Recent Developments of Mathematical Fluid Mechanics [1st ed.] 3034809387, 978-3-0348-0938-2, 978-3-0348-0939-9, 3034809395

Table of contents :
Front Matter....Pages i-ix
The Work of Yoshihiro Shibata....Pages 1-12
Existence of Weak Solutions for a Diffuse Interface Model of Power-Law Type Two-Phase Flows....Pages 13-23
Stationary Solutions for a Navier-Stokes/Cahn-Hilliard System with Singular Free Energies....Pages 25-41
Parabolic Equations on Uniformly Regular Riemannian Manifolds and Degenerate Initial Boundary Value Problems....Pages 43-77
A Generalization of Some Regularity Criteria to the Navier–Stokes Equations Involving One Velocity Component....Pages 79-97
On the Singular p-Laplacian System Under Navier Slip Type Boundary Conditions: The Gradient-Symmetric Case....Pages 99-109
Thermodynamically Consistent Modeling for Dissolution/Growth of Bubbles in an Incompressible Solvent....Pages 111-134
On Unsteady Internal Flows of Bingham Fluids Subject to Threshold Slip on the Impermeable Boundary....Pages 135-156
Inhomogeneous Boundary Value Problems in Spaces of Higher Regularity....Pages 157-173
Blow-Up Criterion for 3D Navier-Stokes Equations and Landau-Lifshitz System in a Bounded Domain....Pages 175-182
Local Regularity Results for the Instationary Navier-Stokes Equations Based on Besov Space Type Criteria....Pages 183-214
On Global Well/Ill-Posedness of the Euler-Poisson System....Pages 215-231
On the Motion of a Liquid-Filled Rigid Body Subject to a Time-Periodic Torque....Pages 233-255
Bounded Analyticity of the Stokes Semigroup on Spaces of Bounded Functions....Pages 257-273
On the Weak Solution of the Fluid-Structure Interaction Problem for Shear-Dependent Fluids....Pages 275-289
Stability of Time Periodic Solutions for the Rotating Navier-Stokes Equations....Pages 291-319
A Weak Solution to the Navier–Stokes System with Navier’s Boundary Condition in a Time-Varying Domain....Pages 321-335
Effects of Fluid-Boundary Interaction on the Stability of Boundary Layers in Plasma Physics....Pages 337-349
On Incompressible Two-Phase Flows with Phase Transitions and Variable Surface Tension....Pages 351-374
On the Nash-Moser Iteration Technique....Pages 375-400
Rate of Convergence to the Stationary Solution of the Navier-Stokes Exterior Problem....Pages 401-410
....Pages 411-442

Citation preview

Advances in Mathematical Fluid Mechanics

Herbert Amann Yoshikazu Giga Hideo Kozono Hisashi Okamoto Masao Yamazaki Editors

Recent Developments of Mathematical Fluid Mechanics

Advances in Mathematical Fluid Mechanics Series editors Giovanni P. Galdi, Pittsburgh, USA John G. Heywood, Vancouver, Canada Rolf Rannacher, Heidelberg, Germany

Advances in Mathematical Fluid Mechanics is a forum for the publication of high quality monographs, or collections of works, on the mathematical theory of fluid mechanics, with special regards to the Navier-Stokes equations. Its mathematical aims and scope are similar to those of the Journal of Mathematical Fluid Mechanics. In particular, mathematical aspects of computational methods and of applications to science and engineering are welcome as an important part of the theory. So also are works in related areas of mathematics that have a direct bearing on fluid mechanics. The monographs and collections of works published here may be written in a more expository style than is usual for research journals, with the intention of reaching a wide audience. Collections of review articles will also be sought from time to time. More information about this series at http://www.springer.com/series/5032

Herbert Amann • Yoshikazu Giga • Hideo Kozono • Hisashi Okamoto • Masao Yamazaki Editors

Recent Developments of Mathematical Fluid Mechanics

Editors Herbert Amann Institut fRur Mathematik University of ZRurich ZRurich, Switzerland Hideo Kozono Department of Mathematics Waseda University Tokyo, Japan

Yoshikazu Giga Graduate School of Mathematical Sciences University of Tokyo Tokyo, Japan Hisashi Okamoto Research Institute for Mathematical Sciences Kyoto University Kitashirakawa Kyoto, Japan

Masao Yamazaki Department of Mathematics Waseda University Tokyo, Japan

ISSN 2297-0320 ISSN 2297-0339 (electronic) Advances in Mathematical Fluid Mechanics ISBN 978-3-0348-0938-2 ISBN 978-3-0348-0939-9 (eBook) DOI 10.1007/978-3-0348-0939-9 Library of Congress Control Number: 2015960758 Mathematics Subject Classification (2010): 35XX, 76XX Springer Basel Heidelberg New York Dordrecht London © Springer Basel 2016 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. Printed on acid-free paper Springer Basel is part of Springer Science+Business Media (www.birkhauser-science.com)

Preface

This volume collects research papers and survey articles of participants in the “International Conference on Mathematical Fluid Dynamics on the Occasion of Yoshihiro Shibata’s 60th Birthday” , which was held from March 5 to 9, 2013, in Nara, the former capital of Japan. We thank all the participants, in particular the invited speakers, who contributed to this volume. Our thanks also go to the referees for their efficient work, and to the secretaries, the staff, and the students who helped us during the meeting. The conference was supported, in part, by the Grant in Aid for Scientific Research of the Japan Society for the Promotion of Science (No. 24224003, 24224004, 24340025) and by the JSPS-DFG Japanese-German Graduate Externship. Zürich, Switzerland Tokyo, Japan Tokyo, Japan Kyoto, Japan Tokyo, Japan

Herbert Amann Yoshikazu Giga Hideo Kozono Hisashi Okamoto Masao Yamazaki

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Contents

The Work of Yoshihiro Shibata .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Herbert Amann, Yoshikazu Giga, Hisashi Okamoto, Hideo Kozono, and Masaso Yamazaki

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Existence of Weak Solutions for a Diffuse Interface Model of Power-Law Type Two-Phase Flows . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Helmut Abels, Lars Diening, and Yutaka Terasawa

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Stationary Solutions for a Navier-Stokes/Cahn-Hilliard System with Singular Free Energies . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Helmut Abels and Josef Weber

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Parabolic Equations on Uniformly Regular Riemannian Manifolds and Degenerate Initial Boundary Value Problems .. . . . . . . . . . . . . . Herbert Amann

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A Generalization of Some Regularity Criteria to the Navier–Stokes Equations Involving One Velocity Component ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Šimon Axmann and Milan Pokorný On the Singular p-Laplacian System Under Navier Slip Type Boundary Conditions: The Gradient-Symmetric Case . .. . . . . . . . . . . . . . . . . . . . H. Beirão da Veiga

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Thermodynamically Consistent Modeling for Dissolution/Growth of Bubbles in an Incompressible Solvent . . . . . . . . . . 111 Dieter Bothe and Kohei Soga On Unsteady Internal Flows of Bingham Fluids Subject to Threshold Slip on the Impermeable Boundary . . . . . . . .. . . . . . . . . . . . . . . . . . . . 135 Miroslav Bulíˇcek and Josef Málek

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Inhomogeneous Boundary Value Problems in Spaces of Higher Regularity .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 157 Robert Denk and Tim Seger Blow-Up Criterion for 3D Navier-Stokes Equations and Landau-Lifshitz System in a Bounded Domain . . . . .. . . . . . . . . . . . . . . . . . . . 175 Jishan Fan and Tohru Ozawa Local Regularity Results for the Instationary Navier-Stokes Equations Based on Besov Space Type Criteria . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 183 Reinhard Farwig On Global Well/Ill-Posedness of the Euler-Poisson System . . . . . . . . . . . . . . . . . 215 Eduard Feireisl On the Motion of a Liquid-Filled Rigid Body Subject to a Time-Periodic Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 233 Giovanni P. Galdi, Giusy Mazzone, and Mahdi Mohebbi Seeking a Proof of Xie’s Inequality: On the Conjecture That m ! 1 . . . 257 John G. Heywood Bounded Analyticity of the Stokes Semigroup on Spaces of Bounded Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 275 Matthias Hieber and Paolo Maremonti On the Weak Solution of the Fluid-Structure Interaction Problem for Shear-Dependent Fluids . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 291 Anna Hundertmark, Mária Lukáˇcová-Medvid’ová, and Šárka Neˇcasová Stability of Time Periodic Solutions for the Rotating Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 321 Tsukasa Iwabuchi, Alex Mahalov, and Ryo Takada Weighted Lp  Lq Estimates of Stokes Semigroup in Half-Space and Its Application to the Navier-Stokes Equations. . . . . .. . . . . . . . . . . . . . . . . . . . 337 Takayuki Kobayashi and Takayuki Kubo On Vorticity Formulation for Viscous Incompressible Flows in R3C . . . . . . . . 351 Humiya Kosaka and Yasunori Maekawa A Weak Solution to the Navier–Stokes System with Navier’s Boundary Condition in a Time-Varying Domain . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 375 Jiˇrí Neustupa and Patrick Penel Effects of Fluid-Boundary Interaction on the Stability of Boundary Layers in Plasma Physics . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 401 Masashi Ohnawa

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On Incompressible Two-Phase Flows with Phase Transitions and Variable Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 411 Jan Prüss, Senjo Shimizu, Gieri Simonett, and Mathias Wilke On the Nash-Moser Iteration Technique . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 443 Paolo Secchi Rate of Convergence to the Stationary Solution of the Navier-Stokes Exterior Problem . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 459 Masao Yamazaki

The Work of Yoshihiro Shibata Herbert Amann, Yoshikazu Giga, Hisashi Okamoto, Hideo Kozono, and Masaso Yamazaki

Abstract Introducing his research carrier, we address Prof. Yoshihiro Shibata’s great contributions to the mathematical analysis. His out-standing influence to the mathematical society is also clarified. Keywords Lp  Lq -estimates • Elastodynamical system • Free boundary value problems • Hyperbolic equations • Local and global well-posedness • Local energy decay • Maximal Lp  Lq regularity • Navier-Stokes equations • Oseen flow • R-boundedness

Yoshihiro Shibata was born in Tokyo on August 28, 1952. After graduating from Azabu Gakuen High School in March 1971, he studied at Tokyo University of Education from April 1971 through March 1977 and at Tsukuba University from April 1977 through March 1978. Afterwards he got a position at the Mathematical Department of Tsukuba University. There he obtained, in October 1981, the degree of a Doctor of Sciences under the supervision of Matsumura Mutsuhide. During that period he studied initial-boundary value problems for linear hyperbolic equations [68, 70, 71] and uniqueness criteria for general partial differential equations with constant coefficients [58, 69, 72, 73]. Then his interest evolved toward initial-boundary value problems for nonlinear hyperbolic equations and systems. In [75] he succeeded to prove global

H. Amann Institut für Mathematik, Universität Zürich, 8057 Zürich, Switzerland e-mail: [email protected] Y. Giga Graduate School of Mathematical Sciences, University of Tokyo, 153-8914 Tokyo, Japan e-mail: [email protected] H. Okamoto Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan e-mail: [email protected] H. Kozono () • M. Yamazaki Department of Mathematics, Waseda University, Tokyo 169-8555, Japan e-mail: [email protected]; [email protected] © Springer Basel 2016 H. Amann et al. (eds.), Recent Developments of Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-0348-0939-9_1

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well-posedness results for nonlinear wave equations with damping terms in Ndimensional exterior domains (N  3). The main step was to establish decay properties of solutions to the linearized equations in exterior domains. To achieve this, he studied the local energy decay near the boundary. In combination with decay estimates for the whole space, this furnishes the required estimates in the exterior domain. To derive the local energy decay, he constructed the solution operator to the corresponding resolvent problem near the origin. Since the latter belongs to the continuous spectrum, he restricted the domain of the solution operator to compactly supported data. In addition, he considered it in local L2 spaces. Thus he constructed the solution operator in a topology which is weaker than the usual one. He is the first mathematician who applied this method, which was originally used in scattering theory by Vainberg [11, 12] and Lax and Phillips [8], for example, to the study of nonlinear evolution equations. Moreover, in [110, 111, 113, 114], together with Yoshio Tsutsumi, he proved global well-posedness results for nonlinear wave equations with Dirichlet as well as Neumann boundary conditions in N-dimensional exterior domains (N  3). The main idea was again to establish decay properties of solutions of wave equations in exterior domains. To treat the high frequency part, they applied a device of B. Vainberg [12], based on the nontrapping condition. For the low frequency part, they applied the technique developed in [75]. In 1985 Yoshihiro Shibata spent 10 months at the University of Paris 6 as a young researcher, supported by the Ministry of Education, Science, and Culture. He participated in the Leray seminar which was then organized by J. Vaillant. During that time, he proved local well-posedness theorems for fully nonlinear wave equations of the form @t u D F.u/; t  0, with Neumann boundary conditions [77]. To attack this problem he introduced a simple linearization procedure: By differentiating the equation with respect to t he obtained the system @t v C A.u/v D 0; F.u/ D v, where A.u/ D DF.u/. The first equation is linear hyperbolic in v, and the second one is a fully nonlinear elliptic equation in u. Then he solved this system by the usual iteration method. Prior to his paper, fully nonlinear hyperbolic equations were treated by employing Nash-Moser iteration. His contribution was to show that this technique can be avoided. Later, Tosio Kato [5] formulated Y. Shibata’s method in an abstract framework. The ideas of [77] were further exploited by Gen Nakamura and Yoshihiro Shibata in [63] to proved the local well-posedness for the elastodynamical system with Neumann type boundary conditions. In the series of papers [42, 78, 79], Y. Shibata provided the detailed proof of the local well-posedness for some fully nonlinear hyperbolic systems. During the period in which he studied nonlinear hyperbolic equations and systems, he was invited to the Bulgarian Academy, in 1987, to work together with Vladimir Georgiev, to Fudan University, in 1988, to collaborate with Song Mu Zheng [109], and, in 1988 and 1989, to the University of Bonn as a member of SFB 256, then chaired by Rolf Leis, to interact with Reinhard Racke. In the late 80s he shifted his interest toward thermoelastic and viscoelastic equations, motivated by the fact that these hyperbolic systems exhibit physically

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reasonable dissipation mechanisms. By combining spectral analysis and multiplicative techniques, he could establish several global well-posedness results for small initial data [40, 41, 57, 65, 66]. In the middle of 1980, Hideo Kozono asked Yoshihiro Shibata to apply his idea, obtained in [75], to the study of Navier-Stokes equations in exterior domains. This question was passed on to Hirokazu Iwashita, who then, under the supervision of Y. Shibata, proved the global well-posedness, for small initial data, of the NavierStokes equations in exterior domains [7]. This result is an extension of one by Kato [4], who considered the full space problem. The key idea consisted in proving Lp Lq decay estimates for the Stokes equations. For this, Iwashita first established the local energy decay by combining the method of [75] with the Bogovski technique to preserve the divergence free condition for the velocity field. Then he combined these findings with Lp -Lq decay estimates for the whole space in order to get the required estimates in exterior domains of RN with N  3. Starting in the middle of 90s, Y. Shibata’s concentrated his interest on the study of the Navier-Stokes equations. The first result he obtained, was the global wellposedness, for small initial data, of the Oseen flow in a three-dimensional exterior domain [45, 91]. Using these results, Galdi, Heywood and Shibata [34] solved a problem, proposed by Finn [2], concerning the Navier-Stokes flow past a moving obstacle that is started from rest. Although this model cannot be handled in the L2 framework, they succeeded to get a solution in L3 . Together with Wakako Dan, Yoshihiro Shibata [20, 24] obtained the optimal decay rate for the non-stationary Stokes equations in 2-dimensional exterior domains. In this case, the fundamental solution of the Stokes resolvent problem has a logarithmic singularity at the origin, whereas it is smooth there if N  3. Thus a new idea, essentially different from the ones used in the case N  3, was needed. They employed a contradiction argument to eliminate the logarithmic singularity by reflection at the boundary. In addition, Y. Shibata studied the Navier-Stokes equations describing the motion of a compressible viscous barotropic fluid in 3-dimensional exterior domains. In collaboration with Takayuki Kobayashi [9, 46], he established the optimal decay rate of the Matsumura-Nishida solutions [57]. For this they used Lp -Lq decay estimates (1 < p  2  q  1) for the linearized equations in exterior domains, thus providing a new approach to the mathematical study of compressible viscous fluids in exterior domains. In 1996, Y. Shibata organized the international workshop “Some applications of real analytic methods to the study of nonlinear partial differential equations in mathematical physics ” at Tsukuba University. It was supported by the Mathematical Society of Japan as a Regional Workshop (RW) and co-organized by Hideo Kozono (Kozono and Shibata [48]). Afterwards, he acted as co-organizer of many further international conferences concerning Navier-Stokes equations. In April 1997, Yoshihiro Shibata moved to Waseda University. From then on, his main interest concerns the study of incompressible and compressible viscous fluid flows. It is one of his achievements in the study of Navier-Stokes equations that he “reintroduced” the pressure term. In fact, starting with the seminal papers

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by Fujita and Kato [3, 6], the Japanese school, including K. Masuda, T. Miyakawa, H. Okamoto, Y. Giga, H. Kozono, M. Yamazaki. . . , contributed a lot to the study of the Navier-Stokes equations by means of the analytic semigroup approach. Thereby, the pressure term is eliminated with the help of the Helmholtz-Leray projection. In contrast, Y. Shibata, and H. Iwashita included the pressure in their studies, using the Bogovski technique to preserve the solenoidal condition for the velocity field. This approach is more in the spirit of PDEs than of functional analysis. It was successfully applied by H. Iwashita [7] to obtain the Lp -Lq decay rate for the Stokes equation in exterior domains, by Kobayashi and Shibata [45] to the Oseen equation in exterior domains, and to the exterior problem for rotating obstacles by T. Hishida and Y. Shibata [37]. Following a suggestion of H. Amann in 2000, Y. Shibata turned to the investigation of free boundary value problems for Navier-Stokes equations by means of maximal regularity theory. This set of problems—in particular the drop and the ocean problem—had already attracted many researchers, predominantly V. A. Solonnikov and his students, T. Beale, T. Nishida, A. Tani, and their Japanese followers, who employed either an L2 framework or Hölder theory. In their paper [133], Yoshihiro Shibata and Senjo Shimizu proved maximal Lp Lq regularity for the Stokes equations with a free boundary condition in a bounded domain. This was based on the R-boundedness of the solution operators in the halfspace model problem, which was then used to obtain the desired result with the help of the Weis operator-valued Fourier multiplier theorem [13]. Y. Shibata got the idea to use R-boundedness during his stay at the University of Konstanz in September 2004, on an invitation of Reinhard Racke, by reading the booklet by Denk, Hieber and Prüss [1]. More recently, he extended this approach to obtain maximal Lp -Lq regularity for the non-stationary Stokes equations with free boundary conditions in very general unbounded domains [103]. Here, he applies the Fourier multiplier theorem of L. Weis to the inverse Laplace transform of the R-bounded solution operator of the generalized Stokes resolvent problem, which acts simultaneously on the right hand side and on the non-homogeneous boundary data. This idea goes back to the work of Reiko Sakamoto [10], who is an academic mother of Yoshihiro Shibata. She used the Plancherel theorem to prove the L2 well-posedness of the initial-boundary value problem for hyperbolic equations satisfying a uniform Shapiro-Lopatinski condition. The method of [103] was further extended to obtain maximal Lp -Lq regularity for the linearized Navier-Stokes equations describing the barotropic motion of compressible viscous fluid flows [32, 33], and also for compressibleincompressible two phase flows [53, 107]. Maximal Lp -Lq regularity with different values of p and q is needed to prove global well-posedness in unbounded domains. In fact, since, unlike in the bounded domain case, only polynomial decay is expected, one has to be able to choose p large to guarantee global integrability in time (cf. Saito and Shibata [67]). At present, he is interested to get global wellposedness results in unbounded domains by combining maximal Lp -Lq regularity with Lp -Lq decay estimates.

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Yoshihiro Shibata has been a steady source of new ideas, and he has influenced many researchers. He supervised more than 10 students who obtained a Doctor Degree (Doctor of Natural Science) either from Tsukuba University or Waseda University. Between 2009 and 2014 Y. Shibata coordinated, together with Matthias Hieber, the Japanese-German Graduate Externship Program for doctor course students between Waseda University and TU Darmstadt. It concerns the theory of mathematical fluid dynamics and is supported by JSPS and DFG. (Since 2014 this program is coordinated by Hideo Kozono and Matthias Hieber.) Also since 2009, he is the team leader of a project in the JST Mathematics Project organized by Yasumasa Nishiura. In 2014 he became the project leader of the Mathematics and Physics Unit of the Top Global University Project of Waseda University, supported by the Ministry of Education, Culture, Sports, Science, and Technology. Together with Hideo Kozono, Tohru Ozawa, Hiroaki Yoshimura, Shinichi Oishi and four professors from the physics department he recently started a new study group and a new education system in the doctor course of the Department of Sciences and Engineering of Waseda University. Its main subject is multi scale analysis, modeling and simulation with interdisciplinary and international background.

References 1. R. Denk, M. Hieber, J. Pruss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 166(788) (2003) 2. R. Finn, Stationary solutions of the Navier-Stokes equations. Am. Math. Soc. Proc. Symp. Appl. Math. 19, 121–153 (1965) 3. H. Fujita, T. Kato, On the Navier-Stokes initial value problem I. Arch. Ration. Mech. Anal. 16, 269–315 (1964) 4. T. Kato, Strong Lp solutions of the Navier-Stokes equation in Rm with applications to weak solutions. Math. Z. 187, 471–480 (1984) 5. T. Kato, Abstract Differential Equations and Nonlinear Mixed Problems. Accademia Nazionale Dei Lincei Scuola Normale Superiore, Lezionei Fermiane, Pisa, 1985 6. T. Kato, H. Fujita, On the non-stationary Navier-Stokes system. Rend. Sem. Math. Univ. Padova 32, 243–260 (1962) 7. H. Iwashita, Lq -Lr estimates for solutions of the non-stationary Stokes equations in an exterior domain and the Navier-Stokes initial value problems in Lq spaces. Math. Ann. 285, 265–288 (1989) 8. P.D. Lax, R.S. Phillips, Scattering theory, in Pure and Applied Mathematics, vol. 26 (Academic, New York/London, 1967) 9. A. Matsumura, T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Commun. Math. Phys. 89(4), 445–464 (1983) 10. R. Sakamoto, Mixed problems for hyperbolic equations I, II. J. Math Kyoto Univ. 10, 349– 373/403–417 (1970) 11. B. Vainberg, On the analytic properties of the resolvent for a certain class of operator pencils, Mat. Sb. (N.S.) 77 (1968); Math. USSR Sb. 6, 241–273 (1968) [English translation]

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12. B. Vainberg, On the short wave asymptotic behaviour of solutions of stationary problems and asymptotic behaviour as t ! 1 of solutions of non-stationary problem. Ushpekhi Mat. Nauk. 30, 3–55 (1975); Russ. Math. Surveys 30, 1–58 (1975) 13. L. Weis, Operator-valued Fourier multiplier theorems and maximal Lp -regularity. Math. Ann. 319, 735–758 (2001)

List of Publications of Yoshihiro Shibata 14. T. Abe, Y. Shibata, On the Stokes and Navier-Stokes flows between parallel planes, in Harmonic Analysis and Nonlinear Partial Differential Equations (Japanese) (1235) (S¯urikaisekikenky¯usho K¯oky¯uroku, Kyoto, 2001), pp. 160–191 15. T. Abe, Y. Shibata, On a generalized resolvent estimate of the Stokes equation on an infinite layer, Part 2  D 0 case. J. Math. Fluid Mech. 5(3), 245–274 (2003) 16. T. Abe, Y. Shibata, On a generalized resolvent estimate of the Stokes equation on an infinite layer, Part 1 jj > 0 case. J. Math. Soc. Jpn. 55(2), 469–497 (2003) 17. T. Akiyama, H. Kasai, Y. Shibata, M. Tsutsumi, On a resolvent estimate of a system of Laplace operators with perfect wall condition. Funkcial. Ekvaj. 47(3), 361–394 (2004) 18. T. Akiyama, Y. Shibata, On an Lp approach to the stationary and non-stationary problems to the Ginzburg-Landau-Maxwell equations. J. Differ. Equ. 243(1), 1–23 (2007) 19. P. D’Ancona, Y. Shibata, On global solvability of nonlinear viscoelastic equations in the analytic category. Math. Meth. Appl. Sci. 17(6), 477–486 (1994) 20. W. Dan, Y. Shibata, On the Lq –Lr estimate of the Stokes semigroup in a two dimensional exterior domain. J. Math. Soc. Jpn. 51(1), 181–207 (1999) 21. W. Dan, T. Kobayashi, Y. Shibata, On the local energy decay approach to some fluid flow in an exterior domain, in Recent Topics on Mathematical Theory of Viscous Incompressible Fluid (Tsukuba, 1996), Lecture Notes Numerical Application Analysis, vol. 16 (Kinokuniya, Tokyo, 1998), pp. 1–51 22. W. Dan, Y Shibata, On a local energy decay of solutions of a dissipative wave equation. Funkcial. Ekvaj. 38(3), 545–568 (1995) 23. W. Dan, Y. Shibata, On the Lp -Lq estimates of the Stokes semigroup in a two-dimensional exterior domain, in Nonlinear Evolution Equations and Their Applications (Japanese) (Kyoto, 1996) (1009) (S¯urikaisekikenky¯usho K¯oky¯uroku, 1997), pp. 79–99 24. W. Dan, Y. Shibata, Remark on the Lq –L1 estimate of the Stokes semigroup in a two dimensional exterior domain. Pacific J. Math. 189(2), 223–239 (1999) 25. R. Denk, R. Racke, Y. Shibata, Lp theory for the linear thermoelastic plate equations in bounded and exterior domains. Adv. Differ. Equ. 14(7–8), 685–715 (2009) 26. R. Denk, R. Racke, Y. Shibata, Local energy decay estimate of solutions to the thermoelastic plate equations in two- and three- dimensional exterior domains. Z. Anal. Anwend. 29(1), 21–62 (2010) 27. Y. Enomoto, Y. Shibata, Local energy decay of solutions to the Oseen equation in the exterior domains. Indiana Univ. Math. J. 53(5), 1291–1330 (2004) 28. Y. Enomoto, Y. Shibata, On the rate of decay of the Oseen semigroup in exterior domains and its application to Navier-Stokes equations. J. Math. Fluid Mech. 7(3), 339–367 (2005) 29. Y. Enomoto, Y. Shibata, On a Stability theorem of the navier-stokes equation in an exterior domain. in Hyperbolic Problems, Theory, Numerics and Applications I (Yokohama Publisher, Yokohama, 2006), pp. 383–389 30. Y. Enomoto, Y. Shibata, On some decay properties of Stokes semigroup of compressible viscous fluid flow in a 2-dimensional exterior domain. J. Differ. Equ. 252(12), 6214–6249 (2012)

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31. Y. Enomoto, Y. Shibata, About compressible viscous fluid flow in a 2-Dimensional exterior domain, in Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations, Operator Theory: Advances and Applications, vol. 221 (Birkhäuser/Springer Basel AG, Basel, 2012), pp. 305–321 32. Y. Enomoto, Y. Shibata, On the R-sectoriality and the initial boundary value problem for the viscous compressible fluid flow. Funkcial. Ekvac. 56(3), 441–505 (2013) 33. Y. Enomoto, L. von Below, Y. Shibata, On some free boundary problem for a compressible barotropic viscous fluid flow. Ann. Univ. Ferrara Sez. VII Sci. Mat., 60(1), 55–89 (2014) 34. G.P. Galdi, J.G. Heywood, Y. Shibata, On the global existence and convergence to steady state of Navier–Stokes flow past an obstacle that is started from rest. Arch. Ration. Mech. Anal. 138(4), 307–318 (1997) 35. D. Gotz, Y. Shibata, On the R-boundedness of the solution operators in the study of the compressible viscous fluid flow with free boundary conditions. Asymptot. Anal. 90(3–4), 207–236 (2014) 36. T. Hishida, Y. Shibata, Globally in time existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle. WSWAS Trans. Math. 5(3), 303–307 (2006) 37. T. Hishida, Y. Shibata, Lp -Lq estimate of the Stokes operator and Navier-Stokes flows in the exterior of a rotating obstacle. Arch. Ration. Mech. Anal. 193(2), 339–421 (2009) 38. M. Hieber, Y. Shibata, The Fujita-Kato approach to the Navier-Stokes equations in the rotational framework. Math. Z. 265(2), 481–491 (2010) 39. H. Iwashita, Y. Shibata, On the analyticity of spectral functions for some exterior boundary value problems. Glasnik Math. Ser. III 23(43, 2), 291–313 (1988) 40. S. Kawashima, Y. Shibata, Global existence and exponential stability of small solutions to nonlinear viscoelasticity. Commun. Math. Phys. 148(1), 189–208 (1992) 41. S. Kawashima, Y. Shibata, On the Neumann problem of one–dimensional nonlinear thermoelasticity with time–independent external force. Czechoslovak Math. J. 45(120, 1), 39–67 (1995) 42. M. Kikuchi, Y. Shibata, On the mixed problem for some quasi–linear hyperbolic system with fully nonlinear boundary condition. J. Differ. Equ. 80(1), 154–197 (1989) 43. T. Kobayashi, H. Pecher, Y. Shibata, On a global in time existence theorem of smooth solutions to nonlinear wave equation with viscosity. Math. Ann. 296(2), 215–234 (1993) 44. T. Kobayashi, Y. Shibata, Exterior problems for the Navier-Stokes equations, in Nonlinear Evolution Equations and their Applications(Japanese) (913) (S¯urikaisekikenky¯usho K¯oky¯uroku, Kyoto, 1994/1995) pp. 185–190 45. T. Kobayashi, Y. Shibata, On the Oseen equation in exterior domains. Math. Ann. 310(1), 1–45 (1998) 46. T. Kobayashi, Y. Shibata, Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in R3 . Commun. Math. Phys. 200(3), 621–659 (1999) 47. T. Kobayashi, Y. Shibata, Remark on the rate of decay of solutions to linearized compressible Navier-Stokes equations. Pacific J. Math. 207(1), 199–234 (2002) 48. H. Kozono, Y. Shibata, Recent topics on mathematical theory of viscous incompressible fluid, in Lecture Notes in Numerical and Applied Analysis, vol. 16 (Kinokuniya, Tokyo, 1998) 49. T. Kubo, Y. Shibata, On some properties of solutions to the Stokes equation in the half-space and perturbed half-space, in Dispersive Nonlinear Problems in Mathematical Physics, Quad. Mat., Dept. Math., vol. 15 (Seconda Univ. Napoli, Caserta, 2004), pp. 149–220 50. T. Kubo, Y. Shibata, On the Stokes and Navier-Stokes equation in a perturbed half-space. Adv. Differ. Equ. 10(6), 695–720 (2005) 51. T. Kubo, Y. Shibata, On the Stokes and Navier-Stokes flows in a perturbed half space, in Regularity and Other Aspects of the Navier-Stokes Equations. Banach Center Publications vol. 70 (Polish Acad. Sci., Warsaw, 2005), pp. 157–167 52. T. Kubo, Y. Shibata, Lp -Lq estimate of the stokes semigroup and its application to navierstokes equation in a perturbed half-space, in Hyperbolic Problems, Theory, Numerics and Applications II (Yokohama Publisher, Yokohama, 2006), pp. 125–132

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53. T. Kubo, Y. Shibata, K. Soga, On the R-boundedness for the two phase problem: compressible-incompressible model problem. Bound. Value Probl. 141, 33 pp. (2014) 54. J. Prüss, Y. Shibata, S. Shimizu, G. Simonett, On well-posedness of incompressible two-phase flows with phase transitions: the case of equal densities. Evol. Equ. Control Theory 1(1), 171– 194 (2012) 55. A. Milani, Y. Shibata, On compatible regularizing data for second order hyperbolic initial– boundary value problems. Osaka J. Math 32(2), 347–362 (1995) 56. A. Milani, Y. Shibata, On the strong well–posedness of quasilinear hyperbolic initial– boundary value problems. Funkcial. Ekvaj. 38(3), 491–503 (1995) 57. J. Muñoz Rivera, Y. Shibata, A linear thermoelastic plate equation with Dirichlet boundary condition. Math. Methods Appl. Sci. 20(11), 915–932 (1997) 58. M. Murata, Y. Shibata, Lower bounds at infinity of solutions of partial differential equations in the exterior of a proper cone. Israel J. Math. 31(2), 193–203 (1978) 59. Y. Naito, Y. Shibata, On the Lp analytic semigroup associated with the linear thermoelastic plate equations in the half-space. J. Math. Soc. Jpn. 61(4), 971–1011 (2009) 60. Y. Naito, R. Racke, Y. Shibata, Low frequency expansion in thermoelasticity with second sound in three dimensions. J. Math. Soc. Jpn. 62(4), 1289–1316 (2010) 61. G. Nakamura, Y. Shibata, On a local existence theorem for quasi–linear hyperbolic mixed problems with Neumann type boundary conditions, Proc. Japan Acad. Ser. A Math. Sci. 62(4), 117–120 (1986) 62. G. Nakamura, Y. Shibata, K. Tanuma, Whispering gallery waves in a neighborhood of a higher order zero of the curvature of the boundary. Publ. RIMS Kyoto Univ. 25(4), 605–629 (1989) 63. G. Nakamura, Y. Shibata, On a local existence theorem of Neumann problem for some quasi– linear hyperbolic systems of 2nd order. Math. Z. 202(1), 1–64 (1989) 64. M. Okamura, Y. Shibata, N. Yamaguchi, A Stokes approximation of two dimensional exterior Oseen flow near the boundary, in Asymptotic Analysis and Singularities–Hyperbolic and Dispersive PDEs and Fluid Mechanics. Advanced Studies in Pure Mathematics, vol. 47 (Mathematical Society, Tokyo, 2007), pp. 273–289 65. R. Racke, Y. Shibata, Global smooth solution and asymptotic stability in one-dimensional nonlinear thermoelasticity. Arch. Ration. Mech. Anal. 116(1), 1–34 (1991) 66. R. Racke, Y. Shibata, S. Mu Zheng, Global solvability and exponential stability in one– dimensional nonlinear thermoelasticity. Quart. Appl. Math. 51(4), 751–763 (1993) 67. H. Saito, Y. Shibata, On the Stokes equations with surface tension and gravity in RNC . J. Math. Soc. Jpn. (to appear) 68. Y. Shibata, A characterization of the hyperbolic mixed problems in a quarter space for differential operators with constant coefficients. Publ. RIMS Kyoto Univ. 15, 357–399 (1979) 69. Y. Shibata, Liouville type theorem for a system fP.D/; Bj .D/; j D 1; : : : ; pg of differential operators with constant coefficients in a half–space. Publ. RIMS Kyoto Univ. 16, 61–104 (1980) 70. Y. Shibata, E –well posedness of mixed initial–boundary value problems with constant coefficients in a quarter space. J. D’Analyse Math. 37, 32–45 (1980) 71. Y. Shibata, E –well posedness of mixed initial–boundary value problem with constant coefficients in a quarter– space II. Proc. Jpn. Acad. Ser. A. 56(7), 318–320 (1980) 72. Y. Shibata, Lower bounds at infinity of solutions of differential equations with constant coefficients in unbounded domains, in Singularities in Boundary Value Problems, ed. by H.G. Garnir. Proceedings of NATO Advanced Institute, Maratea, 1980. NATO Advanced Study Institute Series. Series C: Mathematical and Physical Sciences, vol. 65 (Reidel, Dordrecht, 1981), pp. 213–234 73. Y. Shibata, Lower bounds of solutions of general boundary value problems for differential operators with constant coefficients in a half–space, Japan. J. Math. (N.S.) 8(2), 343–382 (1982)

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74. Y. Shibata, On the global existence of classical solutions of mixed problem for some second order non–linear hyperbolic operators with dissipative term in the interior domain. Funkcial. Ekvac. 25(3), 303–345 (1982) 75. Y. Shibata, On the global existence of classical solutions of second order fully nonlinear hyperbolic equations with first order dissipation in the exterior domain. Tsukuba J. Math. 7(1), 1–68 (1983) 76. Y. Shibata, On a local existence theorem for quasilinear hyperbolic mixed problems with Neumann type boundary conditions, in Hyperbolic Equations (Padua, 1985) Pitman Research Notes in Mathematics Series, vol. 158 (Longman Scientific & Technical, Harlow, 1987), pp. 282–286 77. Y. Shibata, On a local existence theorem of Neumann problem for some quasi–linear hyperbolic equations, in Calcul d’operateurs et fronts d’ondes, ed. by J. Vaillant, Travaux en Cours, vol. 29 (Hermann, Paris, 1988), pp. 133–167 78. Y. Shibata, On the Neumann problem for some linear hyperbolic systems of second order. Tsukuba J. Math. 12(1), 149–209 (1988) 79. Y. Shibata, On the Neumann problem for some linear hyperbolic systems of 2nd order with coefficients in Sobolev spaces. Tsukuba J. Math. 13(2), 283–352 (1989) 80. Y. Shibata, On one-dimensional nonlinear thermoelasticity, in Nonlinear Hyperbolic Equations and Field Theory (Lake Como, 1990), Pitmann Res. Notes Math. Ser., vol. 253 (Longman Scientific & Technical, Harlow, 1992), pp. 178–184 81. Y. Shibata, Neumann problem for one-dimensional nonlinear thermoelasticity, in Partial Differential Equations, Warsaw, 1990, Parts 1, 2, vol. 27 (Banach Center Publications/Polish Academy of Sciences, Warsaw, 1992), pp. 457–480 82. Y. Shibata, Global in time solvability of the initial boundary value problem for some nonlinear dissipative evolution equations. Comment. Math. Univ. Carol. 34(2), 295–312 (1993) 83. Y. Shibata, Neumann problem of one-dimensional nonlinear thermoelastic equations, in Mathematical Analysis of Phenomena in Fluid and Plasma Dynamics Kyoto, 1992. Surikaisekikenkyusho Kokyuroku (Japanese), vol. 824 (1993), pp. 283–296 84. Y. Shibata, On the exponential decay of the energy of a linear thermoelastic plate. Math. Appl. Comput. 13(2), 81–102 (1994) 85. Y. Shibata, Global in time existence of small solutions of nonlinear thermoviscoelastic equations. Math. Methods Appl. Sci. 18(11), 871–895 (1995) 86. Y. Shibata, On a linear thermoelastic plate equation, in Nonlinear Evolution Equations and Their Applications (898) (Japanese) (S¯urikaisekikenky¯usho K¯oky¯uroku, Kyoto ,1993/1995), pp. 149–154 87. Y. Shibata, An initial-boundary value problem for some hyperbolic-parabolic coupled system, in Nonlinear Waves (Sapporo. 1995), GAKUTO International Series Mathematical Sciences Application, vol. 10 (Gakk¯otosho, Tokyo, 1997), pp. 447–450 88. Y. Shibata, An exterior initial-boundary value problem for the Navier-Stokes equation, in Nonlinear waves (Sapporo, 1995), GAKUTO International Series Mathematical Sciences Application, vol. 10 (Gakk¯otosho, Tokyo, 1997), pp. 431–446 89. Y. Shibata, On the decay estimate of the Stokes semigroup in a two dimensional exterior domain. Navier-Stokes Equations and Related Nonlinear Problems(Palanga, 1997) (VSP, Utrecht, 1998), pp. 315–330 90. Y. Shibata, On a Decay Rate of Solutions to One-Dimensional Thermoelastic Equations on a Half Line; Linear Part, ed. by S. Kawashima, T. Yanagisawa. Advances in Nonlinear Partial Differential Equations and Stochastics (World Scientific, Singapore, 1998), pp. 198–291 91. Y. Shibata, On an exterior initial-boundary value problem for Navier-Stokes equations. Quart. Appl. Math. 57(1), 117–155 (1999) 92. Y. Shibata, Global solutions of nonlinear evolution equations and their stability. S¯ugaku, 51(1), 1–17 (1999) (in Japanese). 93. Y. Shibata, On the rate of decay of solutions to linear viscoelastic equation. Math. Methods Appl. Sci. 23(3), 203–226 (2000)

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94. Y. Shibata, On a stability theorem of the Navier-Stokes equation in a three dimensional exterior domain, in Tosio Kato’s Method and Principle for Evolution Equations in Mathematical Physics, Sapporo, 2001 (1234) (S¯urikaisekikenky¯usho K¯oky¯uroku, 2001) pp. 146–172 95. Y. Shibata, On some stability theorems about viscous fluid flow, Quaderni del Seminario Matematico di Brescia (2003) 96. Y. Shibata, Time-global solutions of nonlinear evolution equations and their stability [translation of S¯ugaku 51(1), 1–17 (1999)], in Selected Papers on Analysis and Differential Equations. American Mathematical Society Translations: Series 2, vol. 211 (American Mathematical Society, Providence, RI, 2003), pp. 87–105 97. Y. Shibata, On some stability theorem of the steady flow of compressible viscous fluid with respect to the initial disturbance, in Hyperbolic Problems and Related Topics, Graduate Series Analysis (International Press, Somerville, MA, 2003), pp. 341–357 98. Y. Shibata, On the Oseen semigroup with rotating effect. in Functional Analysis and Evolution Equations (Birkhäuser, Basel, 2008), pp. 595–611 99. Y. Shibata, A stability theorem of the Navier-Stokes flow past a rotating body. in Parabolic and Navier-Stokes Equations. Part 2, vol. 81 (Banach Center Publication, Polish Academy of Science Institute of Mathematics, Warsaw, 2008), pp. 441–455 100. Y. Shibata, On a C0 semigroup associated with a modified Oseen equation with rotating effect, in Advances in Mathematical Fluid Mechanics, (Springer, Berlin, 2010), pp. 513–551 101. Y. Shibata, Generalized resolvent estimates of the Stokes equations with first order boundary condition in a general domain. J. Math. Fluid Mech. 15(1), 1–40 (2013) 102. Y. Shibata, On the R-boundedness of solution operators for the weak Dirichlet-Neumann problem, in RIMS K¯oky¯uroku 1875, Mathematical Analysis of Incompressible Flow (4–6 Feb 2013), ed. by T. Hishida (RIMS, Kyoto University, Kyoto), pp. 1–18 103. Y. Shibata, On the R-boundedness of solution operators for the Stokes equations with free boundary condition. Differ. Integr. Equ. 27(3–4), 313–368 (2014) 104. Y. Shibata, On some free boundary problem of the Navier-Stokes equations in the maximal Lp -Lq regularity class. J. Differ. Equ. 258, 4127–4155 (2015) 105. Y. Shibata, On the global well-posedness of some free boundary problem for a compressible barotropic viscous fluid flow, in The Contemporary Mathematics Series of the American Mathematical Society: Recent Advances in PDEs and Applications, Levico Terme, 17–21 February 2014 [to celebrate the 70th Birthday of Professor Hugo Beirao da Veiga] 106. Y. Shibata, Local well-posedness of compressible-incompressible two-phase flows with phase transitions, To appear in the Proceedings of Levico Conf. on fluid Dyn and Electromagnetism. arXiv:submit/1156284 [math. AP] 10 Jan 2015 107. Y. Shibata, On the R-boundedness for the two phase problem with phase transition: compressible-incompressible model problem. Funk. Ekvaj. (to appear) 108. Y. Shibata, W. Dan, On a local energy decay of solutions of a dissipative wave equation, in Mathematical Analysis of Phenomena in Fluid and Plasma Dynamics (Japanese) (862) (S¯urikaisekikenky¯usho K¯oky¯uroku, Kyoto, 1993/1994) pp. 181–190 109. Y. Shibata, S. Mu Zheng, On some nonlinear hyperbolic system with damping boundary condition. Nonlinear Anal. TMA 17(3), 233–266 (1991) 110. Y. Shibata, Y. Tsutsumi, Global existence theorem for nonlinear wave equations in exterior domain, in Recent Topics in Nonlinear PDE (Hiroshima, 1983), North-Holland Mathematics Studies vol. 98 (North-Holland, Amsterdam, 1984) pp. 155–196 111. Y. Shibata, Y. Tsutsumi, Global existence theorem for nonlinear wave equation in exterior domain. Proc. Japan Acad. A Mat. Sci. 60(1), 14–17 (1984) 112. Y. Shibata, Y. Tsutsumi, Local existence of C1 –solution for the initial–boundary value problem of fully nonlinear wave equation. Proc. Japan Acad. Ser. A Math. Sci. 60(5), 149–152 (1984) 113. Y. Shibata, Y. Tsutsumi, On a global existence theorem of Neumann problem for some quasilinear hyperbolic equations, in Recent Topics in Nonlinear PDE, II (Sendai, 1984), North-Holland Mathematics Studies, vol. 128 (North-Holland, Amsterdam, 1985), pp. 175– 228

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114. Y. Shibata, Y. Tsutsumi, On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain. Math. Z. 191(2), 165–199 (1986) 115. Y. Shibata, Y. Tsutsumi, Local existence of solutions for the initial boundary value problem of fully nonlinear wave equation. Nonlinear Anal. TMA 11(3), 335–365 (1987) 116. Y. Shibata, S.Shimizu, A decay property of the Fourier transform and its application to the Stokes problem. J. Math. Fluid Mech. 3(3), 213–230 (2001) 117. Y. Shibata, S. Shimizu, On a resolvent estimate of the interface problem for the Stokes system in a bounded domain, Harmonic Analysis and Nonlinear Partial Differential Equations (Japanese) (1235) (S¯urikaisekikenky¯usho K¯oky¯uroku, Kyoto, 2001), pp. 132–159 118. Y. Shibata, S. Shimizu, On the Lp and Schauder estimates of solutions to elastostatic interface problems, in Proceedings of the Fourth International Conference on Functional Analysis and Approximation Theory, Potenza, 2000, vol. II; Rend. Circ. Mat. Palermo (2) Suppl. 68(Part II), 821–835 (2002) 119. Y. Shibata, S. Shimizu, On a resolvent estimate of the interface problem for the Stokes system in a bounded domain. J. Differ. Equ. 191(2), 408–444 (2003) 120. P. Secchi, Y. Shibata, On the decay of solutions to the 2D Neumann exterior problem for the wave equation. J. Differ. Equ. 194(1), 221–236 (2003) 121. Y. Shibata, S. Shimizu, On a resolvent estimate for the Stokes system with Neumann boundary condition. Differ. Integr. Equ. 16(4), 385–426 (2003) 122. Y. Shibata, S. Shimizu, Applications of the Fourier transform to some resolvent estimates for the Stokes system, in Progress in Analysis, vols. I, II (Berlin, 2001) (World Science Publisher, River Edge, 2003), pp. 125–134 123. Y. Shibata, S. Shimizu, Lp –Lq maximal regularity and viscous incompressible flows with free surface. Proc. Japan Acad. Ser A, Math. Sci. 81(9), 151–155 (2005) 124. Y. Shibata, S. Shimizu, On the stokes equation with neumann boundary condition, in Regularity and other Aspects of the Navier-Stokes Equations. Banach Center Publication, vol. 70 (Polish Academy of Sciences, Warsaw, 2005), pp. 239–250 125. Y. Shibata, S. Shimizu, On a Free Boundary Problem for the Navier-Stokes Equations. Differ. Integr. Equ. 20(3), 241–276 (2007) 126. Y Shibata, S. Shimizu, Lp -Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain, in Asymptotic Analysis and Singularities–Hyperbolic and Dispersive PDEs and Fluid Mechanics. Advanced Studies in Pure Mathematics, vol. 47 (Mathematical Society of Japan, Tokyo, 2007), pp. 349–362. 127. Y. Shibata, S. Shimizu, Decay properties of the Stokes semigroup in exterior domains with Neumann boundary condition. J. Math. Soc. Jpn. 59(1), 1–34 (2007) 128. Y. Shibata, S. Shimizu, Report on a local in time solvability of free surface problems for the Navier-Stokes equations with surface tension. Appl. Anal. 90(1), 201–214 (2011) 129. Y. Shibata, S. Shimizu, Lp -Lq regularity for the two-phase Stokes equations; model problems. J. Differ. Equ. 251(2), 373–419 (2011) 130. Y. Shibata, S. Shimizu, On the maximal Lp -Lq regularity of the Stokes problem with first order boundary condition; model problems, J. Math. Soc. Jpn. 64(2), 561–626 (2012) 131. Y. Shibata, H. Soga, Scattering theory for the elastic wave equation. Publ. Res. Inst. Math. Sci. 25(6), 861–887 (1989) 132. Y. Shibata, K. Tanaka, On a resolvent problem for the linearized system from the dynamical system describing the compressible viscous fluid motion. Math. Methods Appl. Sci. 27(13), 1579–1606 (2004) 133. Y. Shibata, S. Shimizu, On the Lp -Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain. J. Reine Angew. Math. 615, 157–209 (2008) 134. Y. Shibata, M. Yamazaki, On some stability theorem of the Navier-Stokes equation in the three dimensional exterior domain, in Mathematical Analysis of Liquids and Gases (Japanese) (Kyoto, 1999) (1146) (S¯urikaisekikenky¯usho K¯oky¯uroku, 2000), pp. 73–91 135. Y. Shibata, N. Yamaguchi, Global existence of strong solutions to the micropolar fluid system, in Hyperbolic Problems, Theory, Numerics and Applications II (Yokohama Publisher, Yokohama, 2006), pp. 305–312

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136. K. Tanaka, Y. Shibata, On the steady flow of compressible viscous fluid and its stability with respect to initial disturbance. J. Math. Soc. Jpn. 55(3), 797–826 (2003) 137. Y. Shibata, R. Shimada, On a generalized resolvent estimate for the Stokes system with Robin boundary condition. J. Math. Soc. Jpn. 59(2), 469–519 (2007) 138. Y. Shibata, R. Shimada, On the Stokes equation with Robin boundary condition, in Asymptotic Analysis and Singularities–Hyperbolic and Dispersive PDEs and Fluid Mechanics. Advanced Studies in Pure Mathematics, vol. 47 (Mathematical Society of Japan, Tokyo, 2007) pp. 341– 348 139. M. Yamazaki, Y. Shibata, Uniform estimates in the velocity at infinity for stationary solutions to the Navier-Stokes exterior problem. Japanese J. Math. (N.S.) 31(2), 225–279 (2005) 140. T. Hishida, Y. Shibata, Decay estimates of the Stokes flow around a rotating obstacle, in Kyoto Conference on the Navier-Stokes Equations and Their Applications. RIMS Koky¯uroku Bessatsu, vol. B1 pp. 167–186 (Res. Inst. Math. Sci. (RIMS), Kyoto, 2007) 141. Y. Shibata, S. Shimizu, Free boundary problems for a viscous incompressible fluid, in Kyoto Conference on the Navier-Stokes Equations and Their Applications. RIMS Koky¯uroku Bessatsu, vol. B1 (Res. Inst. Math. Sci. (RIMS), Kyoto, 2007), pp. 343–358 142. Y. Shibata, S. Shimizu, On a resolvent estimate of the Stokes system in a half space arising from a free boundary problem for the Navier-Stokes equations. Mathematische Nachrichten 282(3), 482–499 (2009) 143. Y. Shibata, K. Tanaka, Rate of convergence of non-stationary flow to the steady flow of compressible viscous fluid. Comput. Math. Appl. 53(3–4), 605–623 (2007)

Existence of Weak Solutions for a Diffuse Interface Model of Power-Law Type Two-Phase Flows Helmut Abels, Lars Diening, and Yutaka Terasawa

Dedicated to Professor Yoshihiro Shibata’s 60th birthday

Abstract We first review results about existence of generalized or weak solutions for Newtonian and power-law type two-phase flows. Then we state a recent result by the authors about existence of weak solutions for diffuse interface model of powerlaw type two-phase flows and give a sketch of its proof. The latter part is a summary of Abels et al. (Nonlinear Anal Real World Appl 15:149–157, 2014). Keywords Cahn-Hilliard equation • Diffuse interface model • Free boundary value problems • Lipschitz truncation • Power-law type fluids • Sharp interface model • Two-phase flow

1 Introduction This is a review paper focusing on existence of generalized or weak solutions for Newtonian and power-law type two-phase flows. We also state a recent result by the authors in [5] and give a sketch of its proof. We consider the flow of two macroscopically immiscible, incompressible Newtonian and power-law type fluids. There are models of two types concerning two-phase flows of Newtonian and power-law type fluids. One type are sharp interface models and the other are diffuse interface models. For sharp interface models, a partial mixing of the fluids is not taken into account. One classical sharp interface model is “Two-phase flows with surface tension (without phase transition)”. For two-phase

H. Abels Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany e-mail: [email protected] L. Diening Mathematisches Institut, LMU München, 80333 München, Germany e-mail: [email protected] Y. Terasawa () Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan e-mail: [email protected] © Springer Basel 2016 H. Amann et al. (eds.), Recent Developments of Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-0348-0939-9_2

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flows of Newtonian fluid with surface tension, there are several studies concerning the existence of classical solution of the equations, see, e.g. [14–16, 32]. They treat local in time existence of solutions for arbitrary initial data. Especially in [32], space-time analyticity of solutions and interfaces are shown, together with a complete maximal regularity result of its linearized problem. For global in time behavior of the solutions, see [25]. For two-phase flows of power-law type fluid with surface tension, the global in time existence of a generalized solution, or a measurevalued solution for arbitrary initial data was proven by Abels [1] and Plotnikov [31]. Concerning results about “Two-phase flow of Newtonian fluids with phase transition”, see, e.g. [33, 34]. In diffuse interface models, a partial mixing of the fluids is taken into account in contrast to sharp interface models. This has the advantage that flows beyond the occurrence of topological singularities e.g. due to droplet collision or pinch-off can be described. One well-known model in the case that both densities are the same is the so-called “model H”. It leads to the following system of Navier-Stokes/CahnHilliard type: @t v C v  rv  div S.c; Dv/ C rp D  div.rc ˝ rc/; div v D 0; @t c C v  rc D m;  D  1 .c/  c

(1) (2) (3) (4)

in QT D   .0; T/, where   Rd , d  2, is a bounded domain and T 2 .0; 1/. Here v is the mean velocity, Dv D 12 .rv C rvT /, p is the pressure, c is an order parameter related to the concentration of the fluids e.g. the concentration difference or the concentration of one component, and  is the density of the fluids, which is assumed to be constant. Moreover, S.c; Dv/ is the viscous part of the stress tensor of the mixture to be specified below,  > 0 is a (small) parameter, which is related to the “thickness” of the interfacial region, ˆW R ! R is a homogeneous free energy density and  D ˆ0 and  is the chemical potential. Capillary forces due to surface tension are modeled by an extra contribution rc ˝ rc WD rc.rc/T in the stress tensor leading to the term on the right-hand side of (1). Moreover, we note that in the modeling, diffusion of the fluid components is taken into account. Therefore m is appearing in (3), where m > 0 is a constant mobility coefficient. We close the system by imposing the following boundary and initial conditions vj@ D 0 n  rcj@ D n  rj@ D 0 .v; c/jtD0 D .v0 ; c0 /

on @  .0; T/;

(5)

on @  .0; 1/;

(6)

in :

(7)

Here n denotes the exterior normal at @. The boundary conditions are the most common in the mathematics literature.

Existence of Weak Solutions for a Diffuse Interface Model of Power-Law Type. . .

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In the case of Newtonian fluids, i.e., S.c; Dv/ D .c/Dv for some positive viscosity coefficient .c/, the model was first discussed by Hohenberg and Halperin [22]. Later it was derived in the framework of rational continuum mechanics by Gurtin et al. [21]. The latter derivation can be easily modified to include a suitable non-Newtonian behavior of the fluids. If e.g. S.c; Dv/ is chosen such that S.c; Dv/ W Dv  0, the local dissipation inequality, which yields thermodynamical consistency, remains valid. For results on existence of weak and strong solutions in the case of Newtonian fluids we refer to Starovoitov [36], Boyer [7], and Abels [2]. First analytic results for the system (1)–(4) for Non-Newtonian fluids of powerlaw type were obtained by Kim et al. [23]. The authors proved existence of weak solutions if q  3dC2 , d D 2; 3, where q is the power describing the growth of the dC2 stress tensor with respect to Dv. For this range of q monotone operator techniques can be applied. Moreover, in the case d D 3 and 2  q < 11 the authors prove 5 existence of measure-valued solutions. Grasselli and Pražák [20] discussed the longtime behavior of solutions of (1)–(4) in the case q  3dC2 , d D 2; 3 assuming dC2 periodic boundary conditions and a regular free energy density. For the same range of q results on existence of weak solutions with a singular free energy density ˆ and the longtime behavior were obtained by Bosia [6] in the case of a bounded domain in R3 . Let us review results on the Cahn–Hilliard equation. The Cahn-Hilliard equation is the following equation: @t c D m;

(8)

 D  1 .c/  c

(9)

in QT with the boundary condition n  rcj@ D n  rj@ D 0

on @  .0; 1/;

where m;  > 0 and  D ˆ0 . For ˆ.s/ D .s2  1/2 , Elliott and Zheng [19] proved the existence of global solutions to (8)–(9) in an L2 -setting. Nicolaenko et al. [30] proved the existence of an attractor of (8)–(9) in an L2 - setting with the same ˆ. Results on convergence of solutions to steady states using the Łojasiewicz-Simon inequality with the same ˆ can be found in Rybka and Hoffmann [35]. For the singular potential satisfying Assumption 2.1 stated in Sect. 2, Elliot and Luckhaus [18] first proved the existence and uniqueness of solutions in the case of multi-component mixture. Debussche and Dettori [13] gave another proof in the case of two-component mixture and proved the existence of an attractor and estimated its dimension. Abels and Wilke [4] proved the existence and uniqueness of a solution by solving an abstract Cauchy problem for a suitable Lipschitz perturbation of a suitable monotone operator. Concerning the relation between sharp interface models and diffuse interface models, see, e.g. [3, 12].

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Next we review existence results of weak solutions for single power-law type fluid equations. Ladyzhenskaya [26] is the first to have investigated the NonNewtonian fluid equations mathematically. She proved the existence of weak solutions for power-law type fluid equations for p  11 5 when d D 3: She also proved the uniqueness of weak solutions for p  52 when d D 3: Lions [27] treated the p-Laplacian case and proved the existence of weak solutions when p > 3dC2 dC2 and the uniqueness of weak solutions when p  dC2 where d  2: Later Malék 2 3d et al. [29] proved the existence of weak solutions when p  dC2 ; where d  2; in periodic case. Wolf [37] proved the existence of weak solutions when p > 2dC2 dC2 in an arbitrary domain using L1 truncation method and a careful decomposition of the pressure which is needed since the L1 truncation used does not preserve the divergence freeness of a velocity field. Then Diening et al. [17] proved existence 2d of weak solutions when q > dC2 , d  2 using parabolic Lipschitz truncation method and a decomposition of the pressure as similar to [37]. Recently a solenoidal parabolic Lipschitz truncation method, which keeps divergence free velocity fields divergence free, was developed by Breit et al. [9] and using that, a shorter proof of the existence result of weak solutions in [17] was given. For Serrin type uniqueness theorem and the existence of the global attractor with autonomous external force in the 3D case, see [10]. Recently, Buliˇcek et al. [11] proved existence of weak solutions for Implicitly constituted Incompressible fluids, which relate symmetric gradient of fluids to stress tensor of fluids in an implicit way. For a survey of results before 2006 concerning power-law fluids, see [28]. A recent contribution by the authors [5] employed a solenoidal parabolic Lipschitz truncation method developed in [9] to prove existence of weak solutions 2d to (1)–(7) if S.c; Dv/ is of power law type with an exponent q > dC2 . In Sect. 2, we review that existence result concerning weak solutions of (1)–(7). In Sect. 3, we give a sketch of its proof. For more details on its proof, we refer to [5].

2 Existence of Weak Solutions for the Diffuse Interface Model We use standard notations. The usual Lebesgue spaces with respect to the Lebesgue measure are denoted by Lp .M/, 1  p  1, for some measurable M  RN . Moreover, Lp .MI X/ denotes its Banach space-valued variant and Lp .0; TI X/ D m Lp ..0; T/I X/. The standard Lp -Sobolev space is denoted by Wpm ./. Wp;0 ./ is the 1 m m m m m closure of C0 ./ in Wp ./ and H ./ D W2 ./; H0 ./ D W2;0 ./. Finally L2 ./ is the closure of divergence free C01 ./-vector fields in L2 ./d . For simplicity we assume that  D  D 1 in (1)–(7) , but all results are true for general (fixed) ;  > 0. Moreover, we assume:

Existence of Weak Solutions for a Diffuse Interface Model of Power-Law Type. . .

17

Assumption 2.1 Let   Rd , d D 2; 3, be a bounded domain with C3 -boundary and let ˆ 2 C.Œa; b/ \ C2 ..a; b// be such that  D ˆ0 satisfies lim .s/ D 1;

s!a

 0 .s/  ˛

lim .s/ D 1;

s!b

for some ˛  0. Let m > 0 and let SW Œa; b  Rdd ! Rdd be such that jS.c; M/j  C.j sym.M/jq1 C 1/

(10)

jS.c1 ; M/  S.c2 ; M/j  Cjc1  c2 j.j sym.M/jq1 C 1/ S.c; M/ W M  !j sym.M/j  C1 q

(11) (12)

2d for all M 2 Rdd , c; c1 ; c2 2 Œa; b, and some C; C1 ; ! > 0, q 2 . dC2 ; 1/. Here 1 T dd dd sym.M/ D 2 .M C M /. Moreover, we assume that S.c; /W Rsym ! Rsym is strictly monotone for every c 2 Œa; b. 1 Let v 2 Lq .0; TI Wq;0 ./d / \ L1 .0; TI L2 .//, c 2 L1 .0; TI H 1 .// \ L2 .0; TI H 2 .// with ˆ.c/ 2 L2 .  .0; T//, and  2 L2 .0; TI H 1 .//, where 0 < T < 1. Then .v; c; / is a weak solution of the system (1)–(7) if for any ' 2 C1 .QT /d with div ' D 0 and supp.'/    Œ0; T/ the following equations hold true: Z Z Z  v  @t ' d.x; t/  v ˝ v W D' d.x; t/ C S.c; Dv/ W D' d.x; t/ QT

QT

QT

Z

Z

rc ˝ rc W D' d.x; t/ C

D



QT

and for every

Z c@t QT

(13)

2 C1 .QT / with supp. /    Œ0; T/

Z 

v0  '.0/ dx

Z

d.x; t/  

c0 .0/ dx C

.v  rc/ QT

d.x; t/

Z r  r

D m

d.x; t/;

(14)

QT

 D .c/  c n  rcj@ D 0

in   .0; T/;

(15)

on @  .0; T/:

(16)

Theorem 2.2 Let Assumption 2.1 hold true and let 0 < T < 1. Then for any v0 2 L2 ./ and c0 2 H 1 ./ with c0 .x/ 2 Œa; b almost everywhere there exists a 1 weak solution v 2 Lq .0; TI Wq;0 .// \ L1 .0; TI L2 .//, c 2 L1 .0; TI H 1 .// \ 2 2 2 L .0; TI H .// with ˆ.c/ 2 L .  .0; T//, and  2 L2 .0; TI H 1 .// in the sense above.

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The following theorem, which is a summary of Theorem 2.16 and Corollary 2.17 of [9], is important for the proof of our theorem. Theorem 2.3 Let I0 be an open time interval, let B0 be a ball in Rd , and let Q0 WD q I0  B0 . Let q; 2 .1; 1/ with q; q0 > > 1, where q0 D q1 . Let 2 C01 . 61 Q0 / with 1 Q0   1 Q0 . Let um and Gm satisfy @t um D  div Gm in the sense of 8 6 0 distributions Ddiv .Q0 /, where Ddiv D f' 2 C01 .Q0 /d W div ' D 0g. Assume that um is a weak null sequence in Lq .I0 I Wq1 .B0 //, a strong null sequence in L .Q0 / and bounded in L1 .I0 ; L .B0 //. Further assume that Gm D G1;m C G2;m such 0 that G1;m is a weak null sequence in Lq .Q0 / and G2;m converges strongly to zero

in L .Q0 /. Then there exists a double sequence of open sets Om;k , k; m 2 N, with k 0 lim supm!1 jOm;k j  C 2k 2q 2 for all k 2 N such that for every K 2 Lq . 16 Q0 / ˇZ ˇ ˇ  ˇ  k=q c d.x; t/ˇ  C 2 G1;m C K/ W rum Om;k : lim sup ˇˇ ˇ m!1

The proof of Theorem 2.3 is based on a solenoidal parabolic Lipschitz truncation. In particular, the sets Om;k are level sets of suitable maximal operators defined by um and Gm . It is a modification of the parabolic Lipschitz truncation of [17, 24]. The advantage of Theorem 2.3 is that the pressure can be completely avoided by using a solenoidal Lipschitz truncation. See [8] for a solenoidal Lipschitz truncation in the stationary case.

3 Sketch of Proof of Theorem 2.2 In order to approximate (1)–(7) we consider @t v C div." .v/v ˝ v/  div S.c; Dv/ C rp D ‰" .div.rc ˝ rc// in   .0; T/; div v D 0; @t c C .‰" v/  rc D m;  D .c/  c:

(17)

in   .0; T/;

(18)

in   .0; T/;

(19)

in   .0; T/

(20)

together with (5)–(7), where ‰" w D P . " w/j , " .x/ D "d .x="/, " > 0, is a usual smoothing kernel such that .x/ D .x/ for all x 2 Rd , w is extended by 0 outside of , and P is the Helmholtz projection. Moreover, " .s/ D ."jsj2 / for all s 2 Rd , " > 0 with some  2 C01 .R/ with .0/ D 1.

Existence of Weak Solutions for a Diffuse Interface Model of Power-Law Type. . .

19

We can construct weak solutions of the approximate system (17)–(20) together with (5)–(7) in an appropriate function space with the aid of the Leray-Schauder theorem. The solutions satisfy the energy identity Z tZ 1 kv.t/k2L2 ./ C Emix .c.t// C S.c; Dv/ W Dv dx d 2 0  Z tZ 1 C mjrj2 dx d D kv0 k2L2 ./ C Emix .c0 / 2 0 

(21)

for almost every t 2 .0; T/, where Z Emix .c/ D



jrcj2 dx C 2

Z 

ˆ.c/ dx:

We use those solutions to construct a weak solution of the original system (1)–(7). In the following the solutions of the approximate system are denoted by .v" ; c" ; " / for " > 0. Using a priori estimates for solutions of the approximate system based on (21), we can conclude for a suitable subsequence "i !i!1 0 that Dv"i ! Dv weakly in Lq .QT /dd ; v"i ! v S .c"i ; Dv"i / ! SQ

weakly in Lq

dC2 d

.QT /d ;

0

weakly in Lq .QT /dd ;

Q weakly in Lq v"i ˝ v"i "i .v"i / ! H

dC2 2d

.QT /:

(22)

Moreover, from suitable a priori estimates of a solution of Cahn-Hilliard equations with convection term, we have for a suitable subsequence c"i !i!1 c

in L4 .0; TI W41 .// \ L2 .0; TI Wr2 .//;

"i !i!1 

in L2 .0; TI H 1 .//;

(23)

where .c; / solve (14)–(16). Hence it only remains to prove (13). To this end let K" 2 L2 .QT /dd be such that Z

Z K" W D' d.x; t/ D QT

rc" ˝ rc" W D‰" .'/ d.x; t/ Z

QT

rc ˝ rc W D' d.x; t/

 QT

(24)

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H. Abels et al.

for all ' 2 L2 .0; TI H01 ./d / and kK" kL2 .QT /  Ck div.rc ˝ rc/  ‰" div.rc" ˝ rc" /kL2 .0;TIH 1 / 0

for some C > 0. We can assume that K" is pointwise a symmetric matrix. Then K"i ! 0 strongly in L2 .QT /dd ; due to (23). 2d Since q > dC2 , there exists some 0 > 1 such that q dC2 2d > 0 > 1. Hence, due to (22) we have for some "i !i!1 0, strongly in L2 0 .QT /d

v"i ! v

and v"i ˝ v"i "i .jv" j/ ! v ˝ v strongly in L 0 .QT /dd :

(25) (26)

Q D v ˝ v. Therefore H Then taking the limit of the weak form of the approximate system along the subsequence "i ; we obtain the following limit equation: Z

Z

.SQ  v ˝ v/ W D' d.x; t/

v  @t ' d.x; t/ C

 QT

QT

Z

Z

rc ˝ rc W D' d.x; t/ C

D

(27)



QT

v0  '.0/ dx:

for all ' 2 C1 .QT /d with div ' D 0 and supp.'/    Œ0; T/: By subtracting the above equation from the weak form of the approximate equations, we have the following equation: Z 

Z



.v"  v/  @t ' d.x; t/ C QT

 S.c" ; Dv" /  SQ W D' d.x; t/

QT

Z

Z

.v" ˝ v" " .v" /  v ˝ v/ W D' d.x; t/ C

D QT

K" W D' d.x; t/: QT

Defining u" WD v"  v we can write this as Z

Z u"i  @t ' d.x; t/ D QT

H"i W r' d.x; t/ QT

(28)

Existence of Weak Solutions for a Diffuse Interface Model of Power-Law Type. . .

21

for any ' 2 C1 .QT /d with div ' D 0 and supp.'/    Œ0; T/, where Hi WD H1;i C H2;i with Q H1;i WD S.c"i ; Dv"i /  S; H2;i WD v"i ˝ v"i "i .v"i /  v ˝ v  K"i Then we have the following convergences for suitable "i !i!1 0 u"i ! 0 weakly in Lq .0; tI Vq .//; 2 0

u"i ! 0 strongly in L

.QT /;

u"i ! 0 -weakly in L1 .0; TI L2 .//; 0

(29) (30) (31)

H1;i ! 0 weakly in Lq .QT /;

(32)

H2;i ! 0 strongly in L 0 .QT /

(33)

for some 1 < 0 < min.q; q0 /. Let I0  .0; T/ be a time interval, B0  Rd be a ball such that Q0 WD I0  B0  QT . Let 2 C01 . 16 Q0 / with 1 Q0   1 Q0 . Here 8 6 for c > 0 cQ0 is defined as cI0  cB0 ; where cI0 and cB0 have the same center as I0 and B0 respectively and have c times length and c times radius respectively. Then we can apply Theorem 2.3 with K D SQ  S.c; Dv/ to obtain ˇZ ˇ ˇ  ˇ  ˇ Q c lim sup ˇ .H1;i C S  S.c; Dv// W r.v"i  v/ Oi;k d.x; t/ˇˇ  c 2k=q : i!1

In other words ˇZ  ˇ  ˇ ˇ   k=q ˇ c d.x; t/ˇ  c 2 S.c"i ; Dv"i /  S.c; Dv/ W D.v"i  v/ Oi;k lim sup ˇ : ˇ i!1

Using this estimate, Hölder’s inequality, (23) and Assumption 2.1, we have ˇZ  ˇ  12 ˇ ˇ   k ˇ lim sup ˇ S.c; Dv"i /  S.c; Dv/ W D.v"i  v/ d.x; t/ˇˇ  c 2 2q : i!1

For k ! 1 the right hand side converges to zero. Now the monotonicity of S can be used to prove SQ D S.c; Dv/. Hence we have (13). This ends the proof. Acknowledgements This work was supported by the SPP 1506 “Transport Processes at Fluidic Interfaces” of the German Science Foundation (DFG) through the grant AB 285/4-1. Moreover, T. was supported by JSPS Research Fellowships for Young Scientists and by FMSP, a JSPS Program for Leading Graduate Schools in the University of Tokyo. The supports are gratefully acknowledged.

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References 1. H. Abels, On generalized solutions of two-phase flows for viscous incompressible fluids. Interfaces Free Bound. 9(1), 31–65 (2007) 2. H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities. Arch. Ration. Mech. Anal. 194(2), 463–506 (2009) 3. H. Abels, M. Röger, Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(6), 2403–2424 (2009) 4. H. Abels, M. Wilke, Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal. 67, 3176–3193 (2007) 5. H. Abels, L. Diening, Y. Terasawa, Existence of weak solutions for a diffuse interface model of non-Newtonian two-phase flows. Nonlinear Anal. Real World Appl. 15, 149–157 (2014) 6. S. Bosia, Analysis of a Cahn-Hilliard-Ladyzhenskaya system with singular potential. J. Math. Anal. Appl. 397(1), 307–321 (2013) 7. F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation. Asymptot. Anal. 20(2), 175–212 (1999) 8. D. Breit, L. Diening, M. Fuchs, Solenoidal Lipschitz truncation and applications in fluid. J. Differ. Equ. 253(6), 1910–1942 (2012) 9. D. Breit, L. Diening, S. Schwarzacher, Solenoidal Lipschitz truncation for parabolic PDE’s. M3AS 23(14), 2671–2700 (2014) 10. M. Bulíˇcek, F. Ettwein, P. Kaplický, D. Pražák, Dimension of the attractor for 3D flow of non-Newtonian fluid. Commun. Pure Appl. Anal. 8(5), 1503–1520 (2009) ´ 11. M. Bulíˇcek, P. Gwiazda, J. Málek, A. Swierczevska-Gwiazda, On Unsteady Flows of Implicitly Constituted Incompressible Fluids. SIAM J. Math. Anal. 44(4), 2756–2801 (2012) 12. X. Chen, Global asymptotic limit of solutions of the Cahn-Hilliard equation. J. Differ. Geom. 44, 262–311 (1996) 13. A. Debussche, L. Dettori, On the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal. 24(10), 1491–1514 (1995) 14. I.V. Denisova, A priori estimates for the solution of the linear non stationary problem connected with the motion of a drop in a liquid medium. (Russian) Trudy Mat. Inst. Steklov 188, 3–21 (1990). [translation in Proc. Steklov Inst. Math. 3, 1–24 (1991)] 15. I.V. Denisova, Problem of the motion of two viscous incompressible fluids separated by a closed free interface. Mathematical problems for the Navier-Stokes equations (Centro, 1993). Acta Appl. Math. 37, 31–40 (1994) 16. I.V. Denisova, V.A. Solonnikov, Classical solvability of the problem of the motion of two viscous incompressible fluids. (Russian) Algebra i Analiz. 7(5), 101–142 (1995). [translation in St. Petersburg Math. J. 7(5), 755–786 (1996)] 17. L. Diening, M. R˚užiˇcka, J. Wolf, Existence of weak solutions for unsteady motions of generalized Newtonian fluids. Annali della Scuola Normale Superiore di Pisa Classe di scienze (5) 9(1), 1–46 (2010) 18. C.M. Elliott, S. Luckhaus, A generalized equation for phase separation of a multi-component mixture with interfacial free energy. preprint SFB 256 Bonn No. 195, 1991 19. C.M. Elliott, S. Zheng, On the Cahn-Hilliard equation. Arch. Ration. Mech. Anal. 96(4), 339–357 (1986) 20. M. Grasselli, D. Pražák, Longtime behavior of a diffuse interface model for binary fluid mixtures with shear dependent viscosity. Interfaces Free Bound. 13(4), 507–530 (2011) 21. M.E. Gurtin, D. Polignone, J. Viñals, Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci. 6(6), 815–831 (1996) 22. P.C. Hohenberg, B.I. Halperin, Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435–479 (1977) 23. N. Kim, L. Consiglieri, J.F. Rodrigues, On non-Newtonian incompressible fluids with phase transitions. Math. Methods Appl. Sci. 29(13), 1523–1541 (2006)

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24. J. Kinnunen, J.L. Lewis, Very weak solutions of parabolic systems of p-Laplacian type. Ark. Mat. 40(1), 105–132 (2002) 25. M. Köhne, J. Prüss, M. Wilke, Qualitative behaviour of solutions for the two-phase NavierStokes equations with surface tension. Math. Ann. 356(2), 737–792 (2013) 26. O.A. Ladyzhenskaya, Sur de nouvelles équation dans la dynamique de fluides visqueux et leur resolution globale. Troudi Mat. Inst. Stekloff CII, 85–104 (1967) 27. J.-L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites Non Linéaires. (Dunod, France, 1969) 28. J. Málek, K.R. Rajagopal, Mathematical issues concerning the Navier-Stokes equations and some of its generalizations, in Handbook Differential Equation: Evolutionary equations, vol. II (Elsevier/NorthHolland, Amsterdam, 2005), pp. 371–459 29. J. Málek, J. Neˇcas, M. R˚užiˇcka, On the non-Newtonian incompressible fluids. M3AS 3(1), 35–63 (1993) 30. B. Nicolaenko, B. Scheurer, R. Temam, Some global dynamical properties of a class of pattern formation equations. Commun. Partial Differ. Equ. 14(2), 245–297 (1989) 31. P.I. Plotnikov, Generalized solutions to a free boundary problem of motion of a nonNewetonian fluid. Siberian Math. J. 34(4), 704–716 (1993) 32. J. Prüss, G. Simonett, On the two-phase Navier-Stokes equations with surface tension. Interfaces Free Bound. 12(3), 311–34 (2010) 33. J. Prüss, G. Simonett, R. Zacher, Qualitative behavior of incompressible two-phase flows with phase transition: the case of equal densities. Interfaces Free Bound. 15(4), 405–428 (2013) 34. J. Prüss, S. Shimizu, M. Wilke, Qualitative behavior of incompressible two-phase flows with phase transitions: the case of non-equal densities. Commun. Partial Differ. Equ. 39(7), 1236– 1283 (2014) 35. P. Rybka, K-H. Hoffmann, Convergence of solutions to Cahn-Hilliard equations. Commun. Partial Differ. Equ. 24(5–6), 1055–1077 (1999) 36. V.N. Starovo˘ıtov, On the motion of a two-component fluid in the presence of capillary forces. Mat. Zametki 62(2), 293–305 (1997) 37. J. Wolf, Existence of weak solutions to the equations of non-stationary motion of nonNewtonian fluids with shear rate dependent viscosity. J. Math. Fluid Mech. 9(1), 104–138 (2007)

Stationary Solutions for a Navier-Stokes/Cahn-Hilliard System with Singular Free Energies Helmut Abels and Josef Weber

Dedicated to Yoshihiro Shibata on the occasion of his 60th birthday

Abstract We consider a stationary Navier-Stokes/Cahn-Hilliard type system. The system describes a so-called diffuse interface model for the two-phase flow of two macroscopically immiscible incompressible viscous fluids in the case of matched densities, also known as Model H. We prove existence of weak solutions for the stationary system for general exterior forces and singular free energies, which ensure that the order parameter stays in the physical reasonable interval. To this end we reduce the system to an abstract differential inclusion and apply the theory of multi-valued pseudo-monotone operators. Keywords Cahn-Hilliard equation • Diffuse interface model • Navier-Stokes equation • Two-phase flow

1 Introduction and Main Result We consider the following stationary system of Navier-Stokes/Cahn-Hilliard type v  rv  div.2 .c/Dv/ C rp D rc C fQ

in ;

(1)

div v D 0

in ;

(2)

v  rc D div.m.c/r/

in ;

(3)

in ;

(4)

on @;

(5)

on @:

(6)

 D "c C "1 f 0 .c/ vj@ D 0 n  rcj@ D n  rj@ D 0

H. Abels () • J. Weber Fakultät für Mathematik, Universität Regensburg, Universitätstr. 31, 93053 Regensburg, Germany e-mail: [email protected]; [email protected] © Springer Basel 2016 H. Amann et al. (eds.), Recent Developments of Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-0348-0939-9_3

25

26

H. Abels and J. Weber

Moreover we prescribe 1 jj

Z c.x/dx D m N 2 .1; 1/:

(7)



Here v is the mean velocity, Dv D 12 .rv C rvT /, p is the pressure, c is an order parameter related to the concentration of the fluids (e.g. the concentration difference or the concentration of one component), fQ is an exterior force density,  is a bounded domain with sufficiently smooth boundary, and n is the outer unit normal to @. Moreover, .c/ > 0 is the viscosity of the mixture, " > 0 is a (small) parameter, which will be related to the “thickness” of the interfacial region, and f is a homogeneous free energy density specified below. This system arises as a stationary version of the so-called “model H”, cf. Hohenberg and Halperin [7] and Gurtin et al. [6]: @t v C v  rv  div. .c/Dv/ C rp D rc C fQ div v D 0 @t c C v  rc D div.m.c/r/  D "1 f 0 .c/  "c

in   .0; 1/; in   .0; 1/; in   .0; 1/; in   .0; 1/

together with the same boundary conditions as above. The system describes the motion of two macroscopically immiscible, viscous, incompressible Newtonian fluids. The model takes a partial mixing on a small length scale measured by a parameter " > 0 into account. Therefore the classical sharp interface between both fluids is replaced by an interfacial region and an order parameter related to the concentration difference of both fluids is introduced. Here it is assumed that the densities of both components as well as the density of the mixture are constant and for simplicity equal to one. In the following we set " D 1 for simplicity. But all results hold true for general " > 0. For the instationary system existence of weak solutions and well-posedness were obtained by Starovoitov [10], Boyer [4], Liu and Shen [8], and Abels [1]. Moreover, we refer to Abels, Depner and Garcke [3] for further references and results on similar models in the case of different densities. In the following we will consider a class of singular free energies, which is motivated by the homogeneous free energy of the so-called regular solution models used by Cahn and Hilliard [5]: ‰.c/ D

 c ..1 C c/ ln.1 C c/ C .1  c/ ln.1  c//  c2 ; 2 2

c 2 Œ1; 1;

where 0 <  < c . Mathematically, these singular free energies ensure that the order parameter stays in the physically reasonable interval, which is Œ1; 1 if c is the difference of volume fractions of both fluids. But this leads to singular terms in

Stationary Solutions for a Navier-Stokes/Cahn-Hilliard System with Singular. . .

27

the equation for the chemical potential. In order to deal with these terms we apply techniques, which were developed in Abels and Wilke [2]. More precisely we work with the following assumption: Assumption 1.1 Let   Rd , d D 2; 3, be a bounded domain with C3 -boundary, W Œa; b ! .0; 1/ be continuously differentiable, mW Œa; b ! .0; 1/ be continuous, and f 2 C0 .Œa; b/ \ C2 ..a; b// such that lim f 0 .s/ D 1;

s!a

lim f 0 .s/ D C1;

(8)

s!b

f 00 .s/  

(9)

hold for all s 2 .a; b/ and some constant   0. Let f be as in the previous assumption. Then we can conclude f .s/ D f0 .s/  

s2 2

(10)

with f0 convex and  2 R as in Assumption 1.1. Moreover we can deduce lim f00 .s/ D 1 and lim f00 .s/ D C1. s!a

s!b

Weak solutions of the stationary Navier-Stokes/Cahn-Hilliard equations are defined as follows: Definition 1.2 (Weak Solution) A triple .v; ; c/ with v 2 H01 ./d \ L2 ./;  2 H 1 ./ and c 2 H 1 ./ \ L2.m/ N ./ is called weak solution for the Eqs. (1)–(6) under the constraint (7) if Z Z Z Z .v  rv/  'dx C 2 .c/Dv W D'dx D  rc  'dx C fQ  'dx (11) 







1 ./ and if holds for all ' 2 C0;

Z

Z .v  rc/ 



Z

Z  

holds for all

dx D 

dx D

m.c/r  r dx 

Z

rc  r dx C 

f 0 .c/

(12)

dx

(13)



2 C1 ./ and if f 0 .c/ 2 L2 ./.

Now our main result is the following: Theorem 1.3 (Existence of Weak Solutions) Let , , and f W Œa; b ! R be as in Assumption 1.1. Then for all fQ 2 L2 ./d and m N 2 .1; 1/ there exists a weak solution in the sense of Definition 1.2.

28

H. Abels and J. Weber

2 Preliminaries and Notation Throughout the paper the usual Lebesgue spaces with respect to the Lebesgue measure are denoted by Lq .M/, 1  q  1, for some measurable M  RN . The m standard Lq -Sobolev space is denoted by Wqm ./. Wq;0 ./ is the closure of C01 ./ m m m m m in Wq ./ and H ./ D W2 ./; H0 ./ D W2;0 ./. Furthermore we use the notation Z n o 1 2 2 L.m/ ./ D f 2 L ./ W f .x/dx D m N ; N jj 

1 1 1 H.0/ ./ D H 1 ./ \ L2.0/ ./ and H.0/ ./ WD H.0/ ./0 . Finally L2 ./ is the closure of divergence free C01 ./-vector fields in L2 ./d . For a Banach space X, X 0 denotes its duality space and h:; :iX 0 ;X its duality product. If H is a Hilbert space, .:; :/H denotes its inner product. The projection P0 W L1 ./ ! L1.0/ ./ on the L1 -space with mean value 0 is defined by

1 P0 f WD f  jj

Z f .x/dx 

for every f 2 L1 ./.

3 Subdifferential of F and FQ Let f be as in Assumption 1.1. In this section, we study the functionals F.c/ WD

1 2

1 Q F.c/ WD 4

Z

jrc.x/j2 dx C



Z

Z (14)

f0 .c.x//dx

(15)

 2

Z

jrc.x/j dx C 

f0 .c.x//dx



where we assume c 2 L2.0/ ./, use the decomposition (10) of f and extend f0 to R by f0 .x/ D C1 for all x … Œa; b. Then Q D fc 2 H 1 ./ \ L2.0/ ./ W f0 .c/ 2 L1 ./g: dom.F/ D dom.F/ Q For c … dom.F/, we set F.c/ D C1 and F.c/ D C1.

Stationary Solutions for a Navier-Stokes/Cahn-Hilliard System with Singular. . .

29

In the following, we will collect some statements on the subgradients and Q Therefore we define: subdifferentials of these two functionals F and F. Definition 3.1 (Subgradient, Subdifferential) Let X be a real Banach space and f W X ! .1; 1 a functional on X. A linear functional u0 2 X 0 is called subgradient of f at u if f .u/ ¤ C1 and f .v/  f .u/ C hu0 ; v  uiX 0;X

(16)

holds for all v 2 X. The set of all subgradients of f at u is called subdifferential @f .u/ of f at u. Notation For a functional F W L2.0/ ./ ! .1; C1, its subgradient maps from 1 L2.0/ ./ to P.L2.0/ ./0 /. Since we can also consider F as a functional from H.0/ ./ to .1; C1, we have to distinguish between its different subgradients. Therefore we write @X f for the subgradient of a functional f W X ! .1; C1, where X is a real Banach space. In our specific case, @L2 functional F W

L2.0/ ./

.0/ ./

F respectively @L2 F is the subgradient of the

! .1; C1 and @H 1 1 ./ H.0/

.0/

.0/ ./

F respectively @H 1 F is the .0/

! .1; C1. subgradient of the functional F W Q First of all, we collect some statements on the subdifferentials of F and F: Lemma 3.2 The functionals F; FQ W L2.0/ ./ ! R [ fC1g defined as in (14) Q ¤ ;. and (15) are lower semi-continuous and convex with dom.F/; dom.F/ The proof of this Lemma can be found in [2]. Proposition 3.3 The subgradients @L2 F and @L2 FQ are maximal monotone opera.0/ .0/ tors. Proof Because of Lemma 3.2, F; FQ W L2.0/ ./ ! R [ fC1g are lower semicontinuous functionals with F, FQ 6 C1. The Theorem of Rockafeller, cf. Proposition 32.17 in [11], yields that the operators @L2 F, @L2 FQ W L2.0/ ! P..L2.0/ .//0 / are maximal monotone. t u .0/

.0/

Proposition 3.4 The subgradients @H 1 F and @H 1 FQ are maximal monotone opera.0/ .0/ tors. 1 1 Proof Let .ck /k2N  H.0/ ./ with ck ! c in H.0/ ./. Then it holds ck ! 2 Q Q k /  F.c/ c in L ./ and therefore lim infF.ck /  F.c/ and lim infF.c by k!1

k!1

Lemma 3.2. Thus the Theorem of Rockafeller, cf. Proposition 32.17 in [11], yields the statement. t u

30

H. Abels and J. Weber

Theorem 3.5 Let   Rd , d D 2; 3 be a bounded domain with C3 -boundary and F defined as in (14). Moreover let f00 .x/ D C1 for all x … .a; b/. Then ˚ D.@L2 F/ D c 2 H 2 ./ \ L2.0/ ./j f00 .c/ 2 L2 ./; .0/  f000 .c/jrcj2 2 L1 ./; @n cj@ D 0

(17)

@L2 F.c/ D c C P0 f00 .c/:

(18)

and .0/

Furthermore it holds kck2H 2 ./

C

k f00 .c/k2L2 ./

Z C

f000 .c.x//jrc.x/j2 dx



   C k@L2 F.c/k2L2 ./ C kck2L2 ./ C 1 .0/

for some constant C > 0 independent of c 2 D.@L2 F/. .0/

The proof can be found in [2], cf. Theorem 4.3. Remark 3.6 Here we identify L2.0/ ./0 with L2.0/ ./ in the standard way. But we 1 1 will not identify H.0/ ./ with H.0/ ./ via the Riesz isomorphism. Equation (18) provides a characterization of @L2 F. Later, we will need relations .0/ between @H 1 F and @L2 F. The two following lemmata provide such relations: .0/

.0/

Lemma 3.7 Let F be as in (14). Then for all c 2 D.@L2 F/ we have @L2 F.c/  .0/ .0/ @H 1 F.c/. .0/

Proof Let c 2 D.@L2 F/ and fwg D @L2 F.c/. By definition it holds .0/

.0/

hw; c0  ciL2 ./0 ;L2 ./ D .w; c0  c/L2 ./  F.c0 /  F.c/ 8 c0 2 L2.0/ ./: Thus we can conclude hw; c0  ciH 1 ./;H 1 .0/

.0/ ./

1  F.c0 /  F.c/ 8 c0 2 H.0/ ./:

t u

This yields fwg  @H 1 F.c/. .0/

Lemma 3.8 Let w 2 @H 1 F.c/ and c 2 D.@H 1 F/. Suppose w 2 L2.0/ ./. Then .0/ .0/ c 2 D.@L2 F/ and .0/

w D @L2 F.c/ D c C P0 f00 .c/: .0/

Stationary Solutions for a Navier-Stokes/Cahn-Hilliard System with Singular. . .

31

Proof Let w 2 @H 1 F.c/. Then .0/

hw; c0  ciH 1 ./;H 1 .0/

.0/ ./

1  F.c0 /  F.c/ 8 c0 2 H.0/ ./:

Since w 2 L2.0/ ./, we can conclude .w; c0  c/L2 ./  F.c0 /  F.c/ 8 c0 2 L2.0/ ./: 1 ./ and F.c0 / D C1 for c0 … D.F/. The last conclusion holds since D.F/  H.0/ This yields w 2 @L2 F.c/. Due to (18), we can identify fwg and w, where fwg D .0/ @L2 F.c/. Hence .0/

w D @L2 F.c/ D c C P0 f00 .c/ .0/

t u Finally we state the main result about maximal monotone operators on which the proof of Theorem 1.3 is based: Theorem 3.9 (Browder) 1. Let C be a nonempty, closed and convex subset of a real and reflexive Banach space X. 2. Let A W C ! P.X 0 / be a maximal monotone operator. 3. Let B W C ! X 0 be a pseudo-monotone, bounded and demi-continuous mapping. 4. If the set C is unbounded, then the operator B is A-coercive with respect to b 2 X 0 , i.e., there exists an element u0 2 C \ D.A/ and r > 0 such that hBu; u  u0 iX 0;X > hb; u  u0 iX 0;X for all u 2 C with jjujjX > r. Then the problem b 2 Au C Bu

(19)

has a solution u 2 D.A/ \ C. The proof of this theorem can be found in [9], cf. Theorem 3.42, or [11], cf. Theorem 32.A.

32

H. Abels and J. Weber

4 Existence of Weak Solutions 4.1 Reformulation of the Problem When working with Definition 1.2, some difficulties arise in the analysis of our problem. The most obvious one is that L2.m/ N ¤ 0. But N ./ is no vector space for m w.l.o.g. we can assume m N D 0 in Eq. (7). Otherwise replace c by c0 WD c  m N and f by fmN with fmN .x/ WD f .x C m/ N for all x 2 R. Note that this implies 0 D m N 2 .a; b/. Moreover we will need that  has mean value 0 in the following. Otherwise there will be some difficulties with the coercivity of the operator. But as we will see, we can reduce to this case. To this end we consider the following equations: Z

Z .v  rv/  'dx C



Z 2 .c/Dv W D'dx D



Z 0 rc  'dx C



fQ  'dx

(20)



1 ./ and for all ' 2 C0;

Z

Z .v  rc/  dx D 



Z

Z

m.c/r0  r dx 

0 dx D 

Z

rc  r dx C 

(21)

P0 f 0 .c/ dx

(22)



1 1 for all 2 C1 ./, where v 2 H01 ./d \ L2 ./, 0 2 H.0/ ./, c 2 H.0/ ./ and f as in Assumption 1.1. 1 1 Lemma 4.1 Let .v; 0 ; c/ 2 .H01 ./d \ L2 .//  H.0/ ./  H.0/ ./ be a solution of the Eqs. (20)–(22). Then .v; ; c/ is a weak solution of the stationary NavierStokes/Cahn-Hilliard equations in the sense of Definition 1.2, where  D 0 C R 0 1 f .c/dx. jj 

1 1 ./  H.0/ ./ be a solution of the Proof Let .v; 0 ; c/ 2 .H01 ./d \ L2 .//  H.0/ Eqs. (20)–(22). Since Eq. (20) is fulfilled, we conclude

Z

Z .v  rv/  'dx C



2 .c/Dv W D'dx 

Z

Z

0 rc  'dx C

D 

1 for all ' 2 C0;

./, where  WD

Z  rc  'dx C

 1 jj

R 

f 0 .c/dy.



fQ  'dx

Stationary Solutions for a Navier-Stokes/Cahn-Hilliard System with Singular. . .

33

But this already yields Eq. (11). Equation (12) holds due to Eq. (21). It remains to show that Eq. (13) is fulfilled. Because of Eq. (22) holds, we can deduce Z Z Z  dx D rc  r dx C .P0 f 0 .c/ C / dx: 





Using the definitions of  and P0 yields Eq. (13) and therefore the lemma is proved. u t In Sect. 4.3 we will prove the existence of a solution for the Eqs. (20)–(22) and thus deduce the existence of a solution for the stationary Navier-Stokes/CahnHilliard equations with the help of the previous lemma.

4.2 Abstract Reformulation First of all, we define the following spaces: 1 1 ./  H.0/ ./ X1  X2  X3 X WD .H01 ./d \ L2 .//  H.0/

(23)

C WD X1  X2  fc 2 X3 W c.x/ 2 Œa; b a.e.g X1  X2  C3

(24)

Moreover we define a multivalued mapping A W C ! P.X 0 / by 19 880 ˆ > 0 ˆ ˆ > ˆ < ˆ C= ˆ < B B C 0 @ A> if .v; 0 ; c/ 2 D.A/; A.v; 0 ; c/ WD ˆ ˆ > : Q ˆ @H 1 F.c/ ; ˆ ˆ .0/ ˆ : ; else

(25)

Q and FQ is defined as in (15), and another operator where D.A/ WD X1 X2 D.@H 1 F/ .0/ 0 B W C ! X by hB.v; 0 ; c/; .'; ; /iX 0 ;X Z Z Z WD .v  rv/  'dx C 2 .c/Dv W D'dx  0 rc  'dx 



Z

m.c/r0  r dx C

C 

Z

P0 . c/ dx C

 

1 2

Z



Z

Z .v  rc/ dx 



rc  rdx 

for all .v; 0 ; c/ 2 C and .'; ; / 2 X.

0 dx 

(26)

34

H. Abels and J. Weber

In addition, b 2 X 0 is given by Z hb; .'; ; /iX 0 ;X WD

fQ  'dx



for all .'; ; / 2 X. Because of the Eqs. (25) and (26), the problem b 2 A.v; 0 ; c/CB.v; 0 ; c/ with .v; 0 ; c/ 2 C \ D.A/ can be written as 0

1 0 1 0 1 0 v  rv  div.2 .c/Dv/  0 rc fQ B C @ A @ 0 D 0A  divN .m.c/r0 / C v  rc @ AC 1 Q @H 1 F.c/ 0 0  P0 . c/  2 N c

(27)

.0/

1 1 ./ ! H.0/ ./ is defined as in X 0 Š X10  X20  X30 , where N W H.0/

Z hN u; iH 1 ./;H 1

.0/ ./

.0/

WD

ru  r dx 

for all

1 2 H.0/ ./ and

Z h divN .m.c/r0 /; iH 1 ./;H 1 .0/

.0/ ./

WD

m.c/r0  r dx 

1 for all 2 H.0/ ./. First of all, we suppose that the problem b 2 A.v; 0 ; c/ C B.v; 0 ; c/ has a solution. Then we show that this solution is a weak solution to the NavierStokes/Cahn-Hilliard equations. The existence of such a solution will be proved in the next section.

Lemma 4.2 Let .v; 0 ; c/ 2 C \ D.A/ be such that b 2 A.v; 0 ; c/ C B.v; 0 ; c/ holds. Then .v; 0 ; c/ is a weak solution of the stationary Navier-Stokes/CahnHilliard equations in the sense of Definition 1.2. Proof Let .v; 0 ; c/ 2 C \ D.A/ be a triple such that b 2 A.v; 0 ; c/ C B.v; 0 ; c/ Q exists, i.e. holds. Because of Eq. (27) we see that w 2 @H 1 F.c/ .0/

hw; c0  ciH 1 ./;H 1

.0/ ./

.0/

Q 0 /  F.c/ Q  F.c

1 for all c0 2 H.0/ ./:

Thus we can conclude hw;c0  ciH 1 ./;H 1 .0/

.0/ ./

1 C .rc; rc0  rc/L2 ./ 2

Stationary Solutions for a Navier-Stokes/Cahn-Hilliard System with Singular. . .

35

Q 0 /  F.c/ Q C 1 .rc; rc0  rc/L2 ./  F.c 2 Q 0 /  F.c/ Q C 1 krc0 k2 2  1 krck2 2  1 kr.c  c0 /k2 2  F.c L ./ L ./ L ./ 4 4 4 1 for all c0 2 H.0/ ./. Altogether we obtain

hw; c0  ciH 1 ./;H 1

.0/ ./

.0/

1 C .rc; rc0  rc/L2 ./  F.c0 /  F.c/ 2

1 1 ./. Thus wQ 2 @H 1 F.c/, where wQ 2 H.0/ ./ is defined by for all c0 2 H.0/ .0/

hw; Q iH 1 ./;H 1 .0/

for all holds

.0/ ./

WD hw; iH 1 ./;H 1 .0/

.0/ ./

1 C .rc; r /L2 ./ 2

(28)

1 ./. Since .v; 0 ; c/ 2 C \ D.A/ fulfills Eq. (27) by assumption, it 2 H.0/

1 1 0 C P0 . c/ C N c D w D wQ C N c 2 2

in X30 :

Thus we obtain 0 C P0 . c/ D wQ in X30 . From 0 C P0 . c/ 2 L2.0/ ./ and c 2 D.@H 1 F/ we can deduce with Lemma 3.8 that it holds wQ D N c C P0 f00 .c/ in .0/

X30 . In particular this yields f00 .c/ 2 L2 ./. Altogether 0 C P0 . c/ D N c C P0 f00 .c/

in X30 :

Thus it holds in X30 : 0 C P0 . c/ D N c C P0 f00 .c/ which implies 0 D N c C P0 . f00 .c/   c/ D N c C P0 .f 0 .c//: This means that Eq. (22) holds for all 2 C1 ./\L2.0/ ./. That Eq. (20) holds for 1 ./ and (21) holds for all 2 C1 ./\L2.0/ ./ is a direct consequence all ' 2 C0;

of (26) and (27). Finally, Lemma 4.1 implies the statement. t u

36

H. Abels and J. Weber

4.3 Proof of Existence of Weak Solutions In Lemma 4.2 we considered the existence of a triple .v; 0 ; c/ 2 C \ D.A/ which fulfills b 2 A.v; 0 ; c/ C B.v; 0 ; c/. Then we showed that such a triple is a weak solution to the stationary Navier-Stokes/Cahn-Hilliard equations in the sense of Definition 1.2. Now it remains to show that there really exists such a triple. Proof of Theorem 1.3 According to Lemma 4.2 it is sufficient to show the existence of a solution u 2 C \ D.A/ to the differential inclusion b 2 Au C Bu. To prove the existence of such a solution u, we will apply Theorem 3.9 and show that the conditions (1) – (4) are satisfied. To (1): Obviously C is non-empty and closed. Moreover it is easy to show that C is convex. Since X is a real, reflexive Banach space, C and X fulfill condition (1). Q  C3 . To this end let c 2 D.@H 1 F/. Q Then To (2): First we show that D.@H 1 F/ .0/ .0/ Q Q  dom.F/. Q This yields c.x/ 2 Œa; b a.e. since f0 .x/ D F.c/ ¤ C1 since D.@H 1 F/ .0/

C1 for x … Œa; b. Thus we obtain c 2 C3 . 1 1 ./ ! P.H.0/ .// is Because of Proposition 3.4, the operator @H 1 FQ W H.0/ .0/ maximal monotone. Since A does not depend on v and 0 , the operator A W Q  X ! P.X 0 / is maximal monotone. Due to D.A/ D .X1  X2  D.@H 1 F// .0/ .X1  X2  C3 / D C  X, the operator A W C ! P.X 0 / is maximal monotone as well.

To (3): We split B into several operators .Bi /i2f1;:::;8g with B D B1 C : : : C B8 and will show that all operators Bi W C ! X 0 ; i 2 f1; : : : ; 8g, are pseudo-monotone. Moreover we will show that each operator is locally bounded and demi-continuous as well. Because of (26) we define Z hB1 .v; 0 ; c/; .'; ; /iX 0 ;X WD .v  rv/  'dx; 

Z

hB2 .v; 0 ; c/; .'; ; /iX 0 ;X WD

2 .c/Dv W r'dx; 

hB3 .v; 0 ; c/; .'; ; /i

X 0 ;X

Z

WD  Z

hB4 .v; 0 ; c/; .'; ; /iX 0 ;X WD

0 rc  'dx; 

m.c/r0  r dx; 

Z

hB5 .v; 0 ; c/; .'; ; /iX 0 ;X WD

.v  rc/ dx; 

Stationary Solutions for a Navier-Stokes/Cahn-Hilliard System with Singular. . .

37

Z hB6 .v; 0 ; c/; .'; ; /iX 0 ;X WD 

 dx; 

Z

hB7 .v; 0 ; c/; .'; ; /iX 0 ;X WD 

P0 . c/dx; 

hB8 .v; 0 ; c/; .'; ; /iX 0 ;X

1 WD 2

Z rc  rdx:



for all .'; ; / 2 X. Pseudo-Monotonicity Using the compact embeddings H 1 ./ ,! L4 ./ and 4 L 3 ./ ,! H 1 ./, it is easy to show that the operators B1 ; B3 ; B5 ; B6 ; B7 and B8 are completely continuous and therefore pseudo-monotone. Thus it remains to show the pseudo-monotonicity of the operators B2 and B4 . In the following, we only prove the pseudo-monotonicity of the operator B2 since the proof for the operator B4 is almost the same. To prove the pseudo-monotonicity of B2 , we define another operator BQ 2 W X  X ! X 0 by hBQ 2 .u1 ; u2 /; .'; ; /iX 0 ;X WD

Z 2 .c1 /Dv2 W D'dx; 

where ui D .vi ; 0i ; ci / 2 X, i D 1; 2, respectively u D .v; 0 ; c/ 2 X. Then BQ 2 .u; / is hemi-continuous for every fixed u 2 X. Moreover BQ 2 .u; / is monotone since hBQ 2 .u; u1  u2 /; u1  u2 iX 0 ;X D

Z 2 .c1 /D.v1  v2 / W D.v1  v2 /dx  0: 

Furthermore BQ 2 .; u/ is completely continuous for all u 2 X. To this end we Q Q0 ; c/ consider a sequence .uk /k2N  X such that uk * u in X and uQ D .v; Q 2 X. Then it holds Z   Q Q hB2 .uk ; u/ Q  B2 .u; u/; Q .'; ; /iX 0 ;X D 2 .ck /DvQ  .c/DvQ W D'dx 

Q L2 ./ jj'jjH 1 ./ :  2jj .ck /DvQ  .c/Dvjj Q L2 ./ ! 0, we define To verify jj .ck /DvQ  .c/Dvjj Q .Fc/.x/ WD f .x; c.x// WD .c.x//  Dv.x/:

38

H. Abels and J. Weber

Then Lemma 1.19 in [9] yields that F W L2 ./ ! L2 ./ is continuous and bounded. Since ck ! c in L2 ./, this yields that BQ 2 .; u/ is completely continuous for all u 2 X. Due to Lemma 5.1 below we can conclude that B2 is pseudo-monotone. Boundedness Since the local boundedness is obvious for every operator Bi , i D 1; : : : ; 8, the operator B is locally bounded as well. Demi-Continuity Since the operators B1 ; : : : ; B8 are pseudo-monotone and locally bounded, they are also demi-continuous, cf. Chap. 3, Lemma 2.6 in [9], or Proposition 27.7 in [11]. Thus B is also demi-continuous. To (4): Obviously, the set C is unbounded. We choose u0 .0; 0; 0/ 2 C \ D.A/. Then we have to show that there is some r > 0 such that < B.u/  b; u >X 0;X > 0 for all u 2 C such that jjujjX > r

(29)

For the operator B we can write Z hB.u/; uiX 0;X hb; uiX 0;X D

2 .c/Dv W Dvdx C 

Z

Z Z

c dx 

 



m.c/r  rdx 

1 P0 . c/cdx C 2

Z

Z rc  rcdx 



(30)

fQ  vdx:



Using jc.x/j  max.jaj; jbj/ a.e. and the fact that the mean value of 0 is 0, we can derive the following inequalities: Z

2 .c/Dv W Dvdx  CQ 1 jjvjj2H 1./



Z

m.c/r  rdx  CQ 2 jjjj2H 1 ./



Z

Z

c dx  CQ 3 jjjjH 1./



P0 . c/c dx D  jjcjj2L2 ./  CQ 4



1 2

Z 

rc  rcdx  CQ 5 jjcjj2H 1./ Z 

1 1 fQ  vdx  . jjfQjj2L2 ./ C CQ 1 jjvjj2H 1./ / 2 CQ 1

Stationary Solutions for a Navier-Stokes/Cahn-Hilliard System with Singular. . .

39

for some constants CQ 1 ; CQ 2 ; CQ 3 ; CQ 4 ; CQ 5 > 0. With the help of these inequalities, (30) can be estimated as hB.u/; uiX 0;X  hb; uiX 0;X  CQ 1 jjvjj2H 1./ C CQ 2 jjjj2H 1 ./  CQ 3 jjjjH 1 ./  CQ 4 C CQ 5 jjcjj2H 1./  D

1 Q 2 CQ 1 jjvjj2H 1./ jjf jjL2 ./  2 2CQ 1

CQ 1 jjvjj2H 1./ C jjjjH 1 ./ .CQ 2 jjjjH 1 ./  CQ 3 / 2 ƒ‚ … „ DWh.jjjjH 1 ./ /

C CQ 5 jjcjj2H 1./  CQ 4  Since fQ 2 L2 ./d is fixed, CQ 4 C enough. Moreover we deduce

1 jjfQjj2L2 ./ 2CQ 1

1 Q 2 jjf jjL2 ./ : 2CQ 1

 M1 for a constant M1 > 0 large

lim h.x/ D C1:

x!C1

Thus we can see that the condition (29) is fulfilled for all .v; 0 ; c/ 2 C with jj.v; 0 ; c/jjX large enough. This yields the theorem. t u

Appendix Lemma 5.1 Let X be a real, reflexive Banach space and AQ W X  X ! X 0 such that for all u 2 X: 1. 2.

Q :/ W X ! X 0 is monotone and hemi-continuous. A.u; Q A.:; u/ W X ! X 0 is completely continuous.

Q u/ is pseudo-monotone. Then the operator A W X ! X 0 defined by A.u/ D A.u; Proof Let .uk /k2N  X be such that uk * u in X and lim suphA.uk /; uk  uiX 0;X  0: k!1

We have to show that lim infhA.uk /; uk  wiX 0 ;X  hA.u/; u  wiX 0 ;X k!1

40

H. Abels and J. Weber

Q :/ is monotone, for all w 2 X. Since A.u; Q k ; wt /; uk  wt iX 0 ;X  0; Q k ; uk /  A.u hA.u where we choose wt WD .1  t/u C tv for 0 < t < 1 and v 2 X arbitrary. Now we use that uk  wt D .uk  u/ C t.u  v/ and thus we get two terms. For the first one we use Q k ; uk /  A.u Q k ; wt /; uk  uiX 0;X D lim infhA.uk /; uk  uiX 0;X ; lim infhA.u k!1

k!1

Q wt / is completely continuous for every 0 < t < 1. where we used that A.; This means that for the remaining terms it holds   Q k ; uk /  A.u Q k ; wt /; t.u  v/iX 0 ;X C hA.uk /; uk  uiX 0 ;X  0: lim inf hA.u k!1

Since lim suphA.uk /; uk  uiX 0;X  0 and 0 < t < 1, we can conclude k!1

  Q k ; uk /  A.u Q k ; wt /; u  viX 0 ;X C hA.uk /; uk  uiX 0 ;X  0: lim inf hA.u k!1

This yields   Q k ; wt /; u  viX 0 ;X  0 lim inf hA.uk /; uk  viX 0 ;X  hA.u k!1

Q k ; / is hemi-continuous for every k 2 N and A.; Q u/ is completely Using that A.u continuous for every u 2 X yields the lemma. t u

References 1. H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities. Arch. Ration. Mech. Anal. 194(2), 463–506 (2009) 2. H. Abels, M. Wilke, Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal. 67(11), 3176–3193 (2007) 3. H. Abels, D. Depner, H. Garcke, Existence of weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities. J. Math. Fluid Mech. 15(3), 453–480 (2013) 4. F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation. Asymptot. Anal. 20(2), 175–212 (1999). ISSN 0921–7134 5. J.W. Cahn, J.E. Hilliard, Free energy of a nonuniform system. I. Interfacial energy. J. Chem. Phys. 28(2), 258–267 (1958) 6. M.E. Gurtin, D.Polignone, J. Viñals, Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci. 6(6), 815–831 (1996). ISSN 0218– 2025

Stationary Solutions for a Navier-Stokes/Cahn-Hilliard System with Singular. . .

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7. P.C. Hohenberg, B.I. Halperin, Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435–479 (1977) 8. C. Liu, J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Phys. D 179(3–4), 211–228 (2003). ISSN 0167– 2789 9. M. R˘užiˇcka, Nichtlineare Funktionalanalysis. Eine Einführung (Springer, Berlin, 2004) 10. V.N. Starovo˘ıtov, On the motion of a two-component fluid in the presence of capillary forces. Mat. Zametki 62(2), 293–305 (1997). ISSN 0025–567X 11. E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/B (Springer, New York, 1990), pp. i–xvi and 469–1202. ISBN 0–387–97167–X

Parabolic Equations on Uniformly Regular Riemannian Manifolds and Degenerate Initial Boundary Value Problems Herbert Amann

Dedicated to Professor Yoshihiro Shibata on the occasion of his sixtieth birthday

Abstract In this work there is established an optimal existence and regularity theory for second order linear parabolic differential equations on a large class of noncompact Riemannian manifolds. Then it is shown that it provides a general unifying approach to problems with strong degeneracies in the interior or at the boundary. Keywords Degenerate equations • Linear parabolic boundary value problems • Noncompact Riemannian manifolds • Weighted Sobolev spaces

1 Introduction This paper is devoted to second order initial boundary value problems for linear parabolic equations on a wide class of noncompact Riemannian manifolds, termed ‘uniformly regular’. Important examples are complete Riemannian manifolds with no boundary and bounded geometry.1 In this setting there is already a rich theory for linear parabolic equations—predominantly heat equations—based on kernel estimates. Our main interest concerns, however, noncompact Riemannian manifolds with boundary for which very little is known so far (see the following sections for references). Prototypes of such cases are m-dimensional Riemannian submanifolds of Rn with compact boundary or funnel-like ends (cf. Examples 3.5).

1 Precise definitions of and notations for all terms used in this introduction without further explanation are found in the following sections and the Appendix.

H. Amann () Math. Institut, Universität Zürich, Winterthurerstr. 190, CH 8057 Zürich, Switzerland e-mail: [email protected] © Springer Basel 2016 H. Amann et al. (eds.), Recent Developments of Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-0348-0939-9_4

43

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H. Amann

In order to give the flavor of our main results we consider in this introduction a simplified version of the general problem. Namely, we restrict ourselves to autonomous equations with homogeneous boundary conditions. Let M D .M; g/ be a Riemannian manifold. We set Au WD  div.a q grad u/;

(1)

with a being a symmetric positive definite .1; 1/-tensor field on M which is bounded and has bounded and continuous first order (covariant) derivatives. This is expressed by saying that A is a regular uniformly strongly elliptic differential operator. We assume that @0 M is open and closed in @M and @1 M WD @M n@0 M. Then we put B0 u WD u on @0 M;

B1 u WD . ja q grad u/ on @1 M;

where these operators are understood in the sense of traces and is the inward pointing unit normal vector field on @1 M. Thus B WD .B0 ; B1 / is the Dirichlet boundary operator on @0 M and the Neumann operator on @1 M. Throughout this paper, 0 < T < 1 and J WD Œ0; T. We write MT for the space time cylinder M  J. Moreover, @ D @t is the ‘time derivative’, @MT WD @M  J the lateral boundary, and M0 D M  f0g the ‘initial surface’ of MT . Then we consider the problem @u C Au D f on MT ;

Bu D 0 on @MT ;

u D u0 on M0 :

(2)

The last equation is to be understood as 0 u D u0 with the ‘initial’ trace operator 0 . Of course, @0 M or @1 M or both may be empty. In such a situation obvious interpretations and modifications are to be applied. We are interested in a strong Lp -theory for (2). To describe it we have to introduce (fractional order) Sobolev spaces. We always assume that 1 < p < 1. The Sobolev space Wpk .M/ is defined for k 2 N to be the completion of D.M/, the space of smooth functions with compact support, in L1;loc .M/ with respect to the norm u 7!

k X  j   jr uj 0 p gj

Lp .M/

1=p

:

(3)

jD0

Here r D rg is the Levi-Civita covariant derivative and jjg0j the .0; j/-tensor norm naturally induced by g. Thus Wp0 .M/ D Lp .M/. Moreover,

˚  2 .M/ WD u 2 Wp2 .M/ I Bu D 0 : Wp;B

Parabolic Equations on Uniformly Regular Riemannian Manifolds and. . . 22=p

We also need the space Wp interpolation:

45

.M/ which is defined for p ¤ 2 by real

  Wp22=p .M/ WD Lp .M/; Wp2 .M/ 11=p;p : Then 8˚  ˆ u 2 Wp22=p .M/ I Bu D 0 ; ˆ ˆ 0. This can always be achieved by replacing A by A C ! for a sufficiently large ! > 0. On the surface, this theorem looks exactly the same as the very classical existence and uniqueness theorem for second order parabolic equations on open subsets of Rm with smooth compact boundary (e.g., Ladyzhenskaya et al. [27, Chap. IV] and Denk

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et al. [17]). However, it is in fact a rather deep-rooted vast generalization thereof since it applies to any uniformly regular Riemannian manifold. Closely related to uniformly regular Riemannian manifolds are ‘singular Riemannian manifolds’ which are characterized by a ‘singularity function’  2   C1 M; .0; 1/ . More precisely, let M D .M; g/ be a Riemannian manifold and consider the conformal metric gO WD g=2 on M. Then the basic requirement for M to be a singular Riemannian manifold is that MO WD .M; g/ O be a uniformly regular Riemannian manifold. In Examples 4.1 we present some important instances of singular Riemannian manifolds, most notably the class of m-dimensional Riemannian submanifolds of Rn with finitely many cuspidal singularities. By considering parabolic equations on singular Riemannian manifolds we are naturally led to study degenerate parabolic equations in weighted Sobolev spaces. To be more precise, we now assume  that M D .M; g/ is a singular Riemannian manifold and  2 C1 M; .0; 1/ is a singularity function for it. Then A is said to be a -regular uniformly -elliptic differential operator if 2 a is symmetric, uniformly positive definite, and 2 a and 1 ra are bounded and continuous. Note that this means that A is no longer uniformly strongly elliptic but that the ellipticity condition degenerates if  tends to zero (or to infinity). For  2 R and k 2 N we define the weighted Sobolev space Wpk; .MI / to be the completion of D.M/ in L1;loc .M/ with respect to the norm u 7!

k 1=p X  Cj j   jr uj 0 p : gj Lp .M/ jD0

Then ˚  Wp0; .MI / D Lp .MI / WD u 2 Lp;loc .M/ I  u 2 Lp .M/ : If p ¤ 2, then   Wp22=p; .MI / WD Lp .MI /; Wp2; .MI / 11=p;p : 22=p;

2; .MI / and Wp;B Furthermore, Wp;B 22=p Wp;B .M/,

2 .MI / are defined analogously to Wp;B .M/

resp. Lastly, Wp2 .MI / WD Wp2;0 .MI /, etc. Note that L0p .MI / D and Lp .M/. Using this we can now formulate our main result for degenerate parabolic equations in the present setting. Theorem 1.2 Let M be a singular Riemannian manifold,  a singularity function for it, and p … f3=2; 3g. Suppose A is -regular and -uniformly strongly elliptic. Then (2) has for each   .f ; u0 / 2 Lp J; Lp .M/  Wp22=p;2=p .MI /

Parabolic Equations on Uniformly Regular Riemannian Manifolds and. . .

47

a unique solution     2 u 2 Lp J; Wp;B .MI / \ Wp1 J; Lp .M/ : The map .f ; u0 / 7! u is linear and continuous. 2 Equivalently: Set A WD AjWp;B .MI /. Then A generates a strongly continuous analytic semigroup on Lp .M/ and has the property of maximal regularity. This is a particular instance of Theorem 5.2 and its corollary, both of which apply to general weighted spaces, that is, to  ¤ 0 as well. We should like to point out that we impose minimal regularity requirements on a (within the framework of continuous coefficients). This allows to use Theorems 1.1 and 1.2 (and the more general results below) as a basis for the study of quasilinear equations along well-established lines (e.g., [1, 3]). For the sake of brevity we do not give details in this paper. It should also be noted that only the behavior of   near zero and infinity is of importance. In other words, if Q 2 C1 M; .0; 1/ satisfies Q  , that is, =c  Q  c for some c  1, then Theorem 1.2 remains valid with  replaced by . Q In 2 2 particular, Wp;B .MI / Q equals Wp;B .MI / except for equivalent norms. Now we illustrate the strength of our results by means of relatively simple examples. For this we assume that  is a smooth open subset of Rm with a compact N is a smooth m-dimensional submanifold of Rm . We also smooth boundary, that is,  assume that  is a finite family of compact connected smooth submanifolds  of Rm without boundary and dimension `  m  1 such that the following applies: .i/

if ` D m  1 and  \ @ ¤ ;, then   @:

.ii/

if 1  `  m  2, then   :

S N n f  I  2  g, endowed with the Euclidean metric, is Then M WD  an m-dimensional Riemannian submanifold of Rm whose boundary @M equals S @ n f  I  2  g. For each  2  and x 2 M we denote by ı .x/ the (Euclidean) distance from x to . Then ı is, sufficiently close to , a well-defined strictly positive smooth function. If M contains a neighborhood of infinity in Rm , that is, if  is an exterior domain, then we put ı1 .x/ WD jxj with the Euclidean norm jj in Rm . We also fix ˛  1 and ˛1 2 .1; 0/. Then we choose a function  2 C1 M; .0; 1/ ˛1 near infinitySif  is an exterior satisfying   ı˛ near  2 ,   ı1 domain, and   1 away from the ‘singularity set’ S.M/ WD f  I  2  g and infinity. Then M is a singular Riemannian manifold characterized by the singularity function . Indeed, see Examples 4.1, each  2  is an .˛ ; ` /-wedge and f x 2 M I jxj > R g is for sufficiently large R > 1 diffeomorphic to an infinite ˛1 -cusp (over Sm1 in RmC1 ) if  is an exterior domain. Thus Theorem 1.2 applies to this situation.

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Next we consider some particularly simple subcases which have been treated before in the literature. (a) Suppose  is bounded and S.M/ D @. Thus   ı ˛ for some ˛  1, where ı is the distance to @. In this situation it is shown by Vespri [34] that A generates an analytic semigroup on Lp ./ D Lp .M/. Recently, Fornaro et al. [19] have given a new proof for this generation theorem. (b) Let  be bounded and ` D 0 for each  2 . Then S.M/ consists of finitely many one-point sets fx0 g; : : : ; fxk g lying either in  or on @. We set ıj .x/ WD  Sk N jxxj j for 0  j  k and x 2 M D  ˛j  1 for 0  j  k. jD0 fxj g. Assume   Then Theorem 1.2 implies that, given any  2 C1 M; .0; 1/ satisfying   ˛ 2 ıj j near xj and   1 otherwise, A D AjWp;B .MI / generates a strongly continuous analytic semigroup on Lp .M/ D Lp ./ and has the property of maximal regularity. The only paper known to the author treating the problem of semigroup generation by parabolic equations with strong degeneracies at isolated points is the recent publication of Fragnelli et al. [20]. These authors consider the case where  D .0; 1/ and S.M/ D fx0 g   and show that A generates an analytic semigroup on L2 ./. In none of the above papers it is shown that the maximal regularity property prevails. Furthermore, the proofs given there depend significantly on the fact that second order equations are being considered. In contrast, our approach does not depend on the particular structure of the problem but applies equally well to systems and higher order equations (cf. Amann [7]). Observe that the preceding examples show that a given Riemannian manifold can possess uncountably many non-equivalent singular structures. This is related to and sheds new light on the non-uniqueness results observed by Pozio et al. [30]. Thus, besides being rather general and widely applicable, our approach to highly degenerate parabolic problems via Riemannian manifolds leads to a deeper understanding of such problems as well. In the next section we give the precise definition of a uniformly regular Riemannian manifold. Then we formulate our main result, Theorem 3.1, in the setting of second order equations and trace it back to the much more general propositions in [7]. Note that, besides allowing lower order terms, we prove an optimal regularity theorem in the presence of nonhomogeneous boundary conditions. In addition, we show that we get classical solutions if we impose slightly stronger regularity assumptions on the data. Singular Riemannian manifolds are precisely defined in Sect. 4 and basic examples are presented. Furthermore, weighted function spaces are introduced and their interrelation with non-weighted Sobolev-Slobodeckii spaces on uniformly regular manifolds is established. Section 5 contains our main theorem for second order degenerate parabolic problems involving lower order terms and nonhomogeneous boundary conditions. We attract the reader’s attention to Theorem 5.2 where it is shown that problems with

Parabolic Equations on Uniformly Regular Riemannian Manifolds and. . .

49

homogeneous boundary conditions give rise to generators of analytic semigroups possessing the property of maximal regularity in general weighted spaces Lp .MI / for any  2 R. This generalizes results by Barbu et al. [15], for example, where the case M D , with  a bounded domain on Rm , S.M/ D @, and  D 1 is considered (see also [20]). For the reader’s convenience there is included an Appendix in which some basic facts on tensor bundles over Riemannian manifolds are listed.

2 Function Spaces and Uniformly Regular Manifolds By a manifold we always mean a smooth, that is, C1 manifold with (possibly empty) boundary such that its underlying topological space is separable and metrizable. Thus, in the context of manifolds, we work in the smooth category. A manifold does not need to be connected, but all connected components are of the same dimension. Let M D .M; g/ be a Riemannian manifold with boundary @M and volume measure dv. The metric g on TM gives rise to a vector bundle metric on the tensor bundle V WD T M for ;  2 N, which we denote by g (see the Appendix for more details). In particular, g01 D g and g10 D g , the adjoint (or contravariant) metric on the cotangent bundle T  M. For k 2 N the vector space of all . ; /-tensor fields of class Ck , that is, of all k C sections of V , is denoted by Ck .V /. The Levi-Civita covariant derivative, r,

satisfies r k a 2 C.VCk / for a 2 Ck .V /, where r 0 a WD a. We write D.V / for the space of all smooth sections with compact support in M (which may meet the boundary). As usual, Ck .M/ stands for Ck .V00 /, etc. We fix and  and set V WD V . Then, given k 2 N, we denote by Wpk .V/ the Sobolev space of order k, defined to be the completion of D.V/ in L1;loc .V/ D L1;loc .V; dv/ with respect to the norm u 7!

k X  j  jr uj jD0

 Cj g

p 

Lp .V/

1=p

:

Thus Wp0 .V/ D Lp .V/. We also need fractional order Sobolev spaces, namely the Slobodeckii spaces Wps .V/, for s 2 RC n N. If k < s < k C 1 with k 2 N, then   Wps .V/ WD Wpk .V/; WpkC1 .V/ sk;p ;

(7)

which is the interpolation space between Wpk .V/ and WpkC1 .V/ obtained by means of the real interpolation method with exponent s  k and integrability parameter p. We denote by B.V/ the space ofall  bounded sections of V. It is a Banach space with the norm u 7! kuk1 WD  juj 1 , where kk1 is the maximum norm.

50

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Moreover, BC.V/ WD B.V/ \ C.V/ is a closed linear subspace thereof. For k 2 N we write BCk .V/ for the linear subspace of Ck .V/ consisting of all u satisfying

r j u 2 B.VCj / for 0  j  k. It is a Banach space with the obvious norm. Moreover, T 1 BC .V/ WD k BCk .V/ and bck .V/ is the closure of BC1 .V/ in BCk .V/. Now we define Besov-Hölder spaces Bs1 .V/ for s > 0 by 8  < bck .V/; bckC1 .V/ sk;1 ; Bs1 .V/ WD  : bck .V/; bckC2 .V/ ; 1=2;1

k < s < k C 1; s D k C 1;

(8)

where k 2 N. Besides these isotropic spaces we also need anisotropic versions adapted to parabolic problems. Anisotropic Sobolev-Slobodeckii spaces are introduced for s 2 RC D Œ0; 1/ by     Wp.s;s=2/ .V  J/ WD Lp J; Wps .V/ \ Wps=2 J; Lp .V/ : The second space on the right is a standard Sobolev-Slobodeckii space of Banach ˚ Of course, Wp.0;0/ .V  J/ is naturally identified with space valued distributions onJ.  Lp .V  J/ D Lp V  J; dvdt . Analogously, we define anisotropic Besov-Hölder spaces for s > 0 by     B.s;s=2/ .V  J/ WD B J; Bs1 .V/ \ Bs=2 1 1 J; B.V/ :   Here the second space on the right is a standard Hölder space Cs=2 J; B.V/ if s … 2N, and a Zygmund space for s 2 2N, of Banach space valued functions on J (see Lunardi [28], for example). For k 2 N WD N n f0g we put     BC.k;k=2/ .V  J/ WD C J; BCk .V/ \ Ck=2 J; B.V/ ; recalling that J is compact. Although Sobolev-Slobodeckii spaces, respectively Besov-Hölder spaces, are well-defined for each s 2 RC , respectively s > 0, they are not too useful on general Riemannian manifolds since, for example, the fundamental Sobolev type embedding theorems may not hold in general. Even more importantly, there may be no characterization by local coordinates. For this reason we restrict ourselves to the class of uniformly regular Riemannian manifolds. Loosely speaking, M is a uniformly regular Riemannian manifold if its differentiable structure is induced by an atlas K of finite multiplicity whose coordinate patches are all of comparable size, such that K can be uniformly shrunk to an atlas for M, and the family of all charts in K which intersect @M induces an atlas of the same type for @M. In q q particular, @M D .@M; g/, where g is the Riemannian metric induced by g on @M, is a uniformly regular .m  1/-dimensional Riemannian manifold if M is uniformly regular (see Example 4.1(b)).

Parabolic Equations on Uniformly Regular Riemannian Manifolds and. . .

51

For the precise definition of a uniformly regular Riemannian manifold we introduce some notation and conventions. By c we denote constants  1 whose numerical value may vary from occurrence to occurrence; but c is always independent of the free variables in a given formula, unless a dependence is explicitly indicated. We denote by Hm the closed right half-space RC  Rm1 in Rm , where R0 D f0g. The Euclidean metric on Rm , .dx1 /2 C    C .dxm /2 , is denoted by gm . The same symbol is used for its restriction to an open subset U of Rm or Hm , that is, for  gm , where  W U ,! Rm is the natural embedding. Here and below, we employ standard definitions of pull-back and push-forward operations. On the space of all nonnegative functions, defined on some nonempty set whose specific form will be clear in any given situation, we introduce an equivalence relation  by setting f  g iff there exists c  1 such that f =c  g  cf . Inequalities between vector bundle metrics have to be understood in the sense of quadratic forms. By 1 we denote the constant function s 7! 1, whose domain will always be clear from the context. We set Q WD .1; 1/  R. If  is a local chart for an m-dimensional manifold M, then we write U for the corresponding coordinate patch dom./. A local chart  ˚ the interior of M, whereas is normalized (at q) if .U / D Qm whenever U  M, m m .U / D Q \ H if U \ @M ¤ ; (and .q/ D 0). We put Qm  WD .U / if  is normalized. An atlas K for M has finite multiplicity if there exists k 2 N such that any intersection of more than k coordinate patches is empty. In this case N./ WD f Q 2 K I UQ \ U ¤ ; g has cardinality  k for each  2 K. An atlas is uniformly shrinkable it consists of ˚ if 1 normalized charts and there exists r 2 .0; 1/ such that the family  .rQm /I  2  K is a cover of M. We put KS WD f  2 K I U \ S ¤ ; g for any nonempty subset S of M. Given an open subset X of Rm or Hm and a Banach space X , we write kkk;1 for the usual norm of BCk .X; X /, the Banach space of all u 2 Ck .X; X / such that j@˛ ujX is uniformly bounded for ˛ 2 Nm of length at most k. An atlas K for M is uniformly regular if .i/

K is uniformly shrinkable and has finite multiplicity.

.ii/

kQ ı  1 kk;1  c.k/; ; Q 2 K; k 2 N:

(9)

In (ii) and in similar situations it is understood that only ; Q 2 K with U \ UQ ¤ ; Q are being considered. Two uniformly regular atlases K and KQ are equivalent, K K, if .i/

cardf Q 2 KQ I UQ \ U ¤ ; g  c;  2 KI

.ii/

Q k 2 N: kQ ı  1 kk;1 C k ı Q1 kk;1  c.k/;  2 K; Q 2 K;

(10)

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H. Amann

A uniformly regular structure is a maximal family of equivalent uniformly regular atlases. A uniformly regular manifold is a manifold endowed with a uniformly regular structure. Clearly, on such a manifold all local charts, atlases, etc. under consideration belong to its uniformly regular structure. An m-dimensional Riemannian manifold .M; g/ is a uniformly regular Riemannian manifold if .i/

M is uniformly regularI

.ii/

 g  gm ;  2 KI

.iii/

k gkk;1  c.k/;  2 K; k 2 N;

(11)

for some uniformly regular atlas K for M. Let M be a uniformly regular Riemannian manifold. Then the SobolevSlobodeckii and Besov-Hölder space scales possess all the properties known to hold in the case of the m-dimensional Euclidean space or half-space. In other words, there .s;s=2/ are embedding, interpolation, and trace theorems for Wps .M/ and Wp .M  J/ which are completely analogous to the corresponding theorems for the classical Sobolev-Slobodeckii spaces. In particular, the anisotropic Sobolev-Morrey type embedding theorem .t;t=2/ Wp.s;s=2/ .V  J/ ,! B1 .V  J/;

s C .m C 2/=p > t > 0;

(12)

is valid. In addition, Wps .V/ and Bs1 .V/ can be characterized by means of local coordinates, similarly as in the case of compact manifolds. The spaces BCk .V/ and BC.k;k=2/ .V/ do not belong to either one of these scales. However, they can be arbitrarily well approximated by Besov-Hölder spaces. In fact, given k 2 N , 1 ;s1 =2/ 0 ;s0 =2/ .V  J/ ,! BC.k;k=2/ .V  J/ ,! B.s .V  J/ B.s 1 1

(13)

for 0 < s0 < k < s1 . Note that this implies a corresponding assertion for the isotropic spaces BCk .V/ and Bs1 .V/, since BCk .V/ is naturally identified with the closed linear subspace of BC.k;k=2/ .V/ of all ‘time-independent’ functions therein, etc. Proofs, further results, references to related research, and many more details—in particular spaces of sections of general uniformly regular vector bundles over M— are found in the earlier work [5, 6] of the author (also see [7], [8], as well as [4]).

Parabolic Equations on Uniformly Regular Riemannian Manifolds and. . .

53

3 Parabolic Problems on Uniformly Regular Riemannian Manifolds Let M D .M; g/ be a uniformly regular Riemannian manifold. We consider parabolic initial boundary value problems of the form @u C Au D f on MT ;

Bu D h on @MT ;

u D u0 on M0 :

(14)

In order to reduce the technical apparatus to a minimum we restrict ourselves tothe important  class of second order divergence form problems. Thus we fix ı 2 C @M; f0; 1g and set @j M WD ı 1 .j/ for j 2 f0; 1g. Then @M D @0 M [ @1 M and @0 M \ @1 M D ;. We assume that .A; B/ is of the form Au WD  div.a q grad u/ C .aE j grad u/ C a0 u

(15)

and ( Bu WD

B0 u

on @0 MT ;

B1 u

on @1 MT ;

where B0 u WD  u;

 ˇ  B1 u WD ˇ .a q grad u/ C b0  u:

Here . j / D . j /g WD g.; /, is the (inward pointing) unit normal on @M,  the trace map for @M, and q denotes complete contraction (see the Appendix). More precisely, B0 u D . u/j@0 M, etc. We suppose a 2 C1 .T11 M  J/, aE is a time-dependent vector field, a0 a function on MT , and b0 one on @1 MT . In local coordinates,  1 p Au D  p @i g aij gjk @k u C ai @i u C a0 u: g Hence B is the Dirichlet boundary operator on @0 M and the Neumann or a Robin boundary operator on @1 M. Note that either @0 M or @1 M may be empty. We also allow M to be a manifold without boundary. In this case it is understood throughout the whole paper that all statements, assumptions, and formulas referring explicitly or implicitly to @M are to be unconsidered. For example, problem (14) reduces to the Cauchy problem @u C Au D f on MT ; if @M D ;.

u D u0 on M0

54

H. Amann .2;1/

A function u satisfying (14) is a strong Lp solution if it belongs to Wp .MT /, and a classical solution if it is a member of BC.2;1/ .MT /. The differential operator A is uniformly strongly elliptic on MT if a.; t/ is symmetric and uniformly positive definite, uniformly with respect to t 2 J. Clearly, the latter means that there exists a constant " > 0 such that ˇ   a.q; t/ q X ˇ X g.q/  " jXj2g.q/;

X 2 Tq M;

q 2 M;

t 2 J:

For a concise formulation of the main result we introduce for s  0 the boundary data spaces Wp.sC2•1=p/.1;1=2/ .@MT / WD Wp.sC21=p/.1;1=2/ .@0 MT /  Wp.sC11=p/.1;1=2/ .@1 MT /; whose general point is written h D .h0 ; h1 /. Obvious interpretations apply if either @0 M or @1 M is empty. The total data spaces are then Wp.sC2;.sC2/=2/.MT / WD Wp.s;s=2/ .MT /  Wp.sC2•1=p/.1;1=2/ .@MT /  WpsC22=p .M0 / for s  0. Given Banach spaces E and F, we denote by L.E; F/ the Banach space of bounded linear operators from E into F. We write Lis.E; F/ for the subset of all bijections in L.E; F/. Banach’s homomorphism theorem guarantees that A1 2 L.F; E/ if A 2 Lis.E; F/. Now we can formulate the main existence and uniqueness theorem for problem (14). Theorem 3.1 Let M be a uniformly regular Riemannian manifold and let p … f3=2; 3g. Suppose a 2 BC.1;1=2/ .T11 M  J/; a0 2 L1 .MT /;

aE 2 L1 .TM  J/;

b0 2 BC

.1;1=2/

(16)

.@1 MT /;

.2;1/

and A is uniformly strongly elliptic. Denote by Wp;cc .MT / the vector space of all .2;1/ .f ; h; u0 / 2 Wp .MT / satisfying the compatibility conditions of order zero:  u0 D h0 .; 0/ on @0 M B.; 0/u0 D h.; 0/ on @M

if 3=2 < p < 3; if p > 3:

(17)

Parabolic Equations on Uniformly Regular Riemannian Manifolds and. . . .2;1/

.2;1/

Then Wp;cc .MT / is closed in Wp

55

.MT / and

  .2;1/ .@ C A; B; 0 / 2 Lis Wp.2;1/ .MT /; Wp;cc .MT / :

(18)

Supplement Suppose 0 < s < s < 1 C 3=p with s ¤ 3=p and .T11 M  J/; a 2 B.1Cs/.1;1=2/ 1 a0 2

B.s;s=2/ .MT /; 1

b0 2

aE 2 B.s;s=2/ .TM  J/; 1

(19)

B.1Cs/.1;1=2/ .@1 MT /: 1

.sC2/.1;1=2/

.sC2/.1;1=2/

Let Wp;cc .MT / be the linear subspace of Wp .MT / of all .f ; h; u0 / satisfying, in addition to (17), the first order compatibility condition @h0 .; 0/ C  A.; 0/u0 D  f .; 0/ on @0 M .sC2/.1;1=2/

Then Wp;cc

if s > 2=p:

(20)

.MT / is closed and

  .sC2/.1;1=2/ .@ C A; B; 0 / 2 Lis Wp.sC2/.1;1=2/ .MT /; Wp;cc .MT / : ] E /, using the notations of the Appendix. Proof We set a2 WD a] and a1 WD adiv.a Then we get from (75)

A D a2 q r 2 C a1 q r C a0 :

(21)

We let [ be the unit conormal vector field g[ on @M and set b1 WD [ q  a] . Then B1 D b1 q  r C b0 :

(22)

By means of the characterization of BCk .T  M/ by local coordinates referred to in the preceding section one verifies [ 2 BC1 .T  M/:

(23)

Let (16) be satisfied. Then it is obvious that a2 2 BC.1;1=2/ .T02 M  J/;

a1 2 L1 .TM  J/;

Furthermore, (23) implies   b1 2 BC.1;1=2/ .TM/j@1 M  J :

a0 2 L1 .MT /:

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If (19) applies, then a2 2 B.1Cs/.1;1=2/ .T02 M  J/; 1

a1 2 B.s;s=2/ .TM  J/; 1

a0 2 B.s;s=2/ .MT /; 1 and, once more by (23) and the point-wise multiplier result [5, Theorem 14.3],   .TM/j@1 M  J : b1 2 B.1Cs/.1;1=2/ 1 This shows that .A; B/ satisfies in either case the regularity assumptions of the main theorem of [7]. Since the uniform strong ellipticity of A implies that .@ C A; B/ is a uniformly strongly parabolic boundary value problem the assertion is a very particular consequence of the latter theorem. t u Corollary 3.2 Let (16) be satisfied. Then the initial boundary value problem (14) .2;1/ has for each .f ; h; u0 / 2 Wp;cc .MT / a unique strong Lp solution u on MT . Suppose .m C 2/=p < s < 1 C 3=p with s ¤ 3=p, (19) applies, and .f ; h; u0 / 2 .sC2/.1;1=2/ Wp;cc .MT /, then u is a classical solution. t u

Proof The first assertion is clear and the second one follows from (12). Remarks 3.3

(a) If ı D 0 (Dirichlet boundary value problem), then p D 3 is admissible as well. If ı D 1 (Neumann or Robin boundary conditions), then p D 3=2 can be admitted also. Similarly, (20) is vacuous if ı D 1. (b) If all data are smooth and the compatibility conditions of all orders are satisfied, then u is a smooth solution on MT . (c) We refer to [7] for higher order problems and operators acting on sections of general uniformly regular vector bundles over M. (d) Theorem 3.1 is the basis for establishing results on the existence, uniqueness, and continuous dependence on the data of solutions of quasilinear parabolic problems of the form @u C A.u/u D F.u/ on MT ;

B.u/u D H.u/ on @MT ;

u D u0 on M0 :

Such results are obtained by (more or less obvious) modifications of the proofs in [3]. This is to be carried out somewhere else. t u Of course, Theorem 3.1 applies in particular to autonomous problems. To simplify the presentation we restrict ourselves to the setting of strong Lp solutions. Then (16) reduces to a 2 BC1 .T11 M/; a0 2 L1 .M/;

aE 2 L1 .TM/;   b0 2 BC1 .TM/j@1 M :

(24)

Of particular importance is the case of homogeneous boundary value problems.

Parabolic Equations on Uniformly Regular Riemannian Manifolds and. . .

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  Theorem 3.1 guarantees A 2 L Wp2 .M/; Lp .M/ . Hence A, the restriction of A  2  2 to Wp;B .M/, is a well-defined element of L Wp;B .M/; Lp .M/ . Moreover, A is closed ˚ is a subset of W 2 .M/ (cf. [1, Lemma I.1.1.2]) and densely defined (since D.M/ p;B ˚ is dense in Lp .M/). By means of A we can reformulate the autonomous and D.M/ homogeneous initial boundary value problem (2) as the evolution equation (6). This is made precise by the next theorem for which we rely on semigroup theory and maximal regularity (see Amann [1, Chap. III] and [2], Denk et al. [17], or Kunstmann et al. [26], for example, for information on these concepts). Theorem 3.4 Let M be a uniformly regular Riemannian manifold and let p … f3=2; 3g. Suppose A is autonomous, uniformly strongly elliptic, and conditions (24) are satisfied. Then A generates a strongly continuous analytic semigroup on Lp .M/ and has the property of maximal regularity, that is to say, .@ C A; 0 / belongs to       22=p 2 Lis Lp J; Wp;B .M/ \ Wp1 J; Lp .M/ ; Lp .MT /  Wp;B .M/ : .2;1/

(25)

22=p

Proof In the present setting Wp;cc .MT / D Lp .MT /  Wp;B .M/. Hence (25) is a reformulation of (18). Now the semigroup assertion follows from a result of Dore [18]. t u To indicate the power of these theorems we need to know examples of uniformly regular Riemannian manifolds. This problem is dealt with in [9] where proofs for the following claims are found. Examples 3.5 (a) Every compact Riemannian manifold is a uniformly regular Riemannian manifold. (b) An m-dimensional Riemannian submanifold of Rm possessing a compact boundary is a uniformly regular Riemannian manifold. (c) Rm D .Rm ; gm / and Hm D .Hm ; gm / are uniformly regular Riemannian manifolds. Q g/ Q g/ (d) Let MQ D .M; Q be a Riemannian manifold and ' W .M; g/ ! .M; Q an isometry. Then M is a uniformly regular Riemannian manifold iff MQ is one. (e) A Riemannian manifold has bounded geometry if it has no boundary, a positive injectivity radius, and all covariant derivatives of the curvature tensor are bounded. Every complete Riemannian manifold with bounded geometry is a uniformly regular Riemannian manifold. (f) Suppose S  U  M, where S is closed and U is open in M. An atlas K for U is uniformly regular on S if (9) holds with K replaced by KS . Two uniformly regular atlases K and KQ for U on S are equivalent if (10) applies to KS and KQ S . This defines a uniformly regular structure for U on S. Then U is uniformly regular on S if it is endowed with a uniformly regular structure on S. Lastly, U is a uniformly regular Riemannian manifold on S if (11) is satisfied for U and KS , where K is a uniformly regular atlas for U on S.

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Let Sj  Uj  M and suppose Uj is a uniformly regular Riemannian manifold on Sj for 0  j  `. Let Kj be a uniformly regular atlas for Uj on Sj . Assume (˛) ki ı j1 kk;1  c.k/, .i ; j / 2 Ki;Si  Kj;Sj , 0  i; j  `, k 2 N; (ˇ) M D S0 [    [ S` . Then K WD K0 [    [ K` is a uniformly regular atlas for M and M is a uniformly regular Riemannian manifold. It is obtained by patching together the uniformly regular pieces Uj on Sj . (g) Assume d  m and B is an .m1/-dimensional compact submanifold of Rd1 . For a nonempty subinterval I of .1; 1/ and 0  ˛  1 we set ˚  F˛ .I; B/ WD .t; t˛ y/ I t 2 I; y 2 B  R  Rd1 D Rd ;   Then F˛ .B/ WD F˛ .1; 1/; B , endowed with the Riemannian metric induced by Rd , that is, by gd , is an m-dimensional Riemannian submanifold of Rd with boundary F˛ .@B/, where F˛ .;/ WD ;. It is called ˛-funnel in Rd . Note that a 0-funnel is a cylinder and a 1-funnel a (blunt) cone over  (the basis)  B. Let F D F˛ .B/ be an ˛-funnel in Rd and set S WD F˛ Œ2; 1/; B . Then F is an m-dimensional uniformly regular Riemannian manifold on S. (h) Suppose U is open in M and F D F˛ .B/ an m-dimensional ˛-funnel in Rd . Set F.I/ WD F  ˛ .I; B/. Assume ' W U ! F is a diffeomorphism such that S WD ' 1 FŒ2; 1/ satisfies   (˛) . U n S / \ S D ' 1 F.f2g/ ; (ˇ) ' .g jS/  gF jFŒ2; 1/. Then U is a uniformly regular Riemannian manifold on S, and M is said to have an .˛; B/-funnel-like end in U with representation '. (i) Let U0 ; : : : ; U` be open in M. Suppose (˛) Ui \ Uj D ;; 1  i < j  `; (ˇ) M has an .˛j ; Bj /-funnel-like end in Uj with representation 'j for j  1; () 'j .U0 \ Uj / D Fj .1; 4/, j  1;   S (ı) S0 WD U0 n `jD1 'j1 Fj .3; 1/ is compact. Then M is a uniformly regular Riemannian manifold, a Riemannian manifold with finitely many funnel-like ends. It is obtained by patching together the uniformly regular pieces Uj on Sj for 0  j  `. t u The most elementary situation in which Theorem 3.1 applies is the case in which M is compact. If, notably, M is the closure of a smooth bounded open subset of Rm , then our theorem reduces essentially to a well-known classical result (e.g., [27]). More recently, Grubb [22] has established a general Lp theory for parabolic pseudo-differential boundary value problems acting on sections of vector bundles (also see Sect. IV.4.1 in [23]). It applies to a class of noncompact manifolds, called ‘admissible’ and being introduced in Grubb and Kokholm [24]. It is a subclass of the above family of manifolds with funnel-like ends, namely a family of manifolds with conical ends. Of course, aside from the requirements on the manifold, differential

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boundary value problems of the form considered in the present paper constitute a very particular subcase of Grubb’s general class. However, in order to apply the results of Grubb [22] to (14) we have to require that .A; B/ has C1 coefficients. In contrast, we impose in essence minimal regularity assumptions on .A; B/. This is important for the study of quasilinear equations on the basis of the linear theorems proved here. Now we suppose that M is a noncompact uniformly regular Riemannian manifold not belonging to the class of manifolds with funnel-like ends. This is the case, in particular, if M has no boundary, is complete, and has bounded geometry. There is a tremendous amount of literature on heat equations for such manifolds, most of which is an L2 theory and is concerned with kernel estimates and spectral theory (see, for example, Davies [16] or Grigor’yan [21] and the references therein). There are a few papers dealing with (semilinear) parabolic equations on noncompact complete Riemannian manifolds under various curvature assumptions which are based on heat kernel estimates (e.g., Zhang [35, 36], Mazzucato and Nistor [29], Punzo [31, 32], Bandle et al. [14]). In all these papers either the top-order part is the Laplace-Beltrami operator or smooth leading order coefficients are required. Except for a recent paper by Shao and Simonett [33], the author is not aware of any result on parabolic equations on noncompact manifolds which do not rely on heat kernel techniques, leave alone noncompact manifolds with noncompact boundary. In [33] the authors, building on [5] and [6], establish a Hölder space existence theorem for autonomous nonlinear parabolic equations on uniformly regular manifolds without boundary. As an application they show that the solutions of the Yamabe flow instantaneously regularize and become real analytic in space and time. A prototypical example to which our results apply is furnished by an mdimensional Riemannian submanifold MH D .MH ; gH / of the hyperbolic space Hm represented by the Poincaré model. More specifically, we denote by Bm the open unit ball in Rm with closure BN m and boundary Sm1 , the .m  1/-sphere. Then H D Hm D .Bm ; gH /, where gH D 4gm =.1  jxj2 /2 for x 2 Bm . If @MH is not compact, then we assume that its closure in BN m intersects Sm1 transversally and that this intersection is the boundary of an .m1/-dimensional Riemannian submanifold of Sm1 . Informally expressed this means, in particular, that MH ‘does not collapse at infinity’. Writing divH for divgH , etc., problem (14) is on MH  J given by AH u WD  divH .a q gradH u/ C .aE j gradH u/H C a0 u and ( BH u WD

u ˇ   @MH ˇ .a q gradH u/ H

on

@0 MH ;

on

@1 MH :

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Using the fact that gH is conformal to gm we can express AH and BH in terms of gm , that is, as differential operators on M WD .MH ; gm /. In fact, writing div D divgm , etc., we find with .x/ WD .1  jxj2 /=2 divH .a q gradH u/ D m @i .2m aij ı jk @k u/ D m div.2m a q grad u/ D div.2 a q grad u/  m.a q grad  j grad u/

(26) (27)

and .aE j gradH u/H D .aj E grad u/. Moreover, @MH D  and, consequently, 

ˇ   ˇ  H ˇ .a q gradH u/ H D  ˇ .a q grad u/ :

This shows that the initial boundary value problem @t u C AH u D f on MH  J;

BH u D h on @MH  J;

u D u0 on MH  f0g

can be seen as a degenerate initial boundary value problem on the ‘underlying’ Euclidean manifold M. Note that M is not a uniformly regular Riemannian manifold, even if @M D ;, that is, M D Bm , since it cannot be covered by an atlas K whose coordinate patches are uniformly comparable in size and such that a uniform shrinking of K is still an atlas.

4 Singular Riemannian Manifolds and Weighted Function Spaces Generalizing the preceding example we are led to the concept of singular Riemannian manifolds. Informally  speaking,  such a manifold is characterized by a singularity function  2 C1 M; .0; 1/ such that the conformal metric gO WD g=2 gives rise to a uniformly regular Riemannian manifold MO WD .M; g/. O To be precise: Let M be an m-dimensional uniformly regular manifold. A pair .; K/ is a   singularity datum for M if  2 C1 M; .0; 1/ and K is a uniformly regular atlas such that .i/

k kk;1  c.k/ ;  2 K; k 2 N;   where  WD  .0/ D   1 .0/ I

.ii/

 jU   ;  2 K:

(28)

Q are equivalent, .; K/ .; Q if   Q and Two singularity data .; K/ and .; Q K/ Q K/, Q K K. A singularity structure, S.M/, for M is a maximal  family of equivalent singularity data. A singularity function for M is a  2 C1 M; .0; 1/ such that there exists

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61

an atlas K with .; K/ 2 S.M/. The set of all singularity functions is the singularity type of M. It is convenient to denote it by ŒŒ, where  is one of its representatives. A singular Riemannian manifold of type ŒŒ is a Riemannian manifold .M; g/ such that .i/

M is uniformly regular and endowed with a singularity structure S.M/ of singularity type ŒŒI

.ii/

.M; g=2 / is a uniformly regular Riemannian manifold:

(29)

This definition is independent of the particular choice of  in the following sense: Q .; K/. Then it follows from (11)(ii), (iii) and (28) that .M; g=Q2 / is Let .; Q K/ a uniformly regular Riemannian manifold and g=Q2  g=2 . In [7] it is shown that (28)(i) is equivalent to O d log  2 BC1 .T  M/:

(30)

In [9] there is carried out a detailed study of singular Riemannian manifolds. We refer the reader to that paper for proofs of the following examples. Examples 4.1 (a) A uniformly regular Riemannian manifold is singular of type ŒŒ1, and conversely. (b) Let  be a union of connected components of @M and m  2. We endow  q q q with the induced Riemannian metric g WD   g, where  W  ,! M is the natural embedding. Let K be a uniformly regular atlas for M. For  2 K we set Uq WD q q m1 q @U WD U \ @M D U \  and  WD with 0 W f0g  q 0 ı .q / W U ! R m1 m1 0 0 R ! R , .0; x / 7! x . Then K WD f  I  2 K g is a uniformly regular atlas for , the one induced by K. q Suppose .M; g/ is a qsingular Riemannian manifold of type ŒŒ. We set  WD q q    D  j. q Then .; K/ is a singularity datum for . Thus it defines aq singular q structure S./ for , the one induced by S.M/. Furthermore, .; g=2 / is a q singular Riemannian manifold of type ŒŒ (and dimension m  1). It is always understood that  is endowed with the singular structure induced by the one of M. Q g/ (c) Let .M; Q be a Riemannian manifold and f W M ! MQ an isometric diffeomorphism, that is, gQ D f g. Suppose .M; g/ is singular of type ŒŒ with singularity datum .; K/. Set f K WD f f  I  2 K g. Then the pair .f ; f K/ is a singularity Q g/ datum for .M; Q and the latter is a singular Riemannian manifold of type ŒŒ f . (d) Suppose S  U  M, where S is closed and U is open in M. Assume  2 C1 U; .0; 1/ and K is a uniformly regular atlas for U on S such that (28) holds for KS . Then .; K/ is a singularity structure for U on S. Two such Q are equivalent on S if   Q and K and KQ singularity structures .; K/ and .; Q K/ are equivalent on S. This defines a singularity structure for U on S of type ŒŒ. Then U is a singular Riemannian manifold on S of type ŒŒ if it is endowed

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with a singularity structure on S of type ŒŒ and .U; g=2/ is a uniformly regular Riemannian manifold on S. Assume Sj  Uj  M and Uj is a uniformly regular Riemannian manifold on Sj of type ŒŒj  for 0  j  `. Let .j ; Kj / be a singularity structure for Uj on Sj and assume that (˛) and (ˇ) of Example 3.5(f) apply and i j.Si \ Sj /  j j.Si \ Sj /;

0  i < j  `:

  Then there exists  2 C1 M; .0; 1/ such that  jSj  j jSj for 0  j  ` and .; K/ is a singularity datum for M, where K D K0 [    [ K` . Furthermore, M is a singular Riemannian manifold of type ŒŒ. It is said to be obtained by patching together the singular Riemannian manifolds Uj on Sj of type ŒŒj . (e) Let d  2 and suppose B is a b-dimensional compact Riemannian submanifold of Rd1 . For a nonempty subinterval I of .0; 1/ and ˛  1 we set ˚  C˛d .I; B/ WD .t; t˛ y/ I t 2 I; y 2 B  R  Rd1 D Rd :   We endow C˛d .B/ WD C˛d .0; 1/; B with the metric induced by Rd . It is called model ˛-cusp over (the base) B in Rd . Note that a 1-cusp is a cone. d For ` 2 N we set C˛;` .I; B/ WD C˛d .I; B/ if ` D 0, and d .I; B/ WD C˛d .I; B/  IQ` ; C˛;`

` > 0;

   d d .0; 1/; B .B/ WD C˛;` where IQ` D f tz 2 R` I t 2 I; z 2 Q` . Then C˛;` is a .1 C b C `/-dimensional Riemannian submanifold of Rd  R` D RdC` , a model .˛; `/-wedge over B, also called model `-wedge over C˛d .B/. Thus every d model cusp is a model wedge, a 0-wedge. Every .˛; `/-wedge C D C˛;` .B/ is   d a singular Riemannian manifold on S WD C˛;` .0; 3=4; B of type ŒŒR˛ , where the cusp characteristic R˛ is defined by R˛ .x/ WD t˛ for x D .t; y; z/ 2 C with y 2 Q` .   d d (f) Let U be open in M and set C WD C˛;` .0; 3=4; B with .B/ and S WD C˛;` ` WD m  1  b  0. Suppose that ' W U ! C is a diffeomorphism such that .' g/jS  gC jS. Then U is a singular Riemannian manifold on ' 1 .S/ of type ŒŒ'  R˛ . It is said to be an .˛; `/-wedge represented by '. (g) Let d  m and let B be an .m  1/-dimensional compact submanifold of Rd1 . For a nonempty subinterval I of .1; 1/ and ˛ < 0 we set ˚  K˛d .I; B/ WD .t; t˛ y/ I t 2 I; y 2 B  R  Rd1 D Rd :   Then K˛d .B/ WD K˛d .1; 1/; B is considered as an m-dimensional Riemannian submanifold of Rd , an infinite ˛-cusp over B in Rd . Its cusp characteristic R˛ is given by R˛ .x/ WD t˛ for x D .t; t˛ y/ 2 K˛d .B/. It holds that K˛d .B/ is a singular Riemannian manifold on K˛d Œ2; 1/; B of type ŒŒR˛ .

Parabolic Equations on Uniformly Regular Riemannian Manifolds and. . .

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(h) Let U be open in M and let K WD K˛d .B/ be an infinite ˛-cusp over B in Rd . Set S WD K˛d Œ2; 1/; B . Let ' W U ! K be a diffeomorphism satisfying   (˛) Un' 1 .S/ \ S D ' 1 K˛d .f2g; B/ ; (ˇ) .' g/jS  gK jS. Then U is a singular Riemannian manifold on ' 1 .S/ of type ŒŒ'  R˛ Œ. Furthermore, M is said to have in U an infinite ˛-cusp (more precisely: .˛; B/-cusp) represented by '. (i) Assume that M is an m-dimensional Riemannian submanifold for Rn for N nM, where M N is the closure of M in Rn , is some n  m. Then S.M/ WD M the singularity set of M. It is independent of n since the closure of M in RnQ N also. with nQ > n and Rn D Rn  f0g  RnQ equals M Suppose † is a connected component of S.M/ with the following properties: (˛) it is an `-dimensional compact Riemannian submanifold of Rn without boundary, where ` 2 f0; : : : ; m  1g; (ˇ) there exist ˛  1, a compact .m  `  1/-dimensional Riemannian submanifold B of Rd with d  m  `, and for each p 2 † a normalized chart ˆp for Rn at p such that, setting Vp WD dom.ˆh /, d ˆp .M \ Vp / D C.˛;`/ .B/  f0g  RdC`  Rnd` D Rn

and   ˆp .† \ Vp / D f0g  Q`  f0gI () Up WD M \ Vp is an .˛; `/-wedge represented by 'p WD ˆp jUp . Then M is said to possess a smooth cuspidal singularity of type .˛; `/ (more precisely: .˛; `; B/) near †. Let †  S.M/ and assume M has a smooth cuspidal singularity of type .˛; `/ near †. Also assume that there exist relatively compact open N  V possessing the following neighborhoods V and W of † in Rn with W N \ M. Then there is ˛ 2 properties: set U WD V \ M and S WD W   C1 U; .0; 1/ such that ˛ j.S \ Up /  'p R˛ j.S \ Up /;

p 2 †;

and U is on S a singular Riemannian manifold of type ŒŒ˛ Œ. Loosely speaking: † is then said to be a smooth .˛; `/-wedge, more precisely, a smooth .˛; `; B/-wedge. It is a smooth ˛-cusp, respectively .˛; B/-cusp, if ` D 0. Note that near every smooth .˛; Sm1 /-cusp M looks locally like Rm nf0g near 0. (In this case we choose d D m C 1.)

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(j) Let M be an m-dimensional Riemannian submanifold of Rn for some n  m. Suppose: (˛) S.M/ is compact and for each connected component † of S.M/ there exist ˛†  1, `† 2 f0; : : : ; m  1g, and a compact .m  `  1/-dimensional submanifold B† of Rd , where d  m  `, such that M has a smooth cuspidal singularity of type .˛† ; `† ; B† / near †; (ˇ) there are k 2 N and for each i 2 f1; : : : ; kg an open subset Ui of M, ˛i 2 .1; 0/, an .m1/-dimensional compact Riemannian submanifold Bi of Rd for some d  m, and a diffeomorphism 'i W Uj ! K˛di .Bj / such that M has in Ui an infinite .˛i ; Bi /-cusp represented by 'i ; () Ui \ Uj D ; for 1  i < j  k, M n.U1 [    [ Uk / is relatively compact in Rn , and S.M/n.U1 [    [ Uk / D S.M/. Then M is said to be a manifold with cuspidal singularities. Note that M is relatively compact in Rn if k D 0. Every manifold with cuspidal singularities is a singular Riemannian manifold of type ŒŒŒ, where   ˛ near a smooth .˛; `/-wedge †  S.M/,   'i R˛i on Ui , 1  i  k, and   1 away from the singularities. t u The qualifier ‘smooth’ in the preceding definitions refers to the fact that the bases of the cusps are uniformly regular. If they are singular Riemannian manifolds themselves then we get manifolds with cuspidal corners of various orders. For this we refer to [9] as well. Ammann et al. [11] introduce a class of noncompact Riemannian manifolds, termed Lie manifolds, in order to establish regularity properties of solutions to elliptic boundary value problems on polyhedral domains; also see Ammann et al. [12], Ammann and Nistor [10], and the references therein, as well as the survey by Bacuta et al. [13]. These authors use a desingularization technique by which they introduce conformal metrics g=2 , where  is the distance to the singular set. Let M D .M; g/ be a singular Riemannian manifold of type ŒŒ. Then we can O where we have to use rO WD rgO , of course. apply the results of Sects. 2 and 3 to M, Fortunately, since gO is conformal to g we can express all spaces and operators in terms of g, so that MO does not appear in the final results. Specifically, set V WD V and let  2 R. Then the weighted Sobolev space k; Wp .VI / is for k 2 N the completion of D.V/ in L1;loc .V/ with respect to the norm u 7!

k X  CjC  j p 1=p  jr ujg Cj Lp .V/ : jD0



Weighted Slobodeckii spaces Wps; .VI / are for k < s < k C 1 again defined by interpolation, that is, by replacing Wp` .V/ in (7) by Wp`; .VI / for ` 2 fk; k C 1g. Analogously, B .VI / is the vector all sections u of V such that  space of  C

 jujg 2 B.M/. The norm u 7! C jujg 1 makes it a Banach space. If

Parabolic Equations on Uniformly Regular Riemannian Manifolds and. . .

65

k 2 N, then Bk; .VI / is the Banach space of all u 2 Ck .V/ for which   max CjC  jr j ujg Cj 1

0jk



k; is finite, endowed T withk this norm. k;Furthermore, bc .VI / is the closure of 1; BC .VI / WD k BC .VI / in BC .VI /. Then weighted Besov-Hölder spaces Bs; 1 .VI / are defined by interpolation in complete analogy to (8). Weighted L2 Sobolev spaces of this type have been introduced by Kondrat’ev [25] in the study of elliptic boundary value problems on domains with singular points. Since then they have been used by numerous authors, predominantly in an Euclidean L2 setting. A detailed study of the Lp case and references are found in [6]. We set

Wps .VI / WD Wps;0 .VI /;

BCk .VI / WD BCk;0 .VI /;

Bs1 .VI / WD Bs;0 1 .VI /: In [7] it is proved that : s;m=p : O D O D Wp .VI /; BCk .V/ BCk .VI /; Wps .V/ : s O D B1 .VI /; Bs1 .V/

(31)

: O In [7] it is where D means ‘equal except for equivalent norms’ and VO WD T M.  also shown that .u 7!  u/ belongs to     0 0 0 C 0 .VI /; Bs; Lis Wps; C .VI /; Wps; .VI / \ L is Bs; 1 .VI / 1

(32)

for ; 0 2 R, and .u 7!  u/1 D .v 7!  v/. Thus it suffices to study O and Bs1 .V/ O since by this isomorphism and by (31) we can the spaces Wps .V/ s O O onto Bs; transfer all properties from Wp .V/ onto Wps; .VI / and from Bs1 .V/ 1 .VI /. Alternatively, we can refer directly to [6]. Anisotropic weighted Sobolev-Slobodeckii spaces are defined for s  0 by     Wp.s;s=2/; .V  JI / WD Lp J; Wps; .VI / \ Wps=2 J; Lp .VI / : Analogously, we introduce anisotropic Besov-Hölder spaces for s > 0 by     s=2  .VI // WD B J; Bs; B.s;s=2/; 1 1 .VI / \ B1 J; B .VI / : Again, we omit the superscript  if it equals zero. It is obvious from the above that all embedding, interpolation, and trace theorems, etc. proved in [5] carry over to the present setting using natural adaptions. It is also clear that (31) implies : .s;s=2/;m=p O D Wp.s;s=2/ .V/ Wp .VI /:

(33)

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Furthermore,   0 0 .u 7!  u/ 2 Lis Wp.s;s=2/; C .VI /; Wp.s;s=2/; .VI /

(34)

for ; 0 2 R is a consequence of (32).

5 Degenerate Parabolic Problems In this section we study problem (14) in the case of singular Riemannian manifolds. It turns out that in this situation Theorem 3.1 leads to an isomorphism theorem for degenerate parabolic initial boundary value problems on weighted Sobolev spaces. Let M D .M; g/ be a singular Riemannian manifold of type ŒŒ and  2 R. Similarly as in Sect. 3, we introduce data spaces which are now weighted and dependent. To simplify the presentation we restrict ourselves to the setting of strong Lp solutions. Thus we put q Wp.2•1=p/.1;1=2/;C•C1=p .@MT I /

q q WD Wp.21=p/.1;1=2/;C1=p .@0 MT I /  Wp.11=p/.1;1=2/;C1C1=p .@1 MT I /

and Wp.2;1/; .MT I /

q WD Lp .MT I /  Wp.2•1=p/.1;1=2/;C•C1=p .@MT I /  Wp22=p; .MI /: .2;1/;

Similarly as before, Wp;cc .MT I / is the linear subspace hereof consisting of all .f ; h; u0 / satisfying the compatibility conditions (17). The differential operator (15) is uniformly strongly -elliptic if a.; t/ is symmetric for t 2 J and there exists a constant " > 0 such that ˇ   a.q; t/ q X ˇ X g.q/  "2 .q/ jXj2g.q/;

X 2 Tq M;

q 2 M;

t 2 J:

(35)

Henceforth, we say that .A; B/ is a -regular uniformly -elliptic boundary value problem on MT if A is uniformly strongly -elliptic, a 2 BC.1;1=2/;2 .T11 M  JI /;

aE 2 L1 .TM  JI /;

(36)

a0 2 L1 .MT /; and b0 2 BC.1;1=2/;1 .@1 MT /:

(37)

Parabolic Equations on Uniformly Regular Riemannian Manifolds and. . .

67

If  D 1, then .A; B/ is simply called regularly uniformly elliptic. Note that the first part of (36) implies jajg1  c2 . Using this and the symmetry of a.; t/ we see 1 that (35) is equivalent to the existence of " 2 .0; 1/ with ˇ   (38) "2 .q/ jXj2g.q/  a.q; t/ q X ˇ X g.q/  2 .q/ jXj2g.q/=" for X 2 Tq M, q 2 M, and t 2 J. Now we can formulate the following isomorphism theorem for degenerate parabolic equations. Theorem 5.1 Let M be a singular Riemannian manifold of type ŒŒ and p … f3=2; 3g. Suppose that .A; B/ is a -regular uniformly -elliptic boundary value problem on MT and  2 R. .2;1/; Then Wp;cc .MT I / is closed and   .2;1/; .MT I / : .@ C A; B; 0 / 2 Lis Wp.2;1/; .MT I /; Wp;cc The proof of this theorem is given later in this section. First we derive an analogue of Theorem 3.4. For this we define s; Wp;B .MI /;

s 2 Œ0; 2nf1=p; 1 C 1=pg;

by replacing Wps .M/ in (4) by Wps; .MI /. Theorem 5.2 Let M be a singular Riemannian manifold of type ŒŒ and p … f3=2; 3g. Suppose .A; B/ is an autonomous -regular uniformly -elliptic boundary 2; value problem on MT and  2 R. Set A WD AjWp;B .MI /, considered as an unbounded linear operator in Lp .MI /. Then A generates a strongly continuous analytic semigroup on Lp .MI / and has the property of maximal regularity, that is, .@ C A;  / belongs to   22=p; 2; .MI // \ Wp1 .J; Lp .MI //; Lp .MI /  Wp;B .M/ : Lis Lp .J; Wp;B Proof This follows from Theorem 5.1 by the arguments which led from Theorem 3.1 to Theorem 3.4. t u Corollary 5.3 Set A WD A0 . Then A generates a strongly continuous analytic semigroup on Lp .M/ and has the maximal regularity property on Lp .M/. In order to facilitate the proof of Theorem 5.1 we precede it with a technical lemma. In this connection we identify  with the point-wise multiplication operator u 7!  u. Lemma 5.4 Let .A; B/ be a -regular uniformly -elliptic boundary value problem on MT and  2 R. Then there exists another such pair .A0 ; B 0 / such that .A; B/ ı  D  ı .A0 ; B 0 /:

(39)

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H. Amann

Proof (1) Note that

.A; B/ ı  D  ı .A; B/ C .A; B/; 

(40)

where the commutator

.A; B/;  u WD .A; B/. u/   .A; B/u   is given by ŒA;  ; ŒB;   with   ŒA;  u D 2.a q grad  j grad u/ C .aE j grad  /  div.a q grad  / u and   ŒB;  u D 0; . j.a q grad  //u : For abbreviation, gl WD grad log. Then grad  D 1 grad  D   gl :

(41)

We set aE0 WD 2a q gl ;

  a00 WD  .aE j gl /  div.a q gl /  2 .a q gl  j gl /;  ˇ  b00 WD  ˇ .a q gl / :

Moreover, A0 u WD Au C .aE0 j grad u/ C a00 u;

B 0 u WD Bu C .0; b00  u/:

It follows from (40) and (41) that .A0 ; B 0 / satisfies (39). Hence it remains to show that .aE0 ; a00 ; b00 / possesses the same regularity properties as .a; E a0 ; b0 /. (2) We know from (30) and (31) that : O D d log  2 BC1 .T  M/ BC1 .T  MI /: Using this, grad D g] d, and (71) it follows gl  D g] d log  2 BC1;2 .TMI /:

(42)

It is now an easy consequence of this, the assumptions on a and a, E and of (68) that aE0 2 L1 .TM  J/;

.aj E gl / 2 L1 .MT /:

(43)

Parabolic Equations on Uniformly Regular Riemannian Manifolds and. . .

69

From (42), (72), and (73) we infer j div.a q gl /j D jr.a q gl /jg1 1



1

jrajg2  j gl jg0 C 2 jajg1 2 jr gl jg1 : 1

1

1

1

This guarantees that the second summand of a00 belongs to L1 .MT /. Similarly, j.a q gl  j gl /j  2 jajg1 . j gl jg0 /2 1

1

implies that the third summand lies in L1 .MT / as well. Hence, by the second part of (43), a00 2 L1 .MT /:

(44)

O In local coordinates, Let O be the unit normal vector field of @M. O D .gO11 /1=2 @=@x1 D .g11 /1=2 @=@x1 D  :

(45)

Thus 2 BC1;1 .TMj@M I /. This implies for the conormal field [ D g[ 2 BC1;1 .T  Mj@M I /:

(46)

As above, we derive from (42) .a q gl / 2 BC.1;1=2/ .TMj@M I /: Therefore, by (46), b00 D  [ q .a q gl / 2 BC.1;1=2/;1 .@MT I /:

(47) t u

Now (43), (44), and (47) imply the assertion. Proof of Theorem 5.1 (1) By the definitions of gO and the gradient we get 2 .X j gradgO u/ D .X j gradgO u/gO D hdu; Xi D .X j grad u/

(48)

for any C1 function u and any vector field X on M. From this we obtain grad u D 2 gradgO u:

(49)

We also note that (65)–(67) imply jjgO D   jjg ;

;  2 N:

(50)

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We put aO WD 2 a. Then we infer from (50) jaj O gO1 D 2 jajg1 : 1

(51)

1

Note that r aO D 2 ra  2.d log / q aO (cf. (41)). Hence, see (68),  jr aj O g2  1 jrajg2 C 2 jd log jg1 jaj O g1 1

1

D

1

0

1

jrajg2 C 2 jd log jgO1 jaj O g1 ; 1

1

0

the last equality being a consequence of (50). From this, (30), (51), and the assumption on a we deduce : aO 2 BC.1;1=2/ .T11 M  JI / D BC.1;1=2/ .T11 MO  J/:

(52)

By replacing the index H in (26) by gO and using (48) and (49) we find div.a q grad u/ D div.2 aO q grad u/

(53)

D divgO .aO q gradgO u/ C m.aO q grad  j grad u/ D divgO .aO q gradgO u/ C .maO q 1 gradgO  j gradgO u/gO: Observe that 1 gradgO  D 1 gO] d D gO] d log : Hence, by (68) and (71), O gO1 jd log jgO1 : jaO q 1 gradgO jgO  jaj 1

0

This, (52), and (30) imply aO q 1 gradgO  2 L1 .T MO  J/:

(54)

Furthermore, by (49) and gO D 2 g, .aE j grad u/ D .aE j2 gradgO u/ D .aE j gradgO u/gO: From (50) we derive    jaj E gO 

1

  D 1 jaj E g 1 D kak E L1 .TMJI/ :

(55)

Thus it follows from (54) that dO WD maO q 1 gradgO  C aE 2 L1 .T MO  J/:

(56)

Parabolic Equations on Uniformly Regular Riemannian Manifolds and. . .

71

Now we put O WD  divgO .aO q gradgO u/ C .dO j gradgO u/gO C a0 u: Au Then (53) shows O Au D Au;

: u 2 Wp.2;1/;m=p .MT I / D Wp.2;1/ .MO T /;

(57)

the last equivalence being a consequence of (31). q (2) Recalling (45) and  D  j@M, we find  ˇ   ˇ  ˇ .a q grad u/ D ˇ .aO q gradgO u/ (58)   ˇ  q 2  ˇˇ q D  .aO q gradgO u/ gO D  O ˇ .aO q gradgO u/ gO : q q We denote by r the Levi-Civita connection of @M and set bO0 WD 1 b0 . Then q q q q r bO0 D 1 rb0  .d log /bO0 :

(59)

Relation (50) implies q q q  jd log jgq1 D jd log jgOq1 : 0

0

Since b0 2 BC.1;1=2/;1 .@1 MT I / it holds q kbO0 kL1 .@1 MO T / D k 1 b0 kL1 .@1 MT / < 1;  q   jrb0 j q1  < 1: g L1 .@1 MT /

(60)

0

O Thus we get from (59), (60), Example 4.1(b), and (30) (applied to @1 M D @1 M)  q q that  jr bO0 jgq1 L1 .@1 MT / is finite. This implies 0

: bO0 2 BC.1;1=2/ .@1 MT I / D BC.1;1=2/ .@1 MO T /:

(61)

Now we set  ˇ  BO 1 u WD O ˇ .aO q gradgO u/ gO C bO0  u O B/ O is and BO WD .B0 ; BO 1 /. Then we see from (52), (56), (61), and (35) that .A; O a regular uniformly elliptic boundary value problem on M T . Furthermore, q Bu D .BO 0 u; BO 1 u/:

(62)

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H. Amann .2;1/;m=p

(3) Suppose .f ; h; u0 / 2 Wp;cc

.MT I /. Then, by (34),

q q 1 h1 2 Wp.11=p/.1;1=2/;.m1/=p .@1 MT I /: q More precisely, set hO WD .h0 ; 1 h1 /. Then (34), (31), and (33) imply    .2;1/;m=p .2;1/ O O u0 / 2 Lis Wp;cc .f ; h; u0 / 7! .f ; h; .MT I /; Wp;cc .M T /: .2;1/

In addition, we deduce from (57) and (62) that u 2 Wp .MT I / is a solution .2;1/ O B; O 0 /u D .f ; h; O u0 /. Now Theorem 3.1 of (14) iff u 2 Wp .MO T / and .@ C A; implies the validity of the assertion if  D m=p. (4) Let  ¤ m=p. Lemma 5.4 guarantees the existence of a -regular uniformly -elliptic boundary value problem .A0 ; B 0 / on MT such that  ı .A; B/ D .A0 ; B 0 / ı  : .2;1/;

By (32) and (34) it follows that .f ; h; u0 / 2 Wp;cc

(63)

.MT I / iff

.2;1/;m=p .f 0 ; h0 ; u00 / WD Cm=p .f ; h; u0 / 2 Wp;cc .MT I / .2;1/;

and u 2 Wp

.MT I / iff u0 WD Cm=p u 2 Wp.2;1/;m=p .MT I /:

From (63) we get .A; B; 0 /u D .f ; h; u0 / ” .A0 ; B 0 ; 0 /u0 D .f 0 ; h0 ; u0 /: As the claim holds for  D m=p, the assertion follows.

t u

Remarks 5.5 (a) It is obvious from this proof that there is a straightforward parameter-dependent analogue of the supplement to Theorem 3.1 for degenerate parabolic problems. (b) Remarks 3.3 apply in the present setting also. t u

Appendix: Tensor Bundles Let M be a manifold and V D .V; ; M/ a vector bundle of rank n over it. For a nonempty subset S of M we denote by VjS the restriction  1 .S/ of V to S. If S is a submanifold or a union of connected components of @M, then VjS is a vector bundle

Parabolic Equations on Uniformly Regular Riemannian Manifolds and. . .

73

of rank n over S. As usual, Vp WD Vfpg is the fibre  1 .p/ of V over p. By .S; V/ we mean the RS module of all sections of V (no smoothness). As usual, TM and T  M are the tangent and cotangent bundles of M. Then

T M WD TM ˝ ˝ T  M ˝ is for ;  2 N the . ; /-tensor bundle of M, that is, the vector bundle of all tensors on M being contravariant of order and covariant of order . In particular, T01 M D TM and T10 M D T  M, as well as T00 M D M  R. For 2 N we put J WD f1; : : : ; mg . Then, given local coordinates  D 1 .x ; : : : ; xm / and setting @ @ @ WD i ˝    ˝ i ; @x.i/ @x 1 @x

dx. j / WD dx j1 ˝    ˝ dxj

for .i/ D .i1 ; : : : ; i / 2 J , . j / 2 J , the local representation of a . ; /-tensor field a 2 .T M/ with respect to these coordinates is given by .i/

a D a. j /

@ ˝ dx. j / @x. i /

.i/

with a.j/ 2 RU . We use the summation convention for (multi-)indices labeling coordinates or bases. Thus such a repeated index, which appears once as a superscript and once as a subscript, implies summation over its whole range. Suppose 1 ; 2 ; 1 ; 2 2 N. Then the complete contraction .a; b/ 7! a q b

C1 M/  .T 11 M/ ! .T 22 M/; .T 22C

1

is defined as follows: Given .ik / 2 J k and .jk / 2 Jk for k D 1; 2, we set .i2 I j1 / WD .i2;1 ; : : : ; i2; 2 ; j1;1 ; : : : ; j1;1 / 2 J 2 C1 C1 etc., using obvious interpretations if minf ; g D 0. Suppose a 2 .T 22C

M/ and 1

1 b 2 .T1 M/ are locally represented on U by .i Ij /

a D a.j22 Ii11 /

@ @ ˝ .j / ˝ dx.j2 / ˝ dx.i1 / ; .i / 2 @x @x 1

.i /

b D b.j11 /

@ ˝ dx.j1 / : @x.i1 /

Then the local representation of a q b on U is given by .i Ij /

.i /

a.j22 Ii11 / b.j11 /

@ ˝ dx.j2 / : @x.i2 /

Let g be a Riemannian metric on TM. We write g[ W TM ! T  M for the (fiberwise defined) Riesz isomorphism. Thus hg[ X; Yi D g.X; Y/ for X; Y 2 .TM/, where h; i W .T  M/  .TM/ ! RM is the natural (fiber-wise defined) duality pairing. The inverse of g[ is denoted by g] . Then g , the adjoint Riemannian metric on T  M, is defined by g .˛; ˇ/ WD g.g] ˛; g] ˇ/ for ˛; ˇ 2 .T  M/. In local

74

H. Amann

coordinates g D gij

g D gij dxi ˝ dxj ;

@ @ ˝ j; @xi @x

(64)

Œgij  being the inverse of the .m  m/-matrix Œgij . The metric g induces a vector bundle metric on T M which we denote by g . In local coordinates .i/ .j/

g .a; b/ D g.i/.j/ g.k/.`/a.k/ b.`/ ;

a; b 2 .T M/;

(65)

where g.i/.j/ WD gi1 j1    gi j ;

g.k/.`/ WD gk1 `1    gk `

(66)

for .i/; .j/ 2 J and .k/; .`/ 2 J . Note g01 D g and g10 D g and g00 .a; b/ D ab for a; b 2 .M  R/ D RM . Moreover, jjg W .T M/ ! .RC /M ;

a 7!

p g .a; a/

(67)

is the vector bundle norm on T M induced by g. It follows that the complete contraction satisfies ja q bjg 2  jajg2 C 1 jbjg 1 ; 2

2 C1

1

C1 a 2 .T 22C

M/; 1

b 2 .T 11 M/:

(68)

We define a vector bundle isomorphism T C1 M ! T C1 M, a 7! a] by a] .˛1 ; : : : ; ˛ ; ˛; X1 ; : : : ; X / WD a.˛1 ; : : : ; ˛ ; X1 ; : : : ; X ; g] ˛/

(69)

.i/

for X1 ; : : : ; X 2 .TM/ and ˛; ˛1 ; : : : ; ˛ 2 .T  M/. If a.jIk/ with .i/ 2 J , .j/ 2 J , and k 2 J1 is the coefficient of a in a local coordinate representation, then .iIk/

.i/

.a] /.j/ D gk` a.jI`/ :

(70)

ja] jg C1 D jajg  C1 :

(71)

This implies

The Levi-Civita connection on TM is denoted by r D rg . We use the same symbol for its natural extension to a metric connection on T M. Then the corresponding covariant derivative is the linear map r W C1 .T M/ ! C1 .T C1 M/;

a 7! ra;

Parabolic Equations on Uniformly Regular Riemannian Manifolds and. . .

75

defined by hra; b ˝ Xi WD hrX a; bi for b 2 C1 .T  M/ and X 2 C1 .TM/. It is a well-defined continuous linear map from C1 .T M/ into C.T C1 M/, as follows from its local representation. For k 2 N we define r k W Ck .T M/ ! C.T Ck M/;

a 7! r k a

by r 0 a WD a and r kC1 WD r ı r k . In local coordinates  D .x1 ; : : : ; xm / the volume measure dv D dvg of .M; g/ is  1=2 p p and dx is the Lebesgue represented by  dv D  g dx, where g WD detŒgij  measure on Rm . C1 The contraction C W T C1 M ! T M, a 7! Ca is given in local coordinates by .i/ .iIk/ .Ca/.j/ WD a.jIk/ . It follows jCajg D jajg C1 :

(72)

C1

Recall that the divergence of tensor fields is the map div D divg W C1 .T C1 M/ ! C.T M/;

a 7! div a WD C.ra/:

(73)

If X is a C1 vector field on M, then div X has the well-known local representation 1 @ p i  gX ; p g @xi

X D Xi

@ : @xi

(74)

The gradient, grad u D gradg u, of a C1 function u is the continuous vector field g] du. Suppose a 2 C1 .T11 M/. Then, in terms of covariant derivatives, div.a grad u/ D a] q r 2 u C div.a] / q ru:

(75)

References 1. H. Amann, Linear and Quasilinear Parabolic Problems, Volume I: Abstract Linear Theory (Birkhäuser, Basel, 1995) 2. H. Amann, Maximal regularity for nonautonomous evolution equations. Adv. Nonlinear Stud. 4, 417–430 (2004) 3. H. Amann, Quasilinear parabolic problems via maximal regularity. Adv. Differ. Equ. 10(10), 1081–1110 (2005) 4. H. Amann, Anisotropic Function Spaces and Maximal Regularity for Parabolic Problems. Part 1: Function Spaces. Jindˇrich Neˇcas Center for Mathematical Modeling, Lecture Notes, Prague, vol. 6, 2009. 5. H. Amann, Anisotropic function spaces on singular manifolds (2012). arXiv: 1204.0606 6. H. Amann, Function spaces on singular manifolds. Math. Nachr. 286, 436–475 (2012) 7. H. Amann, Parabolic equations on noncompact Riemannian manifolds (in preparation)

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8. H. Amann, Pseudodifferential boundary value problems (in preparation) 9. H. Amann, Uniformly regular and singular Riemannian manifolds, in Elliptic and Parabolic Equations, Hannover, September 2013, ed. by J. Escher, E. Schrohe, J. Seiler, C. Walker. Springer Proceedings in Mathematics & Statistics (2015), pp.1–43 10. B. Ammann, V. Nistor, Weighted Sobolev spaces and regularity for polyhedral domains. Comput. Methods Appl. Mech. Eng. 196, 3650–3659 (2007) 11. B. Ammann, R. Lauter, V. Nistor, On the geometry of Riemannian manifolds with a Lie structure at infinity. Int. J. Math. Math. Sci. 161–193 (2004) 12. B. Ammann, A.D. Ionescu, V. Nistor, Sobolev spaces on Lie manifolds and regularity for polyhedral domains. Doc. Math. 11, 161–206 (2006) 13. C. Bacuta, A.L. Mazzucato, V. Nistor, Anisotropic regularity and optimal rates of convergence for the finite element method on three dimensional polyhedral domains (2012). arXiv:1205.2128 14. C. Bandle, F. Punzo, A. Tesei, Existence and nonexistence of patterns on Riemannian manifolds. J. Math. Anal. Appl. 387(1), 33–47 (2012) 15. V. Barbu, A. Favini, S. Romanelli, Degenerate evolution equations and regularity of their associated semigroups. Funkcial. Ekvac. 39(3), 421–448 (1996) 16. E.B. Davies, Heat Kernels and Spectral Theory (Cambridge University Press, Cambridge, 1989) 17. R. Denk, M. Hieber, J. Prüss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 166(788) (2003) 18. G. Dore, Lp regularity for abstract differential equations, in Functional Analysis and Related Topics, 1991 (Kyoto). Lecture Notes in Mathematics, vol. 1540 (Springer, Berlin, 1993), pp. 25–38 19. S. Fornaro, G. Metafune, D. Pallara, Analytic semigroups generated in Lp by elliptic operators with high order degeneracy at the boundary. Note Math. 31(1), 103–116 (2011) 20. G. Fragnelli, G. Ruiz Goldstein, J.A. Goldstein, S. Romanelli, Generators with interior degeneracy on spaces of L2 type. Electron. J. Differ. Equ. 2012(189), 1–30 (2012) 21. A. Grigor’yan, Heat Kernel and Analysis on Manifolds (American Mathematical Society, Providence, 2009) 22. G. Grubb, Parameter-elliptic and parabolic pseudodifferential boundary problems in global Lp Sobolev spaces. Math. Z. 218, 43–90 (1995) 23. G. Grubb, Functional Calculus of Pseudodifferential Boundary Problems (Birkhäuser, Boston, 1996) 24. G. Grubb, N.J. Kokholm, A global calculus of parameter-dependent pseudodifferential boundary problems in Lp Sobolev spaces. Acta Math. 171, 165–229 (1993) 25. V.A. Kondrat’ev, Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obšˇc. 16, 209–292 (1967) 26. P.C. Kunstmann, L. Weis, Maximal Lp -regularity for parabolic equations, Fourier multiplier theorems and H 1 -functional calculus, in Functional Analytic Methods for Evolution Equations. Lecture Notes in Mathematics, vol. 1855 (Springer, Berlin, 2004), pp. 65–311 27. O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs (American Mathematical Society, Providence, 1968) 28. A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems ( Birkhäuser, Basel, 1995) 29. A.L. Mazzucato, V. Nistor, Mapping properties of heat kernels, maximal regularity, and semilinear parabolic equations on noncompact manifolds. J. Hyperbolic Differ. Equ. 3(4), 599–629 (2006) 30. M.A. Pozio, F. Punzo, A. Tesei, Criteria for well-posedness of degenerate elliptic and parabolic problems. J. Math. Pures Appl. (9) 90(4), 353–386 (2008) 31. F. Punzo, On well-posedness of semilinear parabolic and elliptic problems in the hyperbolic space. J. Differ. Equ. 251(7), 1972–1989 (2011)

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A Generalization of Some Regularity Criteria to the Navier–Stokes Equations Involving One Velocity Component Šimon Axmann and Milan Pokorný

To Yoshihiro Shibata

Abstract We present generalizations of results concerning conditional global regularity of weak Leray–Hopf solutions to incompressible Navier–Stokes equations presented by Zhou and Pokorný in articles (Pokorný, Electron J Differ Equ (11):1–8, 2003; Zhou, Methods Appl Anal 9(4):563–578, 2002; Zhou, J Math Pure Appl 84(11):1496–1514, 2005); see also Neustupa et al. (Quaderni di Matematica, vol. 10. Topics in Mathematical Fluid Mechanics, 2002, pp. 163–183) We are able to replace the condition on one velocity component (or its gradient) by a corresponding condition imposed on a projection of the velocity (or its gradient) onto a more general vector field. Comparing to our other recent results from Axmann and Pokorný (A note on regularity criteria for the solutions to NavierStokes equations involving one velocity component, in preparation), the conditions imposed on the projection are more restrictive here, however due to the technique used in Axmann and Pokorný (A note on regularity criteria for the solutions to Navier-Stokes equations involving one velocity component, in preparation), there appeared a specific additional restriction on geometrical properties of the reference field, which could be omitted here. Keywords Global regularity • Incompressible Navier–Stokes equations • Regularity criteria

Š. Axmann • M. Pokorný () Charles University in Prague, Faculty of Mathematics and Physics, Sokolovská 83, 186 75 Praha 8, Czech Republic e-mail: [email protected]; [email protected] © Springer Basel 2016 H. Amann et al. (eds.), Recent Developments of Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-0348-0939-9_5

79

80

Š. Axmann and M. Pokorný

1 Introduction We consider the Cauchy problem for the instationary incompressible Navier–Stokes equations in the full three space dimensions 9 @v = C v  rv  v C rp D f in .0; T/  R3 , @t ; divv D 0

(1)

v.0; x/ D v0 .x/ in R3 ; where v W .0; T/R3 ! R3 is the velocity field, p W .0; T/R3 ! R is the pressure, f W .0; T/  R3 ! R3 is the given density of external forces, and > 0 is given kinematic viscosity. For the sake of simplicity, we set D 1 and f 0 in our further considerations. Indeed, the actual value of viscosity does not play any role. It would be also possible to formulate some suitable assumptions on the regularity of f in such a way that our main results remain true. However, it would lead to unnecessary technicalities which we prefer to omit here. The mathematical theory of Navier–Stokes equations has long, interesting history (see e.g. [16]). In the celebrated works of Leray [11, 12] and Hopf [6] the 1;2 existence of weak v 2 L2 .0; T; .Wdiv .R3 /// \  solutions to system (1)2 in space 1 2 3 3 L .0; T; .L R // for any given v0 2 Ldiv .R / was proved; they satisfy energy 1;2 inequality. Further, for v0 2 Wdiv .R3 / the existence of (possibly short) time  interval .0; T / such strong solution in space v 2  that there exists1;2a unique L2 .0; T  ; .W 2;2 R3 // \ L1 .0; T  ; .Wdiv .R3 /// was established (see [9]). The uniqueness and regularity of Leray–Hopf weak solutions is still a challenging open problem [10]. For overview of known results see e.g. [4]. On the other hand, there were established many criteria ensuring the smoothness of the solution under additional assumptions concerning the velocity and its components, the gradient of the velocity and its components, the pressure, the vorticity, or other quantities. During the last decade, an interesting progress was achieved in the field of regularity criteria concerning only one velocity component. The very first result in this direction is criterion proved by Neustupa and Penel [13], which ensures  the regularity for v3 2 L1 0; T; L1 R3 . Similar result for the gradient of      one velocity component rv3 2 Lt 0; T; Ls R3 , 2t C 3s  1, s  3/ is due to He [5]. results were then improved by Neustupa et al. [14]  These  pioneering   (v3 2 Lt 0; T; Ls R3 ; 2t C 3s  12 ; s  2; as local criterion for suitable weak     solution), and Pokorný [15] (rv3 2 Lt 0; T; Ls R3 ; 2t C 3s  32 ; s  2:), observing the equation for vorticity; the same results were obtained also by Zhou [17, 18]. Further improvements were later done via several techniques by Kukavica and Ziane [8], Cao and Titi [2], and finally Zhou and Pokorný [19, 20]. Note that the results in [1] contain generalization of these criteria. However, due to the used technique (multiplicative Gagliardo–Nirenberg inequality which has to be

A Generalization of Some Regularity Criteria to the Navier–Stokes Equations. . .

81

generalized) we get additional geometrical restrictions on the field b which can be avoided in our present paper. Notation   In the whole paper, the standard notation for Lebesgue spaces Lp R3 with the norm kkp will be used. For the sake of brevity, we will denote the norm on Bochner spaces     Lp 0; t; Lq R3 by kkp;q , the length of the time interval will be everywhere clear from the context. We will also use the same notation for scalar spaces X and their vector analogues X N . All generic constants will be denoted by C, although its value may differ from line to line, or even in the same formula. We will use Einstein summation convention over repeated indices.

2 Main Results As we have already mentioned above, our main goal is to generalize the results of Zhou and Pokorný from articles [15, 17, 18] (the latter proved originally for suitable weak solution in [14]). Theorem 1 Let v be a weak Leray–Hopf solution to the Navier–Stokes equations 1;2 corresponding to initial datum v0 2 Wdiv .R3 /. Assume moreover that there exist ı > 0, and a vector field b.t; x/ W .0; T/  R3 7! R3 such that     rb 2 L1 0; T; L1 R3 ;     @b ; r 2 b 2 L1 0; T; L3 R3 ; @t and jb.t; x/j  ı such that the projection of the velocity vb .t; x/ WD b.t; x/  v.t; x/ satisfies either     2 3 1 vb .t; x/ 2 Lt 0; T; Ls R3 , C  , 6  s  1 t s 2 or     2 3 3 rvb .t; x/ 2 Lt 0; T; Ls R3 , C  , 2  s  1: t s 2 Then v is actually a strong solution to the Navier–Stokes equations in the interval Œ0; T. Since the proofs of both cases have lot of similarities, we will prove them simultaneously. It is well known that there exists a unique strong solution to (1)

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Š. Axmann and M. Pokorný

on (possibly short) time interval Œ0; T  /, we will work with this strong solution and show that actually T   T. The result concerning so-called weak-strong uniqueness will then yield the desired result. Let us denote space Y./ WD L1 .0; I L2 .R3 // \ L2 .0; I W 1;2 .R3 //. Our first step in Lemma 1 will be to derive a suitable estimate of b  ! in the norm of space Y, then we will test Eq. (1) by an analogue of the quantity v and get the desired estimate of rv using Lemma 1. Lemma 1 Let v be a strong solution to the Navier–Stokes equations corresponding 1;2 to the initial condition v0 2 Wdiv .R3 /. Suppose that the assumptions of Theorem 1  ! can be estimated on .0; / as follows are satisfied. Let 0 <  < T . Then !b WD b!   k!b k21;2 C kr!b k22;2  k!b .0/k22 C C./ 1 C krvkY. / :

(2)

In particular, if  ! 0C , then C./ ! 0 and if  ! .T  / , then C./ remains bounded.

3 Proof of the Lemma By possible decreasing the value of t we could easily achieve that respectively). Applying the curl operator on (1) we get @!i C v  r!i D !  rvi C !i ; @t

2 t

C

3 s

D

1 2

(or 32 ,

i D 1; 2; 3:

Multiplying these equations by bi .t; x/ bi

summing up

3 P

@!i !  rvi / C .!i /bi ; C bi v  r!i D bi .! @t

, and multiplying the arisen equation by !b D

iD1

3 P iD1

terms which could be rewritten in the following way !b bi

@!i @!b @bi D !b  !b !i ; @t @t @t

bi .v  r!i /!b D .v  r!b /!b  .v  rbi /!i !b ; !  rvi /!b D .! !  rvb /!b  .! !  rbi /vi !b ; bi .! .!i /bi !b D .!b  2rbi  r!i  !i bi / !b I

bi !i , we get four

A Generalization of Some Regularity Criteria to the Navier–Stokes Equations. . .

83

whence @!b !b C v  r!b !b D !  rvb !b C !b !b @t @bi !  rbi /!b  bi !i !b  2.r!i  rbi /!b : !b C !i .v  rbi /!b  vi .! C !i „ @t ƒ‚ … =:I

Integration over the whole R3 with integration by parts gives us 1d k!b k22 C kr!b k22   2 dt

Z

Z

v  r!b !b dx C 3 „R ƒ‚ …

Z R3

!  rvb !b dx C

R3

Idx:

D0

The lower order terms I could be easily estimated using again integration by parts and Hölder’s inequality Z

ˇ @b i ˇ !  rbi /!b C bi !i !b !b  vi .! ˇ!i 3 @t R

ˇ ˇ C 2.!i rbi /  r!b C !i .v  rbi /!b ˇdx "  #  2 2  @b  2 2  C" krvk2   C kbk3 C 2krbk1 C 3" kr!b k22 @t 3 Z ! j jvj j!b j dx: C 2krbk1 j! R3

In the last integral we will use Hölder’s inequality, interpolation, and Young’s inequality Z krbk1

R3

! j jvj j!b j dx Ckrbk1 krvk2 kvk3 k!b k6 j! " kr!b k22 C C" krbk21 krvk22 kvk3 2 ;

where Zt

kvk23 krvk22 d  krvk24;2 kvk24;3  krvk2;2 kvk24;3 krvk1;2 :

0

Now, we will estimate the leading terms, distinguishing two considered cases.

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Š. Axmann and M. Pokorný

1. Assume the projection vb has better integrability properties. Then Z R3

ˇ Z ˇ !  rvb !b dx  ˇˇ

R3

ˇ Z ˇ vb!  r!b dxˇˇ  " kr!b k22 C C" Z

" kr!b k22 Z R3

C 2C"

R3

R3

! j2 vb 2 dx j!

jrvj2 vb 2 dx

vb 2 jrvj2 dx  kvb k2s krvk22s

s2

 kvb k2s krvk

2s s4

krvk2

1 6s

C kvb k2s krvk2 4

6

kvk2s krvk2 6

DCkvb k2s krvk2t kvk2s krvk2 ; (recall

2 t

C

3 s

D 12 ) which gives using assumptions on b

4 6 1d k!b k22 C kr!b k22  C kvb k2s krvk2t kvk2s krvk2 2 dt

C C.b/ krvk22 .1 C kvk23 /: Integrating over time interval .0; /, with usage of Hölder’s inequality then yields

k!b ./k22

Z C

kr!b k22 d

0



k!b .0/k22

Z CC

4

6

kvb k2s krvk2t kvk2s krvk2 d

0

Z C C.b/

.krvk22 C kvk23 krvk22 /d :

0

Recall that from the energy inequality  we  have estimate of v in the spaces 1;2 .R3 /// and L1 .0; T; .L2 R3 //, hence due to the interpolation also L2 .0; T; .Wdiv

A Generalization of Some Regularity Criteria to the Navier–Stokes Equations. . .

85

  in L4 .0; T; .L3 R3 //. Thus k!b ./k22

Z

kr!b k22 d

C 0

 4 6 t s  k!b .0/k22 C C.b/ kvb k2t;s krvk1;2 C 1 krvk22;2 kvk2;2 C C.b/ krvk2;2 kvk24;3 krvk1;2

(3)

which yields the conclusion of Lemma in the first case as 4t C 6s D 1. 2. For a given 2  s  1, we will find 2  p  6, 2  q  3 such that 1 1 1 s C p C q D 1. We will use gradually Hölder’s inequality, interpolation, and Young’s inequality to obtain Z ! kq !  rvb !b dx  krvb ks k!b kp k! R3

6p

3p6

6q

3q6

! k22q k! ! k6 2q  krvb ks k!b k22p k!b k6 2p k! 4p

2

p 6q

2

p 3q6 6Cp

! k2 q 6Cp k! ! k6 q  " kr!b k22 C C krvb ks6Cp k!

2

6p

k!b k2 6Cp :

Altogether we get d k!b ./k22 C kr!b k22 dt 4p

2

p 6q

2

p 3q6 6Cp

! k2 q 6Cp k! ! k6 q  C krvb ks6Cp k!

2

6p

k!b k2 6Cp

C C.b/ krvk22 .1 C kvk23 /; thus using inequality in the form of Theorem 2 from [3]  generalized Gronwall  p6 2p gives us pC6 C 1 D 6Cp 4p

4p

k!b ./k26Cp  k!b .0/k26Cp C

p 3q

4 q 6Cp ! k1;2 C k!

Z

4p

2p

2

p 3q6 6Cp

! k26Cp k! ! k6 q krvb ks6Cp k!

d

0

CC

 Z 0

krvk22 .1 C kvk23 /d

2p  6Cp

:

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Š. Axmann and M. Pokorný

Estimating the second term as above and using the Hölder inequality in the form Z

4p

2p

2

p 3q6 6Cp

! k26Cp k! ! k6 q krvb ks6Cp k!

4p

2p

2

p 3q6

6Cp 6Cp ! k2;2 ! k2;6q 6Cp d  krvb kt;s k! k!

0

we conclude ! kY. / : k!b k21;2  C1 C C2 k! This finishes the proof of Lemma 1.

4 Proof of the Main Theorem Without loss of generality we may assume that jb.; x/j 1. Then for every time

2 ˇ .0; T/, ˇand point x 2 R3 , there exists at least one component bj . ; x/ such that ˇbj . ; x/ˇ > 12 . Since vector field b.; / is continuous on .0; T/  R3 , the sets   ˚ ˚ r; D x 2 R3 j br . ; x/ > 12 a rC3; D x 2 R3 j br . ; x/ <  12 , r D 1; 2; 3 compose at each particular time a covering of R3 by six open sets fr; g6rD1 . For simplicity, we set brC3 WD br , r D 1; 2; 3. Using the partition P of unity (see e.g. [7]) we get functions 'r; 2 C01 .r; /, 0  'r; .x/  1 such that r 'r; D 1, and jr'r; j  C.b/. We will multiply the following equivalent form of Navier–Stokes equations (1)

 1 2 @v !  v  r p C jvj  v D ! @t 2 by the test function 

6 P

@l .'r; br @l v/. Let work with each term separately, for

rD1

illustration only with r D 1: Z  1;

(4)

@v  @l .'1; b1 @l v/dx D @t

Z

1;

Z

 1;

@ @l v  '1; b1 @l vdx @t 1 d '1; jrvj2 dx 4 dt

A Generalization of Some Regularity Criteria to the Navier–Stokes Equations. . .

Z

Z v  @l .'1; b1 @l v/dx D

1;

Z

1;

Z 1;

  v  '1; b1 @l @l v C @l .'1; b1 /@l v dx

1;



87

Z

1 '1; jvj2 dx C 2

v  @l .'1; b1 /@l vdx

1;

„

ƒ‚

DZ11

…



 Z 1 1 @k p C jvj2 @l .'1; b1 @l vk /dx D @l p C jvj2 @k .'1; b1 @l vk /dx 2 2 1;

Z

Z

@l p@k .'1; b1 /@l vk dx C

D 1;

„

ƒ‚

DZ21

…

1;

1 @l jvj2 @k .'1; b1 /@l vk dx 2

„

ƒ‚

DZ31

…

Z !  v/  @l .'1; b1 @l v/ dx .! 1;

Z

Z !  v/  vdx  '1; b1 .!

D 1;

!  v/  @l .'1; b1 /@l vdx .!

1;

„

ƒ‚

DZ41

We rewrite the term in the first integral and get !  v/  v D b1 !2 v3 v1 b1 !1 v3 v2 Cb1 !1 v2 v3 b1 .! b1 !3 v2 v1 Cb1 !3 v1 v2 b1 !2 v1 v3 Cb2 !2 v3 v2 b2 !2 v3 v2 Cb3 !2 v3 v3

b3 !2 v3 v3

b2 !3 v2 v2 Cb2 !3 v2 v2 b3 !3 v2 v3

Cb3 !3 v2 v3 Cb3 !3 v3 v2 Cb2 !2 v2 v3 b3 !3 v3 v2 b2 !2 v2 v3 :

…

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Š. Axmann and M. Pokorný

Thus, !  v/  v D!2 v3 vb  !3 v2 vb b1 .! C!b v2 v3  !b v3 v2

(5)

C!3 vb v2  !2 vb v3 + lower order terms.I41 ; I51 /: Observation The above mentioned equality holds true without additional lower order terms, if the vector b. ; / is constant in space, otherwise we use the following identity Z

Z '1; .!l vm /bl vn dx D

1;

'1; .!l vm /.bl vn /dx 1;

Z



  '1; .!l vm /vn bl C 2'1; rvn  rbl .!l vm / dx:

1;

Similarly, for b2 we get !  v/  v D!3 v1 vb  !1 v3 vb b2 .! C!b v3 v2  !b v1 v3

(6)

C!1 vb v3  !3 vb v2 + lower order terms: For the term with b3 we use (as above) the shifts 1 7! 2, 2 7! 3, 3 7! 1. The case r D 4; 5; 6 is trivial. Summing up

6 P

, and using the definitions of 'r; we get (recall, we use summation

rD1

convention) 1 4

Z R3

1 d jrvj2 dx C dt 2 

Z R3

jvj2 dx

6 X  r  jZ1 j C jZ2r j C jZ3r j C jZ4r j C CI4r C CI5r rD1

C

3 X iD1

jk

jk

jk

"ijk .I1 C I2 C I3 /;

(7)

A Generalization of Some Regularity Criteria to the Navier–Stokes Equations. . .

89

where Z jk

I1 D

Z jk

!k vb vj dx;

I2 D

R3

R3

Z

jk

I3 D

Z

!b vj vk dx; R3

I4r D Z

! j jvj dx; jrvj jr.'r; b/j j!

Z1r D

R3

v  @l .'r; br /@l vdx;

(8)

R3

Z D

! j jvj2 j.'r; b/j dx; j!

R3

Z

I5r D 2 Z2r

!j vk vb dx;

Z

@l p@k .'r; br /@l vk dx;

Z3r

R3

D R3

Z

1 @l jvj2 @k .'r; br /@l vk dx; 2

!  v/  @l .'r; br /@l vdx .!

Z4r D R3

(9) and "ijk is the Levi-Civita tensor, i.e. it is zero unless all indices are different, it is equal to C1 for a positive permutation of 123 and equal to 1 otherwise. Now, we will estimate these integrals in order to finish the proof. At first, we will consider the case, in which we have the additional information about the projection vb itself; we will proceed quite analogously with [18]. Z ˇ ˇ Z Z Z ˇ ˇ   ˇ jk ˇ ˇ ˇ !k vb vj dxd  kvb ks k!k k 2s vj 2 d

ˇI1 ˇ d  s2 0

0 R3

Z C

0 s3 3  kvb ks k!k k2 s kr!k k2s vj 2 d

0

" kvk22;2

Z C C"

2s

kvb kss3 krvk22 d

0 2s

s3 " kvk22;2 C C" .T/ kvb kt;s krvk21;2



t

2s  : s3

90

Š. Axmann and M. Pokorný

Next Z jk

I2 D"jmn

@m vn vk vb dx R3

Z

Z

D  "jmn

@m @l vn vk @l vb dx  "jmn R3

Z

D  "jmn

Z

@m @l vn vk @l vb dx C "jmn R3

„

@m vn @l vk @l vb dx R3

ƒ‚

R3

„

…

@m @l vn @l vk vb dx ƒ‚

jk

…

jk

J1

J2

Z C "jmn

@l @l vk @m vn vb dx: R3

„

ƒ‚

…

jk

J3

Here, Z ˇ ˇ Z ˇ ˇ ˇ jk ˇ 2 ˇ ˇ "jmn @m @l vn vk @l vb dx  " kvk2 C C" vk2 .@l vb /2 dx ˇJ1 ˇ  R3

" kvk22

Z  C"

vb @l @l vb vk2 dx

Z  C"

R3

R3

  vb @l vb @l vk2 dx:

(10)

R3

Let us estimate the first integral on the right hand side of (10): ˇ ˇ  ˇ Z Z ˇZ Z ˇ ˇ 2 ˇ vb @l @l vb vk dxd ˇˇ  jbj jvj jvb j jvj2 dxd

ˇ ˇ ˇ 0 R3 0 R3 „ ƒ‚ … J11

Z

Z

C2

Z Z

2

jrbj jrvj jvb j jvj dxd C 0 R3

„

ƒ‚ J12

…

0 R3

„

jbj jvb j jvj3 dxd : ƒ‚ J13

…

A Generalization of Some Regularity Criteria to the Navier–Stokes Equations. . .

91

Then Z J11 

kvk2 kvb ks kvk2 4s d

s2

0

Z C

kvk2 kvb ks kvk

3s s3

kvk6 d

0

 C kvk2;2 kvb kt;s kvk As

2 4s=.sC6/

C

3 3s=.s3/

4s 3s sC6 ; s3

kvk1;6 :

D 32 , we can interpolate kvk

sC6

s6

4s 3s sC6 ; s3

2s  C.s/ kvk1;2 krvk2;22s :

Thus, using Young’s inequality and energy inequality we get J11  " kvk22;2 C C" kvb k2t;s krvk21;2 : Note that for fixed " > 0, C" ! 0 for  ! 0, uniformly for s 2 Œ6; 1. The lower order terms J12 , and J13 may be bounded as follows: Z J12 

kvb ks krvk2 krbk1 kvk2 4s d

s2

0

Z 

kvb ks krvk2 krbk1 kvk

3s s3

kvk6 d

0

 kvb kt;s krvk1;2 krbk2;1 kvk

4s 3s sC6 ; s3

kvk1;6

C kvb kt;s krvk21;2 ; Z J13 

kvb ks kvk26 kvk

3s s3

 2  r b d

3

0

 kvb kt;s krvk21;2 kvk

4s 3s sC6 ; s3

 2  r b 2;3

 C kvb kt;s krvk21;2 ;   where we have used Hölder’s inequality 1t C 12 C sC6 4s D 1 , the assumptions on b.; /, and the fact that from energy inequality we have estimate of the norm of v in

92

Š. Axmann and M. Pokorný

 4s 3s  space L sC6 .0; T; .L s3 R3 /3 /. Note that C D C./ ! 0 for  ! 0 uniformly for s 2 Œ6; 1. Further, we will estimate the last integral from (10): ˇ  ˇ ˇZ Z ˇ Z Z ˇ ˇ  2 ˇ vb @l vb @l vk dxd ˇˇ  jvb j jrvj2 jvj jbj dxd

ˇ ˇ ˇ 0 R3 0 R3 „ ƒ‚ … DJ14

Z Z C

jvb j jrbj jrvj jvj2 dxd

0 R3

„ Z J14  C

kvb ks krvk23 kvk

3s s3

ƒ‚

…

J15 DJ12

d  C kvb kt;s krvk24;3 kvk

2t 3s t2 ; s3

0

The interpolation inequalities kvk

2t 3s t2 ; s3

2=t

.t2/=t

 C kvk1;2 kvk2;6

1

1

2 2 , and krvk4;3  Ckrvk1;2 ; kvk2;2

and Young’s inequality yield ˇ ˇ  ˇ ˇZ Z ˇ ˇ  2 ˇ vb @l vb @l vk dxd ˇˇ  " kvk22;2 C C" kvb k2t;s krvk21;2 ; ˇ R3 ˇ ˇ 0

jk

which implies the bound on J1

ˇ ˇ  ˇ ˇZ   ˇ ˇ jk ˇ J d ˇ  3" kvk2 C C" kvb k2 C kvb k krvk2 : t;s 2;2 t;s 1;2 1 ˇ ˇ ˇ ˇ 0

Further, Z ˇ Z  Z ˇ ˇ jk jk ˇ ˇJ2 C J3 ˇ d  0

0

Z R3

j@l @m vn @l vk vb j dx C

Z 2

kvk2 kvb ks krvk 0

2s s2

d

R3

 j@l @l vk @m vn vb j dx d

A Generalization of Some Regularity Criteria to the Navier–Stokes Equations. . .

Z

3

s3

C

93

kvb ks krvk2 s kvk2s kvk2 d

0

" kvk22;2 C C" kvb kt;s krvk21;2 ; Z ˇ ˇ Z Z Z ˇ ˇ ˇ jk ˇ 2 j!b j ˇvj ˇ jvk j dxd  " kvk2;2 C C" kvk23 k!b k26 d

ˇI3 ˇ d  0

0 R3

0

" kvk22;2

C

C" kvk21;3

k!b k22;6

:

Using kvk21;3  kvk1;2 kvk1;6  C krvk1;2 , and the information which comes from (3), we get kvk21;3 k!b k22;6 C krvk1;2 .1 C krvkY. / /  C./.krvk1;2 C krvk21;2 C kvk22;2 / C C0 kv0 k21;2 : Recall that C ! 0 for  ! 0C , uniformly for s 2 Œ6; 1. It remains to deduce suitable estimates of the lower order terms with derivatives of b.; /. Z

Z Z jI4r j d



0

R3

0

! j jvj2 j.'r; b/j dxd

j!

Z 

krvk6 kvk3 kvk6 k.'r; b/k3 d

0

" kvk22;2

Z C C"

krvk22 kvk23 k.'r; b/k23 d

0

" kvk22;2 C C" krvk21;2 kvk24;3 k.'r; b/k24;3 Z

Z Z jZ1r j d

0



jvj jr.'r; b/j jrvj dxd

0 R3

" kvk2;2 C C" krvk21;2 kr.'r; b/k22;1

94

Š. Axmann and M. Pokorný

Z jI5r C Z2r C Z3r C Z4r j d

0



Z Z  0

R3

! j jvj C jrpjjr.'r; b/jjrvj jrvj jr.'r; b/j j!  ! j jvj jrvj jr.'r; b/j dxd

C jrvj2 jvjjr.'r; b/j C j!

Z krvk6 krvk2 kvk3 .krbk1 C 1/ d

C 0

 1  C kvk2;2 krvk1;2 kvk4;3 krbk4;1 C .T  / 4  " kvk22;2 C C" krvk21;2 kvk24;3 Collecting all the above estimates together, we see that krvk21;2 C kvk22;2  C0 kv0 k21;2 C C.krvk21;2 C kvk22;2 C 1/;

(11)

where C ! 0 for  ! 0C , uniformly for s 2 Œ6; 1. Therefore, taking  sufficiently small, we get krvk21;2 C kvk22;2  4C0 kv0 k21;2 : Repeating the same estimates on .; 2/ we get that krvk2L1 .;2 IL2 .R3 // C kvk2L2 .;2 IL2 .R3 //  4C0 kv./k21;2 : Therefore, after finite number of steps, we get that the regular solution exists on the whole time interval .0; T/. Let us move to the case where we have information about the gradient of the jk projection and let us estimate all terms from (8). We start with the term I2 : Z jk

I2 D "jmn

@m vn vk vb dx R3

Z D "jmn „

Z R3

@m @l vn vk @l vb dx "jmn ƒ‚ …„ jk

@m vn @l vk @l vb dx ƒ‚ … jk

J1

Z ˇ ˇ ˇ jk ˇ ˇJ1 ˇ  " kvk22 C C"

R3

J2

R3

jvj2 jrvb j2 dx

(12)

A Generalization of Some Regularity Criteria to the Navier–Stokes Equations. . .

95

In the estimate of the right hand side of (12), we will distinguish between two possible cases. For 2  s  3, we get Z R3

jvj2 jrvb j2 dx  krvb k2s kvk2 2s

s2 4s6 s

 krvb k2s krvk2

 2  62s r v s 2

2s  2 " r 2 v2 C C" krvb ks2s3 krvk22 ;

while for s > 3, we will proceed in the following way Z

6s

R3

4s12

jvj2 jrvb j2 dx  krvb ks5s6 krvb k25s6 kvk22 5s6 : 3 s2

Further, due to 2 

2 5s6 3 s2

 6, we can interpolate 4s12

6s

kvk22 5s6  Ckvk25s6 krvk25s6 : 3 s2

Moreover, 4s12

4s12

krvb k25s6  .krvk2 C kvk2 krbk1 / 5s6 ; and using Young’s inequality we have Z

6s

R3

4s12

jvj2 jrvb j2 dx C krvb ks5s6 kvk25s6 jk

  krvk22 C C0 .b/ :

jk

The integrals J2 , and I3 can be estimated in a straightforward way, analogously as in [15] ˇ ˇ 2s  2 ˇ jk ˇ 2 2 ˇJ2 ˇ  krvb ks krvk 2s  " r 2 v2 C C" krvb ks2s3 krvk2 ; s1

ˇ ˇ   2 ˇ jk ˇ  2  4 4 ˇI3 ˇ  r v2 k!b k3 kvk6  " r 2 v2 C "k!b k3 C C" krvk2 :

96

Š. Axmann and M. Pokorný jk

We now return to the term I1 . We have (below, ıij denotes the Kronecker symbol) Z

Z

jk

"ijk I1 D "ijk

R3

!k vb vj dx D "ijk "klm Z

D .ıil ıjm  ıim ıjl /

R3

Z D

R3

R3

@l vm vb vj dx Z

@l vm vb vj dx D

@i vj @l vb @l vj dx C

1 2

R3

Z R3

.@i vj vb vj  @j vi vb vj /dx

.@l vj /2 @i vb dx C

Z R3

vi @j vb vj dx: jk

Therefore these terms can be treated exactly as terms above coming from I2 . Next, we have to estimate the lower order terms. Since they can be treated exactly as in the previous case (additional information about vb ), we skip the details. Altogether we get krvk21;2

 2 C r 2 v2;2  "

Z

k!b k43 d

0

Z  C C"

2s

6s

g.s/ krvb ks2s3 C0 .b/ C krvb ks5s6 0

Ckbk26

C

kvk43

C

krbk41

C

krvk22



krvk22 d ;

where g.s/ D 0, for 2  s  3, and g.s/ D 1, for s > 3. Note that both 6s 4s are less than t D 3s6 . Using the estimate from Lemma 1 we obtain 5s6 Z

2s 2s3

and

k!b k43 d  C.1 C krvk2Y. / /:

0

Choosing " sufficiently small, we can use Gronwall’s inequality in order to conclude that  2 krvk21;2 C r 2 v2;2  C.v0 ; krvb kt;s /: As this inequality holds for any  < T  and C is independent of , the proof of Theorem 1 is complete. Acknowledgements The work of the first author was supported by the grant SVV-2015-260226. The work of the second author was partially supported by the grant No. 201/09/0917 of the Grant Agency of the Czech Republic.

A Generalization of Some Regularity Criteria to the Navier–Stokes Equations. . .

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References 1. Š. Axmann, M. Pokorný, A note on regularity criteria for the solutions to Navier-Stokes equations involving one velocity component (in preparation) 2. C. Cao, E.S. Titi, Regularity criteria for the three-dimensional Navier–Stokes equations. Indiana U. Math. J. 57(6), 2643–2660 (2008) 3. H. El-Owaidy et al., On some new integral inequalities of Gronwall–Bellman type. Appl. Math. Comput. 106, 289–303 (1999) 4. G.P. Galdi, An introduction to the Navier–Stokes initial–boundary value problem, in Fundamental Directions in Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics (Birkhäuser, Basel, 2000), pp. 1–70 5. C. He, Regularity for solutions to the Navier–Stokes equations with one velocity component regular. Electron. J. Differ. Equ. 2002(29), 1–13 (2002) 6. E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4(1), 213–231 (1951) 7. A. Kufner, O. John, S. Fuˇcík, Function Spaces, 1st edn. (Academia, Prague, 1977) 8. I. Kukavica, M. Ziane, One component regularity for the Navier–Stokes equations. Nonlinearity 19(2), 453–469 (2006) 9. O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, 2nd edn. (Gordon and Breach, New York/London/Paris, 1969) 10. O.A. Ladyzhenskaya, Sixth problem of the millenium: Navier–Stokes equations, existence and smoothness. Russ. Math. Surv. 58(2), 251–286 (2003) 11. J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique. J. Math. Pure Appl. 12(2), 1–82 (1933) 12. J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63(1), 193– 248 (1934) 13. J. Neustupa, P. Penel, Regularity of a suitable weak solution to the Navier–Stokes equations as a consequence of regularity of one velocity component, in Applied Nonlinear Analysis, ed. by A. Sequeira et al. (Kluwer Academic/Plenum Pulishers, New York, 1999), pp. 391–402 14. J. Neustupa, A. Novotný, P. Penel, An interior regularity of weak solution to the Navier– Stokes equations in dependence on one component of velocity, in Quaderni di Matematica, ed. by A.O. Eden. Topics in Mathematical Fluid Mechanics, vol. 10 (Dept. Math., Seconda Univ. Napoli, Caserta, 2002), pp. 163–183 15. M. Pokorný, On the result of He concerning the smoothness of solutions to the Navier–Stokes equations. Electron. J. Differ. Equ. 2003(11), 1–8 (2003) 16. R. Temam, Some developments on Navier–Stokes equations in the second half of the 20th century, in Development of Mathematics 1950–2000 (Birkhäuser, Basel, 2000), pp. 1049–1106 17. Y. Zhou, A new regularity criterion for the Navier–Stokes equations in terms of the gradient of one velocity component. Methods Appl. Anal. 9(4), 563–578 (2002) 18. Y. Zhou, A new regularity criterion for weak solutions to the Navier–Stokes equations. J. Math. Pure Appl. 84(11), 1496–1514 (2005) 19. Y. Zhou, M. Pokorný, On a regularity criterion for the Navier–Stokes equations involving gradient of one velocity component. J. Math. Phys. 50(12), 123514 (2009) 20. Y. Zhou, M. Pokorný, On the regularity criterion of the Navier–Stokes equations via one velocity component. Nonlinearity 23(5), 1097–1107 (2010)

On the Singular p-Laplacian System Under Navier Slip Type Boundary Conditions: The Gradient-Symmetric Case H. Beirão da Veiga

Dedicated to Yoshihiro Shibata on the occasion of his 60th birthday

Abstract We consider the p-Laplacian system of N equations in n space variables, 1 < p  2 ; under the homogeneous Navier slip boundary condition without friction. Here, the gradient of the velocity is replaced by the (more physical) symmetric gradient, and the classical non-slip boundary condition is replaced by the Navier slip boundary condition without friction. These combination of circumstances leads to some additional obstacles. We prove W 2; q regularity, up to the boundary, under suitable assumptions on the couple p; q . The singular case  D 0 is covered. Keywords Regularity up to the boundary • Singular p-Laplacian elliptic systems • Slip boundary conditions

Mathematics Subject Classification (2010). 35J57; 35J60; 35J75; 35J92

1 Introduction: The Main Result In the sequel  is a bounded, open, connected, subset of Rn ; locally situated on one side of its boundary, a smooth manifold . We denote by n the outer unit normal to @ : For convenience, we assume that  has not axis of symmetry. The reason will be clear below. By D u D ru C r T u we denote the symmetric gradient. So Di j .u/ D @i uj C @j ui ;

(1)

H. Beirão da Veiga () Dipartimento di Matematica, Università di Pisa, Via Buonarroti 1/C, 56127 Pisa, Italy e-mail: [email protected] © Springer Basel 2016 H. Amann et al. (eds.), Recent Developments of Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-0348-0939-9_6

99

100

H. Beirão da Veiga

where i; j D 1; 2; : : : ; n . Often we simply write D, provided that the vector field under consideration follows from the context. Further, we denote by t.u/ the Cauchy stress vector t.u/ D . D u/  n : So, tj D .@i uj C @j ui / ni ; where (here and in the sequel) we use the summation convention on repeated indexes. In these notes we consider the system   p2  r  .  C j D uj2 / 2 D u D f in  ;

(2)

where u is an N-dimensional vector field,   0 is a given parameter (we are particularly interested in the singular case  D 0 ), and f is a given vector field. The vector field u is subjected to the Navier slip boundary condition without friction, see [22], 

u  n D 0; .t.u// D 0 ;

(3)

where, in general, the subscript  denotes “tangential component”. Equation (2) has been considered by mathematicians mostly with D u replaced by ru : It is worth noting that (2) satisfies the Stokes Principle (see [23, 26], p. 231), a significant physical requirement of isotropy, which does not hold if we replace D u by ru : For a mathematical study of the above boundary condition see, for instance, [7, 25], where this boundary condition is associated to the linear Stokes problem. By Lp ./ and W m;p ./, 1  p  1 , m nonnegative integer, we denote the usual Lebesgue and Sobolev spaces, with the standard norms k  kp and k  km;p , respectively. In notation concerning norms and functional spaces, we do not distinguish between scalar and vector fields. For instance Lp .I RN / D ŒLp ./N , N > 1, is denoted simply by Lp ./. We define Vp D Vp ./ D fv 2 W 1; p ./ W v  n D 0 on  g: The linear space Vp ./ , endowed with one of the following norms 

k v kp C k D v kp

 1p

;



k v kp C k r v kp

 1p

;

k r v kp ;

is a Banach space. The above norms are equivalent in Vp ./ : Further, since we assume that the domain  has not axis of symmetry, k D v kp alone is also an

On the Singular p-Laplacian System Under Navier Slip Type Boundary. . .

101

equivalent norm. Without the above assumption on  , the equivalence does not hold. For a quite complete discussion on this, and related, arguments, we refer to [7]. See in particular the Lemma 2.3, and related results, in this last reference. Note, in particular, that the space V1 ./ considered in Ref. [7] coincides here with V2 ./ , since the space Z , defined in the above reference, is here reduced to f 0 g ; due to the absence of axis of symmetry in  . Further, we remark that in [7] we appeal to (12) instead of appealing to the totally equivalent formulation (11). Our main result is the following. Theorem 1.1 Let 1 < p  2 be fixed. Assume that   0 ; and let f 2 Lq ./ ; where q > n : Let Cq D C.q; / be the constant that appears in the linear estimate (14) below. Assume that .2  p/ Cq < 1 :

(4)

Then, the weak solution u to the problem (2), (3) belongs to W 2;q ./ . Moreover, the following estimate holds kuk2;q



1 p1 :  C k f kq C k f kq

(5)

Roughly speaking, below we tried to follow the main lines of the proof given in [10]. However, the presence of the symmetric gradient in place of the gradient, and that of the Navier slip boundary condition in place of the non-slip boundary condition, lead to additional obstacles. In booth proofs, following [8], the very starting point is the analysis, and subtle utilization, of the inner structure of the equations. An open problem. Let us now consider the absolute vorticity boundary condition 

u  n D 0; !.u/  n D 0 ;

(6)

where N D 3 : This conditions was introduced by Bardos in Ref. [5]. It became very popular, and has been studied by a large number of authors. On flat portions of the boundary it coincides with the boundary condition (3). So, at least at first glance, it seems very likely that proofs and results shown below for (3) also hold for (6), with simple adaptations. We are able to prove -uniform a priori estimates in W 2;q ./ ; but we have not a suitable definition of “weak solution” to the above boundary value problem, even in the full gradient case. Regularity of solutions for systems like (2) has received substantial attention from many authors. We refer, for instance, to [1, 16, 18, 20, 27, 28]. Other related results may be found in [3, 4, 11–13, 15, 21], and references therein. We refer readers particularly interested in studying new connections between different boundary value assumptions, to the recent challenging paper [6]. Finally, we refer the reader to [17], where the authors prove very interesting regularity results for the Stokes

102

H. Beirão da Veiga

problem (insert, as usual, pressure and divergence free conditions in equation (2)), under the boundary value problem (3). The plan of the paper is the following: In Sect. 2 we recall the existence and uniqueness result of weak solutions. In Sect. 3 we introduce an auxiliary linear problem and state (by appealing to well know classical results) the existence of W 2; q ./ solutions to this linear problem. In Sect. 4 we formulate the non-linear problem in a more explicit (formally equivalent) form, in which the non-linearities are (roughly speaking) concentrated in the right hand side (see Eq. (18) below). Further, we appeal to this last formulation to define “strong solution”. In Sect. 5, by assuming  > 0 ; and by appealing to the result stated in Sect. 3 for the auxiliary linear problem, we show that the strong solutions introduced in Sect. 4 exist and belongs to W 2; q ./ ; for each  > 0 : Moreover, the estimates obtained are independent of  : This last property allows us, in Sect. 6, to extend the regularity result to the singular case  D 0 : This is accomplished by passing to the limit in the variational formulation (10), as  tends to zero. Sections 5 and 6 follow arguments developed in [10].

2 Existence and Uniqueness of the Weak Solution Existence and uniqueness of weak solutions follows from well know results. Let us recall some basic points. Set B. D u / D .  C j D uj2 /

p2 2

:

(7)

By appealing to the identity Dij u Dij v D 2 Dij u @j vi ; integration by parts shows that 1 2

Z

Z 

B.D u/ D u  D v dx D 



Z

C 

  r  B.D u/ D.u/  v dx (8)

B.D u/ Π.D u/  v  n  dS :

Hence, 1 2

Z

Z 

B.D u/ D u  D v dx D 

Z

C 



  r  B.D u/ D.u/  v dx

B.D u/ .t.u//  v dS :

Note that on  one has t.u/  v D .t.u//  v ; since v  n D 0 :

(9)

On the Singular p-Laplacian System Under Navier Slip Type Boundary. . .

103

Identity (9) justifies the following definition. Definition 2.1 Let f 2 Vp0 ./. We say that u is a weak solution of problem (2), (3) if u 2 Vp ./ satisfies 1 2

Z

Z B.D u/  D u  D v dx D



f  v dx ;



(10)

for all v 2 Vp ./ . Existence and uniqueness of the weak solution, for each fixed   0 ; follows by appealing to the theory of monotone operators, see Lions, [19, Chap. 2, Theorem 2.1]. Recall that k D v kp is a norm in Vp ./ : Further, take into account that, for y  0 ; .  C y2 /

p 2 2

y2  c yp  c ;

c

if



1  1C 

2 p 2

:

3 An Auxiliary Linear Problem In this section we consider the following linear elliptic equation  r  D u D F in  ;

(11)

under the linear boundary condition (3). We remark that Eq. (11) may be written in the equivalent form   u  r .r  u / D F :

(12)

The following definition coincides with definition 2.1, if p D 2 : Definition 3.1 Let f 2 V20 ./. We say that u is a weak solution to the linear problem (11), (3) if u 2 V2 ./ satisfies 1 2

Z 

Z D u  D v dx D



F  v dx ;

(13)

for all v 2 V2 ./ . Coerciveness of the bilinear form on the left hand side of (13) follows here by appealing to the fact that k D v k is a norm in V2 ./ , since we have assumed that  has not axis of symmetry. Hence, existence, uniqueness, and the standard estimate holds for the above problem.

104

H. Beirão da Veiga

In the next sections we appeal to the following regularity result. Theorem 3.1 Consider the linear boundary value problem (11), (3). Assume that F 2 Lq ./ ; for some q  2 : Then, the weak solution u to the above linear problem belongs to W 2; q ./ : Furthermore, there is a constant Cq D Cq .q; / ; such that k r Du kq  Cq k F kq :

(14)

The W 2; 2 ./ regularity of weak solutions claimed in the theorem may be proved, for instance, by following [25] or [7]. The reader may adapt the argument developed in [25, Sect. 4] or, alternatively, in [7, Sects. 3–7]. In this last reference a particularly complete proof of the W 2; 2 ./ regularity is given. Actually, in [7], we consider a much more involved problem. So it is a quite tedious, but mathematically straightforward exercise, to adapt the proof given in [7] to the simpler problem considered here. This is simply done by cutting out many pieces in the proof. Further, as claimed in [25, Sect. 4], once we have proved W 2; 2 ./ existence and regularity, the deeper W 2; q ./ regularity result, for arbitrarily large exponents q, follows by appealing to the corresponding a priori estimates. For these estimates see the classical references [2] by Agmon et al. or [24] by Solonnikov. Since all the linear maps at stake are bounded, there is a constant CQ q such that k u k2; q  CQ q k F kq :

(15)

Finally, the point-wise estimate jr 2 uj  3 jr D uj  6 jr 2 uj shows that the norms k u k2; q and k r Du kq are equivalent in W 2; q ./ \ Vq ./ : So, (14) holds.

4 The Strong Solution: Definition The main lines followed in this section have their starting point in some ideas introduced, in a more complex context, in [8]. Since r .  C j D uj2 /

p2 2

D

p4 p2 .  C j D uj2 / 2 r . jD uj2 / ; 2

straightforward calculations show that   p2 p2 r  .  C j D uj2 / 2 D u D .  C j D uj2 / 2 r  .Du / C .p  2/ .  C j D uj2 /

p4 2

I.u/

where, by definition, I.u/ D

1 r . jD uj2 /  D u D . Du W rDu /  Du : 2

(16)

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105

The j component of the vector field I.u/ is given by Ij .u/ D

XX k

Dlm .@k Dlm / Dkj :

l; m

By improving an argument already used in [10], we may prove (as in the proof of Lemma 3.4 in [9]) the algebraic relation jI  j  jDj2 jr D uj jj ; for each arbitrary vector field in  2 RN , where jr D uj2 D

X

.@k Dm l /2 :

m;l;k

Consequently, the pointwise estimate jI.u/j  jDj2 jr D uj

(17)

holds. Finally, we introduce the notion of strong solution used in the next section. Definition 4.1 Assume that  > 0 ; and let f 2 Lq ./ be given, q > 1 . We say that u 2 W 2; q ./ is a strong solution of problem (2), (3) if u satisfies (3) in the trace sense and, moreover, the equation  r  D u D .p  2/ G.u/ C .  C j D uj2 /

2p 2

f

(18)

holds almost everywhere in  ; where G.v/ D .  C j D vj2 /1 I.v/ : Note that, by (17), G.v/  j r D v j almost everywhere in  ; independently of  . So, k G.v/ kq  kr D vkq :

(19)

5 Existence of the Strong Solution for  > 0 In this and in the next section we prove the existence of a (unique) strong solution u 2 W 2; q ./ of our problem. In this section the case  > 0 is considered. In the next section the result is extended to the singular case. In these two sections we

106

H. Beirão da Veiga

follow the proofs given in [10]. For the reader’s convenience, we show the main points. For more details, we refer to the original proof, in [10]. O Fix  > 0 ; and let f 2 Lq ./ . Since q > n ; there is a constant C.q; / such that k D vk1  CO kr D v kq ;

(20)

for all v 2 W 2; q ./ \ Vq ./ : Hence, 2 p 2p p k j Dvj2p f kq  k Dv k1 k f kq  CO kr D v k2 k f kq : q

(21)

Further, since .a C b/˛  a˛ C b˛ for nonnegative a and b, and 0 < ˛ < 1 ; it follows that . C jD vj2 /

2p 2

2p 2

C jD vj2p :

(22)

2 p p k f kq C CO k r Dv k2 k f kq : q

(23)

 

From (21) and (22) we show that k . C j D v j2 /

2p 2

f kq  

2p 2

Next we define the convex closed set K D K.R/ D fv 2 W 2; q ./ \ Vq ./ W krD vkq  R g ;

(24)

and consider, for each v 2 K ; the solution u D T.v/ to the problem  r  D u D F.v/ .p  2/ G.v/ C .  C j D vj2 /

2 p 2

f;

(25)

under the boundary conditions (3). We want to prove the existence of a fixed point T.u/ D u 2 K : By appealing to Eqs. (14), (19), and (23), we obtain the estimate k r Du kq  Cq f .2  p/ k r Dv kq C 

2p 2

2 p p k f kq C CO k r Dv k2 k f kq g : q (26)

Next we show that if k r Dvkq  R then the corresponding solution u D T.v/ satisfies the same estimate, namely k r Dukq  R . This shows that T.K/  K : Since v 2 K ; it follows that k r Du kq  

2p 2

Cq k f kq C .2  p/ Cq R C Cq CO

2 p

k f kq R2 p :

By assuming (4), we show that u 2 K.R/ if Π1  .2  p/ Cq  R  

2p 2

2 p Cq k f kq C Cq CO k f kq R2 p :

(27)

On the Singular p-Laplacian System Under Navier Slip Type Boundary. . .

107

This inequality is satisfied if, for instance, its left hand side is larger or equal to two times each of the two terms on the right hand side. This holds for RD

2 p 1 1 2 2p 2 Cq CO  2 Cq k f kq C . / p 1 k f kqp 1 ; ˛ ˛

(28)

where ˛ D 1  C2 .q/ .2  p/ : Hence k r Du kq  R : The inclusion T.K/  K follows. This is the main ingredient to prove the existence of a fixed point in K ; by appealing to the Theorem 3.2 in [10]. For more details we refer to the argument developed in this last reference. The expression of R shows that the uniform estimate (5) holds. Summarizing, we have shown that, for each positive  ; the estimate ku k2;q  C



1 k f kq C k f kqp1

(29)

holds, where u denotes the strong solution related to the particular positive value  .

6 Existence of the Strong Solution for  D 0 Following [10], we appeal here to the uniformity with respect to  > 0 of (29), and to a compactness argument, to pass to the limit, as  tends to zero, in the weak formulation (10). In this way we show that the weak solutions u to the singular problem belong to W 2;q ./ ; and satisfy (5). We start by recalling the definition of weak solution u of problem (2), for   0 ; namely, Z

 

 2

 C jDu j

 p2 2

Z



D u  D v dx D



f  v dx ;

(30)

for all v 2 Vp ./ : This condition is satisfied by the strong solutions u ; for  > 0 ; constructed in the previous section. Since these solutions are uniformly bounded in W 2; q ./ ; suitable sub-sequences, which we continue to denote by u , weakly converge in W 2; q ./ to some u . Let us show, by passing to the limit in (30), that Z j D uj 

p2 2

Z D u  D v dx D



f  v dx ;

(31)

for all v 2 Vp ./ : Consequently, u 2 W 2; q ./ is the solution (know to be unique) for  D 0 : To prove this claim, we have to show that the left hand side of Eq. (30) converges to the left hand side of (31), for each fixed v 2 Vp ./ . Essentially, the proof followed in [10, Sect. 4] applies here. For the reader’s convenience, we repeat the main argument.

108

H. Beirão da Veiga

Since u * u weakly in W 2;q ./ , and q > n ; strong convergence (of suitable subsequences) in W 1;s ./ , for any s ; follows. So, strong convergence in W 1;p ./ holds. Write the integral on the left-hand side of (30) as Z



 C j Du j2



 p2 2 Z



C 

 p2 

Du   C j Du j2 2 Du  Dv dx  C j Duj2

 p2 2

(32)

Du  Dv dx ;

and show that the first integral tends to zero, and the second integral tends to the left hand side of (31). The inequality j . C jAj/p2 A  . C jBj/p2 B j  C jA  Bj . C jAj C jBj / p2 ;

(33)

where C is independent of  (see [14, Eq. (6.8)]), shows that the absolute value of the first integral in Eq. (32) is bounded by Z C 

.  C j D u j C j D u j /p2 j D u  Du j j D vj dx :

Since .  C jDu j C jDu j /p2 j Du  Du j  j Du  Du jp1 ; the absolute value of the first integral in Eq. (32) is bounded by C k D u  D u kpp1 k D v kp ; which tends to zero with  : Finally, by Lebesgue’s dominated convergence theorem, Z



lim

!0C



 CjDuj

2

 p2 2

Z D u  D v dx D



j D u jp2 Du  D v dx :

References 1. E. Acerbi, N. Fusco, Regularity for minimizers of nonquadratic functionals: the case 1 < p < 2. J. Math. Anal. Appl. 140, 115–135 (1989) 2. 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) 3. S. Antontsev, S. Shmarev, Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties for solutions. Nonlinear Anal. 65, 728–761 (2006)

On the Singular p-Laplacian System Under Navier Slip Type Boundary. . .

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4. S. Antontsev, S. Shmarev, Anisotropic parabolic equations with variable nonlinearity. Publ. Mat. 53, 355–399 (2009) 5. C. Bardos, Existence et unicité de la solution de l’équation de Euler en dimension deux. J. Math. Anal. Appl. 40, 769–790 (1972) 6. C. Bardos, F. Golse, L. Paillard, The incompressible Euler limit of the Boltzmann equation with accommodation boundary condition. Commun. Math. Sci. 10, 159–190 (2012) 7. H. Beirão da Veiga, Regularity for Stokes and generalized Stokes systems under non homogeneous slip type boundary conditions. Adv. Differ. Equ. 9(9–10), 1079–1114 (2004) 8. H. Beirão da Veiga, On the regularity of flows with Ladyzhenskaya shear dependent viscosity and slip or non-slip boundary conditions. Commun. Pure Appl. Math. 58, 552–577 (2005). (online: 15th June 2004, doi:10.1002/cpa.20036) 9. H. Beirão da Veiga, On the global regularity for singular p-systems under nonhomogeneous Dirichlet boundary conditions. J. Math. Anal. Appl. 398, 527–533 (2013). doi:10.1016/j.jmaa.2012.08.058 10. H. Beirão da Veiga, F. Crispo, On the global W 2; q regularity for nonlinear N-systems of the p-Laplacian type in n space variables. Nonlinear Anal. Theory Methods Appl. 75, 4346–4354 (2012). doi:10.1016/j.na.2012.03.021 11. H. Beirão da Veiga, F. Crispo, On the global regularity for nonlinear systems of the p-Laplacian type. Discrete Continuous Dyn. Syst. Ser. S 6, 1173–1191 (2013). arXiv:1008.3262v1 [math.AP] 12. E. DiBenedetto, C1C˛ local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7, 827–850 (1983) 13. E. DiBenedetto, J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems. Am. J. Math. 115, 1107–1134 (1993) 14. L. Diening, C. Ebmeyer, M. R˚užiˇcka, Optimal convergence for the implicit space-time discretization of parabolic systems with p-structure. SIAM J. Numer. Anal. 45, 457–472 (2007) 15. M. Fuchs, G. Seregin, Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids. Lecture Notes in Mathematics, vol. 1749 (Springer, Berlin, 2000) 16. C. Hamburger, Regularity of differential forms minimizing degenerate elliptic functionals. J. Reine Angew. Math. 431, 7–64 (1992) 17. P. Kaplický, J. Tichý, Boundary regularity of flows under perfect slip boundary conditions. Cent. Eur. J. Math. 11, 1243–1263 (2013) 18. G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12, 1203–1219 (1988) 19. J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires (Dunod, Paris, 1969) 20. W.B. Liu, J.W. Barrett, A remark on the regularity of the solutions of the p-Laplacian and its application to their finite element approximation. J. Math. Anal. Appl. 178, 470–487 (1993) 21. P. Marcellini, G. Papi, Nonlinear elliptic systems with general growth. J. Differ. Equ. 221, 412–443 (2006) 22. C.L.M. Navier, Mémoire sur les lois du mouvement des fluides. Mémoires de l’Académie Royale des Sciences de l’Institut de France 6, 389–440 (1927) 23. J. Serrin, Mathematical principles of classical fluid mechanics, in Handbuch der Physik, Bd VIII/1, (Springer, Berlin 1959), pp. 125–263 24. V.A. Solonnikov, General boundary value problems for Douglis-Nirenberg elliptic systems. II, in Proceedings of the Steklov Institute of Mathematics, vol. 92 (1966) 25. V.A. Solonnikov, V.E. Šˇcadilov, On a boundary value problem for a stationary system of Navier-Stokes equations. Proc. Steklov Inst. Math. 125, 186–199 (1973) 26. G. Stokes, Trans. Camb. Philos. Soc. 8, 287, 75–129 (1845) 27. P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51, 126–150 (1984) 28. N.N. Ural’tseva, Degenerate quasilinear elliptic systems. Zap. NauK. Sem. LOMI 7, 184–222 (1968) (in Russian)

Thermodynamically Consistent Modeling for Dissolution/Growth of Bubbles in an Incompressible Solvent Dieter Bothe and Kohei Soga

Dedicated to Yoshihiro Shibata on the Occasion of his 60th Anniversary

Abstract We derive mathematical models of the elementary process of dissolution/growth of bubbles in a liquid under pressure control. The modeling starts with a fully compressible version, both for the liquid and the gas phase so that the entropy principle can be easily evaluated. This yields a full PDE system for a compressible two-phase fluid with mass transfer of the gaseous species. Then the passage to an incompressible solvent in the liquid phase is discussed, where a carefully chosen equation of state for the liquid mixture pressure allows for a limit in which the solvent density is constant. We finally provide a simplification of the PDE system in case of a dilute solution. Keywords Entropy principle • Incompressible limit • Mass transfer • Two-phase fluid system

1 Introduction The process of dissolution or growth of gas bubbles in an ambient liquid phase is very common in many situations. In everyday life, we often see bubbles in carbonated mineral water, beer, champagne etc. In particular the dissolution of gases is of huge technological and industrial importance in the context of gas scrubbing. This is, for instance, relevant for CO2 disposal, where gas from a combustion process is injected into a reactive liquid medium. Such processes are usually run under pressure control instead of volume control. Note that the latter is much more

D. Bothe () Technische Universität Darmstadt, Center of Smart Interfaces, 64287 Darmstadt, Germany e-mail: [email protected] K. Soga Department of Mathematics, Faculty of Science and Technology, Keio University, Hiyoshi 3-14-1, Kohoku-ku, Yokohama 223-8522, Japan e-mail: [email protected] © Springer Basel 2016 H. Amann et al. (eds.), Recent Developments of Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-0348-0939-9_7

111

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common in the mathematical analysis of such mass transfer problems, since it allows for a fixed domain in which the mathematical model—usually in the form of a system of partial differential equations—holds. The massive impact of the external pressure is known from the above mentioned everyday life examples, but also can be seen in the medical context. This is the case with decompression sickness or caisson disease, where severe symptoms can be caused by bubble generation in the blood after a fast change of the ambient pressure. There is a large literature on experiments and numerical computation of dissolution/growth of bubbles in a liquid, e.g. Liger-Belair et al. [10], Sauzade and Cubaud [13], Takemura and Yabe [16]. A rigorous mathematical model is necessary for possible theoretical investigations and mathematical analysis on this topic. Based on Continuum Physics, we derive a mathematical model of a two-phase fluid system of type liquid/gas, where both gas and liquid phases are composed of molecularly miscible constituents and the pressure is controlled via a free (upper) surface .t/. The system consists of chemical components A1 ; : : : ; AN . The gas phase is denoted by C .t/, the liquid phase by  .t/ and the movable free interface by †.t/. See Fig. 1 below. In common mathematical models for mass transfer from or to gas bubbles in a liquid phase, the transferred gas is treated as a dilute component in both phases. This allows to use a two-phase Navier-Stokes system together with advection-diffusion equations for passive scalars. If the bubble is composed of a pure gas, this is no longer possible since the dissolution then significantly changes the bubble volume. In this case a much more elaborate modeling is required for both of the bulk phases and the transmission condition at the interface. In particular, the two one-sided limits of the bulk velocities at the interface and the interface’s own velocity need to be distinguished. Since such a more rigorous model accounts for the mass and Fig. 1 The two phase system under pressure control

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volume of individual constituents, an incompressible model for the liquid phase will still lead to non-zero divergence of the barycentric velocity field. Moreover, a thermodynamically rigorous model needs to be developed for compressible bulk phases in the first place. Only then, an incompressible version may be derived as a limit, where the latter depends on the notion of incompressibility which is neither a priori clear nor unique in the mixture context. The novel aspect in the present paper is the idea of an incompressible solvent (associated to AN ) carrying dissolved gas components which add their partial pressure to the total one like being ideal gases. The underlying mixture is supposed to be described by an equation of state according to p D pRN C K.

N1 X k N  1/ C RT; R Mk N kD1

where pRN is a reference pressure and NR a reference density for the solvent, while K is the solvent bulk modulus. The incompressible limit will be attained (formally) by letting K tend to infinity. This leads to the constraint N NR ; i.e. to a constant solvent density. Since the continuity equation for the solvent then reduces to r  vN D 0; it makes sense to employ the solvent momentum balance instead of the one for the mixture. This is attractive, because it leads to a standard incompressible NavierStokes equation for the bulk liquid. Only the diffusive fluxes, which rely on the relative velocity to the barycentric one, become slightly more intricate, but only involving a simple linear relation. The obtained PDE systems still comprise of a compressible gas phase model. Low Mach number approximation seems possible and will be given in a forthcoming paper. Note that the gas phase density in the incompressible limit will still be a function of time, determined by the dynamical mass transfer process.

2 Balance Equations (a) Mass balance For simplicity, we assume that there are no chemical reactions (which could be easily added) and that there is no absorbed mass at the interface, i.e. i† 0 for

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all i D 1; : : : ; N. The partial mass balance in its integral form for a fixed control volume V with the outer normal n reads as Z Z d i dx D  i vi  n do: dt V @V Using the two-phase transport and divergence theorems (see the Appendix), this implies Z

Z @t i dx 

Vn†

Z

†

†V

Z

ŒŒi v  n† do D 

r  .i vi / dx  Vn†

†V

ŒŒi vi  n†  do

with †V WD †.t/ \ V, the surface velocity v † and the surface unit normal n† pointing toward  . Comparison of bulk and interface terms yields the local form 

@t i C r  .i vi / D 0 in C .t/ [  .t/, ŒŒi .vi  v † /  n†  D 0 on †.t/:

(1)

Above, the bracket ŒŒ   denotes the jump of a quantity across the interface (crossing † in the direction opposite to n† ). The mixture is described by the total density  and the barycentric velocity v, given by  WD

N X iD1

i ;

v WD

N X

i vi :

iD1

As a consequence of (1), the mixture obeys the continuity equation 

@t  C r  .v/ D 0 in C .t/ [  .t/, ŒŒ.v  v † /  n†  D 0 on †.t/:

(2)

Let m P ˙ denote the one-sided limits ˙ .v ˙  v † /  n† on †.t/. Then the second equation in (2) becomes m P D m P C , and hence m P WD m P D m P C is well-defined. ˙ ˙ ˙ † Similarly, we introduce m P i WD m P i D i .vi  v /  n† . We define diffusion velocities ui WD vi  v, mass fractions yi WD i = and diffusion mass fluxes ji WD i ui D i .vi  v/. Then we have the following equivalent form of Eqs. (1): 

@t i C r  .i v C ji / D 0 in C .t/ [  .t/, ŒŒ ji  n†  C ŒŒi .v  v † /  n†  D 0 at †.t/;

(3)

.@t yi C v  ryi / C r  ji D 0 in C .t/ [  .t/, P i  D 0 at †.t/: ŒŒ ji  n†  C mŒŒy

(4)

or 

Thermodynamically Consistent Modeling for Dissolution/Growth of Bubbles. . .

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In the common models for mass transfer, the jump condition in (3) or (4) is simplified to read ŒŒji  n†  D 0, assuming m P D 0 which means that the total phase change effect of the mass transfer is neglected; cf. Bothe and Fleckenstein [4] for an assessment of this approximation. (b) Momentum balance The mixture is to be described by a so-called class-I model, where we consider only a single (common) momentum balance. The integral form is d dt

Z

Z v dx D  V

@V

Z v.v  n/ do C

Z

Z

Sn do C @V

b dx C V

@†V

S† ds

† with the bulk PNstress tensor S, the surface stress tensor S and the body force b. Note that b D kD1 k bk with (possibly) individual body forces bk , for instance due to forces in an electrical field. Here is the outer unit normal of the bounding curve @†V of †V , being tangential to †. The transport and (surface) divergence theorems yield the local form



@t .v/ C r  .v ˝ v  S/ D b in C .t/ [  .t/, mŒŒv P  ŒŒSn†  D r†  S† at †.t/:

(5)

We assume non-polar fluids, for which the balance of angular momentum has a simple form without body couples or surface couples. This is equivalent to the assumptions S D ST ;

S† D .S† /T :

This is a constitutive assumption which is made right away. (c) Energy balance The integral form of the total energy balance is d dt

Z .e C V

Z

 @†V

Z C @†V

v2 / dx C 2

Z u† do D 

Z †V

u† v †  ds 

Z

v †  S† ds C

Z

@V

q  n do  @V

@†V

Z v  b dx C V

v2 /v  n do 2 Z q†  ds C v  Sn do

.e C

Z X N V kD1

@V

jk  bk dx

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with the specific internal energy of the bulk e and the internal energy density of the surface u† . After straightforward computations, the local form turns out as 8 

 v2 v2 ˆ ˆ @ / C r  .e C /v C q .e C ˆ t ˆ ˆ 2 2 ˆ ˆ N ˆ X ˆ < jk  bk in C .t/ [  .t/, D r  .Sv/ C v  b C ˆ kD1 ˆ † † ˆ ˆ v2 D u ˆ † † † ˆ C u r†  v C ŒŒ.e C /.v  v /  n†  C ŒŒq  n†  C r†  q† ˆ ˆ Dt 2 ˆ : D ŒŒSv  n†  C r†  .S† v † / at †.t/: Subtracting the balance of kinetic energy derived from (5), one obtains the balance of internal energy as 8 N ˆ X ˆ ˆ ˆ @t .e/ C r  .ev C q/ D rv W S C jk  bk ˆ ˆ ˆ ˆ kD1 ˆ ˆ < in C .t/ [  .t/, † † † 2 (6) 1 D u .v  v / ˆ ˆ C u† r†  v † C mŒŒe  n†  Sn†  P C ˆ ˆ Dt 2  ˆ ˆ ˆ † † ˆ  C r  q  ŒŒ.v  v /  Sn  D r† v † W S† CŒŒq  n † † † k ˆ ˆ : at †.t/; where .vv † /k stands for the tangential projection of .vv † / onto the local tangent plane to †, i.e. .vv † /k D P† .vv † / with the projection tensor P† D I n† ˝n† . Later, we will use the constitutive relation S† D  † P† with a scalar  † . Then we have r† v † W S† D  † r†  v † . (d) Entropy balance Let s denote the density of entropy in the bulk (i.e. s is the specific entropy) and † the area-density of interfacial entropy. The integral form of the entropy balance is Z Z Z d † s dx C  do D  .sv C ˆ/  n do dt V †V @V Z Z Z † † † . v C ˆ /  ds C  dx C  † do  @†V

V

†V

with the bulk entropy flux ˆ and the interfacial entropy flux ˆ† . Hence we obtain the local form 8 C  ˆ < @t .s/ C r  .sv C ˆ/ D  in  .t/ [  .t/, (7) D† † ˆ : C † r†  v † C r†  ˆ† C mŒŒs P C ŒŒˆ  n†  D  † at †.t/: Dt

Thermodynamically Consistent Modeling for Dissolution/Growth of Bubbles. . .

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3 Entropy Principle If the entropy fluxes ˆ and ˆ† in (7) are related to the primitive variables via constitutive relations in such a way that the following entropy principle holds, we speak of a thermodynamically consistent model. Entropy principle. The entropy flux (ˆ; ˆ† ) is such that • The entropy production is a sum of binary X products of “fluxes” times “driving X Fm Dm and  † D Fm†0 D† force”, i.e.  D m0 . m0

m

•   0,  †  0 for any thermodynamical process. •  0 and  † 0 characterizes equilibria of the system. This is a condensed form of the full entropy principle. For more details see Bothe and Dreyer [3], as well as Dreyer [8]. We consider the simplest class of isotropic fluids without mesoscopic forces. This corresponds to the choice of certain primitive variables in modeling the entropy of the material. We assume † D h† .u† /;

s D h.e; 1 ; : : : ; N /;

(8)

where h and h† are concave functions. The concavity is required for thermodynamic stability properties of the mixture. Then we define the (absolute) temperature T, respectively T † of bulk and interface, as well as the bulk chemical potentials i via @h i @h 1 WD ;  WD ; T @.e/ T @i

1 @h† WD † : † T @u

(9)

Next, we compute  and  † from (7)–(9), where we eliminate the time derivatives of i , e, u† by means of the balance equations in (1), (6). This yields the following results. (a) Bulk entropy production  D r  .ˆ 

N N X 1 q X k jk C /  .e  sT  k k /r  v T T T kD1 kD1



N 1 1 X k bk C rv W S C q  r   : jk  r T T kD1 T T We choose the entropy flux as q X k jk  T kD1 T N

ˆD

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and determine further constitutive relations so that the entropy principle holds. We decompose the stress tensor S as S D PI C Sı with the traceless part Sı of S and P D  13 tr.S/. We decompose the pressure P as P D p C …, where … vanishes in equilibrium. This is important, since … can depend on r  v, while p cannot. Hence S is rewritten as S D . p C …/I C Sı : Introducing the Helmholtz free energy 

D  .T; 1 ; : : : ; N / D e  sT;

we change from e as a primitive variable to T (via Legendre transform with @.e/ @.s/ D T). Then  becomes 1 D T



Cp

N X

! k k r  v 

kD1

… 1 1 r  v C rv W Sı C q  r T T T



k bk  : jk  r  T T kD1 N X

Now,   0 for any thermodynamical process implies the Gibbs-Duhem relation p D 

C

N X

k k :

iDk

Thus the entropy production in the bulk reduces to read D



N 1 1 X k bk … r  v C rv W Sı C q  r   : jk  r T T T kD1 T T

Since   0 is required, the simplest closure is linear in the driving P forces and such that a quadratic form is obtained. Note that the constraint NkD1 jk D 0 has P to be accounted for. Hence we eliminate jN , which is chosen as  N1 kD1 jk . For D WD 12 .rv C .rv/T / and its traceless part Dı we have rv W Sı D Dı W Sı and trD D r  v. Then  becomes    1 bk  bN  … 1 X k N r  v C Dı W S ı C q  r   : jk  r T T T kD1 T T N1

 D

(10)

Thermodynamically Consistent Modeling for Dissolution/Growth of Bubbles. . .

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Note that the viscous entropy production can be written as T1 D W Sirr , if we let Sirr WD …I C Sı , i.e. Sirr is the irreversible part of S which produces entropy. (b) Interfacial entropy production We do not consider viscous surface dissipation, hence S† D  † P† . Then it follows from the second equation in (7) and the other balance equations that † D

1 † q† 1 † † † † † .  u C T  /r  v C r  .ˆ  / C q†  r† † † † † † T T T 1 1 1 CŒŒ.  † /.msT P C q  n† / C † ŒŒ.v  v † /k  .Sirr n† / T T T 

N N X m P X k jk  n† .v  v † /2 1   † ŒŒ  n†  Sirr n† : ŒŒ yk k C T T 2  kD1 kD1

We choose the entropy flux as ˆ† D q† =T † and obtain the surface Gibbs-Duhem equation  † D u†  T † † ; which shows that  † is the interfacial free energy. For simplification, we assume from here on that there is no temperature jump at †.t/, i.e. ŒŒT D 0;

Tj† D T † :

Then, with (3), we see that  † becomes  † D q†  r†

1 1 C ŒŒ.v  v † /k  .Sirr n† / T T 

(11)

N 1X .v  v † /2 1  n†  Sirr n† ; m P k ŒŒk C T kD1 2 

P i  D 0 for all i D 1; : : : ; N. where m P i satisfies ŒŒm In the next section, we further determine appropriate constitutive relations such that the entropy principle holds. In addition, one needs a constitutive modeling for the Helmholtz free energy  . This will be constructed from an equation of state for the pressure p and from the chemical potentials i .

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4 Constitutive Modeling Constitutive relations can be derived from the entropy principle in (10) and (11). The standard closure is as follows (cf. de Groot and Mazur [7]; Slattery [15]; Hutter and Jöhnk [9]). (a) Bulk (B1) (B2) (B3) (B4)

… D r  v;  D .T; i /  0 the bulk viscosity, Sı D 2Dı ;  D .T; i /  0 the dynamic viscosity (Newton’s law), q D ˛r T1 ; ˛ D ˛.T; i /  0 the heat conductivity (Fourier’s law),  k N  P N ji D  N1 with a positive (semi-)definite matrix  bk b kD1 Lik r T T ŒLik  D ŒLik .T; 1 ; : : : ; N / of mobilities (Fick’s law for multi-component mixture).

(b) Interface (B5) q† D ˛ † r† T1 , ˛ † D ˛ † .T/  0 the interfacial heat conductivity, (B6) ŒŒvk  D 0, vk˙ D vk† , i.e. continuous tangential velocities, (B7) If i 2 I ˙ WD fi j Ai is only in ˙ g, then m P i D 0 (no transfer) and otherwise 1 .v  v † /2 ; ŒŒi  D ŒŒ n†  Sirr n†   2 or, more general but still neglecting mass transfer cross-effects, .v  v † /2 1  n†  Sirr n† , ˇi D ˇi .T/  0. (B7’) m P i D ˇi ŒŒi C 2  Now we model the Helmholtz free energy  , where we follow Example 2 in Bothe and Dreyer [3]. The free energy can be constructed from an equation of state for the pressure p and from relations for the “chemical part” of the chemical potential i . We consider the gas phase as an ideal mixture of ideal gases and the liquid phase as a solution with AN as the solvent and A1 ; : : : ; AN1 the solutes (i.e. dissolved components). We introduce the following notation: X i ci (molar density); c WD ci ; xi WD (molar fraction); Mi c iD1 N

ci WD

x0 WD .x1 ; : : : ; xN1 /; P where NkD1 xk D 1. We use .; x0 / as a set of primitive variables as well as .1 ; : : : ; N /. Note that .; x0 / 7! .1 ; : : : ; N / is one-to-one with the relations above and i D i .; x0 / WD

Mi xi ; M.x0 /

M.x0 / WD

N X kD1

Mk xk ;

xN WD 1 

N1 X kD1

xk :

Thermodynamically Consistent Modeling for Dissolution/Growth of Bubbles. . .

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Each thermodynamic quantity f is represented as f D f .T; 1 ; : : : ; N / D fQ .T; ; x0 /; where we always suppose the above relations among .T; 1 ; : : : ; N /, .T; ; x0 / and xN . P Now we model the pressure. In the gas phase C .t/, we assume p D NkD1 pk with partial pressures pi according to the ideal gas law pi D Mii RT, namely p D p.T; 1 ; : : : ; N / D

N X RT i ; RT D pQ .T; ; x0 / D Mi M.x0 / kD1

(12)

where i D 0 means that Ai does not exist in C .t/. In the liquid phase  .t/, for the later passage to the incompressible case, we use pN D pRN C K.

N  1/ NR

(13)

with a bulk modulus K D @N pN .NR /NR > 0 and reference quantities pRN and NR . Later we let K ! 1, which leads to N NR . Note that the “1” in (13) can be generalized to an appropriate function of temperature and composition, but then N will not become constant in the incompressible limit. For all other species in the liquid, we assume that they behave as ideal gas components (in the solvent “matrix” instead of a gas volume), namely pi D Mii RT for all i < N. Hence we have p D p.T; 1 ; : : : ; N / D pRN C K. 0

D pQ .T; ; x / D

pRN

CK

N1 X k N  1/ C RT R Mk N kD1

 N1 RT X MN xN  1 C xk ; M.x0 / kD1 NR M.x0 /

(14)

where i D 0 (i < N) means that Ai does not exist in  .t/. The full chemical potential cannot be modeled directly, but needs to be computed from a Helmholtz free energy function  . The modeling of follows the concept laid out in Sect. 13 of Bothe and Dreyer [3] and employs a decomposition of into an “elastic” part el , which takes into account the mechanical (pressure) work, and a “thermal” part th which accounts for the entropy of mixing.

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We start with a fixed temperature T and a reference pressure pR . We have a reference density function  D  .T; x0 / through the equation pQ .T;  ; x0 / D pR : We then define th

.T; x0 / WD Q .T;  .T; x0 /; x0 /;

Note that

el

el

.T; ; x0 / WD Q .T; ; x0 / 

th

.T; x0 /:

.T;  ; x0 / D 0. From the Gibbs-Duhem relation, we obtain 

th

.T; x0 / D pR C

N X

0 k . ; x0 /th k .T; x /;

kD1 0 Q k .T;  .T; x0 /; x0 /: th k .T; x / WD 

The thermal part of the chemical potential needs to be modeled, where we only consider the case of ideal mixtures (only containing entropy of mixing), namely 0 R th i .T; x / D gi .T; p / C

RT ln xi ; i D 1; : : : ; N; Mi

where gi denotes the Gibbs free energy of the pure component Ai in the respective phase. Next we compute el through the relation @ @

el

.T; ; x0 / D

pQ .T; ; x0 / ; 2

inserting pQ modeled in (12) and (14), respectively. For the gas phase, we obtain el

.T; ; x0 / D

Z





 RT pQ .T; ; Q x0 / ln  : d Q D 2 0 Q M.x / 

Hence we have 

D  Q .T; ; x0 / D pR C

  N  X RT   0 R C  . ; x / g .T; p / C ln x k k k   kD1 Mk

 RT ln  : 0 M.x / 

Thermodynamically Consistent Modeling for Dissolution/Growth of Bubbles. . .

123

In order to compute  .T; 1 : : : ; N / D  Q .T; ; x0 .1 ; : : : ; N //, we observe the following relations: N  RT=M.x0 /  p pQ .T; ; x0 / RT X k D D D D ; pQ .T;  ; x0 / RT=M.x0 /  pR pR kD1 Mk

M.x0 .1 : : : ; N // D

 ; c

i =Mi xi D P N ; kD1 k =Mk

i . .T; x0 .Q1 : : : ; QN //; x0 .Q1 : : : ; QN // D 

Mi xi Qi D  : M.x0 / Q

(15) (16)

Direct calculation yields 

D  .T; 1 : : : ; N / D  Q .T; ; x0 .1 ; : : : ; N //

 N N X X RT k k D RT C k gk .T; pR / C ln Mk Mk Mk kD1 kD1 ! N X RT k ln R : CRT Mk p kD1

Hence we obtain for i D 1; : : : ; N the chemical potentials as @. .T; 1 ; : : : ; N // @i 

RT i RT : D gi .T; pR / C ln R Mi p Mi

i D i .T; 1 ; : : : ; N / WD

With the relation i RT=Mi D .RT=Mi /.Mi xi =M.x0 // D pQ .T; ; x0 /xi , we also obtain i D Q i .T; ; x0 / D gi .T; pR / C

RT pQ .T; ; x0 / RT ln C ln xi for i D 1; : : : ; N: Mi pR Mi

(17)

This reproduces the formulas known from the thermodynamical literature; see, e.g., Müller [11].

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For the liquid phase, we obtain el

Z



pQ .T; ; Q x0 / d Q 2 Q  

1  KMN xN 1   C R ln  D . pRN  K/ 0   N M.x /  ! N1 X  RT C xk ln  : 0 M.x / kD1 

.T; ; x0 / D

Hence we have 



N  X RT   0 R C  . ; x / g .T; p / C ln x k k k   kD1 Mk !

 N1   KMN xN RT X  R ln  1 C C xk ln  : C. pN  K/ R  0 0   M.x /  N M.x / kD1

D  Q .T; ; x0 / D pR

Solving pQ .T;  ; x0 / D pR with (15), we get N1 X N  i  D K D C RT R   .T; x0 .1 ; : : : ; N // M N i kD1

! . pR  pRN C K/1 :

Straightforward computation with (15) and (16) yields 

D  .T; 1 : : : ; N / D  Q .T; ; x0 .1 ; : : : ; N // !  N1 X k  N ln   1 C K  pRN D RT CK R M   k N kD1 C



RT k =Mk ; k gk .T; pR / C ln Mk c kD1

N X

where the above = and c still have to be plugged in. Therefore we obtain, for i D 1; : : : ; N  1, i D i .T; 1 ; : : : ; N / WD D gi .T; pR / C

@. .T; 1 ; : : : ; N // @i

RT i =Mi ln Mi c

N1   X k  1 RT N C ln K R C RT R Mi Mk p  pRN C K N kD1

! :

(18)

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For i D N, we obtain N D N .T; 1 ; : : : ; N / WD D gN .T; pR / C

@. .T; 1 ; : : : ; N // @N

RT N =MN ln MN c

N1   X i  1 K N C R ln K R C RT R Mi p  pRN C K N N kD1

! :

(19)

Let us sum up: Up to here, the balance equations (1), (5) and (6) with constitutive relations (B1)–(B7), where the chemical potentials are modeled via (17), (18) and (19), form—up to boundary and initial conditions—a thermodynamically consistent full PDE system for a two-phase gas/liquid multicomponent system with compressible liquid and gas phases and mass transfer. For the non-isothermal case, the temperature dependencies need to be specified and the internal energy balance is usually transformed into a temperature form, i.e. of heat equation type. In the isothermal case, it can be dropped.

5 Incompressible Limit We discuss the passage to an incompressible limit for the liquid solvent. As K ! 1, assuming that the pressure stays bounded, we get N =NR ! 1. After a (formal) computation, the passage K ! 1 yields i ! 1 i , where R 1 i D gi .T; p / C R 1 N D gN .T; p / C

RT ln xi Mi

for i < N;

p  pR RT C ln xN : MN NR

Note that for an incompressible pure substance AN , the Gibbs free energy satisfies gN .T; p/ D gN .T; pR / C

p  pR : N

Hence we have 1 N D gN .T; p/ C

RT ln xN : MN

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Therefore, we obtain the usual formulas for the chemical potential in the limit of K ! 1, except for the fact that the chemical potentials of the solutes do not depend on the pressure. This is not a priori clear. Below, “1” is dropped. P the superscript k Note that N is constant and p D pN C N1 kD1 Mk RT with pN a free primitive variable. In fact, pN acts as a Lagrange multiplier in the liquid phase to account for the constraint r  vN D 0 which results from (1) for i D N. As mentioned in the introduction, we employ the solvent momentum balance in the liquid phase and couple it to the barycentric momentum balance in the gas phase. For this purpose we use the relation vN D v C uN D v C

N1 jN 1 X Dv jk : N N kD1

(20)

Then each mass balance equation in (1) is rewritten with vN , instead of v, in the liquid phase. In particular, the mass transfer transmission conditions ŒŒm P i  D 0 become, for i < N, C    ji C iC .v C  v † /  n† D Ji C i .vN  v † /  n†

on †.t/;

where Ji WD i .vi  vN / D ji C

N1 i X jk N kD1

is the diffusion flux relative to the solvent. For i D N, the transfer condition is rewritten to become a substitution for the second equation in (2) and reads as   † jC P C N  n† C my N  n† D N .vN  v /  n†

on †.t/:

(21)

If the solvent evaporation is neglected, i.e. m P N D 0 and NC D 0, then (21) simplifies to v †  n† D vN  n†

on †.t/:

As for the momentum balance, the standard approach would be to employ the barycentric momentum balance (5). However, this would lead to a velocity field v of non-zero divergence. As an interesting alternative which leads to a divergence free velocity field, we make use of the partial momentum balance for AN . According to Bothe and Dreyer [3], the partial momentum balance for Ai reads as i .@t vi C vi  rvi / D i ri C r  Siirr C i bi  T

N X kD1

fik i k .vi  vk /; (22)

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where Siirr D pi I C Si is the irreversible stress part of Si and fik D fki > 0 are friction coefficients governing the exchange of momentum between the constituents. Comparing (22) to the barycentric momentum balance in dimensionless form, it turns out that the difference of @t vi C vi  rvi to the mixture acceleration @t v C v  rv is negligible againstpthe remaining terms, if the characteristic speed of diffusion is small compared to p= which is about the speed of sound in a gas. The latter is assumed to hold, in which case (22) can be replaced by i .@t vi C vi  rvi / D yi rp C yi r.r  v/ C yi r  Sı C i b: Applied to the solvent (i D N), we obtain 



.@t vN

C

vN



rvN /

D

rp N

 RT

N1 X

 rc k C r.r  v /

kD1

Cr  Sı C  b

(23)

with the standard constraint r  vN 0 in the incompressible limit, where the superscript “” indicates that a quantity refers to the liquid phase. For the momentum transmission, the jump condition in (5) is rewritten with vN and jk , namely    j N  . m P  j  n /.v  /  S n†  .m P C v C  S C n† / † N N N N N D r†  S†

on †.t/:

(24)

In (23) and (24), v  , Sı D 2Dı and S D p I C .r  v  /I C 2Dı are to be rewritten with vN and jk instead of v by means of (20). In order to obtain more detailed information about the diffusive fluxes, we first compute r.i =T/. Since we are finally interested in the isothermal case, we consider constant T from here on. In the gas phase, with C R C i D gi .T; p / C

RT xC RT cC pC RT R ln i R D gC .T; p / C ln i R i Mi p Mi p

for the assumed ideal gas mixture, we obtain the result r

R rcC R C i i D D C riC ; C T Mi ci ci

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where the superscript “C” indicates that a quantity refers to the gas phase. In the liquid phase, we obtain r. i =T/ for i < N and i D N as r

 R rx i i D ; T Mi x i

N1 rp  R rx rp R X  R rx N N N N D r C D C rc C : k  T MN xN MN x TNR TNR NR kD1 N

If these are inserted into the Fickean form of the diffusive mass fluxes, the (molar) mass densities in the denominator only cancel, if the dependence of the phenomenological coefficients Lik on 1 ; : : : ; N has a special structure. To incorporate such structural information, while keeping the derivation as rigorous as possible, we prefer to use the generalized Maxwell-Stefan equations as constitutive relations determining the diffusion fluxes. The Maxwell-Stefan equations read 

N X xk jm  xi jm i

k

Ðik

kD1

D

i i yi ri  rp  .bi  b/ RT RT RT

(25)

with an individual body force bi for Ai and the molar mass fluxes jm i WD

ji D ci .vi  v/: Mi

For a rigorous derivation of (25) see Bothe and Dreyer [3]. There you also find the additional contribution r  Si  yi r  S in the right-hand side of (25). The latter is not included in the classical form of the Maxwell-Stefan equations as given in, e.g., Taylor and Krishna [17] and Bird et al. [1]. For simplicity, we also neglect the effect of diffusion driven by viscous stress. In (25), the Ðik are the so-called MaxwellStefan diffusivities, which are symmetric (cf. [3]). From measurements, one knows that the Ðik depend only weakly on the composition (often as affine functions), in contrast to the Fickean diffusivities. We assume the Ðik to be constant with Ðik D Ðki > 0. Note that the Maxwell-Stefan equations sum up to zero, and hence the N equations are not independent. Concerning the inversion of this equation system, see Bothe [2] and the references cited there. From here on, we assume equal body forces bk b for all components. Insertion of the chemical potential gradients yields for the gas phase 

X xC jm C  xC jm C k i

k¤i

ÐC ik

i

k

D rcC i 

yC i C C rpC D rcC i  yi rc : RT

In the liquid phase, we obtain for i < N 

X x jm   x jm  k i

k¤i

Ð ik

i k

D c rx i 

y i rp : RT

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For i D N, we obtain 

X x jm   x jm  rp y k N N k N   C c rp : D rx  N Ð RT RT Nk

k¤N

We simplify the jump conditions of the chemical potential. Neglecting the viscous and the kinetic effect in (B7), we assume ŒŒi  D 0: See Bothe and Fleckenstein [4] for an assessment of this approximation. For i < N, we have C 0C C R C i .T;  ; x / D gi .T; p / C  0  R  i .T;  ; x / D gi .T; p / C

RT pC xC RT pC R ln R i D gC ln iR ; i .T; p / C Mi p Mi p RT ln x i : Mi

R  R For given T, choose pR D pRi .T/ so that gC i .T; pi .T// D gi .T; pi .T// holds for each i and for a planar interface. Then, neglecting curvature effects via the pressure jump, we obtain

C 0C   0  C i .T;  ; x / D i .T;  ; x / , ln xi D ln

pC i R pi .T/

D ln

iC RT Mi pRi .T/

C R , x i pi .T/ D ci RT:

This is a version of Henry’s law. Thus we obtain the following PDE system for incompressible solvent and compressible gas phase, where we assume equal body forces and a continuous temperature field. Non-dilute solution with incompressible solvent: Gas phase: 8 @t ci C r  .ci v C jm i D 1; : : : ; N; ˆ i / D 0; ˆ ˆ ˆ ˆ ˆ X xk jm  xi jm ˆ ˆ i k ˆ D rci  yi rc; i D 1; : : : ; N;  ˆ ˆ Ðik < k¤i

.@t v C v  rv/ C rp D r.r  v/ C v C b; ˆ ˆ ˆ ˆ ˆ ˆ N N ˆ X X ˆ ˆ ˆ p D cRT D RT c ;  D Mk ck : ˆ k : kD1

kD1

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Liquid phase: 8 ˆ @t ci C r  .ci vN C Jim / D 0; i D 1; : : : ; N  1; ˆ ˆ ˆ ˆ ˆ X xk jm  xi jm ˆ yi ˆ i k ˆ rp; i D 1; : : : ; N  1; D crxi   ˆ ˆ ˆ Ð RT ik ˆ ˆ k¤i ˆ ˆ X xk jm  xN jm ˆ 1  yN ˆ N k ˆ rp; D crxN C ˆ <  ÐNk RT k¤N

N1 N1 ˆ X ˆ i X m m ji ˆ m m ˆ D j C j ; j D ; p D p C ck RT; J ˆ N i i k i ˆ ˆ N kD1 Mi ˆ kD1 ˆ ˆ ˆ N1 ˆ X ˆ ˆ ˆ rck ; .@t vN C vN  rvN / C rpN D r.r  v/ C r  Sı C b  RT ˆ ˆ ˆ ˆ kD1 ˆ : r  vN 0; N NR ;

  where v and Sı D 2Dı D  rv C .rv/T  13 .r  v/I are to be rewritten with vN and jk through (20). Interface: 8 C C † m  † ˆ . jm C cC C c ˆ i i .vN  v //  n† ; i .v  v //  n† D . ji ˆ ˆ ˆ i D 1; : : : ; N  1; ˆ ˆ ˆ ˆ ˆ ˆ C C C †   † ˆ . jC ˆ N  n† C yN  .v  v //  n† D N .vN  v /  n† ˆ ˆ ˆ ˆ < (or v †  n D vN  n† in case of negligible evaporation of AN ); ˆ ˆ C  R ˆ ˆ i D 1; : : : ; N  1; ˆ xi pi .T/ D ci RT; ˆ ˆ ˆ ˆ ˆ j   ˆ N   ˆ P N  j P C v C  S C n† / ˆ N  n† /.vN   /  S n†  .m  .m ˆ   ˆ N N ˆ ˆ : D  † † n† C r†  † ; ˙ ˙ ˙ ı˙ where † D r†  .n† / is the curvature,  S D p I C .r  v/ I C 2D 1 1 ı T with D D 2 rv C .rv/  3 .r  v/I and S is to be rewritten with vN and jk through (20).

6 Dilute Solution with Incompressible Solvent We ignore bulk viscosities in both phases and assume m P N D 0. Note that in the dilute case (xi 1 for i < N), we have Jim jm i ,  N and S SN in the liquid phase. We obtain a simple Fick’s law for i < N, namely   jm D Ð i Ni rci :

Thermodynamically Consistent Modeling for Dissolution/Growth of Bubbles. . .

131

We may approximate c c N . Then Henry’s law becomes c i

cC i

D

c N RT DW Hi : pRi .T/

Thus we obtain the following PDE system for a dilute solution with incompressible solvent and compressible gas phase, where we assume equal body forces and a continuous temperature field. Dilute solution with incompressible solvent: Gas phase: 8 @t ci C r  .ci v C jm i D 1; : : : ; N; ˆ i / D 0; ˆ ˆ ˆ ˆ ˆ X xk jm  xi jm ˆ ˆ i k ˆ D rci  yi rc; ; i D 1; : : : ; N;  ˆ ˆ Ðik < k¤i

.@t v C v  rv/ C RTrc D v C b; ˆ ˆ ˆ ˆ ˆ ˆ N N ˆ X X ˆ ˆ ˆ p D cRT D RT c ;  D Mk ck : ˆ k : kD1

kD1

Liquid phase: 8 @t ci C r  .ci vN C jm i D 1; : : : ; N  1; ˆ i / D 0; ˆ ˆ ˆ ˆ ˆ m ˆ i D 1; : : : ; N  1; ˆ < ji D ÐiN rci ; N1 X ˆ ˆ ˆ  .@ v C v  rv / C rp D  v C  b  RT rck ; ˆ N t N N N N N N N ˆ ˆ ˆ kD1 ˆ : r  vN 0; N NR :

Interface: 8 mC m C † . ji C cC  n† ; i D 1; : : : ; N  1; ˆ i .v  v //  n† D ji ˆ ˆ ˆ v†  n D v  n ; ˆ † ˆ N < c i D Hi ; ˆ cC ˆ i ˆ ˆ j ˆ N  ˆ  n /.v  /  SN n†  .m P C v C  SC n† / D  † † n† C r†  † ; : .j † N N N ı  where SC D pC I C 2DıC and SN D p D p N I C 2N D N I C N .rvN C T  .rvN / / .

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7 Boundary Conditions The mathematical model is to be complemented by appropriate boundary conditions at the fixed walls, called @, and at the free upper surface .t/. Since the derivation of physically sound boundary conditions is a topic on its own (cf., e.g., Bothe et al. [6]), we rest content with the simplest reasonable choice. (a) Boundary conditions at fixed walls The fixed walls are impermeable. Hence v  nw D 0 and ji  nw D 0 at @, where nw is the unit outer normal to the walls. This also implies vN  nw D 0 at @. In order to allow for a movable upper surface, the tangential velocities vk and vN k shall not be assumed to vanish. Instead, we assume a Navier slip condition of the form vk C a.Snw/k D 0, vN k C aN .SN nw /k D 0 at @ with a; aN  0. In the non-isothermal case, we add a Robin-condition for the temperature, i.e. T C ˇrT  nw D Text at @ with ˇ  0. (b) Boundary conditions at the free upper surface The Robin condition for the temperature can also be applied at the free surface. The other conditions are v  n D V and . pext  p/n C Sirr n D    n C r   on .t/

(26)

with the outer unit normal n on .t/ and the curvature  D r  .n /. Let us note that, in the dilute solution limit and for a constant surface tension   , the condition (26) becomes vN  n D V , pext  pN C n  SNirr n D   and n  SNirr n D 0 on .t/.

Thermodynamically Consistent Modeling for Dissolution/Growth of Bubbles. . .

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We assume the mixture composition to be given at .t/ due to local chemical equilibrium with a large and well-mixed external gas phase. Hence  x on .t/ for i < N with xi  0, i D xi

where we assume

PN1 iD1

xi < 1.

(c) Condition at the contact line The free surface .t/ touches the fixed wall in a set of points which forms the socalled contact line C. The modeling of dynamic contact lines is, again, a topic on its own and we refer to Shikhmurzaev [14] and the references therein for detailed information. Here, in order to close the system in the simplest possible manner, we assume a fixed contact angle of =2, i.e. n ? nw on C.

Appendix The derivation of the balance equations is based on standard two-phase transport and divergence theorems: Let V denote an arbitrary fixed control volume with outer normal n. Then Z Z Z d dx D @t dx  ŒŒv †  n† do dt V Vn† †V with †V WD †.t/ \ V, the surface velocity (including tangential part) v † and the surface unit normal n† . Here ŒŒ WD limh!0C ..x C hn† /  .x  hn† // defined for x 2 †. We also have Z Z Z f  ndo D r  fdx C ŒŒf  n† do: @V

Vn†

†V

Since the internal energy and the entropy have surface contributions, we also need the surface transport theorem for  † defined on †. It states that d dt

Z †V

 † do D

Z †V

 Z D†  † C  † r†  v † do   † v †  ds; Dt @†V

which—in this simple form—holds for all fixed V such that its outer normal n satisfies n ? n† on †V , and hence n D , where is tangential to † and normal to the bounding curve @†V . We always choose such control volumes in the integral balances above. For more details and mathematical proofs see, e.g., Slattery [15], Romano and Marasco [12] or the Appendix in Bothe et al. [5].

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Acknowledgements This work was supported by the DFG within scope of the IRTG 1529 “Mathematical Fluid Dynamics” and by the JSPS Japanese-German Graduate Externship. The authors gratefully acknowledge this support.

References 1. R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, 2nd edn. (Wiley, New York 2007) 2. D. Bothe, On the Maxwell-Stefan equations to multicomponent diffusion, in Progress in Nonlinear Differential Equations and Their Applications, vol. 60, ed. by P. Guidotti, C. Walker et al. (Springer, Basel, 2011), pp. 81–93 3. D. Bothe, W. Dreyer, Continuum thermodynamics of chemically reacting fluid mixtures. Acta Mech. 226, 1757–1805 (2015) 4. D. Bothe, S. Fleckenstein, A Volume-of-Fluid-based method for mass transfer processes at fluid particles. Chem. Eng. Sci. 101, 283–302 (2013) 5. D. Bothe, J. Prüss, G. Simonett, Well-posedness of a two-phase flow with soluble surfactant, in Progress in Nonlinear Differential Equations and Their Applications, vol. 64, ed. by M. Chipot, J. Escher (Birkhäuser, Basel, 2005), pp. 37–61 6. D. Bothe, M. Köhne, J. Prüss, On a class of energy preserving boundary conditions for incompressible Newtonian flows. SIAM J. Math. Anal. 45(6), 3768–3822 (2013) 7. S.R. de Groot, P. Mazur, Non-equilibrium Thermodynamics (Dover, New York, 1984) 8. W. Dreyer, On jump conditions at phase boundaries for ordered and disordered phases. WIAS Preprint No. 869 (2003) 9. K. Hutter, K. Jöhnk, Continuum Methods of Physical Modeling (Springer, Heidelberg, 2004) 10. G. Liger-Belair, M. Bourget, S. Villaume, P. Jeandet, H. Pron, G. Polidori, On the losses of dissolved CO2 during champagne serving. J. Agric. Food Chem. 58, 8768–8775 (2010) 11. I. Müller, Thermodynamics (Pitman, London, 1985) 12. A. Romano, A. Marasco, Continuum Mechanics: Advanced Topics and Research Trends (Birkhäuser, Basel, 2010) 13. M. Sauzade, T. Cubaud, Initial microfluidic dissolution regime of CO2 bubbles in viscous oils. Phys. Rev. E 88, 051001(R) (2013) 14. Y.D. Shikhmurzaev, Capillary Flows with Forming Interfaces (Chapman & Hall/CRC, Boca Raton, 2008) 15. J.C. Slattery, Advanced Transport Phenomena (Cambridge University Press, Cambridge, 1999) 16. F. Takemura, A. Yabe, Gas dissolution process of spherical rising gas bubbles. Chem. Eng. Sci. 53(15), 2691–2699 (1998) 17. R. Taylor, R. Krishna, Multicomponent Mass Transfer (Wiley, New York 1993)

On Unsteady Internal Flows of Bingham Fluids Subject to Threshold Slip on the Impermeable Boundary Miroslav Bulíˇcek and Josef Málek

To Yoshihiro Shibata on the occasion of his 60th birthday.

Abstract In the analysis of weak solutions relevant to evolutionary flows of incompressible fluids with non-constant viscosity or with non-linear constitutive equation, it is in general an open question whether a globally integrable pressure exists if the flows are subject to no-slip boundary conditions. Here we overcome this deficiency by considering threshold boundary conditions stating that the fluid adheres to the boundary until certain critical value for the wall shear stress is reached. Once the wall shear stress exceeds this critical value, the fluid slips. The main ingredient in our approach is to look at this type of activated, stick-slip, boundary condition as an implicit constitutive equation on the boundary. We prove the long-time and large-data existence of weak solutions, with integrable pressure, to unsteady internal flows of Bingham and Navier-Stokes fluids subject to such threshold slip boundary conditions. Keywords Bingham fluid • Implicit constitutive theory • Incompressible fluid • Integrable pressure • Long-time and large-data existence • Navier’s slip • NavierStokes fluid • No-slip • Non-Newtonian fluid • Stick-slip • Threshold slip • Unsteady flow • Weak solution

1 Introduction In the analysis of partial differential equations, especially in the theory of weak (distributional or variational) solutions the homogeneous, Dirichlet data are considered to be the simplest boundary conditions to deal with. It is however not the case when This work was initiated with the support of GACR 201/09/0917 financed by the Czech Science Foundation and completed with the support of the ERC-CZ project LL1202 financed by the Ministry of Education, Youth and Sports of the Czech Republic. M. Bulíˇcek • J. Málek () Charles University in Prague, Faculty of Mathematics and Physics, Mathematical Institute, Sokolovská 83, 186 75 Prague 8, Czech Republic e-mail: [email protected]; [email protected]

© Springer Basel 2016 H. Amann et al. (eds.), Recent Developments of Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-0348-0939-9_8

135

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studying the flows of incompressible fluids, in particular, if the flows are varying with time. The no-slip boundary conditions are not compatible with the Helmholtz decomposition and this fact leads to difficulties in the analysis of such flows. Recall that the Helmholtz decomposition splits any vector field into the sum of the field that is divergence free and another field that is the gradient of a suitable function (solution of the Neumann problem for the Laplace equation). In the case of a noslip boundary condition, while the velocity to which the Helmholtz decomposition is applied vanishes on the boundary, its divergence-free counterpart that comes from the Helmholtz decomposition is not zero on the boundary (in general merely its normal component vanishes). Consequently, up to some exceptional cases that will be specified below, it seems impossible (within the context of weak solutions) to identify the pressure field as an integrable function, belonging for example to L1 .0; TI L1 .//, see [35, 36]. This deficiency is however overcome if one requires that only the normal component of the velocity vanishes (thus the boundary is impermeable) and considers Navier’s slip that relates the tangential components of the velocity to the projection of the normal stress to the tangent plane at the boundary (see [1, 2]). Another setting where the question of (global) integrability of the pressure is positively answered represents spatially periodic problems, see for example [34]. Not only is the question of the global integrability of the pressure interesting as a problem itself but it also appears to be a crucial step in the large-data analysis of unsteady flows for heat-conducting incompressible fluids [2, 3], one/two equation turbulence models [5], incompressible fluids with pressure and shear-rate dependent viscosity [1, 3, 26], and finally for the corresponding numerical methods and their analysis. The above discussion about the integrability of the pressure is concerned with the concept of weak solutions.1 In our opinion, such concept of solution is preferable for several reasons. Let us briefly mention a few of them. From the point of view of fluid mechanics, a part of the governing system of equations comes from the balance equations that are based on the balancing the mass, linear and angular momentum, energy, and entropy over any subpart of the flow domain. Thus, the primary form of these balance equations has an integral representation and their classical formulation is derived under additional assumptions on the smoothness (regularity) of the quantities involved. Also, for internal flows, one controls the (internal, kinetic and potential) energy of the system and then also the rate of entropy (that are due to mechanical working, thermal heating, etc.) in terms of the data. These a priori bounds, although rather poor from the point of view of classical formulation, give sense to all terms in the integral (weak) formulation of the problem. On the level of mathematical analysis, weak solutions are in many cases the only concepts (see [19, 24]) that exist globally in time for any size of data measured in reasonable norms (frequently given by the a priori estimates). Furthermore, the concept of weak solution or the integral 1 By weak solution, we mean here a broad class of generalized solutions such as suitable weak solution, renormalized solution, entropy solution, dissipative solution, etc.

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formulation is a point of departure for several numerical approaches - spectral methods, finite volume methods, finite element methods are just the most known examples of methods used intensively in computer simulations. In order to have globally integrable pressure for considered evolutionary problems (and approximate the no-slip boundary conditions as close as possible), we propose to replace no-slip by stick-slip boundary conditions. The stick-slip (or threshold slip) boundary conditions are usually characterized as follows: the fluid exhibits no-slip on the boundary if and only if the magnitude of the projections of the normal stress to the tangent plane is below a certain critical value, called threshold. Once the magnitude of the projection of the normal stress to the tangent plane exceeds this threshold, the fluid slips following the Navier’s slip boundary conditions. Note that this type of boundary condition is observed especially (but not only) for certain non-Newtonian fluids (see [9, 16] for example) and it seems very natural in general. Having said this, we however take a different (purely mathematical) standpoint in this study. We consider the threshold slip as an auxiliary tool for incorporating a no-slip boundary condition into the formulation of the problem following the goal to prove long-time and large-data existence of weak solution, having, in particular, an integrable pressure. The threshold can have only purely auxiliary character and can be any (even very large) number. In some applications this threshold may not even be activated during the processes considered. Also, if one would be capable of proving that the weak solution to the relevant problem involving threshold slip not only exists but also possesses the velocity field that is continuous (up to the boundary), then the problem that is successfully solved in this way is also the problem with no-slip boundary condition provided that the threshold is chosen bigger that the L1 -norm of the velocity. We wish to emphasize that stick-slip boundary conditions lead to a weak formulation and function spaces that are different from those used for the pure no-slip boundary conditions. Furthermore, we look at this type of activated boundary condition as an implicit constitutive relation on the boundary, which leads again to a formulation that is different from those used earlier—we do not need tools such as variational inequalities, penalty methods, etc. To the best of our knowledge, this is a novel feature of our approach. For the sake of completeness, we wish to mention a couple of cases (and conditions) where the global integrability of the pressure is established even for no-slip boundary conditions for incompressible fluid models. First, for fluids with constant viscosity the issue of obtaining an integrable pressure when analyzing unsteady flows subject to the homogeneous Dirichlet boundary condition is solved by means of maximal regularity theory developed for the evolutionary Stokes system (with homogeneous Dirichlet data), see [15, 32]. These results are based on inverting the “evolutionary Stokes operator”, and finding the fundamental solution explicitly. Generalizations of the Lp .0; TI Lq .// maximal regularity theory to more complicated operators seem to be open with the one exception, namely, evolutionary Stokes type problems with the viscosity being a Hölder continuous function of time and the spatial variables, see [33]. Note that these results require, in addition to a Hölder continuous viscosity, smooth domains

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(with the boundary that is at least of the class C2 ) and sufficiently regular right-hand side and the initial value. In other words, the approach presented in this study can be useful for the evolutionary Navier-Stokes equations in situations when the righthand side is only functional (and not an integrable function); also the boundary can be Lipschitz in some cases. Second, in the analysis of time-dependent flows of incompressible generalized Newtonian fluids, it means the fluids where the relation between the deviatoric part of the Cauchy stress and the symmetric part of the velocity gradient is uniformly monotone, there are rather restrictive situations (e.g. the velocity itself has to be at least an admissible test function) in which it is possible to prove (see [20] and [4]) that the time derivative is an integrable function from which then follows easily (by a consequence of de Rham’s theorem) that the pressure itself is integrable as well. To conclude, except these two cases, the question of the existence of weak solution with an integrable pressure to the three-dimensional evolutionary problems with the no-slip boundary condition is unanswered. In order to illustrate our approach (based on replacing no-slip by slip-stick boundary conditions) for interesting problems that generalize those for the incompressible Navier-Stokes equations, yet the basic function space setting is the same as that for the Navier-Stokes system, we consider Bingham fluids in this study. Bingham fluids represents a non-linear (it means non-Newtonian) fluid model described by the following constitutive relation: the fluid responses as a rigid body if and only if the magnitude of the deviatoric part of the Cauchy stress is below a certain critical value for the stress, called the yield stress; once the magnitude of the deviatoric part of the Cauchy stress exceeds the yield stress the fluid responses as a Navier-Stokes fluid. There is a recent observation [28, 29] that this dichotomy description can be included into the setting of implicit constitutive relations, and as recent studies [6, 7, 11, 17] show this leads to a new mathematical setting useful both for the existence, numerical analysis and computer simulation of the relevant problems. We also consider the Navier-Stokes equations with the viscosity dependent continuously on the time and spatial variables. Such a system can be viewed for example as a simplified model for fluids with the viscosity dependent on some quantity like the concentration or the temperature, where such a dependence is somehow known a priori. As mentioned above, if the viscosity is merely continuous the result of Solonnikov [33] is not applicable, hence the question of the existence of globally integrable pressure is open and worth investigating. Although the most of the studies concerning analysis of flows of incompressible fluids described by the Stokes and Navier-Stokes equations concern no-slip boundary conditions, there are several studies analyzing (both steady and unsteady) flows subject to different type of boundary conditions. We mention several studies being aware of the fact that the list is incomplete. (Note that some studies concerning the analysis of problems with Navier’s slip boundary conditions were cited above.) Slip and leak boundary conditions for (mostly steady flows of) Stokes or NavierStokes fluids were studied in the works by Fujita [13, 14] focusing on the question of existence and uniqueness of weak solution. Its regularity properties are investigated

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in Saito [30], and strong solutions are constructed in a recent paper by Kashiwabara [18], where one can find further references to studies focused on numerical (mostly finite element) methods. Threshold slip boundary conditions were investigated in [22] and [23]. Bingham fluids subject to a nonlocal slip boundary condition were studied in Consiglieri [8]. We also refer to [31], and [27] for the analysis of steady flows of Bingham fluids with no-slip boundary conditions using regularity technique. Flows of second grade fluids subject to slip boundary conditions were investigated in [21]. The paper is organized in the following way. In Sect. 2 we describe the problem studied in this paper, specify the class of fluids considered, introduce the function spaces and formulate the results. In Sect. 3 we state and prove the convergence lemma that is suitable for the analysis of the problem. Finally, Sect. 4 contains the proof of the results performed via a suitable approximation scheme based on the truncation of the convective term and modification of the constitutive equations in the bulk and on the boundary in such a way that standard monotone operator theory is directly applicable to this kind of approximations. Then we derive the uniform estimates and take the limit. The assumptions of the convergence lemma are achieved by using a suitable truncation as a test function.

2 Formulation of the Problem and the Main Result 2.1 Formulation of the Problem We consider the following problem: Given the length of the time interval T > 0, a connected bounded open set   Rd , the density % 2 .0; 1/, the density of the dd dd external body forces b W .0; T/   ! Rd , the functions G W Rdd sym  Rsym ! Rsym and g W Rd  Rd ! Rd , and the initial velocity v0 W  ! Rd , find .v; p; S/ W QT WD d .0; T/   ! Rd  R  Rdd sym and s W †T WD .0; T/  @ ! R such that div v D 0 % .v;t C div.v ˝ v//  div S C rp D % b

in QT ; in QT ;

G.S; D/ D O

in QT ;

vnD0

on †T ;

g.s; v / D 0

on †T ;

v.0; / D v0

(1)

in :

In this setting, v represents the velocity, p stands for the mean normal stress (the pressure), S is the deviatoric part of the Cauchy stress T (hence T WD S  pI) and s denotes the projection of the normal stress into the corresponding tangent plane, i.e., s WD .Tn/ . Here, for any vector z defined on @ we set z WD z  .z  n/n, whereas

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n W @ ! Rd denotes the normal vector. Note that .Tn/ D .Sn/ . Finally, the symbol D stands for the symmetric part of the velocity gradient rv defined through D WD 12 .rv C .rv/T /; if needed we use Dv instead of D. Equation (1)1 expresses the fact that the fluid is incompressible (and since the density % is constant the fluid is even homogeneous). The second equation (1)2 is the balance of linear momentum. The third equation (1)3 , characterizing the response of the fluid one deals with, says that the symmetric part of the velocity gradient D (that generalizes the shear rate) and the deviatoric part of the Cauchy stress S are related2 : while the Navier-Stokes fluid is characterized by a linear relation, namely, S D 2  D;

(2)

here we consider general implicit relations opening the possibility of describing, in terms of the same quantities that enter into (2), various phenomena (such as stress thickening or stress thinning, shear thickening or shear thinning, activation phenomena such as the presence of the yield stress, and even normal stress differences) that the Navier-Stokes model (2) cannot capture. Equation (1)4 says that the boundary is impermeable, and Eq. (1)5 describes the result of interactions of the fluid with the boundary. Equation (1)5 states that the tangential components of the velocity are related to the .Tn/ . If this relation is linear, which means that  s D  v 

with  > 0;

(3)

we obtain Navier’s slip boundary condition that can be viewed also as the bridge between no-slip (letting  ! C1) and (perfect) slip (letting  ! 0). The last equation (1)6 is the requirement on the initial datum for the velocity: given v0 should be divergence-free and have vanishing normal component on @, i.e., v0 should satisfy (1)1 and (1)5 .

2.2 Assumptions on Constitutive Functions G and g Next, we shall specify what kind of implicit relations (1)3 and (1)5 we consider in this study. Concerning (1)3 we make the following identification: .S; D/ 2 A

”

G.S; D/ D O;

(4)

2 The reason why only S appears in the constitutive (1)3 is due to the fact that the fluid is incompressible.

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dd and we put the following assumptions on A W QT ! Rdd sym  Rsym :

(A1) A contains the origin: .O; O/ 2 A. (A2) A is a monotone graph: For any .S1 ; D1 /; .S2 ; D2 / 2 A .S1  S2 /  .D1  D2 /  0: dd (A3) A is a maximal monotone graph. Let .S; D/ 2 Rdd sym  Rsym .

Q  .D  D/ Q  0 for all .S; Q D/ Q 2 A; then .S; D/ 2 A: If .S  S/ (A4) A is a 2 graph: There are ˛ 2 .0; 1 and c  0 so that   S  D  ˛ jDj2 C jSj2  c

for any

.S; D/ 2 A:

The last assumption can be relaxed and referring to the setting used in [7] and [6] one can replace (A4) by the condition (A4*) A is a

graph: There are ˛ 2 .0; 1 and c  0 so that

S  D  ˛



.jDj/ C



 .jSj/  c

for any

.S; D/ 2 A;

where is an Orlicz function (convex, even, continuous) and  is its dual (conjugate) function. As mentioned in the Introduction, we restrict ourselves in this study to two cases to which the assumptions (A1)–(A4) are appropriate. First, we consider DD

1 .jSj   /C S 2  jSj

with   0;  > 0;

(5)

where xC D maxfx; 0g. Setting  D 0 in (5) we obtain the Navier-Stokes fluid model (2). If  > 0 the above formula is equivalent to jSj   , D D O

and

jSj >  , S D

 D C 2 .jDj2 /D; jDj

(6)

which is the form in which the response of Bingham fluids is usually described, see for example [12]. Note that (5) fulfills the assumptions (A1)–(A4) above (see Lemma 2.1 in [7] for details). We are also interested in analyzing the model DD

1 S 2.t; x/

with 0 <   .t; x/   in QT ;

(7)

that can be viewed as a simplified model for incompressible fluids where the viscosity changes with another quantity z (such as temperature, concentration,

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electric or magnetic field) and this dependence is somehow known a priori; then .t; x/ D .z.t; x//. It is easy to check that the linear constitutive equation (7) fulfills the assumptions (A1)–(A4). Referring to (1)5 , we introduce the graph B similarly as above through the following identification: .s; v / 2 B

”

g.s; v / D 0;

(8)

and we require that (B1) B contains the origin. .0; 0/ 2 B. (B2) B is a monotone graph. .s1  s2 /  .v1  v2 /  0

for all .s1 ; v1 /; .s2 ; v2 / 2 B:

(B3) B is a maximal monotone graph. Let .s; v/ 2 Rd  Rd . If

 .Ns  s/  .vN   u/  0

for all .Ns; vN  / 2 B

then .s; v/ 2 B:

(B4) B is a 2 graph. There are ˇ > 0 and d  0 such that s  v  ˇ .jv j2 C jsj2 /  d

for all .s; v / 2 B :

Our main interest is to analyze the stick-slip (threshold slip) boundary conditions that are usually written in the form j.Sn/ j   , v D 0; j.Sn/ j >  , .Sn/ D 

v C  v  ; jv j

(9)

where   0 and   0. Recalling the notation s WD .Sn/ we re-write (9) into the equivalent implicit equation (1)5 that takes the form:  v  D 

.jsj   /C s jsj

with   0;   0:

(10)

If  D 0 and  > 0, (10) represents Navier’s slip boundary condition  v C s D 0. If  D  D 0 then we end up with (perfect) slip. As above, Lemma 2.1 from [7] can be again used as the reference for verifying that (10) meets all the assumptions (B1)–(B4). It is important to note that the no-slip boundary condition v D 0 is excluded by (B4).

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2.3 A Priori Estimates and Function Spaces Multiplying (1)2 by v, we obtain 1 @jvj2 2 @t

C div. 12 jvj2 v/  div.Sv/ C S  D D  div.pv/ C b  v :

(11)

Since v  n D 0, integrating (11) over  leads to 1d kvk22 C 2 dt

Z

Z 

S  D dx 

Z @

.Sn/  v dS D



bv:

(12)

The simplest relations, namely (2) and (3), or the assumptions (A4) and (B4) then imply     1d kvk22 C˛ kSk22 C kDk22 C ˇ ksk22;@ C kv k22;@ 2 dt Z Z Z  c 1 C d 1C bv: 

@

(13)



If b is given appropriately (for example b D 0), the last inequality indicates clearly in what function spaces we should look for the solution. We introduce briefly the function spaces needed below. For q 2 Œ1; 1, .Lq ./; k  kq / and .W 1;q ./; k  k1;q / denote the Lebesgue and Sobolev spaces. If X is a Banach space of scalar functions then X d is the space of vector-valued functions dd having d components, each of them belonging to X. Similarly, Xsym is the space of tensor-valued symmetric functions. Next, we set, for any q 2 Œ1; 1/, q

kkq

Ln;div WD fv 2 D./d I div v D 0g

:

The subspaces (and their duals) of vector-valued Sobolev functions from W 1;q ./d which have zero normal component on the boundary are defined in the following way: Wn1;q WD fv 2 W 1;q ./d I v  n D 0 on @g; 1;q

Wn;div WD fv 2 Wn1;q I div v D 0 in g;     1;q 1;q Wn;div WD Wn;div : Wn1;q WD Wn1;q ; By the Helmholtz decomposition we observe that 1;q

Wn1;q D Wn;div ˚ fr'I ' 2 W 2;q ./; r'  n D 0 on @g: Note that this does not hold for .W01;2 .//d .

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2.4 Main Result Definition 2.1 Let v0 2 L2n;div ;

b 2 L2 .0; TI Wn1;2 / :

(14)

We say .p; v; S; s/ is weak solution to the problem (1) if p 2 L1 .QT /; 1;2 /; v 2 Cweak .0; TI L2n;div / \ L2 .0; TI Wn;div 2 dd S 2 L .QT / ; s 2 L2 .†T /d ; limt!0C kv.t/  v0 k22 D 0; hv;t ; wi C .S; Dw/  .v ˝ v; Dw/ C .s; w /@ D hb; wi; C.p; div w/ ; for all w 2 Wn1;1 and a.a. t 2 .0; T/; .S.t; x/; Dv.t; x// 2 A for a.a. .t; x/ 2 QT ; .s.t; x/; v .t; x// 2 B for a.a. .t; x/ 2 †T : We formulate first the following theorem for general G and g fulfilling (A1)– (A4) and (B1)–(B4). Theorem 2.2 Let A satisfy the assumptions (A1)–(A4) and let B satisfy the assumptions (B1)–(B4). Then, for any  2 C 1;1 , T 2 .0; 1/ and for arbitrary v0 and b fulfilling (14), there exists a weak solution to the problem (1). In order to be completely explicit in the construction of the solution in this study we restrict ourselves to equations (5) (or (7)) and (10). We establish the following result. Theorem 2.3 For any  2 C 1;1 and T 2 .0; 1/ and for arbitrary v0 and b fulfilling (14) there exist 1;2 v 2 L1 .0; TI L2 ./3 / \ L2 .0; TI Wn;div /;

S 2 L2 .QT /dd sym ; p1 2 L2 .QT /;

s 2 L2 .0; TI L2 .@/3 /; dC2

dC2

p2 2 L dC1 .0; TI W 1; dC1 .//;

solving, for almost all time t 2 .0; T/ and for all w 2 Wn1;2 , Z hv;t ; wi 

Z 

.v ˝ v/  rw C

Z 

S  Dw C

Z swD @



.p1 C p2 / div w:

Furthermore, S and D fulfill the equation (5) (or (7)) a.e. in QT , and s and v fulfill (10) a.e. in †T .

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In Sect. 4 we prove Theorem 2.3. The proof of Theorem 2.2 follows then from the main theorem proved in [7] and the proof of Theorem 2.3.

3 Convergence Lemma In this section, we establish a convergence lemma that we use to verify the validity of the nonlinear constitutive equations, such as (5) or (10), or more generally (1)3 or (1)5 . The fact that the result is stated locally is useful in applications. Although the lemma is proved in [6], for the sake of completeness, we give a simplified proof here. Lemma 3.1 Let U be an arbitrary measurable set and let a graph A fulfill the assumptions (A2)–(A3). Assume that, for some r 2 .1; 1/, .Sn ; Dn / 2 A Dn * D

U

(15)

weakly in Lr .U/dd ;

(16)

r0

Sn * S Z Z lim sup Sn  Dn  S  D: n!1

almost everywhere in U;

weakly in L .U/dd ;

(17) (18)

U

Then .S; D/ 2 A almost everywhere in U: Proof We start with observing that weakly in L1 .U/:

Sn  Dn * S  D

(19)

To prove this, we first notice that, for arbitrary n, m 2 N and for .Sn ; Dn /, .Sm ; Dm / 2 A, (A2) implies that 0  .Sn  Sm /  .Dn  Dm /

a.e. in U:

Then, it follows from (16)–(18) that lim sup lim sup k.Sn  Sm /  .Dn  Dm /k1 n!1

m!1

Z

Z

D lim sup lim sup n!1

m!1

Z

n!1

 0:

U

U

S  Dn  U

Sn  Dm C U

Z



Z

Sn  D C U



Z

Sm  Dn  U

Z

Sn  Dn 

 lim sup

Z

Sn  Dn 

SD U

Sm  Dm U

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M. Bulíˇcek and J. Málek

Since lim lim k.Sn  Sm /  .Dn  Dm /k1 D 0;

n!1 m!1

we observe that Z lim lim

n!1 m!1 U

for all ' 2 L1 ./:

.Sn  Sm /  .Dn  Dm / ' D 0

(20)

Setting Z L WD lim

`!1 U

.S`  D` /';

we conclude from (20) that Z

Z .S  D / ' 

0 D lim lim

n

n!1 m!1

U



Z D 2 L  .S  D/ ' ;

Z n



Z

.S  D / ' 

n

.S  D / ' C

m

m

U

.S  D / '

n

U

m

m

U

U

which is (19). Next, we take an arbitrary .S ; D / 2 A and arbitrary nonnegative ' 2 L1 .U/. It then follows from (A2), (16), (17) and (19) that Z Z 0  lim .Sn  S /  .Dn  D /' D .S  S /  .D  D /': n!1 U

U

Since '  0 is arbitrary we get 0  .S  S /  .D  D /

almost everywhere in U:

Since .S ; D / 2 A is arbitrary, the property (A3) implies that .S; D/ 2 A

almost everywhere in U:

t u

While the first three assumptions (15)–(17) of the lemma are usually easily available, the key assumption (18) suggests that one needs some kind of energy equalities in order to establish its validity. In applications, when one needs to identify a constitutive equations of the type (1)3 in QT , the verification of (18) is usually straightforward if v, the solutions itself, is an admissible test function in the weak formulation of the problem. If v is not an admissible test function, then one uses some kind of truncation instead of v. Here, in Sect. 4, we apply an L1 -truncation method as this is sufficient to analyze the problem (1) with the constitutive relations (5) and (10). More powerful methods are those based

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on Lipschitz-truncation of Bochner functions having values in Sobolev or OrliczSobolev spaces, we refer to [10] and [7] both for details and the references to earlier studies. Concerning the constitutive equations on the boundary, such as (1)5 or (10), one needs to check that sequences .sn ; vn / 2 B satisfying, for some q > 1, 0

sn * s

weakly in Lq .†T /d ;

vn * v

weakly in Lq .†T /d ;

fulfill also the condition Z

Z sn  vn 

lim sup n!1

†T

†T

s  v:

(21)

Here, however, we do not need to refer to any kind of energy equality as we usually have vn ! v

strongly in L1 .†T /:

Then, by Egorov theorem, for any " > 0 there exists U" such that j†T n U" j  " and vn ! v

strongly in L1 .U" /;

which implies (21). Consequently, by Lemma 3.1 .s; v/ 2 B a.e. in U" . However, " is arbitrary and thus .s; v/ 2 B a.e. in †T . This argument suggests that the maximal monotone graph setting outlined by the assumptions (B1)–(B4) is not necessary as vn is compact on †T ; we will include details in a forthcoming study.

4 Proof of the Main Result We set for simplicity % D 2  D  D 1 in this section. The proof holds true for any % > 0,  > 0 and  > 0 without any essential change.

4.1 Approximations and Their Existence Let n 2 N. We first set

Gn .S; D/ WD D 

 1 .jSj   /C C S; jSj n

(22)

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M. Bulíˇcek and J. Málek

g .s; v/ WD v C n

 1 .jsj   /C C s; jsj n

(23)

and notice that the constitutive equations  1 .jSj   /C C S; jSj n 

1 .jsj   /C C s 0 D v C jsj n

Gn .S; D/ D O

”

gn .s; v / D 0

”

DD

(24) (25)

imply S D SQ n .D/ WD

8 

;

and 8 

;

where SQ n and sQn are continuous monotone functions with linear growth (at infinity). Next, we consider smooth functions Gn W R ! R with jG0n j  2n such that Gn .s/ WD 1 for s  n;

Gn .s/ D 0 for s > 2n:

Then we introduce the approximate problem relevant to (1): for each n 2 N, find .vn ; pn / satisfying

vn;t

div vn D 0

in QT ;

n

C div..v ˝ v /Gn .jv j//  div S .Dv / C rp D b

in QT ;

vn  n D 0

on †T ;

s C sQn .vn / D 0

on †T ;

n

n

n

Qn

n

v .0; / D v0 n

(26)

in :

In what follows, we set Sn WD SQ n .Dvn /

and

sn WD sQn .vn /:

Due to the presence of Gn that truncates the convective term and due to the properties of SQ n and sQn that generate monotone operators, the existence of weak solution vn 2

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1;2 C.Œ0; TI L2n;div / \ L2 .0; TI Wn;div / to the problem (26) that satisfies

hvn;t ; wi Q C .Sn ; Dw/ Q C .div.vn ˝ vn /Gn .jvn j/; w/ Q C .sn ; wQ  /@  hb; wi Q D0

1;2 for all wQ 2 Wn;div and a.a. t 2 .0; T/;

(27)

follows from standard monotone operator theory. For fixed n 2 N a construction of the solution can be done via the Galerkin approximations taking as the basis in 1;2 Wn;div \ W d;2 ./d the eigenfunctions of the following eigenvalue problem Z 

r .d/ v  r .d/ w C rv  rw D 

Z 

vw

1;2 for all w 2 Wn;div \ W d;2 ./d :

We do not go into more details referring to [19, 25] or [1]. Having vn we introduce pn as the solution of the following problem  .rpn ; rz/ D .Sn ; r .2/ z/ C .div.vn ˝ vn /Gn .jvn j/; rz/ C .sn ; .rz/ /@  hb; rzi

(28)

for all z 2 W 2;2 ./ with rz  n D 0 on @ and a.a. t 2 .0; T/: In short, it means that pn WD .N /1 div .div Sn C div.vn ˝ vn /Gn .jvn j/  b/, where N denotes the Laplace Roperator together with homogeneous Neumann boundary conditions. We consider  pn D 0 for a.a. t 2 .0; T/. Applying the Helmholtz decomposition to an arbitrary w 2 Wn1;2 , i.e., w D wQ C rz with z 2 W 2;2 ./; div wQ D 0; w Q  n D rz  n D 0 on @;

(29)

we can write, using (27) and (28), .Sn ;Dw// C .div.vn ˝ vn /Gn .jvn j/; w/ C .sn ; w /@  hb; wi Q C .div.vn ˝ vn /Gn .jvn j/; w/ Q C .sn ; wQ  /@  hb; wi Q D .Sn ; Dw// C .Sn ; r .2/ z/ C .div.vn ˝ vn /Gn .jvn j/; rz/ C .sn ; .rz/ /@  hb; rzi Q  .rp; rz/ D hvn;t ; wi D hvn;t ; wQ C rzi  .rp; w Q C rz/; which finally leads to the following weak formulation of the problem (26): hvn;t ; wi C .Sn ; Dw/ C .div.vn ˝ vn /Gn .jvn j/; w/ C .sn ; w /@ D .pn ; div w/ C hb; wi for all w 2 Wn1;2 and a.a. t 2 .0; T/:

(30)

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4.2 A Priori Estimates Taking vn as a test function in the weak formulation of (26)2 and noticing that the convective term vanishes we obtain Z Z 1d n 2 n n n Q kv k2 C sQn .vn /  vn D hb; vn i0; S .Dv /  Dv C 2 dt  @ which imply that sup kvn .t/k22 C

Z

t2.0;T/

jSn j2 C jrvn j2 C jvn j

2.dC2/ d

Z C †T

QT

 C.kv0 k2 ; kbkL2 .0;TI.W 1;2

n;div /

/

jsn j2 C jvn j2 (31)

/ DW C ;

where C > 0 is a generic constant. In (31), we also incorporated the information coming from interpolating3 vn 2 L1 .0; TI L2 ./d / and vn 2 L2 .0; TI W 1;2 ./d /. Further, we introduce pn2 with zero mean value as a weak solution of the following problem (stated at each time level): Z

Z



rpn2

 r' D  

div..vn ˝ vn /Gn .jvn j//  r'

for all ' 2 W 1;dC2 ./:

Since Z

dC2

j div.vn ˝ vn /Gn .jvn j/j dC1  C ; QT

we observe that Z

T

dC2

kpn2 k dC1 dC2  C :

(32)

1; dC1

0

Next, we set pn1 WD pn  pn2 and find ' with zero mean value solving ' D pn1 in 

and

r'  n D 0 on @:

Then Z

2

2

jr 'j C QT

3

Z

2

Z

jr'j  .0;T/@

QT

2

jpn1 j2 :

d

(33)

We refer to standard interpolation inequality kzk 2.dC2/  kzk2dC2 kzk dC2 valid if d  3, and 2d 2

d

d

kzk 2.dC2/  ckzk2dC2 krzk2dC2 that holds even if d  2. d

d2

On Unsteady Internal Flows of Bingham Fluids Subject to Threshold Slip on. . .

151

Taking then r' for such  as a test function in the weak formulation of (26)2 we arrive at Z Z Z jpn1 j2 D  rpn1  r' D .rpn2  div.vn ˝ vn /Gn .jvn j//  r'C QT

QT

Z

Sn  r 2 ' C QT

Z D

2

QT

QT

sn  r' .0;T/@

Z

Z

S r ' C n

Z

s  r'  C n

.0;T/@

QT

jpn1 j2

 12

:

Finally, from the weak formulation of (26)2 , we directly conclude that kvn;t k.L2 .0;TIWn1;2 \LdC2 ./d //  C :

(34)

4.3 Limit n ! 1 The uniform estimates (31)–(34) together with the Aubin-Lions compactness lemma and the (compact) embedding of the Sobolev spaces into the space of traces suffice to obtain the following convergences (for possibly subsequences that we again denote as the original sequences): vn * v

weakly in L2 .0; TI Wn1;2 /;

Sn * S

weakly in L2 .QT /dd ;

sn * s

weakly in L2 .0; TI L2 .@/d /;

vn ! v

strongly in L2 .QT /d ;

vn ! v

strongly in L2 .0; TI L2 .@/d /;

pn1 * p1

weakly in L2 .QT /;

pn2 * p2

weakly in L dC1 .0; TI W 1; dC1 .//;

vn;t * v;t

weakly in .L2 .0; TI Wn1;2 \ LdC2 ./d // :

dC2

dC2

It is easy to observe that v, p WD p1 C p2 , S and s fulfill the weak formulation of the balance equation (1)2 . It remains to show the validity of the constitutive equations, i.e., to prove that v, S and s fulfill (5) and (10). For the latter, we argue via the Egorov theorem, see the end of Sect. 3. In the remaining part of Sect. 4 we prove the validity of (5).

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n For this purpose, assume that a sequence fkn g1 nD1 is such that 0 < A  k  B < n 1, where A, B and k will be specified later. Next, we take as a test function in the weak formulation of the n-th approximation the function

n wn D Tkn .vn  v/ WD .vn  v/ min 1;

kn o : jvn  vj

Then (see [1] for details) Z Sn  Dwn  pn1 div wn lim sup n!1

QT

Z

D lim sup n!1

QT

hvn;t ; wn i  div.vn ˝ vn /Gn .jvn j/  wn sn  wn  0:

Z b  wn  rpn2  wn C

(35)

†T

Let S 2 L2 .QT / be such that Dv D

.Sj   /C

Then it follows from (35) that Z Z n n .S  S/  Dw  lim sup lim sup n!1

QT

n!1

jSj

S:

jvn vjkn

(36)

kn jpn j.jrvn j C jrvj/; jvn  vj 1

which then leads to Z Z n n lim sup .S  S/  D.v  v/  lim sup n!1

jvn vj 0: Note that I n is uniformly bounded in L1 .QT /. Let N 2 N be arbitrary. We fix A D N and B D N NC1 and define Qni WD f.t; x/ 2 QT I N i  jvn  vj  N iC1 g

i D 1; : : : ; N:

(37)

On Unsteady Internal Flows of Bingham Fluids Subject to Threshold Slip on. . .

153

Since N Z X iD1

Qni

I n  C ;

(38)

there is, for each n 2 N, an index in 2 f1; : : : ; Ng such that Z In  Qnin

C : N

(39)

At this point we return to (37) with kn WD N in and estimate the right-hand side in the following way: Z

kn In D n jvn vjN in jv  vj Z C

Z N in C1 jv n vjN in

jvn vjN in C1

Z 

1 I C N

Z

N in In  vj

(40)

jvn

n

Qnin

N in In jvn  vj

jvn vjN in C1

In 

C ; N

R where we used (39) to bound Qn I n and (38) to note that the last integral in (40) is in bounded by C . Inserting (24) and (36) into (37) and using (40) we obtain !    .Sn   / 1 .Sj  /C C  C n n C S  ; lim sup S S  S  n jS j n N n!1 jSj jv n vjkn Z

which leads to Z lim sup n!1

1 C C lim sup Z  n n N n jv vjk

Z jSn jjSn  Sj 

n

QT

C ; N

where   Z n WD Sn  S 

! .Sn   /C n .Sj   /C S  S  0: jSn j jSj

Since A D N and kn  N it follows from (41) that Z C : Zn  lim sup n N n!1 jv vjN

(41)

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The rest of the argument is the same as in [7], pp. 2787–2788, and will be only briefly sketched. Splitting QT into a union of jvn  vj  N and jvn  vj > N one concludes from above that Z p C lim sup : Zn  N n!1 QT Letting N ! 1 this gives, at least for a subsequence, Zn ! 0

a.e. in QT :

Applying then a biting lemma (see [7] for details), one then concludes that Zn ! 0

strongly in L1 .QT n Ej / ;

where Ej  QT are such that limj!1 jEj j D 0. It follows from the definition of Z n , .Sn  / where we use (36) and replace jSnj C Sn by Dvn  1n Sn , that Z lim sup n!1

1 Sn  .Dvn  Sn / D n QT nEj

Z QT nEj

S  Dv :

Since 1 .Sn ; Dvn  Sn / 2 A ; n Sn * S weakly in L2 .QT /dd ; 1 Dvn  Sn * Dv weakly in L2 .QT /dd ; n by Lemma 3.1 and properties of Ej we conclude .S; Dv/ 2 A a.e. in QT , it means that (5) is proved.

5 Concluding Remarks The main point of this study was to point out that the threshold slip is a possible way for overcoming the difficulties connected with the analysis of unsteady flows subject to homogeneous Dirichlet boundary conditions (no-slip) in particular if one needs to have a globally integrable pressure. The setting presented here is influenced by the framework of implicitly constituted materials. We wish to remark that for implicitly constituted fluids characterized by (A1)– (A3) and (A4*) with .s/  c1 sr and r > 2d=.d C 2/, we can define the concept of weak solution with an integrable pressure and prove its large data existence (see [7]

On Unsteady Internal Flows of Bingham Fluids Subject to Threshold Slip on. . .

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and a forthcoming study by the authors and Luisa Consiglieri for further details). Thus, we obtain a subject for further numerical analysis and computer simulations for large class of non-Newtonian fluids. Acknowledgements The authors acknowledge the membership to the Neˇcas Center for Mathematical Modeling (NCMM) and to the Charles University center for mathematical modeling, applied analysis and computational mathematics (MathMAC).

References 1. M. Bulíˇcek, J. Málek, K.R. Rajagopal, Navier’s slip and evolutionary Navier-Stokes-like systems with pressure and shear-rate dependent viscosity. Indiana Univ. Math. J. 56 51–85 (2007) 2. M. Bulíˇcek, E. Feireisl, J. Málek, A Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients. Nonlinear Anal. Real World Appl. 10, 992– 1015 (2009) 3. M. Bulíˇcek, J. Málek, K.R. Rajagopal, Mathematical analysis of unsteady flows of fluids with pressure, shear-rate, and temperature dependent material moduli that slip at solid boundaries. SIAM J. Math. Anal. 41, 665–707 (2009) 4. M. Bulíˇcek, F. Ettwein, P. Kaplický, D. Pražák, On uniqueness and time regularity of flows of power-law like non-Newtonian fluids. Math. Methods Appl. Sci. 33 1995–2010 (2010) 5. M. Bulíˇcek, R. Lewandowski, J. Málek, On evolutionary Navier-Stokes-Fourier type systems in three spatial dimensions. Comment. Math. Univ. Carol. 52, 89–114 (2011) ´ 6. M. Bulíˇcek, P. Gwiazda, J. Málek, K.R. Rajagopal, A. Swierczewska-Gwiazda, On flows of fluids described by an implicit constitutive equation characterized by a maximal monotone graph, in Mathematical Aspects of Fluid Mechanics, ed. by J.C. Robinson, J.L. Rodrigo, W. Sadowski. London Mathematical Society Lecture Note Series, vol. 402 (Cambridge University Press, Cambridge, 2012), pp. 23–51 ´ 7. M. Bulíˇcek, P. Gwiazda, J. Málek, A. Swierczewska-Gwiazda, On unsteady flows of implicitly constituted incompressible fluids. SIAM J. Math. Anal. 44, 2756–2801 (2012) 8. L. Consiglieri, Existence for a class of non-Newtonian fluids with a nonlocal friction boundary condition. Acta Math. Sin. (Engl. Ser.) 22 523–534 (2006) 9. M.M. Denn, Fifty years of non-newtonian fluid dynamics. AIChE J. 50, 2335–2345 (2004) 10. L. Diening, M. Ružiˇcka, J. Wolf, Existence of weak solutions for unsteady motions of generalized Newtonian fluids. Ann. Sc. Norm. Super. Pisa Cl. Sci. IX, 1–46 (2010) 11. L. Diening, C. Kreuzer, E. Süli, Finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology. SIAM J. Numer. Anal. 51, 984–1015 (2013) 12. G. Duvant, J.-L. Lions, Inequalities in Mechanics and Physics (Springer, Berlin, 1976) 13. H. Fujita, A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions, Mathematical Fluid Mechanics and Modeling (S¯urikaisekikenky¯usho K¯oky¯uroku, Kyoto, 1994), pp. 199–216 14. H. Fujita, H. Kawarada, A. Sasamoto, Analytical and numerical approaches to stationary flow problems with leak and slip boundary conditions, in Advances in Numerical Mathematics; Proceedings of the Second Japan-China Seminar on Numerical Mathematics(Tokyo, 1994), Lecture Notes Numerical Applied Analysis, vol. 14 (Kinokuniya, Tokyo, 1995), pp. 17–31 15. M. Giga, Y. Giga, H. Sohr, Lp estimates for the Stokes system, in Functional Analysis and Related Topics, 1991 (Kyoto), Lecture Notes in Mathematics, vol. 1540 (Springer, Berlin, 1993), pp. 55–67 16. H. Hervet, L.Léger, Flow with slip at the wall: from simple to complex fluids. C. R. Phys. 4, 241–249 (2003)

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17. J. Hron, J. Málek, J. Stebel, K. Touška, A novel view of computations of steady flows of Bingham and Herschel-Bulkley fluids using implicit constitutive relations. Submitted for publication (2015) 18. T. Kashiwabara, On a strong solution of the non-stationary Navier-Stokes equations under slip or leak boundary conditions of friction type. J. Differ. Equ. 254, 756–778 (2013) 19. O.A. Ladyzhenskaya, Modifications of the Navier-Stokes equations for large gradients of the velocities. Zapiski Naukhnych Seminarov Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7, 126–154 (1968) 20. O.A. Ladyzhenskaya, Attractors for the modifications of the three-dimensional Navier-Stokes equations. Philos. Trans. R. Soc. Lond. A 346, 173–190 (1994) 21. C. Le Roux, Existence and uniqueness of the flow of second-grade fluids with slip boundary conditions. Arch. Ration. Mech. Anal. 148, 309–356 (1999) 22. C. Le Roux, Steady Stokes flows with threshold slip boundary conditions. Math. Models Methods Appl. Sci. 15, 1141–1168 (2005) 23. C. Le Roux, A. Tani, Steady solutions of the Navier-Stokes equations with threshold slip boundary conditions. Math. Methods Appl. Sci. 30, 595–624 (2007) 24. J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934) 25. J.-L. Lions, Quelques méthodes de Résolution des Problèmes Aux Limites Non Linéaires (Paris, Dunod, 1969) 26. J. Málek, J. Neˇcas, K.R. Rajagopal, Global analysis of the flows of fluids with pressuredependent viscosities. Arch. Ration. Mech. Anal. 165, 243–269 (2002) 27. J. Málek, M. Ružiˇcka, V.V. Shelukhin, Herschel-Bulkley fluids: existence and regularity of steady flows. Math. Models Methods Appl. Sci. 15, 1845–1861 (2005) 28. J. Málek, K.R. Rajagopal, Compressible generalized Newtonian fluids. Z. Angew. Math. Phys. 61, 1097–1110 (2010) 29. K.R. Rajagopal, A.R. Srinivasa, On the thermodynamics of fluids defined by implicit constitutive relations. Z. Angew. Math. Phys. 59, 715–729 (2008) 30. N. Saito, On the Stokes equation with the leak and slip boundary conditions of friction type: regularity of solutions. Publ. Res. Inst. Math. Sci. 40, 345–383 (2004) 31. V.V. Shelukhin, Bingham viscoplastic as a limit of non-Newtonian fluids. J. Math. Fluid Mech. 4, 109–127 (2002) 32. V.A. Solonnikov, Estimates for solutions of nonstationary system of Navier-Stokes equations. J. Soviet Math. 8, 467–523 (1977) 33. V.A. Solonnikov, Lp -estimates for solutions to the initial boundary-value problem for the generalized Stokes system in a bounded domain. Function theory and partial differential equations. J. Math. Sci. (New York), 105, 2448–2484 (2001) 34. R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 66, 2nd edn. (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1995) 35. R. Temam, Navier-Stokes equations, Theory and numerical analysis (AMS Chelsea Publishing, Providence, RI, 2001). Reprint of the 1984 edition 36. J. Wolf, Existence of weak solutions to the equations of nonstationary motion of nonNewtonian fluids with shear-dependent viscosity. J. Math. Fluid Mech. 9, 104–138 (2007)

Inhomogeneous Boundary Value Problems in Spaces of Higher Regularity Robert Denk and Tim Seger

Dedicated to Yoshihiro Shibata on occasion of his 60 th birthday

Abstract Uniform a priori estimates for parameter-elliptic boundary value problems are well-known if the underlying basic space equals Lp ./. However, much less is known for the Wps ./-realization, s > 0, of a parameter-elliptic boundary value problem. We discuss a priori estimates and the generation of analytic semigroups for these realizations in various cases. The Banach scale method can be applied for homogeneous boundary conditions if the right-hand side satisfies certain compatibility conditions, while for the general case parameter-dependent norms are used. In particular, we obtain a resolvent estimate for the general situation where no analytic semigroup is generated. Keywords Boundary value problems • Higher order Sobolev spaces • Parameterellipticity • Resolvent estimates

1 Introduction For the treatment of nonlinear parabolic equations, a priori estimates in Lp -Sobolev spaces are an important step. Based on the theory of parameter-ellipticity, resolvent estimates have been established for a large class of equations, implying sectoriality of the corresponding operator or even maximal regularity for the non-stationary problem. For a boundary value problem in a domain   Rn , the basic space is usually Lp ./. This leads to a solution in Wpm ./ where m denotes the order of the differential operator. For the boundary traces, one obtains non-integer Besov spaces. The situation becomes more complicated and much less investigated if one is interested in spaces of higher regularity. Here we start with Wps ./, s > 0, as the basic space and expect the solution to be in WpmCs ./. Apart from its own interest, spaces of higher regularity naturally appear in mixed-order systems (Douglis-Nirenberg

R. Denk () • T. Seger Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany e-mail: [email protected]; [email protected] © Springer Basel 2016 H. Amann et al. (eds.), Recent Developments of Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-0348-0939-9_9

157

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systems). Inhomogeneous boundary conditions and non-standard boundary spaces also appear in transmission problems and coupled systems. As an example, we mention the two-phase Stokes equations where the normal component of the velocity jumps across the interface. In the paper [23], Y. Shibata and S. Shimizu have shown maximal Lp -Lq -regularity for this system, introducing a special function space adapted to the inhomogeneous jump conditions. The proofs in this and many other papers in fluid mechanics (see, e.g., Shibata [22]) are based on partial Fourier transform and careful estimates of the solution operators. In the present text, we essentially follow the same approach, however, aiming at uniform a priori estimates where the basic space is Wps ./ instead of Lp ./. We will restrict ourselves to scalar parameter-elliptic equations which can be seen as a first step in the direction of the Stokes system and general mixed-order systems. Let us consider the boundary value problem .A  /u D f Bu D g

in ;

(1)

on @

in a bounded sufficiently smooth domain   Rn . Here A is a scalar differential operator of order m 2 2N, and B is a column of m2 boundary operators, B D .B1 ; : : : ; Bm=2 /T , with ord Bj D mj < m. Classical parameter-elliptic theory states that, under suitable ellipticity and smoothness conditions, a uniform a priori estimate for the solution u holds. More precisely, we have jjjujjjm;p;

m=2   X p  C k f kL ./ C jjjgj jjjmmj  1 ;p;@ : p

(2)

jD1

Here for s > 0 the parameter-dependent norms jjj  jjj are defined by jjjujjjs;p; WD kukWps ./ C jjs=m kukLp ./

.s  0/

(analogously for jjj  jjjs;p;@ ). For s  0, Wps ./ stands for the standard SobolevSlobedeckii space. From the a priori estimate (2), we immediately obtain the resolvent estimate k.  AB /1 kL.Lp .//  C

(3)

for the Lp -realization AB of the boundary value problem .A; B/. This unbounded operator in Lp ./ is defined by D.AB / WD fu 2 Wpm ./ W Bu D 0g and AB u WD A.D/u .u 2 D.AB //. In particular, under suitable parabolicity assumptions, AB is sectorial and generates an analytic semigroup. In fact, AB is even R-sectorial and therefore admits maximal Lp -regularity. A priori estimates of the form (2) are known since long; we refer to the classical works Agmon [1], Agranovich–Vishik [4], Geymonat–Grisvard [13], and Roitberg–Sheftel [20]. Concerning R-sectoriality

Inhomogeneous Boundary Value Problems in Spaces of Higher Regularity

159

and maximal regularity, we mention Denk-Hieber-Prüss [10] and the references therein. In spaces of higher regularity, however, the resolvent estimate (3) in general does not hold. In fact, it is easily seen (cf. Nesensohn [19]) that the DirichletLaplacian D in Wp1 ./ with domain D.D / WD fu 2 Wp3 ./ W uj@ D 0g does not generate an analytic semigroup; its resolvent decays like jj1=21=.2p/ as jj ! 1. The paper Denk-Dreher [8] deals with resolvent estimates for mixedorder systems. Here conditions on the basic space Y  Wps ./ were formulated which are necessary and sufficient for a generation of an analytic semigroup. It was shown that additional conditions have to be included in the basic space; these conditions can be seen as compatibility relations. For scalar equations or systems with the same order in each component, the method of Banach scales developed by Amann in [5] can be applied and gives a rather complete answer to the question of generation of an analytic semigroup. We will comment on this in Sect. 2 below. Generation of analytic semigroups for parabolic equations was also studied by Guidetti in [18] (see also Guidetti [16]). Here in particular mixed-order systems were studied which arise by the reduction of a higher-order system (in time) to a first-order system. A priori estimates in parameter-dependent norms have been studied, e.g., by Faierman and his coauthors in [2, 9, 12]. We also remark that a particular case of an a priori estimate in Wps ./ was used in the second author’s thesis [21] to obtain a compactness property needed for a Schauder-type fixed-point argument in the context of a nonlinear elliptic-parabolic system. In the present paper, we will discuss uniform a priori estimates for the boundary value problem (1) with inhomogeneous boundary conditions. To avoid technicalities, we will restrict ourselves to the model problems in the whole space Rn and the half-space RnC . Here the operators are assumed to have constant coefficients and no lower-order terms. The generalization to bounded sufficiently smooth domains and to variable coefficients by localization and partition of unity is quite standard, and we will not dwell on this. In Sect. 2, we will study the whole space case and the case of homogeneous boundary conditions. Whereas in the whole space the a priori estimates leading to the generation of an analytic semigroup follows quite directly from Michlin’s theorem, the case of homogeneous boundary conditions can be treated by the Banach scale method. In Sect. 3, we will consider the case of inhomogeneous boundary conditions and derive the main a priori estimates of the present text.

2 The Whole Space Case and the Case of Homogeneous Boundary Conditions P Let A.D/ D j˛jDm a˛ D˛ be a linear differential operator in Rn , n  2, of order m 2 2N with constant coefficients a˛ 2 C. Here and in the following, we set D WD i@ and use the standard multi-index notation D˛ WD .i/j˛j @˛x11 : : : @˛xnn . Let L  C

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be a closed sector in the complex plane with vertex at the origin. Without loss of generality, we may assume that L D † with † WD fz 2 C n f0g W j arg zj < g for some  2 .0; . The operator A.D/ is called parameter-elliptic inP † (see [4]) if A./   6D 0 holds for all .; / 2 .Rn  † / n f0g. Here A./ WD j˛jDm a˛  ˛ is the symbol of A.D/. If the latter condition is satisfied with   2 , the operator A is called parabolic. We will consider the realization of the operator A.D/ in different scales of Sobolev spaces. For s 2 R and p 2 .1; 1/, we denote by Hps .Rn / and Bspp .Rn / the standard Bessel potential and Besov space, respectively. The Sobolev-Slobodeckii space Wps .Rn /, s  0, coincides with Hps .Rn / for s 2 N0 and with Bspp .Rn / for s 2 .0; 1/ n N. We recall that a closed linear operator AW X  D.A/ ! X in a complex Banach space X is called sectorial if the domain and the range of A are dense in X and if there exists  2 .0; / such that .A/  † and the set f.  A/1 W  2 † g is bounded in L.X/. In this case, the supremum over all angles satisfying this condition is called the spectral angle A of A. In the following, C stands for a generic constant which may vary from inequality to inequality but is independent of the variables appearing in the inequality (and in particular independent of ). In the whole space, it is easily seen that the realization of the operator A.D/ is sectorial: Lemma 2.1 Let A.D/ be parameter-elliptic in † , and let s 2 R and p 2 .1; 1/. Then for every  2 † n f0g and every f 2 Hps .Rn /, the equation .A.D/  /u D f has a unique solution u 2 HpmCs .Rn /, and the a priori estimate kukHpmCs .Rn / C jj kukHps .Rn /  Ck f kHps .Rn / holds. In particular, if   2 , the operator A.s/ in Hps .Rn /, defined by D.A.s/ / WD HpmCs .Rn /, A.s/ u WD A.D/u .u 2 D.A.s/ //, is sectorial with spectral angle larger than 2 and therefore generates an analytic semigroup. The analog results hold when Hps .Rn / is replaced by the Besov space Bspp .Rn /. Proof This essentially follows from more general results on the existence of a bounded H 1 -calculus, see, e.g., Denk-Saal-Seiler [11]. However, the result can also easily be seen by an application of the Michlin’s theorem. In fact, for each  2 † n f0g, the unique solution u can be written as u D F 1 .A./  /1 F f where F stands for the Fourier transform in Rn . Now it is immediately seen that m./ WD .1 C jj2 /m=2 .A./  /1 satisfies the conditions of Michlin’s theorem (see [24], Sect. 2.2.4). By this and the definition of the Bessel potential spaces, the results in Hps .Rn / follow. The analog results for Besov spaces are obtained by real interpolation. t u

Inhomogeneous Boundary Value Problems in Spaces of Higher Regularity

161

We will now consider boundary value problems where we will again restrict n n ourselves to the model P problem˛in the half-space RC WD fx 2 R W xn > 0g. As before, let A.D/ D j˛jDm a˛ D , m 2 2N, be a differential operator with constant P coefficients and let Bj .D/ D jˇjDmj bjˇ Dˇ , j D 1; : : : ; m2 , be boundary operators with constant coefficients bjˇ 2 C. We assume mj < m throughout this paper. Setting B WD .B1 ; : : : ; Bm=2 /T , we consider the boundary value problem .A.D/  /u D f

in RnC ;

0 B.D/u D g

on Rn1 :

(4)

Here 0 W u 7! ujRn1 denotes the boundary trace operator. The boundary value problem .A.D/; B.D// is called parameter-elliptic in † if A.D/ is parameter-elliptic in † and if the following Shapiro-Lopatinskii condition holds: For all . 0 ; / 2 .Rn1  † / n f0g and all h D .h1 ; : : : ; hm=2 /T 2 Cm=2 , the ordinary differential equation .A. 0 ; Dn /  /v.xn / D 0

.xn > 0/;

B. 0 ; Dn /v.0/ D h; v.xn / ! 0 .xn ! 1/ has a unique solution. If these conditions are satisfied with   2 , the boundary value problem .A.D/; B.D// is called parabolic. Throughout the following, we assume that the boundary value problem .A.D/; B.D// is parameter-elliptic in † . We first discuss the case of homogeneous boundary conditions, i.e., we assume g D 0 in (4), so we discuss the Lp .RnC /realization of .A.D/; B.D// which is given by D.AB / WD fu 2 Wpm .RnC / W 0 B.D/u D 0g and AB u WD A.D/u .u 2 D.AB //. For this, we apply the method of Banach scales (see [5], Chap. V). We recall the main definitions and results. Let X be a Banach space, f; g an exact interpolation functor, and AW X  D.A/ ! X be the generator of a C0 -semigroup. Then for k 2 N0 , the space Ek is defined by Ek WD D.Ak /, and the Ek -realization Ak of Ak1 is iteratively defined by D.Ak / WD fu 2 Ek \ D.Ak1 / W Ak1 u 2 Ek g;

Ak u WD Ak1 u .u 2 D.Ak //:

Here we have set E0 WD X and A0 WD A. For s 2 .0; 1/ n N, we write s D k C  with k 2 N0 and  2 .0; 1/ and define the space Es WD fEk ; EkC1 g and the operator As as the Es -realization of Ak , i.e., D.As / WD fu 2 Es \ D.Ak / W Ak u 2 Es g;

As u WD Ak u .u 2 D.Ak //:

Remark 2.2 In the above situation, Œ.Es ; As / W s  0 defines a scale of Banach spaces in the sense of [5], Definition V.1.1. Moreover, the operator As is again the generator of a C0 -semigroup in Es for all s  0. This follows from [5],

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Theorem V.2.1.3 and Corollary V.2.1.4, after introducing a shift, i.e. considering Q Moreover, we have AQ WD A  ! with ! > 0 sufficiently large such that 0 2 .A/. .As / D .A/ for all s  0, and the resolvent estimate carries over from A to As , see [5], inequality (2.1.16) in Theorem 2.1.3. In particular, if A W E0  E1 ! E0 is sectorial with angle  then the same holds for As W Es  EsC1 ! Es . To apply the above abstract definitions to the boundary value problem, we introduce the following spaces (see Amann [6], Sect. 4.9): Definition 2.3 Assume that the Lp -realization AB of .A.D/; B.D// generates a C0 semigroup. For p 2 .1; 1/ and s 2 Œ0; 1/ n fkm C mj C 1p W k 2 N0 ; j D 1; : : : ; m2 g, s .RnC / as the set of all u 2 Wps .RnC / which satisfy we define the space WpI.A;B/ 0 Bj Ak u D 0 for all k 2 N0 and j D 1; : : : ; m2 with s  mk  mj > 1p . The Banach scale method gives the following result. Theorem 4 Let the boundary value problem .A.D/; B.D// be parabolic, and let p 2 .1; 1/ and s 2 Œ0; 1/ n fkm C mj C 1p W k 2 N0 ; j D 1; : : : ; m2 g. Then for all s .RnC / and all  2 † 2 n f0g the problem .A.D/  /u D f , 0 B.D/u D 0, f 2 WpI.A;B/ mCs has a unique solution u 2 WpI.A;B/ .RnC /, and the a priori estimate kukWpmCs .Rn / C jj kukWps .RnC /  Ck f kWps .RnC /

mCs .u 2 WpI.A;B/ .RnC //

C

(5)

.s/

s holds. In particular, the WpI.A;B/ .RnC /-realization AB given by .s/

mCs D.AB / WD WpI.A;B/ .RnC /;

is sectorial with angle larger than s in WpI.A;B/ .RnC /.

 2

.s/

.s/

AB u WD A.D/u .u 2 D.AB //

and therefore generates an analytic semigroup

Proof We will apply the method of Banach scales as introduced above. We consider the Lp -realization AB and remark that it is well known that AB is sectorial with angle larger than 2 (see, e.g., [2]). km .RnC / D D.AkB /. By (i) First we show that for each k 2 N0 , we have WpI.A;B/ definition, the inclusion “” is obvious. To show the converse inclusion, we have to prove that Ek WD D.AkB / WD fu 2 D.AB / W A` u 2 D.AB / .` D km 0; : : : ; k  1/g is contained in WpI.A;B/ .RnC /. Due to the definition of D.AB /, we have 0 Bj A` u D 0 for all ` D 0; : : : ; k  1 and all j D 1; : : : ; m2 . Therefore, we only have to show that D.AkB /  Wpkm .RnC /. This is done iteratively. As u 2 D.A2B / and 0 Bj u D 0 for all j D 1; : : : ; m2 , we have u 2 Wpm .RnC / and Au 2 Wpm .RnC /. For 0 2 .AB /, the boundary value problem .A  0 ; B/ is regular elliptic in the sense of Triebel [24], Definition 5.2.1/4. By elliptic regularity, we obtain u 2 Wp2m .RnC /. Replacing

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163

u by Au and using A2 u 2 Wpm .RnC /, we can now prove Au 2 Wp2m .RnC /. An iteration gives u 2 Wpkm .RnC /. (ii) We consider .AB /s=m and the scale generated by AB . By real interpolation, we have for k 2 N0 and  2 .0; 1/ the identities   .kC1/m km .RnC /; WpI.A;B/ .RnC / EkC D .Ek ; EkC1 /;p D WpI.A;B/

.kC /m

;p

D WpI.A;B/ .RnC /:

Here the last equality was shown in Amann [6], Corollary 4.9.2, in a more general setting. Due to Amann [5], Theorem 2.1.3, we see that .AB /s=m mCs generates an analytic semigroup and that D..AB /s=m / D E ms C1 D WpI.A;B/ .RnC /. .s/

s -realization AB . In particular, Therefore, .AB /s=m coincides with the WpI.A;B/ .s/

.AB / D .AB /  † 2 n f0g, and the resolvent estimate (5) holds. We remark that the exceptional cases s D km C mj C interpolation results, see the discussion in Amann [6, 7].

1 p

t u

arise due to the real

3 The Case of Inhomogeneous Boundary Conditions Now we consider the boundary value problem (4) for g 6D 0, again restricting ourselves to the model problem in RnC . For this, we will use an explicit representation of the solution. We start with the definition of the basic solutions. Throughout this section, we assume that .A.D/; B.D// is parameter-elliptic in a fixed sector † . Lemma 3.1 For each . 0 ; / 2 .Rn1  † / n f0g and j D 1; : : : ; m2 , we define the basic solution wj D wj . 0 ; xn ; / as the unique stable solution of the ordinary differential equation .A. 0 ; Dn /  /wj .xn / D 0 .xn > 0/; Bk . 0 ; Dn /wj .0/ D ıjk

.k D 1; : : : ;

m /: 2

Then wj can be written in the form wj . 0 ; xn ; / D

Z . 0 ;/

eixn  .A. 0 ; /  /1 Nj . 0 ; ; /d

where . 0 ; / is a smooth contour in the upper complex half-plane which encloses all roots of the polynomial  7! A. 0 ; /   with positive imaginary part. The

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functions Nj and wj are smooth with respect to their arguments and continuous for all . 0 ; / 2 .Rn1  † / n f0g , and we have the quasi-homogeneities Nj . 0 ; ; m / D mmj 1 Nj . 0 ; ; /; wj . 0 ; xn ; m / D mj wj . 0 ; xn ; / for  > 0,  0 2 Rn1 n f0g and  2 † . Proof These assertions are stated in Denk-Faierman-Möller [9], Lemma 2.5. See also Volevich [25] for an explicit construction of Nj . t u To define the solution operators, we will use a parameter-dependent extension k 1p

operator E W Wp

.Rn1 / ! Wpk .RnC / given by

  .E g/.x0 ; xn / WD .F 0 /1 exp  .j 0 j C jj1=m /xn .F 0 g/. 0 /: Here F 0 stands for the partial Fourier transform with respect to the first n  1 variables. This operator was studied in Grubb–Kokholm [15] and in AgranovichDenk-Faierman [2] in connection with the parameter-dependent norms above. It was shown that for all k 2 N, the trace operator    k 1  0 W Wpk .RnC /; jjj  jjjk;p;RnC ! Wp p .Rn1 /; jjj  jjjk 1 ;p;Rn1 p

is continuous and E is a continuous right-inverse to 0 . Here and in the following, we call a linear operator continuous with respect to the parameter-dependent norms if for each 0 > 0 the norm of this operator can be estimated by a constant C D C.0 / which does not depend on  for each  2 † with jj  0 . Let us now consider the boundary value problem .A.D/  /u D 0 0 B.D/u D g

in RnC ; on Rn1

(6)

Qm=2 mCkmj  1p n1 .R /. Following an idea from Volevich [25], we write with g 2 jD1 Wp F 0 u in the form .F 0 u/. 0 ; xn / D

m=2 X

wj . 0 ; xn ; /.F 0 gj /. 0 /

jD1

D

m=2 Z X jD1

D

m=2 X jD1

1 0

i @ h wj . 0 ; xn C yn ; /.F 0 E gj /. 0 ; yn / dyn @yn

  .1/ .2/ F 0 Tj ./E gj C Tj ./.@n E gj / :

Inhomogeneous Boundary Value Problems in Spaces of Higher Regularity .1/

.2/

Here the solution operators Tj ; Tj Z

.1/

.Tj ./'/.x/ WD  Z

.2/

.Tj ./'/.x/ WD 

1 0 1 0

165

are given by

.F 0 /1 .@n wj /. 0 ; xn C yn ; /.F 0 '/. 0 ; yn /dyn ; .F 0 /1 wj . 0 ; xn C yn ; /.F 0 '/. 0 ; yn /dyn :

Lemma 3.2 Let k 2 N0 and p 2 .1; 1/. Then the operators     mCkmj n .1/ .RC /; jjj  jjjmCkmj ;p;RnC ! WpmCk .RnC /; jjj  jjjmCk;p;RnC ; Tj ./W Wp  mCkmj 1 n    .2/ Tj ./W Wp .RC /; jjj  jjjmCkmj 1;p;RnC ! WpmCk .RnC /; jjj  jjjmCk;p;RnC ; j D 1; : : : ; m2 , are continuous with respect to the parameter-dependent norms. .1/

.2/

Proof We only consider Tj , the proof for Tj essentially being the same. Let 0 > 0. We make use of the equivalence of norms jjjujjjmCk;p;RnC

mCk X

X  mCk`   m .D0 /˛0 @˛n u p n : n L .R / C

`D0 j˛jD`

Therefore, we have to estimate .1/

p

jjjTj ./'jjjmCk;p;Rn

C

C

mCk X

XZ

`D0 j˛jD`

1 0

Z  

1 0

.F 0 /1 

mCk` m

0

. 0 /˛ .@˛n n C1 wj /. 0 ; xn C yn ; / p  .F 0 '/. 0 ; yn /dyn  p

L .Rn1 /

C

mCk X

XZ

`D0 j˛jD`

1 0

Z

1 0

dxn

  0 1 .F / Mj;`;˛ . 0 ; xn C yn ; /   Q 0 ; yn / .F 0 '/.

Lp .Rn1 /

p dyn dxn :

Here we have defined 'Q WD .F 0 /1 .j 0 j C jj1=m /mCkmj F 0 ' 2 Lp .RnC / and Mj;`;˛ . 0 ; xn ; / WD .j 0 j C jj1=m /mkCmj 

mCk` m

0

. 0 /˛ .@n˛n C1 wj /. 0 ; xn ; /:

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R. Denk and T. Seger

We will apply Michlin’s theorem to the functions Mj;`;˛ . For this, we abbreviate  WD . 0 ; / WD j 0 j C jj1=m and use the homogeneities stated in Lemma 3.1. In the integral representation for the basic solutions wj in Lemma 3.1, we make the 0 substitution  7! Q D  and use the fact that .  ; m / can be replaced by a contour Q which is independent of  0 and . For ˇ 0 2 N0n1 , we obtain ˇ ˇ ˇ 0 ˇ0 ˇ0 ˇ ˇ. / D 0 Mj;`;˛ . 0 ; xn ; /ˇ Z ˇ  0 ˇ 0 ˇ0 mCk` ˇ0 m D ˇ. /   ˛n C1 eixn D 0 mkCmj . 0 /˛ . 0 ;/

ˇ 0  0   mCk` Z ˇ ˇ  m D ˇ  m

 ˇ ˇ .A. 0 ; /  /1 Nj . 0 ; ; / d ˇ   ˛n C1

. 0 ;/



eixn Hj;˛0 ;ˇ0

 0 

 ˇˇ ;  ; m d ˇ

ˇ 0  0   mCk` Z  0  ˇˇ ˇ ˇ ˛n C1 iQ xn   m 0 0 D ˇ  d Q ˇ   Q e H ; ; Q j;˛ ;ˇ  m m Q

 C exp.Cxn / 

C : xn

Here we have set   0 ˇ0 Hj;˛0 ;ˇ0 . 0 ; ; / WD D 0 mkCmj . 0 /˛ .A. 0 ; /  /1 Nj . 0 ; ; / and used the fact that Hj;˛0 ;ˇ0 is quasi-homogeneous in its arguments of degree j˛ 0 j m  k  1  jˇ 0 j. The two inequalities follow by a compactness argument and by the elementary inequality tet  1 for t  0. Now an application of Michlin’s theorem in Rn1 gives .1/

jjjTj ./'jjjmCk;p;RnC C

mCk X

X Z

`D0 j˛jD`

C

Z

1 0

1

Z

0

1 0

p

p 1=p k'.; Q yn /kLp .Rn1 / dyn dxn xn C yn

k'.; Q yn /kLp .Rn1 / dyn

D Ck'k Q Lp .RnC / :

1=p

Inhomogeneous Boundary Value Problems in Spaces of Higher Regularity

167

Here we have used the continuity of the Hilbert transform in Lp .RC / for the second inequality. Finally, we have     k'k Q Lp .RnC / D .F 0 /1 .j 0 j C jj1=m /mCkmj F 0 ' 

Lp .RnC /

  mCkmj  C k'kmCkmj ;p;RnC C jj m k'kLp .RnC /  Cjjj'jjjmCkmj;p;RnC ; .1/

which shows the continuity of Tj ./.

t u

The last lemma is the main step in the proof of uniform a priori estimates with respect to parameter-dependent norms. We obtain the following result, cf. Agranovich [3], Theorem 3.2.1 for the case p D 2. Theorem 3 Let s 2 Œ0; 1/ and p 2 .1; 1/ with m C s  mj  1p 62 N0 for all j D 1; : : : ; m2 . Then for every  2 † n f0g, all f 2 Wps .RnC / and all Qm=2 mCsmj  1p n g 2 jD1 Wp .R / there exists a unique solution u 2 WpmCs .RnC / of (4). Moreover, the operator .A.D/  ; 0 B.D//W WpmCs .RnC / ! Wps .RnC / 

m=2 Y

mCsmj  1p

Wp

.Rn1 /

(7)

jD1

is an isomorphism of Banach spaces with respect to the parameter-dependent norms. In particular, for every 0 > 0 there exists a constant C D C.0 / such that m=2   X jjjujjjmCs;p;RnC  C jjj f jjjs;p;RnC C jjjgj jjjmCsmj  1 ;p;Rn1 p

(8)

jD1

holds for all  2 † with jj  0 . Proof (i) First we assume s 2 N0 . The case s D 0 is well-known, see, e.g., AgranovichDenk-Faierman [2], Theorem 2.1. In particular, we already know unique solvability of (4) with the solution u being at least in Wpm .RnC /. Moreover, the continuity of the operator in (7) with respect to the parameter-dependent norms is an immediate consequence of the continuity of the derivatives and of the trace operator. Therefore, we only have to show the a priori estimate (8) which also gives the smoothness of the solution. Let rC 2 L.Wps .Rn /; Wps .RnC //; rC f WD f jRnC , denote the operator of restriction onto RnC . Using the fact that there exists a coretraction eC of rC

168

R. Denk and T. Seger

(see Amann [6], Sect. 4.4) with eC 2 L.Wp` .RnC /; Wp` .Rn // for all ` 2 Œ0; s, we see that both rC and eC are continuous with respect to the parameter-dependent norms, too. We write u D rC u1 C u2 , where u1 is the unique solution of .A.D/  /u1 D eC f in Rn . By the explicit representation of u1 (see the proof of Lemma 2.1), we see that jjjrC u1 jjjmCs;p;RnC  jjju1 jjjmCs;p;Rn  CjjjeC f jjjs;p;Rn  Cjjj f jjjs;p;RnC : For u2 we obtain the boundary value problem .A.D/  /u2 D 0

in RnC ;

0 B.D/u2 D gQ

on Rn1

with gQ WD g  0 B.D/rC u1 . By the continuity of rC , B.D/ and 0 (with respect to the parameter-dependent norms), we see that   jjjQgj jjjmCsmj  1 ;p;Rn1  C jjjgj jjjmCsmj  1 ;p;Rn1 C jjj f jjjs;p;RnC : p

p

(9)

Due to Lemma 3.2, we have m=2   X .1/ .2/ Tj ./E gQ j C Tj ./@n E gQ j : u2 D jD1 .1/

.2/

Now the continuity of E ; @n , and Tj ; Tj jjju2 jjjmCs;p;RnC  C

m=2 X

yields

jjjQgj jjjmCsmj  1 ;p;Rn1 p

jD1

which in connection with (9) gives the a priori estimate (8) and the proof for s 2 N0 . (ii) For s 2 .0; 1/ n N with m C s  mj  1p 62 N0 , the result follows by real interpolation. Here we use the fact that for k 2 N0 and  2 .0; 1/ we have 



Wpk .RnC /; WpkC1 .RnC /

;p

D WpkC .RnC /

uniformly in  with respect to the parameter-dependent norms, see GrubbKokholm [15], Theorem 1.1. t u A combination of Theorems 4 and 3 immediately yields the following result.

Inhomogeneous Boundary Value Problems in Spaces of Higher Regularity

169

Corollary 3.4 Let .A.D/; B.D// be parabolic, and let s 2 Œ0; 1/nfkmCmj  1p W k 2 s .RnC / N; j D 1; : : : ; m2 g. Then for all  2 † 2 n f0g, jj  0 > 0, and all f 2 WpI.A;B/ 1 Qm=2 mCsmj  p n1 and g 2 jD1 Wp .R / there exists a unique solution u 2 WpmCs .RnC / of (4), and m=2   X kukWpmCs .Rn / Cjj kukWps .RnC /  C k f kWps .RnC / C jjjgj jjjmCsmj  1 ;p;Rn1 : C

p

(10)

jD1

In particular, this holds for all f 2 Wps .RnC / if s < mj C

1 p

for all j D 1; : : : ; m2 .

Proof We write u D u1 C u2 , where u1 solves .A.D/  /u1 D f ; 0 B.D/u1 D 0 and u2 solves .A.D/  /u2 D 0; 0 B.D/u2 D g, and apply Theorems 4 and 3, respectively. For the application of Theorem 3, we note that by the interpolation inequality, the left-hand side of (10) is not larger than a constant times jjjujjjmCs;p;RnC . The last statement follows directly from the fact that for these s, the spaces Wps .RnC / s and WpI.A;B/ .RnC / coincide. t u Remark 3.5 (a) The results of Theorem 3 and Corollary 3.4 have some connection to results by D. Guidetti [16–18]. In particular, an estimate similar to (10) can be found as Proposition 2.16 in [17] in the Besov space setting, under the additional assumption minj mj < s < minj mj C 1p . We remark that in [17] essentially the same parameter-dependent norms for gj appear as well as some compatibility s conditions. Our assumption f 2 WpI.A;B/ .RnC / enables us to consider a wider range for s. Apart from some restrictions on the smoothness parameter s, the results by Guidetti in the Besov space setting are quite general and include the generation of an analytic semigroup. (b) On the right-hand side of (10) large powers of  may appear, although we only have jj on the left-hand side. The following elementary example in R1C shows that even in the one-dimensional case this cannot be avoided: Consider the boundary value problem u.xn /u00 .xn / D p 0 .xn > 0/; u.0/ D g 2 C. Then for  > 0 the stable solution is u.xn / D exp. xn /g, and a direct calculation shows that for s 2 N0 we have kukWp2Cs .R

C/

 Cjj

2Cs1=p 2

jgj:

The power on the right-hand side is consistent with the exponent .m C s  mj  1 p /=m appearing on the right-hand side of (10). In some sense the parameter-dependent norms in Theorem 3 are natural for parameter-elliptic problems. However, they do not yield resolvent estimates as the parameter  appears on both sides. Moreover, on the left-hand side we have jj.mCs/=m instead of jj. We will now derive a resolvent estimate for the

170

R. Denk and T. Seger

Wps .RnC /-realization. As it was shown in [8], for a decay like jj1 additional conditions on f are necessary. The following result gives a general resolvent estimate. Theorem 6 (a) In the situation of Theorem 3, assume that

WD

max .s  mj  1p / > 0:

(11)

jD1;:::;m=2

Then for all 0 > 0 there exists a constant C D C.0 / > 0 such that m=2   X jjjgj jjjmCsmj  1 ;p;Rn1 : kukmCs;p;RnC C jj kuks;p;RnC  C jj =m k f ks;p;RnC C p

jD1

(12) (b) Let .A.D/; B.D// be parabolic, and define as in (11). For s 2 Œ0; 1/ n fmk C .s/ mj  1p W k 2 N0 ; j D 1; : : : ; m=2g define the unbounded operator AQ B in Wps .RnC / by .s/

D.AQ B / WD fu 2 WpmCs .RnC / W 0 B.D/u D 0g; .s/ .s/ AQ B u WD A.D/u .u 2 D.AQ B //: .s/ Then .AQ B /  † 2 n f0g, and for all 0 > 0 there exists C D C.0 / > 0 such that

 .s/  .AQ  /1  s n B L.W .R p

C //

 Cjj1Cmaxf0; =mg

. 2 † 2 ; jj  0 /:

Proof (a) We write the solution u in the form u D u1 C u2 C u3 . Here u1 solves .A.D/  /u1 D 0; 0 B.D/u1 D g, while u2 WD rC uQ 2 with uQ 2 being the solution of .A.D/  /Qu2 D eC f in Rn . Due to Corollary 3, we obtain ku1 kmCs;p;RnC C jj ku1 ks;p;RnC  C

m=2 X

jjjgj jjjmCsmj  1 ;p;Rn1 : p

jD1

Moreover, by Lemma 2.1, we have ku2 kmCs;p;RnC C jj ku2 ks;p;RnC  Ck f ks;p;RnC :

(13)

Inhomogeneous Boundary Value Problems in Spaces of Higher Regularity

171

The function u3 is the solution of the boundary value problem .A.D/  /u3 D 0

in RnC ;

0 B.D/u3 D 0 B.D/u2

on Rn1 :

We apply Corollary 3 again and get m=2 X

ku3 kmCs;p;RnC C jj ku3 ks;p;RnC  C

jjj0 B.D/u2 jjjmCsmj  1 ;p;Rn1 p

jD1

DC

m=2  X

k0 B.D/u2 kmCsmj  1 C jj

mCsmj 1=p m

p

 k0 B.D/u2 kLp .Rn1 / :

jD1

We estimate the norms on the right-hand side for each j. First, we have k0 B.D/u2 kmCsmj  1 ;p;Rn1  CkB.D/u2 kmCsmj ;p;RnC p

 Cku2 kmCs;p;RnC for all j D 1; : : : ; m2 . If s > mj C 1p , we can estimate jj

mCsmj 1=p m

k0 B.D/u2 kLp .Rn1 /  jj

mCsmj 1=p m

k0 B.D/u2 ksmj  1 ;p;Rn1 p

 Cjj

mCsmj 1=p m

 Cjj

mCsmj 1=p m

kB.D/u2 ksmj ;p;RnC ku2 ks;p;RnC

 Cjj =m jj ku2 ks;p;RnC  Cjj =m k f ks;p;RnC : Here we applied (13) in the last step. If s < mj C 1p , we similarly write jj

mCsmj 1=p m

k0 B.D/u2 kLp .Rn1 /  jj

mCsmj 1=p m

 Cjj =m jj

k0 B.D/u2 k ;p;Rn1

mCsmj  1=p m

ku2 k Cmj C 1 ;p;Rn p

C

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R. Denk and T. Seger

   Cjj =m ku2 kmCs;p;RnC C jj kuks;p;RnC  Cjj =m k f ks;p;RnC ; where we used the interpolation inequality (see Grisvard [14], Theorem 1.4.3.3) for the third inequality. This finishes the proof of the a priori estimate (12) and of part (a). .s/ (b) In the case < 0 the a priori estimate (and the fact that the operator AQ B is sectorial) follows from the last statement in Corollary 3.4. For > 0, we apply part (a) with g D 0. Note that the case D 0 is excluded. t u Remark 3.7 For the Dirichlet-Laplacian D , we have m D 2, m1 D 0, and therefore

D s  1p . For the resolvent in Wps .RnC /, we obtain from Theorem 6 1

k.  D /1 kL.Wps .RnC //  Cjj1C 2  2p : s

For s D 1, we have a decay like jj1=21=.2p/ . It was shown in [19] that this is the exact decay rate.

References 1. S. Agmon, On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems. Commun. Pure Appl. Math. 15, 119–147 (1962) 2. M. Agranovich, R. Denk, M. Faierman, Weakly smooth nonselfadjoint spectral elliptic boundary problems, in Spectral Theory, Microlocal Analysis, Singular Manifolds, Math. Top., vol. 14 (Akademie Verlag, Berlin, 1997), pp. 138–199 3. M.S. Agranovich, Elliptic boundary problems, in Partial Differential Equations, IX, Encyclopaedia Mathematical Science, vol. 79 (Springer, Berlin, 1997), pp. 1–144, 275–281. [Translated from the Russian by the author] 4. M.S. Agranovich, M.I. Vishik, Elliptic problems with a parameter and parabolic systems of general form. Russ. Math. Surv. 19, 53–157 (1964) 5. H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I: Abstract linear theory, Monographs in Mathematics, vol. 89 (Birkhäuser, Boston, 1995) 6. H. Amann, Anisotropic function spaces and maximal regularity for parabolic problems. Part 1: Function spaces, Jind˘rich Ne˘cas Center for Mathematical Modeling Lecture Notes, vol. 6 (Matfyzpress, Prague, 2009) 7. H. Amann, Function spaces on singular manifolds. Math. Nachr. 286(5–6), 436–475 (2013) 8. R. Denk, M. Dreher, Resolvent estimates for elliptic systems in function spaces of higher regularity. Electron. J. Differ. Equ. 109, 12 (2011) 9. R. Denk, M. Faierman, M. Möller, An elliptic boundary problem for a system involving a discontinuous weight. Manuscripta Math. 108(3), 289–317 (2002) 10. R. Denk, M. Hieber, J. Prüss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 166(788), viii+114 (2003) 11. R. Denk, J. Saal, J. Seiler, Bounded H1 -calculus for pseudo-differential Douglis-Nirenberg systems of mild regularity. Math. Nachr. 282(3), 386–407 (2009) 12. M. Faierman, On the resolvent arising in a parameter-elliptic multi-order boundary problem. Math. Nachr. 285(13), 1643–1670 (2012)

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13. G. Geymonat, P. Grisvard, Alcuni risultati di teoria spettrale per i problemi ai limiti lineari ellittici. Rendiconti del Seminario Matematico della Università di Padova 38, 121–173 (1967) 14. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, vol. 24 (Pitman Advanced Publishing Program, Boston, MA, 1985) 15. G. Grubb, N.J. Kokholm, A global calculus of parameter-dependent pseudodifferential boundary problems in Lp Sobolev spaces. Acta Math. 171(2), 165–229 (1993) 16. D. Guidetti, A maximal regularity result with applications to parabolic problems with nonhomogeneous boundary conditions. Rendiconti del Seminario Matematico della Università di Padova 84, 1–37 (1990) 17. D. Guidetti, On elliptic problems in Besov spaces. Math. Nachr. 152, 247–275 (1991) 18. D. Guidetti, On boundary value problems for parabolic equations of higher order in time. J. Differ. Equ. 124(1), 1–26 (1996) 19. M. Nesensohn, Randwertprobleme in Sobolevräumen höherer Ordnung. Diploma thesis, University of Konstanz, 2009 20. J.A. Roitberg, Z.G. Sheftel, Boundary value problems with a parameter for systems elliptic in the sense of Douglis-Nirenberg. Ukrain. Mat. Ž. 19(1), 115–120 (1967) 21. T. Seger, Elliptic-parabolic systems with applications to lithium-ion battery models. Ph.D. thesis, University of Konstanz, 2013 22. Y. Shibata, Generalized resolvent estimates of the Stokes equations with first order boundary condition in a general domain. J. Math. Fluid Mech. 15(1), 1–40 (2013) 23. Y. Shibata, S. Shimizu, On the maximal Lp -Lq regularity of the Stokes problem with first order boundary condition; model problems. J. Math. Soc. Japan 64(2), 561–626 (2012) 24. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators (Deutscher Verlag der Wissenschaften, Berlin, 1978) 25. L.R. Volevich, Solvability of boundary problems for general elliptic systems. Am. Math. Soc. Translat. II. Ser. 67, 182–225 (1968)

Blow-Up Criterion for 3D Navier-Stokes Equations and Landau-Lifshitz System in a Bounded Domain Jishan Fan and Tohru Ozawa Dedicated to Professor Yoshihro Shibata on the occasion of his sixtieth birthday

Abstract In this paper we prove a blow-up criterion for the 3D Navier-Stokes equations in a bounded domain in terms of a BMO norm of vorticity. We will also prove a regularity criterion for the Landau-Lifshitz system in a bounded domain. Keywords BMO-norm • Landau-Lifshitz • Navier-Stokes • Regularity criterion

1 Introduction Let   R3 be a bounded, simply connected domain with smooth boundary @, and be the unit outward normal vector to @. We consider the regularity criterion of the Navier-Stokes equations: div u D 0;

(1)

@t u C u  ru C r D u in   .0; 1/;

(2)

u  D 0; curl u  D 0 on @  .0; 1/;

(3)

u.; 0/ D u0 in   R3 ;

(4)

where u and  denote unknown velocity vector field and pressure scalar of the fluid, respectively. The physical constant   0 is the viscosity and we will take  D 1 for simplicity.

J. Fan () Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, P.R. China e-mail: [email protected] T. Ozawa Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan e-mail: [email protected] © Springer Basel 2016 H. Amann et al. (eds.), Recent Developments of Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-0348-0939-9_10

175

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It is well known that the problem has at least a global-in-time weak solution and a unique local-in-time strong solution [8, 13]. However, the regularity of weak solutions is still a very challenging open problem. On the other hand, the development of blow-up criteria is of great importance for both theoretical and practical purpose. When  WD R3 , many criteria have been obtained, see [1, 3, 4, 6, 11, 12, 15, 16] and the references therein. When  is a bounded domain and the boundary condition is the homogeneous Dirichlet boundary condition, Giga [7] and Kim [10] proved a Serrin type regularity criterion. When  is a bounded domain and the boundary condition is (3), Kang and Kim [9] prove some Serrin type regularity criteria. The aim of this paper is to prove a new regularity criterion for the problem (1)–(4). We will prove that Theorem 1.1 Let u0 2 H 3 ./ and div u0 D 0 in  and u0  D 0; curl u0  D 0 on @. Let u be a unique local smooth solution to the problem (1)–(4). If u satisfies ru 2 L1 .0; TI BMO.// with 0 < T < 1;

(5)

then the solution u can be extended beyond T > 0. Remark 1.2 When  D 0, (5) is proved in [14] for the Euler equations, which is simpler than the case  > 0, since we need to deal with the boundary integrals. To prove Theorem 1.1, we will use the following logarithmic Sobolev inequality [14]: 3 krukL1  C.1 C krukBMO log.e C kukW s;p // with s > 1 C : p

(6)

Next, we consider the 3D Landau-Lifshitz system: dt  d D djrdj2 C d  d; jdj D 1 in   .0; 1/;

(7)

@ d D 0; on @  .0; 1/;

(8)

d.; 0/ D d0 ; jd0 j D 1 in   R : d

(9)

Carbou and Fabrie [2] showed the existence and uniqueness of local smooth solutions. When  WD Rn .n D 2; 3; 4/, Fan and Ozawa [5] prove some regularity criteria. The aim of this paper is to prove a blow-up criterion for the problem (7)–(9) when  is a bounded domain. We will prove Theorem 1.3 Let d0 2 H 3 ./ with jd0 j D 1 in  and @ d0 D 0 on @. Let d be a local smooth solution to the problem (7)–(9). If d satisfies Z

T 0

2q

krdkLq3 q dt < 1 with 3 < q  1 and 0 < T < 1;

(10)

Blow-Up Criterion for 3D Navier-Stokes Equations and Landau-Lifshitz. . .

177

then the solution d can be extended beyond T > 0.

2 Proof of Theorem 1.1 This section is devoted to the proof of Theorem 1.1. We only need to establish some a priori estimates. First, testing (2) by u and using (1) and (3), we have the well-known energy inequality: 1d 2 dt

Z

u2 dx C

Z

jcurl uj2 dx  0;

which gives kukL1 .0;TIL2 / C kukL2 .0;TIH 1 /  C:

(11)

Taking curl to (2) and using (1), we get the well-known equation for the vorticity ! WD curl u W @t ! C u  r!  !  ru D !:

(12)

Testing (12) by ! and using (1) and (6), we see that Z Z 1d j!j2 dx C jcurl !j2 dx 2 dt Z Z D .!  r/u  !dx  krukL1 ! 2 dx Z  C.1 C krukBMO / log.e C kukH 3 /

j!j2 dx;

and therefore Z

Z

2

t

j!j dx C t0

kcurl !k2L2 d  C.e C y/C0 

(13)

provided that Z

t

krukBMO d   2, q > 3 and 2s C 3q D 1. In this review we report on recent results on this problem, considering first of all optimal initial values u.0/ to yield a local in time strong solution, then criteria to prove regularity locally on subintervals of Œ0; T/. Special emphasis is put on results for smooth bounded and also general unbounded domains. Most criteria are based on conditions of Besov space type. Keywords General domains • Initial values • Navier-Stokes equations • Regularity • Serrin’s class • Strong solutions • Uniqueness • Weak solutions

1 Introduction The Navier-Stokes system is the most classical model to describe the flow of a viscous incompressible fluid. Despite of about 80 years of mathematical analysis, since the seminal paper of J. Leray [41] on the existence of global weak solutions in the whole space R3 and corresponding results of E. Hopf [34] for domains, basic

R. Farwig () Fachbereich Mathematik and International Research Training Group (IRTG 1529), Darmstadt-Tokyo, Technische Universität Darmstadt, Schlossgartenstr. 7, 64283 Darmstadt, Germany e-mail: [email protected] © Springer Basel 2016 H. Amann et al. (eds.), Recent Developments of Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-0348-0939-9_11

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questions on uniqueness and regularity of weak solutions are still open. These fundamental problems are also of importance for the general theory of partial differential equations and brought Clay Mathematics Institute in 2000 to classify the issue of regularity as one of the seven Millennium Prize Problem, see C. Fefferman [27]. Given a domain   R3 and a time interval Œ0; T/, 0 < T  1, consider an external force f W   .0; T/ ! R3 and an initial value u0 W  ! R3 . Then we are looking for a velocity field u and an associated scalar pressure function p such that ut  u C u  ru C rp D f ;

div u D 0 in   .0; T/;

u.0/ D u0 ;

u D 0 on @  .0; T/:

(1)

For simplicity the coefficient P of viscosity > 0 has been set to D 1. The nonlinear transport term u  ru D 3jD1 uj @j u can also be written in the form u  ru D div.u ˝ u/   since u is solenoidal; we recall that u ˝ v D .ui vj /3i;jD1 and div .Fij /i;j D 3  P3 iD1 @i Fij jD1 for a matrix field F D .Fij /. In this article we use standard and Bochner    notation for Lebesgue,  Sobolev  spaces, i.e. Lq ./ D Lq ; kkq , W k;q ./ D W k;q ; kkW k;q , and Ls .0; T/I Lq ./ D Ls .Lq //; k  kq;sIT D k  kLs .Lq / , 1  s; q  1, respectively. We do not differ between spaces of scalar-, vector- and matrix-valued functions. The index will denote a subspace of solenoidal vector fields, the subscript 0 a subspace of functions with vanishing trace. Duality products of functions on  and   .0; T/ will be denoted by h; i D h; i and h; i;T , respectively. Definition 1.1 (Definition of Weak and Strong Solutions) 1. A weak solution u (in the sense of Leray-Hopf) is a solution in the sense of distributions, i.e.  hu; wt i;T C hru; rwi;T C hu  ru; wi;T D hf ; wi;T C hu0 ; w.0/i

(2)

  1 ./ , lying in the Leray-Hopf class for all test functions w 2 C01 Œ0; T/I C0;

    u 2 LHT D L1 0; TI L2 ./ \ L2loc Œ0; T/I H01 ./

(3)

and satisfying the strong energy inequality .SEI/ 1 ku.t/k22 C 2

Zt t0

kruk22

1 d  ku.t0 /k22 C 2

for almost all t0 2 Œ0; T/ and for all t 2 Œ0; T/.

Zt hf ; ui d t0

(4)

Local Regularity Results for the Instationary Navier-Stokes Equations Based. . .

185

If u satisfies (4) for t0 and all t > t0 then we say that u satisfies the energy inequality .EI/t0 . To emphasize that .EI/0 holds (only), u is said to satisfy the energy inequality .EI/. 2. A weak solution u 2 LHT is called a strong solution (in the sense of Serrin) if there are exponents s; q such that   u 2 Ls 0; TI Lq ./ ; Under the assumption class.

2 s

C

3 q

s > 2; q > 3;

3 2 C D 1: s q

(5)

  D 1 the space Ls 0; TI Lq ./ is called a Serrin

Let us recall several important results on weak and strong solutions. Remark 1.2 1. Redefining a weak solution u on a subset of .0; T/ of Lebesgue measure equal to 0 we may assume that u W Œ0; T/ ! L2 ./ is weakly continuous,

(6)

  or u 2 Cw0 Œ0; T/I L2 ./ for short. Condition (6) will be tacitly assumed in the following. 3 2. The existence of weak solutions is known  domains   R , initial  for arbitrary 2 1 2 values in u0 2 L ./ and forces f 2 L 0; TI L ./ . More generally, f may be assumed to be a functional of the form f D div F, F 2 L2 0; TI L2 ./ so that in (2) hf ; wi;T WD hF; rwi;T : Under this assumption a weak solution u of (1)   can be extended to t D T and considered as a function u 2 Cw0 Œ0; TI L2 ./ . Then the energy inequality (and the energy equality, if possible) does hold for all t upto t D T.  Functionals of the form f D div F with F 2 L2 0; TI L2 ./ are important since several parts of the theory require the nonlinear term to satisfy u  ru D  div.u ˝ u/ with u ˝ u 2 L2 0; TI L2 ./ . 3. Weak solutions can be constructed by several well-known methods, e.g., by Galerkin’s method. However, in each case the weak solution satisfies the energy inequality .EI/ in (4). The reason for the inequality rather than an equality is the use of approximation techniques and the lower semi-continuity of the norm k  k2 with respect to weak convergence. When  is bounded, the weak solutions even satisfy the strong energy inequality .SEI/. In the case of an exterior domain maximal regularity estimates yielding Ls .Lq /-estimates of the associated pressure p are needed to construct a weak solution satisfying .SEI/, see [46]. In general smooth unbounded domains more special functions spaces are needed to construct (suitable) weak solutions

186

R. Farwig

satisfying the strong energy inequality. For details we refer to [14, 22] and Sect. 2.4. 4. It is an open problem whether each weak solution independent of the way it has been constructed satisfies .EI/ or even .SEI/. 5. One fundamental problem on weak solutions concerns their uniqueness. A classical theorem, the so-called Serrin uniqueness  theorem, see [50], states that a weak solution u lying in a Serrin class Ls 0; TI Lq ./ , see (5), is unique within the class of all Leray-Hopf type weak solutions. We note that this theorem exploits the fact that a Leray-Hopf type weak  solution satisfies .EI/. Uniqueness for a weak solution in the limit class L1 0; TI L3 ./ was proved by KozonoSohr [39] and in [25], see also Theorem 2.11 below. 6. The classical condition on weak solutions to satisfy the energy equality .EE/, i.e. 1 ku.t/k22 C 2

Zt 0

kruk22

1 d D ku0 k22 C 2

Zt hf ; ui d

(7)

0

for all t 2 .0; T/, is given   u 2 L4 0; TI L4 ./

(8)

  or equivalently u ˝ u 2 L2 0; TI L2 ./ . In this case, u may be used as test function w in (2) leading to the L1 -integrability of .u  ru/u and a vanishing integral on   .0; T/ due to the fact that hu  ru; ui.t/ D 0 for a.a. t 2 .0; T/. Note that in the 3D case—in contrast to the 2D case—a weak solution u 2 LHT is not an admissible test function in (2). 7. Assumptions different to (8) to guarantee  .EE/ were  discussed by Shinbrot [51]: it suffices to assume that u 2 Lr 0; TI Lq ./ where 2r C 3q  1 C 1q , q  4. Actually, Shinbrot’s condition together with the Leray-Hopf   integrability   u 2 L1 0; TI L2 ./ implies by Hölder’s inequality that u 2 L4 0; TI L4 ./ . A similar argument can be applied when 2r C 3q  1 C 1r , r  4 together with the   fact that u 2 L2 0; TI L6 ./ . Cheskidov et al. [5] and together with Constantin [4] found much   weaker conditions which concerning their scaling behavior are of the type L3 0; TI L9=2 ./ . 3 D 43 > 54 : Let   R3 be a bounded C2 -domain and u a weak Note that 23 C 9=2   5=12  solution of (1). If u 2 L3 0; TI D A2 ; then u satisfies .EE/; see [5]. The even   1=3 3 3 weaker condition u 2 L 0; TI B3;1 .R / is sufficient to get the same result for 1=3

the whole space  D R3 , see [4]; here B3;1 .R3 / denotes a Besov space. An intermediate result was recently obtained in [13]: It suffices to use a   1=4  condition on a 12 -derivative of u, namely u 2 L3 0; T; D A18=7 . A similar condition is needed for flows in smooth unbounded domains; however, since the Stokes operator A18=7 possibly cannot be defined for an unbounded domain it is

Local Regularity Results for the Instationary Navier-Stokes Equations Based. . .

187

replaced by the Stokes operator AQ 18=7 as in Sect. 2.4. Similar results were found in Lorentz spaces; for details we refer to [13]. 8. The main open problem for weak solutions is the question of regularity, see [27]: is every weak solution u (and an associated pressure p) of class C1 in space and time provided that u0 and @ are of class C1 ? The classical result requires that  s q u lies in Serrin’s class L 0; TI L ./ as in (5); cf. [54, Chap. V, Theorems 1.8.1 and 1.8.2]. There are numerous results on conditional regularity, i.e., a posteriori conditions on a given weak solution u to guarantee its regularity. Most of these criteria are of Serrin type controlling ru, ! D rot u or various  components of u, ru, !; other conditions work with the deformation tensor 12 ru C .ru/T or the pressure p. Rather than trying to summarize and describe these results the focus of this review is on a recent approach to use an optimal initial value condition on u0 and on function values u.t/ of u for all or almost all t. These and further related results, including also unbounded domains, can be found in the articles [7, 8, 11–14, 16, 20, 21, 23– 26, 37, 47]. This article is organized as follows. In Sect. 2 we present the main results, postponing the proofs to Sect. 3. After fixing some notation and preliminaries in Sect. 2.1 the main Theorem 2.1 in Sect. 2.2 describing the optimal initial value condition is the basis for most of the following results and uses a Besov space characterization to be introduced in the beginning of Sect. 2.2. Next, Sect. 2.3 deals with further regularity criteria which either in some sense are beyond Serrin’s condition or need the kinetic energy function. General unbounded domains will be considered in Sect. 2.4. Section 3 contains the proofs or sketches of them; the bounded domain case is described in Sects. 3.1 and 3.2, the case of general smooth domains in Sect. 3.3.

2 Main Results 2.1 Notation First we introduce the Helmholtz projection on Lq ./, 1 Pq W Lq ./ ! Lq ./ D C0;

./

kkq

;

1 ./ D fu 2 C01 ./ W div u D 0g. We recall that Pq is a where 1 < q < 1 and C0;

well-defined bounded projection for bounded and exterior C1 -domains and defines an algebraic and topological decomposition

Lq ./ D Lq ./ ˚ Gq ./

188

R. Farwig q

with L ./ D R.Pq /, the range of Pq , and Gq ./ D frp 2 Lq ./ W p 2 q Lloc ./g D N .Pq /, the kernel of Pq ; for details see [29, 52]. q The Stokes operator Aq D Pq  on L ./, 1 < q < 1, is defined by 1;q

D.Aq / D W 2;q ./ \ W0 ./ \ Lq ./; Aq W D.Aq /  Lq ./ ! Lq ./;

u 7! Aq u D Pq u:

It is well-known, see e.g. [10, 30], that Aq generates a bounded analytic semigroup fetAq I t  0g for bounded and exterior domains of class C1;1 . Since Aq coincides with Ar on D.Aq / \ D.Ar /, 1 < r, q < 1, we simply write A; by analogy, since 1 Pq u D Pr u for u 2 C0;

./, we simply write P. Note that in general non-smooth or general unbounded domains P and A may fail to exist, see [2, 43]. However, for q D 2 and any domain   R3 , Hilbert space methods can be used to define P2 and A2 D P2  with the properties mentioned above. The Stokes operator Aq , 1 < q < 1, being sectorial and generating a bounded analytic semigroup, admits fractional powers A˛q , 1  ˛  1. First we consider the case of a bounded domain   R3 of class C1;1 . Then A˛q , 0  ˛  1, is q an injective, closed and densely defined operator with domain D.A˛q /  L ./ ˇ

and range R.A˛q / D L ./ such that D.Aq /  D.A˛q /  D.Aq /  L ./ for 1 0  ˇ  ˛  1. For 1  ˛ < 0 we define the bounded operators A˛q D .A˛ W q / q ˛ ˛ L ./ ! R.Aq / D D.Aq /. Important properties are the embeddings (cf. [31]) q

kvkq  ckA˛r vkr ;

v 2 D.A˛r /;

krvkq  kA1=2 q vkq ;

1 < r  q < 1; 1;q

v 2 D.A1=2 q / D W0; ./;

2˛ C

q

3 3 D q r

1 < q < 1I

(9) (10)

1=2

1;2 moreover, krvk2 D kA2 vk2 for v 2 W0;

./. From semigroup theory and the fact that .Aq /  .0; 1/ we know that there exist c D c.q; / > 0 and ı D ı.q; / > 0 such that

kA˛q etAq vkq  ct˛ eıt kvkq ;

v 2 Lq ./;

0  ˛  1:

(11)

When   R3 is an exterior domain with @ 2 C1;1 , then inequalities (9)–(11) do still hold under certain restrictions, cf. [3, 33]: In (9) we need that 2˛ C 3q D 3r and either 0˛

1 ; 2

1 0, and, if  is bounded, the term ku0 kq may be omitted [56, Theorem 1.14.5]; cf. the discussion of ku0 kB2=s in Sect. 2.2. q;s

2.2 Optimal Initial Values To describe an optimal condition on initial values u0 2 L2 ./ to allow for a local  s q in time strong solution u 2 L 0; TI L ./ of Serrin type of the Navier-Stokes system (1) it is natural to require that the solution E0 .t/ WD etA u0

Local Regularity Results for the Instationary Navier-Stokes Equations Based. . .

191

of the corresponding linear Stokes system with vanishing external force has the   property E0 2 Ls 0; TI Lq ./ . Actually, this condition which is well-known for the case of the whole space  D R3 yields also a necessary and sufficient condition for smooth bounded and exterior domains, see [8, 11, 21, 24, 37], respectively. The integrability condition on E0 , say Z1

ke A u0 ksq d < 1;

(23)

0

will be explained in terms of Besov spaces. Starting with the classical Besov space 2=s Bq0 ;s0 ./ for a domain   R3 (cf. [56, Chap. 4]) where q0 , s0 are the conjugate exponents to q, s, respectively, and 2s C 3q D 1, solenoidal subspaces ˚ 0 2=s 2=s 2=s Bq0 ;s0 ./ D Bq0 ;s0 ./ \ Lq ./ D v 2 Bq0 ;s0 ./ W div v D 0; N  v j

@

D0



2=s

were defined in [1]. Actually, Bq0 ;s0 ./ coincides with the real interpolation space  q0  q0 L ./; D.Aq0 / 1=s;s0  L ./ [1, Proposition 3.4 (3.18)] yielding an optimal space of initial values u0 such that E0 .t/ D etA u0 satisfies .E0 /t , AE0 2  0 q0 Ls 0; TI L ./ , i.e., E0 is a classical strong solution of the homogeneous Stokes problem with initial value u0 . Here we do need the dual space  2=s  2=s Bq;s ./ WD Bq0 ;s0 ./ :

(24)

By elementary properties of real interpolation and the duality theorem [56, Theorem 1.11.2] 2=s Bq;s ./ D



0

D.Aq0 /; Lq ./



 1=s0 ;s0

  D D.Aq0 / ; Lq ./ 1=s0 ;s

 q0  q since L ./ D L ./. Hence ku0 kB2=s  ku0 kD.A q;s

q0 /



 ;Lq ./

1=s0 ;s

 kA1 u0 kq C

Z1

 kA1 u0 kLq ./;D.A /

kAe A .A1 u0 /ksq d

q

!1=s

11=s;s

(25)

0

where the second equivalence of norms uses the identity hA1 u0 ; A'i D hu0 ; 'i for ' 2 D.Aq0 / and the equivalence k'kW 2;q0  kAq0 'kq0 (for bounded ). The 2=s norm on the right-hand side of (25) is the norm of A1 u0 in Bq;s ./. By [56, Theorem 1.14.5], the interval of integration .0; 1/ may be replaced by any interval

192

R. Farwig 2=s

.0; ı/, 0 < ı < 1, yielding an equivalent norm on Bq;s ./. Finally, for a bounded domain, the term kA1 u0 kq in (25) may be omitted. 2=s Given ı 2 .0; 1 we denote the space Bq;s ./ also by ( q;s Bı ./

!1=s

Zı

D u0 W ku0 kBq;s WD

ke

ı

 A

u0 ksq

d

) 0, are equivalent. For B1 ./ we also write B q;s ./. q;s

ı

Theorem 2.1 Let   R3 be a bounded domain with boundary @ 2 C1;1 , let 0 < T  1, 2 < s < 1, 3 < q < 1 with 2s C 3q D 1 be given. We consider 2 the Navier-Stokes system value  (1) with initial  u0 2 L ./   and an external force f D div F where F 2 L2 0; TI L2 ./ \ Ls=2 0; TI Lq=2 ./ . (i) There exists an absolute constant " D " .q; / > 0 with the following property: If ku0 kBTq;s C kFkq=2;s=2IT  " ;

(27)

  then there exists a unique strong solution u 2 Ls 0; TI Lq ./ of (1). q;s (ii) The condition u0 2 B1 ./ is sufficient and necessary and  for the existence  uniqueness of a local in time strong solution u 2 Ls 0; T 0 I Lq ./ of (1) for some 0 < T 0  T. Let us recall further results and extensions of Theorem 2.1. Remark 2.2 1=4

1. The condition u0 2 D.A2 / is due to Fujita and Kato [28] for a smooth bounded domain. This result bases on L2 -theory and can be generalized to arbitrary bounded and unbounded domains, see [54, Chap. V, Theorem 4.2.2]. Fabes et al. [6] as well as Miyakawa [45] proved that the condition u0 2 Lr ./, r > 3, yields a local strong solution. 1=4 2. The condition u0 2 D.A2 / has the important property of being scaling invariant. Recall that with a solution u of (1) also u .x; t/ D u.x; 2 t/,  > 0, is a solution of (1) with the same Reynolds number (here Re D 1 D 1). Since 2

3

ku kLs .Lq / D 1. s C q / kukLs .Lq / on their respective time intervals and domains, the norm ku kLs .Lq / is -independent if and only if Ls .Lq / is a Serrin class; in this case, the space Ls .Lq / (or its norm) is called scaling invariant. The corresponding condition for the initial value is related to .u0 / .x/ D u0 .x/. Scaling invariant initial value conditions are u0 2 D.A1=4 / with norm kA1=4 u0 k2 and u0 2 L3 ./. The latter case was considered by Kato [35] and Giga [32].

Local Regularity Results for the Instationary Navier-Stokes Equations Based. . .

193

The condition u0 2 L3 ./ can be weakened to assumptions in Lorentz spaces q  s < 1, see [53]. Here we mention the continuous embeddings

L3;s

./ when

1=4

q;s D.A2 /  L3 ./  L3;s

./  B1 ./

(28)

where each space is scaling invariant; for the latter embedding which holds when q;s q;s q  s < 1 we refer to [1, (0.16)]. Replacing B1 ./ by Bı ./, the family of q;s spaces Bı ./ is scaling invariant in the sense that ı must be changed to 2 ı (and  to 1 ) . 3. For a smooth exterior domain   R3 similar results in [8, 37].  were obtained 2 3 It is also shown that the assumption F 2 Ls=2 Lq=2 (with s=2 C q=2 D 2, cf. Theorem 2.1) can be generalized to F 2 Ls .Lq / with s2 C q3 D 2. Then the R1 condition 0 ke A u0 ksq d < 1 is necessary and sufficient for the existence of a local strong solution u 2 Ls .Lq /, 2s C 3q D 1. 4. For a general bounded or unbounded domain   R3 —even with non-smooth boundary—only L2 -theory for the Stokes operator and the Helmholtz projection is available. In this case Theorem 2.1(i) holds with the exponents q D 4, s D 8, cf. [21, Sect. 4]. 5. The largest space of initial values to yield solutions in scaling invariant function spaces was found by Koch and Tataru [36] for the whole space case R3 . Assume that u0 2 BMO1 , i.e., u0 can be written in the form  u0 D div  f with some f 2 BMO. Then there exists a local solution u 2 L2loc R3  Œ0; 1/ such that the scaling invariant norm 1 x2R3 ;R>0 jBR .x/j

Z

R2

Z

juj2 dy d

sup

0

BR .x/

is finite; here BR .x/  R3 denotes the ball with center x and radius R. For simplicity let F D 0 in the following. It suggests itself to use Theorem 2.1(i) not only at t0 D 0, but at all or almost all t0 2 Œ0; T/ to show that a weak solution is a strong one. In view of (27) we need more information on the space B q;s ./ q;s and on functions u W Œ0; T/ ! Bı ./. Although the constant " in (27) cannot be determined precisely, we will fix some " > 0 so that Theorem 2.1(i) can be applied. For the following definition recall that for any nonzero v 2 L2 ./ the function t 7! eA v is nonzero for t  0. Hence V.t/ WD kvkBtq;s is a strictly q;s /. increasing continuous function in t > 0 with range .0; kukB1 Definition 2.3 Let " > 0 be fixed and let 0 ¤ v 2 L2 ./. Then

ı.v/ WD

8 ˆ ˆ0
0. Then the function Œ0; T ! Œ0; 1, t 7! ku.t/kBq;s is   ı q;s lower semi-continuous. If additionally u 2 L1 Œ0; T/I Bı ./ , then  q;s u 2 Cw0 Œ0; TI Bı ./ and kukL1 .0;TIBq;s / D sup ku.t/kBq;s : ı

t2Œ0;T

ı

(29)

(ii) The function ı.t/ WD ı..u.t//, see Definition 2.3, is upper semi-continuous in t. For the problem of local regularity we need the following terminology: s q at • u is left-sided  L .L /-regular  t0 2 .0; T if there exists " D ".t0 / 2 .0; t0 / such s that u 2 L .t0  "; t0 /I Lq ./ • u is right-sided Ls .Lq /-regular at t0 2 Œ0; T/ if there exists " D ".t0 / 2 .0; T  t0 / such that u 2 Ls .t0 ; t0 C "/I Lq ./ • u is Ls .Lq /-regular at t0 2 .0; T/ if u is left- as well as right-sided regular at t0 .

Theorem 2.5 Let u 2 LHT be a weak solution of (1) in a bounded smooth domain   R3 , and let 2 < s < 1, 3 < q < 1, 2s C 3q D 1. (i) Let u satisfy u.t/ 2 B q;s ./ for a.a. t 2 Œ0; T/. Given t1 2 .0; T/ assume that for a.a. t in a left-sided neighborhood of t1 ı.t/  t1  t:

(30)

Then u is left-sided Ls .Lq /-regular at t1 . (ii) Given t1 2 .0; T/ assume that u.t1 / 2 B q;s ./ and that u satisfies .EI/t1 , i.e., the strong energy inequality in t1 . Then u is right-sided Ls .Lq /-regular at t1 . (iii) Let u satisfy u.t/ 2 B q;s ./ for all t 2 Œ0; T/ and condition (30) at t1 2 .0; T/. Then u is  regular at t1. If condition (30) is satisfied at every t1 2 .0; T/, then u 2 Lsloc Œ0; T/I Lq ./ . (iv) Let u satisfy u.t/ 2 B q;s ./ for all t 2 Œ0; T/. Assume that lim ku.t/kBtq;st D 0:

t%t1

1

(31)

q Then u is Ls .L  /-regular att1 . If condition (31) is satisfied at every t1 2 .0; T/, then u 2 Lsloc Œ0; T/I Lq ./ .

Remark 2.6 Replacing the lower bound ı.t/  t1 t in (30) by the weaker condition ı.t/  ˛.t1  t/ for a fixed ˛ 2 .0; 1/, the arguments in the proof of Theorem 2.5 can be used iteratively to get a sequence of instants .tj /, tj D t1  .1  ˛/j .t1  t0 /,   s q converging to t1 ; hence u 2 Lloc t0 ; t1 I L ./ . However, Theorem 2.1 yields no uniform bound of kukq;sI.t0 ;tj / as j ! 1 so that the statement u 2 Ls t0 ; t1 I Lq ./ cannot be guaranteed.

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Corollary 2.7 Let u be a weak solution of the Navier-Stokes system as in Theorem 2.5.     q q (i) Assume that u 2 Lsloc Œ0; t1 /I L ./ , but u … Ls 0; t1 I L ./ . Then there exists an  > 0 such that for all ˛ 2 . 14 ; 12  kA˛ u.t/k2 >  .t1  t/1=4˛

for a.a. t 2 .0; t1 /:

(32)

(ii) Assume that u satisfies .EI/t1 (this condition is e.g. satisfied when t1 is a leftsided regular point of u), but is not right-sided regular at t1 . Then u.t1 / …

B q;s ./ and for each ı > 0 and ˛ 2 14 ; 12 ku.t/k3 ; kA˛ u.t/k2 ; ku.t/kBq;s ! 1 as t & t1 : ı

2.3 Regularity Criteria Beyond Serrin’s Condition and Energy Criteria The regularity criteria of Theorem 2.5 have the disadvantage that the norm of u.t/ q;s in Bt1 t ./ cannot be controlled directly. However, there are many applications of Theorem 2.5 yielding more concrete conditions. Corollary 2.8 Let u 2 LHT be a weak solution of (1) on a bounded smooth domain   R3 .   q;s (i) If there exists ı 2 .0; 1 such that u 2 C0 Œ0; T/I Bı ./ , then u is a strong solution on Œ0; T/. (ii) If there exists ı 2 .0; 1 such kukL1 .0;TIBq;s /  " , then u is a strong solution ı on Œ0; T/. q;s (iii) In (i) and (ii) the space Bı ./ can be replaced by any of the spaces D.A1=4 /, L3 ./ and L3;s

./ (s  q > 3). The next criteria are based on Theorem 2.1, however, in a certain sense work beyond Serrin’s condition at the expense of a further smallness assumption. Theorem 2.9 Let u 2 LHT be a weak solution as in Corollary 2.8. Further let the exponents q, r, s satisfy 2 < s < 1, 3 < q < 1, 2s C 3q D 1 and 1  r  s. (i) Assume that

lim inf ı!0

1 ı 1r=s

Zt1 ku./krq d D 0: t1 ı

  Then u is regular at t1 , i.e., u 2 Ls t1  "; t1 C "I Lq ./ for some " > 0.

(33)

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R. Farwig

(ii) Assume that for 0  t0 < t1 < T and some t1 < T 0  T 1 t1  t0

Zt1

.T 0  /r=s ku./krq d  "Q :

(34)

t0

Then u is regular at t1 . Here "Q > 0 is a constant related to " in (27). A consequence of Theorem 2.9(i) is the well-known fact that a weak solution u 2 LHT is regular almost everywhere (even everywhere in .0; T/ except for a possible set S  .0; T/ of vanishing Hausdorff measure H1=2 .S/ D 0). Actually, since u 2 Rt L2 0; TI L6 ./ , Lebesgue’s differentiation theorem implies that 1ı t11ı kuk26 d ! ku.t1 /k26 t1 -a.e as ı ! 0. Hence the term in (33) (with q D 6, r D 2, s D 4) vanishes t1 -a.e. Finally, we describe a regularity criterion based on the kinetic energy Ek .t/ D

1 ku.t/k22 : 2

(35)

Theorem 2.10 Let u 2 LHT be a weak solution of (1) as in Corollary 2.8. Assume that at t1 2 .0; T/ for ˛ 2 . 12 ; 1 the left-sided ˛-Hölder seminorm ŒEk .t1 /˛ D sup ı>0

jEk .t1  ı/  Ek .t1 /j 0) or that ŒEk .t1 /1=2  " :

(36)

Then u is regular at t1 . Note that for the Hölder exponent ˛ D 12 we do need a smallness condition on the local left-sided Hölder seminorm. Actually, if .t0 ; t1 /  Œ0; T/ is a maximal interval of regularity of a weak solution u, then due to (32) kru./k2 D kA1=2 u./k2  " .t1  /1=4 ;

0 <  < t1 :

Hence the estimate 2c2 

1 ı 1=2

Z

t1 t1 ı

kru./k22 d 

 1  Ek .t1  ı/  Ek .t1 /

ı 1=2

for a.a. ı 2 .t1  t0 ; t1 / shows in this case that the condition (36) with an arbitrary (not sufficiently small) " > 0 does not imply regularity. For the case including an external force we refer to [23]. For further regularity criteria beyond Serrin’s condition see [16].

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Theorem 2.11 Let u 2 LHT be a weak solution of the Navier-Stokes system (1) in a bounded smooth domain   R3 with u0 2 L2 ./. (i) Assume that for some 2 < s < 1, 3 < q < 1, 2s C 3q D 1 there holds u.t0 / 2 B q;s ./ for all t0 2 Œ0; T/, and that u satisfies the energy equality .EE/. Then u is uniquely determined by the initial value u0 within the class of all weak solutions.   (ii) If u 2 L4loc Œ0; T/I L4 ./ or   3;1 u 2 L1 loc Œ0; T/I L ./ ;

(37)

then u even satisfies (EE). Remark 2.12 (i) Let u be a weak solution of (1) as in Theorem 2.11. It is interesting to discuss uniqueness and regularity properties of u satisfying the Serrin condition u 2 Lsloc .Œ0; T/I Lq .//; 2s C 3q D 1 in the limit case s D 1; q D 3. In this case, the arguments in the proof of Lemma 2.4(i) show that u.t/ 2 L3 ./ for each t 2 Œ0; T/. Since L3 ./  L3;1 ./ \ B q;s ./, Theorem 2.11 yields the uniqueness property for u. On the other hand, from   Corollary 2.8(iii) we 3 see that the stronger assumption u 2 C Œ0; T/I L ./ is sufficient to get the

  regularity u 2 Lsloc Œ0; T/I Lq ./ with Serrin exponents s; q.  3 (ii) Furthermore, for u 2 L1 Œ0; T/I L ./ we get the local right-side regularity

loc property for each t 2 Œ0; T/, see Theorem 2.5(ii). Hence Theorem 2.5 is a slightly weaker   result than that in a series of papers on the celebrated L1 0; TI L3 ./ -regularity result of Seregin et al. We refer to [49] for domains with a flat boundary, and to [44] where in domains with curved boundaries some additional condition on the pressure had to be assumed. (iii) In the case of a general domain   R3 , let it be bounded with rough boundary or unbounded, only an L2 ./-theory of the Stokes operator is available. Here we get the following results, cf. [16, 21]: Let u 2 LHT be a weak solution satisfying (SEI). Assume that at t1 2 .0; T/ one of the following scaling invariant conditions is satisfied: there exists 0 < ı < t1 such that 1 ı 1 ı 

Z Z

t1 t1 ı t1 t1 ı

kA1=4 u./k2 d  " ;

(38)

ku./k2 kru./k2 d  " ;

(39)

sup ku./k22

Œt1 ı;t1 

1 Z

t1

ı

t1 ı

kru./k22 d  " ;

(40)

where " in (38)–(40) is an absolute constant independent of the domain. Then u is L8 .L4 /-regular at t1 .

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R. Farwig

2.4 The Navier-Stokes System in General Smooth Domains Definition 2.13 For k 2 N0 and  2 .0; 1 a domain   Rn is called uniform Ck; domain, if there are positive constants ˛; ˇ; K such that for all x0 2 @ there exist— after an orthogonal and an affine coordinate transform—a real-valued function h of class Ck; and a neighborhood U˛;ˇ;h .x0 / of x0 with the following properties: h is defined on the closed ball B0˛ .0/  Rn1 with khkCk;  K and h.0/ D 0 and, if k  1, h0 .0/ D 0; moreover, U˛;ˇ;h .x0 / W D f.y0 ; yn / 2 Rn1  RW jy0 j < ˛; jh.y0 /  yn j < ˇg;  U˛;ˇ;h .x0 / W D f.y0 ; yn / 2 Rn1  RW jy0 j < ˛; h.y0 /  ˇ < yn < h.y0 /g

D  \ U˛;ˇ;h .x0 /; @ \ U˛;ˇ;h .x0 / D f.y0 ; yn / 2 Rn1  RW h.y0 / D yn g: The triple .˛; ˇ; K/ is called the type of  and will be denoted by ./ D .˛; ˇ; K/. For a constant C in some estimate we will write C D C..// if it does depend only on ˛, ˇ and K, but in no other way on . A uniform Ck -domain is defined in an obviously analogous way. Note that bounded and exterior domains are included, as long as the boundary is smooth enough; in these cases the boundary @ is compact, implying that the constants ˛, ˇ, K can be chosen uniformly. Since the Helmholtz projection and the Stokes operator are not necessarily well-defined on a general smooth domain as described in Definition 2.13 we will introduce the spaces ( Lq ./ C L2 ./; if 1  q < 2; LQ q ./ WD (41) Lq ./ \ L2 ./; if 2  q  1: For bounded domains  it holds that LQ q ./ D Lq ./ with equivalent norms by Hölder’s inequality. Note that functions in LQ q ./ locally behave like Lq -functions, but globally like L2 -functions. Moreover, we define for 1 < q < 1, 1    1 the Lorentz spaces ( Lq; ./ C L2 ./; q < 2; q; LQ ./ WD Lq; ./ \ L2 ./; q > 2; letting the case q D 2 undefined. For spaces of Sobolev-type we proceed analogously: For k 2 N and 1  q  1 we let ( k;2 k;q k;q Q ./ WD W ./ C W ./; 1  q < 2; W W k;2 ./ \ W k;q ./; 2  q  1:

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199

Q 1;q ./, 1 < q < 2 and 2  q < 1, based on the Similarly, we define the spaces W 0 1;q classical Sobolev spaces W0 ./ and W01;2 ./. Q k;q ./-spaces have the following properties; for a proof see [47, The LQ q - and W 48]: • Let 1  q < 1. Then .LQ q .// D LQ q ./ • Let 1 < q  r  1. Then kukLQ q  kukLQ r • Let 1  r; p; q  1, 1r D 1p C 1q and let u 2 LQ p , v 2 LQ q . Then uv 2 LQ r and kuvkLQ r  kukLQ p kvkLQ q  • Let 1 < q; r < 1, 0 <  < 1 and let s be defined by 1s D 1 q C r . Then in the sense of complex interpolation spaces 0



LQ q ./; LQ r ./



D LQ s ./

• Let 1  r ¤ q  1, 0 <  < 1, 1    1, and define 1 < s ¤ 2 < 1 by 1 1  s D q C r . Then  q  LQ ./; LQ r ./

;

D LQ s; ./:

For s D 2 and  D 2 we get 

LQ q ./; LQ r ./

 ;2

D L2 ./:

• Let m 2 N, 1  q < 1 and   Rn be a uniform C2 -domain. Then Q m;q ./ ,! LQ r ./ W if either q  r  1 and mq > n, or q  r < 1 and mq D n, or q  r  and mq < n.

nq nmq

Concerning the Helmholtz projection on LQ q ./ for a domain   Rn of uniform type C1 we have the following result, see [17]. We define ( q 2 QLq ./ WD L ./ C L ./; 1 < q < 2 ; q L ./ \ L2 ./; 2  q < 1 equipped with the norm of LQ q ./, and gradient spaces by (

Q q ./ WD Gq ./ C G2 ./; 1 < q < 2; G Gq ./ \ G2 ./; 2  q < 1;

200

R. Farwig

with norm k  kGQ q ./ WD k  kLQ q ./ . Then the space LQ q ./ admits the direct algebraic and topological decomposition Q q ./: LQ q ./ D LQ q ./ ˚ G q The corresponding projection PQ q from LQ q ./ onto R.PQ q / D LQ ./ and kernel Q q ./ has a norm bounded by a constant c D c.q; .//. It satisfies N .PQ q / D G   the relation PQ q D PQ q0 . Using the Helmholtz projection PQ q we can define the Stokes operator AQ q , 1 < q < 1, for a uniform C2 -domain   Rn . Let

( D.AQ q / WD

Dq C D2 ; 1 < q < 2; Dq \ D2 ;

2  q < 1;

q 1;q where Dq WD L ./ \ W0 ./ \ W 2;q ./. Then the Stokes operator AQ q W D.AQ q /  q q LQ ./ ! LQ ./ is defined by AQ q u WD PQ q u and has the following properties, see [22]:   q • AQ q is a densely defined closed operator in LQ ./ satisfying AQ q D AQ q0 . • For all  2 S WD f 2 C n f0gI j arg./j < g, 2 <  < , the resolvent q q q . C AQ q /1 W LQ ./ ! LQ ./ is well-defined. Moreover, for f 2 LQ ./ the q unique solution u 2 LQ ./ to u C AQ q u D f satisfies the estimate

jjkukLQ q ./ C kr 2 ukLQ q ./  Ckf kLQ q ./ with a constant C D C.q; ; ı; .//, as long as jj  ı > 0. • AQ q generates an analytic semigroup etAQ q , t  0, having the bound Q

ketAq f kLQ q ./  Ceıt kf kLQ q ./ q for all f 2 LQ ./ and t  0 with a constant C D C.q; ı; .//, ı > 0.  q • Let 1 < r; q < 1, 0 < T < 1, and let a function f 2 Lr 0; TI LQ ./ as well as an initial value u0 2 D.AQ q / (for simplicity) be given. Then there exists a unique   q solution u 2 Lr 0; TI D.AQ q / \ W 1;r 0; TI LQ ./ of the Stokes system

ut C AQ q u D f ;

u.0/ D u0

in LQ q ./:

It satisfies the maximal regularity estimate kukLr .0;TID.AQ q // C kut kLr .0;TILQ q /  C.ku0 kD.AQ q / C kf kLr .0;TILQ q / /

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with a constant C D C.q; r; T; .//, and can be represented by the variation of constants formula Z t Qq Q tA u.t/ D e u0 C e.t /Aq f ./d; 0  t  T: 0

It is unknown whether the resolvent estimate holds uniformly in  2 S as jj ! 0. Therefore, the semigroup may increase exponentially fast and the maximal regularity estimate is stated only for finite time intervals. A further crucial property of the Stokes operator 1 C AQ q is the fact that it admits bounded imaginary powers, see [40]. Hence complex interpolation methods can be used to describe domains of fractional powers .1 C AQ q /˛ ; 1  ˛  1. To be Q ˛q D D Q ˛q ./ D D..1 C AQ q /˛ /, the domain of the more precise, for 0  ˛  1 let D fractional power .1 C AQ q /˛ , equipped with the norm k.1 C AQ q /˛  kLQ q . If 1  ˛ < 0 Q ˛q as the completion of LQ q ./ in the norm k.1 C AQ q /˛  k Q q . These spaces define D L Q ˛q / D D Q ˛ are reflexive and satisfy the duality relation .D . As special cases we get 0 q 1=2 1;q q 1 1=2 Q Q Q Q Q Q that Dq D D.Aq /, Dq D W0 \ L ./ (with norm k.1 C Aq /  kLQ q equivalent to Q 0q D LQ q ./. Moreover, we obtain the interpolation result k  k Q 1;q ), and D W

./



Q ˇq Q ˛q ; D D



Q q ; DD

when 1  ˛  ˇ  1 and .1  /˛ C ˇ D  ,  2 .0; 1/. This result is the basis to prove the following embedding estimate: Let n  3, 0  ˛  1, 1 < q  r < 1, and 1r D 1q  2˛ n : Then kukLQ r ./  Ck.1 C AQ q /˛ ukLQ q ./

(42)

Q ˛q with a constant C D C../; q; ˛/, see [48]. for all u 2 D Qr Qq Theorem 2.14  (L -L -estimates, [48, Theorem 1]) Let n  3, 1 < q  r < 1; q n 1 and ˛ WD 2 q  1r . Then for every f 2 LQ ./ and t > 0 Q

ketAr f kLQ r ./  Ceıt .1 C t/˛ t˛ kf kLQ q ./ ; Q

kretAr f kLQ r ./  Ceıt .1 C t/

˛C 12

t

˛ 12

kf kLQ q ./

(43) (44)

with a constant C D C../; r; q; ı/ > 0 independent of t and with any ı > 0. We note that the factor .1 C t/˛ in (43), (44) is somehow natural in the context of QLq -spaces. Consider the exterior of a ball in R3 where we may choose ı D 0 in (43). But an estimate of the form ketAQ r f kLQ r  Ct˛ kf kLQ q cannot hold for all t > 0 with

202

R. Farwig

1 a constant C independent of t. Indeed, otherwise, with r D 6, q D 6=5, f 2 C0;

./ we get that Q

Q

ketA2 f kL2  ketA6 f kLQ 6  Ct1 kf kLQ 6=5  Ct1 kf kL2 ; implying that the semigroup etAQ 2 D etA2 has some algebraic decay in t, which is known to be false. Let u 2 LHT be a weak solution of the Navier-Stokes system (2) with initial value u0 2 L2 ./ and—for simplicity—external force f D 0. Regularity results for this case (even including forces of the type f1 C div f2 ) are obtained in the PhD thesis of F. Riechwald [47] and will be discussed in the following. The proofs are based on the theory of very weak solutions. For an introduction to this theory we refer to [1, 9, 15, 18]; a review can be found in [19]. Very weak solutions can be constructed most easily by duality arguments. Therefore, for finite 0 < T < 1 and 1 < s; q < 1, we introduce the function space ˚  0 0 0 Q 1q0 / \ W 1;s0 .0; TI LQ q0 .//W .T/ D 0 T 1;s ;q .T; / WD  2 Ls .0; TI D equipped with the norm kkT 1;s0 ;q0 WD kt kLs0 .0;TILQ q0 / C kkLs0 .0;TIDQ 10 / : q

This reflexive Banach space is related to the backward Stokes problem t  AQ q0  D v; .T/ D 0

0 in LQ q ./

 0 0 0 0 which admits for any v 2 Ls 0; TI LQ q ./ a unique solution  2 T 1;s ;q .TI / satisfying the maximal regularity estimate kkT 1;s0 ;q0  CkvkLQ s0 .0;TILQ q0 / 0

0

with a constant C D C.q; s; T; .//. The dual space to T 1;s ;q .TI / will be denoted by T 1;s;q .T; /, its norm by k  kT 1;s;q . Definition 2.15 Let   Rn be a uniform C2 -domain, 0 < T < 1 and 2 < s < 1, n < q < 1 and 2=s C n=q D 1. For an initial value u0 we define the functional Fu0 by hFu0 ; i D hu0 ; .0/i;

0

0

 2 T 1;s ;q .T; /;

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and assume that Fu0 2 T 1;s;q .T; /. Then we call u 2 Ls .0; TI LQ q / a very weak solution to the Navier-Stokes system with data F , if the conditions hu; t i;T  hu; i;T  hu ˝ u; riT; D hFu0 ; i; div u D 0; u  N D 0 on @ 0

0

hold for all  2 T 1;s ;q .TI /. We note that the conditions div u D 0 in  and u  N D 0 on @ on the normal component of u can be formulated as a variational identity as well. To show that the functional Fu0 in Definition 2.15 is meaningful let  2 0 0 T 1;s ;q .T; / and v D t C AQ q0 . Then from the variation of constants formula applied to  at 0 and the maximal regularity result we get for hFu0 ; i D hu0 ; .0/i that ˇD Z ˇ jhFu0 ; ij D ˇ u0 ; ˇZ ˇ Dˇ  

T 0

Z Z

T 0 T 0

Hence the assumption

RT 0

T 0

E ˇ ˇ Q e.Tt/Aq0 v.T  t/dt ˇ 

˛ ˇˇ ˝ tAQ q u0 ; v.t/  dtˇ e Q

1=s

Q

1=s

ketAq u0 ksLQ q dt ketAq u0 ksLQ q dt

kvkLs0 .0;TILQ q0 / kkT 1;s0 ;q0 :

ketAQ q u0 ksLQ q dt < 1 yields Fu0 2 T 1;s;q .T; /.

In contrast to the case of a bounded domain the semigroup etAQ q cannot shown to be exponentially decreasing. However, we may modify the norm  R T tAQ 1=s q u ks dt to equivalent norms by replacing etAQ q by et.1CAQ q / on 0 LQ q 0 ke .0; T/, 0 < T < 1, and then by integrating even from 0 to 1. Hence, with the exponentially decreasing semigroup et.1CAQ q / , as in the bounded domain case, we are led to the reflexive and complete space of Besov type n Z q;s BQT ./ D u0 W ku0 kBQq;s WD T

T 0

Q

ke A u0 ksLQ q d

1=s

0 with the following property: If u0 2 BQ ./ and T

ku0 kBQq;s  " ; T

  then there exists a unique very weak solution u 2 Ls 0; TI LQ q ./ to the NavierStokes system with initial data u0 . Theorem 2.17 Let   R3 be a uniform C2 -domain and exponents s; q be given 24 2 3 2 such that 16 5  s  16, 7  q  8, and s C q D 1. Moreover, let u0 2 L ./. (i) There exists " D " ../; q; T/ > 0 with the following property: If u0 satisfies ku0 kBQq;s  " T

and

Fu0 2 T 1;4;4 .T; /;

then there exists a unique weak solution u 2 LHT to the Navier-Stokes system with initial data u0 satisfying the energy inequality .EI/. Moreover, q u 2 Ls 0; TI LQ ./ . In particular, every t 2 .0; T/ is a regular point of u.   q q;s (ii) Let u be a weak solution in LHT \ Ls 0; TI LQ ./ . Then u0 2 BQT . The condition Fu0 2 T 1;4;4 .T; / is fulfilled if e.g. u0 2 L2 ./ satisfies 12 " for large j. Hence ı.tj / < ˇ, i.e., ı./ is upper semi-continuous at t. The case ı.t/ D 1 is trivial. t u Proof of Theorem 2.5 (i) Let t1 2 .0; T/. To show that t1 is a left-sided regular point of u we find due to the assumptions (30) and .SEI/ a t 2 .0; t1 / such that ku.t/kBtq;st  1 ku.t/kBq;s  " and that .EI/t holds. Here " > 0 is the constant from ı.t/ Theorem 2.1, see (27).  By Serrin’s  uniqueness theorem and Theorem 2.1 we conclude that u 2 Ls t0 ; t1 I Lq ./ . Hence u is left-sided regular in t1 .   q;s (ii) Since u.t1 / 2 B1 ./, there exists a strong solution v 2 Ls t1 ; t1 C "I Lq ./ , " > 0, of (1) with initial value v.t1 / D u.t1 /. Moreover, by assumption, .EI/t1 holds for u. Hence Serrin’s uniqueness theorem implies that v D u in Œt1 ; t1 C"/, and u is right-sided regular in t1 . (iii) To combine the results from (i) and (ii), in particular to apply (ii), it suffices to prove that u satisfies .EI/t1 at t1 . Let t1 2 .0; T/ be an instant where the validity of the energy inequality is not guaranteed by .SEI/. By (i) t1 is a leftsided regular point for u and, consequently, u 2 Ls .t0 ; t1 I Lq .// for some 0 < t0 < t1 . Therefore, u satisfies .EE/ for all initial times t00 2 .t0 ; t1 /, in particular 1 ku.t1 /k22 C 2

Z

t1 t00

kruk22 d D

1 ku.t00 /k22 : 2

Thus 0lim ku.t00 /k22 D ku.t1 /k22 . Moreover, by .SEI/, there is a sequence tj % t1 t0 %t1

such that

1 ku.t/k22 C 2

Z

t tj

kruk22 d 

1 ku.tj /k22 ; 2

tj  t < T:

Passing to the limit tj % t1 we get that u satisfies .EI/t1 .   q;s Finally, since u.0/ D u0 2 B1 ./, we know that u 2 Ls 0; "0 I Lq ./ for some "0 > 0. Now an elementary compactness argument proves that u 2 Lsloc Œ0; T/I Lq ./ . (iv) Under the assumption (31) for t1 we get for a.a. t in a left-sided neighborhood of t1 that ku.t/kBtq;st < " . Hence ı.t/ > t1  t for these t; moreover, we may 1 assume .EI/t . By the above arguments we conclude that u lies in Serrin’s class Ls .Lq / on the interval .t; t C ı.t//  .t; t1 /, i.e., u is Ls .Lq /-regular at t1 . t u

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209

Proof of Corollary 2.7 (i) Fix ˛ 2 . 14 ; 12  and let q  q0  2 satisfy 2˛ C 3q D get for all t 2 .0; t1 / where .EI/t is satisfied that Z s
1. Hence kA˛ u.t/k2 >  .t1  t/ 4 ˛ for a.a. t 2 .0; t1 /. (ii) Since u.t1 / … B q;s ./, we have ku.t1 /kBq;s D 1 for each ı > 0. The lower ı semi-continuity of the map t 7! ku.t/kBq;s implies that ku.t/kBq;s ! 1 as t & ı ı t1 . Moreover, due to the embeddings (28) we get that ku.t/kBq;s  cku.t/k3  ı cku.t/kD.A1=4 /  cku.t/kD.A1=2 / . t u

3.2 Proofs of Results of Sect. 2.3 Proof of Corollary 2.8

  q;s (i) For 0 < T 0 < T the function u 2 C0 Œ0; T 0 I B ./ is uniformly continuous q;s in t 2 Œ0; T 0  with values in Bı ./. Given " from Theorem 2.1 the uniform continuity allows to find ı 0 2 .0; ı such that ku.t/kBq;s0  " for all t 2 Œ0; T 0 : ı   Then a compactness argument on Œ0; T 0  implies that u 2 Ls Œ0; T 0 I Lq ./ . (ii) From Lemma 2.4 we know that ku.t/kBq;s  kukL1 .0;TIBq;s /  " for all t 2 ı ı Œ0; T/. Now a compactness argument as in (i) completes the proof. (iii) The embeddings (28) and (i), (ii) immediately prove (iii). t u Proof of Theorem 2.9 (i) Assuming (i) we find ı > 0 such that with t0 D t1  ı and T 0 D t1 C ı 1 t1  t0

Zt1

0

.T  / t0

r=s

ku./krq

d 

2r=s ı 1r=s

i.e., (34) is satisfied. Thus it suffices to prove (ii).

Zt1 ku./krq d  "Q ; t0

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(ii) It is sufficient to find t 2 .0; t1 / such that ku.t/kBq;s0  " and that .EI/t is T t satisfied. Indeed, by (34) there exists t 2 .t0 ; t1 / such that .T 0  t/r=s ku.t/krq  "Q or, equivalently, .T 0  t/ku.t/ksq  "Q and that .EI/t holds. Hence, employing q the boundedness of the semigroup e A on L ./, s=r

T Z0 t ke A u.t/ksq d  C.T 0  t/ku.t/ksq  " 0

with an appropriately chosen "Q in (34).

t u . 21 ; 1

Proof of Theorem 2.10 Obviously the condition ŒEk .t1 /˛ < 1 for ˛ 2 implies that ŒEk .t1 /1=2  " (with the supremum taken only for small ı > 0/. Hence it suffices to assume the second condition. With r D 2, q D 6 and s D 4 we get that Zt1

Zt1 ku./krq

t1 ı

d  c

kru./k22 d

t1 ı

   c Ek .t1  ı/  Ek .t1 /

(47)

 c" ı 1=2 provided we choose only those ı > 0 such that .EI/t1 ı holds. We conclude that (33) is satisfied and consequently that u is regular at t1 . t u Proof of Theorem 2.11 (i) Assume that u satisfies .EE/ and u.t0 / 2 B q;s ./ for all t0 2 Œ0; T/. Moreover, let uQ be another weak solution  satisfying.SEI/ for the same initial value u0 2 L2 ./. We obtain that u 2 Ls 0; ıI Lq ./ with some 0 < ı < T. Then Serrin’s uniqueness theorem implies that u.t/ D uQ .t/ for 0  t < ı. Let Œ0; t0 /, 0 < t0  T, be the largest half open interval such that u.t/ D uQ .t/ is satisfied for each t 2 Œ0; t0 /. If t0 < T, then the weak L2 -continuity in time (6) implies that u.t0 / D uQ .t0 /. q;s Since u satisfies u.t0 / 2 B ./ and .EI/t0 we conclude that u 2 Ls t0 ; t0 C ıI Lq ./ . Moreover, since uQ satisfies .SEI/, hence .EI/tj for a sequence .tj / with tj % t0 , and since u D uQ satisfies .EE/ on Œ0; t0 , we conclude that kQu.tj /k2 ! kQu.t0 /k2 . These arguments imply that uQ satisfies .EI/t0 . Consequently, by Serrin’s uniqueness theorem, u D uQ in Œ0; t0 C ı/. This is a contradiction to the construction of t0 .

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(ii) For u 2 L4 .L4 / Remark 1.2(5) yields .EE/. Now assume that u satisfies (37). By Hölder’s inequality in Lorentz spaces [38, Lemma 2.1] and Sobolev’s embedding W01;2 ./  L6;2 ./ [38, Lemma 2.2] we get that kuukL2  ckuukL2;2  ckukL3;1 kukL6;2  ckukL3;1 kukW 1;2 ; where c D c./ > 0. Hence u 2 L4 .L4 /. Now we proceed as above.

t u

3.3 Proofs of Results of Sect. 2.4 Proof of Theorem 2.17 (Sketch) (i) Since u0 2 L2 ./, there exists a (global) weak solution u satisfying the energy inequality (EI). By  Theorem 2.16 we also get the existence of a very weak solution uQ 2 Ls 0; TI LQ q ./ with uQ .0/ D u0 which due to the assumption Fu0 2 T 1;4;4 .T; / can be shown to have the additional property uQ ˝ uQ 2 L2 .0; TI L2 .//. Hence uQ is also a weak solution satisfying even the energy s Qq equality (EE). Now Serrin’s  uniqueness  theorem valid also in the L .L /-setting q s implies that u D uQ 2 L 0; TI LQ ./ . All t 2 .0; T/ are thus regular points.   (ii) Since by assumption u 2 Ls 0; TI LQ q ./ it suffices to show that uQ D u    etAQ q u0 2 Ls 0; TI LQ q ./ as well. Here uQ is a weak solution of the linear Stokes system with initial value 0, but external force F D  div u˝u. Hence it suffices 0 0 to show that F 2 T 1;s;q .T; /. To this aim let  2 T 1;s ;q .T; / so that by q Hölder’s inequality in LQ -spaces jhF; ij D j.u ˝ u; r/T; j  kuk2Ls .LQ q / krkL.s=2/0 .LQ .q=2/0 / : Concerning r we use fractional powers of the operator 1 C AQ q0 and obtain for a.a. 0  t  T that kr.t/kLQ .q=2/0  Ck.1 C AQ q0 /1=2 .t/kLQ .q=2/0  Ck.1 C AQ q0 /3=.2q/ .1 C AQ q0 /1=2 .t/kLQ q0 0  Ck.1 C AQ q0 /1=s .t/kLQ q0 :

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Finally, let v WD t C AQ q0  and use the variation of constants formula to write  in terms of v. Hence, with a constant C D C.T; ı; s/ > 0, .s=2/0

Z T Z

krkL.s=2/0.LQ .q=2/0 /  C

0

t 0

0

Q

k.1 C AQ q0 /1=s e.t /Aq0 v.T  /kLQ q0 d

Z

.s=2/0 dt

.s=2/0 kv.T  /kLQ q0 d dt C .t  /1=s0 0 0

Z T .s=2/0 =s0 s0 C kv.T  /kLQ q0 dt Z

T

t

0

.s=2/0

 CkvkLs0 .LQ q0 / ; where we also used the Hardy-Littlewood inequality. This proves that F 2 T 1;s;q .T; /. u t Acknowledgements This work was completed with the support of the International Research Training Group on Mathematical Fluid Mechanics Darmstadt-Tokyo (IRTG 1529).

References 1. H. Amann, Nonhomogeneous Navier-Stokes equations with integrable low-regularity data, in Nonlinear Problems in Mathematical Physics and Related Topics, II. International Mathematical Series (Kluwer Academic/Plenum Publishing, New York, 2002), pp. 1–28 2. M.E. Bogovski˘i, Decomposition of Lp .; Rn / into the direct sum of subspaces of solenoidal and potential vector fields. Soviet Math. Dokl. 33, 161–165 (1986) 3. W. Borchers, T. Miyakawa, Algebraic L2 decay for Navier-Stokes flows in exterior domains. Acta Math. 165, 189–227 (1990) 4. A. Cheskidov, P. Constantin, S. Friedlander, R. Shvydkoy, Energy conservation and Onsager’s conjecture for the Euler equations. Nonlinearity 21, 1233–1252 (2008) 5. A. Cheskidov, S. Friedlander, R. Shvydkoy, On the energy equality for weak solutions of the 3D Navier-Stokes equations, in Advances in Mathematical Fluid Mechanics (Springer, Berlin 2010), pp. 171–175 6. E.B. Fabes, B.F. Jones, N.M. Rivière, The initial value problem from the Navier-Stokes equations with data in Lp . Arch. Ration. Mech. Anal. 45, 222–240 (1972) 7. R. Farwig, C. Komo, Regularity of weak solutions to the Navier-Stokes equations in exterior domains. Nonlinear Differ. Equ. Appl. 17, 303–321 (2010) 8. R. Farwig, C. Komo, Optimal initial value conditions for strong solutions of the Navier–Stokes equations in an exterior domain. Analysis (Munich) 33, 101–119 (2013) 9. R. Farwig, J. Sauer, Very weak solutions of the stationary Stokes equations on unbounded domains of half space type. Math. Bohemica 140, 81–109 (2015) 10. R. Farwig, H. Sohr, Generalized resolvent estimates for the Stokes system in bounded and unbounded domains. J. Math. Soc. Japan 46, 607–643 (1994) 11. R. Farwig, H. Sohr, Optimal initial value conditions for the existence of local strong solutions of the Navier-Stokes equations. Math. Ann. 345, 631–642 (2009)

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12. R. Farwig, H. Sohr, On the existence of local strong solutions for the Navier-Stokes equations in completely general domains. Nonlinear Anal. 73, 1459–1465 (2010) 13. R. Farwig, Y. Taniuchi, On the energy equality of Navier–Stokes equations in general unbounded domains. Arch. Math. 95, 447–456 (2010) 14. R. Farwig, H. Kozono, H. Sohr, An Lq –approach to Stokes and Navier-Stokes equations in general domains. Acta Math. 195, 21–53 (2005) 15. R. Farwig, G.P. Galdi, H. Sohr, A new class of weak solutions of the Navier-Stokes equations with nonhomogeneous data. J. Math. Fluid Mech. 8, 423–444 (2006) 16. R. Farwig, H. Kozono, H. Sohr, Local in time regularity properties of the Navier-Stokes equations. Indiana Univ. Math. J. 56, 2111–2131 (2007) 17. R. Farwig, H. Kozono, H. Sohr, On the Helmholtz decomposition in general unbounded domains. Arch. Math. 88, 239–248 (2007) 18. R. Farwig, H. Kozono, H. Sohr, Very weak solutions of the Navier-Stokes equations in exterior domains with nonhomogeneous data. J. Math. Soc. Japan 59, 127–150 (2007) 19. R. Farwig, H. Kozono, H. Sohr, Very weak, weak and strong solutions to the instationary Navier-Stokes system, in Topics on Partial Differential Equations, ed. by P. Kaplický, Š. Neˇcasová. Lecture Notes of the J. Neˇcas Center Mathematical Modeling, vol. 2 (MatfyzPress Publishing House/Charles University, Prague, 2007), pp. 1–54 20. R. Farwig, H. Kozono, H. Sohr, Energy-based regularity criteria for the Navier-Stokes equations. J. Math. Fluid Mech. 11, 1–14 (2008) 21. R. Farwig, H. Sohr, W. Varnhorn, On optimal initial value conditions for local strong solutions of the Navier-Stokes equations. Ann. Univ. Ferrara 55, 89–110 (2009) 22. R. Farwig, H. Kozono, H. Sohr, On the Stokes operator in general unbounded domains. Hokkaido Math. J. 38, 111–136 (2009) 23. R. Farwig, H. Kozono, H. Sohr, Regularity of weak solutions for the Navier-Stokes equations via energy criteria, in Advances in Mathematical Fluid Mechanics, ed. by R. Rannacher, A. Sequeira (Springer, Berlin, 2010), pp. 215–227 24. R. Farwig, H. Sohr, W. Varnhorn, Necessary and sufficient conditions on local strong solvability of the Navier-Stokes systems. Appl. Anal. 90, 47–58 (2011) 25. R. Farwig, H. Sohr, W. Varnhorn, Extensions of Serrin’s uniqueness and regularity conditions for the Navier-Stokes equations. J. Math. Fluid Mech. 14, 529–540 (2012) 26. R. Farwig, H. Sohr, W. Varnhorn, Besov space regularity conditions for weak solutions of the Navier-Stokes equations. J. Math. Fluid Mech. 16, 307–320 (2014) 27. Ch. Fefferman, Existence and Smoothness of the Navier–Stokes Equations (2000). http://www. claymath.org/millennium/Navier-Stokes_Equations/navierstokes.pdf 28. H. Fujita, T. Kato, On the Navier-Stokes initial value problem. Arch. Ration. Mech. Anal. 16, 269–315 (1964) 29. D. Fujiwara, H. Morimoto, An Lr -theorem of the Helmholtz decomposition of vector fields. J. Fac. Sci. Univ. Tokyo (1A) 24, 685–700 (1977) 30. Y. Giga, Analyticity of the semigroup generated by the Stokes operator in Lr -spaces. Math. Z. 178, 287–329 (1981) 31. Y. Giga, Domains of fractional powers of the Stokes operator in Lr -spaces. Arch. Ration. Mech. Anal. 89, 251–265 (1985) 32. Y. Giga, Solution for semilinear parabolic equations in Lp and regularity of weak solutions for the Navier-Stokes system. J. Differ. Equ. 61, 186–212 (1986) 33. Y. Giga, H. Sohr, Abstract Lq -estimates for the Cauchy problem with applications to the NavierStokes equations in exterior domains. J. Funct. Anal. 102, 72–94 (1991) 34. E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1950–1951) 35. T. Kato, Strong Lp -solutions of the Navier-Stokes equation in Rm , with applications to weak solutions. Math. Z. 187, 471–480 (1984) 36. H. Koch, D. Tataru, Well-posedness for the Navier-Stokes equations. Adv. Math. 157, 22–35 (2001)

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37. C. Komo, Necessary and sufficient conditions for local strong solvability of the Navier-Stokes equations in exterior domains. Technische Universität Darmstadt, FB Mathematik, J. Evol. Equ. 14, 713–725 (2014) 38. H. Kozono, Uniqueness and regularity of weak solutions to the Navier-Stokes equations, in Recent topics on Mathematical Theory of Viscous Incompressible fluid, Tsukuba, 1996. Lecture Notes in Numerical and Applied Analysis, vol. 16 (Kinokuniya, Tokyo, 1998), pp. 161–208 39. H. Kozono, H. Sohr, Remark on uniqueness of weak solutions to the Navier-Stokes equations. Analysis 16, 255–271 (1996) 40. P.C. Kunstmann, H 1 -calculus for the Stokes operator on unbounded domains. Arch. Math. 91, 178–186 (2008) 41. J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934) 42. P. Maremonti, V.A. Solonnikov, On nonstationary Stokes problem in exterior domains. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24, 395–449 (1997) 43. V.N. Maslennikova, M.E. Bogovski˘i, Elliptic boundary value problems in unbounded domains with noncompact and nonsmooth boundaries. Rend. Sem. Mat. Fis. Milano LVI, 125–138 (1986) 44. A.S. Mikhailov, T.N. Shilkin, L3;1 -solutions to the 3D-Navier-Stokes system in a domain with a curved boundary. Zap. Nauchn. Semin. POMI 336, 133–152 (2006) (Russian). J. Math. Sci. 143, 2924–2935 (2007) (English) 45. T. Miyakawa, On the initial value problem for the Navier-Stokes equations in Lp -spaces. Hiroshima Math. J. 11, 9–20 (1981) 46. T. Miyakawa, H. Sohr, On energy inequality, smoothness and large time behavior in L2 for weak solutions of the Navier-Stokes equations in exterior domains. Math. Z. 199, 455–478 (1988) 47. F. Riechwald, Very weak solutions to the Navier-Stokes equaitons in general unbounded domains. Ph.D. thesis, Technische Universität Darmstadt, Logos-Verlag, Berlin, 2011 48. F. Riechwald, Interpolation of sum and intersection spaces of Lq -type and applications to the Stokes problem in general unbounded domains. Ann. Univ. Ferrara 58, 167–181 (2012) 49. G. Seregin, On smoothness of L3;1 -solutions to the Navier-Stokes equations up to boundary. Math. Ann. 332, 219–238 (2005) 50. J. Serrin, The initial value problem for the Navier-Stokes equations, in Nonlinear Problems, ed. by R.E. Langer (University of Wisconsin Press, Madison, WI, 1963) 51. M. Shinbrot, The energy equation for the Navier–Stokes system. SIAM J. Math. Anal. 5, 948– 954 (1974) 52. C.G. Simader, H. Sohr, A new approach to the Helmholtz decomposition and the Neumann problem in Lq -spaces for bounded and exterior domains, in Advances in Mathematics for Applied Sciences, vol. 11 (World Scientific, Singapore, 1992), pp. 1–35 53. H. Sohr, A regularity class for the Navier-Stokes equations in Lorentz spaces. J. Evol. Equ. 1, 441–467 (2001) 54. H. Sohr, The Navier–Stokes equations, in An Elementary Functional Analytic Approach. Birkhäuser Advanced Texts (Birkhäuser, Basel, 2001) 55. V.A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations. J. Sov. Math. 8, 467–529 (1977) 56. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators (North-Holland, Amsterdam, 1978)

On Global Well/Ill-Posedness of the Euler-Poisson System Eduard Feireisl

Abstract We discuss the problem of well-posedness of the Euler-Poisson system arising, for example, in the theory of semi-conductors, models of plasma and gaseous stars in astrophysics. We introduce the concept of dissipative weak solution satisfying, in addition to the standard system of integral identities replacing the original system of partial differential equations, the balance of total energy, together with the associated relative entropy inequality. We show that strong solutions are unique in the class of dissipative solutions (weak-strong uniqueness). Finally, we use the method of convex integration to show that the Euler-Poisson system may admit even infinitely many weak dissipative solutions emanating from the same initial data. Keywords Dissipative solution • Euler-Poisson system • Weak solution

1 Introduction We consider the Euler-Poisson system of partial differential equations in the form @t n C divx J D 0; 

JJ @t J C divx C rx .nT/ D ˙nV; n

 3 3 J @t .nT/ C divx .TJ/  T C nTdivx D 0; 2 2 n V D rx ˆ; ˆ D n  1:

(1) (2) (3) (4)

In specific applications, n is the density, J the flux, and T the (absolute) temperature of charged particles, driven by the potential volume force proportional

E. Feireisl () Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1, Czech Republic e-mail: [email protected] © Springer Basel 2016 H. Amann et al. (eds.), Recent Developments of Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-0348-0939-9_12

215

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E. Feireisl

to nrx ˆ, see Guo [10], Guo and Pausader [11], Juengel [13], among others. From the mathematical viewpoint, Eqs. (1), (2) represent a hyperbolic Euler system, with the density n and the velocity J=n, coupled with a parabolic “heat equation” (3), and the elliptic Poisson equation (4). To avoid the technical problems caused by the presence of a kinematic boundary, we restrict ourselves to the spatially periodic boundary conditions, specifically all quantities are functions of the time t 2 .0; / and the position x, belonging to the flat torus  3 T D Œ1; 1jf1;1g : Accordingly, the problem is formally closed by prescribing the initial conditions n.0; / D n0 ; J.0; / D J0 ; T.0; / D T0 :

(5)

For smooth and physically relevant initial data, meaning n0 .x/  n > 0; T0 .x/  T > 0 for all x 2 T ;

(6)

the problem (1)–(5) admits a unique regular solution on a maximal existence interval .0; max /, see Alazard [1], Serre [15, 16]. On the other hand, the Euler system (1), (2) being hyperbolic, discontinuities in the form of shock waves are likely to develop in a finite time regardless the smoothness of the initial data, see Guo and Tahvildar-Zadeh [12]. However, as observed by Guo [10], the linearized Euler-Poisson system (1), (2), (4) (with T D const) coincides with the KleinGordon equation, where the dispersive effects due “plasma” oscillations prevents the formation of shocks in small irrotational solutions. In view of these arguments, it is interesting to examine the problem of global existence in the class of weak solutions, satisfying, in addition, certain admissibility criteria that would guarantee well-posedness, that means, existence, uniqueness, and possibly stability for any physically relevant initial data. Motivated by [8], we introduce a relative entropy (energy) functional associated to the system (1)–(5), together with a class of dissipative weak solutions. These are, roughly speaking, the weak solutions satisfying, in addition, the total energy balance. Then we show the weak-strong uniqueness property, namely, any dissipative solution coincides with the strong solution emanating from the same initial data as long as the latter exists. The strong solutions are unique within the class of weak solutions (cf. Berthelin and Vasseur [2], Dafermos [5], Germain [9] for related results). The last part of the paper is devoted to the problem of well-posedness in the class of weak and/or dissipative solutions. Using an extension of the variable coefficients analogue of the results of DeLellis and Székelyhidi [6] developed in [4], we show that the Euler-Poisson system (1)–(5) admits infinitely many global-in-time weak solutions for any smooth initial data. Although one can still hope that some apparently non-physical solutions can be eliminated by imposing the total energy

Euler-Poisson System

217

balance as an admissibility criterion (dissipative weak solutions), we identify a vast class of physically admissible initial data for which the problem possesses infinitely many dissipative weak solutions. The paper is organized as follows. Section 2 contains some preliminary material including proper definitions of the weak and dissipative solutions. In Sect. 3, we show the weak-strong uniqueness property for the dissipative solutions. In Sect. 4, the existence of global-in-time weak solutions is established for any physically admissible smooth initial data. Some examples of ill-posedness within the class of dissipative weak solutions are discussed in Sect. 5. The various concepts of solutions and their basic properties are summarized in Sect. 6.

2 Preliminaries, Weak and Dissipative Solutions We start rewriting (3) as an entropy balance. Specifically, dividing (3) on T and using (1), we arrive at @t .nS.n; T// C divx .S.n; T/J/   log.T/ D jrx log.T/j2 ;

(7)

with the specific entropy

S.n; T/ D log

T 3=2 n

 :

2.1 Weak Solutions We say that Œn; J; T is a weak solution to the problem (1)–(5) in .0; /  T if: R • n > 0, T > 0 a.a. in Œ0; /  T , T .n  1/ dx D 0, n 2 L1 ..0; /  T / \ C.Œ0; I L1 .T //; J D L1 ..0; /  T I R3 / \ Cweak .Œ0; I L2 .T I R3 //; T 2 L1 ..0; /  T / \ C.Œ0; I L1 .T //; rx # 2 L2 ..0; /  T I R3 /I • the equation of continuity (1) is replaced by a family of integral identities Z sZ

Z T

Œn.s; /'.s; /  n0 '.0; / dx D

0

T

Œn@t ' C J  rx ' dx dt

for any s 2 Œ0; / and any test function ' 2 C1 .Œ0;   T /;

(8)

218

E. Feireisl

• the momentum balance (2) is satisfied in the sense that Z ŒJ.s; /  '.s; /  J0  '.0; / dx

(9)

T

Z sZ  J˝J W rx ' C nTdivx ' ˙ nrx ˆ  ' dx dt J  @t ' C D n 0 T holds for any s 2 Œ0; / and any test function ' 2 C1 .Œ0;   T I R3 /; • the heat equation (3) is replaced by a weak form of the entropy balance (7), specifically, the integral identity Z T

ŒnS.n; T/.s; /'.s; /  n0 S.n0 ; T0 /'.0; / dx D

Z sZ

Z sZ C 0

T

0

T

jrx log.T/j2 dx dt (10)

ŒnS.n; T/@t ' C S.n; T/J  rx '  rx log.T/  rx ' dx dt

holds for any s 2 Œ0; / and any test function ' 2 C1 .Œ0;   T /; • the potential ˆ is the (unique) solution of the elliptic equation Z  ˆ.s; / D n.s; /  1 in T ;

T

ˆ.s; / dx D 0 for all s 2 Œ0; /:

(11)

Remark 2.1 The condition Z T

.n  1/ dx D 0

can be replaced by Z T

.n  n/ dx D 0 for a certain n > 0:

Accordingly, we have to take the potential ˆ such that ˆ.s; / D n.s; /  n in T : Remark 2.2 Apparently, replacing the heat equation (3) by the entropy balance (7) may not be an equivalent operation within the framework of weak solutions. On the other hand, however, the density n as well as the temperature T considered in the present paper will be regular enough for (7) to imply (3) and vice versa. The entropy formulation (10) is more convenient for introducing the concept of dissipative solution discussed below.

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219

2.2 Relative Entropy (Energy), Dissipative Solutions Similarly to [8], we introduce the ballistic free energy

H‚ .n; T/ D n

 3 T  ‚S.n; T/ ; 2

together with the relative entropy functional ˇ   ˇ E n; T; JˇN; ‚; V Z " D T

(12)

ˇ ˇ

# @H‚ .N; ‚/ 1 ˇˇ J V ˇˇ2 n  .n  N/  H‚ .N; ‚/ C H‚ .n; T/  dx: 2 ˇn N ˇ @N

Remark 2.3 The relative entropy (12) coincides, modulo the multiplicative factor ‚, with the relative entropy introduced in the context of hyperbolic conservation laws by Dafermos [5]. Thus, correctly speaking, the physical dimension of E is energy rather than entropy.

2.3 Relative Entropy Inequality As we can check by direct manipulation, regular solutions of the system (1)–(5) satisfy the relative entropy inequality ˇ h  itDs Z s Z jrx Tj2 ˇ E n; T; JˇN; ‚; V C ‚ dx dt tD0 T2 0 T

(13)

 1 V .nV  NJ/  @t dx dt N 0 T N



 Z sZ  1 V V .nV  NJ/ ˝ J W rx  nTdivx dx dt C nN N N T 0 Z sZ h     i  n S.n; T/  S.N; ‚/ @t ‚ C S.n; T/  S.N; ‚/ J  rx ‚ dx dt Z sZ



0

T

Z s Z  Z sZ n J rx T C @t .N‚/   rx .N‚/ dx C  rx ‚ dx dt 1 N N T 0 0 T T Z sZ 1 ˙ rx 1 Œn  1 .nV  NJ/ dx dt N 0 T

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for a.a. s 2 Œ0; / and any trio of smooth “test” functions N; ‚; V; N > 0; ‚ > 0;

(14)

cf. [8].

2.4 Dissipative Solutions We say that a trio Œn; J; T is a dissipative solution to the problem (1)–(5) in .0; /T if: R • n > 0, T > 0 a.a. in Œ0; /  T , T .n  1/ dx D 0, n 2 L1 ..0; /  T / \ C.Œ0; I L1 .T //; J D L1 ..0; /  T I R3 / \ Cweak .Œ0; I L2 .T I R3 //; T 2 L1 ..0; /  T / \ C.Œ0; I L1 .T //; rx # 2 L2 ..0; /  T I R3 /I • the relative entropy inequality (13) holds for any choice of smooth test functions N; ‚; V satisfying (14).

3 Weak Strong Uniqueness The concept of dissipative solution in the context of the incompressible Euler system was introduced by DiPerna and Lions (see [14]). It is interesting to note that the dissipative solutions apparently do not satisfy any system of differential equations but just the relative entropy inequality (10). However, the following weak-strong uniqueness property holds: Theorem 3.1 Let Œn; T; J be a dissipative solution of the problem (1)–(5) in .0; / T , with the initial data Œn0 ; T0 ; J0  satisfying (6). Suppose that the problem (1)–(5) Q Q J, admits also a regular solution ŒQn; T; Q @m Q @t J; Q mQ @t nQ ; @t T; Q ; @m xn x T; @x J 2 C.Œ0; /  T /; m D 0; 1; 2; emanating form the same initial data Œn0 ; T0 ; J0 . Then Q J JQ in Œ0; /  T : n nQ ; T T; Q V D JQ as test The proof of Theorem 3.1 is based on taking N D nQ , ‚ D T, functions in the relative entropy inequality (10) and “absorbing” the terms on the

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right-hand side of the resulting expression by means of the Gronwall argument. Since the dissipative solutions introduced in Sect. 2.4 are bounded, the proof of Theorem 3.1 is essentially the same as that of Theorem 6.1 [7, Sect. 6]. Note that the extra term Z sZ 1 ˙ rx 1 Œn  1 .nV  NJ/ dx dt N 0 T in (10) can be handled without any additional difficulty. Finally, note that the existence of local-in-time regular solutions to the problem (1)–(5) ranging in the standard energy Sobolev scale W m;2 was established by Alazard [1], Serre [15], while the existence of possibly global-in-time dissipative solutions remains an outstanding open problem.

4 Existence of Weak Solutions for Physically Relevant Data In the remaining part of the paper, we focus on the class of weak solutions to (1)–(5), and, in particular, on their relation to the dissipative solutions.

4.1 Global-in-Time Weak Solutions We start with a rather striking result concerning the existence of global-in-time weak solutions in the sense specified in Sect. 2.1. Theorem 4.1 Let  > 0 be given. Suppose that the initial data Œn0 ; T0 ; J0 , n0 ; T0 ; J0 2 C3 .T /; satisfy (6). Then the problem (1)–(5) possesses infinitely many weak solutions in Œ0; /  T . In addition, the weak solutions Œn; T; J belong to the class n 2 C2 .Œ0; /  T /; @t T 2 Lp .0; TI Lp .T //; rx2 T 2 Lp .0; I Lp .T I R33 // for any 1  p < 1; J 2 Cweak .Œ0; I L2 .T I R3 // \ L1 ..0; /  T I R3 /; divx J 2 C2 .Œ0; /  T /: Remark 4.2 It is easy to check that any weak solution enjoying the regularity properties specified in Theorem 4.1 satisfies the equation of continuity (1), the entropy balance equation (7) as well as the internal energy equation (3) a.a. in Œ0; /  T . All possible singularities are therefore concentrated on the solenoidal

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component of the flux J. On the other hand, the solutions are neither regular nor dissipative solutions of the problem in agreement with the conclusion of Theorem 3.1. The remaining part of this section is devoted to the proof of Theorem 4.1. 4.1.1 Oscillatory Lemma Similarly to [4], the weak solutions claimed in Theorem 4.1 are obtained by the method of convex integration, in particular, an extension to “variable coefficients” of the following result of De Lellis and Székelyhidi [6, Proposition 3], Chiodaroli [3, Sect. 6, formula (6.9)]: Lemma 4.3 Let ŒT1 ; T2 , T1 < T2 , be a time interval and B  R3 a domain. Let Q 2 R33 be constant fields such that nQ 2 .0; 1/, ZQ 2 R3 , U sym;0 Q < Z; jUj Q < U: 0 < n < nQ < n; jZj Suppose that v 2 Cweak .ŒT1 ; T2 I L2 .B; R3 // \ C1 ..T1 ; T2 /  BI R3 / satisfies a linear system of equations @t v C divx U D 0; divx v D 0 in .T1 ; T2 /  B with some U 2 C1 ..T1 ; T2 /  BI R33 sym;0 / such that " # Q ˝ .v C Z/ Q Q 2 1 jv C Zj 3 .v C Z/ Q max  I  .U C U/ 2 nQ 3 nQ 0 and a sequence fvn g1 nD1  X0 such that vn ! v in Cweak .Œ0; I L2 .T I R3 //; lim inf I" Œvn   I" Œv C ˇ: n!1

Following the arguments of [6] we obtain: (1) cardinality of the space X0 is infinite; (2) the points of continuity of each I" form a residual set in X; (3) the set CD

\˚  v 2 X j I1=m Œv is continuous ; m>1

being an intersection of a countable family of residual sets, is residual, in particular of infinite cardinality; (4) I1=m Œv D 0 for all m > 1 for each v 2 C. The relation (24) implies that



3 3 1 1 jv C rx ‰j2 2 D eŒv D  nTŒv  @t ‰  ˆ C jrx ˆj 2 n 2 2 6

(24)

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for any v 2 C, @t v C divx U D 0 in the sense of distributions in .0; /  T ; where

 1 .v C rx ‰/ ˝ .v C rx ‰/ 1 jv C rx ‰j2 2  I ˙ rx ˆ ˝ rx ˆ  jrx ˆj I : UD n 3 n 3 In other words, v satisfies (19), (20) in the sense of distributions. We have proved Theorem 4.1. Remark 4.9 The reader will have noticed that our construction of the weak solutions enables to prescribe the value of the density also for t D .

5 Well-Posedness in the Class of Dissipative Weak Solutions As we have seen in the previous part, although the problem (1)–(5) admits globalin-time weak solutions, it is not well-posed in this class. On the other hand, the dissipative solutions enjoy the property of weak-strong uniqueness, meaning they coincide with the unique (local) strong solution as long as the latter exists. We introduce an intermediate class of dissipative weak solutions, specifically, the weak solutions satisfying the relative entropy inequality (13). As shown in [7], a weak solution is a dissipative solution as soon as it satisfies the total energy balance: Z  T

Z  3 1 3 1 1 jJj2 1 jJ0 j2 C nT  nˆ .s; / dx D C n0 T0  n0 ˆ0 dx 2 n 2 2 2 n0 2 2 T (25)

for a.a. s 2 .0; /, where we have use the identities Z T

Z rx ˆ  J dx D  Z D T

Z T

ˆdivx J dx D

d1 ˆ@t ˆ dx D dt 2

Z T

T

ˆ@t n dx

jrx ˆj2 dx;

and Z

2

T

jrx ˆj dx D

Z nˆ dx: T

Note that (25) combined with the weak formulation (8)–(11), is, in fact, stronger than the relative entropy inequality (13), where the latter still holds if (25) is replaced

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by an inequality: Z  T

Z  3 1 3 1 1 jJj2 1 jJ0 j2 C nT  nˆ .s; / dx D C n0 T0  n0 ˆ0 dx 2 n 2 2 2 n0 2 2 T

for a.a. s 2 .0; /. A weak solution of the problem (1)–(5) satisfying the energy equality (25) will be termed finite energy weak solution. As a direct consequence of [7, Theorem 6.1] (cf. Theorem 3.1 above) we obtain: Theorem 5.1 Let Œn; T; J be a finite energy weak solution of the problem (1)– (5) in .0; /  T , with the initial data Œn0 ; T0 ; J0  satisfying (6). Suppose that the Q Q J, problem (1)–(5) admits also a regular solution ŒQn; T; Q @m nQ ; @m T; Q Q @t J; Q @m @t nQ ; @t T; x x x J 2 C.Œ0; /  T /; m D 0; 1; 2; emanating form the same initial data Œn0 ; T0 ; J0 . Then Q J JQ in Œ0; /  T : n nQ ; T T; Thus, the stipulation of the total energy conservation (25) seems to eliminate the “non-physical” weak solutions obtained in Theorem 4.1. On the other hand, however, there might still be “irregular” initial data for which the problem (1)–(5) admits infinitely many finite energy weak solution. The precise statement reads: Theorem 5.2 Let  > 0 and the initial data Œn0 ; T0 , n0 ; T0 2 C3 .T /; satisfying (6) be given. Then there exists a flux J0 2 L1 .T I R3 / such that the problem (1)–(5) possesses infinitely many finite energy weak solutions in Œ0; /  T . In addition, the weak solutions Œn; T; J belongs to the class specified in Theorem 4.1. The proof of Theorem 5.2 is essentially the same as that of [4, Theorem 4.2] and we leave it to the interested reader. Clearly, in accordance Theorem 3.1, the initial datum J0 , the existence of which is claimed in Theorem 5.2, cannot be regular. On the other hand, the solutions obtained in Theorem 8 must have a “large” solenoidal part - to be compared with the result of Guo [10] on global existence of smooth solutions with irrotational initial data.

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6 Conclusion Unlike the standard Euler system, the Euler-Poisson system possesses global-intime weak solutions for small irrotational initial data, see Guo [10]. On the other hand, we have shown that large data with a non-zero solenoidal component give rise to infinitely many weak solutions satisfying the global energy balance, see Theorem 5.2. In general, we have introduced several classes of solutions to the Euler-Poisson system (1)–(5), the properties of which can summarized as follows: • Strong (classical) solutions. They are classical (differentiable) solutions of the system (1)–(5), emanating from regular initial data, that exist on a (possibly) short time interval for general (smooth) data Œn0 ; J0 ; T0 . Global-in-time existence is to be expected for small irrotational data, see Guo [10]. • Weak (distributional) solutions. They satisfy (1)–(5) in the sense of distribution. Global-in-time weak solutions do exist for any initial data but they may be “unphysical” in the sense that they produce energy at the initial time. • Dissipative solutions. The satisfy the relative entropy inequality and, consequently, they coincide with the (local) strong solution emanating from the same initial data as long as the latter exists. Global-in-time existence of dissipative solutions for general initial data is an open problem. • Finite energy weak solutions. These are the weak solutions satisfying, in addition, the total energy balance. They are dissipative solutions so they coincide with a strong solutions as long as the latter exists. Global existence of finite energy weak solutions for general initial data is an open problem. On the other hand, there is a vast set of initial data (with an irregular flux J0 ), for which the problem (1)–(5) admits infinitely many global-in-time finite energy weak solutions. Acknowledgements The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC Grant Agreement 320078.

References 1. T. Alazard, Low Mach number flows and combustion. SIAM J. Math. Anal. 38(4), 1186–1213 (2006) (electronic) 2. F. Berthelin, A. Vasseur, From kinetic equations to multidimensional isentropic gas dynamics before shocks. SIAM J. Math. Anal. 36, 1807–1835 (2005) 3. E. Chiodaroli, A counterexample to well-posedness of entropy solutions to the compressible Euler system. J. Hyperbolic Differ. Equ. 11(3), 493–519 (2014) 4. E. Chiodaroli, E. Feireisl, O. Kreml, On the weak solutions to the equations of a compressible heat conducting gas. Ann. Inst. Henri Poincaré Anal Non Linéaire. 32(1), 225–243 (2015) 5. C.M. Dafermos, The second law of thermodynamics and stability. Arch. Ration. Mech. Anal. 70, 167–179 (1979)

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6. C. De Lellis, L. Székelyhidi Jr., On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 195(1), 225–260 (2010) 7. E. Feireisl, Relative entropies in thermodynamics of complete fluid systems. Discrete Cont. Dyn. Syst. Ser. A 32, 3059–3080 (2012) 8. E. Feireisl, A. Novotný, Weak-strong uniqueness property for the full Navier-Stokes-Fourier system. Arch. Ration. Mech. Anal. 204, 683–706 (2012) 9. P. Germain, Weak-strong uniqueness for the isentropic compressible Navier-Stokes system. J. Math. Fluid Mech. 13(1), 137–146 (2011) 10. Y. Guo, Smooth irrotational flows in the large to the Euler-Poisson system in R3C1 . Commun. Math. Phys. 195, 249–265 (1998) 11. Y. Guo, B. Pausader, Global smooth ion dynamics in the Euler-Poisson system. Commun. Math. Phys. 308, 89–125 (2011) 12. Y. Guo, A.S. Tahvildar-Zadeh, Nonlinear partial differential equations, in Contemporary Mathematics (American Mathematical Society, Providence, RI, 1999), pp. 151–161 13. A. Jüengel, Transport Equations for Semiconductors. Lecture Notes in Physis, vol. 773 (Springer, Heidelberg, 2009) 14. P.-L. Lions, Mathematical Topics in Fluid Dynamics: Vol. 1, Incompressible models (Oxford Science Publication, Oxford, 1996) 15. D. Serre, Local existence for viscous system of conservation laws: H s -data with s > 1 C d=2, in Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena, vol. 526. Contemporary Mathematics (American Mathematical Society, Providence, RI, 2010), pp. 339– 358 16. D. Serre, The structure of dissipative viscous system of conservation laws. Phys. D 239(15), 1381–1386 (2010)

On the Motion of a Liquid-Filled Rigid Body Subject to a Time-Periodic Torque Giovanni P. Galdi, Giusy Mazzone, and Mahdi Mohebbi

Dedicated, with friendship and admiration, to Professor Yoshihiro Shibata, on the occasion of his 60th birthday

Abstract In this paper we investigate the existence of time-periodic motions of a system constituted by a rigid body with an interior cavity completely filled with a viscous liquid, and subject to a time-periodic external torque acting on the rigid body. We then show that the system of equations governing the motion of the coupled system liquid-filled rigid body, has at least one corresponding time-periodic weak solution. Furthermore if the size of the torque is below a certain constant, the weak solution is in fact strong. Keywords Navier-Stokes equations • Rigid body motions • Time-periodic solutions

1 Introduction Problems involving the motion of a rigid body with a cavity filled with a viscous liquid are of fundamental interest in several applied research areas, such as dynamics of flight [10, 17], space technology [4, 14], and geophysical problems [16, 18]. Besides its important role in physical and engineering disciplines, the motion of such coupled systems generates a number of mathematical questions, which are intriguing and challenging, mostly due to the different properties of the components of the system: one dissipative (the liquid), the other conservative (the rigid body). However, despite their relevance, basic mathematical questions related to the above

G.P. Galdi () • G. Mazzone Department of Mechanical Engineering and Materials Science, University of Pittsburgh, 3700 O’Hara Street, Pittsburgh, PA 15261, USA e-mail: [email protected]; [email protected] M. Mohebbi Department of Mechanical Engineering, SUNY Korea, 119 Songdo Moonhwa-ro, Yeonsu-gu, Incheon 406–840, Korea e-mail: [email protected] © Springer Basel 2016 H. Amann et al. (eds.), Recent Developments of Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-0348-0939-9_13

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problem in its full generality appear to have received, to date, little or no attention; see, e.g., [8] and the reference cited therein. Objective of this paper is to give a detailed analytical study of one of the above questions, when the coupled system body/liquid moves under the action of a timeperiodic torque. More specifically, let B be a rigid body containing in its interior a hollow cavity, C, completely filled by a viscous liquid, L. We shall assume that the resultant of the external forces applied to the coupled system S D B [ L is identically zero, so that the center of mass, G, of S moves by uniform rectilinear motion with respect to an inertial frame I. As a result, the relevant motion of S reduces to the one occurring around its center of mass, namely, with respect to a frame, G, with the origin in G and axes parallel to those of I. We shall next suppose that on B acts a torque, m, varying in time in a timeperiodic fashion:1 m D fi .t/ ei .t/ ;

(1)

where fei g is a base in G, while fi , i D 1; 2; 3, are given T-periodic scalar functions of time t, that is, fi .t C T/ D fi .t/, for all t 2 R. In the wake of analogous classical problems formulated in absence of liquid, we propose to investigate whether, under the given assumptions, the coupled system S will execute a T-periodic motion in a body-fixed frame S. In order to handle the above question, it appears necessary to impose some restrictions on the functions fi in (1), as we shall show next. To this end, we begin to observe that in the frame S the torque m becomes M D Q> .t/  m D fi .t/ Q> .t/  ei ;

(2)

where Q D Q.t/ is the (unknown) one-parameter family of elements of the special orthogonal group, SO.3/, associated with the change of frame G ! S, and > denotes transpose. We then have 2 3 0 3 2 dQ> D W./  Q> , Q> .0/ D 1 , W./ WD 4 3 0 1 5 ; (3) dt 2 1 0 where  D .1 ; 2 ; 3 / is the angular velocity of B in S, and 1 is the identity matrix; see, e.g., [2, § 6.26]. Assuming the motion of S in the frame S to be T-periodic implies, in particular, that both  and M have to be T-periodic as well. Therefore, from (3) and standard Floquet theory it follows that Q.t/ has the following representation [19, Theorem 1] Q.t/ D P.t/  etS ; t 2 R ; 1

We adopt summation convention over repeated indices.

(4)

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where S is a real, skew-symmetric matrix, and P.t/ 2 SO.3/ ; P.t C T/ D P.t/ ; for all t 2 R :

(5)

From the latter and from (2) we then deduce that for M to be a T-periodic function we must have eTS  ei D ei ; i D 1; 2; 3 ; namely, ei must be parallel to the eigenvectors corresponding to the eigenvalue  D 0 of S. Since, in general,  D 0 is simple, the existence of a T-periodic solution in the body-fixed frame S requires, in general, that m is directed along a constant direction. We thus have m D f .t/ h ;

(6)

where h is a unit, time-independent vector in G, and f is a T-periodic function. We next observe that, denoting by kG the total angular momentum of S with respect to G, the balance of angular momentum in the frame G requires d kG D f .t/ h ; dt from which we at once deduce, in particular, that jkG .t/j is T-periodic if and only if f has a zero average over a period: Z

T 0

f .t/ dt D 0 :

(7)

However, jkG .t/j is left invariant in the frame change G ! S, so that the searched T-periodicity of the motion of S with respect to the frame S requires that f obeys (7). As a consequence of what just shown, we shall then require that the torque m acting on B satisfies (6)–(7). Under these assumptions, the main goal of this paper consists in proving the existence of a motion of the coupled system S that is T-periodic with respect to the body-fixed frame S. It is worth remarking that in S the direction of the torque becomes a function of time given by H.t/ WD Q> .t/  h, and since Q is not known, H becomes a further unknown. From the physical viewpoint, the latter circumstance means that, in order to perform such a periodic motion, the body has to find an “appropriate orientation” with respect to the direction of m. We thus show that, under the hypothesis that f is T-periodic and squaresummable over a period, the problem admits a corresponding (suitably defined) T-periodic weak solution. If, moreover, f is essentially bounded with a sufficiently small norm, then the solution is strong and the relevant equations are satisfied almost everywhere in space-time.

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We would like to complete this introductory section with two further observations. In the first place, it is natural to ask whether the motion of the coupled system S is time-periodic also in the inertial frame G. The answer to this question is, in general, negative. In fact, denoting by v and ! the velocity field of L and the angular velocity of B with respect to G, we have v.x; t/ D Q.t/  V.Q> .t/  x; t/ ; !.t/ D Q.t/  .t/ :

(8)

Therefore, since both V and  are time-periodic, in view of the representation (4)– (5) of Q, we can only conclude that, in general, v and ! are almost periodic functions of time. However, the component of ! along the (fixed) direction h of the torque in G is T-periodic. Actually, by (4)–(5) h D P.t/  etS  H.t/ ; and since h D H.0/, by the T-periodicity of H and P we deduce h D eTS  H.0/ D eTS  h: As a result, being .0/ D !.0/ and using the T-periodicity of  we conclude !.T/  h D eTS  .T/  h D !.0/  eTS  h D !.0/  h : In the light of these considerations it is very likely that if the cavity C possesses suitable geometric symmetry, and the direction of h is constant and chosen along one of the principal axes of inertia of S, the motion of B is periodic also in G. We shall analyze this problem elsewhere. Another interesting question concerns the uniqueness of the solutions. Also here the answer is, in general, negative. It is in fact sufficient to consider the case f 0 and observe that permanent rotations, where  is constant and directed along any principal axis of inertia of S and V D 0, are allowed time-periodic solution of arbitrary period. Therefore, the important issue here becomes the stability of time-periodic motions, aiming at selecting which, among the possibly many, is indeed observable and, as such, physically meaningful. This appears to be a very complicated task that will be the object of future work. The paper is organized as follows. After giving in Sect. 2 the formulation of the problem with respect to the frame S, in the subsequent Sect. 3 we furnish a corresponding weak formulation. The latter, essentially, requires that the relative velocity field, V, of L in S, satisfies the function properties of a solution à la Leray-Hopf (see [5]) while  and H are requested to be only continuous functions of time. Successively, in Sect. 4, we show the existence of a weak solution (see Theorem 4.1). The proof is accomplished by an appropriate combination of the classical Galerkin method with a fixed point argument for triangulable manifolds based on the Lefschetz-Hopf theorem. Finally, in Sect. 5, we prove that if the size of f is suitably restricted, the weak solution constructed in Theorem 4.1 is in fact

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strong, namely, V possesses first time derivative and second spatial derivative that are square summable in space and time, while  and H are suitably differentiable (Theorem 5.2).

2 Formulation of the Problem We keep the notation presented in the introductory section. Assuming that the flow of the viscous liquid is described by the Navier-Stokes equations, we find that the governing equations of the liquid in the frame S attached to the body are given by Kopachevsky and Krein [11] 9 @V P  y C 2  V C V  grad V C grad p  V D 0 = C in D  Œ0; T; @t ; div V D 0 (9) along with the boundary condition V.x; t/ D 0 for .x; t/ 2 @D  Œ0; T .

(10)

In the above equations, D is a bounded, simply connected domain of R3 representing the volume of C, Vand  are relative velocity field of L and angular velocity of B, respectively, referred to S, and > 0 is the coefficient of kinematic viscosity of L. Finally, p WD

1 1 P.x; t/  j  yj2 L 2

where L and P are (constant) density and pressure field of L. As for the body B, during its motion it has to obey the balance of angular momentum that, when referred to S, takes the following form d J C   J   D f .t/ H  dt

Z @D

y  T.V; P/  n ; t 2 Œ0; T ;

(11)

where J is the inertia tensor of B, T.V; P/ D P 1 C L2 .grad V C .grad V/> / is the Cauchy stress tensor, and n is the unit outer normal to @D. Furthermore, H WD Q> .t/  h, which, in view of (3), must satisfy the following (Poisson) equation dH C   H D 0: dt

(12)

It is now convenient to rewrite the system of equations (9)–(11) in a different but equivalent form that is also more significant from the physical viewpoint and more manageable from the mathematical one. To this end, we begin to notice that

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by a direct calculation one shows that (9)–(11) is equivalent to (9), (10) and the following one dA C   A D f .t/ H ; dt

(13)

where A is the angular momentum of the coupled system S, referred to S, given by Z A WD L

D

y V C I  ;

(14)

with I inertia tensor of S. Using (14) we can then eliminate  and write the relevant equations only in terms of the unknowns V; p; A, and H. Thus, observing that DI

1

 A  L I

1

Z





y  V dV D

;

(15)

the system of equations (9), (10), (13), and (12) becomes (see also [11]):

 9   dA @V > C I1   y C 2 I1  A  V > > > @t  >

dtZ  > = 1 y  V dV V 2 I  L in D  Œ0; T ; > D > > > CV  grad V C grad p  V D 0 > > ; div V D 0; V.x; t/ D 0 on @D  Œ0; T ; Z 9  dA > 1 1 C .I  A/  A  L I  y  V dV  A D f .t/H> = dt

ZD  in Œ0; T ; dH > C .I1 A/  H  L I1 y  V dV  H D 0 > ; dt D .1  B/ 

(16)

where BW

2 L1 .D/ 7! B 

2 L1 .D/

(17)

is such that .B 

/. y/ D L I

1

Z





y

dV  y:

(18)

D

Our problem can be then formulated as follows. Given a sufficiently smooth Tperiodic function f , find a corresponding T-periodic solution to (16)–(18). For the resolution of this problem an important role is played by the properties of the operator B. The latter have been studied and determined in [11, Chap. 1, Sect. 7.2.3], and are summarized in the following lemma.

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Lemma 2.1 The operator B defined by (17) and (18) is non-negative and selfadjoint. Moreover, 1  B is a non-negative operator with a bounded inverse.

3 Function Spaces and Weak Formulation In this section we furnish a weak formulation of the time-periodic problem stated in Sect. 2. To this end, we introduce the main function spaces that we shall use throughout. For all the other (standard) notation used in this paper and not mentioned here, we refer to, e.g., [1]. Let D.D/ D f

2 C01 .D/j div

D 0 in Dg:

We denote by H.D/ the closure of D.D/ in the Lebesgue space L2 .D/ and by H k .D/ the closure of D.D/ in the Sobolev space W k;2 .D/. As is well known, the Helmholtz-Weyl decomposition holds (see [6], Theorem III.1.1): L2 .D/ D H.D/ ˚ G.D/; where G.D/ D fv 2 L2 .D/j v D grad p; for some p 2 L1loc .D/g: We indicate by P the orthogonal projection of L2 .D/ onto H.D/. In the space L2 .D/ we shall use the customary notation for inner product and associate norm: Z .u; v/ WD

D

u  v dV;

kuk2 WD .u; v/1=2 ;

defined for any u; v 2 L2 .D/. The following functional h; i W .u; v/ 2 L2 .D/  L2 .D/ 7! hu; ui WD ..1  B/  u; v/;

(19)

defines a scalar product in L2 .D/ with the associate norm kukB WD hu; ui1=2 D ..1  B/  u; u/1=2 ; which is equivalent to the norm kk2 . Indeed, since by Lemma 2.1, B is non-negative and 1  B admits a bounded inverse, we find ..1  B/  u; u/ D k.1  B/  uk22 C .B  u; u/  C2 kuk22

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where, C D C.D/ > 0. Furthermore, again using the fact that B is non-negative, we deduce ..1  B/  u; u/  .u; u/; so that Ckuk2  kukB  kuk2 ;

(20)

for every u 2 L2 .D/. For T > 0, q 2 Œ1; 1, and k 2 N we set q

q

LT .R/ D W fu 2 Lloc .R/j u.t/ D u.t C T/; for a.a. t 2 Rg W f 2 Ck .R/j .t/ D .t C T/; for all t 2 Rg : CTk .R/ D Finally, given a Banach space X with norm k  kX , we denote by Lq .a; bI X/, q 2 .1; 1/ [respectively, C.Œa; bI X/], the class of functions u W Œa; b ! X, with kukLq .a;bIX/ < 1 [respectively, maxt2Œa;b ku.t/kX < 1], where 8 Z 1=q b ˆ ˆ q ˆ ku.t/kX dt ; if 1  q < 1; < a kukLq .a;bIX/ WD ˆ ˆ ˆ if q D 1: : ess sup ku.t/kX ; t2.a;b/

We are now in position to give the definition of T-periodic weak solution to the problem (16)–(18). Definition 3.1 A triple .V; A; H/ is a T-periodic, or simply periodic weak solution to the problem (16)–(18) if the following conditions hold. 1. V 2 L2 .0; TI H 1 .D// \ L1 .0; TI H.D//, A; H 2 CT ; 2. .V; A; H/ satisfies the following equations Z

T 0



Z 1 hV; i C A  I  y

d.t/ dt dt D  

Z Z T Z 1 D  2L y  V dV  I  V 0

Z C2

T

0

Z

Z

dV

D

AI L

D

1

0

 .V 

dV .t/ dt

/ dV .t/ dt

.V  grad ; V/.t/ dt 

2 D.D/,  2 CT1 .R/ ;





T

C

for all



Z 0

T

.grad V; grad /.t/ dt ; (21)

Time-Periodic Motions of a Liquid-Filled Rigid Body

Z

t

A.t/ D A.0/  0

.I

1

 A/  A d C L I Z 0

t

H.t/ D H.0/  0

.I

1

1

Z t Z





y  V dV  A d 0

D

t

C Z

241

f ./H d;

 A/  H d C L I

1



for all t 2 Œ0; T I Z t Z 0

(22)

 y  V dV  H d; D

for all t 2 Œ0; T:

(23)

Remark 3.2 A weak solution has, in fact, more regularity in time than that stated in the above definition. In fact, on the one hand, from (22) and (23) we deduce A; H 2 W 1;r .0; T/ ; provided f 2 LrT .R/, r 2 Œ1; 1. On the other hand, proceeding as in [5, Lemma 2.2], and [7, Remark], one can show that V.; t/ is continuous in Œ0; T weakly in L2 .D/, and strongly in H 1 .D/. As a consequence, with the help of (21) it follows that V is indeed periodic in time in the sense of the above topologies. Remark 3.3 If .V; A; H/ is a periodic weak solution to (16)–(18), then the corresponding angular velocity  is defined via (15) and belongs to CT ; see also Remark 3.2. Remark 3.4 If .V; A; H/ is a periodic weak solution to (16)–(18) and is sufficiently regular then, by a standard procedure one shows that there exists a scalar field p D p. y; t/ such that V; p, and A satisfy (16)1 .

4 Existence of Periodic Weak Solutions Objective of this section is to show the existence of a periodic weak solution to (16)– (18) under suitable assumptions on f . This will be achieved by combining the FaedoGalerkin method with a fixed point argument. Specifically, we have the following. Theorem 4.1 Let f 2 L2T .R/ satisfy (7), and let D be a domain of R3 . Then, there exists at least one periodic weak solution, .V; A; H/, to (16)–(18). Proof Let f n gn2N be a denumerable subset of D.D/ whose linear hull is dense in H 1 .D/, and let us normalize it as h n ; m i D ınm . We look for “approximate solutions” of the type V n . y; t/ WD

n X kD1

cnk .t/

k . y/; An WD

3 X iD1

cQ ni .t/ei ; Hn WD

3 X jD1

cO nj .t/ek ;

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where fei g is a base of S. The coefficients cnk , cQ ni and cO nj are found by solving the following system of ordinary differential equations:  d hV n ; dt

1  r i C An  I

Z

D 2L Z C2

D

Z 

D

 y D

y  V n dV  I

An  I1  .V n 

r/

1

r

dV

Z



 D

Vn 

r

dV C .V n  grad

dAn D .I1  An /  An C L I1  dt

Z

dHn D .I1  An /  Hn C L I1  dt

D

dV

r ; Vn /

 .grad V n ; grad

r /;

 y  V n dV  An C f .t/Hn ;

Z

D

 y  V n dV  Hn ; (24)

which, in terms of cnk , cQ ni and cO nj reads as follows

Z  dQcni dcnr C ei  I1  y  r dV dt dt D

Z 

Z 1 y  k dV  I  D 2L Z C2 C.

D

D k

D

ei  I1  .

 grad

r;

k

 m



r

dV cnk cnm

/ dV cQ ni cnk r

m /cnk cnm

 .grad

k ; grad

dQcni ei D Qcni cQ nj .I1  ei /  ej C cnk cQ ni L I1  dt C f .t/Ocnj ej ; dOcnj ej D Qcni cO nj .I1  ei /  ej C cnk cO nj L I1  dt

(25)

r /cnk ;

Z

y D

k

dV  ei

k

 dV  ej :

Z y D



By replacing the second equation in the first one, we get

dcnr D .I1  ei /  ej  I1  dt 

Z 1  L I  y

Z

D

 f .t/ej  I1 

 y D

k

D



dV  ei  I

Z y



dV cQ ni cQ nj

r

r

 dV cO nj

1

Z





y D

r

dV cnk cQ ni

Time-Periodic Motions of a Liquid-Filled Rigid Body

Z  2L

y

D

C.

k

dV  I

k

D

Z C2



r;

Z

1





m

D

ei  I1  .

 grad

243



dV cnk cnm

r

/ dV cQ ni cnk r

k

m /cnk cnm

 .grad

k ; grad

r /cnk :

Setting

brij WD .I drki

1



 ei /  e j  I



Z 1 WD L I  y

1

k

Z y

k

prkm WD .

k

 grad

r;

ei  I

m /;

1

r

D 1

Z

.

k





y

r

r

Z



 

dV ;

 dV ; m

D 1

dV ;

D

y

dV  I

Z D



Z



srki WD 2

y D

dV  ei  I

frj .t/ WD f .t/ej  I1 

D







D

grkm WD 2L

Z

r/



r

dV ;

dV ;

prk WD  .grad

k ; grad

r /cnk ;

equation (25)1 becomes dcnr D brij cQ ni cQ nj C drki cnk cQ ni C frj .t/Ocnj C grkm cnk cnm C srik cQ ni cnk dt

(26)

C prkm cnk cnm C prk cnk ; where here and in the rest of the proof i; j; ` vary in the set f1; 2; 3g, whereas r; k; m in the set f1; : : : ; ng. Concerning (25)2 and (25)3 , taking the dot product of each side of both with e` , and putting u`ij

WD  .I1  ei /  ej  e` ;

w`kj WD L e`  I

1

Z





y D

k

dV  ej ;

we deduce dQcn` D u`ij cQ ni cQ nj C w`ki cnk cQ ni C f .t/Ocn` ; dt

(27)

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and dOcn` D u`ij cQ ni cO nj C w`kj cnk cO nj ; dt

(28)

respectively. Following [15] we shall next prove that there exist initial data such that the system of ordinary differential equations (26)–(28) admits a corresponding solution .cnr ; cQ n` ; cO n` / such that cnr .0/ D cnr .T/, cQ nr .0/ D cQ nr .T/ and cO nr .0/ D cO nr .T/. To reach this goal, let V n;0 2 spanf

1; : : : ;

ng ;

An;0 2 R3 ; Hn;0 2 S2 ;

and set cnr .0/ D cnr;0 WD hV n;0 ; r i, cQ n` .0/ D cQ n`;0 WD An;0  e` , and cO n` .0/ D cO n`;0 WD Hn;0  e` . Since f ; fr;j 2 CT .R/, by Picard theorem, there exists a unique solution, .cnr ; cQ n` ; cO n` /, to the Cauchy problem associated to (26)–(28) with cnr ; cQ n` ; cO n` 2 C1 .0; T 0 /, r D 1; : : : ; n, ` D 1; 2; 3, where 0 < T 0  T. Multiplying both sides of (26) by cnr , summing over r D 1; : : : ; n, and noticing that the terms corresponding to drki ; grkm ; srki and prkm vanish, we get

1d kV n k2B C kgrad V n k22 D L .I1  An /  An  I1  2 dt  f .t/Hn  I1 

Z Z

D

D

y  V n dV

y  V n dV:

(29)

Multiplying next both sides of (27) by cQ n` , summing over ` D 1; 2; 3, and taking into account that the terms corresponding to u`ij and w`kj vanish, it follows that 1 djAn j2 D f .t/Hn  An : 2 dt

(30)

Finally, multiplying both sides of (28) by cO n` and summing over ` D 1; 2; 3, similarly as above, we get djHn j2 D 0; dt

(31)

which implies jHn .t/j D jHn;0 j D 1. Thus, from (30) and (31) we infer djAn j  j f .t/j dt

(32)

and Z jAn .t/j  jAn;0 j C

T 0

j f ./j d;

for all t 2 Œ0; T 0 /; n 2 N :

(33)

Time-Periodic Motions of a Liquid-Filled Rigid Body

245

Moreover, by Poincaré and Cauchy–Schwarz inequalities, from (29) we deduce, on the one hand, 1 dkV n k2B C C1p kgrad V n k22  C1 jAn j4 C C2 j f .t/j2 ; 2 dt

(34)

and, on the other hand, using one more time Poincaré inequality in conjunction with (20), 1 dkV n k2B C C2p kV n k2B  C1 jAn j4 C C2 j f .t/j2 ; 2 dt

(35)

where Cip D Cip .D; / > 0, while Ci D Ci .D; B; / > 0, i D 1; 2. Using Gronwall’s lemma in (35) furnishes, exp.C3 t/kV n .t/k2B  kV n;0 k2B C C1

Z 0

t

exp.C3 /jAn ./j4 d Z C C2

t 0

exp.C3 /j f ./j2 d

(36)

which, by (33), implies jcnr .t/j2 D kV n .t/k2B  exp.C3 t/kV n;0 k2B Z t C C1 exp.C3 t/ exp.C3 /jAn ./j4 d Z C C2 exp.C3 t/ 

exp.C3 t/kV n;0 k2B

0

t 0

exp.C3 /j f ./j2 d 4

C C4 sup jAn .t/j C C2 t2Œ0;T

Z

T 0

(37)

j f ./j2 d;

with C3 and C4 positive constants depending, at most, on D, B and . As a result, from (31), (33), and (37) we conclude T 0 D T. In order to build our periodic solution, we will use a suitable fixed point argument. To reach this goal, we multiply both sides of (27) by cO n` and sum over ` D 1; 2; 3, then multiply (28) by cQ n` and sum over ` D 1; 2; 3, and finally add the two resulting equations. We get d.Hn  An / D f .t/jHn j2 D f .t/: dt Since f has zero average by assumption, it follows that Hn;0  An;0 D Hn .T/  An .T/:

(38)

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G.P. Galdi et al.

Taking the cross product of (24)2 by Hn on the left, and then that of (24)3 by An on the left, and summing the two equation so obtained, we deduce d.Hn  An / D n  .Hn  An /; dt where n is given by (15). It then follows that 1 1 djHn  An j2 D 2 dt 2

djAn j2 d.Hn  An /2  dt dt

 D0

where we have also used the fact that jHn j D 1. From the last displayed equation and (38), we conclude that jAn;0 j D jAn .T/j:

(39)

We next fix R1 > 0 and take jAn;0 j  R1 . By (39), we obtain jAn .T/j  R1 . Combining (37) and (33), we infer kV n .T/k2B  exp.C3 T/kV n;0 k2B   C C5 jAn;0 j4 C k f k4L1 .0;T/ C k f k2L2 .0;T/

(40)

 exp.C3 T/kV n;0 k2B C C5 R41 C C6 ; where C5 D C5 .D; B; / > 0, and   C6 WD C5 k f k4L1 .0;T/ C k f k2L2 .0;T/ : Thus, choosing R22 

C5 R41 C C6 ; 1  exp.C3 T/

(41)

from (40) we show that if kV n;0 k2B  R22 , then kV n .T/k2B  R22 . Set B WD BR2  BR1  S2 , where BRi denotes the ball of radius Ri in Rn , i D 1; 2, while S2 is the unit sphere in R3 . Let ˆ W B ! B be the map that takes any .V n;0 ; An;0 ; Hn;0 / 2 B to .V n .T/; An .T/; Hn .T// 2 B, where .V n .T/; An .T/; Hn .T// is the solution to (24) at time T. By a straightforward calculation (see e.g. [15]), one shows that ˆ is continuous. Moreover, ˆ is homotopic to the identity by the following homotopy H W B  Œ0; 1 ! B H.V n;0 ; An;0 ; Hn;0 ; s/ WD .V n .sT/; sAn .T/ C .1  s/An .0/; Hn .sT//:

Time-Periodic Motions of a Liquid-Filled Rigid Body

247

Notice that H is well defined, since by similar calculations which lead to the estimate (41), we show that V n .sT/ 2 BR2 for all s 2 Œ0; 1.2 Since the Euler characteristic of B is given by

.B/ D .BR2 / .BR1 / .S2 / D 1  1  2 ¤ 0 (see [3]), by the Lefschetz-Hopf fixed-point theorem (see [3, Chap. IV, Sect. 23], ), ˆ has at least one fixed point, from which it follows that there exist .V n;0 ; An;0 ; Hn;0 / such that the solution to (24), .V n ; An ; Hn /, starting from .V n;0 ; An;0 ; Hn;0 / satisfies V n .; T/ D V n;0 ./;

An .T/ D An;0 ;

Hn .T/ D Hn;0 :

Now, multiplying (24)1 by  2 CT1 .R/ and integrating over Œ0; T, we show that V n satisfies the following Z

T 0



Z



d.t/ dt dt D  

Z Z T Z D 2L y  V n dV  I1  Vn  1  r i C An  I

hV n ;

0

Z 2

T

Z

0

Z

y

r

D

D

dV

D

An  I1  .V n 

 / dV .t/ dt r Z

T

 0

.V n  grad

r

 dV .t/ dt

r ; V n /.t/ dt 

T 0

.grad V n ; grad

/.t/ dt ; r (42)

CT1 .R/.

for all r D 1; : : : ; n, and all  2 Likewise, integrating (24)2;3 over Œ0; t, t 2 Œ0; T, we deduce that An and Hn satisfy Z An .t/ D An .0/ 

t 0

.I1  An /  An d C L I1 

Z t Z 0

D

Z C

0

 y  V n dV  An d t

f ./Hn d;

(43)

The same argument does not work for An , i.e. the solution map does not necessarily lie in BR1 for all times t 2 Œ0; T. This is the reason for which we have used the linear homotopy for the An component. 2

248

G.P. Galdi et al.

and Z Hn .t/ D H n .0/ 

t 0

.I1  An /  Hn d C L I

1



Z t Z 0

D

 y  V n dV  Hn d;

(44)

respectively. Let us investigate the convergence properties of .V n ; An ; Hn /. By (26) and (37), together with Schwarz inequality, (33) and the fact that jHn j D 1, we .r/ can conclude that the sequence of functions Gn WD hV n .t/; r i is uniformly bounded and equicontinuous. Then, by Ascoli-Arzelà theorem, we can extract a .r/ .r/ subsequence of fGn gn2N , which we will again denote by fGn gn2N , which is .r/ uniformly convergent to a continuous function G .t/. By a Cantor diagonalization .r/ argument, we can make that subsequence independent of r, so that fGn gn2N r converges to G , for all r 2 N, uniformly with respect to t 2 .0; T. Moreover, by (37) and by Banach-Alaoglu Theorem, we can also conclude that there exists V.t/ 2 H.L/ such that lim hV n .t/  V.t/;

n!1

ri

D 0 uniformly on Œ0; T and for all r 2 N:

From the latter and (20), we can thus prove that fV n .t/gn2N converges weakly in L2 .D/, uniformly with respect to time, i.e. lim hV n .t/  V.t/; wi D 0; uniformly in t 2 Œ0; T; for all w 2 L2 .D/:

n!1

Concerning An and Hn , since they are finite-dimensional functions, starting from (27) and (28), respectively, again by the Ascoli-Arzelà theorem and the Cantor diagonalization argument, we can find A; H 2 CT such that lim j.An .t/  A.t//j D 0;

n!1

lim j.Hn .t/  H.t//j D 0;

n!1

respectively, uniformly with respect to t 2 Œ0; T. From (37) and the weak compactness of L2 .0; TI H.D//, we can conclude that V 2 L2 .0; TI H 1 .D// \ L1 .0; TI H.D//. Finally, by applying Friederichs inequality (see [6]) we conclude that Vn ! V

strongly in L2 .0; TI H.D// :

Summarizing, we have proved the existence of subsequences of fV n gn2N , fAn gn2N and fHn gn2N , and functions .V; A; H/ such that

Time-Periodic Motions of a Liquid-Filled Rigid Body

249

V 2 L2 .0; TI H 1 .D// \ L1 .0; TI H.D//; A; H 2 C.Œ0; T/; V n ! V strongly in L2 .0; TI H.D//; V n ! V weakly in L2 .0; TI H 1 .D//; V n .t/ ! V.t/ weakly in L2 .D/; uniformly in t 2 Œ0; T; An ! A uniformly in t 2 Œ0; T; Hn ! H uniformly in t 2 Œ0; T:

(45)

Since An .0/ D An .T/ and Hn .0/ D Hn .T/, from (45) we can conclude also that A; H 2 CT . Finally, again employing (45) and taking into account the properties of f n gn2N , we can then pass to the limit in (42)–(44) and conclude that .V; A; H/ possesses all the properties of a periodic weak solution to (16)–(18). t u Remark 4.2 For future reference, we wish to observe that the choice of the radius R1 in the proof of the previous theorem is completely arbitrary, provided, of course, R2 is taken appropriately according to (41). This follows from the fact that jAn .t/j is T-periodic regardless of the choice of R1 ; see (39).

5 Existence of Strong Periodic Solutions In this section we will show existence of a strong solution, whenever the magnitude of the torque “sufficiently small”. To this end, we need the following Gronwall-type Lemma. Lemma 5.1 Let y 2 C1 ..a; b// \ C.Œa; b/, a < b, y  0, satisfy the following inequality y0 .t/  c1 y.t/ C c2 y˛ .t/ C c3 ; t 2 .a; b/

(46)

where c1 , c2 , c3 are positive constants and ˛ > 1. There exists ı > 0 such that if Z

t

y.a/ C

y.s/ ds  ı;

for all t 2 Œa; b;

(47)

a

then y.t/
0. Combining the latter with (56) we then conclude Z  D

.V n  grad V n / 

@V n dV  K1 kgrad V n k62 @t     @V n 2   C" kPV n k22 C   @t  ; 2

(57)

where K1 D K1 .D; "/ > 0. Taking into account (52)–(55), the last displayed equation, and (20), from (52) we deduce that

C

D2R1 ;F

 

 @V n 2 d  C  kgrad V n k22  C6  .R1 C F/  3"  @t 2 2 dt 2

C K1 kgrad V n k62 C "kPV n k22 :

(58)

As a result, if we choose " < C=6 and take R1 F (see Remark 4.2), from (58) it follows that there is a constant 1 D 1 .D/ > 0 such that if k f kL1 .0;T/  1 ;

(59)

then    @V n 2  C d kgrad V n k2  C7 C 2K1 kgrad V n k6 C 2"kPVn k2 :  C 2 2 2 @t 2 dt

(60)

In order to “absorb” the term "kPV n k2L2 , let us multiply (25)1 by r cnr and sum over r D 1; : : : ; n. Proceeding in a way completely analogous to that leading to (53)–(55), and (57), we can show that, under an assumption similar to (59), it follows kPV n k2L2

    @V n 2  C kgrad V n k6 ;   C8 C K2  2 @t  2

(61)

Time-Periodic Motions of a Liquid-Filled Rigid Body

253

where K2 D K2 .D; / > 0 and C8 depends also on . Let us multiply both sides of (61) by 4", and then add side by side the resulting inequality and inequality (60). If we take " sufficiently small we thus arrive at     @V n 2  d 2   K3 kPV n k2 C  C kgrad V n k22  C9 C K4 kgrad V n k62  @t L2 dt

(62)

where, Ki D Ki .D; / > 0, i D 3; 4, and C9 depends also on . At this point, we observe the following facts. In the first place, in view of classical estimates for the Stokes problem [6, Theorem IV.6.1], we get k grad V n k2  c kPV n k2 ; with c D c.D/ > 0. In the second place, recalling that R1 k f kL1 .0;T/ and that f satisfies (59), from (41) we deduce that the radius R2 may be chosen to become only a function of D, B, L, and T. As a consequence, the constant C9 in (62) depends only on the same quantities and L . In view of all the above, from (62), we conclude that y WD kgrad V n k22 satisfies the following differential inequality y0  c1 y C c2 y3 C c3 where the constants ci , i D 1; 2, depend, at most, on D, B, L, and T. Our next objective is to show that y D y.t/ obeys the hypothesis (47) of Lemma 5.1 with a suitable choice of a and b, provided f satisfies a restriction of the type (49). To this end, we observe that integrating both sides of (34) between 0 and T and using the T-periodicity of V n , along with (33) we show Z

T 0

  kgrad V n .t/k2L2 dt  k1 jAn;0 j4 C k f k4L1 .0;T/ C kf k2L2 .0;T/    k2 R41 C kf k4L1 .0;T/ C kf k2L1 .0;T/    k3 kf k4L1 .0;T/ C kf k2L1 .0;T/ ;

(63)

where ki D ki .D; B; L; T/ > 0, i D 1; 2; 3, and in the last step we used the fact that R1 D k f kL1 .0;T/ . From (63), the integral mean-value theorem, and the T-periodicity of V n we deduce kgrad V n .Nt/k2L2

Z C Nt

NtCT

  kgrad V n .t/k2L2 dt  2k3 kf k4L1 .0;T/ C kf k2L1 .0;T/ ; (64)

254

G.P. Galdi et al.

for some Nt 2 .0; T/. From (64), Lemma 5.1 and again the T-periodicity of V n we then derive that there is 2 D 2 .D; B; L; T/ > 0 such that if k f kL1 .0;T/  2

(65)

it follows k grad V n .t/k2  2

c3 for all t 2 Œ0; T. c1

(66)

Moreover, integrating (62) over a period, and taking into account (66) we also conclude Z

T 0

kPV n .t/k22 dt C

Z

T 0

   @V n .t/ 2    @t  dt  k4 : 2

(67)

where k4 D 4 .D; B; L; T/ > 0. Therefore, we conclude that setting  D minf1 ; 2 g, 1 , 2 defined in (59) and (65), respectively, under the hypothesis (49) the approximating T-periodic solutions .V n ; An ; Hn / constructed in the proof of Theorem 4.1 satisfy, in addition, the uniform bounds (50), (51), (66) and (67). As a consequence, the limiting fields .V; A; H/ defined through (45) satisfy all the properties stated in the theorem,3 which is thus completely proved. t u Acknowledgements This work is partially supported by NSF grant DMS-1311983.

References 1. R.A. Adams, J.J. Fournier, Sobolev Spaces. Pure and Applied Mathematics, 2nd edn. (Elsevier/Academic, Amsterdam, 2003) 2. V.I. Arnol0 d, Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, 2nd edn. (Springer, New York, 1989) 3. G.E. Bredon, Topology and Geometry (Springer, New York, 1993) 4. F.L. Chernousko, Motion of a Rigid Body with Cavities Containing a Viscous Fluid. (NASA Technical Translations, Moscow, 1972) 5. G.P. Galdi, An Introduction to the Navier-Stokes Initial-Boundary Value Problem. Fundamental Directions in Mathematical Fluid Mechanics (Birkhäuser, Basel, 2000), pp. 1–70 6. G.P. Galdi, An Introduction to the Mathematical Theory of Navier-Stokes Equations: SteadyState Problems. Springer Monographs in Mathematics, 2nd edn. (Springer, New York, 2011) 7. G.P. Galdi, A.L. Silvestre, Existence of time-periodic solutions to the Navier-Stokes equations around a moving body. Pac. J. Math. 223, 251–267 (2006) 8. G.P. Galdi, G. Mazzone, P. Zunino, Inertial motions of a rigid body with a cavity filled with a viscous liquid. C. R. Méc. 341, 760–765 (2013)

3 The stated continuity property of grad V follows from classical interpolation results; see, e.g., [13, Théorème 2.1].

Time-Periodic Motions of a Liquid-Filled Rigid Body

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9. J.G. Heywood, The Navier-Stokes equations: on the existence, regularity and decay of solutions. Indiana Univ. Math. J. 29, 639–681 (1980) 10. B.G. Karpov, Dynamics of Liquid-Filled Shell: Resonance and Effect of Viscosity. Ballistic Research Laboratories, Report no. 1279 (1965) 11. N.D. Kopachevsky, S.G. Krein, Operator Approach to Linear Problems of Hydrodynamics, vol. 2: Nonself-Adjoint Problems for Viscous Fluids (Birkhäuser Verlag, Basel/Boston/Berlin, 2000) 12. O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, revised 2nd edn. (Gordon and Breach Science Publisher, New York, 1969) 13. J.L. Lions, Espaces intermédiaires entre espaces Hilbertiens et applications. Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine 2, 419–432 (1958) 14. N.N. Moiseyev, V.V. Rumiantsev, Dynamic Stability of Bodies Containing Fluids (Springer, Berlin, 1968) 15. G. Prouse, Soluzioni periodiche dell’Equazione di Navier-Stokes. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 8, 443–447 (1963) 16. P.H. Roberts, K. Stewartson, On the motion of a liquid in a spheroidal cavity of a precessing rigid body II. Proc. Camb. Philos. Soc. 61, 279–288 (1965) 17. W.E. Scott, The Free Flight Stability of Liquid-Filled Shell, Part 1a. Ballistic Research Laboratories, Report no. 120 (1960) 18. K. Stewartson, P.H. Roberts, On the motion of a liquid in a spheroidal cavity of a precessing rigid body. J. Fluid Mech. 17, 1–20 (1963) 19. F.S. Van Vleck, A note on the relation between periodic and orthogonal fundamental solutions of linear systems II. Am. Math. Mon. 71, 774–776 (1964)

Seeking a Proof of Xie’s Inequality: On the Conjecture That m ! 1 John G. Heywood

Dedicated to Yoshihiro Shabata on his 60th birthday

Abstract I pursue an argument of Wenzheng Xie, as furthered in several of my papers, to prove a particular point-wise bound for solutions of the threedimensional steady Stokes problem. If proven, it will provide the basis for an existence and regularity theory for the non-stationary Navier-Stokes equations, free of assumptions about the regularity of the boundary of the flow region. It will be valid for flow in an arbitrary open set. In his doctoral thesis, Xie proved an analogous bound for solutions of the Poisson problem for the Laplacian, considering it as a model problem. His proof carries over to the Stokes problem except at one point where the maximum principle is invoked. Subsequently, I’ve proposed a variant of Xie’s argument that circumvents the maximum principle, but requires instead a proof that a certain sequence of functions introduced in Xie’s argument tends to become singular. I’ve expressed this as a further conjecture, which is studied here for both the Stokes problem and for the Poisson problem, the latter being considered as a model problem. Keywords Estimates for the stokes and poisson problems • Navier-stokes equations • Non-smooth boundaries

1 Introduction ˚  Let  be an arbitrary open subset of R3 : Let D ./ D ' 2 C1 0 ./ W r  ' D 0 : Let J ./ and J0 ./ be the completions of D ./ in the L2 -norm kk and the Dirichlet norm krk ; respectively. Then, given u 2 J0 ./ ; there is at most one f 2 J ./ such that .ru; r'/ D  .f; '/ ; for all ' 2 D ./ : If such a function Q Q referred to as the f exists, it is denoted by 4u; thus defining an operator 4; “Stokes operator”, from a subspace of J0 ./ to J ./ : Under these assumptions

J.G. Heywood () Department of Mathematics, University of British Columbia, Vancouver, BC, Canada e-mail: [email protected] © Springer Basel 2016 H. Amann et al. (eds.), Recent Developments of Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-0348-0939-9_14

257

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it is conjectured that sup juj2  

  1 Q  kruk 4u : 3

(1)

The constant has been proven to be optimal, by Wenzheng Xie in [6]. It cannot be improved, even for particular choices of the domain. In attempting to prove (1), Xie [4, 5] succeeded in proving sup juj2  

1 kruk k4uk 2

(2)

O 1 ./ of C1 ./ in the Dirichlet for functions u that belong to the completion H 0 0 norm krk ; and have a square-integrable Laplacian 4u; defined here as a function f 2 L2 ./ such that .ru; r'/ D  .f ; '/ ; for all ' 2 C01 ./ : Again, the domain  is taken to be an arbitrary open subset of R3 ; and the constant is optimal. The inequality is valid for either scalar or vector-valued functions, the vector-valued case being a simple corollary of the scalar case. In our paper [3] with Xie, we have based an existence and regularity theory for the vector Burgers equation on the inequality (2). It is valid for solutions defined in arbitrary three-dimensional open sets. Once the inequality (1) is proven, this theory for the Burgers equation will serve as a model for a similar existence and regularity theory for the Navier-Stokes equations. In [6], Xie showed that the argument he used to prove (2) in [4, 5] carries over to a proof of (1), except for his use of the maximum principle in proving that for every point y in a smoothly bounded domain ;     G .; y/ 2  g .; y/ 2 3 ; ./ L L .R /

(3)

where G is the Green’s function for the Helmholtz operator  4 C; and g is the corresponding fundamental singularity. Although he conjectured an analogous Q inequality for the spectral Stokes operator 4C; it has yet to be proven. However, we have proven in [1], even for the spectral Stokes operator, that the ratio of the two sides of (3) tends to 1 as  ! 1: This will suffice in place of (3) for the proof of either (1) or (2), provided the relevant values of  tend to infinity. Xie’s argument in [4, 5] involves a sequence fum g of finite sums of eigenfunctions. And his argument in [6] for the Stokes operator involves an analogous sequence fum g, defined here in Lemma 2.1 below. The values m of  that need to be considered provide a measure of the regularity of the functions um ; which must be shown to become increasingly singular. Specifically, what remains to be shown in this approach to the problem is that    Q 2 2 m 4u m  = krum k ! 1;

as m ! 1:

(4)

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259

In this paper we take several new steps towards a proof of (4), furthering our confidence that it will ultimately be proven, and along with it (1). In Sect. 2, we will review the main steps already taken in attempting to generalize the argument for (2), given in [4] and [5], to a proof of (1). These begin with the results of Xie’s paper [6], which provide a complete proof of (1) modulo Xie’s conjecture that an analogue of (3) holds for the Stokes equations. Then we state our result from [1], which provides a means of circumventing Xie’s conjecture if our conjecture (4), that m ! 1; can be proven. In Sect. 3, we seek information about m by showing that it is a root of an equation fm .; y; e/ D 0; numbered as (21) below. For a fixed point y and unit vector e; the function fm .; y; e/ is an expression involving the first m eigenvalues of the Stokes operator along with the projections e  'n .y/ of the associated eigenfunctions 'n : The functions fm .; y; e/ are shown to converge to a limit f1 .; y; e/ as m ! 1: If it can be shown that f1 .; y; e/ < 0 for all   0; then the desired conclusion (4) follows easily. Thus we are led to a further conjecture, that f1 .; y; e/ < 0 for all   0: These results along with this new conjecture are analogous to ones we have given in [2] for the Poisson problem. The conjectures, for both the Stokes and Poisson problems, are formulated below as (27) and (28), respectively. Even though (2) has been proven using (3), the conjecture (28), like (27), remains unproven, and seems an appropriate model problem to consider in trying to prove (27). The remainder of this paper concerns our attempt to prove (28). The conjecture (28) concerns a function f1 .; y/ that is expressed in terms of the eigenvalues and eigenfunctions of the Laplacian. In Sect. 4 we show that f1 .; y/ can also be expressed in terms of certain integrals of Green’s functions, and an identity is proven for one of these integrals that greatly simplifies its evaluation. In Sect. 5, we begin by using the results of Sect. 4 to prove that the inequality (28) holds if  is a ball and y is its center. This is shown by a precise calculation, resulting in a remarkable identity, (46) below. This result is then extended to centrally located open subregions of points y within nearly spherical domains ; using the maximum principle. It follows for such domains and points, that m ! 1 for the sequence of functions considered in Xie’s proof of (2). We have every expectation that these results are generic, for all smoothly bounded domains, and all points within them, and for the Stokes operator as well as for the Laplacian. If confirmed, the conjecture (4) and the inequality (1) will be proven, for all domains. [4, 5], Xie showed that if  D R3 ; then the functions u .x/ D  In ˛jxj =˛ jxj satisfy sup juj2 D .1=2/ kruk k4uk for every ˛ > 0: 1e As ˛ ! 1; the functions u become increasingly singular at the origin, in that  k4uk2 = kruk2 ! 1; while becoming very small and nearly constant away from the origin. Thus, if  is a domain with boundary, and the origin is interior to ; the functions u can be truncated near the boundary with arbitrarily small changes in ; provided ˛ is large enough. This was the argument Xie used to prove that the constant 1=2 in (2) cannot be improved, even for particular choices of : Xie also showed, in [7], that this ˛-dependent family of functions u is unique (up to multiples or shifts in position) in possessing the property of equalizing the two sides of (2). That suggests that any function that nearly equalizes the two sides of (2) must be close to one of these ˛-dependent functions. If the domain has a boundary, that will

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only be possible for large ˛ (to minimize the effects of truncation) and thus large : It is this reasoning that led us to conjecture (4), and hope for a proof by contradiction. Of course, the existence of this ˛-dependent family of functions also shows that one must expect the proof of (4) to be quite subtle. Indeed, if  D R3 ; it provides functions that equalize the two sides of (2) for which one has, not  ! 1; but even  ! 0; by letting ˛ ! 0: Everything said in this paragraph carries over to the Stokes equations, using results provided by Xie in [6] and [8]. In particular, in [8], an ˛-dependent family of functions u that equalizes the two sides of (1) was found and proven to be unique, up to shifts in position, magnitude, and a choice of direction.

2 Background Results from [6] and [1] We begin by reviewing the results of Xie’s paper [6]. He began by making two reductions of the problem. First, it was shown that if the inequality (1) holds for all smoothly bounded domains, with a fixed constant independent of the domain, then it holds for all domains (meaning for arbitrary open sets). This was proved by considering an expanding sequence of smoothly bounded domains. The second reduction concerns eigenfunction expansions in smoothly bounded domains. The orthonormal eigenfunctions f'n g and associated eigenvalues fn g of the Stokes operator satisfy Q n D n 'n ; 4'

'n j@ D 0 ;

(5)

and linear combinations of the eigenfunctions are dense in J ./ and J0 ./ : Xie showed that if (1) holds for every finite linear combination um D

m X

cn ' n

nD1

Q 2 J ./ : The proofs of the eigenfunctions, then it holds for all u 2 J0 ./ with 4u of these reductions are elegant, but irrelevant to what follows. Now let e denote an arbitrary three-dimensional unit vector. Clearly sup juj2 D sup sup .e  u .y//2 : 

y2

(6)

e

In attempting to prove (1), Xie went on to consider the ratio of its left to right sides, beginning with the following lemma. P Lemma 2.1 For any fixed choice of y; e and m; let um D m nD1 cn 'n maximize R .y; e; mI c1 ;   ; cm /

.e  um .y//2 :  Q  krum k 4u m

(7)

Xie’s Inequality

261

among all functions of the form um D

Pm

nD1 cn 'n

Then

 m p X .e  um .y//2 e  'n .y/ 2  D 4 m  Q  n C m krum k 4u nD1 m

(8)

where

m D

   Q 2 4um  krum k2

:

(9)

To prove this, note first that R is smooth and homogeneous as a function of the variables .c1 ;   ; cm / 2 R3 n f0g : Being homogeneous, it is constant along rays from the origin, and thus assumes all of its values on any sphere about the origin. Therefore it has a maximum and attains it at a point .c1 ;   ; cm / where R is smooth. At such a point @R=@cn D 0; for n D 1;   m: With a bit of work these equations can be reassembled to obtain (8). In the next step, the terms summed on the right of (8) were identified as the squares of the Fourier coefficients of a Green’s function. It was proven that: Lemma 2.2 Let G .x; yI ; e/ be the Green’s function for the spectral Stokes problem .4  / G C rP D ı .x  y/ e;

r  G D 0;

Gj@ D 0:

(10)

Then Z 

jG .x; yI ; e/j2 dx D

 1 X e  'n .y/ 2 nD1

n C 

:

(11)

Next, the rather complicated fundamental singularity g .x; yI ; e/ for the problem (10) was found explicitly and its square was integrated. Lemma 2.3 The fundamental singularity g .x; yI ; e/ for the problem (10) satisfies Z R3

jg .x; yI ; e/j2 dx D

1 p : 12 

(12)

At this point in the corresponding argument for the Poisson problem, Xie proved (3) using the maximum principle. Along with analogues of the preceding lemmas, that completed the proof of (2). To circumvent this use of the maximum principle, which is not available for the Stokes equations, we proved the following lemma in [1].

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Lemma 2.4 There holds R jG .x; yI ; e/j2 dx ! 1; R 2 R3 jg .x; yI ; e/j dx

as  ! 1:

(13)

We mention that the analogue of this for the scalar Helmholtz operator  4 C is easily proven, because the fundamental singularity decays exponentially. But in the case of the Stokes equations, g .x; yI ; e/ has terms that decay only like 1=r3 ; considerably complicating the proof of (13). Combining Lemmas 2.1–2.4, we have  m p X .e  um .y//2 e  'n .y/ 2  D 4 m  Q  n C  m krum k 4u nD1 m p Z  4 m jG .x; yI m ; e/j2 dx 

1 ; ! 3

(14)

provided m ! 1 as m ! 1; which is needed to apply Lemma 2.4 in the last step. Since the left side of ( 14) increases with m; (14) implies the desired inequality (1). Thus it only remains to show that m ! 1; as m ! 1.

3 An Equation fm .; y; e/ D 0 Satisfied by  D m Due to the use of (3) rather than (13) in Xie’s proof of (2), the values of m became irrelevant, and were even eliminated by cancellation. In trying to get some information about them, we have discovered an equation that they satisfy. We will now provide the full details of this for the Stokes operator, paralleling an analogous argument we have given for the Laplacian in [2]. To determine m ; we reconsider the maximization of

.e  um .y//2   Q  m P krum k 4u m nD1

2

m P

cn e  'n .y/

nD1

n c2n

1=2

m P

2n c2n

1=2 ;

(15)

nD1

for fixed y; e and m; using the method of Lagrange multipliers. Since the function to be maximized is a homogeneous function of the variables c1 ;   ; cm ; it is constant on lines through the origin in the space of these variables. Thus, corresponding to any maximizing point, there will be another, on the same line through the

Xie’s Inequality

263

origin, satisfying e  um .y/ D 1: Thus, we may revisit Lemma 2.1 above and make this an additional condition to be satisfied by um : Henceforth we will assume that Lemma 2.1 has been so modified. Hence the maximizing choice of um ; with coefficients .c1 ;   ; cm / ; must minimize the denominator, or the square of the denominator, ! m ! m X X 2 2 2 f .c1 ;   ; cm /

(16) n c n n c n nD1

nD1

subject to the constraint g .c1 ;   ; cm /

m X

cn e  'n .y/ D 1:

(17)

nD1

Therefore there exists a Lagrange multiplier  such that " cn n

m X

2k c2k

C

2n

kD1

m X

# k c2k

D e  'n .y/ ;

for n D 1;   ; m :

(18)

kD1

Multiplying this by cn and summing over n; from 1 to m; one obtains m X

2f .c1 ;   ; cm / D

cn e  'n .y/ D  :

nD1

To eliminate ; multiply this by e  'n .y/ ; getting 2e  'n .y/ f .c1 ;   ; cm / D e  'n .y/ ; and substitute for the right side the left side of (18) to obtain " 2e  'n .y/ f .c1 ;   ; cm / D cn n

m X

2k c2k

C

2n

kD1

m X

# k c2k

:

kD1

Solving this for cn gives

2e  'n .y/ cn D n

m P kD1

m P

k c2k



kD1

2k c2k

m P

kD1

C 2n

m P

kD1

2k c2k

k c2k

 :

(19)

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J.G. Heywood

Letting Am D krum k2

   Q 2 and Bm D 4u m ;

(19) can be rewritten as cn D

2e  'n .y/ Am Bm : Bm n C Am 2n

(20)

Lemma 3.1 The ratio m D Bm =Am is a root of the equation fm .; y; e/

 m X .e  'n .y//2   1 D 0: . C n /2 n nD1

(21)

Proof Square (20), multiply by n and sum to get m P .e  'n .y//2 n  2 nD1 Bm n C Am 2n

 m .e  ' .y//2 P m n : D 4Am Bm 2 n nD1 .m C n /

Am D 4A2m B2m

(22)

Similarly, square (20), multiply by 2n and sum to get m P .e  'n .y//2 2n  2 nD1 Bm n C Am 2n m .e  ' .y//2 P n D 4B2m : 2 nD1 .m C n /

Bm D 4A2m B2m

(23)

Multiplying (23) by Am =Bm gives Am D 4Am Bm

m X .e  'n .y//2 nD1

.m C n /2

:

(24)

Subtracting (24) from (22) and dividing by 4Am Bm gives the desired result,  m X .e  'n .y//2 m  1 D 0: fm .m ; y; e/

.m C n /2 n nD1

(25)

The next lemma follows immediately from the definition of fm .; y; e/ : Lemma 3.2 Each function fm .; y; e/ is negative on the interval 0   < 1 and positive on the interval m <  < 1: Hence 1  m  m :

Xie’s Inequality

265

The next lemma follows immediately from the convergence of the series on the right of (11). Lemma 3.3 The infinite series f1 .; y; e/

 1 X .e  'n .y//2   1 . C n /2 n nD1

(26)

is uniformly and absolutely convergent for  in any bounded subinterval of Œ0; 1/ : From these lemmas we reach the following conclusion. Proposition 3.1 Consider a fixed point y of ; and a fixed unit vector e: Suppose   Q  1 f1 .; y; e/ < 0 for all   0: Then the inequality je  u .y/j2  3 kruk 4u  Q This holds for every u 2 J0 ./ that has a square-integrable Stokes operator 4u: implies (1) if the hypothesis holds for every y 2 ; and every unit vector e: Proof Since the sequence of functions fm .; y; e/ is uniformly convergent to the negative function f1 .; y; e/ on every bounded subinterval of Œ0; 1/ ; the roots of fm .; y; e/ are pushed out further and further to the right as m increases. Thus m ! 1 as m ! 1; and this is all that remained to be shown at the end of the previous section. The final conclusion (1) is reached remembering (6). It follows from the preceding results that if the following conjecture is proven, then the inequality (1) will be proven. Conjecture 1 For every smoothly bounded three-dimensional domain ; and for every y 2 ; and every unit vector e; we suspect that f1 .; y; e/

 1 X .e  'n .y//2   1 < 0; . C n /2 n nD1

for all  2 Œ0; 1/ :

(27)

In trying to prove (27), we have looked again to the corresponding problem for the Laplacian. Let fn g be the L2 -orthonormal eigenfunctions of the Laplacian 4; satisfying  4 n D n n ; n j@ D 0: We will attempt to prove the following conjecture as a model problem. If proven, it will provide a proof of (2) that is independent of (3). Conjecture 2 For every smoothly bounded three-dimensional domain ; and for every y 2 ; we suspect that f1 .; y/

1 X

n2 .y/

nD1

. C n /2



  1 < 0; n

for all  2 Œ0; 1/ :

(28)

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J.G. Heywood

4 Green’s Function Formulation of f1 .; y/ < 0 The inequality (28) can be expressed in terms of Green’s functions. For a smoothly bounded domain   R3 ; and arbitrary point of singularity y 2 ; the Green’s functions for the Laplacian 4 and for the Helmholtz operator  4 C are 1 C w0 .x; y/ G0 .x; y/ D 4r

p

e r C w .x; y/ and G .x; y/ D 4r

(29)

respectively, where r jx  yj and the functions w0 .x; y/ and w .x; y/ satisfy 8 < 4w0 .; y/ D 0 : w0 .x; y/jx2@ D 

1 4r

8 < 4w .; y/ D w .; y/ p ˇ e r : : w .x; y/ˇ D x2@ 4r

and

(30)

Lemma 4.1 For all y 2  and  2 Œ0; 1/ ; Z f1 .; y/ D



 2 G .x; y/ G0 .x; y/ dx  2 G .; y/ :

(31)

Proof Since n satisfies . 4 C/ n D .n C / n ;

n j@ D 0;

it can be represented using the Green’s function for the Helmholtz operator as Z n .y/ D



G .x; y/ .n C / n .x/ dx:

It follows that the nth Fourier coefficient of G .; y/ is n .y/ D  C n

Z 

G .x; y/ n .x/ dx:

Hence, simply rearranging (28), we have f1 .; y/ D

1 P

n2 .y/



1 P

n2 .y/

. C n /2 nD1 . C  n /2 1 P n2 .y/ n2 .y/ n2 .y/   D 2 2 . C n / nD1 n . C n / nD1 . C n / 2 2 1 1 P P n .y/ n .y/ 2 D 2  . C n / nD1 . C  n / 2 RnD1 n  D  G .x; y/ G0 .x; y/ dx  2 G .; y/ : n nD1  1 P

(32)

Xie’s Inequality

267

Lemma 4.2 For all x; y 2 ; there holds Z 



G .z; y/ G0 .z; x/ dz D G0 .x; y/  G .x; y/ :

(33)

Proof This identity is a representation of G0 .x; y/  G .x; y/ ; considered as a function of x with y fixed, in terms of the Green’s function G0 for the Laplacian. For fixed y and x ¤ y; it follows from (29) and (30) that the function G0 .x; y/G .x; y/ satisfies    4 G0 .x; y/  G .x; y/ D G .x; y/

(34)

and vanishes on @: Thus we need only check that it is sufficiently regular to justify the representation (33). In fact, the singularity of G0  G at x D y is not very badp since the difference between the fundamental singularities, u .r/ 1=4r  e r =4r; satisfies lim u .r/ D

r!0

p

=4;

(35)

 3 2 R ; since it can be approxiand is therefore continuous. Furthermore, u 2 Hloc mated by a sequence of functions un .r/

8
0 and M > 0 such that  M; kR.; A/kL.L1

.//

 2 †=2C" ;

(3)

where A denotes the generator of the Stokes semigroup on L1

./ defined above. Here R.; A/ D .  A/1 for all  2 %.A/ and %.A/ denoting the resolvent set of A. As already mentioned above, the above estimates (3) generalizes to a certain extend the corresponding estimates for p 2 .1; 1/ due to Borchers and Sohr [5] to the situation of L1

./. Let us now turn our attention to uniform L1 -estimates for rT.t/ or r 2 T.t/ with respect to t > 0. Our second main result then reads as follows. Theorem 1.3 Let T be the Stokes semigroup on L1

./. Then there exists a constant M > 0 such that t1=2 1 2

t C1

C krT.t/kL.L1

.//

t kr 2 T.t/kL.L1  M; t > 0:

.// tC1

(4)

Remark 1.4 Let us point out that the above estimate (4) is sharp in the following sense: it is impossible to obtain estimates of the form t˛ krT.t/kL.L1 .///  M; t > 0

or

(5)

tˇ kr 2 T.t/kL.L1 .//  M; t > 0;

(6)

for some ˛; ˇ > 0. In order to see this, we follow the arguments employed in [14]. Note first that following e.g. [12], the solution u.t/ D T.t/u0 of the Stokes problem (1) satisfies n

kru.t/kLq ./  Mt 2p ku0 kLp ./ ; q  p  n;

t > 0:

(7)

Moreover, in [12], it is proved that estimate (7) is sharp, in the sense that for all " > 0 and M > 0, there exists a u0 and t > 1 such that n

kru.t/kLq ./ > Mt 2p " ku0 kLp ./ :

(8)

278

M. Hieber and P. Maremonti

If (5) would be true for some for some ˛ > 0 and all u0 2 fe u0 2 C./ and je u0 .x/j ! 0 for jxj ! 1g  L1

./; then, for all r 2 Œn; 1/; by the semigroup property of T n

kru.t/kL1 ./  Ct˛ ku. 2t /kL1 ./  Ct˛ 2r ku0 kr ; t > 0; for all u0 2 Lr ./: An application of the Riesz-Thorin theorem yields kru.t/kLq .˝/  Mt˛  2p ; n

1rp r pn 1 D ; D ; t > 0; q prn p rn

(9)

n

for all p 2 Œn; r. This contradicts the fact that t 2p is the sharp decay rate for t > 1 for krT.t/kL.Lp ./;Lq .//. Similarly, also the estimate for kr 2 TkL.L1 .// is sharp. Indeed, it follows e.g. from Lemma 8.1 of [12] that kr 2 u.t/kL n2 ./  Mt1 ku0 k

n

L 2 ./

;

t > 0:

(10)

Again, if (6) would be true for some ˇ > 0, then, the same arguments give n

kr 2 u.t/kLq ./  Mtˇ  2p ku0 kLp ./ ;

1 q

D

2 .rp/ ; p 2rn

D

r 2pn ; p 2rn

t > 0;

(11)

for all p 2 Πn2 ; r. By interpolation we would obtain 1

1

ˇ

kru.t/kq  kr 2 u.t/kq2 ku.t/kq2  Mt 4q  2p  2 ku0 kLp ./ ; n

n

t > 0;

which for p 2 . nr.1Cˇ/ nC2ˇr ; r/, however, contradicts the optimal decay rate of ru.t/ given in (7). The plan of this paper is as follows. We begin in Sect. 2 by collecting existence, uniqueness and regularity results for the Stokes equation. Subsequentially, in Sect. 3, we present the proofs of our main results, Theorems 1.1 and 1.3. For further information regarding maximum modulus estimates for bounded domains and L1 estimates for the Stokes semigroup in the half space RnC we refer to [6] and [8, 16].

2 Results on the Stokes Equations: Existence, Uniqueness, Regularity We start this section with the following result due to Abe and Giga, saying in particular that, for u0 2 L1

./ the solution of the Stokes equation (1) is governed by an analytic semigroup on L1

./.

Bounded Analyticity of the Stokes Semigroup

279

Theorem 2.1 ([2, Theorem 1.5]) Let u0 2 L1

./. Then the Stokes equation (1) admits a unique solution .u; rq/ 2 C2;1 ..0; T/C..0; T// for some T > 0 and there exists a constant C > 0, independent of u0 , such that sup kN.u; q/k1 .t/  Cku0 k1 ;

0tT

where N.u; q/.x; t/ D ju.t; x/j C tjut .t; x/j C t1=2 jru.t; x/j C tjr 2 u.t; x/j C tjrq.t; x/j; for x 2  and t 2 .0; T. For the rest of this chapter, we assume that the initial data u0 is smooth in the following sense. More precisely, set C./ WD fu 2 C./ \ L1

./ W u D 0 on @g; and C./ WD C./ \ C./: We then assume in the following that u0 2 C./ \ C1 ./. In order to prove Theorem 1.1 we decompose .u; q/ as u D z C v;

qu D z C v ;

where z is the solution of the following Stokes problem on Rn 8 < @t z  z D 0 in J  Rn ; div z D 0 in J  Rn ; : z.0/ D uQ0 in Rn ;

(12)

and where uQ0 is defined as the trivial extension by zero of u0 to Rn . Moreover, .v; v / is defined as the solution of the problem 8 ˆ ˆ @t v  v C rv D < div v D ˆ vD ˆ : v.0/ D

0 0  z 0

in J  ; in J  ; on J  @; in ;

(13)

where z denotes the trace of z, where z is the solution of (12). We look for a solution .v; v / of the form v D w C F;

v D w C P;

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where the pair .w; w / solves the equation 8 @t w  w C rw D ˆ ˆ < div w D ˆ wD ˆ : w.0/ D

Ft C G 0 0 0

in J  ; in J  ; on J  @; in ;

(14)

and where F is a suitable extension of  z to J   defined precisely as well as G in the following lemma. Lemma 2.2 Let z be the unique solution of (12),  z.t/ be its trace on @ for all t 2 J. Let p 2 .1; 1/ and k 2 N. Then there exists an extension F 2 C./ \ Ck ./ of  z on .0; T/   having compact support in  satisfying div F D 0 for all t > 0, as well as the following properties: (i) (ii) (iii) (iv) (v) (vi)

kDkt F.t/k1  Ctk ku0 k1 ; t > 0; k  0, 1 kDkt F.t/k1  CtkC 2 kru0 k1 ; t > 0; k  0, F D rP C G on J   for some G 2 C2 ./ having compact support in , limt!0 kF.t/k1 D limt!0 tkFt .t/k1 D 0, kDkt G.t/k1  Ctk ; t > 0; k  0, 1 tkrFt .t/kp C tkD3t F.t/kp C krG.t/kp  Ct 2 ku0 k1 ; t > 0;

for some C > 0 independent of u0 . For a proof of the above properties we refer to [14, Corollary 1 and 2]. We now consider regularity properties of the inhomogeneous Stokes equation (14). To this end, for  > 0, q 2 Œ1; 1 and .t; x; L; u0 / 2 RC  Rn  RC  L1 .Rn / we set 1

Mq .t; x; L; u0 / WD ku0 kLq .B.x;L// t 2q C ku0 k1 t=2 .L C t 2 / : n

Then, by Maremonti [14, Theorem 5.4], the following result concerning the inhomogenous equation (14) holds true. Proposition 2.3 Let F and G as above, s 2 .1; 2/ and p 2 .1; 1/. Given T > 0, p there exist a unique w 2 C.Œ0; TI L .// and a unique rw 2 Ls .0; TI Lp .// with n 0 s p w 2 L .0; TI L .// solving Eq. (14). Moreover, for p 2 . n2 ; 1/ and q 2 . n2 ; 1 there exists a constant C > 0 such that 2 X j˛j t t t kw0 .t/kp C / 2 kD˛ w.t/kp C krw .t/kp . tC1 tC1 tC1 j˛jD0

 C sup Mq .t; ; L; u0 /; t > 0; L > 0: 2@

Moreover, tw0 ; w 2 C.Œ0; T/I Lp .// and limt!0 tkw0 .t/kp D limt!0 kw.t/kp D 0.

Bounded Analyticity of the Stokes Semigroup

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We remark that Theorem 5.4. of [14] does not state explicitly the Lp -continuity of tw0 and the related limit property; however both follow directly from the step 2 in the proof of Theorem 5.4 of [14]. Next, for T > 0 we set e J D . 2t ; 1/ for t  T. Consider then the initial boundary value problem 8 @t e w  e w C r e ˆ wD ˆ < div e wD ˆ e wD ˆ : t e w. 2 / D

Ftt C Gt 0 0 0

in e J  ; in e J  ; on e J  @; in ;

(15)

where F and G are the same as in (14). Then the following holds. Lemma 2.4 Assume that F and G are given as in Lemma 2.2 and let r; p 2 .1; 1/. p Then, given T > 0, there exist a unique e w 2 C.e JI L .// and a unique r e w 2 n r e p 0 r e p L .JI L .// with e w 2 L .JI L .// solving Eq. (15). Moreover, for p 2 . n2 ; 1/ there exists C > 0 such that 2 X j˛j s s s ke w0 .s/kp C / 2 kD˛ e kr e . w.s/kp C w .s/kp sC1 sC1 sC1 j˛jD0

 Cs1 ku0 k1 ;

s > t=2:

We only sketch a proof following the lines of [14]. In fact, taking into account Lemma 2.2 (i) and (v) and following the arguments given in the third step of the proof of Theorem 5.4 in [14], we obtain the estimate ke w.s/kp C ske ws .s/kp  Cs1 ku0 k1 ; s >

t ; 2

(16)

for some C independent of u0 . The Lp -theory of the steady Stokes problem yields kD2e w.s/kp C kr e ws .s/kp C ke w.s/kp /; w kp  C.ke

s>

t : 2

(17)

and as a consequence of estimate (16) 2 kD2e w.s/kp C kr e C s1 /ku0 k1 ; w kp  C.s

s>

t : 2

(18)

Combining the Gagliardo-Nirenberg inequality with (16)–(18) we obtain 1

1

3

kre w.s/kp  CkD2e w.s/kp2 ke w.s/kp2  C.s 2 C s1 /ku0 k1 ; s >

t ; 2

(19)

which finally proves the lemma. Next, we state a result which was proved in [14], Theorem 6.1. To this end, for % > d WD diam Rn n set % WD  \ B.0; %/.

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Proposition 2.5 Let u0 2 C./ \ C1 ./ and let .u; u / be the solution to equation n (1). Then, for p 2 . n2 ; 1/; q 2 . n2 ; 1 and % > d there exists a constant C > 0 such that 2 X j˛j t t kut .t/kLp .% / C / 2 kD˛ u.t/kLp .% /  . tC1 tC1 j˛jD0

C sup Mq .t; y; L; u0 /;

t > 0; L > 0:

(20)

y2B.0;%/

Moreover, limt!0 ku.t/  u0 k1 D 0 and for  2 .0; 12 / there exists a constant C > 0 such that .

t 1 / 2 C ju .t; x/j tC1

 C sup Mq .t; y; L; u0 /.1 C jxj/2n ; .t; x/ 2 .0; T/  ; L > 0:

(21)

y2B.0;%//

The constant C is independent of t; L and u0 . In addition, 

.u; u / 2 C2; ./  C1; ./ and ut ; r 2 u 2 C0; 2 ..0; T/  /: We finally recall from Maremonti [13] Theorem 3.2 a results concerning the Stokes initial boundary value problem 8 @t u  u C ru D 0 in J  ; ˆ ˆ < div u D 0 in J  ; (22) ˆ uD 0 on J  @; ˆ : .u.0/; '.0// D . 0 ; '/ ' 2 C0 ./ for 0 2 C01 ./. Denote by J 1;p ./ the completion of C0 ./ with respect to the W 1;p -norm. Proposition 2.6 Let 0 2 C01 ./,  > 0 and q 2 .1; 1/. Then Eq. (22) admits a unique solution . ;  / satisfying 2 C.Œ0; T/I Lq .// \ Lq .; TI W 2;q ./ \ J 1;q .// and r ;

t

2 Lq .; TI Lq .//:

Moreover, for q 2 .1; 1 there exists a constant C > 0, independent of u0 , such that 1 n .1  /; 2 q

k .t/kq  Ck

0 k1 t

1

; t > 0;

1 D

k t .t/kq  Ck

0 k1 t

2

; t > 0;

2 D 1 C 1

In addition, limt!0 . .t/; '/ D .

0 ; '/

for any ' 2 C0 ./.

Bounded Analyticity of the Stokes Semigroup

283

For the duality arguments in the following chapter we also use the following lemma which can be found in [14]. p0

p

Lemma 2.7 Let be v 2 C.Œ0; T/I L .//, 2 C.Œ0; T/I L .// and Assume that limt!0 . .t/; '/ D . 0 ; '/ for all ' 2 C./. Then, lim .v.t  "/; ."// D .v.t/;

"!0

0 /;

0

2 C0 ./.

t 2 .0; T/:

3 Proof of the Main Results We subdivide the proof of Theorem 1.1 into three steps. Step 1: The case of smooth initial data, i.e. u0 2 C./ \ C1 ./. Taking into account Theorem 2.1 we only have to prove the estimate (2) for arbitrary t > T. To this end, note first that the solution z of the Stokes problem on Rn satisfies kz0 .t/k1 

C ku0 k1 ; t

t > 0:

(23)

for some C > 0 since z.t/ D et uQ0 . Secondly, Lemma 2.2 implies that kF 0 .t/k1 

C ku0 k1 ; t

t > 0:

(24)

It thus remains to prove kw0 .t/k1  Ct1 ku0 k1 for all t > T: To this end, we multiply the first equation of (14) by t and differentiate then with respect to t. We then obtain the equation 8 ˆ ˆ .tw/tt  .tw/t C .trw /t D .tFt  tG/t C wt in J  ; < div w D 0 in J  ; ˆ w D 0 on J  @; ˆ : w.0/ D 0 in :

(25)

We continue by considering the solution . ;  / to problem (22) with initial data 0 2 C01 ./. Its existence is ensured by Proposition 2.6. We then multiply the first equation of (25) by .t  / for  2 .0; t/ and t > T and integrate by parts on .0; t/  . By virtue of Lemma 2.7 and taking into account that

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lims!0 kw.s/kp D lims!0 skw0 .s/kp D 0, the following integral relation holds: Z .twt .t/;

0/

t

D 0

Z

@ . .F .//; .t  //d C @

Z

t

.

0

@ .G.//; .t  //d @

t

C 0

.w ./; .t  //d  .w.t/;

0 /:

Lemma 2.2 (ii) implies that limt!0 t.Ft .t/; .t  // D 0 and Proposition 2.3 yields limt!0 kw.t/kp D 0. Thus, integrating by parts yields Z j.twt .t/;

0 /j 

Z

tT=2 0

j.F ./;

 .t  //jd C

tT=2

j.w./;

 .t

0

 //jd

T /.Ft ..t  T=2/; .T=2//j C j.w.t  T=2/; .T=2//j C j.w.t/; 2 Z t Z t @ @ j. .G.//; .t  //jd C j. .F .//; .t  //jd C @ 0 tT=2 @

Cj.t 

Z C

t

j.w ./; .t  //jd DW

tT=2

8 X

0 /j

Ii .t/:

iD1

Let us start with estimating the term I1 CI2 . By Lemma 2.2 (i) and Proposition 2.3 kF ./k1 C kw./kp  Cku0 k1 ;

t > 0:

Further, since F has compact support, Hölder’s inequality with r 2 .1; n2 / and r0 its conjugate exponent as well as Proposition 2.6 yield Z I1 .t/ C I2 .t/  Cku0 k1

tT=2

k 0

 .t

 /kr0 d  Cku0 k1 k

0 k1 ;

t > T;

for some C C.T/ > 0. Next, estimating the terms I3 and I4 similarly as above, we obtain I3 .t/ C I4 .t/  Cku0 k1 k .T=2/kr0  Cku0 k1 T  2r k n

0 k1

t > T;

for some C C.T/ > 0. Concerning I5 , note that Proposition 2.3 together with Sobolev’s embedding ensures that kw.t/k1  C.t1 C 1/k 0 k1 . Hence, I5 .t/  kw.t/k1 k

0 k1

 C.t1 C 1/ku0 k1 k

0 k1 ;

t > 0:

Bounded Analyticity of the Stokes Semigroup

285

Next, splitting the integral at T=2 we obtain by Hölder’s inequality Z I6 .t/D

@ j. .G.//; .t  //jd C @

tT=2 0

Z

tT=2

k

 0

Z

t

k tT=2

Choosing p >

n 2

Z

t

j. tT=2

@ .G.//; .t  //jd @

@ .G.//k1 k .t  /k1 d C @

@ .G.//kp k .t  /kp0 d: @ and employing Lemma 2.2 (v) we obtain I6 .t/  Cku0 k1 k

0 k1 ;

t > T;

for some C C.T/ > 0. Finally, we turn our attention to I7 and I8 . Since F has compact support, choosing p > n=2 and using Hölder’s inequality as well as Proposition 2.3 we obtain Z I7 .t/ C I8 .t/  C Z

t

k

tT=2

@ .F .//kp k .t  /kp0 d C @

t

kw ./kp k .t  /kp0 d tT=2 n

 C.T 1 C 1/T 1 2p ku0 k1 k

0 k1 ;

t > T:

Summing up, we deduce that given T > 0 there exists a constant C.T/ > 0 such that tkwt .t/k1  C.T/ku0 k1 ;

t > T;

(26)

with C.T/ independent of u0 . Finally, combining the estimates (23), (24) with estimate (26), we see that kut k1 D kzt C Ft C wt k1 

C.T/ ku0 k1 ; t

t > T:

(27)

Step 2: The case of u0 2 L1

./ We start the second step with an approximation result due to Abe and Giga, see [2], Lemma 5.1. Lemma 3.1 Let  be an exterior domain with Lipschitz boundary. Then, for u 2 1 L1

./, there exists a sequence .um /  C ./ \ C./ and a constant C > 0, independent of u, such that kum k1  Ckuk1 ;

m 2 N;

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and um ! u a.e. in . In particular, kum  ukLp .% / D 0 for p 2 Œ1; 1/ and  > 0. m 1 Assume now that u0 2 L1

./ and let .u0 /  C./ \ C ./ be a sequence converging to u0 a.e. in  according to Lemma 3.1. By Proposition 2.5 there exists a sequence of solutions .um ; um / to Eq. (1) enjoying the estimates (27) and (21). The estimate (20) implies further that for all q 2 . n2 ; 1/ and all % > d D diam Rn n , 2 X j˛j t t p . / C kukt .t/  um / 2 kD˛ uk .t/  D˛ um .t/kLp .% / .t/k . L % t tC1 tC1 j˛jD0

 C sup Mq .t; y; L; uk0  um 0 /; t > 0; L > 0: B.0;%/

Moreover, by (21), for each  2 .0; 12 / 1

2n j k .t; x/   m .t; x/j  C.t 2  C 1/ sup Mq .t; y; L; uk0  um ; 0 /.1 C jxj/ B.O;e %/

with C independent of k; m 2 N and L. Following [14] we then see that (i) for all Œ; T0   .0; T/ and all compact %  , .um / is a Cauchy sequence in C.Œ; T0   % /, (ii) for all t > 0 and all  > 0, .um / is a Cauchy sequence in W 2;p .% / and .um t / is a Cauchy sequence in Lp .% /, (iii) . m / is a Cauchy sequence in C..0; T/  /. Taking the limit with respect to the above family of seminorms yields that the limit .u; u / satisfies Eq. (1). Moreover, ut is a Hölder continuous function in space and in time. Thus, the estimate (2) follows from the estimate (27) proved for smooth p data. Indeed, since the sequence .um t / converges in L .% / for all t > 0, we find mk a subsequence .ut / converging to ut .t; x/ for a.e. .t; x/ 2 ftg  . Hence, by estimate (27) mk k jut .t; x/j  jut .t; x/  um t .t; x/j C jut .t; x/j 1 k  jut .t; x/  um t .t; x/j C Ct ku0 k1 :

Thus, we deduce (2) by letting mk ! 1 for such .t; x/. Finally, since ut is Hölder continuous in x, we may extend the latter estimate pointwise. Step 3: Bounded analyticity of T For u0 2 L1

./ let T.t/u0 WD u.t/, where .u; / is the solution of the Stokes equation (1). Then T is the semigroup on L1

./ and we denote its generator by

Bounded Analyticity of the Stokes Semigroup

287

A. Taking into account Steps 1 and 2, it follows from Corollary 3.7.12 of [4] that fz 2 C W Rez > 0g  %.A/ and that sup kR.; A/kL.L1 < 1:

.//

Re>0

Thus, there exists ı > 0 and M > 0 such that kR.; A/k  M;

 2 †=2Cı :

(28)

Thus A generates a bounded analytic semigroup on L1

./. The proof of Theorem 1.1 is complete. In order to prove Theorem 1.3 consider the time derivative of .w; w /, which by (14) then is a solution of 8 @ss w  ws C rws D Fss C Gs ˆ ˆ < div ws D 0 ˆ ws D 0 ˆ : ws . 2t / D wt . 2t /

in e J  ; e in J  ; on e J  @; on f 2t g  :

(29)

We are looking for a solution of this problem in the form ws .s/ D e w.s/ C b w.s/;

w .s/ D wQ .s/ C b .s/ ; s > w

t ; 2

(30)

where the pair .e w; wQ / is the solution of problem (15) and .b w; b / is the solution w of the following problem 8 @sb w  b w C rb D0 ˆ w ˆ < div b wD 0 ˆ b wD 0 ˆ : b w. 2t / D wt . 2t /

in e J  ; in e J  ; on e J  @; on ftg  :

(31)

The Lp -theory for the Stokes problem (see e.g. [5, 10, 12, 15]) combined with estimate (26) for the initial data ut . 2t / yields for p > n t t t (32) w.s/kp  Ckwt . /kp  Ct1 ku0 k1 ; s > ; .s  /1 krb 2 2 2 ( 1 if s 2 . 2t ; 2t C 1/; for some C > 0 and 1 D 2n By (30), (19) and (32) we t 2p if s  2 C 1 : obtain t t krws .s/kp  c.s  /1 t1 ku0 k1 ; s > : 2 2

(33)

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Considering problem (14) as a stationary Stokes boundary value problem with data Ft C G  wt , we obtain via well known estimates in exterior domains for the Stokes problem (see [9])   kD3 w.s/kp C kD2 w kp  C krws .s/kp C krFs .s/kp C krG.s/kp C kw.s/kp provided s > 2t . Now, combining Lemma 2.2 (vi) with Lemma 2.4 for kw.s/kp and with (33) for krw.s/kp , respectively, yields t t 3 kD3 w.s/kp  CŒ.s  /1 t1 C s 2 C s1 C 1ku0 k1 ; s > : 2 2

(34)

By virtue of the Gagliardo-Nirenberg inequality for non-homogeneous boundary values, see [7], and assuming p > n, we get kD2 w.s/k1  CkD3 w.s/kap kw.s/kp1a ; a D

n C 2p : 3p

Hence, employing estimates (34) and Lemma 2.4 for kw.s/kp and also assuming s D t we see that kD2 w.t/k1  C.tı C 1/ku0 k1 ; t  T for some ı > 0. Since the solution of the Stokes problem (1) is given by u D z C F C w and by virtue of estimate Lemma 2.2 (vi), we deduce that kD2 u.t/k1  C.tı C 1/ku0 k1 ; t  T: Taking into account Theorem 2.1 by Abe and Giga, this completes the proof of Theorem 1.3 for the second derivatives. We finally note that the case of first derivatives follows by means of the interpolation.

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) 3. K. Abe, Y. Giga, M. Hieber, Stokes resolvent estimates in spaces of bounded functions. Ann. Sci. Éc. Norm. Supér. 48, 537–559 (2015) 4. W. Arendt, C. Batty, M. Hieber, F. Neubrander, in Vector-Valued Laplace Transforms and Cauchy Problems. Birkhauser Monographs in Mathematics, vol. 96, 2nd edn. (Birkhäuser, Basel, 2012) 5. W. Borchers, H. Sohr, On the semigroup of the Stokes operator for exterior domains in Lq spaces. Math. Z. 196, 415–425 (1987)

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6. T. Chang, H. Choe, Maximum modulus estimates for the solution of the Stokes equation. J. Differ. Equ. 254, 2682–2704 (2013) 7. F. Crispo, P. Maremonti, An interpolation inequality in exterior domains. Rend. Semin. Mat. Univ. Padova 112, 11–39 (2004) 8. W. Desch, M. Hieber, J. Prüss, Lp -theory of the Stokes equation in a half space. J. Evol. Equ. 1, 115–142 (2001) 9. G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, 2nd edn. (Springer, New York, 2011) 10. Y. Giga, H. Sohr, Abstract Lp -estimates for the Cauchy problem with applications to the NavierStokes equations in exterior domains. J. Funct. Anal. 102, 72–94 (1991) 11. T. Kato, Strong Lp -solutions of the Navier-Stokes equations in Rm , with applications to weak solutions. Math. Z. 187, 471–480 (1984) 12. P. Maremonti, V.A. Solonnikov, On nonstationary Stokes problem in exterior domains. Ann. Sc. Norm. Sup. Pisa 24, 395–449 (1997) 13. P. Maremonti, A remark on the Stokes problem with initial data in L1 . J. Math. Fluid Mech.13, 469–480 (2011) 14. P. Maremonti, On the Stokes problem in exterior domains: the maximum modulus theorem. AIMS J. A 34, 2135–2171 (2014) 15. Y. Shibata, R. Shimada, On a generalized resolvent estimate for the Stokes system with Robin boundary conditions. J. Math. Soc. Jpn. 59, 469–519 (2007) 16. V.A. Solonnikov, On nonstationary Stokes problem and Navier-Stokes problem in a half-space with initial data nondecreasing at infinity. J. Math. Sci. 114, 1726–1740 (2003)

On the Weak Solution of the Fluid-Structure Interaction Problem for Shear-Dependent Fluids Anna Hundertmark, Mária Lukáˇcová-Medvid’ová, and Šárka Neˇcasová

Abstract In this paper the coupled fluid-structure interaction problem for incompressible non-Newtonian shear-dependent fluid flow in two-dimensional timedependent domain is studied. One part of the domain boundary consists of an elastic wall. Its temporal evolution is governed by the generalized string equation with action of the fluid forces by means of the Neumann type boundary condition. The aim of this work is to present the limiting process for the auxiliary .; "; k/-problem. The weak solution of this auxiliary problem has been studied in our recent work (Hundertmark-Zaušková, Lukáˇcová-Medvid’ová, Neˇcasová, On the existence of weak solution to the coupled fluid-structure interaction problem for non-Newtonian shear-dependent fluid, J. Math. Soc. Japan (in press)). Keywords Existence of weak solution • Fluid-structure interaction • Hemodynamics • Non-Newtonian fluids • Shear-thickening fluids • Shear-thinning fluids

1 Problem Definition The problem of a fluid interaction with a moving or deformable structure is important in many applications like biomechanics, hydroelasticity, aeroelasticity, sedimentation, modeling of blood flow, etc. We consider a two-dimensional fluid motion governed by the momentum and the continuity equation @t v C  .v  r/ v  div C r D 0;

div v D 0;

(1)

with  denoting the constant density of fluid, v D .v1 ; v2 / the velocity vector,  the pressure and  the shear stress tensor.

A. Hundertmark • M. Lukáˇcová-Medvid’ová Institute for Mathematics, Johannes Gutenberg University, Staudingerweg 9, Mainz, Germany e-mail: [email protected]; [email protected] Š. Neˇcasová () Academy of Sciences of Czech Republic, Žitná 25, Praha, Czech Republic e-mail: [email protected] © Springer Basel 2016 H. Amann et al. (eds.), Recent Developments of Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-0348-0939-9_16

291

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A. Hundertmark et al.

Let us first specify the shear-dependent fluids that will be considered in this paper. We assume that  D .e.v// D 2.je.v/j/e.v/;

where e.v/ D

1 .rv C rvT / 2

is the symmetric deformation tensor. Moreover we assume that there exists a potential U 2 C2 .R22 / of the stress tensor , such that for some 1 < p < 1; C1 ; C2 > 0 we have for all ;  2 R22 sym and i; j; k; l 2 f1; 2g, cf. [11] @U./ D  ij ./; @ij

U.0/ D

@U.0/ D 0; @ij

@2 U./ mn rs  C1 .1 C jj/p2 jj2 ; @mn @rs ˇ 2 ˇ ˇ @ U./ ˇ p2 ˇ ˇ ˇ @ @ ˇ  C2 .1 C jj/ : ij kl

(2) (3) (4)

One particular example satisfying the above properties is a stress tensor, which contains shear-dependent viscosity obeying the power-law model, cf. [8, 11, 12, 16] .je.v/j/ D .1 C je.v/j2 /

p2 2

p > 1:

(5)

For p < 2 the viscosity is a decreasing function of the shear rate, i.e., shear-thinning. For p > 2 we have shear-thickening property and this model is an analogy of the so-called Ladyzhenskaja’s fluid; for p D 3 it yields the Smagorinskij model of turbulence. In numerical simulations presented in our recent papers [8, 10] the shear-thinning model of Carreau has been used in order to model blood flow in compliant vessels. For the simplicity of presentation we will consider here only the case of shear-thickening fluids, i.e. p  2. The generalization for shear-thinning fluids may be done in an analogous way as here, using an appropriate techniques for shear-thinning fluids, see results of Diening, RVužiˇcka and Wolf [5, 15]. The two-dimensional deformable computational domain ..t// f.x1 ; x2 /I 0 < x1 < L; 0 < x2 < R0 .x1 / C .x1 ; t/g ; 0 < t < T is given by a reference radius function R0 .x1 / and the unknown free boundary function .x1 ; t/ describing the domain deformation. The fluid and the geometry of the computational domain are coupled through the following Dirichlet boundary condition on the deformable part of the boundary w .t/ 

@.x1 ; t/ ; v.x1 ; R0 .x1 / C .x1 ; t/; t/ D 0; @t

(6)

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where w .t/ D f.x1 ; x2 /I x2 D R0 .x1 / C .x1 ; t/; x1 2 .0; L/g. The normal component of the fluid stress tensor Tf n and the outside pressure Pw provide the forcing terms for the boundary displacement , that is modeled by the generalized string equation: i @2  @2  @5  @2 R0 g h Q Qw I nQ  e2 on w0 :  a C b C c  a D C P T f 2 4 2 Q @t2 @x1 @t@x1 @x1 E

(7)

Here w0 WD w .t/jtD0 , Œ.TQ f C PQw I/n.Q Q x/ D Œ.Tf CPw I/n.x/; x 2 w .t/; xQ 2 w0 and Tf D   I. Moreover, n; nQ denote the unit outward normals on wp .t/; w0 , respectively and njnj D .@x1 .R0 C /; 1/T . The coefficient g D

.R0 C/ R0

1C.@x1 .R0 C//2

p

1C.@x1 R0 /2

arises from the transformation from the Eulerian frame of the fluid forces into the Lagrangian formulation of the string. Equation (7) is equipped with the following boundary and initial conditions .0; t/ D x1 .0; t/ D .L; t/ D x1 .L; t/ D .x1 ; 0/ D

@ .x1 ; 0/ D 0: @t

(8)

Q a; b; c appearing in (7) are given as follows [8], Positive coefficients E; EQ D w „;

j z j E aD  2 2 ; b D .R0 C /R0 ; 0 1 C @R @x1

c > 0;

where E is the Young modulus, „ the wall thickness, w the density of the vessel wall tissue, the coefficient c D =.w „/,  positive constant. j z j D G is the longitudinal stress,  D 1 is the Timoshenko’s shear correction factor and G is the shear modulus, equal to G D E=2.1 C / with D 1=2 for incompressible materials. Note that the coefficients a; b are non-constant, however, according to the assumption (16) below they are upper- and down-bounded. In what follows, we E linearize the term b D .R0 C/R by  ER2 and for the sake of simplicity we work with 0 w 0 constant coefficients a; b; c. Equation (7) can be transformed as follows. 

@2  @2  @5  @2 R0 E  a 2 C b C c  a 2 .x1 ; t/ D @t2 @x1 @t@x41 @x1 i h  Tf njnj  e2  Pw .x1 ; R0 .x1 / C .x1 ; t/; t/; p x1 2 .0; L/. Here E D EQ 1 C .@x1 R0 /2 . We assume that E is bounded.

(9)

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We complete the system (1) with the following boundary and initial conditions: on the inflow part of the boundary, which we denote in , we set v2 .0; x2 ; t/ D 0;



@v1  2.je.v/j/   C Pin  jv1 j2 .0; x2 ; t/ D 0 @x1 2

(10)

for any 0 < x2 < R0 .0/, 0 < t < T and for a given function Pin D Pin .x2 ; t/. On the opposite, outflow part of the boundary out , we set 

@v1  2 v2 .L; x2 ; t/ D 0; 2.je.v/j/   C Pout  jv1 j .L; x2 ; t/ D 0 @x1 2

(11)

for any 0 < x2 < R0 .L/, 0 < t < T and for a given function Pout D Pout .x2 ; t/. Note that we require that the so-called kinematic pressure is prescribed on the inflow and outflow boundary. This implies that the fluxes of kinetic energy on inflow and outflow boundary will disappear in the weak formulation. Finally, on the remaining part of the boundary, c , we set the flow symmetry condition v2 .x1 ; 0; t/ D 0 ;

.je.v/j/

@v1 .x1 ; 0; t/ D 0 @x2

(12)

for any 0 < x1 < L, 0 < t < T. The initial conditions read v.x1 ; x2 ; 0/ D 0

for any 0 < x1 < L; 0 < x2 < R0 .x1 /:

(13)

The problem defined in (1)–(13) is a generalization of the problem for Newtonian fluid previously studied by Filo and Hundertmark in [6, 17]. Here the original generalized string model of Quarteroni [13], Quarteroni and Formaggia [14] with a regularization term txx has been used. The iterative process with respect to the domain deformation, cf. item 3 below and Sect. 4, has been be completed only for the .; "/-approximation of the original problem and the convergence with respect to domain deformation was an open problem. In the present paper, similarly as in [4], we use a modified model for the structure equation having a viscoelastic term txxxx . For this model we show global existence in time of weak solution of unsteady, fully coupled fluid-structure problem. The existence result holds until a contact of the elastic boundary with a fixed boundary part. The question of existence of weak solution of fully coupled fluid-structure interaction problem with the original Quarteroni’s generalized string model for generalized Newtonian fluids is still an open problem. The main result of this paper is formulated in Theorem 1.2. For the existence proof a suitable approximation of the problem (1)–(13), see Sect. 2, is constructed. 1. "-approximation (22): the space of solenoidal functions on a moving domain is approximated by the artificial compressibility approach,

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2. -approximation (20), (21): the boundary conditions (6)–(7) has been splitted and the deformable boundary becomes semi-pervious for  < 1, 3. h-approximation: the domain deformation is assume to be given by a sufficiently smooth function ı.x1 ; t/; the weak formulation on a deformable domain .ı.t// DW .h.t// is transformed to a reference domain D D .0; L/  .0; 1/ using the known radius h WD R0 C ı, see (23). Letting " ! 0,  ! 1 and finally the fixed point procedure for the domain deformation complete the proof. In [9] the above fluid-structure interaction problem has been studied and the existence of weak solution for fixed parameters ; " and given deformation ı, such that h D R0 C ı, i.e., to the .; "; h/-approximation of the problem (1)–(13) has been proven in details. In this work we only present the limiting processes for " ! 0;  ! 1 and the fixed point procedure with respect to the domain deformation regarding the geometric nonlinearity of our problem.

1.1 Weak Formulation In this section our aim is to present the weak formulation of the problem (1)–(13). Assuming that  is enough regular (see below) and taking into account the results from [4] we can define the functional spaces that gives sense to the trace of velocity from W 1;p ...t/// and thus to define the weak solution of the problem. We assume that R0 2 C02 .0; L/. Definition 1.1 (Weak Formulation) We say that .v; / is a weak solution of (1)– (13) on Œ0; T/ if the following conditions hold – – – –

v 2 Lp .0; TI W 1;p ...t//// \ L1 .0; TI L2 ...t////,  2 W 1;1 .0; TI L2 .0; L// \ H 1 .0; TI H02 .0; L//, div ˇ v D 0 a.e. on ..t//, ˇ vˇw .t/ D .0; t / for a.e. x 2 w .t/; t 2 .0; T/, v2 ˇin [out [c D 0, Z

T

Z

0

n ..t//

Z

Z

 v 

  jv1 j2 '1 .L; x2 ; t/ dx2 dt 2 0 0 Z T Z R0 .0/    Pin  jv1 j2 '1 .0; x2 ; t/ dx2 dt  2 0 0 Z TZ L @2 R0 Pw '2 .x1 ; R0 .x1 / C .x1 ; t/; t/  a 2  dx1 dt C @x1 0 0 Z TZ L @3  @2  @ @ @ @ C Cc 2  Ca C b  dx1 dt D 0 2 @t @t @x @x @t @x 1 @x1 0 0 1 1

C

T

R0 .L/ 

2 X @' @vj o C 2.je.v/j/e.v/e.'/ C  vi ' dx dt @t @xi j i;jD1

Pout 

(14)

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for every test functions '.x1 ; x2 ; t/ 2 H 1 .0; TI W 1;p ...t//// such that div ' D 0 a.e on ..t//; ˇ ˇ ˇ '2 ˇw .t/ 2 H 1 .0; TI H02 .w .t///; '2 ˇin [out [c D '1 ˇw .t/ D 0

(15)

and

.x1 ; t/ D E '2 .x1 ; R0 .x1 / C .x1 ; t/; t/: Theorem 1.2 (Main Result: Existence of a Weak Solution) Let p  2. 0 Assume that the boundary data fulfill Pin 2 Lp .0; TI L2 .0; R0 .0///, Pout 2 0 0 Lp .0; TI L2 .0; R0 .L///, Pw 2 Lp .0; TI L2 .0; L//, 1p C p10 D 1. Furthermore, assume that the properties (2)–(4) for the viscous stress tensor hold. Then for T  T  , T  depending on the data R0 ; Pin ; Pout ; Pw ; K; ˛; cf. (16), and 1 ˛  minfRmin ; Rmin CR g; Rmin  R0  Rmax there exists a weak solution .v; / of max the problem (1)–(13) such that (i) v 2 Lp .0; TI W 1;p ...t//// \ L1 .0; TI L2 ...t////;  2 W 1;1 .0; TI L2 1 2 .0; ˇ L// \ H .0; TI H0 .0; L//, ˇ (ii) vˇw .t/ D .0; t / for a.e. x 2 w .t/; t 2 .0; T/, v2 ˇin [out [c D 0, (iii) v satisfies the condition div v D 0 a.e on ..t// and (14) holds. Remark 1.3 Let us point out that T  denotes a time when the elastic boundary reaches the bottom boundary. If T  D 1 we have an existence of the global weak solution, otherwise the existence of the weak solution holds until the elastic boundary reaches the bottom boundary, see Sect. 4 for further details.

2 Auxiliary Problem: .; "; h/-Approximation In what follows we will formulate a suitable approximation of the original problem (1)–(13). First of all we approximate the deformable boundary w by a given function h D R0 C ı, ı 2 H 1 .0; TI H02 .0; L// \ W 1;1 .0; TI L2 .0; L//, R0 .x1 / 2 C2 Œ0; L satisfying for all x1 2 Œ0; L 1

0 < ˛  h.x1 ; t/  ˛ ;

ˇ Z Tˇ ˇ ˇ ˇ @h.x1 ; t/ ˇ2 ˇ @h.x1 ; t/ ˇ ˇC ˇ ˇ ˇ ˇ ˇ @t ˇ dt  K< 1 ˇ @x 1 0

(16)

h.0; t/ D R0 .0/; h.L; t/ D R0 .L/: We look for a solution .v; ; / of the following problem 

@v C .v  r/v D div   r @t

in .h.t//;

(17)

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and for all x1 2 .0; L/, see (9), 0 < t < T 

@2  @5  @2 R0 @2  .x1 ; t/ D  a C b C c  a @t2 @x21 @t@x41 @x21     @v1 @h @v2 @v2   C Pw .Nx; t/; C C2 .je.v/j/  @x1 @x2 @x1 @x2 

@ v.Nx; t/ D 0; .x1 ; t/ ; @t

 E

(18) (19)

xN D .x1 ; h.x1 ; t//. Furthermore, in the analysis of problem (1)–(13) the boundary condition (6)–(7), cf. (18)–(19), is splitted in the following way, see [6]     @v2 @v1 @h @v2 .je.v/j/    C Pw .Nx; t/ C C2 @x1 @x2 @x1 @x2  h @ i   @h  v2 v2 .Nx; t/  .x1 ; t/ D  .x1 ; t/  v2 .Nx; t/ 2 @t @t

(20)

and  2 h @ i @2  @5  @2 R0 @ .x .x  a ; t/ D  ; t/  v .N x ; t/ E 2  a 2 C bCc 1 1 2 @t @t @x1 @t@x41 @x21 with   1:

(21)

We will show later, that the approximation with  is reasonable. One of the possible physical interpretations for introducing finite  comes from the mathematical modeling of semi-pervious boundary, where this type of boundary condition occurs. In our case, the boundary w seems to be partly permeable for finite , but letting  ! 1 it becomes impervious. In fact, we prove the existence of solution if  ! 1 and thus we get the original boundary condition (18)–(19). Furthermore, we overcome the difficulties with solenoidal spaces by means of the artificial compressibility. We approximate the continuity equation similarly as in [6] with

"

 @"  " C div v" D 0 in .h.t//; t 2 .0; T/ @t

@" D 0; on @.h.t//; t 2 .0; T/; @n

" .0/ D 0 in .h.0//

" > 0:

(22)

By letting " ! 0 we show that v" ! v, where v is the weak solution of (1). For fixed ", due to the lack of solenoidal property for velocity, we have the additional term in momentum equation 2 vi div v, which we include into the convective term, see (27).

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Our approximated problem is defined on a moving domain depending on function h D R0 C ı. Now we will reformulate it to a fixed rectangular domain. Set def

u.y1 ; y2 ; t/ D v.y1 ; h.y1 ; t/y2 ; t/ q.y1 ; y2 ; t/ D 1 .y1 ; h.y1 ; t/y2 ; t/ def def

.y1 ; t/ D

@ .y1 ; t/ @t

(23)

for y 2 D D f.y1 ; y2 /I 0 < y1 < L; 0 < y2 < 1g, 0 < t < T. We define the following space ˚  V w 2 W 1;p .D/ W w1 D 0 on Sw ; w2 D 0 on Sin [ Sout [ Sc ; Sw D f.y1 ; 1/ W 0 < y1 < Lg;

Sin D f.0; y2 / W 0 < y2 < 1g;

Sout D f.L; y2 / W 0 < y2 < 1g;

Sc D f.y1 ; 0/ W 0 < y1 < Lg:

(24)

Let us introduce the following notations def

divh u D

@u1 y2 @h @u1 1 @u2  C ; @y1 h @y1 @y2 h @y2

 Z  y2 @h @q @q @ h  a1 .q; / D @y h @y @y @y 1 1 2 1 D

   y2 @h @q @h @q @ 1 @q dy; C  y2  h @y2 @y1 @y1 h @y1 @y2 @y2 viscous term

(25)

Z

..u; // D

h ij .Oe.u//Oeij . /dy;

(26)

D

1  ij .Oe.u// D 21 .jOe.u/j/Oeij .u/; eO ij .u/ D .@O i .uj / C @O j .ui //; 2 

y 1 @ @h @ @ 2 ; @O 2 D  ; @O 1 D @y1 h @y1 @y2 h @y2 convective term

  Z h y2 @h @z @z @z hu1 C u2  C z  divh u dy  b.u; z; / D @y1 h @y1 @y2 @y2 2 D Z 1 Z 1 1 1 R0 u1 z1 1 .L; y2 / dy2 C R0 u1 z1 1 .0; y2 / dy2  2 0 2 0 Z 1 L  u2 z2 2 .y1 ; 1/ dy1 : (27) 2 0

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Remark 2.1 Note, that the transformed stress tensor  ij D 21 .jOe.u/j/Oeij .u/ from (26) with .jOe.u/j/ defined in (5) also satisfies (2)–(4). Remark 2.2 Since the terms defined in (25)–(27) are dependent on the domain deformation h, it will be sometimes useful to denote this explicitly, e.g., b.u; z; / D bh .u; z; /; eO .u/ D eO h .u/. Definition 2.3 (Weak Solution of .; "; h/-Approximate Problem) Let u 2 Lp .0; TI V/ \ L1 .0; TI L2 .D//, q 2 L2 .0; TI H 1 .D// \ L1 .0; TI L2 .D// and

2 L1 .0; TI L2 .0; L// \ L2 .0; TI H02 .0; L//: A triple w D .u; q; / is called a weak solution of the regularized problem (1)–(13) if the following equation holds  @.hu/ ; dt D  @t 0  Z TZ @h @.y2 u/   C b.u; u; /  h q divh dy C ..u; // dt @t @y2 0 D Z TZ 1 h.L; t/qout 1 .L; y2 ; t/  h.0; t/qin 1 .0; y2 ; t/ dy2 dt C Z

T

0



0

 1 @h C  .u2  / qw C u2 C 2 @t 0 0  Z T @.hq/ ;  dt C" @t 0 Z

T

Z

L

2

.y1 ; 1; t/ dy1 dt (28)

 Z @h @.y2 q/ C "  C "a1 .q; / C h divh u  dy dt @t @y2 0 D Z TZ L " @h .y1 ; t/q.y1 ; 1; t/ dy1 dt C C 2 0 0 @t Z t Z TZ L @2 @2  @ @ @

 Cc 2 2 Ca

.y1 ; s/ds C @t @y @y @y @y 1 0 1 0 0 1 1  Z t 2 @ R0  Cb

.y1 ; s/ds   a 2  C .  u2 /  .y1 ; t/ dy1 dt E @y1 0 Z

T

for every . ; ; / 2 H01 .0; TI V/  L2 .0; TI H 1 .D//  L2 .0; TI H02 .0; L//. p Here we remind E D EQ 1 C .@y1 R0 /2 : For simplicity and without lost of generality we assume in what follows that E; a; b; c are constant, cf. (16).

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2.1 Existence of .; "; h/-Approximate Weak Solution For the proof of weak solution to the auxiliary problem defined in Definition 2.3 following properties of viscous and convective forms are useful1 ; for their proofs see [9, Sect. 3.1]. Lemma 2.4 (Coercivity of the Viscous Form) The viscous form defined in (26) Q satisfies for any 2  p < 1 the following estimates: there exists ıQ D ı.K; ˛/ > 0 such that p 2 Q Q 1/ ..u; u//  ıkuk 1;p C ıkuk1;2 ;

2/ ..u1 ; u1  u2 //  ..u2 ; u1  u2 //; Z  ıQ jOe.u1 /  eO .u2 /j2 C jOe.u1 /  eO .u2 /jp ; D

3/ ..u1 ; u1  u2 //  ..u2 ; u1  u2 //  0: Lemma 2.5 (Boundedness of the Viscous Form) Let u; v 2 V, then for 2  p < 1 it holds p1

..u; v//  Ckuk1;p kvk1;p CC0 kuk1;p kvk1;p ;

C0 > 0:

(29)

Lemma 2.6 (Nonlinear Convective Term b.u; z; /) For the trilinear form b.u; z; /, defined in (27) the following properties hold 1 1 B.u; z; /  B.u; ; z/; 2 2

  Z y2 @h @z @z @z hu1 C u2  B.u; z; /

 @y1 h @y1 @y2 @y2 D b.u; z; / D

where

(30) dy:

Moreover for p  2 we have jB.u; z; /j  ckuk1;p kzk1;p k k1;p : In our recent work [9] the following result has been proved. Theorem 2.7 (Existence of .; "; h/-Approximate Weak Solution) Let "; ; be fixed. Assume (2)–(4), a given function h, such that (16) holds, qin ; qout 2 0 0 Lp .0; TI L2 .0; 1//; qw 2 Lp .0; TI L2 .0; L//: Then there exists a weak solution of

1

We use here notations k  kp WD k  kLp .D/ ; k  k1;p WD k  kW 1;p .D/ .

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301

the .; "; h/-approximated problem transformed to the fixed domain, in the sense of integral identity (28). Moreover, 8 p0 L .0; TI V  / for 2 < p < 1; @.hu/ < p0 @.hq/ 2 L .0; TI V  / ˚ L4=3 ..0; T/  D/; 2 L2 .0; TI H 1 .D//; : @t @t for p D 2; such that Z 0

T

D @.hu/ @t

;

E

Z

T

Z

dt D 

Proof See [9, Sects. 3 and 4].

hu  0

D

@ dy dt: @t t u

3 Problem with " D 0;  D 1 The weak solution from Theorem 2.7 depends on the parameters ";  and h. Passing to the limit with " ! 0;  ! 1 we will obtain the weak solution of the original problem (1)–(13) defined on .h/. By this procedure we will prove the existence of the weak solution for the domain deformation h. We realize the limiting process by passing to the limit in both parameters at once, taking  D "1 and letting  ! 1. In what follows we point out the dependence of weak solution on parameters "; : u ; q ;  . In this section, we omit the notation of the dependence on h. In analogy to the estimate [9, Sect. 4.1, Estimate (4.7)] we obtain the following a priori estimate by testing (28) with .u ; q ;  /, using the lemmas from Sect. 2.1 and after analogous manipulations as in [9]. Note, that the right hand side is independent on "; . Z Z   E L 2 2 max h.t/ ju j C "jq j .t/dy C j  .t/j2 dy1 (31) 0tT D 2 0 Z L ˇ 2 ˇ2 Z TZ ˇ @  ˇ 2˛" p 2 Q jrq j dy C E c ˇˇ 2 ˇˇ dy1 dt ıjru j C C 2 2CK @y1 0 0 D ˇZ t ˇZ t ˇ2 ˇ2 Z L ˇ bE ˇˇ aE ˇˇ @  .s/ ˇˇ dsˇ C

 .s/sˇˇ dy1 C ˇ ˇ 2 0 @y1 2 0 0  2 2 Z TZ L Z T  @ R0  2 p0  Q C 2 j   u2 j dy1 dt  M P C c1   @y2  dt : 0 0 0 1 L2 .0;L/ Q D M.p; Q K; ˛/ and P WD kqin kL2 .0;1/ C kqout kL2 .0;1/ C Here c1 D c1 .p; E; a; c/; M . kqw kL2 .0;L/

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3.1 Limiting Process  D "1 ! 1 First of all let us investigate the solenoidal property of the weak solution in the limiting case, i.e., for  D 1. The estimate (31) implies the weak convergence of p "q ;  / * .u; qQ ; /

.u ;

2

(32) 1

2

2

in L .0; TI V/  L .0; TI H .D//  L .0; TI H .0; L// p

as  ! 1: Moreover, after inserting test functions .0; ; 0/ into (28) for sufficiently smooth  we obtain Z

T

Z hdiv u 

0

(33)

D

p p "Ck "q kL2 .0;TIH 1 .D// .kkL2 .0;TIH 1 .D// C k@t kL2 ..0;T/D/ /; Using the boundedness of we get

p "q in L2 .0; TI H 1 .D// and letting " D  1 ! 0 in (33)

divh u D 0

a:e: on .0; T/  D:

This fact allows us to confine later the space of test functions to the solenoidal space, i.e. divh D 0 a.e. on D. In the limiting process for  ! 1 we cannot use the Lions-Aubin lemma in order to obtain strong convergences in appropriate spaces for .u ;  / ! .u; /, since the estimates of the time derivatives @t u ; @t  , depend on , see [9, Sect. 4.1]. In fact, we have to use another argument to obtain the strong convergence. We follow the lines of [6, Sect. 8] and use the equicontinuity in time as in Alt, Luckhaus cf. [1, Lemma 1.9]. It can be shown that Z

T

Z

0

j.hu /.t C /  .hu /.t/j2 C "j.hq /.t C /  .hq /.t/j2 dydt D

Z

T

Z

L

C 0

0

j.h  /.t C /  .h  /.t/j2 dy1 dt  C.K; ˛/;

(34)

where C is a positive constant independent on ; ; ". The proof of (34) can be found in [9, Sect. 5.1] and we omit its presentation here. The estimate (34) and the compactness argument from [1, Lemma 1.9] imply the following strong convergences for  ! 1 u ! u in L1 ..0; T/  D/;

 ! in L1 ..0; T/  .0; L//:

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303

Using the standard interpolations of spaces Lr .QT / and Ls .ST /, QT D .0; T/  D; ST D .0; T/  .0; L/ and boundedness of u; in L4 .QT /; L6 .ST /, respectively, we obtain u ! u in Lr ..0; T/  D/;

 ! in Ls ..0; T/  .0; L//;

where 1  r < 4; 1  s < 6 for  ! 1. Now let us consider test functions 2 Lp .0; TI X/; where ˇ X D f 2 Vdiv I 2 ˇSw 2 H02 .0; L/g;

.T/ D 0;  D E

ˇ ˇ

2 Sw ,

(35)

Vdiv WD f f 2 V; divh f D 0 a:e: on Dg; cf. (24) in (28). With this choice of test functions the boundary terms with  are canceled. Now, we can pass to the limit in  ! 1 inp(28), where  D "1 . We use the weak convergences of u in Lp .0; TI Vdiv /, "q in L2 .0; TI H 1 .D//,  in L2 .0; TI H 2 .0; L//, see (32) and strong convergence of hu in Lr ..0; T/  D/; 0  r < 4. The convergence of the viscous term follows from the monotonicity of the viscous operator and the Minty-Browder theorem, see also [9, Sect. 4.1]. Analogous arguments in order to obtain convergence in the viscous term when k ! 1 will be presented in Sect. 4. The convergence of the convective term for 2 H 1 .0; TI X/ can be obtained for all p > 2 in following way. For case p D 2 see [6, Sect. 8]. In order to obtain RT RT RT b.u ; u ; / ! 0 b.u; u; / one needs to show that 0 jB.u  u; u ; /j ! 0 R T 0; 0 jB.u; u  u ; /j ! 0. Indeed, using the Hölder inequality and imbedding 2p

L p2 .D/ ,! W 1;p .D/ we have Z T Z jB.u  u; u ; /j  C.K; ˛/ 0

T 0

ku  uk2 ku k1;p k k

2p p2

(36)

 C.K; ˛/k kH 1 .0;TIW 1;p .D// ku  ukL2 ..0;T/D/ ku kLp .0;TIW 1;p .D// : RT RT Thus 0 jB.u  u; u ; /j ! 0: Further 0 jB.u; u  u ; /j ! 0 due to the weak convergence of u in Lp .0; TI Vdiv /. p The convergence of the terms containing "q can be realized by the weak RT R convergence in the corresponding spaces. The term 0 D hq divh is canceled due to the solenoidal test functions. Finally, after the limiting process ˇ  ! 1 in (28) using above considerations for all 2 H01 .0; TI X/ and  D E 2 ˇSw we arrive at Z

T 0

Z  hu  D

Z 0

T



@h @.y2 u/ @ C  @t @t @y2

 dy D

..u; //h C bh .u; u; /

(37)

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Z

1

C Z

0 L

C Z

0 L

C 0

h.L/qout .y2 ; t/

1

 1 @h u2 qw C 2 @t

.L; y2 ; t/  h.0/qin .y2 ; t/ 2

1

.0; y2 ; t/ dy2

.y1 ; 1; t/ dy1

Z t @2 @2  @ @ @ Cc 2 2 Ca 

.y1 ; s/ds @t @y1 0 @y1 @y1 @y1  Z t 2 @ R0 a 2  Cb

.y1 ; s/ds .y1 ; t/ dy1 dt: @y1 0

In order to investigate the meaning of the left hand side of the above equality we define the ALE-type time derivative @Nt @.hu/ @h 1 @.y2 hu/  : @Nt .hu/ WD @t @t h @y2

(38)

  y y y2 @ denotes in fact the time Note that @Nt .hu/ D h@t u, where @t WD @t@  @h @t h @y2 derivative transformed to the rectangle domain D, i.e., in coordinates .y1 ; y2 /. The right hand side of (37) is bounded for every 2 M, M D f! 2 Lp .0; TI X/ for p > 2I

(39)

4

! 2 L .0; TI X/ \ L ..0; T/  D/ for p D 2g: p

Thus it can be identified with some functional 2 M . Then using integration by parts with respect to y2 on the left hand side, backward transformation from D to the moving domain .h.t// and the separation of variables it can be shown that 0

D @N t .hu/ 2 Lp .0; TI X  /, see [9, Appendix A] for more details. Thus we can replace Z

T 0

Z  D

@h @.y2 u/ @ C hu   @t @t @y2



Z

T

dy dt D  0

˛ ˝ @N t .hu/; X :

Finally, we transform (37) from the rectangle D to the moving domain .h.t// and obtain the existence of a weak solution to our original problem (1)–(13) with the Dirichlet boundary condition @t  D v2 jw .h.t// for a prescribed domain deformation h. Theorem 3.1 (Existence of Weak Solution for " D 0;  D 1) Assume that h 2 H 1 .0; TI H02 .0; L//\W 1;1 .0; TI L2 .0; L// satisfies (16). Let the boundary data fulfill 0 0 qin ; qout 2 Lp .0; TI L2 .0; 1//; qw 2 Lp .0; TI L2 .0; L//. Furthermore, assume that

Weak Solution of the FSI Problem for Shear-Thickening Fluids

305

the properties (2)–(4) for the viscous stress tensor hold. Then there exists a weak solution .v; / of the problem (1)–(13), such that (i) .u; / 2 ŒLp .0; TI V/  H 1 .0; TI H02 .0; L// \ ŒL1 .0; TI L2 .D//  W 1;1 .0; TI L2 .0; L//, where u is defined in (23), 0 (ii) the time derivative @Nt .hu/ 2 Lp .0; TI X  / for p > 2 and @Nt .hu/ 2 p0  4=3 L .0; TI X / ˚ L ..0; T/  D/ for p D 2, Z

T

Z 

0

D

@h @.y2 u/ @ C hu   @t @t @y2



Z dy dt D  0

T

D

@N t .hu/;

E dt;

@.y2 hu/ where @N t .hu/ D @.h@tu/  1h @h D h@t u, for every test function 2 @t @y2 1 M \ H0 .0; TI X/, (iii) v satisfies the condition div v D 0 a.e on .h.t//, v2 .x1 ; h.x1 ; t/; t/ D @t .x1 ; t/ for a.e. x1 2 .0; L/; t 2 .0; T/ y

and the following integral identity holds Z

T

Z

0

n

.h.t//

Z

Z

 v 

2 X @vj o @' C 2.je.v/j/e.v/e.'/ C  vi ' dx dt @t @xi j i;jD1

   Pout  jv1 j2 '1 .L; x2 ; t/ dx2 dt 2 0 0 Z T Z R0 .0/    Pin  jv1 j2 '1 .0; x2 ; t/ dx2 dt  2 0 0 

Z T Z L  @h '2 .x1 ; h.x1 ; t/; t/ dx1 dt Pw  v2 v2  C 2 @t 0 0 Z TZ L @3  @2  @ @ @ @ Cc 2  Ca dx1 dt C 2 @t @t @x @x @t @x 1 @x1 0 0 1 1 Z TZ L @2 R0 C a 2  C b  dx1 dt D 0 @x1 0 0 C

T

R0 .L/

for every test functions '.x1 ; x2 ; t/ D

x1 ;

2 H01 .0; TI V/; div ' D 0 and

 x2 ;t such that h.x1 ; t/ ˇ 1 2 ˇ 2 Sw 2 H0 .0; TI H0 .0; L//;

a:e: on .h.t//;

.x1 ; t/ D E '2 .x1 ; h.x1 ; t/; t/:

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Note that the structure equation is fulfilled in a slightly modified sense, 

@2  @5  @2 R0 @2  .x1 ; t/ D  a C b C c  a @t2 @x21 @t@x41 @x21 i h  .Tf C Pw I/njnj  e2 C @t .@t   @t h/ .x1 ; h.x1 ; t/; t/ 2 a.e. on .0; T/  .0; L/; compare (9): E

4 Fixed Point Procedure Until now we have proved the existence of weak solution of the original problem in a domain given by a known deformation function ı, i.e. h D R0 C ı, ı 2 H 1 .0; TI H02 .0; L// \ W 1;1 .0; TI L2 .0; L//, R0 .x1 / 2 C2 Œ0; L. In this section we show the existence of the weak solution of (14), which implies, that the domain deforms according to the function .x1 ; t/, i.e. h D R0 C . This will be realized with the use of the Schauder fixed point theorem. First, applying the compactness argument based on the equicontinuity in time we obtain that bounded sequence .v.k/ ; .k/ / defined on .ı .k/ / for some sequence ı .k/ ! ı converges to the limit .v; / defined on .ı/. Consequently, the Schauder fixed point argument implies, that the weak solution  is associated with the time dependent domain ./. Finally we obtain the main result: existence of weak solution for a fully coupled fluid structure interaction problem (1)–(13). Let us denote the space Y D ˇ H 1 .0; TI L2 .0; L//. For each test function 2 p L .0; TI X/; .T/ D 0,  D E 2 ˇSw , recalling (35), and for any h D R0 C ı 2 Y, such that (16) holds, we construct solutions u;  of the following problem defined on the reference domain D, . D @t /, Z T ˝ ˛  @N t .hu/; dt (40) 0

Z

T

 ..u; //h C bh .u; u; /

D 0

Z

1

C Z

0

L

C 0

˝

h.L/qout .y2 ; t/

1

 1 @h 

qw C 2 @t ˛

C @t ;  C

Z 0

L

.L; y2 ; t/  h.0/qin .y2 ; t/ 2

1

.0; y2 ; t/ dy2

.y1 ; 1; t/ dy1

Z t @2 @2  @ @ c 2 2 Ca

.y1 ; s/ds @y1 0 @y1 @y1 @y1  Z t 2 @ R0 a 2  Cb

.y1 ; s/ds .y1 ; t/ dy1 dt : @y1 0

Weak Solution of the FSI Problem for Shear-Thickening Fluids

307

Further, let the ball B˛;K be defined by n B˛;K D ı 2 YI kıkY  C.˛; K/; 0 < ˛  R0 .y1 / C ı.y1 ; t/  ˛ 1 ; ˇ @ı.y ; t/ ˇ ˇ ˇ 1 ˇ ˇ  K; ı.y1 ; 0/ D 0; 8y1 2 Œ0; L; 8t 2 Œ0; T; @y1 ˇ Z Tˇ o ˇ @ı.y1 ; t/ ˇ2 ˇ ˇ ˇ @t ˇ dt  K; 8y1 2 Œ0; L ; 0 where C.˛; K/ is a suitable constant large enough with respect to K; ˛ and the data. By choosing ı 2 B˛;K the following energy estimate holds for all 2  p < 1 uniformly in ı, kuk2L1 .0;TIL2 .D// C kukLp .0;TIW 1;p .D// p

(41)

Ckt k2L1 .0;TIL2 .0;L// C kt k2L2 .0;TIH 2 .0;L// C kk2L1 .0;TIH 1 .0;L//   p0  c.T; p; K; ˛/ kPkLp0 .0;T/ C kR0 k2C2 Œ0;L : This estimate is obtained by multiplying (40) by D u and  D Eu2 jSw D Et , cf. (31). Now, let us define the following mapping by (40), F W B˛;K ! YI F .ı/ D ;

.ı D h  R0 /:

Our aim is to apply the Schauder fixed point theorem and prove that the mapping F has at least one fixed point. This implies the existence of the weak solution to our problem (14). First we check that F .B˛;K /  B˛;K : Note that our a priori estimate (41) yields ky1 kC.Œ0;TŒ0;L/  K, kt kL2 .0;TICŒ0;L/  K and kkY  C.˛; K/ for given data Pin ; Pout ; Pw ; R0 , given K; ˛I ˛ < Rmin WD miny1 2Œ0;L R0 .y1 / and for Q Moreover, since H 1 .0; TI H 2 .0; L// ,! C.0; TI C1 Œ0; L/ sufficiently small time T. and .y1 ; 0/ D 0, there exist a maximal time Tmax , such that .i/ kk1 WD kkC.Œ0;Tmax Œ0;L/  Rmin  ˛. This yields that mint2.0;Tmax / miny1 2.0;L/ .R0 C /  Rmin  kk1  ˛. Thus we can avoid a contact of the deforming wall with the solid bottom. .ii/ Further, we require that the domain deformation is bounded from above, kR0 C k1  ˛ 1 . Having .i/, the condition .ii/ is satisfied if Rmin  ˛  ˛ 1  Rmax . Thus, for instance if ˛ 1  Rmin C Rmax . Consequently, F .B˛;K /  B˛;K as far as

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Q for given data Pin ; Pout ; Pw ; R0 ; K and ˛ such that t  T  WD minfTmax ; Tg 1 ˛  minfRmin ; Rmin CR g. max Secondly, we verify the relative compactness of the mapping F in Y. Let us .k/ consider a sequence fı .k/ g1 and .k/ F .ı .k/ / the weak kD1 in B˛;K . Denote by u .k/ .k/ solution of (40) for h D h WD R0 C ı . Note, that due to the apriori estimate (41) we have weak convergences of .k/ ; u.k/ in the corresponding spaces. In Sect. 4.1, cf. Lemma 4.1, we show the equicontinuity in time, which implies the strong convergences of .k/ in Y as well as the strong convergence u.k/ in L2 ..0; T/  D/. Finally, in Sect. 4.2 we check the continuity of the mapping F with respect to the strong topology in Y.

4.1 Relative Compactness of the Fixed Point Mapping F In this section we verify the relative compactness of the mapping F using the integral equicontinuity in time and the Riesz-Fréchet-Kolmogorov compactness argument. We prove Lemma 4.1, which provides the integral equicontinuity of .k/ and additionally of u.k/ , that holds independently on k. To this end we need to find suitable divergence free test functions in order to control difference of velocity at different time instances. In order to obtain such test functions we follow a construction presented in [4], see also the reference [16] therein. We introduce, in analogy to [4, Lemma 3], the following extensions of the domain and the weak solution. Let BM be a box domain BM .0; L/  .0; M/ 2 R2

(42)

for some M > ˛ 1 specified later. Let us consider a sequence fı .k/ g1 kD1 in B˛;K and h.k/ WD R0 C ı .k/ . We define an extension of solution u.k/ .y; t/ D v.k/ .x; t/ of (40) where h D h.k/ into BM , ( in .h.k/ .t// v.k/ .k/ (43) vN D .k/ .0; t / in BM n.h.k/ .t//: Further, for  > 1 and any function f .x1 ; x2 / we define f  as follows f  .x1 ; x2 / D . f1 .x1 ;  x2 /; f2 .x1 ;  x2 //: Note that if f is divergence free, then f  is divergence free, too. In what follows we will use the following property, which is valid for any f 2 p C  1 H .BM /, for  D 1 C ˛ and M  2˛ 1 , see [9, Lemma 9.2] jf   f j  jf .x1 ;  x2 ; t/  f .x1 ; x2 ; t/j C .  1/jf .x1 ;  x2 ; t/j:

(44)

Weak Solution of the FSI Problem for Shear-Thickening Fluids

309

.k/

Lemma 4.1 For the weak solution .v.k/ ; t / D .u.k/ ; .k/ / of the problem (40), where h D h.k/ , it holds Z

Z T

0

.k/

t jvN .k/ .t C /  vN .k/ .t/j2 C BM

Z L T

Z 0

0

.k/

.k/

jt .t C /  t .t/j2  C. 1=p C  1=2 /:

(45)

.k/

Here t denotes the characteristic function of .h.k/ .t//. The constant C D C.K; ˛/ does not depend on k. Proof We recall that h.k/ D R0 C ı .k/ , but for the sake of simplicity we omit the superscript .k/ in this proof and we denote h WD R0 C ı .k/ ; vN WD vN .k/ ;  WD .k/ . To prove the statement of this lemma, we will use following two properties. 2 H 1 .0; TI X/, cf. (35),

1. For each

Z

Q

 0

Z

Q

D

˝

@Nt .hu/;

Z

0

˛

.T/ D 0 it holds

dt

(46)

@h @.y2 u/ @ C hu dydt  @t @t @y2 D

Z hu.; Q y/ .Q ; y/dy: D

For classical time derivative, this property is clear. For our distributive derivative @N it can be proven using test function D .y; t/' .t/, where 2 H 1 .0; TI X/; ' .t/ D maxf0; minf1; QCt gg and passing  ! 0, cf.[17].  2. By inserting any time independent test function D .y/ into (46) and subtracting (46) for Q D t C ; and Q D t we obtain Z

tC

 t

Z

tC

˝

˛ @t v; '.x; t/ X ds

Z

D t

D

@h @.y2 u/ .y/dyds  @t @y2

(47) Z Œhu.t C /  hu.t/ .y/dy: D

Here the integral on the left hand side has been transformed into .h.t//, .y/ D '.x/; y 2 D; x 2 .h.t// and the space X is defined as

D

˚ X D ' 2 W 1;p ./I div ' D 0 a.e. on ;

 '2 jw 2 H02 .0; L/; '1 jw D '2 jin [out [c D 0 ;

 D .h.t//:

R T Now, let us integrate (47) over 0 dt. The first term on the right hand side R T (integrated over 0 ) can be bounded with C independently on k for test

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functions (52) specified later . The second term on the right hand of (47) can be rewritten due to the transformation to the .h.t// Z

T

Z

Z v.xtC ; t C /'.xtC /dx 

0

.h.tC //

v.xt ; t/'.xt / dx dt:

(48)

.h.t//

Note, that the space coordinate xt x.t/ 2 .h.t// depends on time, hence the test functions ' implicitly depend on time, which is pointed out above. Using the previously defined extensions of the solution vN and some further manipulations we can rewrite (48) as follows Z

Z

T 0

v.x N tC ; t C /'.xtC /  v.xt ; t/'.xt /dx

.h.t//

Z

. tC  t /v.x N tC ; t C /'.xtC / dx dt D

C

(49)

BM

Z

Z T

ŒvN .xtC ; t C /  v.xt ; t/'.xt / C Œ'.xtC /  '.xt /v.x N tC ; t C / ƒ‚ … „ ƒ‚ …

.h.t//„

0

(I)

Z C BM

(II)

. tC  t /v.x N tC ; t C /'.xtC / dx dt: „ ƒ‚ … (III)

Here t ; tC are the characteristic functions of .h.t//; .h.t C //, respectively. In what follows we estimate the term (II) for any test function ' 2 Lp .0; TI X /. Further, we concentrate on the terms (I), (III) using specific test functions. From the imbeddings in one dimension we have ı 2 C0;1=2 .Œ0; TI H 1 .0; L//; cf. (58), thus p kı.t C /  ı.t/kL1 ..0;T/.0;L//  C :

(50)

Using (50) we can estimate the term (II): T Z

Z (II)  0

Z

.h.t// T

j'.xtC /  '.xt /j2 dx

Z

D 0

Z

.h.t// T

 0

Z

ˇZ ˇ ˇ

x2 .tC /

x2 .t/

2

1=2 kvk N L2 ..h.t//kdt

ˇ2 ˇ @s '.x1 ; s/dsˇ dx

jr'j dxjx2 .t C /  x2 .t/j BM

2

!1=2

1=2

kvk N L2 ..h.t//kdt kvk N L2 ..h.t//kdt

 k'kL2 .0;TIH 1 .BM // kı.t C /  ı.t/kL1 ..0;T/.0;L// kvk N L2 ..0;T/BM / p  C :

(51)

Weak Solution of the FSI Problem for Shear-Thickening Fluids

311

Now we specify proper test functions, that will be used in what follows. For xt D x.t/ 2 .h.t//,  > 1 and fixed t;  we set '.xt / D vN  .xtC ; t C /  vN  .xt ; t/;

(52)

.x1 / D E.@t .x1 ; t C /  @t .x1 ; t//: Note that since v is divergence-free, the test function ' is also divergence-free.2 p Moreover, taking into account (50), for   1 C C ˛  and x2 2 w .t/ the coordinate  x2 exceeds the moving domain .h.s//, since we have .R0 C ı.s//  R0 C ı.s/ C kı.t C /  ı.t/k1 ; s D t; t C . According to the construction, such a test function fulfill the boundary condition E'.x1 ; R0 .x1 / C ı.x1 ; t// D E.0; @t .x1 ; t C /  @t .x1 ; t// .0; .x1 //: Let us estimate now the term (III). Since @t  is bounded in L1 .0; TI L2 .0; L// independently on k, we have Z

Z

2

j tC  t j D BM

L 0

2

jı.t C /  ı.t/j D

Z

L

0

ˇZ ˇ ˇ ˇ

tC t

ˇ2 ˇ @t ı.s/dsˇˇ  C:

(53)

Thus, the term (III) can be bounded for ' from (52) as follows. Z

T

(III)  0

p k tC  t kL2 .BM / kvk N L4 .BM / k'kL4 .BM / dt  C :

(54)

For the test functions from (52) the term (I) equals Z

T

Z

(I) D Z

0 T

D 0

Z

.h.t//

.h.t//

Œv.t N C /  v.t/Œ N vN  .t C /  vN  .t/dxdt Œv.t N C /  v.t/ N 2C „ ƒ‚ …

(55)

(Ia/

  N C /  ŒvN  .t/  v.t/ N dxdt Œv.t N C /  v.t/: N ŒvN  .t C /  v.t „ ƒ‚ … (Ib)

For the simplicity we used shorter notations here, e.g., vN .t C / WD v.x N tC ; t C /. The term (Ia) appears on the left hand side of the assertion of this lemma; the term

Since '.xtC / D vN  .xtC2 ; t C 2 /  vN  .xtC ; t C  /, we have to integrate over estimate of the term (II), or we define '.xtC / D 0 if t C  > T.

2

R T2 0

dt in the

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(Ib) need to be estimated from above. We illustrate the estimate of some chosen terms of (Ib) as follows. Estimates of p other terms are analogous. In the sequel we take  D 1 C C ˛  and M  2˛ 1 . For these parameters we have according to lemma (44), Z

Z

T 0

.h.t//

Z

p C˛ 

T

0

v.t N C /ŒvN  .t/  v.t/dxdt N  p kv.t N C /kL2 .BM / kv.t/k N H 1 .BM / dt  C˛ :

To complete the proof, the remaining terms coming from the fluid equations, i.e., the convective term, the viscous term, boundary terms and the equation for  have to be estimated. We illustrate here only the calculations for the nonlinear viscous term and omit tedious but standard calculations for other terms, previously performed also in [6]. R tC Rt After subtracting the weak formulation (40) for 0 ds  0 ds, inserting test functions constructed above (independent on s) into (40) and integrating over R T dt we obtain from the viscous term 0 Z

T

Z

0

tC

Z .h.s//

t

ij .eŒv.s//:eŒvN  .t C /  vN  .t/dx ds dt:

(56)

For the simplicity, we set ! WD vN  .t C / or ! WD vN  .t/ in (56). Using the fact, that jij .e.v//j  C5 .1 C je.v/j/p1 ; which can be derived from (2), (4), cf. [11, Lemma 1.19], (56) can be bounded as follows, Z

T

 0

Z

tC t

.h.s//

Z

 C.K; ˛/

Z

T

Z

0

 C.K; ˛/

tC

p1

k1 C rv.s/kLp ..h.s///kr!kLp ..h.s///ds dt

t T Z tC

Z  C.K; ˛/

C5 .1 C jeŒv.s/j/p1 eŒ!dx ds dt

k1 C 0 1 p

t

Z

T 0

1

k1 C

 p1 p

p rv.s/kLp ..h.s///ds

 p1 Z p

p rv.s/kLp ..h.s///ds

p1

T 0

1

kr!kLp .BM /  p dt kr!kLp .BM / dt 1

 C.K; ˛/ p k1 C rvkLp .0;TILp ..h//kr!kL1 .0;TILp .BM //  C.K; ˛/ p : The estimates of remaining terms on the right hand side can be obtained using the so-called Steklov averages analogously as in e.g., [9, Sect. 5] or [6, Sect. 8] and we leave them to the valued reader. The proof of the lemma is now completed. u t

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313

Due to the (53) it is also easy to obtain from (45) that Z 0

Z T

.k/

.k/

j tC vN .k/ .t C /  t vN .k/ .t/j2 C

BM

Z L T

Z 0

0

.k/

.k/

jt .t C /  t .t/j2

 C. 1=p C  1=2 /:

(57)

.k/

This result implies that t vN .k/ .t/, and consequently vN .k/ .t/ is relatively compact in L2 ..0; T/  BM /. Consequently, the Riesz-Fréchet-Kolmogorov compactness argument [2, Theorem IV.26] based on (57) implies the relative compactness of @t .k/ ; vN .k/ in L2 .0; TI L2 .0; L//; L2 .0; TI L2 .BM //; respectively. Additionally, the standard interpolations give us the compactness of vN .k/ in Lr ..0; T/  BM /; 1  r < 4 and @t .k/ in Ls ..0; T/  .0; L//; 1  s < 6.

4.2 Continuity of the Mapping F As already shown above .k/ converges strongly to some  in Y as k ! 1: In this section we show by limiting process for k ! 1 in (40) that for any convergent subsequence ı .k/ 2 B˛;K ; ı .k/ ! ı in Y we have F .ı .k/ / D .k/ ! F .ı/ and that  F .ı/: First, we know that .k/ !  in H 1 .0; TI L2 .0; L//. Due to the boundedness of  from apriori estimate (41) and the imbeddings in one dimension we have even stronger result—the uniform convergence of @y1 .k/ in C.Œ0; T  Œ0; L/. Indeed, L1 .0; TI H 2 .0; L// \ W 1;1 .0; TI L2 .0; L//

(58)

,! C0;1ˇ .0; TI H 2ˇ .0; L// for 0 < ˇ < 1. From the continuous imbedding of H 2ˇ .0; L/ into H 2ˇ .0; L/ and the Arzelá-Ascoli Lemma we conclude that a subsequence of .k/ converges strongly in C.Œ0; TI H s .0; L//; 0 < s < 2. Since for s > 3=2 we also have continuous imbedding H s .0; L/ ,! C1 Œ0; L, we can conclude, that .k/ !  strongly in C.0; TI C1 Œ0; L/: Let us summarize available convergences u.k/ * u weakly in Lp .0; TI W 1;p .D//; vN .k/ ! vN strongly in Lr ..0; T/  BM /; 1  r < 4; u.k/ ! u strongly in Lr ..0; T/  D/; 1  r < 4; .k/ *  weakly in H 1 .0; TI H 2 .0; L//;

(59)

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.k/ *  weakly* in L1 .0; TI L2 .0; L// .k/ !  uniformly in C.0; TI C1 Œ0; L/; @t .k/ ! @t  strongly in Ls ..0; T/  .0; L//; 1  s < 6:  p0 L .0; TI V  / for 2 < p < 1 @Nt .hu/.k/ * weakly in 0 Lp .0; TI V  / ˚ L4=3 ..0; T/  D/ for p D 2: The last statement of (59) follows from (41) and from the boundedness of the functional @Nt .hu/.k/ . Let us present the estimation of nonlinear terms on the right hand RT side. Indeed, from Lemma 2.5 it follows 0 ..u.k/ ; //  C.K; ˛/k kLp .0;TIW 1;p .D/ . The non-linear convective term can be estimated using Lemma 2.6, the Hölder 2p

inequality and the imbedding of W 1;p .D/ into L p2 .D/ for p > 2 as follows Z

T

0

.k/

Z

.k/

Bh.k/ .u ; u ; /  C.K; ˛/

T 0

ku.k/ k1;p ku.k/ k2 k k

2p p2

 C.K; ˛/ku.k/ kL1 .0;TIL2 .D// ku.k/ kLp .0;TIW 1;p .D// k kLp0 .0;TIW 1;p .D// ; which is bounded due to (41) for all 2 Lp .0; TI V/. Analogously the term RT .k/ .k/ 3 0 Bh.k/ .u ; ; u / is bounded, which leads to Z

T 0

bh.k/ .u.k/ ; u.k/ ; /  C.K; ˛/k kLp .0;TIW 1;p .D// for p 2 .2; 1/:

(60)

Further estimates of the remaining terms on the right hand side conclude the proof of (59)8. In what follows we have to verify, that F .ı .k/ / ! F .ı/ and that the limit  from (59) is the weak solution associated with ı, and thus F .ı/ D :

4.2.1

Limiting Process k ! 1

Now let us consider (40) with u.k/ instead of u, h.k/ instead of h and .k/ D @t .k/ instead of : First of all we have to realize, that due to the solenoidal property, which depends on h.k/ , the test functions are also implicitly dependent on k. This fact presents a difficulty when we pass with k ! 1. Nevertheless we can construct sufficiently smooth test functions Q .y; t/ D '.x; Q t/, which are independent on k and divergence free in .h/ (i.e. divh Q D 0). They are also well defined on infinitely many

3

For p D 2 this estimate is valid for

2 Lp .0; TI V/ \ L4 ..0; T/  D/, cf. [6].

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approximate domains .h.k/ / and dense in the space of admissible test functions Lp .0; TI X/, cf. (35). Such a test functions 'Q can be constructed on .0; T/  BM as algebraic sum, see [4, Remark 3] 'Q D '0 C '1 ; where '0 is a smooth function with compact support in .h/, div'0 D 0 on .h/ and '0 is extended by 0 to .0; T/  BM . Further, having  2 H 1 .0; TI H02 .0; L// we def

define '1 D .0; .x1 /=E/ on BM nB˛ , B˛ D .0; L/  .0; ˛/ 2 R2 , the constant E comes from (15). Note that div'1 D 0 on BM nB˛ . Moreover, '1 such that R R˛ 1 RL  1 @B˛ '1  n D 0 '1 .L; x2 ; t/  '1 .0; x2 ; t/dx2 C 0 E .x1 ; t/dx1 D 0 can be extended into B˛ by a divergence-free extension, whereas remaining boundary conditions on in ; out ; c are preserved, see e.g., [7, p. 144]. Note, that due to the uniform convergence of .k/ we have that sup '0  .h.N/ / for sufficiently large N and the function '0 is well defined on each .h.N/ / for such N. Moreover '1 is defined on .h.k/ / for each k. For more details on this construction we refer a reader to [3, Sect. 7, pp. 35–36], compare [4]. Having Q .y; t/ D Q .x1 ; h.xx12;t/ ; t/ D '.x; Q t/; x 2 .h/; y 2 D, let us construct the set of admissible test functions .k/

.k/

.y1 ; y2 ; t/ WD Q .x1 ;

by transformation of 'Q from .h.k/ / into D,

x2 ; t/ .k/ h .x1 ; t/

D '.x Q 1 ; x2 ; t/;

(61)

x 2 .h.k/ /; y 2 D: The test functions (61) have the following property .k/

W D ! R2 I divh.k/

.k/

D 0;

E

! Q; eO h.k/ . .k/ / ! eO . Q / .k/



.k/ 2 .y1 ; 1; t/

D .y1 ; t/;

and

uniformly on .0; T/  D:

This property follows from the special construction of ', Q the property (63) below and the uniform convergence of h.k/ and @y1 h.k/ that follows from (58). The test functions (61) satisfy the boundary conditions on Sin ; Sout ; Sc , cf. (24) as well. Thus it is enough to consider test functions D Q , which are independent on k and smooth enough. The limiting process in the test functions follows afterwards using the uniform convergence .k/ and eO . .k/ /. In the following lines we will present the limiting process for k ! 1 in chosen nonlinear terms. Let us first consider the convective term and show Z

T 0



 bh.k/ .u.k/ ; u.k/ ; /  bh .u; u; / dt ! 0:

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Recalling (30), the following terms appear in the above expression Z

T 0

Bh .u; u.k/  u; / C Bh.k/ .u.k/  u; u.k/ ; / C B.h.k/ h/ .u; u.k/ ; /dt:

To show the convergence of above integrals, we restrict ourselves only to the terms containing @y1 h.k/, convergence of terms with h.k/ is analogous. Let us consider Z

T 0

Z

 @u @u.k/ @h @u.k/  u1  C  @y2 @y2 @y1 @y2 D 

.k/ @h @u.k/ @h .k/ u1 dy dt:   @y2 @y1 @y1

 @h.k/  .k/ u1  u1 @y1

The convergence of the first term is obvious due to the weak convergence of u.k/ 0 in Lp .0; TI W 1;p .D//. The strong convergence of u.k/ in Lp ..0; T/  .D// and the uniform convergence of @y1 h.k/ imply the convergence in the remaining two terms. Now we denote eO .k/ WD eO h.k/ ; eO WD eO h , cf. (26) and Remark 2.2. The limiting process in the viscous term will be realized as follows. Z

T 0

..u.k/ ; //h.k/  ..u; //h dt

Z

T

Z

D 0

D

(62)

h i .k/ h ij .Oe.k/ .u.k/ //Oeij . /  ij .Oe.u//Oeij . /

.k/ C h.k/  h ij .Oe.k/ .u.k/ //Oeij . / dy dt Z TZ h i .k/ D h ij .Oe.k/ .u.k/ // eO ij . /  eO ij . / dy dt „0 D ƒ‚ … Z

TZ

C „0 Z

.I/



h ij .Oe.k/ .u.k/ //  ij .Oe.u// eO ij . / dy dt D ƒ‚ … .II/

TZ



C „0

D

.k/ h.k/  h ij .Oe.k/ .u.k/ //Oeij . / dy dt : ƒ‚ … .III/

It is easy to see that the term .III/ goes to zero. Using the fact that eO h .u/ D ruF.h/ C .ruF.h// 2 T

R22 sym I

1 F.h/ D 2

"

1

0

y2 @h 1 h @y1 h

# (63)

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and due to the uniform convergence of h.k/ in C.0; TI C1 Œ0; L/ the convergence in all components of F is obvious and we obtain that .I/ ! 0. In order to show the convergence in the term .II/, we will use the Minty- Trick. Let us denote for better readability  k WD eO h.k/ .u.k/ /  WD eO h .u/ and  WD eO . /. 0 Define the operator A; A W Lp ..0; T/  D/ ! Lp ..0; T/  D/, ˛ ˝ A. k /;  WD

Z

T

Z

0

h ij . k / dy dt: D

˝ ˛ Lemma 2.4 implies the monotonicity of operator A, A. k /  A./;  k    0: 2 Further, from assumptions (16) on h.k/ and Lemma 2.5 we obtain for u; Lp .0; TI W 1;p .D// and for any k ˇ˝ k ˛ˇ ˇ A. /;  ˇ  c.K; ˛/kkLp .0;TILp .D// : 0

Thus A is bounded in Lp ..0; T/  D/ and consequently A. k / * f . Moreover, from the weak convergence of ru.k/ and the uniform convergence of h.k/ in C.0; TI C1 Œ0; L/ we obtain the weak˝ convergence  k * . ˛ k k Now ˝ we prove that limk!1 A. /;  D h f ; i, i.e., we show that ˛ limk!1 A. k /;  k   D 0. To this end, we limit in the rest terms of the weak formulation (40) with u.k/ ; h.k/ ; .k/ instead of u; h; using test functions D u.k/  u. In what follows we present the limiting process in the nonlinear convective term. Recalling (30) we can write Z

T

0

bh.k/ .u.k/ ; u.k/ ; u.k/  u/ dt D

1 2

Z

T 0

Bh.k/ .u.k/ ; u.k/ ; u.k/  u/

Bh.k/ .u.k/ ; u.k/  u; u.k/ / dt:

(64)

We can estimate the first term on the right hand side using the Hölder and the 1=2 1=2 interpolation inequality k'k4  ck'k1;2 k'k2 , cf. [9, Lemma 3.1] Z

T 0

Bh.k/ .u.k/ ; u.k/ ; u.k/  u/dt 

 3 1 C.K; ˛/ku.k/  ukL2 .L2 / ku.k/ kL22 .W 1;2 / C ku.k/ kL2 .W 1;2 / kukL22 .W 1;2 / ; here L2 .W 1;2 / WD L2 .0; TI W 1;2 .D//. Thus, the strong convergence of u.k/ in L2 ..0; T/  D/ implies, that the first term on the right hand side of (64) tends to 0. Further, we can rewrite the second term as Bh.k/ .u.k/ ; u.k/  u; u.k/ / D Bh .u; u.k/  u; u/ C B.h.k/ h/ .u; u.k/  u; u/ CBh.k/ .u.k/  u; u.k/  u; u.k/ / C Bh.k/ .u; u.k/  u; u.k/  u/:

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Due to the weak convergence of ru.k/ , uniform convergence of h.k/ and the strong RT convergence of u.k/ , cf. (59) we obtain also for the second term 0 Bh.k/ .u.k/ ; u.k/  u; u.k/ /dt ! 0. This concludes the proof of convergence in the convective term (64). The limiting process in the remaining terms of˝ (40) is obvious and we omit it here. ˛ Consequently, we have obtained limk!1 A. k /;  k D h f ; i and the MintyTrick argument implies that f D A./, i.e. A. k / * A./

and thus

˝

˛ A. k /;  ! hA./; i

for any  2 Lp ..0; T/  D/ as k ! 1: This concludes the limiting process in (40) and the Sect. 4.2. We have found out that F .ı .k/ / ! F .ı/ as k ! 1 and that  is the weak solution of (40) associated with the limit ı, (h D R0 C ı), thus F .ı/ D . Finally, using the continuity of the mapping F , its relative compactness in Y and the property F .B˛;K /  B˛;K we deduce from the Schauder fixed point theorem, that there exists at least one fixed point of the mapping F defined by the weak formulation (40), F ./ D . Thus, we obtain the existence of at least one weak solution (14) of the original unsteady fluid-structure interaction problem (1)–(13). The proof of the Theorem 1.2 is now completed.  Remark on the Global Existence Result Let us point out that we have obtained the existence of weak solution until some time T  . We remind that this time is obtained in order to achieve the fixed point of the mapping F and to avoid the contact of the elastic boundary w .t/ with the fixed boundary for given data Pin ; Pout ; Pw ; R0 and ˛; K. Similarly as in [4, Grandmont et al.], we can prolongate the solution in time and even obtain the global existence until the contact with the solid bottom. Indeed, we can construct a non-decreasing sequence of times fT  D 1   T1 ; : : : ; Tm1 ; Tm ; : : :g, such that for given ˛; K; ˛  minfRmin ; Rmin CR g, starting max  from initial data in time Tm1 , we have the existence of weak solution for some  time Tm1 C T WD Tm . We distinguish between two situations. Either sup Tm D 1, which means, that the contact with the solid bottom never happens and we obtain global existence. Otherwise sup Tm WD T  < 1 for given ˛. In this case we can decrease ˛. If the time interval of the existence cannot be prolongated for chosen ˛, we have to decrease ˛ again. This is repeated until ˛ reaches 0. The later represents N where TN  0. the contact with the solid boundary at some time T C T, Acknowledgements The present research has been financed by the DFG project ZA 613/1-1, the Neˇcas Centrum for Mathematical Modelling LC06052 (financed by MSMT) and the Grant of the Czech Republic, No. P201/11/1304. It has also been partially supported by the 6th EU-Framework Programme under the Contract No. DEASE: MEST-CT-2005-021122 and the DST-DAAD project based personnel exchange program with Indian Institute of Science, Bangalore. We would like to thank Ján Filo (Comenius University, Bratislava) for fruitful discussions on the topic.

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References 1. H.W. Alt, S. Luckhaus, Quasilinear elliptic-parabolic differential equations. Math. Z. 183, 311– 341 (1983) 2. H. Brezis, Analyse Fonctionelle- Théorie et Applications (Masson, Paris, 1983) ˇ c, B. Muha, Existence of a weak solution to a nonlinear fluid-structure interaction 3. S. Cani´ problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls. Arch. Ration. Mech. Anal. 207(4), 919–968 (2013) 4. A. Chambolle, B. Desjardin, M.J. Esteban, C. Grandmont, Existence of weak solutions for unsteady fluid-plate interaction problem. J. Math. Fluid. Mech. 4(3), 368–404 (2005) 5. L. Diening, M. RVužiˇcka, J. Wolf, Existence of weak solutions for unsteady motions of generalized Newtonian fluids. Ann. Sc. Norm. Super. Pisa Cl. Sci. 9(1), 1–46 (2010) 6. J. Filo, A. Zaušková, 2D Navier-Stokes equations in a time dependent domain with Neumann type boundary conditions. J. Math. Fluid Mech. 12(1), 1–46 (2010) 7. G.P. Galdi, An Introduction to the Theory of Navier-Stokes Equations I (Springer, New York, 1994) 8. A. Hundertmark-Zaušková, M. Lukáˇcová-Medvid’ová, Numerical study of shear-dependent non-Newtonian fluids in compliant vessels. Comput. Math. Appl. 60, 572–590 (2010) 9. A. Hundertmark-Zaušková, M. Lukáˇcová-Medvid’ová, Š. Neˇcasová, On the existence of weak solution to the coupled fluid-structure interaction problem for non-Newtonian shear-dependent fluid. J. Math. Soc. Japan (in press) 10. M. Lukáˇcová-Medvid’ová, G. Rusnáková, A. Hundertmark-Zaušková, Kinematic splitting algorithm for fluid-structure interaction in hemodynamics. Comput. Methods Appl. Mech. Eng. 265, 83–106 (2013) 11. J. Málek, J. Neˇcas, M. Rokyta, M. RVužiˇcka, Weak and Measure-Valued Solutions to Evolutionary PDEs (Chapman and Hall, London, 1996) 12. J. Málek, K.R. Rajagopal, Mathematical issues concerning the Navier-Stokes equations and some of its generalizations, in Handbook of Differential Equations, ed. by C.M. Dafermos, E. Feireisl (North-Holland Publishing Company, Amsterdam, 2005) 13. A. Quarteroni, Mathematical and numerical simulation of the cardiovascular system, in Proceedings of the ICM, Beijing, vol. 3, 2002, pp. 839–850 14. A. Quarteroni, L. Formaggia, Computational models in the human body, in Handbook of Numerical Analysis, vol. XII, ed. by P.G. Ciarlet, Guest Editor N. Ayache (Elsevier/North Holland, Amsterdam, 2004) 15. J. Wolf, Existence of weak solution to the equations of non-stationary motion of nonNewtonian fluids with shear rate dependent viscosity. J. Math. Fluid Mech. 9(1), 104–138 (2007) 16. K.K. Yeleswarapu, Evaluation of continuum models for characterizing the constitutive behavior of blood. Ph.D. Thesis, University of Pittsburgh, Pittsburgh, 1996 17. A. Zaušková, 2D Navier–Stokes equations in a time dependent domain. Ph.D. Thesis, Comenius University, Bratislava, 2007

Stability of Time Periodic Solutions for the Rotating Navier-Stokes Equations Tsukasa Iwabuchi, Alex Mahalov, and Ryo Takada

Dedicated to Professor Yoshihiro Shibata on his sixtieth birthday

Abstract We consider the stability problem of time periodic solutions for the rotating Navier-Stokes equations. For the non-rotating case, it is known that time periodic solutions to the original Navier-Stokes equations are asymptotically stable under the smallness assumptions both on the time periodic solutions and on the initial disturbances. We shall treat the high-rotating cases, and prove the asymptotic stability of large time periodic solutions for large initial perturbations. Keywords Asymptotic stability • The rotating Navier-Stokes equations • Time periodic solutions

1 Introduction Let us consider the time periodic problems for the rotating Navier-Stokes equations, describing the motion of viscous incompressible fluids in the rotational framework: 8 @v ˆ <  v C e3  v C .v  r/v C rq D f @t ˆ :div v D 0

t 2 R; x 2 R3 ;

(1)

3

t 2 R; x 2 R ;

T. Iwabuchi Department of Mathematics, Osaka City University and OCAMI, Sumiyoshi-ku, Osaka 558-8585, Japan e-mail: [email protected] A. Mahalov School of Mathematical and Statistical Science, Arizona State University, PO Box 871804, Tempe, AZ 85287, USA e-mail: [email protected] R. Takada () Mathematical Institute, Tohoku University, Sendai 980–8578, Japan e-mail: [email protected] © Springer Basel 2016 H. Amann et al. (eds.), Recent Developments of Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-0348-0939-9_17

321

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where v D v.t; x/ D .v1 .t; x/; v2 .t; x/; v3 .t; x// and q D q.t; x/ denote the unknown velocity field and the unknown pressure of the fluid at the point .t; x/ 2 R  R3 , respectively, while f D f .t; x/ D .f1 .t; x/; f2 .t; x/; f3 .t; x// denotes the given time periodic external force. Here  2 R represents the speed of rotation around the vertical unit vector e3 D .0; 0; 1/, which is called the Coriolis parameter. In [12], Koh et al. showed the existence and uniqueness of periodic in time solutions to (1) in the class P s;p .R3 //3 v 2 BC.RI W

with

3 1

s

1

sup kv.t/kWP s;p 6 Cjj 2 . 3 C 3 / p

(2)

t2R

for 0 < s < 3=5; 1=3 C s=9 6 1=p < min f1=3 C s=3; 1  sg and some constant C D C.s; p/. The purpose of this paper is to show the stability of time periodic solutions to (1) in the class (2). If v.0; x/ is initially perturbed by a, the perturbed flow w.t; x/ is governed by the following initial value problem for the rotating Navier-Stokes equations: 8 @w ˆ ˆ  w C e3  w C .w  r/w C r D f ˆ < @t

t > 0; x 2 R3 ; t > 0; x 2 R3 ;

ˆ div w D 0 ˆ ˆ : w.0; x/ D v.0; x/ C a.x/

(3)

x 2 R3 :

In this paper, we shall prove that for every time periodic solution v 2 P s;p .R3 //3 and every initial disturbance a 2 H P s .R3 /3 with 1=2 < s < 3=5 L1 .RI W and 1=3 C s=9 6 1=p < 1  s, there exists a unique global mild solution w to (3) such that the global space-time integral Z

1 0

kw.t/  v.t/kWP s;p dt

for some  D .s; p/ with 2 <  < 1

converges to zero with algebraic decay rates as jj ! 1, provided that the speed of rotation is sufficiently high. Note that in the high-rotating case we do not impose the smallness assumptions either on time periodic solutions or on initial disturbance. Introducing a new unknown vector u WD w  v, we can reduce the original equations (3) to the following ones with the initial disturbance a: 8 @u ˆ  u C e3  u ˆ ˆ ˆ @t ˆ ˆ < C.u  r/v C .v  r/u C .u  r/u C rp D 0 ˆ ˆ ˆ ˆ div u D 0 ˆ ˆ : u.0; x/ D a.x/

t > 0; x 2 R3 ; 3

t > 0; x 2 R ; x 2 R3 :

(4)

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323

Hence our problem on the stability of time periodic solutions v can be reduced to an investigation into the unique existence of the global solution u to (4) and its asymptotic behavior. Let us review the known results on the existence of time periodic solutions to (1) and its stability. In the non-rotating case  D 0, (1) corresponds to the original Navier-Stokes equations and there are a number of literatures concerning our problem. We shall only mention the results on time periodic solutions in unbounded domains. Maremonti [17, 18] considered the whole space R3 and the half space R3C , respectively, and proved the unique existence of time periodic solution and its asymptotic stability under small perturbations. Kozono and Nakao [14] introduced the notion of mild solutions to the time periodic problems for (1), and proved the existence and uniqueness of time periodic mild solutions in the whole space Rd , the half space RdC with d > 3 and exterior domains in Rd with d > 4. The asymptotic stability of the periodic solutions constructed in [14] was proved by Taniuchi [19]. Yamazaki [20, 21] treated the 3-dimensional exterior domains and generalized the results of Kozono and Nakao [14], Taniuchi [19] for Morrey spaces, respectively. We remark that all of the stability results are obtained under the assumption that both time periodic solutions and initial perturbations are small enough. In the rotating case  2 R, following the idea in [14] several authors obtained the existence results of time periodic mild solutions to (1). Konieczny and Yoneda [13] proved the existence of stationary mild solutions to (1). Kozono et al. [16] showed the uniform solvabilities of (1) and (4) and generalized the results of Kozono and Nakao [14], Taniuchi[19] for all  2 R. In the high-rotating case jj  1, as is already known in [2–6], the Coriolis force e3 u exhibits a dispersion phenomenon which is closely related to dispersive estimates for the operator D3

e˙it jDj f .x/ WD

Z R3

e



3 ix˙it jj

fO ./d;

see Lemma 2.2 in Sect. 2. Based on such estimates, the authors in [9, 12] proved the unique existence of time periodic solutions to (1) for large external forces provided that the speed of rotation is sufficiently large. Now let us state our main result. In order to solve (4), we consider the following integral equations: Z u.t/ D T .t/a  Z

0

T .t  /Pr  Œv./ ˝ u./ d

t

 Z

t

T .t  /Pr  Œu./ ˝ v./ d

0 t

 0

T .t  /Pr  Œu./ ˝ u./ d;

(5)

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where P D .ıjk CRj Rk /16j;k63 denotes the Helmholtz projection onto the divergencefree vector fields and T ./ denotes the semigroup corresponding to the linear problem of (4), which is given explicitly by   

3 3 2 2 T .t/g D F 1 cos  t ejj t I gO ./ C sin  t ejj t R./Og./ jj jj for t > 0 and the divergence-free vector field g. Here fRj g3jD1 denote the Riesz transforms, I is the identity matrix in R3 and R./ is the skew-symmetric matrix related to the symbol of the Riesz transforms, which is defined by 1 0 0 3 2 1 @ R./ WD 3 0 1 A ; jj 2 1 0

 2 R3 n f0g:

For the derivation of the explicit form of the semigroup T .t/, we refer to Babin et al. [1–3] and Hieber and Shibata [8]. We call that u is a mild solution to (4) if u satisfies (5) in some appropriate function space. Our theorem on the stability of time periodic solutions now reads: Theorem 1.1 Let s; p and  satisfy 1 3 3=4  3=.2p/ and jj  1, (8)

Stability of Time Periodic Solutions

325

yields that the perturbed flow w behaves like the time periodic flow v in the sense that kw  vkL .0;1IWP s;p / D O.jj

n  o 3  1  34  2p

/

as jj ! 1:

Remark 1.3 In the non-rotating case  D 0, the stability of time periodic solutions to the original Navier-Stokes equations was proved under the smallness assumptions both on the time periodic solutions and on the initial disturbances (see [15, 19, 20]). On the other hand, since the powers of jj in (7) are positive, we can obtain the unique global solution to (4) for large time periodic solutions v and initial perturbations a provided  satisfies jj > max



  1 2s 1   21 ı11 kvkL1 .RIWP s;p / 3f. 3 C 3 / p g ; ı21 kakHP s s 2 :

Remark 1.4 The size conditions (7) to time periodic solutions and initial disturbances are closely related to the scaling invariance to (4). Indeed, if .u; v; p/ solves (4) with the parameter  then so does .u ; v ; p / with  for all  > 0, where u .t; x/ WD u.2 t; x/;

v .t; x/ WD v.2 t; x/;

p .t; x/ WD 2 p.2 t; x/;

 WD 2 :

The size conditions (7) are invariant under the above scalings. This property could be regarded as a counterpart of the scaling invariance to the original Navier-Stokes equations such as ku .0; /k P 12 3 D ku.0; /k P 12 3 for all  > 0. H .R /

H .R /

In the periodic case T3 , it seems to be difficult to obtain the characterization (7) for the size condition on initial data and time periodic solutions because of the presence of resonant domains and the lack of dispersion effect. Concerning the existence theorem and the global regularity results on T3 , we refer to Babin et al. [1– 3], and also Flandoli and Mahalov [7] for the stochastic case.

Remark 1.5 In the non-rotating case  D 0, the solutions to (4) constructed in [15, 19, 20] satisfy the integral equation (5) in the weak form. This difficulty is caused by the estimate for the convection terms .u  r/v C .v  r/u: Since a time periodic flow does not satisfy the time global estimates like supt>0 t1=2 krv.t/kL3 .R3 / < 1, the standard Kato iteration scheme [11] does not work well. On the other hand, our solution in Theorem 1.1 satisfies (5) in the strong sense. This is due to the use of dispersive effects of the Coriolis force and the space-time integral estimates. This paper is organized as follows. In Sect. 2, we recall the dispersive estimates due to the Coriolis force and prepare the linear estimates for the semigroup T .t/. In Sect. 3, we prove the bilinear estimates associated with the convection terms. In Sect. 4, we give the proof of Theorem 1.1.

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Throughout this paper, we denote by C the constants which may differ from line to line. In particular, C D C .; : : : ; / will denote the constant which depends only on the quantities appearing in parentheses.

2 Linear Estimates Let S .R3 / be the and let S 0 .R3 / be the space of tempered ˚ Schwartz class, 3 distributions. Let 'j j2Z  S .R / be a smooth partition of unity in the frequency space satisfying the following properties: 0 6 '0 ./ 6 1 for all  2 R3 ; ˇ  ˚ supp '0   2 R3 ˇ 1=2 6 jj 6 2 ;

  'j ./ D '0 for all j 2 Z 2j and X

'j ./ D 1 for all  2 R3 n f0g:

j2Z

For j 2 Z and f 2 S 0 .R3 /, we define the frequency localized projection operator j by

b

j f ./ WD 'j ./fO ./: Then, we recall the definition of the homogeneous Besov space BP sp;q .R3 /. Definition 2.1 For s 2 R and 1 6 p; q 6 1, the homogeneous Besov space BP sp;q .R3 / is defined to be the set of all tempered distributions f 2 S 0 .R3 / such that ˚       kf kBP sp;q WD  2sj j f Lp j2Z  q < 1: `

D3

Next, we define the operators e˙it jDj by the Fourier integral D3

e˙it jDj f .x/ WD

Z R3



e

3 ix˙it jj

fO ./d;

t 2 R; x 2 R3 :

Stability of Time Periodic Solutions

327

Then, we can rewrite the semigroup T .t/ as   

3 3 2 2 T .t/f D F 1 cos  t ejj t I fO ./ C sin  t ejj t R./fO ./ jj jj D

D3 1 it D3 t 1 e jDj e .I C R/f C eit jDj et .I  R/f 2 2

(9)

for t > 0 and  2 R, where et denotes the standard heat semigroup and R denotes the matrix of singular integral operators defined by 0

1 0 R3 R2 R WD @ R3 0 R1 A : R2 R1 0 D3

The operators e˙it jDj represent the oscillation parts of T .t/. Let us first recall the D3

dispersive estimates for e˙it jDj . Lemma 2.2 ([12, Lemma 2.2]) For 2 6 p 6 1, there exists a positive constant C D C.p/ such that   D3  ˙it jDj  f e

BP sp;q

2

6 C.1 C jtj/.1 p / kf k sC3.1 2p /

for all t 2 R; s 2 R; 1 6 q 6 1, and f 2 BP p0 ;q

sC3.1 p2 /

BP p0 ;q

.R3 /, where 1=p C 1=p0 D 1.

By the explicit formula (9) of the semigroup T .t/, Lemma 2.2 and the smoothing properties of the heat semigroup yield the following Lq -Lp estimates for T .t/. Lemma 2.3 ([12, Lemma 3.2]) Let ˛ 2 .N [ f0g/3 , and let p and q satisfy 2 6 p < 1 and 1 < q 6 p0 , where 1=p C 1=p0 D 1. Then there exists a positive constant C D C.˛; p; q/ such that  ˛  j˛j 2 3 1 1 @ T .t/f  p 6 C.1 C jjt/.1 p / t 2 . q  p / 2 kf kLq x L for all t > 0;  2 R and f 2 Lq .R3 /3 . Using the TT  method and Lemma 2.3, we have the following homogeneous and inhomogeneous space-time estimates for our semigroup T .t/. Lemma 2.4 ([12, Lemma 3.3]) Let 2 < p < 6, and let  satisfy   3 1 5 3 1 5  6 < min ;  : 4 2p  2 4 2p

328

T. Iwabuchi et al.

Then there exists a positive constant C D C.p; / such that kT ./f kL .0;1WLp / 6 Cjj

n o 3  1 . 34  2p /

kf kL2

(10)

for all  2 R n f0g and f 2 L2 .R3 /3 . In particular, in the critical case 1= D 3=4  3=2p, (10) holds for all  2 R. Lemma 2.5 ([12, Lemma 3.4]) Let p; q and  satisfy 2 < p < 3;

1

1 1 1 1 6 < C p q p 3

and 

 

 1 3 1 1 2 1 1 3 1 1 max 0;    1 < 6   ; 2 2 q p p  2 2 q p respectively. Then there exists a positive constant C D C.p; q; / such that Z t     T .t  /Prf ./d    0

L .0;1ILp /

6 Cjj

n o  12  32 . 1q  1p / 1

kf k



L 2 .0;1ILq /

(11)



for all  2 R n f0g and f 2 L 2 .0; 1I Lq .R3 //. In particular, in the case 1= D 1=2  3.1=q  1=p/=2, (11) holds for all  2 R. In order to estimate the convection terms .u  r/v C .v  r/u, we first prepare the following inhomogeneous space-time estimate, which is a variant of Lemma 2.5. Lemma 2.6 Let p and q satisfy 2 < p < 3 and 1

1 1 1 1 6 < C ; p q p 3

7 1 1 1 1  < < C : 3p 3 q p 3

Then there exists a positive constant C D C.p; q/ such that Z t     T .t  /Prf ./d    0

L .0;1ILp /

6 Cjj

 32

n

1 1 3Cp



 1q

o

kf kL .0;1ILq /

for all  2 R n f0g; 1 6  6 1 and f 2 L .0; 1I Lq .R3 //. Proof Since P is bounded in Lq .R3 /, it follows from Lemma 2.3 that Z t     T .t  /Prf ./d    0

Z 6C Lp

t 0

k .t  /kf ./kLq d;

Stability of Time Periodic Solutions

329

where 2

3 1

1

1

k .t/ WD f1 C jjtg.1 p / t 2 . q  p / 2 : Hence by the Hausdorff-Young inequality we have Z t     T .t  /Prf ./d    0

L .0;1ILp /

Z t     q 6 C k .t  /kf ./k d  L   0

L .0;1/

6 C kk kL1 .0;1/ kf kL .0;1ILq / : Here, by the change of variable t ! t=jj, we see that Z kk kL1 .0;1/ D

1 0

D jj

1

1

.1 C jjt/  32

n

1 1 3Cp



 1q

1 2p

o

Z

t

3 1 1 1 2 . q  p /C 2

1 0

dt

1 .1 C t/

1 1 2p

t

3 1 1 1 2 . q  p /C 2

dt:

(12)

By the assumptions on p and q, the last integral in (12) converges. This completes the proof of Lemma 2.6. t u

3 Nonlinear Estimates In this section, we shall prove several bilinear estimates. First, we recall the following bilinear estimates in the Sobolev spaces of fractional order. Lemma 3.1 ([10, Lemma 2.7]) Let s; p and q satisfy 0 6 s < 3;

1 1 s s < < C ; 3 p 2 6

1 2 s D  : q p 3

Then there exists a positive constant C D C.s; p; q/ such that kfgkWP s;q 6 Ckf kWP s;p kgkWP s;p P s;p .R3 /. for all f ; g 2 W Combining Lemmas 2.5 and 3.1, we have the following bilinear estimates for the nonlinear term .u  r/u. Lemma 3.2 ([12, Lemma 4.2]) Let s; p and  satisfy 3 0 0. Since 1= < 5=8  3=2p C s=4 < 5=4  3=p C s=2, the time integrals on the right hand side of (17) converge and

Z

t 0

Z

t 0

.t  / 3

. 3p C 2s C 14 / 0

1

 0

 10 d

.t  /. p C 2 C 4 /. 2 / d s





1 . 2 /0

5

3

1

D Ct. 4  p C 2 /  ; D Ct

s

n o 3 2 . 58  2p C 4s / 1

:

P s .R3 /3 for t > 0. Similarly, we see that u 2 This shows that u.t/ 2 H s 3 3 P C.Œ0; 1/I H .R // . This completes the proof of Theorem 1.1. u t Acknowledgements T. Iwabuchi is partially supported by JSPS Grant-in-Aid for Young Scientists (B) #25800069. A. Mahalov is partially supported by NSF DMS grant 1419593. R. Takada was partially supported by JSPS Grant-in-Aid for Research Activity Start-up #25887005.

Stability of Time Periodic Solutions

335

References 1. A. Babin, A. Mahalov, B. Nicolaenko, Long-time averaged Euler and Navier-Stokes equations for rotating fluids, in Advanced Series in Nonlinear Dynamics, vol. 7 (World Scientific Publishing, River Edge, NJ, 1995), pp. 145–157 2. A. Babin, A. Mahalov, B. Nicolaenko, Regularity and integrability of 3D Euler and NavierStokes equations for rotating fluids. Asymptot. Anal. 15, 103–150 (1997) 3. A. Babin, A. Mahalov, B. Nicolaenko, Global regularity of 3D rotating Navier-Stokes equations for resonant domains. Indiana Univ. Math. J. 48, 1133–1176 (1999) 4. A. Babin, A. Mahalov, B. Nicolaenko, 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity. Indiana Univ. Math. J. 50, 1–35 (2001) 5. J.-Y. Chemin, B. Desjardins, I. Gallagher, E. Grenier, Anisotropy and dispersion in rotating fluids, in Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, Paris, 1997/1998, vol. XIV. Studies in Mathematics and Its Applications, vol. 31, (North-Holland, Amsterdam, 2002), pp. 171–192 6. J.-Y. Chemin, B. Desjardins, I. Gallagher, E. Grenier, Mathematical Geophysics. Oxford Lecture Series in Mathematics and Its Applications, vol. 32 (The Clarendon Press/Oxford University Press, Oxford, 2006) 7. F. Flandoli, A. Mahalov, Stochastic three-dimensional rotating Navier-Stokes equations: averaging, convergence and regularity. Arch. Ration. Mech. Anal. 205, 195–237 (2012) 8. M. Hieber, Y. Shibata, The Fujita-Kato approach to the Navier-Stokes equations in the rotational framework. Math. Z. 265, 481–491 (2010) 9. T. Iwabuchi, R. Takada, Time periodic solutions to the Navier-Stokes equations in the rotational framework. J. Evol. Equ. 12, 985–1000 (2012) 10. T. Iwabuchi, R. Takada, Global solutions for the Navier-Stokes equations in the rotational framework. Math. Ann. 357, 727–741 (2013) 11. T. Kato, Strong Lp -solutions of the Navier-Stokes equation in Rm , with applications to weak solutions, Math. Z. 187, 471–480 (1984) 12. Y. Koh, S. Lee, R. Takada, Dispersive estimates for the Navier-Stokes equations in the rotational framework. Adv. Differ. Equ. 19, 857–878 (2014) 13. P. Konieczny, T. Yoneda, On dispersive effect of the Coriolis force for the stationary NavierStokes equations. J. Differ. Equ. 250, 3859–3873 (2011) 14. H. Kozono, M. Nakao, Periodic solutions of the Navier-Stokes equations in unbounded domains. Tohoku Math. J. (2) 48, 33–50 (1996) 15. H. Kozono, T. Ogawa, On stability of Navier-Stokes flows in exterior domains. Arch. Ration. Mech. Anal. 128, 1–31 (1994) 16. H. Kozono, Y. Mashiko, R. Takada, Existence of periodic solutions and their asymptotic stability to the Navier-Stokes equations with the Coriolis force. J. Evol. Equ. 14, 565–601 (2014) 17. P. Maremonti, Existence and stability of time-periodic solutions to the Navier-Stokes equations in the whole space. Nonlinearity 4, 503–529 (1991) 18. P. Maremonti, Some theorems of existence for solutions of the Navier-Stokes equations with slip boundary conditions in half-space. Ricerche mat. 40, 81–135 (1991) 19. Y. Taniuchi, On stability of periodic solutions of the Navier-Stokes equations in unbounded domains. Hokkaido Math. J. 28, 147–173 (1999) 20. M. Yamazaki, The Navier-Stokes equations in the weak-Ln space with time-dependent external force. Math. Ann. 317, 635–675 (2000) 21. M. Yamazaki, Solutions in the Morrey spaces of the Navier-Stokes equation with timedependent external force. Funkcial. Ekvac. 43, 419–460 (2000)

Weighted Lp  Lq Estimates of Stokes Semigroup in Half-Space and Its Application to the Navier-Stokes Equations Takayuki Kobayashi and Takayuki Kubo

Dedicated to Professor Yoshihiro Shibata on the occasion of his sixtieth birthday

Abstract We consider the Navier-Stokes equations in half-space and in Lp space with Muckenhoupt weight and show the Lp  Lq estimates of Stokes semigroup with hx0 is1 hxn isn type weight. Finally as the application of the weighted Lp  Lq estimates, we shall obtain the weighted asymptotic behavior of the solution to the Navier-Stokes equations. Keywords Half-space • Navier-Stokes equations • Weighted Lp  Lq estimates

1 Introduction Let n  2. In this paper, we shall consider the Navier-Stokes equations in half-space RnC defined by RnC D fx D .x0 ; xn / 2 Rn j xn > 0g and in weighted Lp space. The Navier-Stokes equations are given by 8 ˆ @t u  u C .u  r/u C r D 0; ˆ ˆ ˆ 0g. The weighted Lp space with Muckenhoupt weight w 2 Ap is defined by o n Lpw .RnC / D u 2 L1loc .RnC / j kukLpw .Rn / D kuw1=p kLp .RnC / C

(5)

for 1 < p < 1 (see [8] for the definition of Muckenhoupt class Ap ). In order to consider the Navier-Stokes equations by Kato’s argument [16], we need the Lp  Lq estimates of Stokes semigroup in the weighted Lp space. In [9, 10], Fröhlich proved that the Stokes operator ARnC is defined by ARnC D PRnC  with

340

T. Kobayashi and T. Kubo tA

n

Helmholtz decomposition PRnC and it generates an analytic semigroup e RC . In this paper, we fix w.x/ D wsp .x/ D hx0 is1 p hxn isn p as the weight function and assume 

n1 1 < s1 < .n  1/ 1  ;  p p



1 1 < sn < 1  : p p

(Cp )

We notice that w.x/ belongs to the Muckenhoupt class Ap .1 < p < 1/ when the exponent s D .s1 ; sn / of the weight function satisfies (Cp ). The main result is the following Lp  Lq estimates of Stokes semigroup with weight w.x/. Theorem 2.1 (Weighted Lp  Lq Estimates) Let n  2 and 1 < p  q < 1. Let s D .s1 ; sn / and s0 D .s01 ; s0n / satisfy 

n1 1 0  < s1  s1 < .n  1/ 1  ; q p



1 1 < s0n  sn < 1  q p

(Cp;q )

and let w.x/ D wsp .x/. (i) For a 2 Lpw .RnC /, the following estimate holds: for t > 0, n 1

0

.s1 Csn /.s01 Cs0n / 2

1

kws etA akLq .RnC /  Ct 2 . p  q / .1 C t/

kws akLp .RnC / :

(6)

(ii) For a 2 Lpw .RnC /, the following estimate holds: for t > 0, n 1

0

1

1

kws retA akLq  Ct 2 . p  q / 2 .1 C t/

.s1 Csn /.s01 Cs0n / 2

kws akLp .RnC / :

(7)

We next consider the application of weighted Lp  Lq estimates to NavierStokes equations (NS). By applying the Helmholtz projection PRnC to (NS), we can rewrite (NS) as follows: @t u C ARnC C PRnC Œ.u  r/u.t/ D 0;

u.0/ D u0 :

By Duhamel’s principle, we obtain the integral equations: u.t/ D e

tARn

C

Z u0 

t

e 0

.t /ARn

C

PRnC Œ.u  r/u./d:

(IE)

By using the weighted Theorem 2.1, we can obtain the following theorem: Theorem 2.2 Let n  2 and let s D .s1 ; sn / satisfy 0  s1 < .n  1/.1  1n / and 0  sn < 1  1n . Then there exists ı > 0 such that if a 2 Lnw; satisfied kws u0 kLn < ı,

Weighted Lp  Lq Estimates of Stokes Semigroup in Half-Space and Its. . .

341

then (IE) admits a unique solution u.t/ 2 BC.Œ0; 1/I Lnw; / satisfying s0

kw u.t/kLq s0

kw ru.t/k



.s Csn /.s01 Cs0n / n  12 C 2q  1 2 ; DO t

(8)



.s Csn /.s01 Cs0n / n 1C 2q  1 2 DO t

(9)

Lq

for n  q < 1; .n  1/=q < s01  s1 and 1=q < s0n  sn as t ! 1.

3 Preliminaries In this section, we shall introduce some definition and some lemmas which play important role for our proof. We first introduce the lemma concerning the weight function in Muckenhoupt class Ap . The weight w 2 Ap has the important property that regular singular integral operators are continuous on Lpw .Rn / into itself (see [22] for detail). Lemma 3.1 Let 1 < p < 1, w 2 Ap and let T be a regular singular integral operator. Then T is bounded on Lpw .Rn /. More precisely, there is a positive constant C such that for all f 2 Lpw .Rn /, we have kTf kLpw  Ckf kLpw : Remark 3.2 By Lemma 3.1, the Riesz transform Rj f and the partial Riesz transform Sj f defined by Rj f WD F1 Sj f WD

F1 0

 

ij Fx Œf ./ ; jj

ij 0 Fx0 Œf . ; xn / ; j 0 j

j D 1; : : : ; n;

(10)

j D 1; : : : ; n  1;

(11)

are continuous on Lpw .Rn / and Lpw .RnC / into itself respectively. Here Fx and Fx0 denote the Fourier transform with respect to x and the partial Fourier transform with respect to x0 . We next introduce the Helmholtz decomposition of the weighted space Lpw .RnC / .1 < p < 1/. The following Helmholtz decomposition is proved by Fröhlich[9]: p P w1;p .Rn /, where Lpw .RnC / D Lw; .RnC / ˚ r H C Lpw; .RnC / D fu 2 C01 j r  u D 0

in RnC g

kkLp .Rn

P w1;p .RnC / D f 2 L1loc .RnC / j r 2 Lpw .RnC /g: H

w

C

/

;

342

T. Kobayashi and T. Kubo p

Their result implies that the Helmholtz projection PRnC from Lpw .RnC / to Lw; .RnC / p is bounded. By the Helmholtz projection, the Stokes operator ARnC in Lw; .RnC / is p defined by ARnC D PRnC  with domain D.ARnC / D fu 2 Ww2;p \ Lw; .RnC / j n u D 0 on @RC g. Moreover in [10], he proved that the Stokes operator generates an tA

n

analytic semigroup e RC . For simplicity, we often use the abbreviations A for ARnC and ARn and P for PRnC and PRn if there is no confusion.

4 Proof of Theorem 2.1 In this section, we consider the weighted Lp  Lq estimates of Stokes semigroup in half-space. First step is the following lemma which says the weighted Lp  Lq estimates for the whole space. Lemma 4.1 Let n  2, 1 < p  q < 1 and let s D .s1 ; sn / and s0 D .s01 ; s0n / satisfy the condition (Cp;q ). Then the following estimate holds. 0

kws @kt r ˛ etA Pf kLq .Rn / n 1

1

 Ct 2 . p  q /

j˛j 2 k

.1 C t/

.s1 Csn /.s01 Cs0n / 2

kws f kLp .Rn /

(12)

for k 2 N0 D N [ f0g, ˛ 2 Nn0 and t > 0. Proof We recall the following estimate proved by Murata [21]. Z

r

E.x  y/hyir dr  M.1 C t/ 2 Rn

.0  r < n/:

(13)

jxj2

where En .x/ D En .t; x/ D .4t/ 2 e 4t . The following estimates are obtained from (13): Z r E1 .xn  yn /hyn ir dyn  M.1 C t/ 2 .0  r < 1/; (14) n

R

Z

Rn1

r

En1 .x0  y0 /hy0 ir dy0  M.1 C t/ 2

.0  r < n  1/:

(15)

We shall prove (12) by using these estimates. Since we can prove the case j˛j ¤ 0 or k > 0 in the same way as j˛j D k D 0, we shall prove only the case where j˛j D 0; k D 0. We first prove the case where 0  s01  s1 < .n  1/.1  1p / and 0  s0n  sn  1  1p . Taking the fact that the Stokes semigroup in Rn is described q by using heat kernel etA f D E f for f 2 Lw; .Rn / into account, it is sufficient to

Weighted Lp  Lq Estimates of Stokes Semigroup in Half-Space and Its. . .

343

estimate E f . Since we see j.E f /.x/j ˇ ˇZ   1r  1q  ˇ p sqr 1 ˇ .ws .y/f .y//p Enr .x  y/ q dyˇˇ jws .y/f .y/j1 q w qr .y/Enr .x  y/ D ˇˇ n R

Z

p

 kw

s

for 1 C

1 f kLp q

1 q

D

sqr

w.y/

 qr

Rn

1 p

Enr .x

 y/dy

 1r  1q Z

jw

s

Rn

.y/f .y/jp Enr .x

 y/dy

 1q

C 1r , we obtain

0

q

kws .E f /kLq qp

 kws f kLp  qr 1 Z Z Z qr  qr r s s0  .w .y/w .x// En .x  y/dy Rn

Rn

Z

qr

q

D kwf kLp sup

Rn

x

.w.y/w0 .x// qr Enr .x  y/dy

Rn

 jws f .y/jp Enr .x  y/dy dx

 qr 1 Z

Enr .x/dx:

We notice that Z

qr

0

Rn

.ws .y/ws .x// qr Enr .x  y/dy

Z Z qr qr 0 0 r D .hy0 is1 hx0 is1 / qr En1 .x0  y0 /dy0 .hyn isn hxn isn / qr E1r .xn  yn /dyn : Rn1

R

Here we know that hzia En .z/  Ct 2 .1 C t/ 2 e n

Z Rn

hyia Enr .x  y/dy 

a

Z Rn

jzj2 8t

for a  0 and

hyia En .x  y/Enr1 .x  y/dy0

 Mt 2 .r1/ .1 C t/ 2 n

a

for 0  a < n2 by (14). Since hxia  C.hx  yia C hyia / for a > 0, by (14) and (15) we have Z qr 0 r .hy0 is1 hx0 is1 / qr En1 .x0  y0 /dy0 Rn1

Z

C

hy i Rn1

 Ct

s qr

1 0  qr

s01 qr 2.qr/

Z Rn1

 s01 qr s01 qr 0 0 qr 0 qr r hx  y i En1 C hy i .x0  y0 /dy0 s1 qr

r hy0 i qr EQ n1 .x0  y0 /dy0

344

T. Kobayashi and T. Kubo

Z

hy0 i

CC

.s1 s01 /qr qr

Rn1

 Ct

s01 qr s1 qr n1 2 .r1/C 2.qr/  2.qr/

D Ct

.s1 s01 /qr n1 2 .r1/ 2.qr/

where EQ n .x/ D .4t/ 2 e n

Z

jxj2 8t

C Ct

.s1 s01 /qr n1 2 .r1/ 2.qr/

;

. Therefore we see qr

0

Rn

r En1 .x0  y0 /dy0

.ws .y/ws .x// qr Enr .x  y/dy

 Ct

.s1 s01 /qr n1 2 .r1/ 2.qr/

D Ct 2 .r1/ n

1

Ct 2 .r1/

.s1 s01 /qr .sn s0n /qr 2.qr/  2.qr/

.sn s0n /qr 2.qr/

;

which implies 0

1

1

kws .E f /kLq  Ct 2 .r1/. r  q / n

1

D Ct 2 .1 r / n

n 1

1

.s1 Csn /.s01 s0n / 2

.s1 Csn /.s01 Cs0n / 2

D Ct 2 . p  q /

.s1 Csn /.s01 Cs0n / 2

1

t 2 . q  q / kws f kLp n r

kws f kLp kws f kLp :

Therefore we obtain the estimate for 0  s01  s1 < .n  1/.1  1p / and 0  s0n  sn < 1  1p . Since we can obtain the other cases in a similar way, we may omit the proof for the other cases. Therefore we obtain the desired result. t u By using Lemma 4.1, we shall prove Theorem 2.1. Proof of Theorem 2.1 In half-space Rn , we have the solution formula obtained by Ukai [23]. Let Rj and Sj be the Riesz transform and the partial Riesz transform defined by (10) and (11). And let rf D f jRnC and let  f j@RnC and e0 be zero extension operator from RnC to Rn with value 0. Finally, let E.t/ be the solution operator for the heat equations in RnC , which is derived from E0 .t/ by odd extension from RnC to Rn . Then the solution .u.t/; .t// of the non-stationary Stokes equations in RnC is u.t/ D WE.t/Vu0 ;

.t/ D Dr .@n E.t/V1 u0 / ;

where

 I SU WD ; 0 U

 V1 VD V2

Weighted Lp  Lq Estimates of Stokes Semigroup in Half-Space and Its. . .

345

with S Dt .S1 ; : : : ; Sn1 /;

R0 Dt .R1 ; : : : ; Rn1 /;

U D R0  S.R0  S C Rn /e0 ;

V1 a D a0 C San ;

V2 a D S  a0 C an :

and D is the Poisson operator for the Dirichlet problem of the Laplace equation in RnC . By Remark 3.2, we can reduce to the whole space case, so that we can obtain the following estimate for the half-space. t u

5 Proof of Theorem 2.2 In this section, we shall show the proof of Theorem 2.2. Our method is based on the contraction mapping principle with Theorem 2.1. Since this method is well-known, we shall describe the outline of the proof (see [7] for example). As the underlying space, we set ˚ I" D u 2 BC.Œ0; 1/ W Lnw; .RnC // j

 lim fŒu./  an:0;t C Œup;.p/;t C Œrun; 1 ;t g D 0; jjjujjjt < " ; 2

t!0C

where p 2 .n; 1/ and s 2 Œ0; n  1/ are fixed numbers, .p/ D small number determined late and

1 2



n 2p ,

" > 0 is a

  Œup;`;t D sup  ` kws u.; /kLq ; 0 0, as follows. ˚ 3 XT D f 2 C.Œ0; T/I .L 2 .R3C //3 / j kf kXT D sup kf .t/k ˚

0 0 the heat capacity, and l./ D ŒŒ 0 ./ D ŒŒ./ the latent heat. Further di ./ > 0 denotes the coefficient of heat conduction in Fourier’s law, i ./ > 0 the viscosity in Newton’s law, 1 ; 2 > 0 the constant, positive densities of the phases, and > 0 the (coefficient of) surface tension. In the sequel we drop the index i, as there is no danger of confusion; we just keep in mind that the coefficients depend on the phases. In the previous papers [14, 15] we have mathematically analyzed the following problem with sharp interface: Find a family of closed compact hypersurfaces f.t/gt0 contained in  and N ! Rn , and ;  W RC   N ! R such appropriately smooth functions u W RC   that .@t u C u  ru/  div T D 0 T D 2./D.u/  I;

div u D 0

u.0/ D u0

./.@t  C u  r/  div .d./r/  2jD.u/j22 D 0 l./j  ŒŒd./@   D 0; @  D 0 on @; ŒŒ ./ C ŒŒ

in  n .t/;

1 ŒŒu D ŒŒ j  

1 ŒŒ j2   ŒŒT   D H  ;  u D 0 on @;

in  n .t/;

ŒŒ D 0 .0/ D 0

1 2 T     D 0 j  ŒŒ 2 2  V  D u    D u   

on .t/; in : in  n .t/; on .t/;

(1) (2)

in : on .t/;

1 j 

on .t/;

(3)

.0/ D 0 : This model is explained in more detail in [12]; see also [1, 6]. It is thermodynamically consistent in the sense that in absence of exterior forces and heat sources, the total mass and the total energy are preserved, and the total entropy is nondecreasing.

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This model is in some sense the simplest non-trivial sharp interface model for incompressible Newtonian two-phase flows taking into account phase transitions driven by temperature. Note that in this model neither kinetic undercooling nor temperature dependence of the surface tension , i.e. Marangoni forces, have been taken into account. It is the aim of the present paper to remove these shortcomings. Here we concentrate on the case of constant densities i > 0 which are not equal, i.e. ŒŒ ¤ 0. To achieve this goal, the model has to be adjusted carefully. Surface tension . / here is precisely the free surface energy density. Therefore we define, in analogy to the bulk case, the surface energy density  , the surface entropy density  , the surface heat capacity  , and surface latent heat l by means of the relations  . / D . / C   . /;  . / D  0 . /  . / D 0 . / D  00 . /; l . / D  0 . /: Then total surface energy will be Z E D



 . /d;

and total surface entropy reads Z ˆ D



 . /d:

Note that in case D const we have E D jj and ˆ D 0. We point out that experiments have shown that in certain situations surface heat capacity cannot be neglected, see [2]. In case  is not identically zero (i.e. if is not a linear function of  ), there will also be a non-trivial surface heat flux q which we assume to satisfy Fourier’s law, that is q D d . /r  ; with coefficient of surface heat diffusivity dR . / > 0. In analogy to the bulk, the surface heat flux induces the contribution  .q  r /=2 d to the production of surface entropy. R Kinetic undercooling with coefficient . / > 0 produces surface entropy  ./j2 = d. Following the derivation in [12], this leads to the following three modifications of the system (1)–(3). The momentum balance on the interface becomes 1 ŒŒ j2   ŒŒT   D . /H  C 0 . /r  : 

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The energy balance on the interface reads  . /

D  C div q D ŒŒd./@   C l./j C l . /div u C . /j2 ; Dt

and the Gibbs-Thomson law changes to ŒŒ ./ C ŒŒ

T    1 2  D . /j : j  ŒŒ 2 2 

Here D=Dt denotes the Lagrangian derivative coming from the velocity u of the interface, and we employ the symbol P for the orthogonal projection onto the tangent bundle of . Note that  D j is the trace of  on  as ŒŒ D 0. We assume the tangential part of u to be continuous across , i.e. ŒŒP u D 0, and P u D P uj . Then the quantities j , V and u can be expressed as j D ŒŒu   =ŒŒ1=;

V D ŒŒu   =ŒŒ;

u  D P  u C V   :

The complete extended model now reads as follows. In the bulk  n .t/: .@t u C u  ru/  2div../D.u// C r D 0; 2D.u/ D ru C ŒruT ;

div u D 0;

(4)

./.@t  C u  r/  div.d./r/ D 2./jD.u/j22 : On the interface .t/: ŒŒP u D 0; P u D P uj ; ŒŒu    D ŒŒ1=j ; ŒŒ D 0;  D j ; ŒŒ1=j2   2ŒŒ./D.u/   C ŒŒ  D . /H  C 0 . /r  ; 

D   div .d . /r  / D Dt

(5)

D ŒŒd./@  C l./j C . /j2 C l . /div u ; ŒŒ ./ C ŒŒ1=22 j2  2ŒŒ./D.u/    = C ŒŒ= D . /j ; V D u    j =: On the outer boundary @: u D 0;

@  D 0:

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Initial conditions: .0/ D 0 ;

u.0/ D u0 ;

.0/ D 0 :

This model has conservation of total mass and total energy, and total entropy is non-decreasing. Indeed, along smooth solutions we have d .ˆb .t/ C ˆ .t// D dt

Z Z



C 

Œ2./jD.u/j22 = C d./jrj2 = 2 dx

Œd . /jr  j2 =2 C . /j2 = d  0;

R

where ˆb D  ./dx. For the sake of well-posedness we assume that i and are strictly concave. Experiments show that is also strictly decreasing and positive for low temperatures. Therefore, has precisely one zero c > 0 which we call the critical temperature. As the problem in the range  > c is no longer well-posed, we restrict our attention to the interval  2 .0; c /. In all of the paper we impose the following assumptions. (a) Regularity. i ; di ; d ;  2 C2 .0; c /;

i;

2 C3 .0; c /:

(b) Well-posedness. i ;  ; i ; di ; d ; > 0;   0 in .0; c /;

N 0 < 0 < c in :

(c) Compatibilities. div u0 D 0 in  n 0 ; 2P0 ŒŒ.0 /D.u/ 0  C 0 .0 /r0 0 D 0;

P0 u .0/ D P0 u0 j0

P0 ŒŒu0  D 0; ŒŒ0  D 0;  .0/ D 0 j0 ; u0 j@ D 0; @ 0 j@ D 0: Below we present a rather complete analysis of problem (4), (5) which parallels that in [15] where the simpler case > 0 constant and  0 has been studied. We obtain local well-posedness of the problem in an Lp -setting, prove that the stability properties of equilibria are the same as in [15], and we show that any bounded solution that does not develop singularities converges to an equilibrium as t ! 1 in the state manifold SM which is the same as in [15]. We refer to [11, 13] for related results, and to [3, 4] for background material on maximal regularity for parabolic boundary value problems.

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2 Total Entropy and Equilibria (a) As we have seen in Sect. 1, the total mass M D Z E D E.u; ;  ; / WD

n

R n

 dx and the total energy

f.=2/juj2 C "./g dx C

Z 

 . / d

are conserved, and the total entropy Z ˆ D ˆ.;  ; / WD

Z ./ dx C

n



 . / d

is nondecreasing along smooth solutions. Even more, ˆ is a strict Lyapunov functional in the sense that it is strictly decreasing along smooth solutions which are non-constant in time. Indeed, if at some time t0  0 we have d ˆ.u.t0 /; .t0 // D 0; dt then Z 

Œ2. /jD.u/j2 = Cd. /jr j2 = 2 dxC

Z 

Œd . /jr j2 = 2 C. /j2  d D 0;

which yields D.u.t0 // D 0 and r.t0 / D 0 in , as well as r  .t0 / D 0 and j .t0 / D 0 on .t0 /. As in [15] this implies u.t0 / D 0 and .t0 / D const D  .t0 / in . From the equations we see that .V .t0 /; j .t0 // D .0; 0/ and therefore  is also constant in the components of the phases, and ŒŒ D . /H ; ŒŒ . / C ŒŒ= D 0:

These relations show that the curvature H is constant over all of  , and it determines the values of the pressures in the phases, in particular  is constant in each phase, not only in its components. Since  is bounded, we may conclude that .t0 / is a union of finitely many, say m, disjoint spheres of equal radius, i.e. .u.t0 /; .t0 /; .t0 // is an equilibrium. Therefore, the limit sets of solutions in the state manifold SM , to be defined below, are contained in the .mn C 2/dimensional manifold of equilibria ED

˚

0;  ;

[ 1lm

 SR .xl / W  2 .0; c /; BN R .xl /  ;  BN R .xl / \ BN R .xk / D ;; k ¤ l ;

(6)

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where SR .xl / denotes the sphere with radius R and center xl . Here R > 0 is uniquely determined by the total mass and by the number m of spheres, and  is uniquely given by the total energy. (b) Another interesting observation is the following. Consider the critical points of the functional ˆ.u; ;  ; / with constraint M D M0 , E.u; ;  ; / D E0 , say on N n /nC1 ;  2 MH2 ./;  2 C./g; f.u; ;  ; / W .u;  / 2 BUC.

see below for the definition of MH2 ./. So here we do not assume from the beginning that  is continuous across  , and  denotes surface temperature. Then by the method of Lagrange multipliers, there are constants ;  2 R such that at a critical point .u ;  ;  ;  / we have ˆ0 .u ;  ;  ;  / C M0 . / C E0 .u ;  ;  ;  / D 0:

(7)

The derivatives of the functionals are given by hˆ0 .u; ;  ; /j.v; #; # ; h/i D .0 . /j#/ C .0 . /j# /  .ŒŒ. / C  . /H./jh/ ; hM0 ./jhi D .ŒŒjh/ ;

with H./ WD H , and hE0 .u; ;  ; /j.v; #; # ; h/i D .ujv/ C . 0 . /j#/ C .0 . /j# /  .ŒŒ.=2/juj2 C . / C  . /H./jh/ :

Setting first .u; # ; h/ D .0; 0; 0/ and varying # in (7) we obtain 0 . / C  0 . / D 0

in ;

and similarly varying # yields 0 . / C 0 . / D 0 on  :

The relations . / D  0 . / and . / D . /   0 . / imply 0 D 00  . /.1 C  /, and this shows that  D 1= is constant in , since . / D  00 . / > 0 for all  2 .0; c / by assumption. Similarly on  we obtain  D 1= constant as well, provided  . / > 0, hence in particular    .

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Next varying v implies u D 0, and hence u D 0 as  ¤ 0. Finally, varying h we get .ŒŒ. / C  . /H /  ŒŒ  .ŒŒ. / C  . /H. // D 0 on  :

This implies with the above relations ŒŒ . / C . /H. / D ŒŒ :

Since  is constant and assuming  2 .0; c /, we see with  > 0 that H. / is constant. Therefore, as  is bounded,  is a sphere whenever connected, and a union of finitely many disjoint spheres of equal size otherwise. Thus the critical points of the entropy functional for prescribed energy are precisely the equilibria of the problem (4), (5). (c) Going further, suppose we have an equilibrium e WD .0;  ;  ;  / where the total entropy has a local maximum w.r.t. the constraints M D M0 and E D E0 . Then D WD Œˆ C M C E00 .e / is negative semi-definite on the kernel of .M0 ; E0 /.e /, where  and  are the fixed Lagrange multipliers found above. The kernel of M0 .e/ is given by .1jh/ D 0, as ŒŒ ¤ 0, and that of E0 .e/ is determined by the identity .ujv/ C .. /j#/ C . . /j# /  .ŒŒ. / C  . /H./jh/ D 0;

which at equilibrium yields . j#/ C . j# / D 0;

(8)

where  WD . /,  WD  . /, and  D . /. On the other hand, a straightforward calculation yields with z D .v; #; # ; h/ hD zjzi D .vjv/ C

1 . #j#/ C . # j# /    .H 0 . /hjh/ : 

(9) As  and  are positive, we see that the form hDzjzi is negative semi-definite as soon as H 0 . / is negative semi-definite. We have H 0 . / D .n  1/=R2 C  ;

where  denotes the Laplace-Beltrami operator on  and R means the radius of an equilibrium sphere. To derive necessary conditions for an equilibrium e to be a local maximum of entropy, we suppose that  is not

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connected, i.e.  is a finite union of spheres k . Set # D # D 0, and let P k h D hk be constant on  with k hk D 0. Then the constraint (8) holds, and with the volume !n of the unit sphere in Rn hDzjzi D .   /..n  1/=R2 /!n Rn1 

X

h2k > 0;

k

hence D cannot be negative semi-definite in this case, as  > 0. Thus if e is an equilibrium with locally maximal total entropy, then  must be connected, and hence both phases are connected. On the other hand, if  is connected then H 0 . / is negative semi-definite on functions with mean zero, hence in this case D is in fact positive semidefinite. We will see below that connectedness of  is precisely the condition for stability of the equilibrium e . (d) Summarizing, we have shown • The total energy is constant along smooth solutions of (4), (5). • The negative total entropy is a strict Lyapunov functional for (4), (5). • The equilibria of (4), (5) are precisely the critical points of the entropy

functional with prescribed total energy and total mass. • If the entropy functional with prescribed energy and total mass has a local maximum at e D .0;  ;  ;  /, then  is connected.

It should be noted that we are using the term equilibrium to describe a stationary solution of the system, while a thermodynamic equilibrium would—by definition—require the entropy production to be zero. Our results show that stable equilibria are precisely those that guarantee the system to be in a thermodynamic equilibrium.

3 Local Well-Posedness and the Semiflow  is not a real system variable as it is the trace of  on the interface. For analytical reasons it is a useful quantity while we consider the linear problem and the nonlinear problem on the reference manifold † after a Hanzawa transform. Local well-posedness of Problem (4), (5) is based on maximal Lp -regularity of its principal linearization and on the contraction mapping principle.

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3.1 Principal Linearization The principal part of the linearized problem reads as follows @t u  0 .x/u C r D fu div u D gd P† ŒŒu C c.t; x/r† h D P† gu 2ŒŒ0 .x/D.u/ † CŒŒ †  0 .x/† h †  1 .x/r† † D g uD0 u.0/ D u0 .x/@t   d.x/ D .x/f 0 .x/@t †  d0 .x/† † D 0 .x/g ŒŒ D 0; .0/ D 0

in ;

† D j†

in  n †;

in  n †; in  n †; on †; on †; on @; in : (10)

on †; on †;

@  D 0

on @;

† .0/ D 0 j†

on †;

ŒŒ@t h  ŒŒu  †  C b.t; x/  r† h D ŒŒfh

(11)

on †;

2ŒŒ.0 .x/=/D.u/ †  †  C ŒŒ= D gh

on †;

h.0/ D h0

on †:

(12)

Here 0 ; 0 , d0 , 0 , d0 , 0 and 1 are functions of x, which are realized by 0 .x/ D .0 .x// and so on. The difference between the linear problem for variable surface tension (10)–(12) and the linear problem for constant surface tension [15, Sect. 3.1] is the fourth equation of (10) and the second equation of (11). Observe that (11) decouples from the remaining problems (10)–(12). Maximal Lp -regularity of (11) has been proved in [16], while maximal Lp -regularity of (10) and (12) has been considered in [15]. Therefore we obtain the maximal Lp -regularity of (10)–(12). N i /, 0 , d0 , 0 , Theorem 3.1 Let p > n C 2, i > 0, 2 ¤ 1 , 0i ; 0i ; d0i 2 C.

1 2 C.†/, 0i ; 0i ; d0i > 0, 0 ; d0 ; 0 ; 1 > 0, i D 1; 2, .b; c/ 2 Wp11=2p .JI Lp .†//nC1 \ Lp .JI Wp21=p .†//nC1 ;

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where J D Œ0; a. Then the coupled system (10)–(12) admits a unique solution .u; ; ; † ; h/ with regularity .u; / 2 Hp1 .JI Lp .//nC1 \ Lp .JI Hp2 . n †//nC1 DW Eu; .J/; P p1=p .†//; ŒŒu  †  2 Hp1 .JI W

P p1 . n †// DW E .J/;  2 Lp .JI H

† 2 Hp1 .JI Wp1=p .†// \ Lp .JI Wp21=p .†// DW Etr .J/; i WD j@i 2 Wp1=21=2p .JI Lp .†// \ Lp .JI Wp11=p .†// DW Etr .J/; i D 1; 2; h 2 Wp21=2p .JI Lp .†// \ Hp1 .JI Wp21=p .†// \ Lp .JI Wp31=p .†// DW Eh .J/; if and only if the data .fu ; f ; g ; gd ; P† gu ; g; fh ; gh ; u0 ; 0 ; 0 j† ; h0 / satisfy the following regularity (a) .fu ; f / 2 Lp .JI Lp .//nC1 , 1=p (b) g 2 Lp .JI Wp .†//, 1 P p1 .// \ Lp .JI Hp1 . n †//, (c) gd 2 Hp .JI H 1=21=2p

11=p

.JI Lp .†//nC1 \ Lp .JI Wp .†//nC1 , (d) .g; gh / 2 Wp 11=2p 21=p (e) .P† gu ; fh / 2 Wp .JI Lp .†//n \ Lp .JI Wp .†//n , 22=p 23=p 32=p nC1 (f) .u0 ; 0 ; 0 j† ; h0 / 2 X WD Wp . n †/  Wp .†/  Wp .†/, and compatibility conditions: (g) div u0 D gd .0/ in  n †, (h) P† ŒŒu0  C c.0; /r† h0 D P† gu .0/ on †, (i) P† ŒŒ0 ./.ru0 C Œru0 T / †   1 ./r† † D P† g.0/ on †. The solution map Œ.fu ; f ; g ; gd ; P† gu ; g; fh ; gh ; u0 ; 0 ; 0 j† ; h0 / 7! .u; ; ; † ; h/ is continuous between the corresponding spaces. Remark 3.2 X is the time trace space of the solution space E.J/ with E.J/ WD Eu .J/  E .J/  Etr .J/  Eh .J/:

3.2 Local Existence As in [15], the basic result for local well-posedness of Problem (4), (5) in an Lp -setting is the following theorem, which is proved by the contraction mapping principle.

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Theorem 3.3 Let p > n C 2, 1 ; 2 > 0, 1 ¤ 2 . We assume the conditions (a), (b), (c) in Sect. 1 and the regularity conditions .u0 ; 0 / 2 Wp22=p . n 0 /nC1 ;

0 2 Wp32=p :

Then there exists a unique Lp -solution of Problem (4), (5) on some possibly small but nontrivial time interval J D Œ0; . 32=p

Here the notation 0 2 Wp means that 0 is a C2 -manifold, such that its 22=p (outer) normal field 0 is of class Wp .0 /. Therefore, the Weingarten tensor 12=p L0 D r0 0 of 0 belongs to Wp .0 / which embeds into C˛C1=p .0 /, with ˛ D 1  .n C 2/=p > 0 since p > n C 2 by assumption. For the same reason we N i .0//n , 0 2 C1C˛ . N i .0//, i D 1; 2, and V0 2 C1C˛ .0 /. also have u0 2 C1C˛ . The notion Lp -solution means that .u; ; ;  ; / is obtained as the push-forward N N† ; h/ of the transformed problem, which means that N ; of an Lp -solution .Nu; ; N N .Nu;  ;  ; h/ belongs to E.J/. The regularity of the pressure is obtained from the equations.

3.3 Time-Weights For later use we need an extension of the local existence result to spaces with time weights. For this purpose, given a UMD-Banach space Y and  2 .1=p; 1, we define for J D .0; t0 / s Kp; .JI Y/ WD fu 2 Lp;loc .JI Y/ W t1 u 2 Kps .JI Y/g;

where s  0 and K 2 fH; Wg. It has been shown in [10] that the operator d=dt in Lp; .JI Y/ with domain 1 D.d=dt/ D 0 H 1p; .JI Y/ D fu 2 Hp; .JI Y/ W u.0/ D 0g

is sectorial and admits an H 1 -calculus with angle =2. This is the main tool to extend Theorem 3.3 to the time weighted setting, where the solution space E.J/ is replaced by E .J/, and z 2 E .J/ , t1 z 2 E.J/: The trace spaces for .u; ; h/ for p > 3 are then given by .u0 ; 0 / 2 Wp22=p . n †/nC1 ; 0 j† 2 Wp23=p .†/; h0 2 Wp2C2=p .†/; h1 WD @t hjtD0 2 Wp23=p .†/;

(13)

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where for the last trace we need in addition  > 3=2p. Note that the embeddings N i //nC1 ; E;u; .J/ ,! C.JI C1 .

E;h .J/ ,! C.JI C2C˛ .†// \ C1 .JI C1 .†//

with ˛ D 1=2  n=p > 0 require  > 1=2 C .n C 2/=2p, which is feasible since p > n C 2 by assumption. This restriction is needed for the estimation of the nonlinearities. For these time weighted spaces we have the following result. Corollary 3.4 Let p > n C 2,  2 .1=2 C .n C 2/=2p; 1, 1 ; 2 > 0, 1 ¤ 2 . We assume that the conditions (a), (b), (c) in Sect. 1 and the regularity conditions .u0 ; 0 / 2 Wp22=p . n 0 /nC1 ;

0 2 Wp2C2=p

are satisfied. Then the transformed problem admits a unique solution z D .u; ; † ; h/ 2 E .0; / for some nontrivial time interval J D Œ0; . The solution depends continuously on the data. For each ı > 0 the solution belongs to E.ı; /, i.e. it regularizes instantly.

4 Linear Stability of Equilibria 1. We call an equilibrium non-degenerate if the balls making up 1 .t/ do neither touch each other nor the outer boundary; this set is denoted by E. To derive the full linearization at a non-degenerate equilibrium e WD .0;  ; † ; †/ 2 E, note that the quadratic terms u  ru, u  r, jD.u/j22 , ŒŒuj , and j2 give no contribution to the linearization. Therefore we obtain the following fully linearized problem for .u; ; h/, the relative temperature # D .   /= and #† D #j† . @t u   u C r D fu div u D gd P† ŒŒu D P† gu 2P† ŒŒ D.u/ †    0 r† #† D P† g 2ŒŒ D.u/ †  †  C ŒŒ C  A† h 

 0 H #†

D g  †

in  n †; in  n †; on †; on †; on †;

uD0

on @;

u D u0

in ; (14)

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 @t #  d # D  f ŒŒ# D 0  @t #† d † #† .l = /j† ŒŒd @ † # 0 div† u†

in  n †; on †;

D  g

@ # D 0

on †; on @;

# D #0 2ŒŒ D.u/ †  † = C ŒŒ= C l #† C  j† D gh

on †;

@t h  ŒŒu  † =ŒŒ D fh

on †;

h.0/ D h0

on †;

in ; (15) (16)

where  D . /,  D . /, d D d. /,  D . /, l D l. /,  D . /, l D l . /,  D  . /, d D d . /, and A† D H 0 .0/ D .n  1/=R2  † ;

H D .n  1/=R :

Here we used that l = D 0 D 0 . /. Finally, u† denotes the transformed velocity field u , and j† is given by j† WD ŒŒu  † =ŒŒ1=: The time-trace space E of E.J/ is given by .u0 ; #0 ; #0 j† ; h0 / 2 E D ŒWp22=p . n †/nC1  Wp23=p .†/  Wp32=p .†/; and the space of right hand sides is ..fu ; f /; gd ; .fh ; P† gu /;.g; g ; gh // 2 F.J/ WD Fu; .J/  Fd .J/  Fh .J/nC1  F .J/nC2 ; where Fu; .J/ D Lp .J  /nC1 ;

P p1 .// \ Lp .JI Hp1 .//; Fd .J/ D Hp1 .JI H

and F .J/ D Wp1=21=2p .JI Lp .†// \ Lp .JI Wp11=p .†//; Fh .J/ D Wp11=2p .JI Lp .†// \ Lp .JI Wp21=p .†//:

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As the terms .l = /j† and 0 div† u† are lower order, the remaining system is triangular, where the equations for  decouple. Therefore, as in Sect. 3, it follows from the maximal regularity results in [5, 9, 15] and a standard perturbation argument that the operator L defined by the left hand side of (14)–(16) is an isomorphism from E into F  E . The range of L is determined by the natural compatibility conditions. If the time derivatives @t are replaced by @t C !, ! > 0 sufficiently large, then this result is also true for J D RC . 2. We introduce a functional analytic setting as follows. Set X0 D Lp; ./  Lp ./  Wp1=p .†/  Wp21=p .†/; where the subscript means solenoidal, and define the operator L by L.u; #; #† ; h/ D   . =/u C r=; .d = /#;

  .1= /.d † #† C .l = /j† C ŒŒd @ † # C 0 div† u† /; ŒŒu  † =ŒŒ :

To define the domain D.L/ of L, we set X1 D f.u; #; #† ; h/ 2 Hp2 . n †/nC1  Wp21=p .†/  Wp31=p .†/ W div u D 0 in  n †; P† ŒŒu D 0; ŒŒ# D 0 on †; u D 0; @ # D 0 on @g; and D.L/ D f.u; #; #† ; h/ 2 X1 W 2P† ŒŒ D.u/ †  C  0 r† #† D 0 on †g:  is determined as the solution of the weak transmission problem .rjr=/2 D .. =/ujr/2;

 2 Hp10 ./;  D 0 on †;

ŒŒ D   A† h C  0 #† H C 2ŒŒ .D.u/ † j † /;

on †;

ŒŒ= D 2ŒŒ. =/.D.u/ † j † /  l #   ŒŒu  † =ŒŒ1=

on †:

Let us introduce solution operators Tk , k 2 f1; 2; 3g, as follows 1 r D T1 .. =/u/ C T2 .  A† h C  0 #† H C 2ŒŒ .D.u/ † j † //  C T3 .2ŒŒ. =/.D.u/ † j † /  l #   ŒŒu  † =ŒŒ1=/:

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We refer to Köhne et al. [7] for the analysis of such transmission problems. The linearized problem can be rewritten as an abstract evolution problem in X0 . zP C Lz D f ;

t > 0;

z.0/ D z0 ;

(17)

where z D .u; #; #† ; h/, f D .fu ; f ; f† ; fh /, z0 D .u0 ; #0 ; #0 j† ; h0 /, provided .gd ; gu ; g; g ; gh / D 0. The linearized problem has maximal Lp -regularity, hence (17) has this property as well. Therefore, by a well-known result, L generates an analytic C0 -semigroup in X0 ; see for instance Proposition 1.1 in [8]. Since the embedding X1 ,! X0 is compact, the semigroup eLt as well as the resolvent . C L/1 of L are compact as well. Therefore, the spectrum .L/ of L consists of countably many eigenvalues of finite algebraic multiplicity, and it is independent of p. 3. Suppose that  with Re   0 is an eigenvalue of L. This means u   u C r D 0

in  n †;

div u D 0

in  n †;

P† ŒŒu D 0

on †;

2P† ŒŒ D.u/ †    0 r† #† D 0

on †;

2ŒŒ D.u/ †  †  C ŒŒ C  A† h   0 H #† D 0

on †;

uD0

on @;

 #  d # D 0 ŒŒ# D 0;

(18)

# D #†

 #†  d † #†  .l = /j†  ŒŒd @ † #  0 div† u† D 0 @ # D 0 2ŒŒ D.u/ †  † = C ŒŒ= C l #† C  j† D 0 on †;

in  n †; on †; on †; on @; (19)

ŒŒh  ŒŒu  †  D 0 on †: Observe that on † we may write uk D P† u C h † C j† † =k D u† C j† † =k ;

k D 1; 2:

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By this identity, taking the inner product of the problem for u with u and integrating by parts we get 0 D j1=2 uj22  .div T.u; ;  /ju/2 1=2

D j1=2 uj22 C 2j D.u/j22 C .ŒŒT.u; ;  / † jP† u C h † /† C .ŒŒT.u; ;  / †  † =jj† /† 1=2 N † hjh/† C l .#jj† /† D j1=2 uj22 C 2j D.u/j22 C  .A 0 N C  jj† j2†   0 H .#jh/ †    .r† #jP† u/† ;

since ŒŒT.u; ;  / †  D  A† h †   0 #† H †   0 r† #† and, moreover, ŒŒT.u; ;  / †  † = D l # C  j† . On the other hand, the inner product of the equation for # with # by an integration by parts and ŒŒ# D 0 leads to 1=2

0 D j. /1=2 #j22 C jd r#j22 C .ŒŒd @ † #j#/† 1=2

1=2

1=2

D .j. /1=2 #j22 C j #† j2† / C jd r#j22 C jd r† #† j2†  l .j† j#/† = C 0 .P† ujr† #† /† C  0 H .hj#/ where we employed ŒŒd @ † # D  #†  d † #†  .l = /j†  0 div† u† and u† D P† u C .u†  † / † . Adding the first identity to the second multiplied by  and taking real parts yields the important relation 1=2

0 D Re j1=2 uj22 C 2j D.u/j22 C  Re .A† hjh/† 1=2

C  .Re j. /1=2 #j22 C jd r#j22 / 1=2

(21)

1=2

C  jj† j2† C  .Re j #† j2† C jd r† #† j2† /: On the other hand, if Im  ¤ 0, taking imaginary parts separately we get 0 D Im j1=2 uj22  Im   .A† hjh/† C Im l .#jj† /† N  0 H .#jh/† g  Im  0 .r† #jP† u/†  Im f   1=2

0 D  Im .j. /1=2 #j22 C j #† j2† /  Im l .j† j#/† C Im f 0 H .hj#/† g C Im  0 .P† ujr† #/† ; hence 1=2

 .A† hjh/† D j1=2 uj22   .j. /1=2 #j22 C j #† j2† /:

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Inserting this identity into (21) leads to 1=2

1=2

1=2

0 D 2Re j1=2 uj22 C 2j D.u/j22 C  jd r#j22 C  jj† j2† C  jd r† #† j2† : This shows that if  is an eigenvalue of L with Re   0 then  is real. In fact, otherwise this identity implies # D const D #† , D.u/ D 0 and j† D 0, and then u D 0 by Korn’s inequality and the no-slip condition on @, as well as .#; #† ; h/ D .0; 0; 0/ by the equations for # and h, since  ¤ 0. 4. Suppose now that  > 0 is an eigenvalue of L. Then we further have Z 

Z hd† D †

†

.uk  †  j† =k /d† D k1

Z †

j† d† D 0;

R R as † uk  † d† D k div uk dx. Hence the mean values of h and j† both vanish since the densities are non-equal. Integrating the equations for # and #† , we obtain from this the relation Z Z  #† d† C  # dx D 0: †



Since A† is positive semidefinite on functions with mean zero in case † is connected, by (21) we obtain .u; #; h/ D .0; 0; 0/, i.e. in this case there are no positive eigenvalues. On the other hand, if † is disconnected, there is at least one positive eigenvalue. To prove this we need some preparations. 5. First we consider the heat problem  #  d # D 0

in  n †;

ŒŒ# D 0

on †;

#† D #j† D g

on †;

@ # D 0

(22)

on @;

H and define DH  g D ŒŒd @ † # on †, where D denotes the Dirichlet-toNeumann operator for this heat problem. The properties of DH  are stated in [16]. Then the solution # of (19) can be expressed by 0 0 .   d † C DH  /#†  .l = /j† C   H h   div† P† u D 0;

where we made use of the identity div† u† D div† P† u  H u†  † D div† P† u  H h:

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6. Next we solve the asymmetric Stokes problem u   u C r D 0

in  n †;

div u D 0

in  n †;

P† ŒŒu D 0

on †;

ŒŒT.u; ;  / †  †  D g1

on †;

ŒŒT.u; ;  / †  † = D g2

on †;

P† ŒŒT.u; ;  / †  D g3

on †;

uD0

(24)

on @;

to obtain the output ŒŒu  † =ŒŒ D S11 g1 C S12 g2 C S13 g3 ; ŒŒu  † =ŒŒ1= D S21 g1 C S22 g2 C S23 g3 ; P† u D S31 g1 C S32 g2 C S33 g3 : Note that g1 , g2 , .S g/1 , .S g/2 2 L2 .†/ are scalar functions, while g3 and .S g/3 are vectors tangent to †, i.e., g3 and .S g/3 2 L2 .†I T†/, with T† being the tangent bundle of †. For this problem we have Proposition 4.1 The operator S for the Stokes problem (24) admits a bounded extension to L2 .†/2  L2 .†I T†/ for   0 and has the following properties. (i) If u denotes the solution of (24), then Z .S gjg/L2 D 



juj2 dx C 2

Z 

 jD.u/j22 dx; 0; g2L2 .†/2  L2 .†I T†/:

(ii) S 2 B.L2 .†/2  L2 .†I T†// is self-adjoint, positive semidefinite, and compact; in particular S11 D ŒS11  ;

S22 D ŒS22  ;

S33 D ŒS33 

S12 D ŒS21  ;

S13 D ŒS31  ;

S23 D ŒS32  :

(iii) For each ˇ 2 .0; 1=2/ there is a constant Cˇ > 0 such that jS jB.L2 / 

Cˇ ; .1 C /ˇ

  0:

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(iv) jS jB.L2 ;H 1 /  C uniformly for   0. 2 (v) S11 ; S22 W L2;0 .†/ ! H21 .†/ \R L2;0 .†/ are isomorphisms, for each   0, where L2;0 .†/ D fu 2 L2 .†/ j † u d† D 0g. Proof The assertions follow from similar arguments as in the proof of [15, Proposition 4.3]. t u 7. The following lemma is needed in the proof of the main result of this section. Lemma 4.2 Let H, V be Hilbert spaces. Let B be a positive definite operator on H, A W D.A/  H ! V be a closed, densely defined operator such that AD.B1=2 /  D.A /. Then A A C B is a self-adjoint positive definite operator with D.A A C B/ D D.B/ and jA.A A C B/1 A jB.V/  1: In addition, if A.A A C B/1 A v D v, then v D 0. Proof By the closed graph theorem A W D.B1=2 / ! V is bounded. The closedness of A in turn implies that the operator A A W D.B1=2 / ! H is closed. Another application of the closed graph theorem then shows that A A W D.B1=2 / ! H is bounded as well. This implies that A A is a lower order perturbation of B and also that A A C B is self-adjoint and positive definite. Therefore, .A A C B/1 W H ! D.B/ exists and is bounded. For v 2 D.A / we have with K WD ACA WD A.A A C B/1 A jKvj2V D .CA v j A ACA v/H D .CA v j A v/H  .CA v j BCA v/H D .ACA v j v/V  .BCA v j CA v/H D .Kv j v/V  .Bw j w/H with w D CA v D .A A C B/1 A v. Since B is positive definite, there exists ˇ > 0 such that jKvj2V  jKvjV jvjV  ˇjwj2H  jKvjV jvjV

(25)

which shows jKjB.V/  1. Moreover if Kv D v, then jvj2  jvj2  ˇjwj2 holds from (25). Hence w D 0, and consequently v D Kv D Aw D 0. t u Now suppose that  > 0 is an eigenvalue of L. We set 0 1 1 0  0 H #†   A† h   A† h g D @ l #†   j† A D @  j† A C  Q#†  0 r† #† 0

(26)

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with Q D . 0 H ; .l = /; 0 r† /T to obtain .h; j† ; P† u/T D S g;

ij

S D .S /1i;j3 :

(27)

We recall (23). Since .l = /j† C  0 H h 0 div† P† u D Q S g D  Q S Q#† Q S .  A† h;  j† ; 0/T ; (23) is equivalent to   T .   d † C DH  C  Q S Q/#† D Q S .  A† h;  j† ; 0/ :

(28)

 Observing that    d † C DH  C  Q S Q is injective for   0, we solve the equation above for #† to the result

#† D L Q S .  A† h;  j† ; 0/T  1 with L D .   d † C DH  C  Q S Q/ . We set

S Q D .u1 ; u2 ; u3 /T : Combining (26)–(28), we obtain 1 1 1 0 0 h

 A† h

 A† h @ j† A D  S QL .S Q/ @  j† A  S @  j† A P† u 0 0 0 1

 A† h D .S   S QL Q S / @  j† A : 0 0

(29)

In order to obtain the positivity of S   S QL Q S , we symmetrize it as 1=2

1=2

1=2

1=2

1=2

1=2

1=2

1=2

S   S QL Q S D S .I   S QL Q S /S DW S .I  K/S ; with K D  S QL Q S D A.A A C B/1 A : 1=2 1=2

Here we have set A D  S Q with domain D.A/ D H21 .†/ and B D    2 d † C DH  with domain D.B/ D H2 .†/. Furthermore, H D L2 .†/ and V D 2 L2 .†/  L2 .†I T†/.

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By Lemma 4.2, we know jKj  1, therefore it holds that 1=2

1=2

1=2

.S v   S QL Q S v j v/ D jS vj2  .KS v j S v/ 1=2

1=2

 jS vj2  jKjjS vj2  0; which shows the positivity of S   S QL Q S . Writing the upper left 2  2 block of S   S QL Q S as S0 WD

1  R R ; R R2

S0 is also positive. Then by (29) it holds that



  1   

 R 0 h R

 A† h 0 h 0  A† h C D C D : CS 2 R R C 1=  j†  j† j† 0 0 0 (30) The following lemma is needed to solve (30). Lemma 4.3 (Schur) Let H be a Hilbert space, S; T; R 2 B.H/, S D S , T D T  and suppose that T is invertible. If

S R R T

 0

on H  H;

then S  R T 1 R  0 on H. Proof For x fixed, we set .x; y/T D .x; T 1 Rx/T . Then

0

S R R T

  ˇ   x ˇˇ x D ..S  R T 1 R/x j x/; y ˇ y t u

and this proves the assertion.

Since S0 is positive, R1 and R2 are positive as well. This implies, in particular, that .R2 C "/ is positive definite for any " > 0. Thus (30) is equivalent to the equation h C .R1  R .R2 C

1 1 / R /  A† h D 0: 

We first show that R1  R .R2 C

1 1 / R W L2;0 .†/ ! L2;0 .†/ 

(31)

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433

is injective. Let

S R R T



R R1 WD 2 R R C 21

!

and observe that the assertion of Lemma 4.3 holds true for this matrix with H D L2;0 .†/. Suppose .R1  R .R2 C 1 /1 R /h D 0 for some h 2 L2;0 .†/. Then 0 D ..R1  R .R2 C

1 1 / R /hjh/ 

D ..S  R T 1 R/hjh/ C

1 1 1 .T 1 .T C / RhjRh/  cjRhj2  0: 2 2

Thus, Rh D 0, and then also Sh D 0. (That is, R1 h D 0). This implies 1=2

1=2

0 D .S0 .h; 0/j.h; 0// D ..I  K/S .h; 0; 0/jS .h; 0; 0//:

(32)

1=2

We conclude that .I  K/S .h; 0; 0/ D .0; 0; 0/, and Lemma 4.2 then yields 1=2 S .h; 0; 0/ D .0; 0; 0/. Therefore, S .h; 0; 0/ D .0; 0; 0/, and in particular S11 h D 0. We can now, at last, infer from Proposition 4.1(v) that h D 0. An analogous argument shows that R1 W L2;0 .†/ ! L2;0 .†/ is injective as well. Indeed, if R1 h D 0 for some h 2 L2;0 .†/, then (32) holds, and the proof proceeds just as above. We note that R1 admits a representation R1 D S11 .I  C / on L2;0 .†/, with C a compact operator. Since R1 is injective on L2;0 .†/, .I C / must be so as well. Consequently, .I  C / is a bijection as it has Fredholm index zero. Proposition 4.1(v) then implies R1 2 Isom.L2;0 .†/; L2;0 .†/ \ H21 .†//; i.e., R1 is an isomorphism between the indicated spaces. A similar argument now shows that R1  R .R2 C

1 1 / R 2 Isom.L2;0 .†/; L2;0 .†/ \ H21 .†//: 

Setting T WD ŒR1  R .R2 C

1 1 1  / R  ,

Eq. (31) can be written as

T h C  A† h D 0: This equation can be treated in the same way as in [15]. As a conclusion, B WD T C  A†

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has a nontrivial kernel for some 0 > 0, which implies that L has a positive eigenvalue. Even more is true. We have seen that B is positive definite for large  and B0 D  A† has  .n  1/=R2 as an eigenvalue of multiplicity m  1 in L2;0 .†/. Therefore, as  increases to infinity, m1 eigenvalues k ./ of B must cross through zero, this way inducing m  1 positive eigenvalues of L. 8. Next we look at the eigenvalue  D 0. Then (21) yields 1=2

1=2

1=2

2j D.u/j22 C  jj† j2† C  .jd r#j22 C jd r† #† j2† / D 0; hence # is constant, D.u/ D 0 and j† D 0 by the flux condition for #. This further implies that ŒŒu D 0, and therefore Korn’s inequality yields ru D 0 and then we have u D 0 by the no-slip condition on @. This implies further that the pressures are constant in the phases and ŒŒ D   A† h C  0 H #† ; as well as ŒŒ= D l #. Thus the dimension of the eigenspace for eigenvalue  D 0 is the same as the dimension of the manifold of equilibria, namely mn C 2 if 1 has m  1 components. We set † D [1km †k . Hence, † consists of m spheres †k , k D 1; : : : ; m, of equal radius with †k \ †j D ;, l ¤ j, †k  , k D 1; : : : ; m. The kernel of L is spanned by e#;#† D .0; 1; 1; 0/, eh D .0; 0; 0; 1/, eik D .0; 0; 0; Yki / with the spherical harmonics Yki of degree one for the spheres †k , i D 1; : : : ; n, k D 1; : : : ; m. To show that the equilibria are normally stable or normally hyperbolic, it remains to prove that  D 0 is semi-simple. So suppose we have a solution of L.u; #; h/ D P i;k ˛ik eik C ˇe;#˙ C ıeh . This means  u C r D 0

in  n †;

div u D 0

in  n †;

P† ŒŒu D 0 ŒŒT.u; ;  / †  C

 A† h †  0 H #† †  0 r† #†

D0

uD0 d # D  ˇ ŒŒ# D 0 d † #†  .l = /j†  ŒŒd @ † #  0 div† u† D  ˇ @ # D 0

in †; on @:

in  n †; on †; on †;

(33)

(34)

on @:

ŒŒT.u; ;  / †  † = C l #† C ı j† D 0 X ˛ik Yki C ı ŒŒu  † =ŒŒ D i;k

on †;

on †; on †:

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We have to show ˛ik D ˇ D ı D 0 for all i; k. By div u D 0, we have ıj†j D ıj†j C

Z

X

˛ik

†

i;k

which implies ı D 0. Since

R

† j†

Z Yki d† D 

ŒŒu  † =ŒŒ d† D 0;

d† D 0 we have

ˇŒ. j1/ C  j†j D  0 D

†

Z

0 H

†

div† u† d† D 0 H

Z

†

Z †

u†  † d†

ŒŒu  † =ŒŒ d† D 0;

which implies ˇ D 0. Integrating and adding up the first equation of (33) multiplied by u, the fourth equation of (33) multiplied by u† , the first equation of (34) multiplied by  #, the third equation of (34) multiplied by  #† , and the first equation of (35) multiplied by j† , we obtain 1=2

1=2

1=2

2j D.u/j22 C  jj† j2† C  .jd r#j22 C jd r† #† j2† / C  .A† hj †  u† /† D 0: Next we observe that .A† h j †  u† /† D .A† h j ŒŒu  † =ŒŒ/† D 

X

˛ik .A† h j Yki /† D 0;

i;k

since A† is self-adjoint and the spherical harmonics Yki are in the kernel of A† . We can now conclude that u† D u  j† † = D 0, and hence 0 D u †  † D 

X

˛ik Yki :

i;k

Thus ˛ik D 0 for 1  i  n, 1  k  m, as the spherical harmonics Yki are linearly independent. Therefore, the eigenvalue  D 0 is semi-simple. 9. Let us summarize what we have proved. Theorem 4.4 Let L denote the linearization at e WD .0;  ;  j† ; †/ 2 E as defined above. Then L generates a compact analytic C0 -semigroup in X0 which has maximal Lp -regularity. The spectrum of L consists only of eigenvalues of finite algebraic multiplicity. Moreover, the following assertions are valid. (i) The operator L has no eigenvalues  ¤ 0 with nonnegative real part if and only if † is connected. (ii) If † is disconnected, then L has precisely m  1 positive eigenvalues. (iii)  D 0 is an eigenvalue of L and it is semi-simple.

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(iv) The kernel N.L/ of L is isomorphic to the tangent space Te E of the manifold of equilibria E at e . Consequently, e D .0;  ;  j† ; †/ 2 E is normally stable if and only if † is connected, and normally hyperbolic if and only if † is disconnected.

5 Nonlinear Stability of Equilibria 1. We look at Problem (4), (5) in the neighborhood of a non-degenerate equilibrium e D .0;  ;  / 2 E. Employing a Hanzawa transform with reference manifold † D  as in [15, Sect. 3], the transformed problem becomes @t u   u C r D Fu .u; ; #; h/; div u D Gd .u; h/ P† ŒŒu D Gu .u; h/ 2P† ŒŒ D.u/ †    0 r† #† D G .u; #; #† ; h/; 2ŒŒ D.u/ †  †  C ŒŒ C  A† h   0 H #† D G .u; #; h/ C G ; 2ŒŒ D.u/ †  † = C ŒŒ= C l #† C  j† D Gh .u; #; h/; u D 0; u.0/ D u0 ; (36) with G D G .#† ; h/, where  D . /,  D . /, 0 D 0 . /, l D l. /,  D . /, and A† D H 0 .0/ D .n  1/=R2  † : For the relative temperature # D .   /= we obtain  @t #  d # D F .u; #; h/

in  n †;

 @t #†  d † #†  .l = /j†  ŒŒd @ † #  0 div† u† D G .u; #; #† ; h/

on †;

ŒŒ# D 0

on †;

@ # D 0

on @;

#.0/ D #0

in ;

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437

with  D . /, d D d. /,  D  . /, d D d . /. Finally, the evolution of h is determined by @t h  ŒŒu  † =ŒŒ D Fh .u; h/

on †;

h.0/ D h0 :

(38)

Here Fu , Gd , Gu , G , F , Fh are the same as in the case where is constant; see [15, Sect. 3], and G .u; #; #† ; h/ D 2P† ŒŒ../  . //D.u/ †   2P† ŒŒ./D.u/M0 .h/r† h  P† ŒŒ./.M1 .h/ru C ŒM1 .h/ruT /. †  M0 .h/r† h/ C ŒŒ./..I  M1 /ru C Œ.I  M1 /ruT / . †  M0 r† h/  † M0 .h/r† h 

 . 0

0

 . .† /=ˇ.h//r† #†

  . 0 .† /=ˇ.h//P† .I  P .h/M0 .h//r† #† C . 0 .† /=ˇ.h//.P .h/M0 .h/r† #†  † /M0 .h/r† h; G .#† ; h/ D .† /H.h/  . /H 0 .0/h   0 H #† C  0 .† /=ˇ.h/.P .h/M0 .h/r† #†  M0 .h/r† h/; G .u; #; #† ; h/ D .   .† //@t #†   .† /u†  P .h/M0 .h/r† #†  .d  d .† //† #†  .d .† /† #†  trfP .h/M0 .h/r† .d .† /P .h/M0 .h/r† #† /g/  ŒŒd @ † # C ŒŒd./.I  M1 .h//r† #†     . 0 div u†  .† = / 0 .† /trfP .h/M0 .h/r† u† g/  .1= /.l. /j†  l.† /j / C ..† /= /j2 ; Gh .u; #; h/ D 2ŒŒ../  . //D.u/ †  † =  2.ŒŒ./D.u/ †  † =  ŒŒ./D.u/    =/  ŒŒ./.M1 .h/ C ŒM1 .h/T /    =  .ŒŒ ./ C  ŒŒ

0

. /   ŒŒ

0

. /#† /

 ../j  . /j† /  ŒŒ.1=22 /j2 ;

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cf. [15, Sect. 2] for the definition of M0 .h/, M1 .h/, P .h/, and ˇ.h/. Moreover,  D ˇ.h/. †  M0 .h/r† h/;

j D ŒŒu   =ŒŒ1=;

j† D ŒŒu  † =ŒŒ1=:

The nonlinearities are C1 from E to F, satisfying .Fi0 .0/; G0k .0// D .0; 0/ for all i 2 fu; ; hg and k 2 fd; u; ; ; ; h; g. The state manifold locally near the equilibrium e D .0;  ;  / reads as n SM WD .u; #; h/ 2 Lp ./nC1  C2 .†/ W

(39)

.u; #/ 2 Wp22=p . n †/nC1 ; h 2 Wp32=p .†/; div u D Gd .u; h/ in  n †;

u D 0; @  D 0 on @;

P† ŒŒu D Gu .u; #; h/; ŒŒ# D 0 on †;

o  2P† ŒŒ D.u/ †    0 r† # D G .u; #; #† ; h/ on † :

Note that only .u; #; h/ need to be considered as state manifold for the flows, as #† is the trace of #, and the pressure  can be recovered from the flow variables, as in [15]. Due to the compatibility conditions this is a nonlinear manifold. By parameterizing this manifold over its tangent space n SX WD .u; #; h/ 2 Lp ./nC1  C2 .†/ W .u; #/ 2 Wp22=p . n †/nC1 ; h 2 Wp32=p .†/; div u D 0 in  n †;

u D 0; @  D 0 on @;

o P† ŒŒu D 0; ŒŒ# D 0; 2P† ŒŒ D.u/ †    0 r† # D 0 on † ;

the nonlinear problem (36)–(38) is written in the form studied in [15], where the generalized principle of linear stability is proved for the problem with constant surface tension. An adaption of this proof implies the following result. Theorem 5.1 Let p > n C 2, 1 ; 2 > 0, 1 ¤ 2 , and suppose the assumptions (a), (b), (c) in Sect. 1. Then in the topology of the state manifold SM we have: (i) .0;  ;  / 2 E is stable if and only if  is connected. (ii) Any solution starting in a neighborhood of a stable equilibrium exists globally and converges to a possibly different stable equilibrium in the topology of SM. (iii) Any solution starting and staying in a neighborhood of an unstable equilibrium exists globally and converges to a possibly different unstable equilibrium in the topology of SM.

Incompressible Two-Phase Flows with Phase Transitions

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6 Qualitative Behaviour of the Semiflow In this section we study the global properties of problem (4), (5), following the approach of [15, Sect. 6]. Recall that the closed C2 -hypersurfaces contained in  form a C2 -manifold, which we denote by MH2 ./ as in [15, Sect. 2]. The charts are the parameterizations over a given hypersurface †, and the tangent space consists of the normal vector fields on †. We define a metric on MH2 ./ by means of dMH2 .†1 ; †2 / WD dH .N 2 †1 ; N 2 †2 /; where dH denotes the Hausdorff metric on the compact subsets of Rn . This way MH2 ./ becomes a Banach manifold of class C2 . Let d† .x/ denote the signed distance for †. We may then define the level function '† by means of '† .x/ D .d† .x//;

x 2 Rn ;

where .s/ D s.1  .s=a// C sgn s .s=a/;

s 2 R:

It is easy to see that † D '†1 .0/, and r'† .x/ D † .x/, for each x 2 †. Moreover,  D 0 is an eigenvalue of r 2 '† .x/, the remaining eigenvalues of r 2 '† .x/ are the principal curvatures j of † at x 2 †. If we consider the subset MH2 .; r/ of MH2 ./ which consists of all closed hyper-surfaces  2 MH2 ./ such that    satisfies the ball condition with N defined by ˆ./ D ' fixed radius r > 0 then the map ˆ W MH2 .; r/ ! C2 ./ 2 N is an isomorphism of the metric space MH .; r/ onto ˆ.MH2 .; r//  C2 ./. 2 s Let s  .n  1/=p > 2; for  2 MH .; r/, we define  2 Wp .; r/ if ' 2 Wps ./. In this case the local charts for  can be chosen of class Wps as well. A subset A  Wps .; r/ is (relatively) compact, if and only if ˆ.A/  Wps ./ is (relatively) compact. As an ambient space for the state manifold SM of Problem (1)–(3) we consider the product space Lp ./nC1  MH2 ./ and set n SM WD .u; ; / 2 Lp ./nC1  MH2 ./ W .u; / 2 Wp22=p . n /nC1 ; 0 <  < c ;  2 Wp32=p ; div u D 0 in  n ;

u D @  D 0 on @;

P ŒŒu D 0; ŒŒ D 0 on ;

o 2P ŒŒ./D.u/   C 0 ./r  D 0 on  :

440

J. Prüss et al.

Charts for these manifolds are obtained by the charts induced by MH2 ./, followed by a Hanzawa transformation. Applying Theorem 3.3 and re-parameterizing the interface repeatedly, we see that (1)–(3) yields a local semiflow on SM . Theorem 6.1 Let p > n C 2, ; 1 ; 2 > 0, 1 ¤ 2 , and suppose the assumptions (a), (b), (c) in Sect. 1. Then problem (4), (5) generates a local semiflow on the state manifold SM . Each solution .u; ; / of the problem exists on a maximal time interval Œ0; t /, where t D t .u0 ; 0 ; 0 /. Again we note that the pressure  as well as the phase flux j and  are dummy variables which are determined for each t by the principal variables .u; ; /. In fact, j D ŒŒu   =ŒŒ1=;  is the trace of , and  is determined by the weak transmission problem .rjr=/L2 ./ D .21 div../D.u//  u  rujr/L2 ./ ;

 2 Hp10 ./;  D 0 on ;

ŒŒ D 2ŒŒ./D.u/    C . /H  ŒŒ1=j2 on ; ŒŒ= D 2ŒŒ../=/D.u/      ŒŒ1=.22 /j2  ŒŒ ./  . /j on ; Concerning such transmission problems we refer to [7].

6.1 Convergence There are several obstructions against global existence: • regularity: the norms of either u.t/, .t/, or .t/ become unbounded; • geometry: the topology of the interface changes; or the interface touches the boundary of . • well-posedness: the temperature leaves the range 0 < .t/ < c . Note that the compatibility conditions, div u.t/ D 0 in  n .t/;

u D 0; @  D 0 on @;

P ŒŒu.t/ D 0; ŒŒ D 0; P ŒŒ./.ru C ŒruT /   C 0 . /r  D 0 on .t/; are preserved by the semiflow. Let .u; ; / be a solution in the state manifold SM with maximal interval Œ0; t /. By the uniform ball condition we mean the existence of a radius r0 > 0

Incompressible Two-Phase Flows with Phase Transitions

441

such that for each t, at each point x 2 .t/ there exists centers xi 2 i .t/ such that Br0 .xi /  i and .t/ \ BN r0 .xi / D fxg, i D 1; 2. Note that this condition bounds the curvature of .t/, prevents parts of it to touch the outer boundary @, and to undergo topological changes. Hence if this condition holds, then the volumes of the phases are preserved. With this property, combining the local semiflow for (4), (5) with the Lyapunov functional, i.e., the negative total entropy, and compactness we obtain the following result. Theorem 6.2 Let p > n C 2, 1 ; 2 > 0, 1 ¤ 2 , and suppose the assumptions (a), (b), (c) in Sect. 1. Suppose that .u; ; / is a solution of (4), (5) in the state manifold SM on its maximal time interval Œ0; t /. Assume that the following conditions hold on Œ0; t /: (i) sup0 0 sufficiently small, and 0  1 sufficiently large, both chosen independently of ˛, such that for all k D 0; : : : ; n  1, and for all integer m 2 Œm0 ; ˛Q  maxfr; r0 g, one has ke00k kYm  C ı 2 k

L.m/1

k ;

(22)

where L.m/ WD maxfm C m0 C r  2˛I .m C r0  ˛/C C 2m0  2˛g. Proof The substitution error given in (21) may be written as Z 1 e00k D d2 F .Sk uk C .I  Sk /uk /.ıuk ; .I  Sk /uk / d : 0

As in the calculation for the quadratic error, we first show that we can apply (6) for ı sufficiently small. Then, the estimate (22) follows from .Hn1 /, (16)–(18). t u Adding (20), (22) gives the estimate for the sum of errors defined in (10): Lemma 3.5 Assume that ˛  m0 C 1 also satisfies ˛  m0 C r0  r C 1. There exist ı > 0 sufficiently small, and 0  1 sufficiently large, both chosen independently of ˛, such that for all k D 0; : : : ; n  1 and all integer m 2 Œm0 ; ˛Q  maxfr; r0 g, one has kek kYm  C ı 2 k

L.m/1

k ;

(23)

where L.m/ is defined in Lemma 3.4. The preceding lemma immediately yields the estimate of the accumulated error En defined in (11): Lemma 3.6 Assume that ˛  m0 C 1 also satisfies ˛  m0 C r0  r C 1. Let ˛Q D 2˛ C maxfr; r0 g C 1  m0  r. There exist ı > 0 sufficiently small, 0  1 sufficiently large, both chosen independently of ˛, such that kEn kYp  C ı 2 nL.p/ ; where we have set p WD ˛Q  maxfr; r0 g.

(24)

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P. Secchi

Proof For the estimate in Yp of the accumulated error we choose p to be as large as possible, namely p D ˛Q  maxfr; r0 g. Moreover ˛Q is taken sufficiently large so that L.p/  1. Then it follows from (23) that kEn kYp  C ı 2

n1 X

L.p/1

k

k  C ı 2 nL.p/ ;

kD0

which gives (24). We can check that L.p/  1 if ˛Q  minf2˛ C maxfr; r0 g C 1  m0  r; 3˛ C maxfr; r0 g C 1  2m0  r0 g D 2˛ C maxfr; r0 g C 1  m0  r ; which explains our choice for ˛. Q

t u

3.4 Estimate of fn Going on with the iteration scheme, the next lemma gives the estimates of the source term fn , defined by Eq. (12): Lemma 3.7 Let ˛ and ˛Q be given as in Lemma 3.6. There exist ı > 0 sufficiently small and 0  1 sufficiently large, both chosen independently of ˛, such that for all integer m 2 Œm0 ; ˛Q C s, one has ˚  k fn kYm  C n nm˛s1 k f kY˛Cs C ı 2 nL.m/1 :

(25)

Proof From (12) we have fn D .Sn  Sn1 /f  .Sn  Sn1 /En1  Sn en1 : Using (8c) gives m˛s1 k f kY˛Cs k.Sn  Sn1 /f kYm  C n1 n1

(26)

for all m  0. Using (8c), (24) gives k.Sn  Sn1 /En1 kYm  C n1 ı 2 n1

mp1CL.p/

 C n1 ı 2 n1

L.m/1

;

(27)

because m  p C L.p/  L.m/ for all m  0. Moreover, from (8a), (23) we get kSn en1 kYm  C n1 ı 2 n1

L.m/1

;

(28)

On the Nash-Moser Iteration Technique

for all m  m0 . Finally, using n1  n  yields (25).

453

p 2n1 and n1  3 n in (26)–(28) t u

3.5 Proof of Induction We now consider problem (13), that gives the solution ıun . Lemma 3.8 Let ˛  m0 C maxfr; r0 g C maxfs; s0 g C 1, and let ˛Q be given as in Lemma 3.6. If ı > 0 is sufficiently small, 0  1 is sufficiently large, both chosen independently of ˛, and kf kY˛Cs =ı is sufficiently small, then for all m 2 Œm0 ; ˛, Q one has kıun kXm  ı nm˛1 n :

(29)

Proof Let us consider problem (13). By (17a) Sn un satisfies kSn un kXm0  Cı : So for ı sufficiently small we may apply (7) in order to obtain   kıun kXm  C k fn kYmCs C k fn kYm0 kSn un kXmCs0 :

(30)

Estimating the right-hand side of (30) by Lemma 3.7 and (17a) yields ˚  kıun kXm  C n nm˛1 k f kY˛Cs C ı 2 nL.mCs/1 ˚  .mCs0 ˛/C C C n nm0 ˛s1 k f kY˛Cs C ı 2 nL.m0 /1 ı n :

(31)

Q the One checks that, for ˛  m0 C maxfr; r0 g C maxfs; s0 g C 1, and m 2 Œm0 ; ˛, following inequalities hold true: L.m C s/ < m  ˛ ; m0  ˛  s C .m C s0  ˛/C  m  ˛ ; L.m0 / C .m C s0  ˛/C < m  ˛ :

(32)

From (31), we thus obtain   kıun kXm  C k f kY˛Cs C ı 2 nm˛1 n ; and (29) follows, for ı > 0 and kf kY˛Cs =ı sufficiently small.

(33) t u

The crucial point of the method is seen in (32): the quadratic nature of the errors is reflected in the estimate (23) by the presence of the term “2˛”, while the tame

454

P. Secchi

nature of the estimates contributes linearly in m (with jL0 .m/j  1). It is the “2˛” term which allows (for ˛ sufficiently large) to get (32) and close the induction. Lemma 3.8 shows that .Hn1 / implies .Hn / provided that ˛  m0 C maxfr; r0 g C maxfs; s0 g C 1, ˛Q D 2˛ C maxfr; r0 g C 1  m0  r, ı > 0 is small enough, kf kY˛Cs =ı is small enough, and 0  1 is large enough. We fix ˛, ˛, Q ı > 0, and 0  1, and we finally prove .H0 /. Lemma 3.9 If kf kY˛Cs =ı is sufficiently small, then property .H0 / holds. Proof Let us consider problem (13) for n D 0: dF .0/ ıu0 D S0 f : Applying (7) gives .m˛/C

kıu0 kXm  C kS0 f kYmCs  C 0

kf kY˛Cs :

Then kıu0 kXm  ı 0m˛1 0 ;

m0  m  ˛; Q

provided kf kY˛Cs =ı is taken sufficiently small.

t u

3.6 Conclusion of the Proof of Theorem 2.4 i) Given an integer ˛  m0 C maxfr; r0 g C maxfs; s0 g C 1, in agreement with the requirements of Lemma 3.8, we take ˛Q D 2˛ C maxfr; r0 g C 1  m0  r as in Lemma 3.6. If ı > 0 and kf kY˛Cs =ı are sufficiently small, 0  1 is sufficiently large, then .Hn / holds true for all n. Let us set m0 D ˛  1. In particular, from .Hn / we obtain X kıun kXm0 < C1 ; (34) n0

so the sequence fun g converges in Xm0 towards some limit u 2 Xm0 . From (15) we have F.unC1 /  f D .Sn  I/f C .I  Sn /En C en : Using (8b), (23), (24) we can pass to the limit in the right-hand side in Ym0 Cs and get limn!1 F .unC1 / D F .u/ D f : Therefore u is a solution of (1), and the proof of Theorem 2.4 i) is complete.

On the Nash-Moser Iteration Technique

455

Remark 3.10 In view of (7) with a loss of regularity of order s from g, given f 2 Ym0 CsC1 we could wish to find a solution u 2 Xm0 C1 instead of u 2 Xm0 as above. The regularity of u follows from the condition m0 < ˛ for the convergence of the series (34). Working with spaces Xm with integer index the condition yields m0  ˛  1; in spaces with real index it would be enough m0  ˛  , for all  > 0, and we would get u 2 Xm0 C1 .

3.7 Additional Regularity of the Solution Constructed Let us now prove assertion ii) of the Nash-Moser theorem. Let us assume that f 2 Ym00 CsC1 , with m00 > m0 . Let us set ˛ 0 D m00 C 1 and define ˛Q 0 accordingly, ˛Q 0 D 2˛ 0 C maxfr; r0 g C 1  m0  r. The proof is obtained by finite induction. For it we shall use the estimate .Hn / which is now true for all n, and the estimates that can be obtained from it. We consider again (31) and remark that the exponents of n of the terms not involving f are strictly less than m  ˛  1, as shown in (32). On the other hand, the terms in (31) involving f come from (25), or more precisely from (26). Using the fact that f is now more regular, we can substitute (26) by k.Sn  Sn1 /f kYm  C n nm˛s2 k f kY˛CsC1 ; and, accordingly, instead of (33) we find   kıun kXm  C k f kY˛CsC1 C ı 2 nm˛2 n  C nm˛2 n ;

8n  0:

(35)

Starting from these new estimates instead of .Hn /, we can revisit the proof of assertion i). Note that in e0k ; e00k there is at least one factor involving ıun in each term. Estimating this factor by (35) gives L.m/2

kek kYm  C ı k

k :

Going on with the repetition of the proof we obtain kEn kYpC1  C ı nL.p/ ; ˚  k fn kYm C1  C n nm˛s1 k f kY˛CsC1 C ı 2 nL.m/1 ;   kıun kXmC1  C kf kY˛CsC1 C ı nm˛1 n :

456

P. Secchi

This gives the gain of one order. After a finite number of iterations of the same procedure we find 0

kıun kXm  C nm˛ 1 n

for all m 2 Œm0 ; ˛Q 0 :

The conclusion of the proof of assertion ii) follows as for (34).

4 Simplified Case To understand better the role of parameters in the induction of the proof, let us assume for simplicity that m0 D 0; r D r0 D s D s0 D 1. Then estimate (23) holds with L.m/ D m C 1  2˛. The number p D ˛Q  1 in (24) is chosen such that L.p/ D 1 which yields p D 2˛; ˛Q D 2˛ C 1. To close the induction we choose ˛ from (32) that now reads m C 2  2˛ < m  ˛ ; ˛  1 C .m C 1  ˛/C  m  ˛ ; 1  2˛ C .m C 1  ˛/C < m  ˛ : Here it is sufficient to take ˛ > 2, i.e. ˛  3, and .Hn / will hold for all m 2 Œ0; 2˛ C 1. The quadratic nature of the errors with the presence of the term “2˛”allows (for ˛ sufficiently large) to close the induction. Thus, the same nonlinearity of the equation is exploited for the convergence of the approximating sequence.

References 1. S. Alinhac, Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels. Commun. Partial Differ. Equ. 14, 173–230 (1989) 2. S. Alinhac, P. Gérard, Opérateurs pseudo-différentiels et théorème de Nash-Moser (InterEditions, Paris, 1991) 3. V.I. Arnol’d, Small denominators. I. Mapping the circle onto itself. Izv. Akad. Nauk SSSR Ser. Mat. 25, 21–86 (1961) 4. V.I. Arnol’d, Small denominators and problems of stability of motion in classical and celestial mechanics. Usp. Mat. Nauk 18, 91–192 (1963) 5. G.-Q. Chen, Y.-G. Wang, Existence and stability of compressible current-vortex sheets in threedimensional magnetohydrodynamics. Arch. Ration. Mech. Anal. 187, 369–408 (2008) 6. J.F. Coulombel, P. Secchi, Nonlinear compressible vortex sheets in two space dimensions. Ann. Sci. École Norm. Sup. 41(4), 85–139 (2008) 7. I. Ekeland, An inverse function theorem in Fréchet spaces. Ann. Inst. Henri Poincaré Anal. Non Linéaire 28, 91–105 (2011) 8. M. Günther, On the perturbation problem associated to isometric embeddings of Riemannian manifolds. Ann. Glob. Anal. Geom. 7, 69–77 (1989) 9. R.S. Hamilton, The inverse function theorem of Nash and Moser. Bull. Am. Math. Soc. (N.S.) 7, 65–222 (1982)

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10. L. Hörmander, The boundary problems of physical geodesy. Arch. Ration. Mech. Anal. 62, 1–52 (1976) 11. L. Hörmander, Implicit function theorems. Stanford University Lecture Notes (1977) 12. T. Kato, Abstract Differential Equations and Nonlinear Mixed Problems. Lezioni Fermiane [Fermi Lectures] (Scuola Normale Superiore, Pisa, 1985) 13. A.N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton’s function. Dokl. Akad. Nauk SSSR (N.S.) 98, 527–530 (1954) 14. J. Moser, A new technique for the construction of solutions of nonlinear differential equations. Proc. Natl. Acad. Sci. U.S.A. 47, 1824–1831 (1961) 15. J. Moser, A rapidly convergent iteration method and non-linear differential equations II. Ann. Scuola Norm. Sup. Pisa 20(3), 499–535 (1966) 16. J. Nash, The imbedding problem for Riemannian manifolds. Ann. Math. 63(2), 20–63 (1956) 17. J. Schwartz, On Nash’s implicit functional theorem. Commun. Pure Appl. Math. 13, 509–530 (1960) 18. P. Secchi, Y. Trakhinin, Well-posedness of the plasma-vacuum interface problem. Nonlinearity 27, 105–169 (2014) 19. Y. Trakhinin, The existence of current-vortex sheets in ideal compressible magnetohydrodynamics. Arch. Ration. Mech. Anal. 191, 245–310 (2009)

Rate of Convergence to the Stationary Solution of the Navier-Stokes Exterior Problem Masao Yamazaki

Dedicated to Professor Yoshihiro Shibata on the occasion of his sixtieth birthday

Abstract This paper is concerned with the nonstationary Navier-Stokes equation in two-dimensional exterior domains with stationary external forces, and provides the rate of convergence of solutions to the stationary solution under the smallness condition of the stationary solution. Keywords Exterior problem • Navier-Stokes equations • Rate of convergence • Stationary solutions

1 Introduction Let  be an exterior domain in the two-dimensional space R2 with C3C -boundary  with some  > 0. We are concerned with stability of the stationary solution of the following nonstationary Navier-Stokes equation on  with stationary external force and stationary inhomogeneous boundary data:   @u .x; t/  u.x; t/ C u.x; t/  r u.x; t/ C r pQ .x; t/ D f .x/ in   .0; C1/; @t r  u.x; t/ D 0

(1)

in   .0; C1/; (2)

u.x; t/ D a.x/ on   .0; C1/; (3)

M. Yamazaki () Department of Mathematics, Faculty of Science and Engineering, Waseda University, Shinjuku, Tokyo 169-8555, Japan e-mail: [email protected] © Springer Basel 2016 H. Amann et al. (eds.), Recent Developments of Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-0348-0939-9_24

459

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M. Yamazaki

u.x; t/ ! 0

as jxj ! 1;

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

(4) (5)

where the vector-valued unknown function u.x; t/, the scalar-valued unknown function pQ .x; t/, the vector-valued given function a.x/, u0 .x/ and f .x/ stand for the velocity, the pressure, the stationary boundary data, the initial data and the stationary external force satisfying the outflow condition Z 

a.x/  n.x/ ds.x/ D 0

(6)

respectively, where n.x/ denotes the outer normal vector at x 2 . Finn and Smith, in the pioneering work [9], started the study on the stationary Navier-Stokes equation   w.x/ C w.x/  r w.x/ C r.x/ D f .x/ in ; r  w.x/ D 0

(7)

in ;

(8)

w.x/ D a.x/ on ;

(9)

w.x/ ! w1 as jxj ! 1;

(10)

in two-dimensional exterior domains, first in the case with definite velocity w1 ¤ 0 at infinity with no external force. This problem was considered by Gilbarg and Weinberger [15, 16] and Amick [1, 2]. In particular, Amick [1, 2] considered the case where the domain is symmetric with respect to the direction of w1 . Then Galdi and Simader [12] considered the problem for external force f .x/ with little regularity, and Galdi and Sohr [13], Sazonov [24] and Russo [22] obtained precise asymptotic behavior of w.x/ and .x/. (For more complete references, see Galdi [11].) In order to consider the case w1 D 0, Galdi [10, Sect. 3] and Pileckas and Russo [21] posed the assumption The domain  is invariant under the mappings .x1 ; x2 / 7! .x1 ; x2 /;

.x1 ; x2 / 7! .x1 ; x2 /

(D4)

with a specific coordinate variables .x1 ; x2 /, and assumed that the external force   f .x/ D f1 .x/; f2 .x/ satisfies the condition (

f1 .x1 ; x2 / D f1 .x1 ; x2 /; f2 .x1 ; x2 / D f2 .x1 ; x2 /; f1 .x1 ; x2 / D f1 .x1 ; x2 /;

f2 .x1 ; x2 / D f2 .x1 ; x2 /;

(U4)

  and find a solution w.x/ D w1 .x/; w2 .x/ satisfying (U4), which is exactly the same as [10, (3.18)]. In particular, Pileckas and Russo [21] proved the existence

Rate of Convergence to the Stationary Solution of the Navier-Stokes Exterior Problem

461

of a weak solution with finite Dirichlet integral satisfying (U4) for external force satisfying (U4) belonging to a suitable class. For related works, see Russo [23] and the references cited therein. On the other hand, in this case there seems to be few results on strong solutions corresponding to the physically reasonable solutions in the three-dimensional case. There are two points of difficulty. For one thing, typical decay order of stationary solutions of exterior problems are the same as that of the fundamental solutions of the Laplacian due to the net force exerted to the obstacle. (See Novotny and Padula [20], Borchers and Miyakawa [4], Kozono et al. [18].) Here we limit ourselves to the domains satisfying (D4) above, and the external force and the boundary data satisfying (U4) above in order to eliminate the net force. In this case the typical decay order of the solutions is the same as the derivative of the fundamental solution, which we call critically decaying solutions. However, in the case n D 2, the product of these functions belongs only in the weak-L1 space, in which almost no functional analytic tools are available in finding function spaces where we can apply the Banach fixed point theorem, of which fact the second difficulty consists. Hence we must rely on pointwise estimates for the time being. By this method Yamazaki [26, 27] showed the existence of solutions of the nonlinear equation, for  D R2 and for exterior domains respectively, under assumptions on symmetry stronger than (D4) and (U4). Yamazaki [27] also showed that, if the external force decays faster, then the solutions decay faster than the derivative of the fundamental solution, which we call supercritically decaying solutions. Nakatsuka [19] recently showed that, if the solution constructed in [27] is sufficiently small, then it coincides with the on in [21] under symmetry condition weaker than (U4). In this article we are concerned with the stability of the stationary solutions above. Putting v.x; t/ D u.x; t/  w.x/ and p.x; t/ D pQ .x; t/  .x/, we see that  v.x; t/; p.x; t/ enjoys the system:   @v .x; t/  v.x; t/ C w.x/  r v.x; t/ @t     C v.x; t/  r w.x/ C v.x; t/  r v.x; t/ C rp.x; t/ D 0

in   .0; C1/; (11)

r  v.x; t/ D 0

in   .0; C1/; (12)

v.x; t/ D 0

on   .0; C1/; (13)

v.x; t/ ! 0

as jxj ! 1;

v.x; 0/ D v0 .x/ in ; where v0 .x/ D u0 .x/  w.x/.

(14) (15)

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M. Yamazaki

Then Galdi and Yamazaki [14] showed the global stability of the stationary solutions above if they are sufficiently small. Namely, we consider the following two cases: (C)  is an exterior domain satisfying (D4), w.x/ is a sufficiently small critically decaying stationary solution satisfying (U4), and v0 .x/ 2 L2 ./ is a function satisfying (U4). (S)  is an arbitrary exterior domain, w.x/ is a sufficiently small supercritically decaying stationary solution, and v0 .x/ 2 L2 ./ is arbitrary. Then there uniquely exists a time-global solution v.t/ of (11)–(15) belonging to a q suitable class. Moreover, v.; t/ tends to 0 in L ./ for every q 2 Œ2; 1/ and in  1 2 H0 ./ as t ! 1. Precise conditions, statement and notations are introduced in the next section.  2 The purpose of this paper is to show that v.; t/ tends to 0 also in L1 ./ , and to obtain the decay rate of kv.; t/kq for 2 < q  1 and krv.; t/k2 as t ! 1. For this purpose we show some estimates of the resolvent of the perturbed operator by employing the representation of the Neumann series expansion in Kozono and Yamazaki [17] as well as the resolvent estimate by Borchers and Varnhorn [5] and Dan and Shibata [7, 8]. We need also the estimate for fractional powers of the resolvent as well, due to the fact that the coerciveness holds only for more limited exponents. This paper is organized as follows. In Sect. 2 we introduce some function spaces and state the main result. In Sect. 3 we recall some preliminary results concerning inequalities and the Stokes semigroup. Section 4 is devoted to the treatment of the semigroup generated by perturbed operators by the aforementioned manner. Finally the main result is proved in Sect. 5.

2 Notations and Main Result We first introduce some function spaces. For a domain   R2 , let C01 ./ denote the set of infinitely differentiable functions on  supported by a compact 1 subset of , and let C0;

./ denote the set of vector-valued functions '.x/ D   1 2  '1 .x/; '2 .x/ 2 C0 ./ such that div ' 0. Next, for a domain  in R2 and q 2 Œ1; 1, let Lq ./ denote the standard Lebesgue spaces equipped with the norm kkq . Next, for q 2 .1; 1/ and s > 0, let Hqs .R2 / denote the Sobolev space consisting of the functions satisfying   kukHqs .R2 / D F 1 Œhis FŒu./q < 1; p P qs .R2 / denote the set of functions modulo where hi D 1 C jj2 , and let H polynomials such that   kf kHP qs .R2 / D F 1 Œjjs FŒu./q < 1:

Rate of Convergence to the Stationary Solution of the Navier-Stokes Exterior Problem

463

P q2 .R2 / has a unique canonical representative in the space Every modulo class in H L2q=.2sq/ .R2 / if s < 2=q, and has a unique canonical representative modulo 2 P1 constants in the space H 2q=.2sq/ .R / if 2=q  s < 1 C 2=q. We next introduce the Littlewood-Paley decomposition. Let .t/ be a monotonedecreasing smooth function on R such that .t/ 1 holds for t < 4=3 and that

.t/ 0 holds for t > 5=3. Then we put ˆ./ D .jj/ for  2 R2 ;     'j ./ D 2j jj  21j jj for j 2 Z and  2 R2 n f0g: Suppose that q, r 2 Œ1; 1 and s 2 R. For a tempered distribution f 2 S 0 .R2 /, put 

 r kf kBsq;r D F 1 ˆ./F Œf ./ q C 1 X



 r 2 F 1 'j ./F Œf ./ q

!1=r

sjr 

;

jD1

0 kf kBP sq;r D @

1 X

2



sjr 

F

1



11=r

 r 'j ./F Œf ./  A q

jD1

for 1  r < 1, and (



 kf kBsq;1 D max F 1 ˆ./F Œf ./ q ; 

 sup 2 F 1 'j ./F Œf ./ q

)

sj 

j1



 kf kBP sq;1 D sup 2sj F 1 'j ./F Œf ./ q : j2Z

Then we introduce the Besov space by Bsq;r .R2 / D ff 2 S 0 j kf kBP sq;r < 1g and the homogeneous Besov space by BP sq;r .R2 / D ff 2 S 0 =P j kf kBP sq;r < 1g, where S 0 and

P denote the set of tempered distributions on R2 and the set of polynomials of x1 and x2 respectively. The spaces B0q;r .R2 / and BP 0q;r .R2 / are contained in Lq1 .R2 /CLq2 .R2 / for every q1 , q2 such that q1 < q < q2 . In particular, if r  minfq; 2g, these spaces are contained in Lq .R2 /. If either 0  s < 2=q or s D 2=q and r D 1, every modulo class of BP sq;r .R2 / possesses a canonical representative in BP 02q=.2Cqsq/;r .R2 /, and if 2=q  s < 2=q C 1 and 1 < r 2 1 or 2=q < s  2=q C 1 and r D 1, every modulo class in BP sq;r .R2 / possesses a canonical representative modulo constants in

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BP 12q=.2Cqsq/;r .R2 /. (See Bourdaud [6].) We also observe that the spaces Bs2;2 .R2 / P 2s .R2 / respectively for s  0, and the and BP s2;2 .R2 / coincide with H2s .R2 / and H spaces Bs1;1 .R2 / and BP s1;1 .R2 / coincide with the Hölder space Cs .R2 / and the homogeneous Hölder space CP s .R2 / respectively for s > 0 and s … N. For a domain   R2 with C3C -boundary with some  > 0 and for X D s Ps Hq , Hq , Bsq;r and BP sq;r with s such that 0 < s  3, let X./ denote the set of the restrictions of the elements of X.R2 / equipped with the norm ˇ ˚  kukX./ D inf kQukX.R2 / ˇ uQ j D u ; and let X0 ./ denote the closure of the space C01 ./ in X./. If q D 2, we P qs ./, H s ./ and HP s ./ to H s ./, H P s ./, H s ./ and abbreviate Hqs ./, H q;0 q;0 0 P 0s ./ respectively. Then we have the following properties: Suppose that  is a H domain with compact boundary  of class C3C with some  > 0. Then every 11=q function f 2 Hq1 ./ has the trace f j 2 Bq;q ./, and there exists a constant C such that the trace estimate  Ckf kHq1 ; kf j kB11=q q;q

(16)

11=q

where the Besov space Bq;q ./ is defined by coordinate patch form onedimensional Besov spaces. (See Bergh-Löfström [3] or Triebel [25].) We next introduce the Helmholtz decomposition. Let  be either the whole plane, the half plane, a bounded domain or an exterior domain with C3C boundary  2 . For every q 2 .1; 1/, the space Lq ./ admits the direct sum decomposition q L ./ ˚ Gq ./, where o n  2 ˇˇ Lq ./ D u 2 Lq ./ ˇ div u D 0 in ; n  u D 0 on  and n o  2 ˇˇ q Gq ./ D grad f 2 Lq ./ ˇ f 2 Lloc ./ :  2 q Let Pq denote the projection from Lq ./ onto L ./ with respect to the direct  2 sum decomposition above. Then we have Pq D Pr on Lq ./ \ Lr ./ . From this fact we can abbreviate Pq to P. Define the Stokes operator A by P. In order to state our main results, we introduce some function spaces. For a positive number b, let X .b/ denote the set of continuous functions w.x/ on  satisfying kwkX .b/ D supx2 .1 C jxj/b jw.x/j < 1. Then the space X .b/ becomes a Banach space with norm kkX .b/ . In the case that  satisfies the condition (D4),  2 let X4 .b/ denote the set of vector-valued functions w 2 X .b/ satisfying the condition (U4).

Rate of Convergence to the Stationary Solution of the Navier-Stokes Exterior Problem

465

Remark 2.1 The following assertion is proved in [27]. Suppose that 1  b  2 and that the exterior domain  satisfies a symmetry condition stronger than (D4). 11=q If the boundary datum a.x/ is sufficiently small in Bq;q ./ with some q > 2 and satisfies the outflow condition (6), and a symmetry condition stronger than (U4), and if the external force f .x/ is given by a finite sum of the derivatives of sufficiently small potential functions and satisfies the same symmetry condition   as is posed on a.x/, then there uniquely exists a stationary solution w.x/; .x/ of (7)–(10) such  2 that w 2 X .b/ and that kwkX .b/ is sufficiently small, and rw is sufficiently small 4  in L2 ./ .   Suppose that w.x/; .x/ is a solution of the stationary equations (7)–(10) such   P 1 ./ 2 holds with some b  1. We consider the stability of that w 2 X .b/ \ H w.x/ by considering the system (11)–(15) for v0 2 L2 ./. Applying the operator P to (11) and making use of (12)–(14), we have    

dv .t/ D Av.t/  P .w  r/v.t/ C v.t/  r w C v.t/  r v.t/ : dt

(17)

Then (17) with (15) is formally equivalent to the integral equation Z v.t/ D exp.tA/v0 

t 0

  exp .t  /A

   

P .w  r/v./ C v./  r w C v./  r v./ d:

(18)

In the following we assume either one of the following assumptions, which are precise statement of those given in the introduction. (C)  is an exterior domain with C3C boundary satisfying (D4), the pair of   functions w.x/; .x/ is a solution of (7)–(10) such that w 2 X4 .1/ and that kwkX .1/ and krwk2 are sufficiently small, and v0 .x/ 2 L2 ./ satisfies (U4). (S)  is an arbitrary exterior domain with C3C boundary, the pair of functions    2 w.x/; .x/ is a solution of (7)–(10) such that w 2 X .b/ with some b > 1 such that kwkX .b/ and krwk2 are sufficiently small, and v0 .x/ 2 L2 ./ is arbitrary. Then Galdi and Yamazaki [14] showed the following theorem. Theorem 2.2 (C) or (S), there uniquely exists a solution  Under the assumption  v.; t/ 2 BC Œ0; 1/; L2 ./ of the integral equation (18) such that v.x; 0/ D v0 .x/   1 2  1=2 and that t v.; t/ 2 BC .0; 1/; H0 ./ . This solution belongs to the class      2   ; C Œ0; 1/; L2 ./ \ C1 .0; 1/; L2 ./ \ C .0; 1/; H 2 ./

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and is a solution of the abstract differential equation (17). Furthermore, the function kv.; t/k2 is monotone-decreasing with respect to t, and v.; t/ enjoys the decay properties kv.; t/kq ! 0 as t ! 1 for every q 2 Œ2; 1/ and krv.; t/k2 ! 0 as t ! 1. Remark 2.3 The condition (6) is necessary for the existence of the stationary solution w.x/ 2 X .b/ with b > 1. Indeed, for sufficiently large R > 0, we have Z u.x/  jxjDR

x ds.x/ D O.R1b /; R

which tends to 0 if b > 1. Hence the divergence theorem yields (6). Our main result is the following theorem, which provides the decay rate of v.t/ with various norms. In particular, we give the decay of kv.t/k1 . Note that even its boundedness was not shown in [14]. Theorem 2.4 In the same assumptions as in Theorem 2.2, we have kv.t/kq D       p o t1=q1=2 for q 2 Œ2; 1/, krv.t/k2 D o t1=2 and kv.t/k1 D o t1=2 log t as t ! 1.

3 Preliminary Results on the Stokes Semigroup In this section we list some results needed in the proof of the main results. The first one is the Sobolev embedding theorem and the Gagliardo-Nirenberg inequality. Lemma 3.1 Suppose that q, r, s 2 .1; 1/ satisfy r > s and 0 1, and every u.x/ 2 H01 ./ satisfying (U4) in the case b D 1, we have w.x/  u.x/ 2 L2 ./ with the estimate kw.x/  u.x/k2  Cb kwkX .b/kruk2 : We next recall some properties of the Stokes operator and Stokes semigroup proved by Borchers and Varnhorn [5] and Dan and Shibata [7, 8], and add some observations needed later. Proposition 3.5 Put D D f 2 C j ¤ 0; j arg j  3=4g. Then the Stokes operator A satisfies the following assertions: (1) For every q and r such that 1 < q  r  1, there exists a positive constant Cq;r such that, for every 2 D, the operator . C A/1 is a bounded operator  2 q from L ./ to Lr ./ satisfying the estimate   . C A/1 u  Cq;r j j1C1=q1=r kuk q r q

for every u 2 L ./ and every 2 D. In particular, if q  r < 1, the function . C A/1 u belongs to Lr ./. (2) For every q and r such that 1 < q  r  2, There exists a positive constant Cq;r such that, for every 2 D, the operator r. C A/1 is a bounded operator  4 q from L ./ to Lr ./ satisfying the estimate   r. C A/1 u  Cq;r j j1=2C1=q1=r kuk q r q

for every u 2 L ./ and every 2 D. This proposition immediately implies the following theorem. (See [5, 7, 8].) Theorem 3.6 For every q 2 .1; 1/, the operator A generates a bounded q analytic C0 -semigroup exp.tA/ on the space L ./, and it satisfies the following estimates:

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M. Yamazaki

(1) Suppose that q  r  1 and that ˛  0. Then there exists a positive constant Cq;r;˛ such that the estimate kA˛ exp.tA/ukr  Cq;r;˛ t1=r1=q˛ kukq q

holds for every u 2 L ./. (2) Suppose that q  r  2. Then we can choose Cq;r;1=2 so large that the estimate kr exp.tA/ukr  Cq;r;1=2 t1=r1=q1=2 kukq q

holds for every u 2 L ./ as well. Proof Put  D 1 C 2 C 3 , where ˇ    i ˇ ˇ 1 3 e ˇˇ 3 ˇ 2 D   ; 1 D re ˇ 1 > r  jzj ; jzj ˇ 4 4 ˇ   ˇ 3i=4 ˇ 1 3 D re ˇ jzj  r < 1 : 

3i=4

Then, for z 2 C such that z ¤ 0 and j arg zj  =12, we can write exp.tA/u D

1 2i

Z 

et . C A/1 u d :

Hence Assertion (1) with q  r < 1 and ˛  0, Assertion (1) with r D 1, ˛ D 0 and Assertion (2) follow from Proposition 3.5. Assertion (2) for r D 1 and ˛ > 0 follows from the estimate   t   t     kA˛ exp.tA/uk1  exp  A A˛ exp  A u 2 2 1  t 1=2q   t     ˛  C2q;1;0 A exp  A u 2 2 2q  t 1=2qC.1=qC1=2q˛/  C2q;1;0 Cq;2q;˛ kukq : 2 This completes the proof.

t u

In order to obtain the estimate for higher order derivative, we make use of operator we have the following lemma. Lemma 3.7 We have the following assertions:    2 1. For v 2 D.A1=2 / D L2 ./ \ H01 ./ , we have the identity krvk2 D A1=2 v 2 .  2 exists 2. For v 2 D.A/ D L2 ./ \ H01./ \ H 2 ./ , there    a constant C such  that we have the estimate r 2 v 2  C kAvk2 C A1=2 v 2 .

Rate of Convergence to the Stationary Solution of the Navier-Stokes Exterior Problem

469

Proof We first prove Assertion (1). For v 2 D.A/, we have  2 krvk2 2 D .v; v/ D .Av; v/ D A1=2 v 2 : Since D.A/ is dense in D.A1=2 / and both sides are well-defined for v 2 D.A1=2 /, we obtain the conclusion.  2 We turn to the proof of Assertion (2). For v 2 L2 ./ \ H01 ./ \ H 3 ./ and R > 0 such that @  B.0; R/, we have the identity 2 X

.rj rk v; rj rk v/

j;kD1 2 I X

D

j;kD1 2 X

C

Z @

.rj rk v; nj rk v/ ds.x/ C

2 1



I jxjDrR

.rj rk v; nj rk v/ ds.x/dr

(19)

.rk v; rk v/:

kD1

 8  4 Since rj rk v 2 L2 ./ and rk v 2 L2 ./ , the Schwarz inequality yields ˇZ ˇ ˇ ˇ

2

I

1

jxjDrR

ˇ ˇ .rj rk v; nj rk v/ ds.x/drˇˇ

Z

jrj rk v.x/j2 dx

 Rjxj2R

1=2 Z

jrk v.x/j2 dx

1=2

:

Rjxj2R

The right-hand side of this inequality tends to 0 as R ! 1. Hence, substituting this fact and into (19), we obtain 2 X

.rj rk v; rj rk v/ D

j;kD1

2 I X j;kD1 @

.rj rk v; nj rk v/ ds.x/ C

2 X

.rk v; rk v/:

kD1

Since the identity rk v D rk v D rk Av holds, it follows that 2 I 2 X X  2 2 r v  D .r r v; n r v/ ds.x/ C .rk Av; rk v/: j k j k 2 j;kD1 @

(20)

kD1

In the same way we have .Av; v/ D

2 I X kD1

@

 .Av; nk rk v/ ds.x/ C .rk Av; rk v/ :

(21)

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M. Yamazaki

Subtracting (21) from (20) and making use of the trace estimate (16) and Assertion (1), we obtain  2  r v 

2

2

   j.Av; v/j C Cr 2 vj@ H 1=2 krvj@ kH 1=2 C CkAvj@ kH 1=2 krvj@ kH 1=2   2 1 1  kAvk2 r 2 v 2 C r 2 v 2 C 4C2 krvk2 2 C kAvk2 2 C 4C2 krvk2 2 4 4    3 3 2 2  r 2 v 2 C kAvk2 2 C 8C2 A1=2 v 2 : 4 4

We thus conclude   2  2  r v   3kAvk 2 C 32C2 A1=2 v  2 2 2 2  2 for v 2 L2 ./ \ H01 ./ \ H 3 ./ . Since this space is dense in D.A/ and both sides are well-defined for v 2 D.A/, we obtain the conclusion. t u From this estimate we have the following proposition. Proposition 3.8 We have the following assertions: (1) Suppose that 1 < q  2. Then there exists a constant Cq0 such that, for every q u 2 L ./ and every t > 0, the function exp.tA/u belongs to the space  1

2 H0 ./ \ H 2 ./ , and satisfies the estimate  2  r exp.tA/u  C0 t1=q .1 C t1=2 /kuk : q q;s 2  2 (2) There exists a constant Cs00 such that, for every u 2 L2 ./ \ H01 ./ , the function exp.tA/u satisfies the estimate  2  r exp.tA/u  C00 .1 C t1=2 /kruk : 2 s 2 Proof We first show Assertion (1). Since exp.tA/ is a bounded analytic C0 semigroup on L2 ./, we have  t   j=2   j=2    A exp.tA/u  C t  u exp 2 2 2 2 C

 t j=2C1=21=q 2

for j D 1, 2. Now the conclusion follows from Lemma 3.7, (2).

kukq

Rate of Convergence to the Stationary Solution of the Navier-Stokes Exterior Problem

471

Assertion (2) follows from the estimate   j=2   A exp.tA/u  Ct.j1/=2 A1=2 u 2 2 for j D 1, 2 and Lemma 3.7, (1).

t u

This proposition immediately implies the following corollary. Corollary 3.9 Suppose that 1  s < 3=2. Then we have the following assertions: 0 such that, for every (1) Suppose that 1 < q  2. Then there exists a constant Cq;s q u 2 L ./ and every t > 0, the function exp.tA/u belongs to the space  s 2 H0 ./ , and satisfies the estimate 0 1=q t .1 C t.s1/=2 /kukq : kexp.tA/ukHP s  Cq;s

 2 (2) There exists a constant C00 such that, for every u 2 L2 ./ \ H01 ./ , the function exp.tA/u satisfies the estimate kexp.tA/ukHP s  Cs00 .1 C t.s1/=2 /kruk2 : Proof Assertion (1) with s D 1 coincides with Theorem 3.6 with r D 2, and Assertion (2) with s D 1 follows immediately from Lemma 3.7, (1).  2 .0; 1/, let X denote the complex interpolation space  1For every P 01 ./ \ H P 2 ./ . Then it follows from Assertion (1) and Proposition 3.8 P 0 ./; H H  that 0 1=q t .1 C t=2 /kukq kexp.tA/ukX  Cq;s

(22)

kexp.tA/ukX  Cs00 .1 C t=2 /kruk2

(23)

q

for u 2 L ./, and

 2 for u 2 L2 ./ \ H01 ./ . If 0 <  < 1=2, P 01 ./ \ H 1C ./, which coincides For every  2 .0; 1/, we see that X D H 1C 1 P P with H0 ./ \ H0 ./ if 0 <  < 1=2. Hence the conclusion follows from (22) and (23). t u

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M. Yamazaki

4 Perturbation of the Stokes Operator In this section we introduce a perturbation of the operator A, and show some  2 properties. Suppose that w satisfies w 2 X .b/ with some b  1 and rw 2 4  2 L ./ , and put ˚  BŒu D P .w  r/u C .u  r/w : 1 ./ \ Hq2 ./ with 1 < q  2, we Then, for every u 2 D.A/ D L ./ \ Hq;0 4  2  q have ru 2 L ./ , which implies .w  r/u 2 Lq ./ . We moreover have   2 2q=.2q/ ./ if 1 < q < 2 and u 2 L1 ./ if q D 2, which imply .u  r/w 2 u 2 L

2  q L ./ in both cases. Hence the operator Lw Œu D Au C BŒu is well-defined on u 2 D.A/. In the sequel we obtain the resolvent estimate of this operator. For this end we first define the fractional power . C A/1=2 . Let ./ denote the spectral measure associated with the operator A on L2 ./. Then we can write Z 1 Z 1 1 1 d./; . C A/1=2 D . C A/1 D p d./ C C 0 0 q

for 2 D on L2 ./. Then the operator . C A/1=2 is holomorphic in the interior of D with values in bounded linear operators on L2 ./. Here p we note that 2 D implies C 2 D for every   0, and hence the branch of C  is well-defined. ˚ 2 It is easy to see that . C A/1=2 D . C A/1 . For the operator . C A/1=2 we have the following lemma. Lemma 4.1 We have the following assertions: (1) There exists a positive constant C2 such that, for every 2 D and every u 2 L2 ./, we have . C A/1=2 u 2 D.A1=2 / with the estimates   . C A/1=2 u  C2 j j1=2 kuk ; 2 2

  r. C A/1=2 u  C2 kuk : 2 2

(2) For every r satisfying 2 < r < 1, there exists a constant Cr such that we have the estimate   . C A/1=2 u  Cr j j1=r kuk 2 r for every 2 D and every u 2 L2 ./. (3) For every q satisfying 1 < q < 2, there exists a constant Cq such that we have the estimate   . C A/1=2 u  Cq j j1C1=q kuk q 2 for every 2 D and every u 2 L2 ./ \ L ./. q

Rate of Convergence to the Stationary Solution of the Navier-Stokes Exterior Problem

473

Proof We first show Assertion (1). Fix u 2 L2 ./. Then, for every 2 D, we have     . C A/1=2 u 2 D . C A/1=2 . C A/1=2 u; u 2 1 0 Z 1 1 C B d./u; uA : q D@ 0 . C /. C / p Since the inequality j C j D j C j  j j= 2 holds for every 2 D and   0, we can estimate   . C A/1=2 u

2 2



2 kuk2 2 : j j

Next, for u 2 D.A1=2 / and 2 D, Lemma 3.7, (1) yields     r. C A/1=2 u 2 D A1=2 . C A/1=2 u 2 2 2 1 0 Z 1  C B d./; uA : D@ q 0 . C /. C / p Since the inequality j C j D j C j  = 2 holds for every 2 D and   0, we can estimate  1=2  A . C A/1=2 u

2

2

 2kuk2 2

for u 2 D.A1=2 / and 2 D. From these inequalities we see that the required inequalities hold for u p 2 D.A1=2 /. Since D.A1=2 / is dense in L2 ./, we obtain the conclusion with C2 D 2. Assertion (2) follows from Assertion (1) and Lemma 3.1, and Assertion (3) follows from Assertion (2) and duality argument. t u From this lemma we can prove the following estimate.  2  4 Lemma 4.2 Suppose that w 2 X .b/ with some b  1 and rw 2 L2 ./ . Suppose also that 2 C n f0g satisfies j arg j  3=4. Then the operator . C A/1=2 B. C A/1=2 is bounded in L2 ./, and it satisfies the estimate 

 . C A/1=2 B . C A/1=2 u   Ckwk X .b/ kuk2 ; 2 where C is a constant depending only on .

474

M. Yamazaki

1 Proof Suppose that ' 2 C0;

./. In view of the equalities r  w D 0 and r  . C 1=2 A/ u D 0, we have

ˇ ˚   ˇ ˇ '; . C A/1=2 P .w  r/. C A/1=2 u C . C A/1=2 u  r w ˇ ˇ  ˇ D ˇ r. C A/1=2 '; w. C A/1=2 u ˇ      r. C A/1=2 ' 2 w. C A/1=2 u2 :

(24)

Then Lemma 4.1 implies   r. C A/1=2 '   Ck'k 2 2

(25)

with a constant C depending only on . On the other hand, in view of the fact  2 . C A/1=2 u 2 D.A1=2 / D L2 ./ \ H01 ./ ; Proposition 3.4 and Lemma 4.1 imply     1=2   w. C A/1=2 u  Ckwk u 2  CkwkX .b/ kuk2 ; X .b/ r. C A/ 2

(26)

where the constant C depends only on . Substituting (25) and (26) into (24) and 1 observing the denseness of C0;

./ in L2 ./, we obtain the conclusion. t u For this operator we have the following proposition. Proposition 4.3 For every q, r such that 1 < q  2  r < 1, there exist positive  2  4 numbers A and Aq;r such that, for every w 2 X .b/ satisfying rw 2 L2 ./ and kwkX .b/  A, we have the estimates   . C Lw /1 u  Aq;r j j1C1=q1=r kuk ; q r   1 1C1=q r. C Lw / u  Aq;2 j j kukq 2 q

for every u 2 L ./ and every 2 D. Proof Suppose that kwkX .b/  1=2C. Then Lemma 4.2 implies that the series 1 X ˚ j . C A/1=2 B. C A/1=2 jD0

converges to an operator T. This operator is bounded on L2 ./ uniformly in 2 D, and it satisfies . C A/1=2 T. C A/1=2 D . C A/1

1 X ˚ j B. C A/1 : jD0

Rate of Convergence to the Stationary Solution of the Navier-Stokes Exterior Problem

475

It follows that . C A/1=2 T. C A/1=2 D . C A C B/1 D . C Lw /1 :

(27)

For q and r as in the assumption, Lemma 4.1 implies   . C A/1 u  Cq j j1C1=q kuk q 2

(28)

for u 2 L ./ \ L2 ./, and q

  . C A/1 u  Cr j j1=r kuk ; 2 r

  r. C A/1 u  C2 kuk 2 2

for u 2 L2 ./. Hence the required estimates follow from (27)–(29).

(29) t u

From this proposition we have the following theorem. Theorem 4.4 Let w be the same as in Proposition 4.3. Then the operator Lw generates a bounded analytic C0 -semigroup exp.tLw / on L2 ./. Furthermore, for every q and r such that 1 < q  2  r < 1, there exists a constant Bq;r such that we have the estimates kexp.tLw /ukr  Bq;r t1=qC1=r kukq ;

kr exp.tLw /uk2  Bq;2 t1=q kukq

q

for every u 2 L ./ and t > 0.

  Proof Putting q D r D 2, we have . C Lw /1 u2  Cj j1 kuk2 for u 2 L2 ./ and 2 D. Let 1 , 2 , 3 and  be the same as in the proof of Theorem 3.6. Then, for z 2 C such that z ¤ 0 and j arg zj  =12, we introduce the operator T.z/ on L2 ./ by T.z/u D

1 2i

Z 

ez . C Lw /1 u d :

Then we have 2=3  arg z  5=6 for 2 1 . It follows that jez j  ejzjj j=2 . Hence we have   Z Z 1  1  1 A2;2 z 1   e . C L / u d  erjzj=2 kuk2 dr w  2i  2 r 1=jzj 1 2 Z A2;2 kuk2 1 e=2  d  CA2;2 kuk2 : 2  1

(30)

In the same way we have   Z   1 z 1   CA2;2 kuk :  e . C L / u d w 2   2i 3

2

(31)

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M. Yamazaki

We finally have   Z Z  1  A2;2 kuk2 3=4 z 1   e . C Lw / u d   d  A2;2 kuk2 :  2i 2 3=4 2 2

(32)

It follows from (30)–(32) that T.z/ defines a bounded analytic semigroup on L2 ./. Since D.Lw / D D.A/ is dense in L2 ./, the family of operators T.t/ coincides with the bounded analytic C0 -semigroup exp.tLw / on L2 ./. q Next, fix q such that 1 < q  2, and consider a function u 2 L2 ./ \ L ./. Then, for every r such that 2  r < 1, we have   Z Z 1  1  1 z 1   e . C L / u d  erjzj=2 A2;2 kukq r1C1=q1=r dr w  2i  2 1=jzj 1 r Z A2;2 kukq 1=r1=q 1 e=2  jzj d  CA2;2 jzj1=r1=q kuk2 : 2  1 Arguing for 2 and 3 in the same way, we conclude kexp.tLw /ukr  CA2;2 jzj1=r1=q kuk2 : In the same way we obtain kr exp.tLw /uk2  CA2;2 jzj1=q kuk2 : Since L2 ./ \ L ./ is dense in L ./, we obtain the conclusion. q

q

t u

5 Asymptotic Behavior: Proof of Theorem 2.4 Theorem 2.2 implies that, for every " > 0, there exists a positive number T0 such that, for every t  T0 we have kv.t/k2 < ", kv.t/k4 < " and krv.t/k2 < ". Next, for T1 such that T0 < T1 < 1, we put  ˚ ˛.T1 / D sup max .t  T0 /1=4 kv.t/k4 ; .t  T0 /1=2 krv.t/k2 : T0 tT1

Then the function ˛.T1 / is continuous and monotone-increasing. For t 2 ŒT0 ; T1 , we can write Z t     

 v.t/ D exp .t  T0 /Lw v.T0 / C exp .t  /Lw P v./  r v./ d: T0

Rate of Convergence to the Stationary Solution of the Navier-Stokes Exterior Problem

477

From this we can estimate kv.t/k4  B2;4 .t  T0 /1=4 kv.T0 /k2 Z t C C4=3 B4=3;4 .t  /1=2 kv./k4 krv./k2 d T0

 B2;4 .t  T0 /1=4 " C C4=3 ˛.t/2

Z

t

B4=3;4 .t  /1=2  3=4 d

T0

 2 where C4=3 denotes the operator norm of the projection P from L4=3 ./ to 4=3 L ./. This implies .t  T0 /1=4 kv.t/k4  B2;4 " C C4=3 B4=3:4 B

 1 1 ; ˛.T1 /2 : 2 4

(33)

In the same way, from the estimate krv.t/k2  B2;2 .t  T0 /1=2 " C C4=3 ˛.t/2

Z

t

B4=3;2 .t  /3=4  3=4 d

T0

it follows that .t  T0 /1=2 krv.t/k2  B2;2 " C C4=3 B4=3:2 B

 1 1 ; ˛.T1 /2 : 4 4

(34)

Hence, putting  

1 1 1 1 ; ; C4=3 B4=3:2 B ; ; C1 D max C4=3 B4=3:4 B 2 4 4 4 

C2 D maxfB2;4 ; B2;2 ; 1g and taking the maximum of (33) and (34), we see that ˚  max .t  T0 /1=4 kv.t/k4 ; .t  T0 /1=2 krv.t/k2  C1 ˛.T1 /2 C C2 ": Taking the supremum for t 2 ŒT0 ; T1 , we see that ˛.T1 / satisfies ˛.T1 /  C1 ˛.T1 /2 C C2 ":

(35)

We suppose that " < 1=4C1 C2 . Then there exists two distinct roots of the equation C1 X 2  X C C2 " D 0. Let f ."/ denote the smaller one; namely, f ."/ D

1

p 2C2 " 1  4C1 C2 " p : D 2C1 1 C 1  4C1 C2 "

478

M. Yamazaki

Then we have f ."/ > . Hence we have ˛.T1 / < f ."/ if we take T1 sufficiently close to T0 . From this we see ˛.T1 /  f ."/ for every T1 > T0 . Indeed, if ˛.T1 / > f ."/ holds for some T1 , the intermediate theorem implies that f ."/ < ˛.T2 /