coincide with the half-lines arg./ D  ˙ ˛; respectively, for some fixed ˛ 2 .0; =2/ and large jj. Both the curves 1 and 3 lie in the half-plane f 2 CI Re  < 0g. Since Spess .L / touches the

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imaginary axis at point 0; the curve 2 WD f 2 CI  D as 2 C .s02  s 2 / C is for  s0  s  s0 g (where a > 0 and s0 > 0 are appropriate fixed positive numbers) extends into the half-plane f 2 CI Re  > 0g. If  ! 0C; then 2 approaches 20 WD f 2 CI  D as 2 C is for  s0  s  s0 g; and consequently,   approaches  0 D 1 [ 20 [ 3 . (Number a > 0 is chosen so that Sp.L / lies on the left from  0 ; with the exceptionof point 0.) In order to verify inequality (138) in condition (B), one has to estimate eLt i; j . Since one can prove that the range of L  jD.˝/ (the adjoint operator to L reduced to D.˝/) is dense in D01;2 .˝/; in order to satisfy (137), one can choose functions j in the form j WD L  0j ; where 0j 2 D.˝/ (for j D 1; : : : ; N ). Then Z  Lt    1 et .I  L/1 i ; j d  e i ; j D 2i   Z   1 t .I  L/1 i ; .L   I C I / 0j d  e D 2i Z Z     1 1 t 0 e et  .I  L/1 i ; 0j d : (140) D i ; j d  C 2i   2i   As the integrand in the first integral on the right-hand side depends on  only through et; one can consider the limit for  ! 0C and show that the integral equals the integral on the curve  0 . A simple calculation yields that the integral on  0 ; as a function of t; is in L1 .0; 1/ \ L2 .0; 1/. (Here, it is important that  0  f 2 CI Re   0g and  0 touches the imaginary axis only at the point 0.) The treatment of the second integral on the right-hand side of (140) is much more complicated. It is necessary to derive a series of estimates of u WD .L  I /1 i; which satisfies the equation .A C B1 C B3  I /u D i:

(141)

This equation can be treated as the perturbed Oseen resolvent equation with the resolvent parameter . Especially the estimates for  62 Sp.L / in the neighborhood of 0 (hence also in the neighborhood of Spess .L /) are very subtle. They finally enable  limit for  ! 0C and show that the integral of  one to pass to the 0 t 1 e  .I  L / i ; j on curve  0 is, as a function of t; in L1 .0; 1/ \ L2 .0; 1/. The factor  plays a decisive role because it allows one to control integrand for  on the critical part of curve  0 ; i.e., near  D 0. Note that the assumption on the nonzero translational motion of body B in fluid (i.e.,  6D 0) is important because it enables one to apply the theory of Oseen equation and to obtain appropriate estimates of function u .

the the the 

Remark 9. A result similar to Theorem 22 was stated by L.I. Sazonov [98]. There, the main theorem on stability claims that the steady solution v0 is asymptotically stable in the L3 -norm if L; as an operator in H3 .˝/; does not have eigenvalues in the half-plane Re  > 0. However, the proofs of the fundamental estimates of the

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Oseen semigroup as well as of the main theorem do not contain all the necessary details, which makes it difficult to assess the validity of author’s arguments.

10.5.2 The Case T 6D 0 The results of [93] are generalized to the case T 6D 0 (i.e., B is allowed to spin at constant rate) involving the rotational motion of body B; in the paper [34] by G. P. Galdi and J. Neustupa. The steady solution v0 is assumed to satisfy the properties v0 2 L3 .˝/; @j v0 2 L3 .˝/ \ L3=2 .˝/ (j D 1; 2; 3) and the estimate jrv0 .x/j  C jxj1 for x 2 ˝. The existence of such a solution is known for a large class of body forces f; provided  6D 0; see Lemma 9 and Theorem 6. The main theorem on stability of the zero solution of equation (102) is analogous to Theorem 21, that is why it is not repeated here. The presence of the term T B2 v in the operator L defined in equation (102) causes a series of new problems that one has to face and overcome. For example, unlike the case T D 0; the time derivative of v need not be an element of H .˝/. However, one can show that .d v=dt /  B1 v  T B2 v 2 H .˝/ and Z   dv d 1  B1 v  T B2 v  v d x D kvk22 ; dt dt 2 ˝ Z   dv d 1 2  B1 v  T B2 v  Av d x D  jvj : dt dt 2 1;2 ˝ These identities play an important role in the proof of the theorem on stability. Another important step is to show that the functions ri and i (i D 1; : : : ; N ) are square-integrable with the weight jxj2 in ˝. This enables one to estimate the norm kB2 vk2 by C jvj1;2 ; which is again a crucial property in the proof of the stability result. All details can be found in [52]. Open Problem. The question whether – in analogy with the case T D 0 and paper [93] – the stability of the zero solution of equation (102) can be determined by the location of the eigenvalues of operator L is open. The difficulties related to this problem are generated by the fact that now, being T 6D 0; the operator L is no longer sectorial. Thus, even if all the eigenvalues have negative real parts, one cannot express the eLt i by a formula similar to (139), where the curve   coincides with the half-lines arg./ D  ˙ ˛ (for some ˛ 2 .0; =2/ and large jj) and touches or intersects the half-plane CC only in a small neighborhood of 0. On the contrary, the curve   must lie at the right of infinitely many points ikT (k 2 Z) on the imaginary axis, and even if one formally passes to the limit  ! 0 in order to obtain a curve  0 in the half-plane f 2 CI Re   0g; then  0 must pass through the points ikT (k 2 Z). Consequently, the integral on the right-hand side of (139) cannot be treated and estimated in the same way as in the case T D 0.

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Conclusion

The chapter is an updated survey of important known qualitative properties of mathematical models of viscous incompressible flows past rigid and rotating bodies. The models are based on the Navier-Stokes equations. Greatest attention is paid to steady problems, as well as problems that are quasi-steady in the sense that the transformed equations describing the motion of the fluid around a rotating body are steady in the body-fixed frame. The presented results concern the existence, regularity, and uniqueness of solutions (see Sects. 4–6). Section 7 deals with the spatial asymptotic properties of steady solutions, like the questions of the presence of a wake behind the body and the decay of velocity and vorticity in or outside the wake in dependence on the distance from the body. Here, the case T 6D 0 (i.e., the case when the body rotates with a nonzero constant angular velocity) is much more difficult than the case T D 0; and the relevant results are therefore of a relatively recent date. The structure of the set of steady solutions for arbitrarily large given data is studied in Sect. 8 by means of tools of nonlinear analysis, like the theory of proper Fredholm operators, corresponding mod 2 degree, etc. One of the results asserts that, to a given nonzero translational velocity and angular velocity, the solution set is generically finite and has an odd number of elements. Sect. 9 analyzes sufficient and necessary conditions for bifurcations of steady or time-periodic solutions from steady solutions. The corresponding theorems provide a theoretical explanation of the well-known phenomenon, i.e., that the properties and shape of a steady solution may considerably change if some characteristic parameters of the flow field vary. The longtime behavior of unsteady perturbations of a given steady solution v0 is finally studied in Sect. 10. This section also brings some necessary results on the existence and uniqueness of solutions. The core of the section are (1) the results on the stability of v0 under the assumption that v0 is in some sense “sufficiently small” and (2) the results that do not need any condition of smallness of v0 . Instead, they use either an assumption on a “sufficiently fast” time decay of a certain finite family of functions related to v0 or an assumption on the position of eigenvalues of a certain associated linear operator. (Here, one has to overcome the difficulties following from the presence of the essential spectrum, having a nonempty intersection with the imaginary axis.) The readers find a series of references to related papers or books inside each section. The chapter also brings the formulation of altogether eight open problems that concern the discussed topics and represent a challenge for future research.

12

Cross-References

 Large Time Behavior of the Navier-Stokes Flow  Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes

Flow  Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions  Stokes Semigroups, Strong, Weak, and Very Weak Solutions for General Domains

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Acknowledgements The authors acknowledge the partial support of NSF grant DMS-1614011 (G.P.Galdi) and the Grant Agency of the Czech Republic, grant No. 13-00522S, and Academy of Sciences of the Czech Republic, RVO 67985840 (J.Neustupa). This work was also partially supported by the Department of Mechanical Engineering and Materials Sciences of the University of Pittsburgh that hosted the visit of J. Neustupa in Spring 2015.

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8

Stokes Semigroups, Strong, Weak, and Very Weak Solutions for General Domains Reinhard Farwig, Hideo Kozono, and Hermann Sohr

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q q -Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Stokes Operator on L q Q 2.1 L -Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q q -Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Stokes Resolvent on L 2.3 Maximal Regularity and Bounded Imaginary Powers of the Stokes Q q -Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operator on L 2.4 The Stokes Operator with Robin Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . 3 The Navier-Stokes System in General Unbounded Domains . . . . . . . . . . . . . . . . . . . . . . . 3.1 Strong and Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Very Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Regularity of Mild, Weak, and Very Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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R. Farwig () Department of Mathematics, Darmstadt University of Technology, Darmstadt, Germany International Research Training Group Darmstadt-Tokyo (IRTG 1529), Darmstadt, Germany e-mail: [email protected] H. Kozono Department of Mathematics, Waseda University, Tokyo, Japan Japanese-German Graduate Externship Program, Japan Society of Promotion of Science, Tokyo, Japan e-mail: [email protected] H. Sohr Faculty of Electrical Engineering, Informatics and Mathematics, Department of Mathematics, University of Paderborn, Paderborn, Germany e-mail: [email protected]; [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_8

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Abstract

To solve the (Navier-)Stokes equations in general smooth domains   Rn , the Q q ./ defined as Lq \L2 when 2  q < 1 and Lq CL2 when 1 < q < 2 spaces L have shown to be a successful strategy. First, the main properties of the spaces Q q ./ and related concepts for solenoidal subspaces, Sobolev spaces, Bochner L spaces, and the corresponding Helmholtz projection and Stokes operator will be discussed. Then these concepts are used to construct and analyze very weak, weak, mild, and strong solutions to the instationary (Navier-)Stokes equations in general domains. In particular, the strategy allows to find weak solutions of the (Navier-)Stokes system satisfying the localized energy inequality and the strong energy inequality which are important in the context of Leray structure theorem and partial regularity results.

1

Introduction

One of the basic functional analytic tools in the analysis of the instationary NavierStokes system is the Stokes operator A D P  where P is the Helmholtz projection mapping a vector field onto its solenoidal part such that gradient fields will be canceled. To be more precise, let 1 < q < 1 and   Rn be a smooth domain like a bounded or exterior one. Then Pq W Lq ./ ! Lq ./;

u 7! u0 D Pq u;

is the uniquely defined bounded projection vanishing on gradient fields such that in the weak sense, div u0 D 0 and u0  N D 0 on @. Here N D N .x/ denotes the q exterior normal at x 2 @ and L ./ is the space of solenoidal Lq -vector fields 1 ./ D fu 2 with vanishing normal on @ defined as the closure in Lq ./ of Cc; q 1 n Cc ./ W div u D 0g. With its range R.Pq / D L ./ and its kernel N .Pq / D G q ./ where G q ./ is the homogeneous Sobolev space of gradient fields, q

G q ./ D frp 2 Lq ./n W p 2 Lloc ./g; the Helmholtz projection yields the direct and topological decomposition Lq ./n D Lq ./ ˚ G q ./;

1 < q < 1:

(1)

The gradient part rp in the Helmholtz decomposition u D u0 C rp of (1) is related to the Neumann problem p D div u in  with boundary condition N  rp D u  N on @ so that formally N  u0 D 0 on @. Rigorously, rp is the solution of a weak Neumann problem, i.e., of the variational problem .rp; r'/ D .u; r'/

0

for all r' 2 G q ./:

(2)

8 Stokes Semigroups, Strong, Weak, and Very Weak Solutions for General Domains

421

Given Pq , the Stokes operator A D Aq (in the case of Dirichlet boundary conditions) is defined by 1;q

D.Aq / D W 2;q ./ \ W0 ./ \ Lq ./;

Aq u D Pq u:

With the help of the Stokes operator and the Helmholtz projection Pq , the instationary Navier-Stokes system ut  u C div .u ˝ u/ C rp D div u D uD u.0/ D

f 0 0 u0

in in on at

.0; T /  ; .0; T /  ; .0; T /  @; t D 0;

(3)

can be rewritten as the abstract evolution problem ut C Aq u C Pq div .u ˝ u/ D Pq f in .0; T / at t D 0: u.0/ D u0 q

(4)

in the space L ./. On the other hand, the pressure gradient is recovered from (4) via rp D .I  Pq /.f C u  div .u ˝ u//. Note that in this chapter the coefficient of viscosity  is put equal to 1. However, due to counterexamples, see Bogovski˘i and Maslennikova [9, 48]; the Helmholtz decomposition does not hold for all q > 1 when   R2 is an infinite cone with “smoothed vertex” at the origin and of opening angle larger than . Similar counterexamples can be found also for n > 2: This implies that the classical strategy considering the instationary Navier-Stokes system (3) as q an abstract evolution problem (4) solved by analytic semigroup theory in L ./ breaks down. Actually, as shown by Geißert, Heck, Hieber, and Sawada [37], the existence of the bounded Helmholtz projection, i.e., the well definedness of the weak Neumann problem (2), does not only imply the definition of the Stokes operator Aq D Pq , but even the maximal regularity of Aq , a fundamental property in the analysis of linear and nonlinear equations involving the Stokes problem; for details, we refer to chapter “ The Stokes Equation in the Lp -Setting: Well-Posedness and Regularity Properties”. A similar result due to Abels [1] even shows existence of so-called bounded imaginary powers of the Stokes operator supposing that the Helmholtz decomposition exists and pressure functions enjoy a certain extension property; see also [2]. On the other hand, Hilbert space methods allow for an L2 -approach to the Helmholtz decomposition and the Stokes operator A2 D P2  for any bounded and unbounded domain. Actually, A2 is a positive self-adjoint operator generating a bounded analytic semigroup e tA2 ; t  0; for details, see the monograph of Sohr [56, Ch. III.2]. Moreover, typical constants as in resolvent estimates, semigroup, and maximal regularity estimates are independent of the domain when q D 2.

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Since an Lq -theory with 1 < q ¤ 2 < 1 of the Stokes problem is necessary to deal with the nonlinear term u  ru in weak, strong, and very weak solutions of the Navier-Stokes problem, and, in particular, for the construction of weak solutions satisfying the localized energy inequality and the strong energy inequality, the idea Q q -spaces. This concept allows to combine elements of L2 - and Lq -theory is to use L to work locally with Lq -regularity and Lq -Sobolev spaces, but globally, as jxj ! 1, with L2 -theory. This handbook article is organized as follows. In Sect. 2, the Stokes operator Q q D PQ q  on the space L Q q ./ for the class of uniform C 2 -domains   Rn (see A Definition 1) will be investigated. The properties of the Helmholtz projection PQ q and Q q -spaces are recalled in Sect. 2.1. The Stokes resolvent of Sobolev spaces based on L is analyzed in Sect. 2.2 (cf. Theorem 2), and the main ideas of the proof are given. Crucial results on the maximal regularity of the Stokes operator (see Theorem 3) are Q q are mentioned; discussed in Sect. 2.3 where also bounded imaginary powers of A Q they are important to deal with domains of fractional powers of Aq and their relation Q q; ,  2 Œ0; 1/, to Sobolev spaces. The final Sect. 2.4 considers the Stokes operator A 3 on domains of uniform C type (see Assumption 1) with Robin and Navier slip boundary condition u  N D 0, .1   /u C .T .u; p/N / D 0 on @ (cf. (44)); here the subscript  denotes the tangential part of the Cauchy stress vector T .u; p/N along @. The nonlinear Navier-Stokes system is analyzed in Sect. 3. First, strong and mild Q q .//, where 2 C n D 1, are solutions u contained in Serrin’s class Ls ..0; T I L s q constructed; see Theorem 6 when n < q < 1 and Theorem 7 for the limit case when q D n, s D 1 in which the Fujita-Kato iteration has to be used. Then the focus will be on the theory of very weak solutions which is based on duality to strong solutions of the Stokes system; see Sect. 3.2 in general and Theorem 9 for the main result on existence and uniqueness. Q q -spaces was the theory of weak solutions in the The starting point for the use of L sense of Leray-Hopf: Although weak solutions satisfying the basic energy inequality can be constructed for any domain of R2 and R3 (cf. [56, Ch. V.3]), there was the open problem to get weak solutions satisfying the localized energy inequality (suitable weak solutions) and the strong energy inequality. The construction of such weak solutions is described in Sect. 3.3. The final Sect. 3.4 uses results from the previous parts of Sect. 3 to discuss the regularity of mild, weak, and very weak solutions.

2

The Stokes Operator on LQ q -Spaces

2.1

LQ q -Spaces

Definition 1. A domain   Rn is called uniform C k -domain, k 2 N0 D N [ f0g if the following holds: There are constants ˛; ˇ; K > 0 such that for all x0 2 @

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there exists a function h W B˛0 .0/ ! R of class C k (on the closed ball B˛0 .0/  Rn1 with center 0 and radius ˛) and a neighborhood U˛;ˇ;h .x0 / of x0 such that khkC k  K; h.0/ D 0 and, if k  1; h0 .0/ D 0I U˛;ˇ;h .x0 / W D f.y 0 ; yn / 2 Rn1  RW jy 0 j < ˛; jh.y 0 /  yn j < ˇg;  .x0 / W D f.y 0 ; yn / 2 Rn1  RW jy 0 j < ˛; h.y 0 /  ˇ < yn < h.y 0 /g U˛;ˇ;h

D  \ U˛;ˇ;h .x0 /; @ \ U˛;ˇ;h .x0 / D f.y 0 ; yn / 2 Rn1  RW h.y 0 / D yn g W here an orthogonal and a translational transform depending on x0 2 @ is used to map points .x 0 ; xn / 2  to y D .y 0 ; yn /-coordinates in Rn . The triple .˛; ˇ; K/ is called a type of  and, although not uniquely determined, will shortly be denoted by  D .˛; ˇ; K/. For a constant C in some estimate, we will write C D C . / if its dependence on  depends only on ˛, ˇ, and K. By analogy, we define uniform C k; -domains, k 2 N0 , 0 <  1. Note that bounded and exterior domains are uniform C k -domains as long as the boundary is smooth enough. Q q -spaces is as follows: Let The basic definition of L (

q 2 Q q ./ WD L ./ C L ./; 1 < q < 2 ; L Lq ./ \ L2 ./; 2  q < 1

(5)

Q q ./ equipped with the norm Obviously, L ˚  kukLQ q D inf ku1 kLq C ku2 kL2 W u D u1 C u2 ; u1 2 Lq ./; u2 2 L2 ./ when 1 < q < 2, but kukLQ q D maxfkukLq ; kukL2 g when 2  q < 1 is a Banach Q q 0 ./ where q 0 D q denotes the conjugate exponent to Q q .// D L space, and .L q1 q; cf. Bergh-Löfström and Triebel [8, 60]. By the latter duality, ( kukLQ q D sup

) hu; vi 0 Q q ./ : W0¤v2L kvkLQ q0

Q q ./ locally behave as Lq -functions but that It is easy to see that functions u 2 L their global behavior at space infinity is mainly given by L2 -functions. In particular, Q q ./ D Lq ./ with equivalent norms when  is bounded. L

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Spaces of Sobolev type are introduced analogously to the definition in (5): For k 2 N and 1  q  1, let WQ

k;q

( ./ WD

W k;2 ./ C W k;q ./;

1  q < 2;

W k;2 ./ \ W k;q ./;

2  q  1:

(6)

1;q Similarly, the spaces WQ 0 ./, 1 < q < 2, and 2  q < 1, based on the classical 1;q Sobolev spaces W0 ./ and W01;2 ./, are defined. Q q - and WQ k;q .-spaces have the following properties; for a proof, see The L Riechwald [50, Ch. 1], [51]:

• Let 1  q < r  1. Then kukLQ q  kukLQ r . Q p ./, v 2 L Q q . Then uv 2 L Q r ./ • Let 1  r; p; q  1, 1r D p1 C q1 and let u 2 L and kuvkLQ r  kukLQ p kvkLQ q . • Let m 2 N, 1  q  r < 1 and   Rn be a uniform C 2 -domain. Then m;q Q r ./ ./ ,! L WQ nq if either r  1 and mq > n, or r < 1 and mq D n, or r  nmq and mq < n. In general, the latter embedding estimate kukLQ r  ckukWQ m;q cannot be improved to a homogeneous estimate of kukLQ r by krukLQ q because of the L2 -norm inherent in Q q -spaces. the definition of L Q q ./ for a domain   Rn Concerning the Helmholtz decomposition of L 1; of uniform type C , 0 <  1, the following spaces are needed; see [21, Theorem 1.2] of the authors of this article: Let

( Q q ./ L

WD

q

L ./ C L2 ./;

1 0. Q q generates an analytic semigroup e t AQ q ; t  0; in (iii) The Stokes operator A Q q  ./ with bound L Q

ke t Aq f kLQ q   M e ıt kf kLQ q  ;

Q q  ./; t  0; f 2L

(13)

where M D M .q; ı;  / > 0. It is unknown whether the usual resolvent estimate for the infinitesimal generator Q q of the analytic semigroup e t AQ q holds uniformly in the resolvent parameter A  2 C as jj ! 0. Therefore, the semigroup may increase exponentially fast and the maximal regularity estimate in Theorem 3 below is stated only for finite time Q q has often to be replaced by I C A Qq intervals. For the same reason, the operator A in the following. Proof of Theorem 2. The proof of Theorem 2 consists of several steps. 1. Auxiliary L2 - and Lq -estimates in bounded domains Q q for bounded, uniform C 1;1 -domains with a constant 2. Resolvent estimates in L depending on  when 2 < q < 1 Q q for bounded, uniform C 1;1 3. Duality arguments to get resolvent estimates in L domains with a constant depending on  when 1 < q < 2

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4. Exhaustion of an unbounded, uniform C 1;1 -domain  by a sequence of bounded uniform C 1;1 -domains k and passage to the limit k ! 1 Step 1. As consequence of Definition 1, it is easily shown that there exists a covering of  by open balls Bj D Br .xj / of fixed radius r > 0 with centers xj 2 , such that with suitable functions hj 2 C 1;1 of type  B j  U˛;ˇ;hj .xj / if xj 2 @;

B j   if xj 2 :

(14)

Here j runs from 1 to a finite number N D N ./ 2 N if  is bounded, and j 2 N if  is unbounded. Related to the covering fBj g, there exists a partition of unity f'j g; 'j 2 C01 .Rn /; such that

0  'j  1; supp 'j  Bj;

and

N X

'j D 1 or

j D1

1 X

'j D 1 on :

j D1

(15) The functions 'j may be chosen so that jr'j .x/j C jr 2 'j .x/j  C uniformly in j and x 2  with C D C . /: Moreover, as an important implication, the covering fBj g of  may be constructed in such a way that no more than a fixed number N0 D N0 . / 2 N of these balls have a nonempty intersection. If  is unbounded, then  can be represented as the union of an increasing sequence of bounded uniform C 1;1 -domains k  ; k 2 N; 1  : : :  k  kC1  : : : ;

D

1 [

k ;

(16)

kD1

where each k is of the same type .˛ 0 ; ˇ 0 ; K 0 /; see Heywood [41, p. 652]. Without loss of generality, assume that ˛ D ˛ 0 ; ˇ D ˇ 0 ; K D K 0 : To deal with the pressure and the divergence condition, the following Lemma 1 on local results is needed; its proof is found for parts (i), (ii) in the monograph of Galdi [35, III, Theorems 3.1 and 3.2], [21, Lemma 2.1], and for (iii) in [31, Theorem 3.1, (i)]. Here H denotes a smooth set of type  (for each j  N or  .0/ \ Br .0/ such that B r .0/  U˛;ˇ;h .0/ and r D j 2 N) defined as H D U˛;ˇ;h ˚  R q r. / > 0. Also the space L0 .H / WD u 2 Lq .H / W H u dx D 0 of functions with vanishing mean value on H is indispensable. Lemma 1. Let 1 < q < 1. q

1;q

(i) There exists a bounded linear operator R W L0 .H / ! W0 .H / such that q div ı R D I on L0 .H / and a constant C D C . ; q/ > 0 such that

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kRf kW 1;q  C kf kLq .H / for all f 2 L0 .H /:

(17)

  q 1;q 2;q Moreover, R L0 .H / \ W0 .H /  W0 .H / and kRf kW 2;q  C kf kW 1;q .H / q 1;q for all f 2 L0 .H / \ W0 .H /. q (ii) There exists C D C . ; q/ > 0 such that for every p 2 L0 .H / kpkq  C krpkW 1;q D C sup

n jhp; div vij krvkq 0

1;q 0

W 0 ¤ v 2 W0

o .H / :

(18)

(iii) For any f 2 Lq .H /,  2 S" where 0 < " < 2 , let u; p satisfy the Stokes resolvent equation u  u C rp D f , div u D 0 in H, u D 0 on @H . Moreover, assume that supp u [ supp p  Br .0/: Then there are constants 0 D 0 .q;  / > 0, C D C .q; ";  / > 0 such that kukLq .H / C kukW 2;q .H / C krpkLq .H /  C kf kLq .H /

(19)

if jj  0 . Next, several results on Sobolev embedding estimates for a bounded C 1;1 -domain will be mentioned. Note that the short notation k  kq may be used for the classical Lq ./-norm k  kLq ./ when the underlying domain  is known from the context. Lemma 2. Let   Rn be a bounded C 1;1 -domain, 0 < "  1 and 1 < q < 1. Then there is a constant C" D C .q; ";  / > 0 such that for all u 2 W 2;q ./ krukLq  "kr 2 ukLq C C" kukLq ;   kukLq  "kr 2 ukLq C C" kr 2 ukL2 C kukL2 ; 1 kukW 2;q  kukLq C kAq ukLq  C1 kukW 2;q C1

(20) 2  q < 1;

.u 2 D.Aq //:

(21) (22)

2;q

Proof. The proofs of (20) and (21) are reduced to the case u 2 W0 .0 / with 0  using an extension operator, the norm of which is shown to depend only on q and  : A further tool is the trivial interpolation inequality kvkr  .1="/1= kvk2 C .1   /"1=.1/ kvkq ;

(23)

with  2 .0; 1/, 1r D 2 C 1 . q For (22), see [25, Lemma 3.1]. Actually, the proof uses the same ideas as that of the resolvent estimate for uniform C 1;1 -bounded domains; see Step 2. t u The estimates (9) and (22) will be used to change between a formulation of the Stokes problem in .u; p/ (cf. (11)) and the abstract one using the Stokes operator

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Q q . The same holds for the instationary Stokes problem formulated as (31) and (34) A below; see also (4) and (3) for the nonlinear case. Recall the resolvent estimate of the Stokes operator Aq D Pq  W D.Aq / ! q L ./, 1 < q < 1, on a bounded C 1;1 -domain (cf. Farwig-Sohr [31], Giga [38] q and Solonnikov [57]): For any  2 S" , 0 < " < 2 , and f 2 L ./, the Stokes resolvent problem u C Aq u D f , div u D 0 has a unique solution u 2 D.Aq / such that kukq C kAq ukq  C kf kq ;

C D C ."; q; / > 0I

(24)

however, the precise dependence of the constant C D C ."; q; / > 0 on  remains unclear. On the other hand, if q D 2, then the Stokes resolvent problem has a unique solution u 2 D.A2 / satisfying the estimate kuk2 C kA2 uk2  C kf k2

(25)

with constant C D 1 C 2= cos " independent of ; in particular, (25) even holds for general unbounded domains. Moreover, A2 is self-adjoint and hA2 u; ui D 1

kA22 uk2L2 D kruk2L2 for all u 2 D.A2 /, cf. [56]. Step 2. For  2 †" , 0 < "
0; (26) with a constant C D C .q; "; ı;  / > 0 depending on  only through  . As in Step 1, the finite partition of unity .'j /, the sets B j  Uj WD U˛;ˇ;hj .xj / if xj 2 @ and B j D Uj   if xj 2 , cf. (14), are used. Furthermore, define   2;q wj WD R .r'j /  u 2 W0 .Uj /, and choose a constant Mj D Mj .p/ such that q p  Mj 2 L0 .Uj /. Then the pair .'j u  wj ; 'j .p  Mj // is a solution of the local equation .'j u  wj /  .'j u  wj / C r.'j .p  Mj //

(27)

D 'j f C wj  2r'j  ru  .'j /u  wj C .r'j /.p  Mj /: Concerning the term wj , we apply a Sobolev embedding, Lemma 1(i), and the interpolation estimate (23) to get for M 2 .0; 1/ that kwj kLq .Uj /  M kukLq .Uj / C C kukL2 .Uj / :

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Moreover, kr 2 wj kLq .Uj /  C krukLq .Uj / . For p  Mj (18) and (21), imply that  n jhu; vij o 1;q 0 kp  Mj kLq .Uj /  C kf ; rukLq .Uj / C sup W 0 ¤ v 2 W0 .Uj / krvkq 0    C kf ; rukLq .Uj / C kukL2 .Uj / C M kukLq .Uj / : Finally, the estimate (19) with  replaced by  C 00 where 00  0 is sufficiently large such that j C 00 j  0 for jj  ı will be applied to the local resolvent equation (27). Combining these estimates, the local inequality k'j ukLq .Uj / C k'j r 2 ukLq .Uj / C k'j rpkLq .Uj /

(28)

   C kf kLq .Uj / C kukLq .Uj / C krukLq .Uj / C kukL2 .Uj / C M kukLq .Uj / with C D C .M; q; ı; ";  / > 0 is proved. Then raise each term in (28) to the qth power, take the sum over j D 1; : : : ; N , and use the crucial property of the integer N0 introduced in Step 1 to get the global estimate (29) kukLq ./ C kr 2 ukLq ./ C krpkLq ./    C kf kLq ./ C kukLq ./ C krukLq ./ C kukL2 ./ C M kukLq ./ with C D C .M; q; ı; ";  / > 0, jj  ı. For the proof of (29), also the reverse  P q 1=q  P 2 1=2 Hölder inequality  for the reals aj D kukL2 .Uj / valid j aj j aj for q  2 is exploited. With (20) and M sufficiently small, the terms krukLq ./ and kukLq ./ from the right-hand side in (29) can be removed by absorption. Then the term kukLq ./ is removed with the help of (21) implying that   kukq C kr 2 ukq C krpkq  C kf kq C kuk2 C kuk2 C kr 2 uk2 : Finally, this inequality is combined with the L2 -estimate (25) for jj  ı and (22) with q D 2 is applied. This proves the estimate (26) for 2  q < 1. q

q

Step 3. Let 1 < q < 2. Consider for f 2 L2 ./ C L ./ D L ./ and  2 S" , jj  ı, the resolvent equation (11), and its unique solution u 2 D.Aq / C Q q , Pq D PQ q . The proof D.A2 / D D.Aq /, rp D .I  PQ q /u. Note that Aq D A q0 is a simple duality argument using the resolvent estimate on L ./ from Step 2. To see that the constant in the resolvent estimate depends on  only through  , note that 

jhu; gij 0 sup I 0 ¤ g 2 Lq \ L2 kgkLq0 \L2



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431

q

defines a norm on L C L2 which is equivalent to the norm k  kLq CL2 with constants related to the norm of PQ q 0 and thus depending only on q and  . Step 4. Given the sequence of bounded subdomains j  , j 2 N, of uniform Q q ./ and fj WD PQ q f C 1;1 -type as in (16), let f 2 L j . Then consider the j

solution .uj ; rpj / of the Stokes resolvent equation uj  PQ q uj D uj  uj C rpj D PQ q fj ;

  rpj D I  PQ q uj

in j :

By Steps 2 and 3, there holds the uniform estimate kuj kLQ q  .j / C kuj kWQ 2;q . / C krpj kLQ q .j /  C kf kLQ q  ./ j

(30)

with jj  ı > 0, C D C .q; ı; ";  / > 0. Extending uj and rpj by 0 to vector fields on , there exist, suppressing subsequences, weak limits Q q  ./; in L

u D w lim uj j !1

rp D w lim rpj j !1

Q q ./n in G

solving (11) in , satisfying u 2 DQ q and the a priori estimates (12). Uniqueness is proved by an elementary duality argument and the above result of existence for both q 0 and q. Now Theorem 2 is proved. t u

2.3

Maximal Regularity and Bounded Imaginary Powers of the Stokes Operator on LQ q -Spaces

Theorem 3 ([20,25]). Let   Rn be a uniform C 2 -domain and let 1 < s; q < 1, Q q .// and an initial value 0 < T < 1. Given an external force f 2 Ls .0; T I L Q q / (for simplicity), there exists a unique vector field u 2 Ls .0; T I D.A Q q //\ u0 2 D.A q 1;s Q W .0; T I L .// solving the Cauchy problem Q q u D f; ut C A

u.0/ D u0 :

(31)

It can be represented by the variation of constants formula Q

u.t / D e t Aq u0 C

Z

t

Q

e .t /Aq f ./ d

for a.a. 0  t  T

(32)

0

and satisfies maximal regularity estimate   kukLs .0;T ID.AQ q // C kut kLs .0;T ILQ q /  C ku0 kD.AQ q / C kf kLs .0;T ILQ q / with a constant C D C .q; s; T;  / > 0.

(33)

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Proof of Theorem 3. The proof follows the lines of the proof of Theorem 2. The Q q /-estimate on the bounded domain H D H˛;ˇ;hIr replacing crucial point is an Ls .L Lemma 1 (iii) from Step 1 above. Lemma 3. Let 0 < T < 1; u0 2 D.Aq / and f 2 Ls .0; T I Lq .H // be given. Assume that u 2 Ls .0; T; D.Aq //, p 2 Ls .0; T I W 1;q .H // solves the system ut  u C rp D f;

div u D 0;

u.0/ D u0

(34)

and satisfies supp u0 [ supp u.t / [ supp p.t/  Br .0/ for a.a. t 2 Œ0; T . Then there is a constant C D C .q; s;  ; T / > 0 such that kut kLs .0;T ILq .H // C kukLs .0;T IW 2;q .H // C krpkLs .0;T ILq .H //

(35)

   C ku0 kW 2;q .H / C kf kLs .0;T ILq .H // : Proof. This estimate follows from [58, Theorem 1.1]; see also [57, Theorem 4.1, (4.2) and (4:210 )] and Maremonti-Solonnikov [47, Theorem 1.4]. A careful inspection of the proofs shows that the constant C in (35) depends only on  and on q; T ; actually, it suffices to assume the boundary regularity C 1;1 since only the boundedness of second-order derivatives of functions locally describing the boundary is used. t u For simplicity, let u0 D 0. Moreover, it suffices to consider the case s D q since for bounded as well as for unbounded domains, an abstract extrapolation argument shows that the validity of (33) with s D q immediately extends to all s 2 .1; 1/; see Amann [3, p. 191] and Cannarsa-Vespri [15, (1.12)], where A has to be replaced Q q C ıI with ı > 0. by A Step 2 of the proof (with s D q  2) analyzes the unique solution u 2 Lq .0; T I D.Aq // of (31) given by (32) (cf. Giga-Sohr [39, 57]). It remains to prove that u satisfies the estimate (33) with a constant C depending only on T; q and  : For this reason, use – as in Step 2 of the proof of Theorem 2 – the system of functions fhj g; 1  j  N , the covering of  by balls fBj g, and the partition of unity f'j g as well as the bounded sets Uj  Bj . On Uj , define w D R..r'j /  u/ 2 2;q Lq .0; T I W0 .Uj //, and let Mj D Mj .p/ be the constant depending on t 2 .0; T / q such that p  Mj 2 Lq .0; T I L0 .Uj //; see Lemma 1. Since div w D .r'j /  u and div wt D .r'j /  ut for a.a. t 2 .0; T /; , the pair .'j u  w; 'j .p  Mj // solves in Uj the local equation   .'j u  w/t  .'j u  w/ C r 'j .p  Mj / (36) D 'j f  wt C w  2r'j  ru  .'j /u C .r'j /.p  Mj /: Due to the estimates (23), (17), and (18) and Sobolev embeddings with embedding constants c D c.q; r;  / > 0 independent of j , and the equations wt D R..r'j /  ut / and rp D f  ut C u, for " 2 .0; 1/, there holds

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kwt kLq .Lq .Uj //  C kut kLq .L2 .Uj // C "kut kLq .Lq .Uj // ;   kr 2 wkLq .Lq .Uj //  C kukLq .Lq .Uj // C krukLq .Lq .Uj // ;

(37)

  kp  Mj kLq .Lq .Uj //  C kf; rukLq .Lq .Uj // C kut kLq .L2 .Uj // C "kut kLq .Lq .Uj // with C D C .q; T; ";  / > 0 for all j 2 N. Then an application of the local estimate (35)–(37) implies that k'j ut kLq .Lq .Uj // C k'j ukLq .Lq .Uj // C k'j r 2 ukLq .Lq .Uj // C k'j rpkLq .Lq .Uj //    C kf kLq .Lq .Uj // C kukLq .W 1;q .Uj // C kut kLq .L2 .Uj // C "kut kLq .Lq .Uj // with C D C .T; q; ";  / > 0. Raise this inequality to its qth power, sum over j D 1; : : : ; N , and exploit the crucial property of the number N0 to get the estimate q

kut ; u; r 2 u; rpkLq .Lq .// Z

T

Z

T

Z

D Z

0



 q q0

 CN0

j

X

q q0

 0

! ˇq ˇ X ˇq ˇ X ˇq ˇX ˇq ˇ X ˇ ˇ ˇ ˇ ˇ ˇ ˇ 2 ˇ 'j ut ˇ C ˇ 'j uˇ C ˇ 'j r uˇ C ˇ 'j rp ˇ dx dt ˇ N0

X

q q0

C "N0

j

j'j ut jq C

X

j

kf

j

q kLq .0;T ILq .Uj // C

j

X

j

j'j ujq C

X j

q kut kLq .0;T ILq .Uj //

X

j

j'j r 2 ujq C

j q kukLq .0;T IW 1;q .Uj // C

X

X

 j'j rpjq dx dt

j

 q kut kLq .0;T IL2 .Uj //

j

:

j

With a sufficiently small " > 0, the absorption principle and again the property of the number N0 help to simplify the above inequality to the Lq .0; T I Lq .//estimate kut kLq .0;T ILq / C kukLq .0;T ILq / C kr 2 ukLq .0;T ILq / C krpkLq .0;T ILq /    C kf kLq .0;T ILq / C kukLq .0;T ILq / C kut kLq .0;T IL2 /

(38)

where C D C .q;  / > 0. Here we also used the reverse Hölder inequality to deal with the sum of the terms kut kLq .0;T IL2 .Uj // . The term kukLq .0;T ILq / can be absorbed exploiting (21) with " > 0 sufficiently small. Finally, the L2 ./-maximal regularity estimate for kut kLq .0;T IL2 / (see [56, Ch.IV 2.5]) admits to absorb this term in (38) as well. Step 3 deals with the bounded domain case when 1 < s D q < 2. As in Step 3 of the proof of Theorem 2, the main tools are duality arguments and, in particular, the duality

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 0 0 0 0 Lq .0; T I Lq .// \ Lq .0; T I L2 .// D Lq .0; T I Lq .// C Lq .0; T I L2 ./ with a norm equivalence depending on q and  only. Step 4 considers the passage to the limit of solutions .uj ; rpj / in j to a weak limit identified as the unique solution .u; rp/ of (31) and (34), satisfying the a priori estimate (33). Now the proof of Theorem 3 is complete. t u Q q admits A further crucial property of the Stokes operator is the fact that I C A Q q /is ; s 2 R (see Kunstmann [42]) so that complex bounded imaginary powers .I C A interpolation methods can be applied to describe domains of fractional powers .I C Q q /˛ ; 1  ˛  1. For 0  ˛  1, let DQ q˛ D DQ q˛ ./ D D..I C A Q q /˛ / denote the A ˛ Q q /˛  k Q q . Q domain of the fractional power .I C Aq / equipped with the norm k.I C A L q ˛ Q  ./ in the norm k.I CA Q q /˛ k Q q . For 1  ˛ < 0, define DQ q as the completion of L L Qq As in Sects. 2.2 and 2.3, it is unknown whether similar results will hold for A Q q. instead of I C A Proposition 1. Let   Rn n  2; be a uniform C 1;1 -domain of type  , and let Q q ./. In particular, the Q q has a bounded H1 -calculus in L 1 < q < 1. Then I C A spaces DQ q˛ are reflexive and satisfy the duality relation .DQ q˛ / Š DQ q˛ 0 . Moreover,

DQ q˛ ; DQ qˇ



D DQ q ;

(39)

1=2 when 1  ˛  ˇ  1 and .1  /˛ C ˇ D  , 2 .0; 1/. Finally, DQ q D 1;q q Q  ./ with norm k.1 C A Q q /1=2  k Q q equivalent to k  k 1;q : WQ 0 ./ \ L L Q ./ W

The proof is found in [42, Theorem 1.1, Corollary 1.2] and [43]. These results imply the following Sobolev embedding and decay estimates   ([51, Proposition 3, Theorem 1]): Let n  3, 1 < q  r < 1, and ˛ WD n2 q1  1r > 0. Then kukLQ r ./  t AQ  e r f  r Q ./ L  t AQ  re r f  r Q ./ L  t AQ  e r PQ r div F  r Q ./ L

Q q /˛ uk Q q ;  C k.1 C A L ./  C e ıt t ˛ kf kLQ q ./ ; 1

 C e ıt t ˛ 2 kf kLQ q ./ ; 1

 C e ıt t ˛ 2 kF kLQ q ./ ;

0  ˛  1;

(40) (41) (42) (43)

Q q ./, or matrix field F 2 L Q q ./, respectively, for any for every u 2 DQ q˛ , f 2 L t > 0 and ı > 0; here C D C . ; r; q; ı/ > 0. Note that in (43) the operator e t AQ r PQ r div must be defined by duality to the operator re t AQ r 0 in (42).

8 Stokes Semigroups, Strong, Weak, and Very Weak Solutions for General Domains

2.4

435

The Stokes Operator with Robin Boundary Condition

The Navier boundary condition or more general the Robin boundary condition reads as follows: u  N D 0;

.1   /u C .T .u; p/N / D 0

on @;

(44)

with  2 .0; 1 . Here N denotes the exterior normal, and the subscript  denotes the tangential part of a vector along @, i.e., u WD u  .u  N /N . Moreover, T D T .u; p/ D pI C 2D.u/;

D.u/ D

1 .ru C .ru/> / 2

are the Cauchy stress tensor and the symmetric part of the velocity gradient, respectively. In the limit  D 1, we are left with the Navier condition of pure slip. The analysis of the Stokes resolvent problem and of maximal regularity follows the Dirichlet case in Sects. 2.2 and 2.3. However, there are crucial differences. In Sect. 2.2, a sequence of bounded uniform C 1;1 -subdomains j of  exhausting  has been introduced; see (16); then the Dirichlet condition uj D 0 on @j yields in the weak limit the boundary condition u D 0 on @. For the Robin boundary condition, the sequence fj gj must be chosen more carefully. Assumption 1. A uniform C 3 -domain   Rn of type  D .˛; ˇ; K/ in Sect. 2.4 is assumed to have the following property: There exists a sequence fj gj 2N of bounded uniform C 3 -domains of type  D .˛; ˇ; K/ such that I I I

S j  j C1 for all j 2 N and  D 1 j D1 j , j WD @j \ @ 6D ; for all j 2 N,S j  j C1 for all j 2 N and @ D 1 j D1 j :

Note that typical layer-like domains or tubular-like domains with finitely and even countably many exits at infinity satisfy Assumption 1. The same holds for the smooth vertex domain from [9, 48]. To define the Stokes operator with Robin boundary condition, the Sobolev space ˚ W2;q ./ D u 2 W 2;q ./ W .1   /u C .T .u; p/N / D 0 2;q

 on @ ;

1 < q < 1, is needed. The boundary condition for the space W ./ is understood locally in the sense of usual traces. For a bounded domain , the domain of the q 2;q Stokes operator Aq; D Pq  is given by D.Aq; / D L ./ \ W ./: However, for general unbounded domains, its domain is defined as

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(

Q q; / D D.Aq; / \ D.A2; /; DQ q; D D.A D.Aq; / C D.A2; /;

2  q < 1; 1 < q < 2:

(45)

Then, with the help of the Helmholtz projection PQq , the Stokes operator with Navier boundary condition for a general uniformly smooth domain is given by Q q  ./ ! L Q q  ./: Q q; D PQ q  W DQ q;  L A

(46)

Q q; reads Now the main result on the Stokes resolvent problem for the operator A as follows: Q q; ). Let 1 < q < 1, 0 < " <  , ı > 0. Theorem 4 (Resolvent problem for A 2 n 3 Let   R , n  2, be a uniform C -domain of type  and let Assumption 1 be satisfied. Then the following assertions hold: Q q; . To be more precise, for (i) The sector †" is contained in the resolvent set of A q Q Q f 2 L  ./, there exists a unique u D . C Aq; /1 f of the abstract Stokes Q q; u D PQq f satisfying the estimate resolvent problem u C A kukLQ q ./ C kukWQ 2;q ./  C kf kLQ q ./

(47)

for all  2 †" with jj  ı, where C D C .q; "; ı;  / > 0. Q q ./ and  2 †" , the Stokes resolvent system in .u; p/ has (ii) For a given f 2 L Q q ./ defined by u D . C A Q q; /1 PQ q f a unique solution .u; rp/ 2 DQ q;  L Q and rp D .I  Pq /.f C u/, satisfying kukLQ q ./ C kukWQ 2;q ./ C krpkLQ q ./  C kf kLQ q ./

(48)

for jj  ı with C D C .q; "; ı;  / > 0. Q q  ./ is a densely defined closed Q q; W DQ q; ! L (iii) The Stokes operator A Q operator, and Aq; generates an analytic semigroup satisfying the estimate Q

ke t Aq; f kLQ q ./  C e ıt kf kLQ q ./

(49)

Q q  ./, t  0, where C D C .q; ı;  / > 0. for f 2 L Q q 0 ; , hA Q q; u; vi D Q 0q; D A (iv) For the adjoint operator, the duality relations A Q q 0 ; vi for all u 2 DQ q; , v 2 DQ q 0 ; , hold. hu; A The proof is similar to the proof of Theorem 2. One main difference concerns the resolvent estimate with Robin boundary condition in the set H (or for a bent half space) obtained by Shibata-Shimada [54] (cf. Lemma 1) (iii) above for the Dirichlet case. It is necessary to check that the proofs in [54] yield constants C D C .q; ı;  /;

8 Stokes Semigroups, Strong, Weak, and Very Weak Solutions for General Domains

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see also Farwig-Rosteck [29, Prop. 2.4, Lemma 2.5]. The partition of unity for @ in the context of the Robin boundary condition also necessitates the analysis of nonhomogeneous Robin boundary conditions for bent half spaces. A second technical difference is the Friedrichs inequality replacing the Poincaré inequality. R 1;q Actually, if u 2 W0 .H / or u 2 W 1;q .H /, H u D 0, then kukLq .H /  C .q;  /krukLq .H / : In the case of vector fields u 2 W 1;q .H / satisfying u  N D 0 on @H , a similar estimate holds (cf. [29, Lemma 2.2] and [30]). Corollary 1. [[29], Corollaries 1.3, 1.4] Let 1 < q < 1, and let   Rn , n  2, be a domain satisfying Assumption 1. (i) The norms Q q;  k Q q ; k  k Q q C k.1 C A Q q; /  k Q q ; k  kWQ 2;q ./ ; k  kLQ q ./ C kA L ./ L ./ L ./ Q q; /  k Q q k.1 C A L ./ are equivalent on DQ q; with a constant depending on  only through  : (ii) The estimate Q

ke t Aq; ukLQ r ./  C t ˛ e ıt kukLQ q ./ ;

Q q  ./; t > 0; u2L

(50)

with a constant C D C .ı; q;  / > 0, holds true under the following conditions (a) and (b):   nq (a) q < n2 and q  r  n2q , where 0  ˛ D n2 q1  1r  1, (b) q  n2 and q  r, where 1  ˛  1  qr  0. Q q; ). Let 1 < q; s < 1, 0 < T < 1. Let Theorem 5 ( Maximal regularity for A n   R , n  2, be a domain satisfying Assumption 1. Then the following assertions hold: Q q  .// and u0 2 DQ q; , there exists a unique solution (i) For every f 2 Ls .0; T I L s Q Q q  .// of the evolution equation u 2 L .0; T I Dq; / with ut 2 Ls .0; T I L Q q; u D f; ut C A

u.0/ D u0 ;

satisfying the estimates Q q; uk s kut kLs .0;T ILQ q .// C kukLs .0;T ILQ q .// C kA Q q .// L .0;T IL    C kf kLs .0;T ILQ q .// C ku0 kDQ q; with a positive constant C D C . ; T; q; s/.

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Q q .// and every u0 2 DQ q; , the instationary (ii) For every f 2 Ls .0; T I L Stokes system in .u; p/ has a unique solution .u; rp/ 2 Ls .0; T I DQ q; /  Q q .// with ut 2 Ls .0; T I L Q q  .//, defined by ut C A Q q; u D PQ q f , Ls .0; T I G u.0/ D u0 , as well as rp.t/ D .I  PQ q /.f C u/.t /, satisfying kut kLs .0;T ILQ q .// C kukLs .0;T IWQ 2;q .// C krpkLs .0;T ILQ q .//    C kf kLs .0;T ILQ q .// C ku0 kDQ q; with a positive constant C D C . ; T; q; s/. The proof is similar to the proof in the Dirichlet case (cf. Theorem 3). The key lemma is the maximal regularity estimate for bent half spaces and the set H which are allowed to depend on the domain through  only. This result can be extracted from Shimada [55] where – starting with the half space case and nonhomogeneous boundary data as well as a nonzero divergence – the bent half space and finally the case H are analyzed (cf. Rosteck [52, Lemmata 3.9, 3.10 and 3.11]).

3

The Navier-Stokes System in General Unbounded Domains

Throughout this section, let   Rn , n  3, be a uniform C 1;1 -domain of type  . Then the instationary Navier-Stokes system (3) on a finite time interval Œ0; T / is solved for strong, mild, very weak, and weak solutions.

3.1

Strong and Mild Solutions

Using the variation of constants formula, mild solutions of (3) are given as solutions of the nonlinear integral equation u.t / D e

Qq t A

Z

t

u0 C

 Q  e .t /A PQ f ./  div .u ˝ u/. / d ;

0  t  T:

(51)

0

A mild solution u is called a strong solution if u is contained in Serrin’s class Q q .// where 2 C n D 1, q  n; in particular, u 2 Ls .0; T I Lq .//. Ls .0; T I L s q Theorem 6 (Strong Solutions). Let s > 2, q > n satisfy 2s C nq D 1; let f be given Q q=2 .// and, for simplicity, u0 2 DQ q . in the form f D div F with F 2 Ls=2 .0; T I L  Then there exists 0 < T  T such that the Navier-Stokes system (3) has a unique Q q .//. strong solution u 2 Ls .0; T  I L

8 Stokes Semigroups, Strong, Weak, and Very Weak Solutions for General Domains

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The proof is a simple consequence of Banach’s fixed-point theorem. The crucial Q q .// is the estimate of the estimate to get in (51) a self-map on Ls .0; T  I L n 1 .0/ nonlinear term: By (43) and with ˛ D 2q D s 0 , u .t / D e t AQ q u0 , Z

.0/

ku.t /kLQ q  ku .t /kLQ q C C

t

0

 0 .t  /1=s kF . /kLQ q=2 C ku. /k2LQ q d

for a.a. t 2 Œ0; T /. Then the Hardy-Littlewood-Sobolev inequality implies for every 0 < T   T that   kukLs .0;T  ILQ q /  ku.0/ kLs .0;T  ILQ q / C C2 kF kLs=2 .0;T  ILQ q=2 / C kuk2Ls .0;T  ILQ q / with a constant C2 independent of T  . Choosing T  sufficiently small, Banach’s fixed-point theorem completes the proof. Next, the classical Fujita-Kato method will be modified to work for general unbounded domains (cf. [27]). For simplicity, let f D 0. Q n ./ be an initial velocity. Then Definition 2. Let 0 < T < 1 and let u0 2 L 1 n 1=2 1 Q Q n .// is called mild Fujitau 2 L .0; T I L .// with t ru.t / 2 L .0; T I L Kato solution to the Navier-Stokes system with initial velocity u0 if it solves the integral equation u.t / D e

Qn t A

Z

t

u0 

Q e .ts/An=2 PQ n=2 .u.s/  ru.s// ds

(52)

0

for almost all 0  t < T . Q n ./, and Theorem 7 (Mild Fujita-Kato Solutions). Let 0 < T < 1, u0 2 L n < q < 1. (i) There is a constant  D . ; q/ > 0 such that the condition Q

Q

sup t .1n=q/=2 ke t An u0 kLQ q ./ C sup t 1=2 kre t An u0 kLQ n ./  

0t 0 with the following property: If for a point t 2 .0; T / lim inf ı!0C

where ˛ D



r0 2 2 r0

C

3 q0

1 ı˛

Z

t

tı

r0 ku. /kL d  ; Q q0 ./

(87)

  1 , then t is a regular point in the sense that u 2

Q q ./). In particular, the left-side Serrin condition u 2 Lr .t  Lr .t  ı; t C ıI L q Q ı; t I L .// for some ı > 0 is sufficient for t to be a regular point. • There is a constant  D . ; T / with the following property: If for a point t 2 .0; T / and some > 0 the kinetic energy Ekin .t / D 12 ku.t /k2L2 satisfies the left-sided 12 -Hölder continuity jEkin .t /  Ekin .t  ı/j  ; ı 1=2 0 2 and Z.x/ D 0 for jxj < 1, and Z w.y; s/ D R3

ry

1 rz .z/  U0 .z; s/ d z: 4 jy  zj

(126)

One can show that W is differentiable in y and s, T periodic, divergence-free, U0  W 2 L1 .0; T I L2 .R3 //, and, for R0 , sufficiently large,

and

kW kL1 .0;T ILq .R3 //  C "0 ;

(127)

kW kL1 .0;T IL4 .R3 //  c.R0 ; U0 /;

(128)

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kLW kL1 .0;T IH 1 .R3 //  c.R0 ; U0 /:

(129)

Decompose U D W C V with the difference V satisfying a perturbed Leray system, LV C .W C V /  rV C V  rW C rP D R.W /;

div V D 0;

(130)

where the source term is R.W / WD LW C W  rW:

(131)

To get a solution of (130) satisfying the local energy inequality, consider mollified systems in R3 , LV C .W C " V /  rV C V  rW C rP D R.W /;

div V D 0;

(132)

where " , 0 < "  1, are standard mollifiers. Because Z

V  rV  W  kW kL1 Lq kV k2H 1 

1 kV k2H 1 8

(133)

for sufficiently small "0 and large R0 by (127), and 1 j.R.W /; V /j  .kLW kH 1 C kW k2L4 /kV kH 1  C2 C jjV jj2H 1 ; 8

(134)

where C2 D C .kLW kH 1 C kW k2L4 /2 , for both (130) and (132) one can obtain the a priori bound d 1 1 jjV jj2L2 C jjV jj2L2 C jjrV jj2L2  C2 : ds 2 2

(135)

Using this bound, one can first construct time-periodic Galerkin approximations V";k , k 2 N, for (132). They are uniformly bounded in the usual energy norms and converge weakly to a periodic solution V" of (132) as k ! 1 up to a subsequence, with kV" kL1 .0;T IL2 .R3 // C krV" kL2 .R3 Œ0;T /  C:

(136)

At this stage one can construct the pressure P" associated to V" for (132) by Riesz transforms as in [55, 70], with uniform bound kP" kL5=3 .R3 Œ0;T /  C kjV" j C jW jk2L10=3 .R3 Œ0;T / :

(137)

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499

One then passes limit " ! 0 to get a weak solution V; P of (130) satisfying the corresponding local energy inequality. Then U D W C V and P then become a suitable weak solution of the Leray equations (77). A DSS solution of the NavierStokes equations (14) is obtained by the similarity transform (76). An interesting scenario is the following. For the usual Leray-Hopf weak solutions, it is well known that the hypothetical singular set is contained in a compact subset of space-time. Such property can be called eventual regularity. The eventual regularity of local Leray solutions is unclear: if a -DSS solution v is singular at some point .x0 ; t0 /, it is also singular at .k x0 ; 2k t0 / for all integers k. Since v is regular if it is SS or if  is close to one, the possibility of a non-compact singular set for some local Leray solutions is suggested only by Theorem 12. When the initial data is self-similar, the above approach gives a third construction for self-similar solutions of (14). One has the following result: Theorem 13. Let u0 be a self-similar divergence-free vector field in R3 which is in L3loc away from the origin. Then, there exists a local Leray solution u to (14)–(15) which is self-similar and satisfies ku.t /  e t u0 kL2 .R3 /  C0 t 1=4 for any t 2 .0; 1/ and a constant C0 D C0 .u0 /.

6

Possible Nonuniqueness for Large Forward Self-Similar Solutions

As is well known, topological argument gives existence but not uniqueness of solutions. In fact in the case of steady Navier-Stokes equations, where the existence proof is similar in spirit, there is nonuniqueness [69] through nontrivial bifurcations. In [31, 32] the authors conjectured nonuniqueness for general large-scale-invariant initial data based on bifurcations from large-scale-invariant solutions. Consider a scale-invariant initial condition u0 which is smooth away from the origin. For   0, at first taken sufficiently small so that uniqueness holds, let u .x; t / D p1 t U . px t / be the unique scale-invariant solution to NSE with the initial data  u0 . The field U satisfies U C

x 1  rU C U  U  rU C rP D 0 2 2

(138)

1 in R3 , with jU .x/   u0 .x/j D o. jxj / as x ! 1. U can be viewed as a steady state to the time-dependent equation

@s U D U C

x 1  rU C U  U  rU C rP; 2 2

(139)

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with the boundary condition jU .x; s/  u0 j D o.

1 /; jxj

(140)

as x ! 1, in R3  Œ0; 1/. The linearization of Eq. (139) around the steady state U can be written as s D L ;

(141)

where L  WD  C

1 x  r C   U  r    rU C rP; 2 2

(142)

and the function P is chosen so that L  is divergence-free. Also, P is assumed to have a suitable decay at 1 (so that it is uniquely determined, perhaps up to a constant). Clearly, the behavior of the linearized equation (142) depends on the spectral property of the operator L . To study the spectrum of the linearized operator, one needs to set up some notations. Let X WD f 2 L2 \ L4 .R3 / W div  D 0g;

(143)

with the natural norm kkX WD kkL2 .R3 / C kkL4 .R3 / :

(144)

The choice of the space X is dictated by the need to preserve the boundary condition (140) for the perturbed solution and local regularity considerations. Define the domain of L as D WD f 2 X W @j ; @ij  2 X; and x  r 2 X g:

(145)

It was shown in [32] that the spectrum of L is contained in the set 1 † WD f 2 C W Re    g 4

(146)

and K  WD U  r C   rU C rP is relatively compact with respect to L. Thus the spectrum of L D L  K is the union of one part contained in † and some isolated eigenvalues in the set f 2 C W 

1 < Re g: 4

9 Self-Similar Solutions to the Nonstationary Navier-Stokes Equations

501

When  is small, one can solve Eq. (138) by a perturbation argument (and the solution is unique). Moreover, still for small , the set of eigenvalues of L can be shown to remain away from the imaginary axis, with Re < 0. It is not hard to show that as long as 0 is not an eigenvalue of L , one can use perturbation arguments to continue the solution curve U as a regular function of , thus there is no confusion in the definition of U and L . If one increases  further, some eigenvalues of L might cross the imaginary axis. The following crucial spectral assumption will play an important role: (A) For some   > 0, the operator L associated with forward self-similar solution p1t U px t has eigenvalues with positive real part, all of which lie to the

left of fz 2 C W Re z D 18 g. A few comments are in order on the condition that fz 2 C W Re z < 18 g: One of course expects the eigenvalues to have small real part when  has just crossed the critical value. The restriction on the size of the unstable eigenvalues plays a crucial role in the localization of the large forward self-similar solutions. Under spectral assumption (A) one has the following result. Theorem 14. Suppose that for some 1-homogeneous initial data u0 smooth away from origin and  > 0, the L , associated with the forward self-similar  operator  1 x solution u .x; t / D pt U pt as above, satisfies the spectral assumption (A), and then there exist two different solutions u and uQ  to the Navier-Stokes equations with the same initial data  u0 . uQ  is given by  1 x ; log t ; uQ  .x; t / D u .x; t / C p   t

(147)

k.; s/kX  C e ıs ; for all s  0:

(148)

and  satisfies

Remark. A proof will not be given here. The reader is referred to the paper [32]. The spectral assumption is slightly different, but the proof of Theorem 4.1 in [32] with obvious modifications works here as well. The main idea is to construct an unstable manifold around the steady state U , and then any nontrivial solution  on the unstable manifold will suffice. A large part of the work [32] deals with the spectral properties of L and their implications on the growth rate of the semigroup e L s . These properties are of course essential in the perturbation analysis around the steady state U . Due to slow decay at spatial 1, solutions u ; uQ  do not belong to the energy space. It is natural to localize these solutions to obtain nonuniqueness for LerayHopf weak solutions. This is done in [32], by perturbing u ; uQ  in L4 , so that one can preserve the singularity at the origin while cutting off the slowly decaying tail.

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In the end, one can obtain sharp nonuniqueness for Leray-Hopf weak solutions, provided spectral assumption (A) can be verified. An interesting feature of the perturbation argument is that the resulting equation contains critically singular scale-invariant lower-order terms, for which the classical well-posedness theory is not applicable. Consider, for example, @t    C a.x; t /  r C b.x; t/ D 0; where a.x; t / D

1 p A. px t / t

(149)

and b.x; t/ D 1t B. px t /. For simplicity, assume A; B 2 5=2

Cc1 .R3 /. Classical parabolic theory tells us if a 2 L5x;t ; b 2 Lx;t then Eq. (149) is well-posed in any Lp , p > 1. The coefficients just miss such space. The wellposedness (in Lp , say) is then not clear from the general parabolic theory. It is observed in [32] that the well-posedness of Eq. (149) in Lp is decided by the spectrum of the operator T  D  C

y  r  A  r  B: 2

(150)

3 g then More precisely if the spectrum of T is contained in f W Re  <  2p p Eq. (149) is well-posed in L . On the other hand if there are eigenvalues of T 3 with real part greater than  2p , Eq. (149) is ill-posed in Lp through appearance of nontrivial solution in the form u.x; t / D t  ‰. px t / with trivial initial data. Here ‰ is an eigenfunction corresponding to eigenvalue . Applying this observation to the Navier-Stokes equations, it is found that if the spectrum of L contains unstable eigenvalues but these eigenvalues are all contained in f W Re  < 18 g, then the perturbed Navier-Stokes would be ill-posed for 1homogeneous initial data, but still well-posed for initial data in L4 . More precisely one has the following theorem: Denote

Y D a 2 L1 .R3 /j div a D 0;    and sup .1 C jxj/1Cj˛j jr ˛ a.x/j < 1; for j˛j D 0; 1; 2: : x

equipped with the natural norm kakY WD

sup



 .1 C jxj/1Cj˛j jr ˛ a.x/j :

(151)

x2R3 ;j˛jD0;1;2

Theorem 15. Let a 2 Y be such that 18 > ˇ, which is the maximum of real part of any eigenvalue of L  K.a/. Denote a.x; Q t / D p1 t a. px t /. Let

9 Self-Similar Solutions to the Nonstationary Navier-Stokes Equations

x 1 Q b.x; t/ WD p b. p ; log t /; for x 2 R3 ; t 2 .0; 1/: t t

503

(152)

Suppose sup .kb.; s/kX C krb.; s/kX /

D

(153)

s2.1;0/

is sufficiently small depending on kakY ; ˇ. Let T D T .kakY ; ˇ; ku0 kL4 .R3 / / > 0 be sufficiently small. Then there exists a unique solution u 2 ZT to the generalized Navier-Stokes system with singular lower-order terms 9 @t u  u C aQ  ru C u  r aQ C bQ  ruC = in R3  .0; T /; (154) Cu  r bQ C u  ru C rp D 0 ; div u D 0 with initial data u.; 0/ D u0 2 L4 .R3 / in the sense that lim ku.; t /  u0 kL4 .R3 / D 0:

t!0C

(155)

In the above,

4 3 ZT WD u 2 L1 t Lx .R  .0; T // W

and

1 2

1

sup t 2 kru.; t /kL4 .R3 / < 1; : t2.0;T /



lim t kru.; t /kL4 .R3 / D 0 ;

t!0C

equipped with the natural norm kukZT WD sup



 1 ku.; t /kL4 .R3 / C t 2 kru.; t /kL4 .R3 / :

(156)

t2.0;T /

Moreover, u satisfies 1

C sup t 2 kru.; t /kL4x .R3 /  C .ˇ; kakY /ku0 kL4x .R3 / : kukL1 4 3 t Lx .R .0;T //

(157)

t2.0;T /

Using the above result, one can show Theorem 16. Assume the spectral condition (A) holds. Then there exist two different Leray-Hopf weak solutions which are smooth in R3  .0; 1/ with the same compactly supported initial data u0 2 C 1 .R3 nf0g/. The initial data obeys 1 u0 .x/ D O. jxj / bound near the origin.

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Conclusion

Classically, the study of self-similar solutions has played an important role in the understanding of singularities for many elliptic and parabolic equations. Backward self-similar singularities for Navier-Stokes equation can be excluded, as discussed in Sect. 3. However, many questions remain open. For example, one can combine the scaling symmetries with the rotational symmetries and consider solutions invariant by one-parameter subgroups of this larger group. Not much is known about the existence/nonexistence of such singularities. The study of forward self-similar solutions is closely related to the problem of optimal well-posedness results, as discussed in Sect. 4. The forward self-similar solutions seem to provide one of the natural ways to “test” the borderline between the regime where the viscosity term dominates and the regime where the nonlinear term dominates. It is conceivable that further study of these solutions can also shed light on the problem of uniqueness of the Leray-Hopf weak solutions with initial data in L2 .

8

Cross-References

 Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes

Flow  Regularity Criteria for Navier-Stokes Solutions  Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions  Vorticity Direction and Regularity of Solutions to the Navier-Stokes Equations

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Giovanni P. Galdi and Mads Kyed

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Bounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Existence of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Existence of Strong Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Lq Estimates for the Linearized Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Exterior Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Existence of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Existence of Strong Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 New Approach to Lq Estimates for the Linearized Problem in the Whole Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Fundamental Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Weighted Estimates for the Linearized Problem in the Whole Space . . . . . . . . . . . . 4.7 Lq Estimates for the Linearized Problem in Exterior Domains . . . . . . . . . . . . . . . . . 4.8 Existence of Lq Strong Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Asymptotic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Flow in a Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

510 513 515 515 520 523 527 529 532 533 534 537 539 545 548 550 560 565 572 575 575 576

The work of G.P. Galdi was partially supported by the NSF grant DMS-1614011 G.P. Galdi () Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA, USA e-mail: [email protected] M. Kyed Fachbereich Mathematik, Technische Universität Darmstadt, Darmstadt, Germany e-mail: [email protected]; [email protected] © Springer International Publishing AG, part of Springer Nature 2018 Y. Giga, A. Novotný (eds.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, https://doi.org/10.1007/978-3-319-13344-7_10

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Abstract

The Navier-Stokes equations with time-periodic data are investigated with respect to solutions of the same period. In the physical terms, such a system models the flow of a viscous liquid under the influence of a time-periodic force. The three most relevant types of flow domains, from a physical point of view, are considered: a bounded domain, an exterior domain, and an infinite pipe. Methods to show existence of both weak and strong solutions are introduced. Moreover, questions regarding regularity, uniqueness, and asymptotic structure at spatial infinity of solutions are addressed.

1

Introduction

An object performing a time-periodic interaction with a viscous liquid is one of the most frequently occurring mechanical systems in nature. In such systems, the object exerts a time-periodic force on the liquid, and it can be expected that the resulting flow undergoes a motion of the same period. The mathematical analysis of such motions naturally leads to the study of time-periodic Navier-Stokes equations. If the region of flow is a domain   Rn , n  2, the Navier-Stokes equations governing the motion of a liquid subjected to a body force f W R   ! Rn can be written as 

@t u C u  ru D u  rp C f div u D 0

in R  ; in R  ;

(1)

where u W R   ! Rn denotes the Eulerian velocity field, p W R   ! R the pressure field, and  the constant coefficient of kinematic viscosity of the liquid. As is natural for time-periodic problems, the time axis is taken to be the whole of R. If the domain  has a boundary, that is,  ¤ Rn , a boundary condition u D u

on R  @

(2)

is added to the system. In case  is unbounded, an asymptotic value of the velocity field at spatial infinity lim u.t; x/ C u1 D 0

jxj!1

(3)

is prescribed. If the data f , u , and u1 are all time-periodic, say of period T > 0, the condition that the liquid undergoes a motion of the same period 8.t; x/ 2 R   W u.t C T; x/ D u.t; x/

(4)

completes the system. In the following, a mathematical analysis of (1), (2), (3), and (4) will be carried out for a fixed period T .

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Interaction with a moving boundary is often the driving force in a fluid flow. As typical examples, one may think of an object performing a motion in a fluid or the motion of a fluid in a container with the flow driven by momentum flux on the fluidstructure boundary due to an alteration of the surface geometry. In both cases, the domain  changes in time. In all relevant applications, a body force in the fluid is typically absent or, at the most, is potential-like and, therefore, can be absorbed in the pressure term in (1)1 . In order to mathematically investigate the corresponding equations of motion, however, such a system is usually rewritten in a fixed (in time) domain. In the new system, an artificial body force then appears. If the motion of the boundary, and hence the domain, is time-periodic, the artificial body force will be too. Consequently, the mathematical analysis of (1), (2), (3), and (4) in a fixed domain  is imperative for understanding more natural time-periodic fluid flows. In this article, basic questions concerning existence, regularity, uniqueness, and asymptotic properties of solutions to (1), (2), (3), and (4) will be addressed. The aim is to comprehensively introduce the most basic methods that can deliver answers to these questions. Focus will be on the following types of spatial domains: (i)  a bounded domain (Sect. 3). (ii)  an exterior domain, in which case a time-periodic velocity u1 .t / is prescribed at spatial infinity (Sect. 4). (iii)  a pipe, in which case a Poiseuille flow is prescribed at spatial infinity of each outlet (Sect. 5). Models based on bounded domains cover a large class of important physical systems. The motion of a liquid in a container with a time-periodic inflow (and outflow) is a classic example. The flow of a liquid in the gap between two rotating concentric spheres or cylinders is another. In fact, many intrinsic properties of viscous fluid flows are traditionally observed and studied in the setting of a timeperiodic flow in a bounded domain. The exterior domain case is just as interesting, as (1), (2), (3), and (4) in this case models a fluid flow past an object moving with time-periodic velocity u1 through a liquid. Also highly relevant from a physical point of view is the time-periodic fluid flow in a pipe. The cardiovascular system, for example, is essentially a fluid flow in a piping system with a prescribed timeperiodic flow rate. Generally, time-periodic fluid flows occurring in nature fall into one of the categories (i)–(iii). The investigation of time-periodic Navier-Stokes equations was initiated in a short note by Serrin [43], who suggested to study (1) as a dynamical system and identify a time-periodic solution as a periodic orbit. Serrin made the proposition that for time-periodic data f and any initial value, the solution u.t; x/ to the corresponding initial-value problem tends to a periodic orbit as t ! 1. However, as remarked by Serrin himself, the assumptions he makes are very stringent and, certainly at the time when the paper appeared, not sustained by any known result. The first complete results on existence of time-periodic solutions are due to Prodi [39] and Yudovich [53]. These authors considered the Poincaré map that takes an initial value into the state described by the solution to the initial-value problem

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at time T . A time-periodic solution can then be obtained via a fixed point of this map. The two originally proposed methods were subsequently used by several authors to build a mathematical foundation for the time-periodic Navier-Stokes equations. Of particular importance is the work [41] of Prouse, in which the celebrated method introduced by Hopf in [21] for the initial-value Navier-Stokes problem was adapted to the time-periodic setting for the first time. Although Prouse focused on weak solutions in bounded domains, the approach of Hopf based on a Galerkin approximation can be employed, as will be shown in the following, more broadly. The first to treat strong solutions were Kaniel and Shinbrot [23], followed shortly after by Takeshita [46]. Time-dependent domains were treated by Morimoto [38] and Miyakawa and Teramoto [37]. The first results for unbounded domains are due to Maremonti [34, 35], who treated the half- and whole-space problem. More general unbounded domains were investigated by Maremonti and Padula [36] by combining the Galerkin approximation with the “invading domain” technique. An important contribution was given by Kozono and Nakao [25], who, for the first time, proposed a direct representation formula for a time-periodic solution. Yamazaki [52] employed this formula to treat the case of a three-dimensional exterior domain. Further results for exterior domains were obtained by Galdi and Sohr [7] and Taniuchi [47]. Another direct representation formula was introduced by Kyed [27, 28] based on the Fourier transform on the locally compact abelian group R=T Z  Rn . This idea further lead to the concept of a time-periodic fundamental solution [30], maximal Lp regularity in Rn [29] for the linearization of (1), and an analysis by Lemarié-Rieusset [32] of time-periodic whole-space Navier-Stokes equations in critical spaces. Maximal Lp regularity was established in the twoand three-dimensional exterior domain by Galdi [11, 12] and Galdi and Kyed [14], respectively. Investigation of time-periodic Navier-Stokes equations in cylindrical domains was initiated by da Veiga [1] and continued by Galdi [9]. After the basic questions of existence, regularity, and uniqueness of solutions have been addressed, another critical issue emerges in the case of unbounded domains, namely, the inquiry into the asymptotic structure of solutions at spatial infinity. In the exterior domain case in particular, where u1 .t / in (3) describes the velocity of an object moving through a liquid, does the asymptotic structure of a solution reveal important physical properties. For u1 D 0, the leading term in an asymptotic expansion was identified by Kang, Miura, and Tsai [22] to be the same as the one found for the corresponding steady-state equations, namely, the Landau solution. Also for u1 ¤ 0, it has been shown [12, 14, 15, 27] that the leading term in the time-periodic case coincides with leading term found in the steady-state case. Other results concerning the asymptotic properties of time-periodic Navier-Stokes flow include a technique developed by Baalen and Wittwer [48] and the investigation by Silvestre [44] of flows with finite kinetic energy. The references given above concern results directly related to (1), (2), (3), and (4). Over the years, related systems have been investigated. Although out of the scope of this article, the system of equations governing the flow past a rigid body moving freely in a liquid under the action of a time-periodic force deserves mentioning. In order to mathematically investigate this problem, it is necessary to

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rewrite the equations of motion for the liquid, that is, the Navier-Stokes system, in a frame of reference attached to the body. The result is a system of time-periodic Navier-Stokes equations in a frame of reference that is not necessarily an inertial frame. This special type of time-periodic Navier-Stokes problem was investigated by Galdi and Silvestre [16,17], who extended a famous result of Weinberger [50,51] and Serre [42] to the time-periodic case. A general approach to time-periodic fluid flow problems was developed recently by Geissert, Hieber, and Nguyen [18]. A comprehensive treatment of time-periodic partial differential equations, including the Navier-Stokes equations, can be found in the books of Vejvoda [49] and Lions [33].

2

Notation

In the following,   Rn will always denote a domain, namely, an open connected set. Points in R   are generally denoted by .t; x/, with t being referred to as time and x as the spatial variable. Differential operators act only in the spatial variable unless otherwise indicated. In particular, @j D @xj for j D 1; : : : ; n. The notation BR refers to a ball in Rn centered at 0 with radius R > 0. Moreover, R B WD Rn n BR and BR1 ;R2 WD BR2 n BR1 . Additionally R WD  \ BR , R1 ;R2 WD  \ BR1 ;R2 , and R WD  \ B R . Einstein’s summation convention, that is, implicit summation over all repeated indices, is employed throughout. Constants in capital letters in the proofs and theorems are global, while constants in small letters are local to the proof in which they appear. For vector fields f ; g and second-order tensor fields F ; G on a domain   Rn , the notation 

 f ; g WD

Z

Z f  g dx D 

fi  gi dx;



 F ; G WD



Z

Z F W G dx D



Fij W Gij dx 

is used to denote their inner products. Classical Lebesgue spaces with respect to spatial domains are denoted by Lq ./ and with respect to time-space domains by Lq ..0; T /  /. If no confusion can arise, the norm is denoted by kkp in both cases. The norm in Lq ..0; T /  / is normalized so that  kf kLq ..0;T // WD

1 T

Z

T 0

Z

jf .t; x/jq dxdt

 q1 :



  The Lebesgue space Lq ./ is treated as the subspace of functions in Lq .0; T / that are time independent. This identification is made without furthernotification. With the normalization above, the Lq ./ norm coincides with the Lq .0; T /   norm on the subspace of time-independent functions.

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Classical Sobolev spaces are denoted by W m;q ./ and their norms by kkm;q . The subspaces of Sobolev functions vanishing on the boundary are denoted by kkm;q

m;q

m;q

W ./ WD C01 ./ . The dual space of the latter is denoted by W0 ./ WD  0 m;q 0 0 W0 ./ and its norm by kk1;2 . Sobolev spaces over time-space domains are introduced for j; k 2 N0 and q 2 Œ1; 1/ in the form     W j;k;q .0; T /   WD ff 2 L1loc .0; T /   j kf kj;k;q < 1g; 1 q1 0 X X q q kf kj;k;q WD @ k@˛t f kLq ..0;T // C k@ˇx f kLq ..0;T // A : j˛jj

(5)

0 0):

kvkX q

8 1 1 2 ˆ 2q C 4 krvk 4q C k@1 vkq C kr vkq ˆ 2 kvk 2q ˆ 4q ˆ ˆ ˆ ˆ 0 such that R n  BR0 , and recall for q 2 Œ1; 1/ that

hpi1;q

ˇZ ˇ WD krpkq C ˇ

R0

Z ˇ ˇ p.x/ dx ˇ; hpi0;1;q WD krpkq C

T 0

ˇZ ˇ ˇ

R0

ˇ ˇ p.t; x/ dx ˇdt (63)

1 .R  /, respectively. Classical and timedefines a norm on C01 ./ and C0;per periodic homogeneous Sobolev spaces can thus be introduced as the Banach spaces hi1;q

D 1;q ./ WD C01 ./

;

hi0;1;q

1;q 1 Dper .R  / WD C0;per .R  /

;

respectively. It is easy to verify that thelatter space coincides with the canonical  T -periodic extension of functions in Lq .0; T /I D 1;q ./ . Both characterizations, as well as the subspace 1;q

1;q Dper;? .R  / WD P? Dper .R  /;

are used in the following.

4.2

Existence of Weak Solutions

Existence of weak solutions to (61) in the case  is a three-dimensional exterior domain can be established without any restrictions on the “size” of the data. The approach employed below is sometimes referred to as the invading domain technique. It is based the Galerkin approximation used in the proof of Theorem 1.

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Firstly, a definition of a weak solution to (61) is needed. Compared to Definition 1 in the case a bounded domain , the concept of a weak solution in the exterior domain case has to incorporate also the decay property limjxj!1 u.t; x/ D 0 of the solution. The definition below is a simple extension of a similar definition in the steady-state case [10, Definition X.1.1].   Definition 2. Let   R3 be an exterior domain. Let f 2 L2per RI D01;2 ./ . A   vector field u 2 L2per RI D 1;2 ./ is called a weak time-periodic solution to (61) if div u D 0, u D u1 on R  @ in the trace sense, the identity Z

T

       u; @t ' C  ru; r' C .u  u1 /  ru; '  hf ; 'i dt D 0

(64)

0 1 .R  /, and u satisfies for almost all t 2 R the decay holds for all ' 2 C0;;per property Z 1 lim ju.t; x/j d .x/ D 0: (65) R!1 j@BR j @BR

In order to employ a Galerkin approximation to establish existence of a weak solution, it is necessary to first “lift” the non-homogeneous boundary values in (61), that is, find a suitable extension of the boundary values and subtract it from u to obtain an equivalent system with homogeneous boundary data. It is critical that the terms in which the extension appears in the new system can be suitably estimated in each step of the Galerkin approximation. A well-known method, which goes back to Leray and Hopf, can be modified to construct an appropriate extension. 1;2 Lemma 2. Let   R3 be an exterior domain of class C 0;1 . Let u1 2 Wper .R/.   1;2 2;2 For every " > 0, there is a vector field W " 2 Wper RI W ./ satisfying W " D u1 on @, div W " D 0 in ,

 C3 ku1 k1;2 ; kW " kWper 1;2 .RIW 2;2 .//

(66)

and 1 8' 2 C0;per .R  / W

Z

 

 '.t; x/  rW " .t; x/; '.t; x/ dx  "kr'.t /k22 : (67)

Proof. Let " 2 C01 .I R/ be a “cutoff” function with of @. It follows directly that

"

D 1 in a neighborhood

1  x3  u12 .t / W " .t; x/ WD r  @ " .x/  x1  u13 .t /A " .x/  x2  u11 .t / 0

" .x/

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satisfies all the desired properties except (67). To obtain (67), " needs to choose in t u a certain way. For example, one may choose " as in [10, Lemma III.6.2]. Remark 6. Lemma 2 can be extended to yield solenoidal extensions for a much larger class of time-periodic boundary values u .t; x/, .t; x/ 2 R@. For example, [10, Lemma X.4.1] can be modified to include time-periodic boundary values, which would then produce such a class. All results in this section continue to hold if the boundary condition u D u1 in (61) is replaced with u D u for a time-periodic vector field u that has a solenoidal extension with the properties from Lemma 2. With the lemma above, existence of a weak solution to (61) can be shown with the invading domain technique. The main idea is to apply a Galerkin approximation to show existence of a solution on the bounded domain  \ BR .0/. After securing a priori estimates independent on R, a weak solution is found by a limiting process R ! 1. 1;2 .R/ Theorem 7. Let   R3 be an exterior domain of class C 0;1 . If u1 2 Wper   1;2 2 and f 2 Lper RI D0 ./ , then there is a weak time-periodic solution u 2   L2per RI D 1;2 ./ to (61) in the sense of Definition 2.

Proof. Let W " be the extension field from Lemma 2. Existence of a solution to (61) on the u Dw C W " can be obtained by finding a solenoidal vector field  form 1;2 w 2 L2per RI D0; ./ satisfying Z T          w; @t ' C  rw; r' C w  rw; ' C .W "  u1 /  rw; ' dt 0 (68) Z T   e ; 'i dt  w  rW " ; ' C hf D 0 1 for all ' 2 C0;;per .R  /, where

    e ; 'i WD hf ; 'i  @t W " C .W "  u1 /  rW " ; '   rW " ; r' : hf

(69)

Let fRk g1 1 as k ! 1. Put k WD  \ BRk .0/. kD1  R be a sequence with Rk !   1;2 2 .k / to (68) with respect to test Since k is bounded, a solution wk 2 Lper RI W0; 1 functions ' 2 C0;;per .Rk / can be obtained as in the proof of Theorem 1. For this purpose, choose "  12  and employ (67) in order to deduce an identity equivalent to (11). By repeating the rest of the argument from the proof of Theorem 1, a solution wk that satisfies the equivalent to (15), which due to term w  rW " ; ' on the right-hand side in (68) becomes an inequality kwk kL2

1;2 per .RID0; .k //



2 

Z

T 0

e ; wk i dt; hf

10 Time-Periodic Solutions to The Navier-Stokes Equations

537

is then obtained. Provided the vector field is extended by 0 on R3 n k , it  wk 1;2  1 2 follows that fwk gkD1 is bounded in Lper RI D0; ./ . Consequently, there is a that converges weakly in this space to a vector field subsequence of fwk g1  kD1 1;2 2 w 2 Lper RI D0; ./ . To ensure that w satisfies (68), it suffices to verify that wk   converges strongly in L2per RI L2 .R / for any R > R0 . This will in particular   ensure convergence of the nonlinear term wk  rwk ; r' . The verification can   1;2 be done by deducing from (68) that @t wk is bounded in L1per RI D0; .R / , and   utilize that wk thereby lies in a space that embeds compactly into L2per RI L2 .R / . Finally, since  is a three-dimensional exterior domain, the decay property (65) 1;2 follows directly from the fact that w.t / 2 D0; ./ and W " .t / 2 W 2;2 ./; see [10, Lemma II.6.3]. Remark 7. A similar result is open in the two-dimensional case n D 2. Although the arguments in the proof of Theorem 7 that ensure existence of a vector field satisfying (64) are all valid also when n D 2, it is not clear if this field satisfies the decay property (65). The same problem has been open for the corresponding steady-state problem for decades. Recall that the steady-state problem is a special case of the time-periodic problem.

4.3

Existence of Strong Solutions

Compared to the bounded domain case, integrability properties are a more delicate matter in unbounded domains such as exterior domains, since they describe not only local regularity but also decay properties as jxj ! 1 of the solution. Consequently, different characterizations of strong solutions transpire. Below, a class (71), (72), and (73) of strong solutions is introduced that emerges from adaptation of the methods from Sect. 3.3 to the exterior domain case. By modification of the Galerkin approximation from the proof of Theorem 7, existence of a strong solution of this type to (61) for data f and u1 sufficiently restricted in “size“ is established in the three-dimensional case n D 3. Later, in Sect. 4.8, a different class of strong solutions based on Lq estimates for the linearization of (61) is treated. In comparison, the method based on Lq theory for the linearized problem yields better decay properties of the solution, while the method based on Galerkin approximation is more versatile when it comes to the admissible structure of u1 . 2 Theorem 8. Let   R3 be an exterior There is a constant  of class  domain  C . 1;2 2 2 2 "2 .; ; T / > 0 such that if f 2 Lper RI L ./ \ Lper RI D0 ./ and u1 2 1;2 Wper .R/ satisfy

kf kL2per .RIL2 .// C kf kL2

1;2 .// per .RID0

C ku1 k1;2  "2 ;

(70)

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G.P. Galdi and M. Kyed

then there is a solution .u; p/ to (61) that satisfies     2 u 2 L2per RI D 1;2 ./ ; ru 2 L1 per RI L ./ ;     r 2 u 2 L2per RI L2 ./ ; @t u 2 L2per RI L2 ./ ;     P? u 2 L2per RI L2 ./ ; p 2 L2per RI D 1;2 ./ :

(71) (72) (73)

Proof. Again, the “lifting” field W " from Lemma 2 is utilized. If w is a solution to 1 (68) for all ' 2 C0;;per .R  /, then u WD w C W " is a solution to (61). Since W " satisfies (71), (72), and (73), it suffices to verify that also w satisfies (71), (72), and (73). To obtain a vector field w with the desired properties, one proceeds as in the proof of Theorem 7 and employs a Galerkin approximation to first solve (68) on an ascending sequence of bounded domains k WD  \ BRk , limk!1 Rk ! 1. On each Rk , a solution wk with kwk kL2 .RID 1;2 .// bounded independently on k is per 0; thereby obtained. Furthermore, the argument from the proof of Theorem 3 can be reused to deduce that wk is a strong solution. One may verify, using, for example, a simple scaling argument, that the constants c0 and c1 in the proof of Theorem 3 are independent on k. By repeating the proof of Theorem 3 up till (37) and taking into consideration the additional terms in (68) containing the “lifting” field W " , which is not present in the proof of Theorem 3, one obtains the estimate krwk .t /k22



e k2 e k2 2  c0 kf C kf 1;2 L .0;T IL2 .// .// L2 .0;T ID 0

Z

t

C s0

Z



 kW " .s/k22;2 C ju1 .s/j2 krwk .s/k22 ds

t

C s0

krwk .s/k42

C

krwk .s/k62

(74)

 ds ;

e defined as in (69) and the constant c0 independent on k. Now take " WD "2 , with f with "2 still to be chosen. Then (66) and the “smallness” assumption (70) furnish krwk .t /k22



2

Z

t

 c1 "2 C s0

2

"2 krwk .s/k22

C

krwk .s/k42

C

krwk .s/k62

 ds ; (75)

again with the constant c1 independent on k. Based on the inequality (75), the , argument following (37) in the proof Theorem 3 yields that krwk kL1 2 per .RIL .// kr 2 wk kL2per .RIL2 .// , and k@t wk kL2per .RIL2 .// are bounded independently on k, provided "2 is chosen sufficiently small. At this point, it is therefore possible to let k ! 1. After possibly passing to a subsequence, one finds as a weak limit of fwk g1 kD1 in the spaces (71) and (72) a solenoidal vector field w. Clearly, u WD 1 w C W " satisfies (61) with respect to test functions ' 2 C0;;per .R  /. By a

10 Time-Periodic Solutions to The Navier-Stokes Equations

539

standard method from the well-known analysis of the corresponding initial-value problem, or alternatively described in Remark 2, the existence  by the approach  of a pressure p 2 L2per RI D 1;2 ./ that renders .u; p/ a solution to (61) follows.   It remains to show P? u 2 L2per RI L2 ./ . For this purpose, expand u into a   P i 2 2 T ht , and deduce from @t u 2 L2 Fourier series u.t / D per RI L ./ h2Z uh e that fhuh gh2Z 2 `2 .L2 .//. Consequently, fuh gh2Znf0g 2 `2 .L2 .//. Since the latter sequence  can be recognized as the Fourier coefficients of P? u, it follows that P? u 2 L2per RI L2 ./ . t u

4.4

New Approach to Lq Estimates for the Linearized Problem in the Whole Space

While the strong solutions to (61) established in Theorem 8 are locally very regular, little information is revealed about their integrability at spatial infinity, that is, the rate of decay as jxj ! 1. Only when the asymptotic behavior at spatial infinity is known can it be determined whether the fluid flow described by a solution is meaningful from a physical point of view. At the outset, no such information is available for the solution from Theorem 8. To gain more insight, global Lq estimates of solutions to an appropriate linearization of (61) in terms of the data are needed. Such estimates can be used to extract information from a solution to (61) directly or to establish, for example, by a fixed point argument, existence of a (strong) solution to (61) for which the decay as jxj ! 1 is better understood. An Lq theory for an exterior domain problem is usually derived via Lq estimates for the corresponding whole-space problem. In the following, the linearized time-periodic Navier-Stokes system in the whole-space 8 @t u  u  @1 u C rp D F ˆ ˆ ˆ ˆ ˆ ˆ < div u D 0 u.t C T; x/ D u.t; x/; ˆ ˆ ˆ ˆ ˆ ˆ : lim u.t; x/ D 0;

in R  Rn ; in R  Rn ; (76)

jxj!1

shall be investigated. Here,  0 is a constant. In the context of Lq estimates for (76), one has to distinguish between the two cases D 0 and ¤ 0. In the former case, the system is referred to as a timeperiodic Stokes problem and in the latter as a time-periodic Oseen problem. It is well known from the corresponding steady-state problem that Lq estimates for the Stokes and Oseen problem are different. From a physical point of view, the discrepancy is not surprising as the Oseen equations model a fluid flow past a moving body, which therefore should exhibit a wake region behind the body, whereas a Stokes flow describes a flow without a wake region around a stationary body .

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A simple but important step toward optimal Lq estimates for (76) lies in the decomposition of (76) by the complementary projections P and P? introduced in (62). If the velocity field and pressure term are expressed as u D Pu C P? u DW v C w;

p D Pp C P? p DW p C ;

one easily verifies that .v; p/ is a solution to the steady-state problem 8  v  @1 v C rp D PF ˆ ˆ ˆ < div v D 0 ˆ ˆ ˆ : lim v.x/ D 0;

in Rn ; in Rn ;

(77)

jxj!1

and .w; / a solution to the time-periodic problem 8 @t w  w  @1 w C r D P? F ˆ ˆ ˆ ˆ ˆ ˆ < div w D 0

in R  Rn ; in R  Rn ; (78)

w.t C T; x/ D w.t; x/; ˆ ˆ ˆ ˆ ˆ ˆ : lim w.t; x/ D 0: jxj!1

Since (78) resembles (76), not much insight seems to have been won by this decomposition. It turns out, however, that in the context of Lq estimates, system (78) has some remarkable characteristics. Since both the data P? F and the solution .w; / have vanishing time average over the period, that is, they are purely oscillatory, an analysis of (78) can be carried out in subspaces of Lq consisting entirely of oscillatory functions. As a result, much better Lq estimates materialize for .w; / than can be shown for a solution to the original problem (76). An optimal Lq theory for the time-periodic problem (76) can be established by combining these estimates for .w; / with well-known Lq estimates for the solution .v; p/ to the steady-state problem (77). Below, it is described how to establish the Lq estimates for .w; / by means of Fourier analysis. Alternatively, one can also establish the estimates by extending the method used in the proof of Theorem 6. In comparison, the approach below is more direct as it is based on a direct representation of the solution in terms of a Fourier multiplier. Moreover, it leads naturally to the concept of a fundamental solution for the time-periodic problem. The main idea is to reformulate (78) as partial differential equation on the locally compact abelian group G WD R=T Z  Rn and analyze the problem with the corresponding Fourier transform FG . q

Theorem 9. Let q 2 .1; 1/. For every F 2 Lper;? .R  Rn /, there is a solution 1;2;q

1;2;q

1;q

.w; / 2 Wper;? .R  Rn /  Wper;? .R  Rn /  Dper;? .R  Rn /

10 Time-Periodic Solutions to The Navier-Stokes Equations

541

to (78). The solution satisfies the estimate kwk1;2;q C kr kq  C4 P . ; T / kF kq ;

(79)

where P . ; T / is a polynomial in and T and C4 D C4 .; n; q/. If for some 1;2;r 1;r r 2 .1; 1/ .e w; e / 2 Wper;? .R  Rn /  Dper;? .R  Rn / is another solution, then wDe w and .t; x/ D e .t; x/ C d .t / for some T -periodic function d W R ! R. Some nomenclature from abstract harmonic analysis is needed to sketch a proof of Theorem 9. A topology and an appropriate differentiable structure on the group G WD R=T Z  Rn are inherited from R  Rn . More precisely, G becomes a locally compact abelian group when equipped with the quotient topology induced by the canonical quotient mapping q W R  Rn ! R=T Z  Rn ;

q.t; x/ WD .Œt ; x/:

(80)

The restriction … WD qjŒ0;T /Rn is used to identify G with the domain Œ0; T /  Rn ; … is clearly a (continuous) bijection. Via …, one can identify the Haar measure dg on G as the product of the Lebesgue measure on Œ0; T / and the Lebesgue measure on Rn . The Haar measure is unique up to a normalization factor, which in the following is chosen such that Z u.g/ dg D G

1 T

Z

T

Z u ı ….t; x/ dxdt:

0

Rn

For the sake of convenience, the symbol … in integrals of G-defined functions with respect to dxdt shall be omitted. By C 1 .G/ WD fu W G ! R j u ı q 2 C 1 .R  Rn /g;

(81)

the space of smooth functions on G is defined. For u 2 C 1 .G/, derivatives are defined by 8.˛; ˇ/ 2 Nn0  N0 W

ˇ

ˇ @t @˛x u WD @t @˛x .u ı q/ ı …1 : ˇ

(82)

It is easy to verify for u 2 C 1 .G/ that also @t @˛x u 2 C 1 .G/. The subspace C01 .G/ denotes the compactly supported smooth functions. With a differentiable structure available on G, the space of tempered distributions can be defined. For this purpose, recall the Schwartz-Bruhat space of generalized Schwartz functions (see, e.g., [2]) given by S.G/ WD fu 2 C 1 .G/ j 8.˛; ˇ;  / 2 Nn0  Nn0  N0 W ˛;ˇ; .u/ < 1g; 

˛;ˇ; .u/ WD sup jx ˛ @ˇx @t u.t; x/j: .t;x/2G

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G.P. Galdi and M. Kyed

˚ Equipped with the semi-norm topology of the family ˛;ˇ; j .˛; ˇ;  / 2 Nn0  Nn0   N0 , S.G/ becomes a topological vector space. The dual space S 0 .G/ equipped with the weak* topology is referred to as the space of tempered distributions on G. ˇ For a tempered distribution u 2 S 0 .G/, distributional derivatives @t @˛x u 2 S 0 .G/ are defined by duality as in the classical case. Similarly, tempered distributions on b are introduced. By associating each .k; / 2 Z  Rn with the G’s dual group G 2 character W G ! C; .t; x/ WD eix Ci T kt on G (it is standard to verify that all b D Z  Rn . By characters are of this form), one can characterize the dual group as G b WD fw 2 C .G/ b j 8k 2 Z W w.k; / 2 C 1 .Rn /g; C 1 .G/ b is introduced. The Schwartz-Bruhat space on the space of smooth functions on G b is given by the dual group G b j 8.˛; ˇ;  / 2 Nn0  Nn0  N0 W O˛;ˇ; .w/ < 1g; b WD fw 2 C 1 .G/ S.G/ ˇ

O˛;ˇ; .w/ WD sup j ˛ @ k  w.k; /j; .k; /2b G and equipped with the canonical semi-norm topology. The Fourier transform associated to the locally compact abelian group G is denoted by FG . It is explicitly given by b FG W L1 .G/ ! C .G/;

FG .u/.k; / WD

1 T

Z

T

Z

u.t; x/ eix i

2 T

kt

dxdt:

Rn

0

b is the product of the counting measure on Z and the Since the Haar measure on G n Lebesgue measure on R , the inverse Fourier transform is formally defined by 1 b F1 G W L .G/ ! C .G/;

F1 G .w/.t; x/ WD

XZ

w.k; / eix Ci

2 T

kt

d :

Rn

k2Z

b is a homeomorphism with F1 It is standard to verify that FG W S.G/ ! S.G/ G as the actual inverse, provided the Lebesgue measure d is normalized appropriately. b By duality, FG extends to a homeomorphism S 0 .G/ ! S 0 .G/. q 1;2;q 1;q Similar to the spaces Lper .R  Rn /, Wper .R  Rn /, and Dper .R  Rn /, Lebesgue and Sobolev spaces with respect to the domain G are defined by Lq .G/ WD C01 .G/ and

kkq

;

W 1;2;q .G/ WD C01 .G/

D 1;q .G/ WD C01 .G/

hi0;1;q

kk1;2;q

;

;

where kkq denotes the Lq -norm with respect to the Haar measure dg, the norm kk1;2;q defined as in (5), and the norm hi0;1;q as in (63). It is standard to verify

10 Time-Periodic Solutions to The Navier-Stokes Equations

543

that W 1;2;q .G/ D fu 2 Lq .G/ j kuk1;2;q < 1g. As in (62), the time-averaging projection P and its complement P? are introduced on functions defined on G. 1;2;q 1;q Thus, the subspaces W? .G/ WD P? W 1;2;q .G/, D? .G/ WD P? D 1;q .G/, and q q L? .G/ WD P? L .G/ can be defined. It is now possible to formulate (78) as a system of partial differential equations on G in a context that permits utilization of the Fourier transform FG . A representation formula for the solution in terms of a Fourier multiplier can then be obtained. The proof of Theorem 9 below is based on such a representation. Proof of Theorem 9. Since the topology and differentiable structure on G is inherited from R  Rn , the T -time-periodic problem (78) can be formulated equivalently as a system of G-defined vector fields (

@t w  w  @1 w C r D P? F

in G;

div w D 0

in G

(83)

with unknowns w W G ! Rn , W G ! R, and data F W G ! Rn . In this formulation, the periodicity condition is not needed anymore. Indeed, all functions defined on G are intrinsically T -time-periodic. Since (83) can be interpreted as a system of equations in S 0 .G/, the Fourier transform

be applied to obtain FG can a formula for w. An easy calculation shows that FG P? F D .1  ıZ .k//FG F , where ıZ denotes the Dirac delta distribution on Z, which is simply the function ıZ .0/ WD 1 and ıZ .k/ WD 0 for k ¤ 0. Formally, application of the Fourier transform FG in (83) yields    ˝ FG F ; I wD j j2 C i . 2 k  1 / j j2 T     1 i 1 i D FG  FG F D FRn  FR n F : j j2 j j2 F1 G



1  ıZ .k/

(84)

The Fourier multiplier b ! C; M .k; / W G

M .k; / WD

1  ıZ .k/   j j C i 2 k  1 T 2

(85)

b and M 2 C 1 .G/, b which can easily is bounded and smooth, that is, M 2 L1 .G/ be seen by observing that the numerator of M vanishes in a neighborhood of the only zero .0; 0/ of the denumerator. It can be shown that M is an Lq .G/ multiplier in the sense that the mapping f ! F1 G M FG Œf  extends from a mapping S.G/ ! S 0 .G/ into a bounded operator Lq .G/ ! Lq .G/. Since multiplier theorems like the ones of Mikhlin, Lizorkin, or Marcinkiewicz are only available in an Euclidean setting and not in the general setting of group multipliers, a proof has to rely on a so-called transference principle. Originally introduced by de Leeuw [4], a

544

G.P. Galdi and M. Kyed

transference principle for Fourier multipliers on local compact abelian groups makes it possible to study the properties of M via a corresponding multiplier defined on RnC1 . The transference principle (see, e.g., [5, Theorem B.2.1]) states that M is an b ! RnC1 such that Lq .G/ multiplier if there is a continuous homomorphism ˆ W G q nC1 M D m ı ˆ for some L .R / multiplier m. Moreover, the norm of the Lq .G/ operator corresponding to M coincides with the norm of the Lq .RnC1 / operator corresponding to m. To identify such an m in the particular case above, let be a “cutoff” function with 2 C01 .RI R/, ./ D 1 for jj  12 , and ./ D 0 for jj  1. Define m W R  R ! C; n

m.; / WD

1

T   2

j j2 C i .  1 /

:

(86)

  b ! RnC1 , ˆ.k; / WD 2 k; . Clearly, ˆ is a continuous Further, let ˆ W G T homomorphism and M D m ı ˆ. To show that m is a Lq .RnC1 / multiplier, a standard multiplier theorem can be applied. Since m is a rational function with nonvanishing denominator away from .0; 0/, it is easy to verify that all functions of type "

.; / ! 1"1    n"n "nC1 @"11    @"nn @nC1 m.; /

(87)

stay bounded as j.; /j ! 1. Consequently, it follows from Marcinkiewicz’s multiplier theorem (see, e.g., [45, Chapter IV, §6]) that m is an Lq .RnC1 / multiplier. A more careful analysis of the bounds obtained for the functions in (87) shows that the norm of the corresponding operator is bounded by a polynomial P . ; T / in

and T . It therefore follows that 8f 2 Lq .G/ W



kF1 G M FG Œf  kq  P . ; T /kf kq :

Now return to (84). Observe that I  ˝ 2 is the symbol of the Helmholtz-Weyl j j projection, that is, the projection PH W Lq .Rn /n ! Lq .Rn /n onto the subspace q L .Rn / of solenoidal vector fields. It is well known that PH is continuous on Lq .Rn / for any q 2 .1; 1/. The projection PH extends trivially to a continuous projection on Lq .G/. Now define .w; / by (84). Clearly, .w; / is a solution in the sense of distributions S 0 .G/ to (83). Moreover, 

   M .k; /F ŒP F  kwkq D F1   P . ; T / kPH F kq  P . ; T / kF kq : G H G q

The argument above can be repeated for @t w and @˛x w for j˛j  2. More precisely, one may verify that also .k; / ! kM .k; / and .k; / ! ˛ M .k; / are Lq .G/ multipliers for j˛j  2. It thus follows that kwk1;2;q  P . ; T / kF kq . Based on (84), it is standard to show kr kq  c kF kq . Since ıZ is the Fourier symbol of P,

10 Time-Periodic Solutions to The Navier-Stokes Equations

545

it follows directly from (84) that P? w D w and P? D . Thus a solution .w; / 2 1;2;q 1;q W? .G/D? .G/ to (83) that satisfies (79) is obtained. Via the canonical quotient 1 map q, the spaces C0;per .R  Rn / and C01 .G/ are isometrically isomorphic in the norm kk1;2;q . By the definition of Sobolev spaces as completions of theses spaces in 1;2;q 1;2;q the norm kk1;2;q , it follows that also W? .RRn / and W? .G/ are isometrically 1;q 1;q isomorphic. The same is clearly true for Dper;? .R  Rn / and D? .G/ as well q q n as for Lper;? .R  R / and L? .G/. It follows that .w ı q; ı q/ is a solution in 1;2;q

1;q

Wper;? .R  Rn /  Dper;? .R  Rn / to (78) that satisfies (79). It remains to establish the desired uniqueness property. It suffices to do so for the 1;r system (83). Assume .e w; e / is another solution in W?1;2;r .G/  D? .G/. Employ in (83) first the Helmholtz projection P and then the Fourier transform to deduce  

H 2 2 that i 2 F w  e w D 0. Since the polynomial j j k C j j 

i C i 2 k 1 G T  T

that supp F w  e w  f.0; 0/g.

1 vanishes only at .k; / D .0; 0/, it follows G

However, since P w  e w D 0 also ıZ .k/F w .k; / D 0, whence G w

e .0; 0/

… supp FG we w . Consequently, supp FG we w D ;. It follows that FG we w D 0 and thus w D e w. By (83), .t; x/ D e .t; x/ C d .t / for some T -periodic function d W R ! R. t u To complete the Lq estimate for a solution to (76), also the steady-state part .v; p) needs to be addressed. However, the Lq theory for the Stokes/Oseen system (77) is well known; see, for example, [10, Theorem IV.2.1 and VII.4.1]. As the steady-state Lq estimates for .v; p/ are completely decoupled from the time-periodic nature of (76), they are omitted here.

4.5

Fundamental Solution

The reformulation of the linear T -time-periodic system (76) on the group G carried out in the proof of Theorem 9 motivates the introduction of a time-periodic fundamental solution. In the setting of tempered distributions S 0 .G/, a fundamental solution to (76) or rather to the equivalent system on the group G (

@t u  u  @1 u C rp D F

in G;

div u D 0

in G;

(88)

can be defined as a tensor field 0

 TP

1 TP  TP 11 : : :  1n B :: : : :: C B : : C WD B : C 2 S 0 .G/.nC1/n @ TP : : :  TP A n1 nn 1TP : : : nTP

(89)

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G.P. Galdi and M. Kyed

that satisfies (

@t 

TP

ij

  TP ij  @1 

TP

ij

C @i jTP D ıij ıG ; @i 

TP

ij

D 0;

(90)

where ıij and ıG denote the Kronecker delta and delta distribution, respectively. For a sufficiently regular right-hand side, say F 2 S.G/n , a solution to (88) is then given by component-wise convolution over the group G with the fundamental solution:   u WD  TP  F : (91) p The ability to identify a solution in terms of such a direct expression offers many advantages. To the extent that pointwise information can be obtained for  TP , a similar type of information can be obtained for (91). In particular, knowledge of the asymptotic structure of  TP at spatial infinity can be used to analyze the pointwise behavior of u.t; x/ at as jxj ! 1. As already observed, the projection P can be expressed as Pf WD F1 G ıZ FG Œf  . From this expression, it is seen that P can be extended to a projection in the context of distributions P W S 0 .G/ ! S 0 .G/. The same is true for P? . Consequently, the idea from the previous section to use P and P? to decompose (90) into a steady-state and oscillatory part can be reused. As a result, it is possible to identify  TP as a sum of a well-known fundamental solution to the corresponding steady-state system and an oscillatory fundamental solution. It turns out that the oscillatory fundamental solution has significantly better decay properties as jxj ! 1. The structure of the fundamental solution differs depending on whether

D 0 or ¤ 0. This phenomenon is well known from the steady-state case, from which one may recall that a fundamental solution . ;  / 2 S 0 .Rn /nn S 0 .Rn /n to (



ij

 @1 

ij

C @i j D ıij ıRn ; @i 

ij

D 0;

(92)

in the Stokes case D 0 is given by the Stokes fundamental solution . Stokes ;  / and in the Oseen case ¤ 0 by the Oseen fundamental solution . Oseen ;  /; the pressure  is the same in two cases. Explicit expressions for both are well known; see, for example, [10, Chapter IV.2] and [10, Chapter VII.3] for the Stokes and Oseen fundamental solution, respectively. Theorem 10. Let n  2. Put (  WD

 Stokes

if D 0

(Stokes case);

Oseen

if ¤ 0

(Oseen case):



10 Time-Periodic Solutions to The Navier-Stokes Equations

547

Then  TP WD  ˝ 1R=T Z C  ? ; 

TP

(93)

WD  ˝ ıR=T Z ;

(94)

with 

?

WD

F1 G



  1  ıZ .k/ ˝ 2 S 0 .G/nn  I  2 j j2 C i 2 k 

j j 1 T

(95)

defines a fundamental solution  TP 2 S 0 .G/.nC1/n to (88) on the form (89) satisfying (90) and   nC2 8q 2 1; W  ? 2 Lq .G/nn ; n h n C 2

W @j  ? 2 Lq .G/nn 8q 2 1; nC1

(96) .j D 1; : : : ; n/;

8r 2 Œ1; 1/ 8" > 0 9C > 0 8jxj  " W k ? .; x/kLr .R=T Z/ 

(97) C ; jxjn

8r 2 Œ1; 1/ 8" > 0 9C > 0 8jxj  " W k@j  ? .; x/kLr .R=T Z/ 

C jxjnC1

(98) ; (99)

8q 2 .1; 1/ 9C > 0 8F 2 S.G/n W k

?

 F kW 1;2;q .G/  C kF kLq .G/ ; (100)

where 1R=T Z 2 S 0 .R=T Z/ denotes the constant 1. Proof. Apply in (90) first the projections P and P? and subsequently the Fourier transform to deduce P TP D  ˝ 1R=T Z ;

P? 

TP

D F1 G



  1  ıZ .k/ ˝ I    j j2 C i 2 k  1 j j2 T

and  TP D F1 G



i j j2



D F1 Rn



i j j2

 ˝ ıR=T Z :

It thus follows that . TP ;  TP / given by (93) and (94) defines a fundamental solution  TP 2 S 0 .G/.nC1/n to (88) on the form (89). Recall (85) and the observation made

548

G.P. Galdi and M. Kyed

b \ C 1 .G/ b to see that the right-hand in the proof of Theorem 9 that M 2 L1 .G/ 0 ? side in definition (95) of  is well defined. The properties (96), (97), (98), and (99) can be shown by a lengthy but direct computation and subsequent estimate of the inverse Fourier transform in (95); see [15, 30]. Finally, (100) is just a reiteration of the statement in Theorem 9. Remark 8. It is well known (see again [10, Chapter IV.2 and Chapter VII.3]) that both  Stokes and  Oseen have a pointwise decay rate as jxj ! 1 that is slower than jxjn . Estimate (98) therefore implies that the oscillatory part  ? of  TP decays faster than the steady-state part. Consequently, the asymptotic behavior as jxj ! 1 of a solution u to (88) given by (91) will be dominated by the steady-state part Pu.

4.6

Weighted Estimates for the Linearized Problem in the Whole Space

The properties obtained for the fundamental solution  TP in Theorem 10 can be  ˇ used to establish pointwise weighted estimates, with weights of type 1 C jxj , of solutions to the linearized system (76). Weighted estimates of this type are well known for the corresponding steady-state problem; see, for example, [10, Lemma V.8.2]. Thus, if a time-periodic solution is once again decomposed into a steadystate and oscillatory part, weighted estimates need only be established for the oscillatory part. As in the case of the Lq estimates in Theorem 9, better estimates in terms of decay at spatial infinity materialize for the oscillatory part. For ˇ 2 Œ1; 1/, let Xˇ denote the Banach space ˚  Xˇ WD ' 2 L1loc .Rn / j ŒŒ'ˇ < 1 ;

 ˇ ŒŒ'ˇ WD ess sup 1 C jxj j'.x/j: x2Rn

(101) It is illustrated below how to obtain  estimates for oscillatory  q-generalized solutions to (76) in spaces L1 WD P? L1 per RI Xˇ . These estimates can be per;? RI Xˇ augmented with Lq estimates. For this purpose, recall the interpretation from Sect. 4.4 of T -time-periodic vector fields as functions defined on the group G, and let 1

W?2

;1;q

˚  .G/ WD w 2 Lq .G/ j kwk 1 ;1;q < 1; Pw D 0 ; 2      1  12  kwk 1 ;1;q WD F jkj C j j C 1 FŒw  2

q

denote the Sobolev space of oscillatory functions with “half” a derivative in time and one derivative in space belonging to Lq .G/. The corresponding space 1

;1;q

2 Wper;? .R  Rn / is defined canonically via the quotient mapping q.

10 Time-Periodic Solutions to The Navier-Stokes Equations

549

Theorem 11. Let n  2, 2 Œ0; 1/ and ˇ 2 Œ1; n C 1. For every F D divG with nn , there is a solution .w; / to (76) with w 2 L1 G D L1 per;? RI Xˇ per;? RI Xˇ and     1   n r n 2 ;1;r .R  Rn /; 2 L1 ; 1 ; 1 W w 2 Wper;? 8r 2 max per;? RI L .R / ˇ (102) which satisfies kwkL1  C5 kG kL1 ; per .RIXˇ / per .RIXˇ /

(103)

    with C5 D C5 .ˇ; ; ; n/ and for all r 2 max ˇn ; 1 ; 1 r n r n kwk 1 ;1;r C k kL1  C6 kG kL1 per .RIL .R // per .RIL .R //

(104)

2

  w; e / is another solution with e w 2 L1 with C6 D C6 .r; ; ; n/. If .e RI X˛ for per;?   s n some ˛ 2 Œ1; 1/ and 2 L1 per;? RI L .R / for some s 2 .1; 1/, then .w; / D .e w; e /. Proof. From the integrability of the fundamental solution  ? stated in Theorem 10, recall (97), and the integrability of G , it follows that the convolution integral Z Z 1 T wi .t; x/ WD  @k  ? ij .t  s; x  y/ G kj .s; y/ dyds .i D 1; : : : ; n/ T 0 Rn (105) n is well-defined as an element of, say, L1 per .R  R /. From the definition (95) and the interpretation of T -time-periodic vector fields as functions defined on the group G, it is clear that w together with

WD

F1 G



 i j j j2



FG G ij



D

F1 Rn



 i j j j2



FRn G ij



is a solution to (76). It can be shown with the same approach as in the proof of 1

;1;r

2 Theorem 9 that w 2 Wper;? .R  Rn / and satisfies (104). It follows directly from n the definition of above that also satisfies (104). Since w 2 L1 per .R  R /, it is enough to establish the weighted estimate (103) for sufficiently large jxj. Consider for this purpose an jxj > 2 and decompose the integral in (105) as

1 wi .t; x/ D  T Z C

Z

T

Z

Z

Z

C 0

B 2jxj

Bjxj=2

C Bjxj=2;2jxj nB1 .x/

B1 .x/

@k  ? ij .t  s; x  y/ G kj .s; y/ dyds

DW I1 .t; x/ C I2 .t; x/ C I3 .t; x/ C I4 .t; x/:

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G.P. Galdi and M. Kyed

On the strength of estimate (99) in Theorem 10, it follows that Z jI1 .t; x/j  Bjxj=2

ˇ k@k  ? dy kG kL1 ij .; x  y/kL2per .R/ .1 C jyj/ per .RIXˇ /

Z

 c0 Bjxj=2

jx  yj.nC1/ .1 C jyj/ˇ dy kG kL1 per .RIXˇ /

 c1 jxjˇ kG kL1 ; per .RIXˇ / where the last inequality is valid due to the assumption that ˇ  nC1. The integrals I2 and I4 are estimated in a similar fashion. The summability property (97) from Theorem 10 is needed to deduce that @k  ? is integrable over B1 , which then leads to an estimate of I3 . It follows from the estimates of I1 , I2 , I3 , and I4 that jw.t; x/j  c2 jxjˇ kG kL1 for jxj > 2, which implies (103). The uniqueness property per .RIXˇ / can be obtained as in the proof of Theorem 9. t u The solution in Theorem 11 is called q-generalized since it possesses, at the outset, only “half” a derivative in time and one derivative in space belonging some 1

;1;q

2 Lq space, that is, it belongs to Wper .R  Rn /. Weighted estimates for strong   1;2;q solutions in Wper .R  Rn / corresponding to data F 2 L1 per;? RI Xˇ can be obtained in a similar fashion based on the estimates of the fundamental solution  ? available in Theorem 10.

4.7

Lq Estimates for the Linearized Problem in Exterior Domains

Consider the following linearization of (61) with homogeneous boundary values: 8 @t u  u  @1 u C rp D F ˆ ˆ ˆ ˆ ˆ ˆ div u D 0 ˆ ˆ ˆ < uD0 ˆ ˆ ˆ u.t C T; x/ D u.t; x/; ˆ ˆ ˆ ˆ ˆ ˆ : lim u.t; x/ D 0;

in R  ; in R  ; on R  @;

(106)

jxj!1

where  0 is a constant. The Lq estimates established in the whole-space case can be extended to solutions to the exterior domain problem (106). As in the whole-space case, the projections (62) can be used to decompose (106) into a steadystate and oscillatory problem. The following Lq estimate holds for the oscillatory problem:

10 Time-Periodic Solutions to The Navier-Stokes Equations

551

Theorem 12. Let   Rn (n D 2; 3) be an exterior domain of class C 2 . Let q q 2 .1; 1/ and 2 Œ0; 0 . For any vector field F 2 Lper;? .R  /, there is a 1;2;q

1;q

solution .u; p/ 2 Wper;? .R  /  Dper;? .R  / to (106) which satisfies kuk1;2;q C krpkq  C7 kF kq ;

(107)

1;2;r 1;r u; e p/ 2 Wper;? .R/Dper;? .R/ with C7 D C7 .q; ; ; 0 /. If r 2 .1; 1/ and .e is another solution, then e u D u and e p D p C d .t / for some T -periodic function d W R ! R.

A proof of Theorem 12 is given below. Beforehand, the Lq estimates for (106) that follow by combining Theorem 12 with well-known Lq estimates for the corresponding steady-state problem are manifested. Only the case ¤ 0 is included. A similar statement can be made in the case D 0; see also Remark 9. Corollary 1. Let   Rn (n D 2; 3) be an exterior domain of class C 2 and 2 .0; 0 . Let q 2 .1; 32 / if n D 2, and q 2 .1; 2/ if n D 3. Define q

q

X ./ WD fv 2 X ./ j div v D 0; v D 0 on @g;   Z D1;q ./ WD v 2 D 1;q ./ j p dx D 0 BR0

q

as subspaces of X ./ and D 1;q ./, defined in Sect. 4.1, respectively. Moreover, let r 2 .1; 1/ and 1;2;q;r

Wper;? .R  / ˚  1;2;q 1;2;r WD w 2 Wper;? .R  / \ Wper;? .R  / j div w D 0; w D 0 on @ equipped with the norm kwk1;2;q;r WD kk1;2;q C kk1;2;r . Additionally let 1;q;r Dper;? .R

  / WD 2

1;q Dper;? .R

 / \

1;r Dper;? .R

 / j

dx D 0 BR0

with norm hi0;1;q;r WD hi0;1;q C hi0;1;r and q;r

q



Z

Lper;? .R  / WD Lper;? .R  / \ Lrper;? .R  /

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G.P. Galdi and M. Kyed

with norm kkq;r WD kkq C kkr . Then the T -time-periodic Oseen operator q

1;2;q;r

AOseen W X ./ ˚ Wper;? .R  / 1;q;r

q;r

 D 1;q ./ ˚ Dper;? .R  / ! Lq ./ ˚ Lper;? .R  /;

(108)

AOseen .v C w; p C / WD @t w  .v C w/  @1 .v C w/ C r.p C / only on n, q, r, , , and . If q 2 is with kA1 Oseen  a3homeomorphism   k depending  1; 2 in the case n D 3 or q 2 1; 65 in the case n D 2, then kA1 Oseen k depends only on the upper bound 0 and not on itself. Proof. It is well known that the steady-state Oseen operator, that is, the Oseen operator from (108) restricted to time-independent functions, is a homeomorphism q as a mapping AOseen W X ./  D1;q ./ ! Lq ./; see, for example, [10, Theorem VII.7.1]. By Theorem 12, it follows that also the time-periodic Oseen operator is 1;2;q;r 1;q;r a homeomorphism as a mapping AOseen W Wper;? .R  /  Dper;? .R  / ! q;r Lper;? .R  /. Since clearly P and P? commute with AOseen , it further follows that AOseen is a homeomorphism in the setting (108). The dependency of kA1 Oseen k on the various parameters follows from [10, Theorem VII.7.1] and Theorem 12. t u Remark 9. In Corollary 1, the function space that yields maximal Lq regularity for the time-periodic Oseen operator, that is, the function space that is mapped q homeomorphically onto Lper .R/ by AOseen , is identified. As will be demonstrated later, this function space constitutes a suitable setting for investigation of the fully nonlinear problem in the Oseen case ¤ 0. More specifically, it is possible to choose the parameters in (108) in such a way that for any vector field u 2 q 1;2;q;r X ./ ˚ Wper;? .R  /, the corresponding nonlinear term u  ru belongs to the range of AOseen . The Stokes case D 0 is different. Although maximal Lq regularity for a certain range of exponents q can be obtained also in this case, one q can simply use [10, Theorem V.4.8] to identify the appropriate space X0 ./ that q is mapped homeomorphically onto L ./ by the steady-state Stokes operator and q use this space instead of X ./ in Corollary 1; the setting based on this space is not well suited for the investigation of the nonlinear problem when D 0. The weighted spaces introduced in Sect. 4.6 constitute a better alternative in this case. It is common practice to establish Lq estimates for an exterior domain problem by decomposing the solution into a solution to a bounded domain problem and a whole-space problem, respectively, and then employ the Lq theory available for these simpler cases. The decomposition is typically done by multiplying the solution with a “cutoff” function. In the case of the (linearized) Navier-Stokes system (106), this “cutoff” technique produces zero-order terms for the pressure on the “right-hand side” of the new equations. It is particularly challenging to estimate these terms. For

10 Time-Periodic Solutions to The Navier-Stokes Equations

553

this purpose, the following lemma, which was established for a two-dimensional and three-dimensional exterior domain in [12, 14], respectively, is needed: Lemma 3. Let  and be as in Theorem 12. Moreover, let R0 > 0 be a constant such that Rn n   BR0 and s 2 .1; 1/. There is a constant C8 D C8 .R0 ; ; s/ 1;2;r 1;r .R  /  Dper;? .R  / is a solution to (106) with data such that if .u; p/ 2 Wper;? R r F 2 Lper;? .R  / for some r 2 .1; 1/ and satisfying R p dx D 0, then 0

kp.t; /k n s;R0 n1   s1 1 s s  C8 kF .t; /ks C kru.t; /ks;R0 C kru.t; /ks; kru.t; /k 1;s;R R 0

0

(109)

for, a.e., t 2 R. Moreover, for every  > 0 with Rn n   B , there is a constant C9 D C9 .; ; s/ such that   krp.t; /ks;  C9 kF .t; /ks C kp.t; /ks;

(110)

for, a.e., t 2 R. Proof. For the sake of simplicity, the t -dependence of functions is not indicated in the proof. All norms are taken with respect to the spatial variables only. Consider an 1 arbitrary ' 2 C01 ./. Observe that for any 2 Cper .R/ holds Z

T

Z

Z @t u  r' dx 

0

which implies

R 

T

Z

dt D

div u  '  @t

dxdt D 0;



0

@t u  r' dx D 0 for, a.e., t . Moreover, Z

Z @1 u  r' dx D 



div u  @1 ' dx D 0: 

Hence it follows from (106)1 that p is a solution to the weak Neumann problem for the Laplacian: 8' 2

C01 ./

Z

Z

W

rp  r' dx D 

F  r' C u  r' dx: 

Recall that p2

L1loc ./

^

r

rp 2 L ./

Z ^

p dx D 0: R0

(111)

554

G.P. Galdi and M. Kyed

It is well known (see, e.g., [10, Section III.1]) that the weak Neumann problem for the Laplacian is uniquely solvable in the class (111). One can thus write p as a sum p D p1 C p2 of two solutions (in the class above) to the weak Neumann problem Z

8' 2 C01 ./ W

Z rp1  r' dx D

F  r' dx





and 8' 2 C01 ./ W

Z

Z rp2  r' dx D



u  r' dx; 

respectively. The a priori estimate 8q 2 .1; 1/ W krp1 kq  c0 kF kq

(112)

is well known. An estimate of p2 shall nowRbe established. Consider for this purpose an arbitrary function g 2 C01 .R0 / with R g dx D 0. The existence of a vector 0 field h 2 C01 .R0 / with div h D g and 8q 2 .1; 1/ W khk1;q  c1 kgkq is well known; see, for example, [10, Theorem III.3.3]. Let ˆ be a solution to the weak Neumann problem for the Laplacian: 8' 2

C01 ./

Z

Z

W

rˆ  r' dx D 

h  r' dx: 

By classical theory, such a solution exists with 8q 2 .1; 1/ W ˆ 2 C 1 ./ ^ krˆk1;q  c2 khk1;q  c3 kgkq : Since ˆ is harmonic in Rn n BR0 , an asymptotic expansion of ˆ implies rˆ.x/ D  n O jxj . The regularity and decay of ˆ ensure the validity of a computation (see [14] for the details) that yields Z p2 g dx D 

Z 

  ru W rˆ ˝ n  n ˝ rˆ d : @

Apply first Hölder’s inequality and then a classical trace inequality [10, Theorem II.4.1] to deduce ˇZ ˇ ˇ ˇ s ˇ p2 g dx ˇ  c4 kruks;@ krˆk s1 ;@ 

ns ns  c5 kruks;@ krˆk1; ns.n1/ ;R0  c6 kruks;@ kgk ns.n1/ ;R0 :

10 Time-Periodic Solutions to The Navier-Stokes Equations

Since

R R0

555

p2 dx D 0, it follows that

kp2 k

n n1 s;R0

D

sup g2C01 .R /;kgk D1 ns 0 ns.n1/

R

R

0

ˇZ ˇ ˇ ˇ ˇ p2 g dx ˇ  c7 kruks;@ : 

g dxD0

Another application of the trace inequality [10, Theorem II.4.1] now implies   s1 1 s s n kruk :  c C kruk kruk kp2 k n1 8 s;R0 s;R0 s;R 1;s;R 0

0

By Sobolev’s embedding theorem and (112), it can finally be concluded that n n kpk n1 s;R0  c9 krp1 ks;R0 C kp2 k n1 s;R0   s1 1 s s  c10 kF ks C kruks;R0 C kruks;R kruk1;s;R 0

0

and thus (109). To show (110), one can introduce an appropriate “cutoff” function

2 C 1 .Rn I R/ and analyze the weak Neumann problem satisfied by WD p in a similar manner; see again [14]. t u Also needed for the proof of Theorem 12 are the following embedding properties of time-periodic Sobolev spaces: n 1 Lemma 4. Let   R

(n  2/ be an exterior domain of class C and q 2 .1; 1/. Assume that ˛ 2 0; 2 and p0 ; r0 2 .q; 1 satisfy

8 2q ˆ ˆ r0  ˆ ˆ 2  ˛q < r0 < 1 ˆ ˆ ˆ ˆ : r0  1

if ˛q < 2; if ˛q D 2; if ˛q > 2;

8 nq ˆ p0  ˆ ˆ n  .2  ˛/q < p0 < 1 ˆ ˆ ˆ : p0  1

if .2  ˛/q < n; if .2  ˛/q D n; if .2  ˛/q > n; (113)

and that ˇ 2 0; 1 and p1 ; r1 2 .q; 1 satisfy 8 2q ˆ ˆ r1  ˆ ˆ 2  ˇq < r1 < 1 ˆ ˆ ˆ ˆ : r1  1

if ˇq < 2; if ˇq D 2; if ˇq > 2;

8 nq ˆ p1  ˆ ˆ n  .1  ˇ/q < p1 < 1 ˆ ˆ ˆ : p1  1

if .1  ˇ/q < n; if .1  ˇ/q D n; if .1  ˇ/q > n: (114)

556

G.P. Galdi and M. Kyed

Then 1;2;q 8u 2 Wper .R  / W

kukLrper0 .RILp0 .// C krukLrper1 .RILp1 .//  C kuk1;2;q : (115) t u

Proof. See [14].

On the strength of the estimates for the pressure in Lemma 3 and the embedding properties in Lemma 4, a proof of Theorem 12 can be provided: q

1 .R  / in Lper;? .R  /, it suffices to Proof of Theorem 12. By density of C0;per;? 1 consider only F 2 C0;per;? .R  /. The starting point will be the solution .u; p/ 2 1;2;2 1;2 .R  /  Dper;? .R/ from Theorem 8. By adding a function that depends Wper;? R only on time to p, one may assume without loss of generality that R p dx D 0. 0 The solution .u; p/ shall be decomposed by multiplication of a “cutoff” function. For this purpose, fix three constants 0 < R <  < R0 such that Rn n   BR . For convenience, the notation T for the time-domain Œ0; T / is used in the scope of the proof. Two fundamental estimates shall be established. To show the first one, a cutoff function 1 2 C 1 .Rn I R/ is introduced with 1 .x/ D 1 for jxj   and 1 .x/ D 0 for jxj  R . Let  L denote the fundamental solution to the Laplace operator in Rn and put

P W R  Rn ! R;

  V WD r L Rn r 1  u ;  P WD  L Rn Œ@t    @1 .r

w W R  Rn ! Rn ;

w.t; x/ WD

V W R  Rn ! Rn ;

W R  Rn ! R;

.t; x/ WD

1 .x/ u.t; x/ 1 .x/ p.t; x/

1

  u/ ;

(116)

 V .t; x/;

 P .t; x/:

Then .w; / is a solution to the whole-space problem 8 @t w  w C @1 w C r D ˆ ˆ < 1 F  2r 1  ru   ˆ ˆ : div w D 0

1u

C @1

1u

Cr

1p

in R  Rn ; in R  Rn : (117)

The precise regularity of .w; / is not important at this point. It is enough to observe that w and belong to the space of tempered time-periodic distributions S0per .R  Rn /, which is easy to verify from the definition (116) and the regularity of u and p. It is not difficult to show (see [28, Lemma 5.3]) that a solution w to (117) is unique in the class of distributions in S0per .R  Rn / satisfying Pw D 0. Consequently, w coincides with the solution from Theorem 9 corresponding to the right-hand side in (117) and therefore satisfies

10 Time-Periodic Solutions to The Navier-Stokes Equations

557

kwk1;2;s  c0 k 1 F  2r 1  ru   1 u C @1 1 u C r 1 pks    c1 kF ks C kuks;T C kruks;T C kpks;T for all s 2 .1; 1/. Clearly,   krV ks C kr 2 V ks  c2 kuks;T C kruks;T : Since u D w C V for x 2  , the estimates above imply kruks;T C kr 2 uks;T    c3 kF ks C kuks;T C kruks;T C kpks;T for all s 2 .1; 1/. For a similar estimate on u itself, first turn to (106) and apply Lemma 3 to deduce   k@t uks;T  c4 kF ks C kuks;T C k @1 uks;T C krpks;T    c5 kF ks C kuks;T C kruks;T C kpks;T : Since Pu D 0, Poincaré’s inequality yields kuks;T  c6 k@t uks;T . It thus follows that   kuk1;2;s;T  c7 kF ks C kuks;T C kruks;T C kpks;T

(118)

for all s 2 .1; 1/. Now a similar estimate for u over the bounded domain T   shall be established. To this end, a “cutoff” function 2 2 C 1 .Rn I R/ is introduced with 2 .x/ D 1 for jxj   and 2 .x/ D 0 for jxj  R0 . Let V be a vector field with 1;2;2 V 2 Wper;? .R  Rn /;

8s 2 .1; 1/ W kVk1;2;s Since

supp V  R  ;R0 ; div V D r 2  u;    c kuks;T;R0 C kruks;T;R0 C k@t uks;T;R0 : (119) Z

Z r

2  u dx D

;R0

div



R0

 2 u dx D

Z u  n d D 0; @R0

the existence of a vector field V with the properties above can be established by the same construction as the one used in [10, Theorem III.3.3]; see also [27, Proof of Lemma 3.2.1]. Now let w W R  Rn ! Rn ; W R  R ! R; n

w.t; x/ WD .t; x/ WD

2 .x/ u.t; x/ 2 .x/ p.t; x/:

 V.t; x/;

(120)

558

G.P. Galdi and M. Kyed

1;2;2 1;2 Then .w; / 2 Wper;? .R  R0 /  Dper;? .R  R0 / is a solution to the problem

8 @t w  w C r D 2 F C 2 @1 u  2r ˆ ˆ ˆ ˆ ˆ < C r 2 p C @t V  V

2

 ru  

2u

in R  R0 ;

ˆ div w D 0 ˆ ˆ ˆ ˆ : wD0

in R  R0 ; on R  @R0 :

By Theorem 6, it follows that kwk1;2;s  c8 k 2 F C 2 @1 u  2r 2  ru   2 u C r 2 p C @t V  Vks    c9 kF ks C kuks;TR0 C kruks;TR0 C kpks;T;R0 C kVk1;2;s for all s 2 .1; 1/. Since ;R0   , (119) and (118) can be combined to estimate   kuk1;2;s;T  c10 kF ks C kuks;T C kruks;T C kpks;T : In combination with (118), the estimate above implies   kuk1;2;s  c kF ks C kuks;TR0 C kruks;TR0 C kpks;TR0

(121)

for all s 2 .1; 1/. The estimate (121) has been established for all s 2 .1; 1/, but it is not actually known at this point whether the right-hand side is finite or not. At the outset, it is only known that the right-hand side is finite for s 2 .1; 2. A bootstrap argument can be used to show that it is also finite for s 2 .2; 1/. For this purpose, the embedding 1;2;s .R/ stated in Lemma 4 are employed to show the implication properties of Wper ns

8s 2 Œ2; 1/ W

1;2;s n1 u 2 Wper .R  / ) u; ru 2 Lper .R  /:

(122)

Now turn to estimate (109) of the pressure term. By Hölder’s inequality, Z

T 0



s1

sn  n1

1

s s kru.t; /ks; kru.t; /k1;s; R R 0

dt

0

Z  0

T

n.s1/s s.n1/n

kru.t; /ks;R

0

 s.n1/n s.n1/ dt

n1 n kuk1;2;s :

By Lemma 4, utilized this time with ˇ D 1, the right-hand side above is finite for 1;2;s all s 2 Œ2; 1/ provided u 2 Wper .R  /. Due to the normalization of the pressure p carried out in the beginning of the proof, Lemma 3 can be applied to infer from (109) that

10 Time-Periodic Solutions to The Navier-Stokes Equations

559 ns

8s 2 Œ2; 1/ W

1;2;s n1 u 2 Wper .R  / ) p 2 Lper .R  R0 /:

(123)

By (121) and the implications (122) and (123), it follows that 8s 2 Œ2; 1/ W

ns 1;2; n1

1;2;s u 2 Wper .R  / ) u 2 Wper

.R  /:

(124)

Starting with s D 2, the implication (124) can be bootstrapped a sufficient number 1;2;s of times to deduce u 2 Wper .R  / for any s 2 .2; 1/. It follows that the righthand side of (121) is finite for all s 2 .1; 1/. At this point, it is now possible to use interpolation and with Young’s inequality in (121) to show  (109) in combination  kuk1;2;s  c11 kF ks C kuks;TR0 for all s 2 .1; 1/. It then follows directly from (106) that   kuk1;2;s C krpks  c12 kF ks C kuks;TR0 (125) for all s 2 .1; 1/. 1;2;r 1;2;r 1;r If .U ; P/ 2 Wper;? .R  /  Wper;? .R  /  Dper;? .R  / (106) with homogeneous right-hand side, then U D rP D 0 follows by a duality argument. 1 More specifically, since for arbitrary ' 2 C0;per;? .R  / existence of a solution 0

0

1;2;r 1;r .R  /  Dper;? .R  / to (106) with ' as the right-hand side .W ; …/ 2 Wper;? has been established, the computation

Z

T

Z

0D

.@t U C @1 U  U C rP/  ' dxdt 0



Z

T

Z

Z

T

Z

U  .@t W C @1 W  W C r…/ dxdt D

D 0



U  ' dxdt 0



(126) is valid. It follows that U D 0 and in turn, directly from (106), that also rP D 0. Now return to the estimate (125). Owing to the fact that a solution to (106) with homogeneous right hand is necessarily zero, which was just shown above, a standard contradiction argument (see, e.g., [12, Proof of Proposition 2]) can be used to eliminate the lower-order term on the right-hand side in (125) to conclude kuk1;2;q C krpkq  c13 kF kq :

(127)

q

1 It is easy to verify that C0;per;? .R  / is dense in Lper;? .R  /. By a density 1;2;q

1;q

argument, the existence of a solution .u; p/ 2 Wper;? .R  /  Dper;? .R  / to q (106) that satisfies (127) therefore follows for any F 2 Lper;? .R  /. 1;2;r 1;r .R  /  Dper;? .R  / is another solution Finally, assume .e u; e p/ 2 Wper;? to (106) with r 2 .1; 1/. Applied to the difference .u  e u; p  e p/, the duality argument used in (126) yields u D e u and rp D re p. The proof of theorem is thereby complete. t u

560

4.8

G.P. Galdi and M. Kyed

Existence of Lq Strong Solutions

The question now arises as to whether or not existence of a solution to the fully nonlinear problem (61) can be established on the basis of the Lq estimates and corresponding function spaces from Sect. 4.7. Compared to the class of strong solutions introduced in Sect. 4.3, more information on the asymptotic structure at spatial infinity could be derived for such a solution. If the T -time-periodic velocity u1 .t / 2 Rn is directed along a single axis, say u1 .t / D u1 .t /e 1 , and its net RT motion over a period is non-zero, that is, 0 u1 .t / dt ¤ 0, the question can be RT answered affirmatively. If 0 u1 .t / dt D 0, the Lq estimates from Sect. 4.7 are not adequate. In this case, the fully nonlinear problem (61) can be treated in weighted function spaces based on the estimates introduced in Sect. 4.6. Below, the case RT 0 u1 .t / dt ¤ 0 is investigated more closely. The projections P and P? introduced in (62) can be employed to decompose u1 .t / into a constant WD Pu1 and a oscillatory part P? u1 . A linearization of (61) around leads to (106). In the case ¤ 0, Corollary 1 in combination with a fixed point argument can be used to show existence of a solution to the fully nonlinear problem (61) for data sufficiently restricted in “size.” For this purpose, it is convenient to put nq

1;2;q; nq

Wper;? 1;q;

1;2;

1;2;q

nq

.R  / WD Wper;? .R  / \ Wper;?nq .R  /;

nq

1;q

1;

nq

nq .R  /: Dper;?nq .R  / WD Dper;? .R  / \ Dper;?

The resolution of (61) then reads: 2 Theorem 13. Let   Rn (n D 2; 3) be an exterior domain of class

C . Assume 6 4 u1 .t / D u1 .t /e 1 with WD Pu1 > 0. If n D 3, let q 2 5 ; 3 . If n D 2, let   q 2 1; 65 . There is an 0 > 0 such that for all 2 .0; 0 , there is an "3 > 0 such q 1;1 that for all f 2 Lper .R  / and u1 2 Wper .R/ satisfying

kf kq C kP? f k

nq nq

C kP? u1 k1;1  "3 ;

(128)

there is a solution q

nq

1;2;q; nq

.u; p/ 2 X ./ ˚ Wper;?

1;q;

nq

.R  /  D 1;q ./ ˚ Dper;?nq .R  /

(129)

to (61).

Proof. Consider first the case n D 3 and q 2 65 ; 43 . In order to “lift” the boundary value u1 in (61), that is, rewrite the system as one of homogeneous boundary

10 Time-Periodic Solutions to The Navier-Stokes Equations 3q

1;2;q; 3q

values, a solution .W; …? / 2 Wper;?

561 1;q;

3q

.R  /  Dper;?3q .R  / to

8 W C r…? D W ˆ ˆ < div W D 0 ˆ ˆ : W D P ? u1

in R  ; in R  ;

(130)

on R  @;

is introduced. One can use standard theory for elliptic systems to solve (130) in T -time-periodic function spaces and obtain a solution that satisfies 8r 2 .1; 1/ W kWk1;2;r C kr…? kr  c0 kP? u1 k1;1 ;

(131)

where c0 D c0 .r; q; ; /. Furthermore, classical results for the steady-state Oseen q problem [10, Theorem VII.7.1] ensure existence of a solution .V; …s / 2 X ./  D 1;q ./ to 8 V  @1 V C r…s D 0 in ; ˆ ˆ < div V D 0 in ; (132) ˆ ˆ : V D on @; which satisfies 8r 2 .1; 2/ W kVkX r ./ C kr…s kr  c1 ;

(133)

where c1 D c1 .r; ; /. Focus will now be on finding a solution .u; p/ to (61) on the form u D v C V C w C W;

p D p C …s C C …? ;

(134)

q

where .v; p/ 2 X ./  D 1;q ./ is a solution to the steady-state problem 8 ˆ ˆ v  @1 v C rp D R1 .v; w; V; W/ < div v D 0 ˆ ˆ : vD0

in ; in ;

(135)

on @;

with R1 .v; w; V; W/ WD  v  rv  v  rV  V  rv  V  rV







 P w  rw  P w  rW  P W  rw  P W  rW



C P P? u1 @1 w C P P? u1 @1 W C Pf;

562

G.P. Galdi and M. Kyed 3q

1;2;q; 3q

and .w; / 2 Wper;?

1;q;

3q

.R  /  Dper;?3q .R  / a solution to

8 @t w  w  @1 w C r D R2 .v; w; V; W/ in R  ; ˆ ˆ < div w D 0 in R  ; ˆ ˆ : wD0 on R  @;

(136)

with







R2 .v; w; V; W/ WDP? w  rw P? w  rW P? W  rw P? W  rW  v  rw  v  rW  w  rv  w  rV  V  rw  V  rW  W  rv  W  rV CP? u1 @1 vCP? u1 @1 V

CP? P? u1 @1 w CP? P? u1 @1 W  @t W  W C @1 W C P? f: The systems (135) and (136) appear as the result of inserting (134) into (61) and subsequently applying first P then P? to the equations. Recalling the function spaces introduced in Corollary 1 to define the Banach space q

1;2;q;

q

3q

1;q;

3q

K .R  / WD X ./ ˚ Wper;? 3q .R  /  D 1;q ./ ˚ Dper;?3q .R  /; (137) one can obtain solutions .v; p/ and .w; / to (135) and (136), respectively, as a fixed point of the mapping q

q

N W K .R  / ! K .R  /;   N .v C w; p C / WD A1 Oseen R1 .v; w; V; W/ C R2 .v; w; V; W/ I q;

3q

3q .R  /. note that R1 .v; w; V; W/ 2 Lq ./ and R2 .v; w; V; W/ 2 Lper;? More specifically, one can show that N is a contracting self-mapping on ball of sufficiently small radius. For this purpose, let  > 0 and consider some q .v C w; p C / 2 K \ B . Suitable estimates of R1 and R2 in combination with a smallness assumption on "3 from (128) are needed to guarantee that N has the desired properties. Regarding the estimates, one can employ Hölder’s inequality, Sobolev embedding, and basic interpolation to obtain

kv  rvkq  kvk

2q 2q

 krvk2  c2 

3q3 q

kvk2X q  c2 

3q3 q

2 :

(138)

10 Time-Periodic Solutions to The Navier-Stokes Equations

563

This step requires q 2 65 ; 43 . The other terms in the definition of R1 can be estimated in a similar fashion to conclude in combination with assumption (128) that  3q3 1 1 3 kR1 .v; w; V; W/kLq ./  c3  q 2 C 4  C 2 C 2 C 2  C "3 C "3 2 C "3 : 3q

q

3q An estimate of R2 is required both in the Lper .R  / and Lper .R  / norm. Observe that

kP? u1 @1 vkLqper .R/  c4 kP? u1 k1 k@1 vkq  c4 1 "3 :

(139)

The other terms in R2 can be estimated, in part with the help of the embedding properties from Lemma 4, to obtain   kR2 .v; w; V; W/kLqper .R/  c5 1 "3  C 2 C "3 C "3 2 C  C "3 C "3 : 3q 3q .R  / estimate of R2 . For Lemma 4 can also be used to establish an Lper;? example,

kw  rwk

 c6 kwk

3q 3q

krwk

3q 3q

3q

3q .// L1 per .RIL

Lper .RIL1 .//

Lper .R/

 c7 2 ;

where Lemma 4 is utilized with ˛ D 0 and ˇ D 1 in the last inequality. For this utilization of Lemma 4, it is required that q  65 . Further note that kP? u1 @1 vk

3q 3q

 "3 kvkX q  "3 ;

3q which explains the choice of the exponent 3q in the setting of the mapping N . The rest of the terms in R2 can be estimated to conclude

   c8 2 C "3 C "3 2 C "3 C "3 :

kR2 .v; w; V; W/k

3q 3q Lper

.R/

Now choose 0  1 and deduce by Corollary 1; recall that kA1 Oseen k does not depend on , the estimate kN .v C w; p C /k

q K





1

 kAOseen k  kR1 k

Lq ./

C kR2 k

3q q; 3q

Lper



3q3  q

 C10

.R/

 1 3 2 C 1 "3  C 4  C 2 C "3 2 C "3 :

564

G.P. Galdi and M. Kyed

In particular, N becomes a self-mapping on B if  3q3  1 3 C10  q 2 C 1 "3  C 4  C 2 C "3 2 C "3  : One may choose "3 WD 2 and  WD to find the above inequality satisfied for sufficiently small . For such choice of parameters, one may further verify that N is also a contraction. By the contraction mapping principle, existence of a fixed point for N follows. This concludes the proof in the case  n D 3. The proof in the case n D 2 and q 2 1; 65 follows along the same lines. To ensure the existence of a solution to (132) satisfying an estimate like (133) with a constant independent of , one cannot use [10, Theorem VII.7.1], but can instead use [10, Theorem XII.5.1] to obtain a solution to (132) that satisfies 8r 2 .1; 6=5/ W kVkX r ./ C h…s i1;r  c9

2r2 r

jlog j1 ;

(140)

with c9 D c9 .r; ; /. The estimate corresponding to (138) in the case n D 2 reads kv  rvkq  c10 

3q2 q

kvk2X q  c10 

323 q

2 I

(141)



see, for example, [10, Lemma XII.5.4]. The rest of the proof in the case n D 2 follows by simple adjustments to the proof for n D 3 above. Remark 10. It is possible to establish higher-order regularity for the solution in Theorem 13 by a bootstrap argument based on the linear theory from Theorem 12. More specifically, if the data possesses higher-order regularity, say m;q f 2 Wper;? .R  /, one can put the nonlinear term u  ru on the right-hand side in (61) and iteratively apply Theorem 12 after taking partial derivatives on both sides. With such an argument, it is possible to establish a degree of regularity for .u; p/ ˇ q 1;2;q corresponding to the regularity of the data f , that is, @˛t @x u 2 X ./ ˚ Wper;? .R  / for j˛j C jˇj  m. For more details on such a result, see [28, Theorem 2.4]. Alternatively, higher-order regularity can be obtained via regularity theory for the initial-value problem as mentioned in Remark 5. Remark 11. The solution u in Theorem 13 possess enough summability at spatial infinity for an adaptation of the uniqueness argument from the proof of Theorem 2 to be carried out. More precisely, given a weak solution U in the sense of Definition 2, it is possible to insert u as a “test function” in the weak formulation (64) for U . In addition, after multiplication by U in the system (61) satisfied by u, the summability of both, in particular the latter, is adequate to integrate by parts. These are the two main steps in proof of Theorem 2. Provided therefore that both u and U satisfy an appropriate energy inequality corresponding to (61) and the data is sufficiently restricted in “size,” it can be shown that u D U . In other words, the strong solution in Theorem 13 can be shown to be unique in a class of weak solutions satisfying an energy inequality. See [28, Theorem 2.3] for more details on such a result.

10 Time-Periodic Solutions to The Navier-Stokes Equations

4.9

565

Asymptotic Structure

Important physical properties of a solution u to (61) are related to its asymptotic structure at spatial infinity. The asymptotic structure is best exposed by an asymptotic expansion u.t; x/ D A.t; x/ C R.t; x/ into an explicitly known leading term A and a remainder term R that decays faster to 0 as jxj ! 1 than A. The task of identifying such an expansion shall now be addressed for u1 .t / D u1 .t /e 1 , that is, for functions u1 directed along a single axis. The case Pu1 ¤ 0 is considered first. A strong solution in the class (129) is singled out for investigation. Theorem 13 yields existence of a solution in this class, so it is a reasonable starting point. By nature of the function space in (129), the steady-state part Pu of such a solution enjoys, at the outset, different Lq q summability properties than the oscillatory part P? u. In fact, since Pu 2 X ./ 1;2;q and P? 2 W .R  /, better spatial decay is available for P? u in the sense of summability, that is, the range of exponents q for which P? u.t; / 2 Lq ./ is lower than the range of exponents q for which Pu 2 Lq ./. This suggests that the leading term in an asymptotic expansion of u is dominated by Pu; a key observation that underpins the analysis below. Recall that the Oseen fundamental solution  Oseen introduced in Sect. 4.5 satisfies 8q0 2 Œ1; 2 W



Oseen

… Lq0 .Rn n Br / for any r > 0I

(142)

see, for example, [10, Chapter VII.3]. On the strength of this information, the theorem below yields an asymptotic expansion of a solution to (61) with the decay of the remainder term characterized in the sense of summability as described above. Theorem 14. Let   Rn (n D 2; 3) be an exterior domain of class C 2 . Assume 1 u1 .t / D u1 .t /e 1 with WD Pu1 > 0. Moreover, assume f 2 C0;per .R  /. 6 4  6 If n D 3, let q 2 5 ; 3 . If n D 2, let q 2 1; 5 . There is an "4 > 0 such that if 1 .R/ satisfies kP? u1 k1  "5 , then a solution .u; p/ to (61) in the class u1 2 Cper (129) satisfies u.t; x/ D  Oseen .x/  F C R.t; x/;

(143)

where F WD

1 T

Z

T

Z S.u; p/  n d C

0

@

1 T

Z

T

Z f .t; x/ dxdt

0

(144)

Rn

and  8q0 2

 nC1 ;1 W n

  q0 R 2 L1 per RI L ./ :

(145)

566

G.P. Galdi and M. Kyed

  Here, S.u; p/ denotes the Cauchy stress tensor S.u; p/ WD  ru C ru>  pI . Remark 12. As previously described, a solution to (61) describes the fluid flow around an object moving with constant velocity u1 . In physical terms, the quantity F in (144) equals the total force exerted upon the fluid by the moving object and the external force f combined over a full time-period. Remark 13. In light of (142), the decomposition (143) indicates that the velocity field u possesses the asymptotic structure of the Oseen fundamental solution  Oseen . From a physical point of view, this implies the existence of a wake behind the moving object; see again [10, Chapter VII.3]. Another interesting observation is that the leading term in the expansion (143) is independent of time, which suggests that oscillatory characteristics of the fluid flow appear only locally around the body. Proof of Theorem 14. Only the case n D 3 is considered. The case n D 2 can be treated by a similar approach. Since the data f and u1 are assumed to be smooth, 1 1 recall from Remark 10 that also u 2 Cper .R  / and p 2 Cper .R  /. Although not strictly necessary, the smoothness of the solution simplifies the proof as local regularity becomes a nonissue. Let 2 C01 .R3 I R/ be a “cutoff” function with

D 0 on B1 and D 1 on R3 n B2 . Put R .x/ WD Rx for some R > R0 . Since Z

Z r R .x/  u.t; x/ dx D

BR;2R

r R .x/  u.t; x/ dx 2R

Z

R .x/  div u.t; x/ dx D 0;

D 2R

1 .R  R3 / satisfying both supp U  R  BR;2R there is a vector field U 2 Cper and div U D r R  u. The construction of U is well known in the case of timeindependent vector fields; see [10, Theorem III.3.3]. The same construction can be applied in the above case of time-periodic vector fields; see [27]. Let e u WD R u  U and e p WD R p. Then

8 u  e u  @1e u C re p D P? u1 @1e u e u  re u C R f C h @te ˆ ˆ < ˆ ˆ :

in R  R3 ; in R  R3 ;

dive uD0

e u.t C T; x/ D e u.t; x/ (146)

1 .R  R3 / of bounded support supp h  R  B2R . Put v WD Pe u and with h 2 Cper w WD P?e u. At the outset q

3q

1;2;q; 3q

.v C w; p/ 2 X .R3 / ˚ Wper;?

1;q;

3q

.R  R3 /  D 1;q .R3 / ˚ Dper;?3q .R  R3 /: (147)

10 Time-Periodic Solutions to The Navier-Stokes Equations

567

However, it is possible to show that 8q0 2 .1; q W

q

1;2;q

v 2 X 0 .R3 /; w 2 Wper;?0 .R  R3 /:

(148)

For this purpose, put MR WD R re u and identify .e u; e p/ as a fixed point of the 2 mapping q

q

N W K .R  R3 / ! K .R  R3 /;   N .e u; e p/ WD A1 u  MRe u C R f C h ; Oseen P? u1 @1e q

where K is defined as in (137). Provided R is chosen sufficiently large and kP? u1 k1 sufficiently small, it can be shown, by estimates very similar to those made in the proof of Theorem 13, that N as mapping q

q

q

q

N W K 0 .R  R3 / \ K .R  R3 / ! K 0 .R  R3 / \ K .R  R3 / is a contraction for all q0 2 .1; q. By the contraction mapping principle, N therefore has a unique fixed point. Uniqueness of the fixed point implies (148). From the embedding properties in Lemma 4, it immediately follows from (148) that   q0 3 8q0 2 .1; 1/ W w 2 L1 per RI L .R / :

(149)

Focus is now shifted to v. Applying P to the system (146), one finds that v is a solution to the steady-state problem 8  v  @1 v C rp ˆ ˆ
0; 8q0 2 Œ1; 3/ W

q

0  Oseen 2 Lloc .R3 /

(152)

(see, e.g., [10, Chapter VII.3]), it follows as a consequence of (148) that the convolution on the right-hand side in (151) is well defined and thus a solution

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to (150). A standard uniqueness argument implies that this solution coincides with v. An asymptotic expansion of v can be derived from (151). To this end, recall (see again [10, Chapter VII.3]) that 8q0 2 .4=3; 1/ W

r Oseen 2 Lq0 .R3 n Br / for any r > 0;

8q0 2 Œ1; 3=2/ W

0 r Oseen 2 Lloc .R3 /:

(153)

q

Consequently, the summability properties from (148) in combination with Young’s inequality imply  

Oseen

 



 P P? u1 @1 w  v  rv  P w  rw 



D 

Oseen

i







 div P P? u1 w ˝ e 1  v ˝ v  P w ˝ w

 



Oseen D @k  ij  P P? u1 w ˝ e 1  v ˝ v  P w ˝ w

i

2 Lq0 .R3 /

jk

(154) for all q0 2 

Oseen

4 3

 ; 1 . Moreover, due to (153), it is standard to show

 P R f C h  

Oseen

Z  R3



P R f C h dx 2 Lq0 .R3 /

(155)

  for all q0 2 43 ; 1 . It remains to compute the integral in (155). For this purpose,

isolate P R f C h in (146), and compute Z



1 P R f C h dx D lim R!1 T R3

Z 0

T

Z BR

@te u  e u  @1e u

C re p  P? u1 @1e u Ce u  re u dxdt Z TZ

1  div S.e u; e p/ C div .e u  u1 / ˝ e u dxdt D lim R!1 T 0 BR Z TZ

1 D lim S.u; p/  n C .u  u1 / ˝ u  n d dt R!1 T 0 @BR Z TZ Z Z 1 T 1 S.u; p/  n d dt C f dxdt; D T 0 @ T 0  (156) where the change of sign in the last integral is due to n denoting the outer normal on @. The identity above together with (149), (154), and (155) concludes the proof. t u

10 Time-Periodic Solutions to The Navier-Stokes Equations

569

Under certain conditions, the asymptotic expansion in Theorem 14 can also be established with a pointwise decay estimate of the remainder term. Below, a sketch of a proof is given in the case Pu1 ¤ 0 and P? u1 D 0. The proof is based on the pointwise estimates of the fundamental solution in Sect. 4.5. The assumption P? u1 D 0 is equivalent to the requirement that u1 is constant. Only the case of a three-dimensional exterior domain is included. A similar result is not available in the two-dimensional case. Theorem 15. Let   R3 be an exterior domain of class C 2 . Assume u1 D e 1 1 with > 0 a constant. Let f 2 C0;per .R  /. Then a solution .u; p/ to (61) in the class (129) satisfies (143) with 3

kR.; x/k1  C jxj 2 C" :

8" > 0 9C > 0 8jxj > 1 W

(157)

p WD R p are introduced Proof. As in the proof of Theorem 14, e u WD R u  U and e and the system (146) investigated. As in the proof of Theorem 9, the interpretation of (146) as a system of partial differential equations on the group G is employed. One can proceed as in the proof of Theorem 14 to deduce (148), which implies that the convolution of the time-periodic fundamental solution  TP from Sect. 4.5 with the right-hand side in (146) is well defined in the classical sense and thus constitutes a solution to (146). By a uniqueness argument similar to the one made in the proof of Theorem 9, it therefore follows that e uD

TP

   e u  re u C R f C h :

Compared to the right-hand side in (146), the term P? u1 @1e u is missing above since u1 D e 1 implies P? u1 D 0. The structure of  TP identified in Theorem 10 now implies for v WD Pe u and w WD P?e u the identities 

vD

Oseen

wD

?







  v  rv  P w  rw C P R f C h ; 





(158) 

  v  rw  w  rv  P? w  rw C P? R f C h ;

(159)

where in the first identity, it is used that  Oseen ˝ 1R=T Z  F D  Oseen  PF and in the second the property  ?  P? F D  ?  F for all sufficiently regular F . Inspired by the proof in [10, Chapter X.8] of the asymptotic expansion of a solution to the corresponding steady-state problem, one can show Z 8" > 0 W

1 jrvj dx C T R3 nBR 2

Z 0

T

Z R3 nBR

jrwj2 dxdt  CR1C" :

(160)

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G.P. Galdi and M. Kyed

The above estimate is crucial. Together with the pointwise estimate (98) of  ? , it delivers the foundation for a pointwise estimate of all terms on the right-hand side of (158) and (159). The resulting estimates for v and w (see [15] for the details) are 8" > 0 W 8" > 0 W

jv.x/j  C jxj1C" ; (161)   3 2 jw.t; x/j  C jxj 2 C jxj 5 C" .kvkL1 .B jxj=2 / C kwkL1 .RB jxj=2 / / : (162)

Now insert (161) into (162). After two bootstra