Mathematical Theory of Compressible Viscous Fluids: Analysis and Numerics [1st ed.] 331944834X, 978-3-319-44834-3, 978-3-319-44835-0

Table of contents :
Front Matter....Pages i-xii
Preliminaries, Notation, and Spaces of Functions....Pages 1-22
Front Matter....Pages 23-23
Mathematical Model....Pages 25-30
Weak Solutions....Pages 31-38
A Priori Bounds....Pages 39-47
Weak Formulation Revisited....Pages 49-53
Weak Sequential Stability....Pages 55-77
Front Matter....Pages 79-79
Numerical Method....Pages 81-102
Stability of the Numerical Method....Pages 103-109
Consistency....Pages 111-133
Convergence....Pages 135-154
Front Matter....Pages 155-155
Weak Solutions with Artificial Pressure....Pages 157-164
Strong Convergence of the Approximate Densities....Pages 165-176
Concluding Remarks and Suggestions for Further Reading....Pages 177-178
Back Matter....Pages 179-186

Citation preview

Lecture Notes in Mathematical Fluid Mechanics

Eduard Feireisl Trygve G. Karper Milan Pokorný

Mathematical Theory of Compressible Viscous Fluids Analysis and Numerics

Advances in Mathematical Fluid Mechanics Lecture Notes in Mathematical Fluid Mechanics Editor-in-Chief: Galdi, Giovanni P Series Editors Bresch, D. John, V. Hieber, M. Kukavica, I. Robinson, J. Shibata, Y.

Lecture Notes in Mathematical Fluid Mechanics as a subseries of ‘Advances in Mathematical Fluid Mechanics’ is a forum for the publication of high quality monothematic work as well lectures on a new field or presentations of a new angle on the mathematical theory of fluid mechanics, with special regards to the NavierStokes equations and other significant viscous and inviscid fluid models. In particular, mathematical aspects of computational methods and of applications to science and engineering are welcome as an important part of the theory as well as works in related areas of mathematics that have a direct bearing on fluid mechanics. More information about this series at http://www.springer.com/series/15480

Eduard Feireisl • Trygve G. Karper • Milan Pokorný

Mathematical Theory of Compressible Viscous Fluids Analysis and Numerics

Eduard Feireisl Institute of Mathematics CAS Praha, Czech Republic

Trygve G. Karper Department of Mathematical Sciences Norwegian University of Science & Tech. Trondheim, Norway

Milan Pokorný Charles University Faculty Mathematics and Physics Charles University Praha, Czech Republic

ISSN 2297-0320 ISSN 2297-0339 (electronic) Advances in Mathematical Fluid Mechanics ISSN 2510-1374 ISSN 2510-1382 (electronic) Lecture Notes in Mathematical Fluid Mechanics ISBN 978-3-319-44834-3 ISBN 978-3-319-44835-0 (eBook) DOI 10.1007/978-3-319-44835-0 Library of Congress Control Number: 2016951227 © Springer International Publishing Switzerland 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 This book is published under the trade name Birkhäuser, www.birkhauser-science.com The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Despite the concerted effort of generations of excellent mathematicians, the fundamental problems in partial differential equations related to continuum fluid mechanics remain largely open. Solvability of the Navier–Stokes system describing the motion of an incompressible viscous fluid is one in the sample of Millennium Problems proposed by Clay Institute, see [32]. In contrast with these apparent theoretical difficulties, the Navier–Stokes system became a well-established model serving as a reliable basis of investigation in continuum fluid mechanics, including the problems involving turbulence phenomena. Mathematicians developed an alternative approach to problems in continuum fluid mechanics based on the concept of weak solutions. As a matter of fact, the balance laws, expressed in classical fluid mechanics in the form of partial differential equations, have their origin in integral identities that seem to be much closer to the modern weak formulation of these problems. Leray [67] constructed the weak solutions to the incompressible Navier–Stokes system as early as in 1930, and his “turbulent solutions” are still the only ones available for investigating large data and/or problems on large time intervals. Recently, the real breakthrough in the theory is the work of Lions [68] who generalized Leray’s theory to the case of barotropic compressible viscous fluids (see also Vaigant and Kazhikhov [94]). The quantities playing a crucial role in the description of density oscillations as the effective viscous flux were identified and used in combination with a renormalized version of the equation of continuity to obtain first large data/large time existence results in the framework of compressible viscous fluids. The main goal of this book is to present the mathematical theory of compressible barotropic fluids in the framework of weak solutions proposed by Lions [68], including the extensions developed in [34], in a close connection with the associated numerical analysis based on the scheme proposed in [58]. Our main aim was to present the material in a concise and, at the same time, elementary way accessible to graduate students, engineers and researchers interested in mathematical fluid mechanics. The book consists of three major parts. After a short introduction to the mathematical model, we first focus on the crucial question of stability of a family of weak solutions that is the core of the abstract theory, with the relevant implications v

vi

Preface

to the numerical analysis and the associated real world applications. For the sake of clarity of presentation, we discuss first the case, where the pressure term grows sufficiently fast for large value of the density yielding strong energy bounds. The main novelty of our approach is a detailed existence proof performed in the second part, where the approximate solutions are constructed by means of a mixed finite element finite volume numerical scheme. In particular we show that the numerical solutions, up to a subsequence, converge to a weak solution of the (compressible) Navier–Stokes system at least for the pressure that increases sufficiently fast for large values of the density. The final third part of the book is devoted to the mathematical theory with (nowadays) optimal restriction on the pressure function. Praha, Czech Republic Trondheim, Norway Praha, Czech Republic

Eduard Feireisl Trygve G. Karper Milan Pokorný

Acknowledgements

The research of Eduard Feireisl 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. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by ˇ (Czech RVO:67985840. The work of Milan Pokorný was supported by the GACR Science Foundation) project No. 16-03230S. The authors thank Martin Michálek for preparing the graphic material and Radim Hošek and Martin Michálek for critical reading and valuable comments on the final version of the manuscript.

vii

Contents

1

Preliminaries, Notation, and Spaces of Functions . .. . . . . . . . . . . . . . . . . . . . 1.1 Notation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.1 Numbers.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.2 Topological Objects, Domains, and Boundaries . . . . . . . . . . 1.1.3 Vectors and Algebraic Operations . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Derivatives and Differential Operators . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.1 Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.2 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.3 Other Differential Operators . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.1 Continuous Functions . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.2 Integrable Functions .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Sobolev Spaces .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.1 Nonlinear Compositions . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.2 Dual Spaces to Sobolev Spaces . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.3 Embeddings .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.4 Sobolev–Slobodeckii Spaces . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.5 Trace Theorem for Sobolev Functions and Green’s Formula . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.6 Poincaré’s Inequality . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.7 Functions of Bounded Variations . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 Fourier Transform .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5.1 Pseudodifferential Operators .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5.2 Hörmander–Mikhlin Theorem . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.6 Weak Convergence of Integrable Functions.. . . .. . . . . . . . . . . . . . . . . . . . 1.7 Fixed-Point Theorem .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 1 2 2 3 4 4 5 5 5 7 10 12 12 13 14 15 16 16 17 18 20 20 22

ix

x

Contents

Part I

Mathematics of Compressible Fluid Flows

2

Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Balance of Momentum .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Constitutive Relations and Navier–Stokes System . . . . . . . . . . . . . . . . . 2.4 Spatial Domain and Boundary Conditions .. . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Initial Conditions and Well Posedness . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

25 26 27 28 29 30

3

Weak Solutions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Equation of Continuity: Weak Formulation .. . . .. . . . . . . . . . . . . . . . . . . . 3.1.1 Weak–Strong Compatibility.. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.2 Weak Continuity in Time .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.3 Total Mass Conservation, Positive Density, and Vacuum .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Balance of Momentum: Weak Formulation .. . . .. . . . . . . . . . . . . . . . . . . . 3.2.1 Weak Continuity in Time .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.2 Admissibility Criteria . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

31 32 32 32

4

A Priori Bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Total Mass Conservation .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Total Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Pressure Estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

39 39 40 43

5

Weak Formulation Revisited . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Regularity Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Renormalized Equation of Continuity.. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 Momentum Equation .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 Energy Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

49 50 50 51 52

6

Weak Sequential Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Uniform Bounds .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1.1 Energy Bounds.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1.2 Pressure Estimates . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Limit Passage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.1 Compactness in Convective Terms . . . .. . . . . . . . . . . . . . . . . . . . 6.2.2 Limit in the Equation of Continuity . . .. . . . . . . . . . . . . . . . . . . . 6.2.3 Limit in the Momentum Equation . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Strong Convergence of the Densities . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.1 Amplitude of Density Oscillations. . . . .. . . . . . . . . . . . . . . . . . . . 6.3.2 Effective Viscous Flux . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.3 Compactness via Div–Curl Lemma.. . .. . . . . . . . . . . . . . . . . . . .

55 56 56 58 60 61 62 64 65 65 68 70

35 36 37 38

Contents

Part II

xi

Existence of Weak Solutions via a Numerical Method

7

Numerical Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 81 7.1 Time Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 82 7.2 Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 83 7.2.1 Spatial Domain, Triangulation, and Mesh .. . . . . . . . . . . . . . . . 83 7.2.2 Finite Elements, Finite Volumes, and Function Spaces .. . 86 7.2.3 Discretization of Convective Terms, Upwind . . . . . . . . . . . . . 87 7.3 Numerical Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 90 7.4 Discrete Renormalization . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 91 7.5 Energy Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 94 7.5.1 Pressure Potential .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 95 7.5.2 Time Derivative of the Total Energy .. .. . . . . . . . . . . . . . . . . . . . 95 7.5.3 Convective Term .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 95 7.5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 98 7.6 Well Posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 99 7.6.1 Positivity of the Numerical Densities . .. . . . . . . . . . . . . . . . . . . . 99 7.6.2 Solvability of the Numerical Scheme . .. . . . . . . . . . . . . . . . . . . . 101

8

Stability of the Numerical Method. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1.1 Approximation by Smooth Functions .. . . . . . . . . . . . . . . . . . . . 8.1.2 Discrete Sobolev Embeddings . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Stability Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2.1 Total Mass Conservation . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2.2 Energy Estimates . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2.3 Numerical Dissipation .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

103 103 105 106 107 108 108 109

9

Consistency .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 Preliminaries and Useful Estimates . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1.1 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1.2 Interpolation Inequalities .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1.3 Trace Estimates . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 Consistency Formulation of the Continuity Method.. . . . . . . . . . . . . . . 9.3 Consistency Formulation of the Momentum Method.. . . . . . . . . . . . . . 9.3.1 Discretized Time Derivative . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3.2 Upwind in the Momentum Equation . .. . . . . . . . . . . . . . . . . . . . 9.3.3 Momentum Consistency: Conclusion .. . . . . . . . . . . . . . . . . . . .

111 111 112 113 114 114 122 123 125 133

10 Convergence .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1 Refined Pressure Estimates . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2 Convergence in Field Equations, Convective Terms, and Time Derivatives .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2.1 Convergence in Convective Terms . . . . .. . . . . . . . . . . . . . . . . . . . 10.2.2 Time Derivatives .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2.3 Limit in Fields Equations . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

135 135 139 140 143 144

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Contents

10.3 Strong Convergence of the Numerical Densities . . . . . . . . . . . . . . . . . . . 10.3.1 Effective Viscous Flux Identity . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3.2 Density Convergence: Conclusion .. . . .. . . . . . . . . . . . . . . . . . . . 10.4 Energy Inequality and Convergence of Numerical Solutions . . . . . . 10.5 Weak Solutions to the Navier–Stokes System . .. . . . . . . . . . . . . . . . . . . . Part III

145 149 151 152 153

Existence Theory for General Pressure

11 Weak Solutions with Artificial Pressure . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.1 Uniform Bounds .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.2 Energy Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.3 Pressure Estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.4 Weak Limit of the Sequence of Approximate Solutions . . . . . . . . . . .

157 158 159 159 163

12 Strong Convergence of the Approximate Densities .. . . . . . . . . . . . . . . . . . . . 12.1 Effective Viscous Flux . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.2 Oscillations Defect Measure .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.2.1 Oscillations Defect Measure and Renormalization . . . . . . . 12.2.2 Oscillations Defect Measure and Effective Viscous Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.3 Existence of Weak Solutions . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

165 165 167 168 171 173

13 Concluding Remarks and Suggestions for Further Reading . . . . . . . . . . 177 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 179 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 185

Chapter 1

Preliminaries, Notation, and Spaces of Functions

This chapter introduces notation as well as the basic mathematical tools used in the book such as the function spaces, embedding theorems, and elementary inequalities. We suppose the reader to be familiar with this material and will refer to it throughout the text without further specification.

1.1 Notation Unless otherwise indicated, the symbols and the basic notation will be used as stated below.

1.1.1 Numbers The symbols Z, N, and C denote the sets of integers, positive integers, and complex numbers, respectively. The symbol R denotes the set of real numbers, and RN is the N-dimensional Euclidean space. The symbol const, c, or ci will be used for a generic positive constant. These constants may have different values at different parts of the book. We will also write
denotes also the scalar product in X. The symbol spanfMg, where M is a subset of a vector space X, denotes the space of all finite linear combinations of vectors contained in M.

1.3.1 Continuous Functions For Q  RN , the symbol C.Q/ denotes the set of continuous functions on Q. For a bounded set Q, the symbol C.Q/ denotes the Banach space of functions continuous on the closure Q endowed with norm kgkC.Q/ D sup jg.y/j: y2Q

6

1 Preliminaries, Notation, and Spaces of Functions

Similarly, C.QI X/ is the Banach space of vectorial functions continuous in Q and ranging in a Banach space X with norm kgkC.Q/ D sup kg.y/kX : y2Q

The symbol Cweak .QI X/ denotes the space of vector-valued functions on Q ranging in a Banach space X continuous with respect to the weak topology. More specifically, g 2 Cweak .QI X/ if the mapping y 7! kg.y/kX is bounded and y 7!< f I g.y/ >X IX is continuous on Q for any linear form f belonging to the dual space X  . We shall say that gn ! g in Cweak .QI X/ if < f I gn >X  IX !< f I g >X  IX in C.Q/ for all f 2 X  : The symbol Ck .Q/, Q  RN , where k is a nonnegative integer, denotes the space of functions on Q that are restrictions of k-times continuously differentiable functions on RN . Ck; .Q/,  2 .0; 1/ is the subspace of Ck .Q/ of functions having their kth derivatives -Hölder continuous in Q. Ck;1 .Q/ is a subspace of Ck .Q/ of functions whose kth derivatives are Lipschitz on Q. For a bounded domain Q, the spaces Ck .Q/ and Ck; .Q/,  2 .0; 1 are Banach spaces with norms kukCk .Q/ D max sup j@˛ u.x/j j˛jk x2Q

and j@˛ u.x/  @˛ u.y/j ; jx  yj j˛jDk .x;y/2Q2 ; x¤y

kukCk; .Q/ D kukCk .Q/ C max

sup

P where @˛ u stands for the partial derivative @˛x11 : : : @˛xNN u of order j˛j D NiD1 ˛i . The k spaces Ck; .QI RM / are defined in a similar way. Finally, we set C1 D \1 kD0 C . We recall the standard Arzelà–Ascoli Theorem: Theorem 1 Let Q  RM be compact and X a compact topological metric space endowed with a metric dX . Let fvn g1 nD1 be a sequence of functions in C.QI X/ that is equi-continuous, meaning, for any " > 0 there is ı > 0 such that i h dX vn .y/; vn .z/  " provided jy  zj < ı independently of n D 1; 2; : : : :

1.3 Function Spaces

7

Then fvn g1 nD1 is precompact in C.QI X/, that is, there exists a subsequence (not relabeled) and a function v 2 C.QI X/ such that h i sup dX vn .y/; v.y/ ! 0 as n ! 1: y2Q

See Kelley [60, Chap. 7, Theorem 17] for the proof. For Q  RN an open set and a function g W Q ! R, the symbol suppŒg denotes the support of g in Q, specifically, suppŒg D closure Œfy 2 Q j g.y/ ¤ 0g : The symbol Cck .QI RM /, k 2 f0; 1; : : : ; 1g denotes the vector space of functions belonging to Ck .QI RM / and having compact support in Q. If Q  RN is an open set, the symbol D.QI RM / will be used alternatively for the space Cc1 .QI RM / endowed with the topology induced by the convergence: 'n ! ' 2 D.Q/ if suppŒ'n   K; K  Q a compact set; 'n ! ' in Ck .K/ for any k D 0; 1; : : : : (1.3) We write D.Q/ instead of D.QI R/. The dual space to Cc .Q/ is the space M.Q/ of Radon measures on an open set Q. The dual space D0 .QI RM / is the space of distributions on Q with values in RM . Continuity of a linear form belonging to D0 .Q/ is understood with respect to the convergence introduced in (1.3).

1.3.2 Integrable Functions The Lebesgue measure of a set Q  RM is denoted by jQj. We denote Z ˝

v dx

the Lebesgue integral of a measurable function v D v.x/ over a measurable set ˝  R3 , Similarly, we write Z Q

v dx dt; Q  R  R3 ;

8

1 Preliminaries, Notation, and Spaces of Functions

if v D v.t; x/. We denote Z ˝

v.t; x/ dx

tD2

Z

Z

D ˝

tD1

v.t2 ; x/ dx 

˝

v.t1 ; x/ dx;

and the convolution Z .u v/.x/ D

u.x  y/v.y/ dy: Q

The Lebesgue spaces L p .QI X/ are spaces of (Bochner) measurable functions v ranging in a Banach space X such that the norm Z p

kvkL p .QIX/ D

p

Q

kvkX dy is finite; 1  p < 1:

Similarly, v 2 L1 .QI X/ if v is (Bochner) measurable and kvkL1 .QIX/ D ess sup kv.y/kX < 1: y2Q

p

The symbol Lloc .QI X/ denotes the vector space of locally L p -integrable functions, meaning p

v 2 Lloc .QI X/ if v 2 L p .KI X/ for any compact set K in Q: We write L p .Q/ for L p .QI R/. Let f 2 L1loc .Q/, where Q is an open set. A Lebesgue point a 2 Q of f in Q is characterized by the property 1 r!0C jB.a; r/j

Z f .x/dx D f .a/:

lim

(1.4)

B.a;r/

For f 2 L1 .Q/ the set of all Lebesgue points is of full measure, meaning its complement in Q is of zero Lebesgue measure. A similar statement holds for vector valued functions f 2 L1 .QI X/, where X is a Banach space (see Brezis [8]). If f 2 C.Q/, then identity (1.4) holds for all points a in Q. Linear functionals on L p .QI X/ are characterized as follows: Theorem 2 Let Q  RN be a measurable set, X a Banach space that is reflexive and separable, 1  p < 1. Then any continuous linear form  2 ŒL p .QI X/ admits a unique representation 0 w 2 L p .QI X  /, Z < I v >.Lp .QIX//IL p .QIX/ D

< w .y/I v.y/ >X IX dy for all v 2 L p .QI X/; Q

1.3 Function Spaces

9

where 1 1 C 0 D 1: p p Moreover the norm on the dual space is given as kkŒL p .QIX/ D kw kL p0 .QIX / : Accordingly, the spaces L p .QI X/ are reflexive for 1 < p < 1 as soon as X is reflexive and separable. See Gajewski et al. [45, Chap. IV, Theorem 1.14, Remark 1.9]. Identifying  with w , we write 0

ŒL p .QI X/ D L p .QI X  /; kkŒL p .QIX/ D kkL p0 .QIX / ; 1  p < 1: This formula is known as Riesz representation Theorem. If the Banach space X in Theorem 2 is merely separable, we have p0

ŒL p .QI X/ D Lweak./ .QI X  / for 1  p < 1; where p0

Lweak./ .QI X  / ˇ n ˇ

 W Q ! X  ˇ y 2 Q 7!< .y/I v >X IX measurable for any fixed v 2 X; 0

y 7! k.y/kX  2 L p .Q/

o

(see Edwards [27], Pedregal [80, Chap. 6, Theorem 6.14]). Hölder’s inequality reads kuvkLr .Q/  kukL p .Q/ kvkLq .Q/ ;

1 1 1 D C ; 1  p; q; r  1 r p q

for any u 2 L p .Q/, v 2 Lq .Q/, Q  RN (see Adams [1, Chap. 2]). Interpolation inequality for L p -spaces reads .1/

kvkLr .Q/  kvkL p .Q/ kvkLq .Q/ ;

 1 1 D C ; 1  p < r < q  1;  2 .0; 1/ r p q

for any v 2 L p \ Lq .Q/, Q  RN (see Adams [1, Chap. 2]).

10

1 Preliminaries, Notation, and Spaces of Functions

Jensen’s inequality reads Z

 Z v dy  ˚.v/ dy

˚ Q

Q

whenever ˚ is convex on the range of v and Q  RM , jQj D 1, see, e.g., Ziemer [99, Chap. 1, Sect. 1.5]. Finally, we report Gronwall’s Lemma: Lemma 1 Let a 2 L1 .0; T/, a  0, ˇ 2 L1 .0; T/, b0 2 R, and Z b./ D b0 C



ˇ.t/ dt

0

be given. Let r 2 L1 .0; T/ satisfy Z r./  b./ C 0



a.t/r.t/ dt for a.a.  2 Œ0; T:

Then Z r./  b0 exp

 0

 Z a.t/ dt C

 0

Z ˇ.t/ exp



 a.s/ ds dt

t

for a.a.  2 Œ0; T. See Carroll [13].

1.4 Sobolev Spaces A domain ˝  RN is of class C if for each point x 2 @˝, there exist r > 0 and a mapping  W RN1 ! R belonging to a function class C such that—upon rotating and relabeling the coordinate axes if necessary—we have 9 ˝ \ B.xI r/ D fy j .y0 / < yN g \ B.x; r/ = @˝ \ B.xI r/ D fy j .y0 / D yN g \ B.x; r/

;

;

where y0 D .y1 ; : : : ; yN1 /:

In particular, ˝ is called Lipschitz domain if  is Lipschitz. If A  WD @˝ \ B.xI r/,  is Lipschitz and f W A ! R, then one can define the surface integral v u N1 u X  @ 2 0 0 t f dSx WD f .y ; .y // 1 C dy0 ; @y i A ˚ .A/ iD1

Z

Z

1.4 Sobolev Spaces

11

where ˚ W RN 7! RN , ˚ .y0 ; yN / D .y0 ; yN  .y0 //, whenever R the (Lebesgue) integral at the right-hand side exists. If f D 1A , then SN1 .A/ D A dSx is the surface measure on @˝ of A that can be identified with the .N  1/-Hausdorff measure on @˝ of A (cf. Evans and Gariepy [31, Chap. 4.2]). In the general case of A  @˝, R one can define A f dSx using a covering B D fB.xi I r/gM iD1 , xi 2 @˝, M 2 N of @˝ by balls of radii r and subordinated partition of unity F D f'i gM iD1 , and set Z f dSx D A

M Z X iD1

i

'i f dSx ;

i D @˝ \ B.xi I r/;

see Neˇcas [75, Section I.2] or Kufner et al. [64, Sect. 6.3]. A Lipschitz domain ˝ admits the outer normal vector n.x/ for a.a. x 2 @˝. Here a:a: refers to the surface measure on @˝. A differential operator @˛ of order j˛j can be identified with a distribution < @˛ vI ' >D0 .Q/ID.Q/ D .1/j˛j

Z

v@˛ ' dy Q

for any locally integrable function v. The Sobolev spaces W k;p .QI RM /, 1  p  1, k a positive integer, are the spaces of functions having all distributional derivatives up to order k in L p .QI RM /. The norm in W k;p .QI RM / is defined as

kvkW k;p .QIRM / D

9 8P P 1=p p M ˛ > ˆ k@ v k if 1  p < 1 = < p i L .Q/ iD1 j˛jk ˆ :

˛

max1iM; j˛jk fk@ vi kL1 .Q/ g if p D 1

> ;

;

where the symbol @˛ stands for any partial derivative of order j˛j. If Q is a bounded domain with boundary of class Ck1;1 , then there exists a continuous linear operator which maps W k;p .Q/ to W k;p .RN /; it is called extension operator. For 1 < p < 1 the extension operator exists even for the boundary of class C0;1 . If 1  p < 1, then W k;p .Q/ is separable and the space Ck .Q/ is its dense subspace. The space W 1;1 .Q/, where Q is a bounded domain, is isometrically isomorphic to the space C0;1 .Q/ of Lipschitz functions on Q. k;p The symbol W0 .QI RM / denotes the completion of Cc1 .QI RM / with respect to 0;p the norm k  kW k;p .QIRM / . In what follows, we identify W 0;p .˝I RN / D W0 .˝I RN / with L p .˝I RN /. R P 1;p .Q/ D W 1;p .Q/\LPp .Q/. We denote LP p .Q/ D fu 2 L p .Q/ j Q u dy D 0g and W N 1;p P .Q/ can be viewed as closed If Q  R is a bounded domain, then LPp .Q/ and W subspaces of Lp .Q/ and W 1;p .Q/, respectively. For basic properties of Sobolev functions, see Adams [1] or Ziemer [99].

12

1 Preliminaries, Notation, and Spaces of Functions

1.4.1 Nonlinear Compositions Let Q  RN be an open set, 1  p  1 and v 2 W 1;p .Q/. Then we have: (a) ŒvC , Œv 2 W 1;p .Q/ and @xj ŒvC D

9 8 < @xk v a.a. in fv > 0g = :

;

0 a.a. in fv  0g

; @xj Œv D

9 8 < @xk v a.a. in fv < 0g = :

0 a.a. in fv  0g

;

;

j D 1; : : : ; N, where ŒvC D maxfu; 0g denotes a positive part and Œv D minfv; 0g a negative part of v. (b) If f W R ! R is a Lipschitz function and v 2 W 1;p .Q/, then f ı v 2 W 1;p .Q/ and @xj Œf ı v.x/ D f 0 .v.x//@xj v.x/ for a.a. x 2 Q: For more details see Ziemer [99, Sect. 2.1].

1.4.2 Dual Spaces to Sobolev Spaces Theorem 3 Let ˝  RN be a domain, and let 1  p < 1. Then the dual space k;p ŒW0 .˝/ is a proper subspace of the space of distributions D0 .˝/. Moreover, any k;p linear form f 2 ŒW0 .˝/ admits a representation < f I v >ŒW k;p .˝/ IW k;p .˝/ D 0

0

XZ j˛jk

0

where w˛ 2 L p .˝/;

˝

.1/j˛j w˛ @˛ v dx;

1 1 C 0 D 1: p p

The norm of f in the dual space is given as

k f kŒW k;p .˝/ D 0

8 n o 1=p0 ˇ P ˇ p0 ˆ ˆ inf kw k satisfy (1.5) w ˇ 0 ˛ ˛ ˆ j˛jk ˆ L p .˝/ ˆ ˆ ˆ ˆ ˆ ˆ ˆ < for 1 < p < 1I ˇ n o ˆ ˆ ˇ ˆ ˆ 1 inf max fkw k g ˆ ˛ L .Q/ ˇ w˛ satisfy (1.5) j˛jk ˆ ˆ ˆ ˆ ˆ ˆ : if p D 1:

(1.5)

1.4 Sobolev Spaces

13

The infimum is attained in both cases. See Adams [1, Theorem 3.8], Mazya [72, Sect. 1.1.14]. 0 k;p The dual space to the Sobolev space W0 .˝/ is denoted as W k;p .˝/. k;p The dual to the Sobolev space W .˝/ admits formally the same representation formula as (1.5). However it cannot be identified as a subspace of distributions on ˝. A typical example is the linear form Z < f I v >D

˝

wf  rx v dx; with divx wf D 0

that vanishes on D.˝/ but generates a nonzero linear form when applied to v 2 W 1;p .˝/.

1.4.3 Embeddings We report the celebrated Rellich–Kondrachov Embedding Theorem: Theorem 4 Let ˝  RN be a bounded Lipschitz domain. (i) Then, if kp < N and p  1, the space W k;p .˝/ is continuously embedded in Lq .˝/ for any 1  q  p D

Np : N  kp

Moreover, the embedding is compact if k > 0 and q < p . (ii) If kp D N, the space W k;p .˝/ is compactly embedded in Lq .˝/ for any q 2 Œ1; 1/. (iii) If kp > N then W k;p .˝/ is continuously embedded in CkŒN=p1; .˝/, where Œ   denotes the integer part and

D

8 ˆ
0 and q < 1 satisfy q>

p Np ; where p D if kp < N; p  1 N  kp q > 1 for kp D N;

or q  1 if kp > N: 0

Then the space Lq .˝/ is compactly embedded into the space W k;p .˝/, 1=p C 1=p0 D 1.

1.4.4 Sobolev–Slobodeckii Spaces The Sobolev–Slobodeckii spaces W kCˇ;p .Q/, 1  p < 1, 0 < ˇ < 1, k D 0; 1; : : :, where Q is a domain in RL , L 2 N, are Banach spaces of functions with finite norm 0

1 1p Z Z ˛ ˛ p X j@ v.y/  @ v.z/j p W kCˇ;p .Q/ D @kvkW k;p .Q/ C dy dzA ; jy  zjLCˇp Q Q j˛jDk

see, e.g., Neˇcas [75, Sect. 2.3.8]. Let ˝  RN be a bounded Lipschitz domain. Referring to the notation introduced at the beginning of Sect. 1.4, we say that f 2 W kCˇ;p .@˝/ if .'f / ı .I0 ;  / 2 W kCˇ;p .RN1 / for any D @˝ \ B with B belonging to the covering B of @˝ and ' the corresponding term in the partition of unity F containing M components. The space W kCˇ;p .@˝/ is a Banach space endowed with an equivalent norm k  kW kCˇ;p .@˝/ , where p

kvkW kCˇ;p .@˝/ D

M X

k.v'i / ı .I0 ;  /kW kCˇ;p .RN1 / : p

iD1

In the above formulas .I0 ;  / W RN1 ! RN maps y0 to .y0 ; .y0 //. For more details see, e.g., Neˇcas [75, Sect. 3.8]. In the situation when ˝  RN is a bounded Lipschitz domain, the SobolevSlobodeckii spaces admit similar embeddings as classical Sobolev spaces. Namely,

1.4 Sobolev Spaces

15

the embeddings W kCˇ;p .˝/ ,! Lq .˝/ and W kCˇ;p .˝/ ,! Cs .˝/ Np , and s D 0; 1; : : : ; k, are compact provided .k C ˇ/p < N, 1  q < N.kCˇ/p .k  s C ˇ/p > N, respectively. The former embedding remains continuous (but not Np . compact) at the border case q D N.kCˇ/p

1.4.5 Trace Theorem for Sobolev Functions and Green’s Formula Theorem 6 Let ˝  RN be a bounded Lipschitz domain. Then there exists a linear operator 0 with the following properties: Œ0 .v/.x/ D v.x/ for x 2 @˝ provided v 2 C1 .˝/; k0 .v/k

W

1 p1 ;p

.@˝/

 ckvkW 1;p .˝/ for all v 2 W 1;p .˝/; 1;p

kerŒ0  D W0 .˝/ provided 1 < p < 1. Conversely, there exists a continuous linear operator 1

` W W 1 p ;p .@˝/ ! W 1;p .˝/ such that 1

0 .`.v// D v for all v 2 W 1 p ;p .@˝/ provided 1 < p < 1. In addition, the following formula holds: Z

Z ˝

@xi uv dx C

Z ˝

u@xi v dx D

@˝

0 .u/0 .v/ni dSx ; i D 1; : : : ; N;

0

for any u 2 W 1;p .˝/, v 2 W 1;p .˝/, where n is the outer normal vector to the boundary @˝. See Neˇcas [75, Theorems 5.5, 5.7]. 1 1 The dual ŒW 1 p ;p .@˝/ to the Sobolev–Slobodeckii space W 1 p ;p .@˝// D

1

;p

W p0 .@˝/ is denoted by W

 p10 ;p0

.@˝/.

16

1 Preliminaries, Notation, and Spaces of Functions

If ˝  RN is a bounded Lipschitz domain, then we have the interpolation inequality 1 kvkW ˛;r .˝/  ckvkW ˇ;p .˝/ kvkW ;q .˝/ ; 0    1;

(1.6)

for 0  ˛; ˇ;   1; 1 < p; q; r < 1; ˛ D ˇ C .1  /;

 1 1 D C r p q

(see Sects. 2.3.1, 2.4.1, 4.3.2 in Triebel [90]).

1.4.6 Poincaré’s Inequality Theorem 7 Let 1  p < 1, and let ˝  RN be a bounded Lipschitz domain. Then the following holds: (i) For any A  @˝ with the nonzero surface measure there exists a positive constant c D c. p; N; A; ˝/ such that kvkL p .˝/

  Z  c krvkL p .˝IRN / C jvj dSx for any v 2 W 1;p .˝/: A

(ii) There exists a positive constant c D c. p; ˝/ such that Z   1   v dx p  ckrvkL p .˝IRN / for any v 2 W 1;p .˝/: v  L .˝/ j˝j ˝ The above theorem can be viewed as a particular case of more general results, for which we refer to Ziemer [99, Chap. 4, Theorem 4.5.1].

1.4.7 Functions of Bounded Variations The symbol BV.Q/ denotes the space of functions in L1 .Q/, with distributional derivatives belonging to the space of measures M.Q/. Functions belonging to BVŒ0; T possess well-defined right and left-hand limits and as such can be defined at any t 2 Œ0; 1. More generally, we may define the space BV.Œ0; TI X/ of functions of bounded variation from a real interval Œ0; T into a metric space X endowed with a metric d, VX Œv D

sup

m X

0t1 for any g 2 S.RN /:

(1.10)

It is a continuous linear operator defined on S 0 .RN / onto S 0 .RN / with the inverse 1 F!x , 1 1 < F!x Œ f ; g >D< f ; F!x Œg >;

f 2 S 0 .RN /; g 2 S.RN /:

(1.11)

We recall formulas @k Fx! Œ f  D Fx! Œixk f ;

Fx! Œ@xk f  D ik Fx! Œ f ;

(1.12)

where f 2 S 0 .RN /, and     Fx! Œ f g D Fx! Œ f   Fx! Œg ; where f 2 S.RN /, g 2 S 0 .RN / and denotes convolution.

1.5.1 Pseudodifferential Operators A partial differential operator D of order m, DD

X j˛jm

a˛ @˛ ;

(1.13)

1.5 Fourier Transform

19

can be associated with a Fourier multiplier in the form Q D D

X

a˛ .i/˛ ;  ˛ D 1˛1 : : : N˛N

j˛jm

in the sense that 2 1 4 DŒv.x/ D F!x

X

3 a˛ .i/˛ Fx! Œv./5 ; v 2 S.RN /:

j˛jm

The operators defined through the right-hand side of the above expression are called pseudodifferential operators. Various pseudodifferential operators used in the book are identified through their Fourier symbols: • Riesz transform: Rj 

ij ; j D 1; : : : ; N: jj

• Inverse Laplacian: ./1 

1 : jj2

• The “double” Riesz transform: fRgNi;jD1 ; R D 1 rx ˝ rx ; Ri;j 

i j ; i; j D 1; : : : ; N: jj2

• Inverse divergence: Aj D @xj 1 

ij ; j D 1; : : : ; N: jj2

We also denote AWR

3 X i;jD1

Ai;j Ri;j ; RŒvi

3 X jD1

Ri;j Œvj ; i D 1; 2; 3:

20

1 Preliminaries, Notation, and Spaces of Functions

1.5.2 Hörmander–Mikhlin Theorem Theorem 9 Consider an operator L defined by means of a Fourier multiplier m D m./,

1 m./Fx! Œv./ ; LŒv.x/ D F!x where m 2 L1 .RN / has classical derivatives up to order ŒN=2 C 1 in RN n f0g and satisfies j@˛ m./j  c˛ jjj˛j ;  ¤ 0; for any multiindex ˛ such that j˛j  ŒN=2 C 1, where Œ   denotes the integer part. Then L is a bounded linear operator on L p .RN / for any 1 < p < 1. See Stein [86, Chap. 4, Theorem 3].

1.6 Weak Convergence of Integrable Functions Let X be a Banach space, BX the (closed) unit ball in X, and BX  the (closed) unit ball in the dual space X  . (i) Here are some known facts concerning weak compactness: • BX is weakly compact only if X is reflexive. This is stated in Kakutani’s theorem, see Theorem III.6 in Brezis [9]. • BX  is weakly-(*) compact. This is Banach–Alaoglu–Bourbaki theorem, see Theorem III.15 in Brezis [9]. • If X is separable, then BX  is sequentially weakly-(*) compact, see Theorem III.25 in Brezis [9]. • A nonempty subset of a Banach space X is weakly relatively compact only if it is sequentially weakly relatively compact. This is stated in Eberlein– Shmuliyan–Grothendieck theorem, see Kothe [62, Paragraph 24, 3.(8); 7]. (ii) In view of these facts: • Any bounded sequence in L p .Q/, where 1 < p < 1 and Q  RN is a domain, is weakly (relatively) compact. • Any bounded sequence in L1 .Q/, where Q  RN is a domain, is weakly-(*) (relatively) compact. (iii) Since L1 is neither reflexive nor dual of a Banach space, the uniformly bounded sequences in L1 are in general not weakly relatively compact in L1 . On the other hand, the property of weak compactness is equivalent to the property of sequential weak compactness.

1.6 Weak Convergence of Integrable Functions

21

Weak compactness in the space L1 : Theorem 10 Let V  L1 .Q/, where Q  RM is a bounded measurable set. Then the following statements are equivalent: (i) any sequence fvn g1 nD1  V contains a subsequence weakly converging in L1 .Q/; (ii) for any " > 0 there exists k > 0 such that Z fjvjkg

jv.y/j dy  " for all v 2 VI

(iii) for any " > 0 there exists ı > 0 such for all v 2 V Z jv.y/j dy < " M

for any measurable set M  Q such that jMj < ıI (iv) there exists a nonnegative function ˚ 2 C.Œ0; 1//, lim

z!1

˚.z/ D 1; z

such that Z ˚.jv.y/j/ dy  c:

sup

v2V

Q

See Ekeland and Temam [28, Chap. 8, Theorem 1.3], Pedregal [80, Lemma 6.4]. Condition (iii) is termed equi-integrability of a given set of integrable functions and the equivalence of (i) is Dunford–Pettis theorem (cf., e.g., Diestel [25, p. 93]). Condition (iv) is called De la Vallé–Poussin criterion, see Pedregal [80, Lemma 6.4]. The statement “there exists a nonnegative function ˚ 2 C.Œ0; 1//” in condition (iv) can be replaced by “there exists a nonnegative convex function on Œ0; 1/.” For a sequence vn ! v weakly in L1 .Q/; we denote b.v/, b.vn / ! b.v/ weakly in L1 .Q/

22

1 Preliminaries, Notation, and Spaces of Functions

a weak limit of the superpositions with a function b, provided such a limit exists. Note that the weakly converging sequence fvn g1 nD1 may admit several subsequences fb.vnk /g1 converging to different limits. kD1

1.7 Fixed-Point Theorem We shall use the following version of the Schauder fixed-point Theorem, known also as Schaeffer’s fixed point Theorem, see, e.g., [30, Theorem 9.2.4]. Theorem 11 Let T W Z ! Z be a continuous mapping defined on a finitedimensional space Z. Suppose that the set n

ˇ o ˇ z 2 Z ˇ z D T .z/;  2 Œ0; 1

is bounded. Then there exists z 2 Z such that T .z/ D z:

Part I

Mathematics of Compressible Fluid Flows

Chapter 2

Mathematical Model

As this book is focused on purely mathematical aspects of the theory of compressible viscous fluids, we omit a detailed derivation of the model in terms of classical continuum mechanics. The interested reader may consult the monographs of Batchelor [6], Lamb [65], or the more recent treatment by Gallavotti [47]. We suppose that the motion of a fluid is described by means of two basic fields: the mass density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . % D %.t; x/, the velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . u D u.t; x/, both being (numerical) functions of the time t 2 R and the spatial position x 2 R3 . This is the so-called Eulerian description, where the trajectories of hypothetical fluid particles—streamlines—can be obtained by solving the system of ordinary differential equations dX.t/ D u.t; X.t//: dt

(2.1)

In the Eulerian description, the values of all relevant quantities are evaluated with respect to the coordinate system attached to a fixed physical domain occupied by the fluid. The spatial distribution of the mass density %.t; / characterizes the state of the fluid at each instant t, while the velocity field u describes its motion. Such a description may be incomplete from the point of view of physics as it ignores the changes of the internal energy manifested through another state variable—the temperature. However, as we shall see below, considering % and u as the only relevant state variables leads to mathematically well-posed problems acceptable as useful simplification in numerous real world applications, where the effect of temperature changes is either negligible or without substantial impact on the fluid motion, cf. Truesdell and Rajagopal [93].

© Springer International Publishing Switzerland 2016 E. Feireisl et al., Mathematical Theory of Compressible Viscous Fluids, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-319-44835-0_2

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26

2 Mathematical Model

2.1 Mass Conservation In order to describe the time evolution of a fluid, we need a mathematical formulation of certain basic physical principles expressed in terms of conservation or balance laws. As a model example, we recall the classical argument leading to the mathematical formulation of the physical principle of mass conservation, see, e.g., Chorin and Marsden [19]. Consider a volume B  R3 containing a fluid of the mass density %. The change of the total mass of the fluid contained in B during a time interval Œt1 ; t2  is given by Z

Z %.t2 ; x/ dx 

%.t1 ; x/ dx:

B

B

One of the basic laws of physics incorporated in continuum mechanics as the principle of mass conservation asserts that mass is neither created nor destroyed. Accordingly, the change of the fluid mass in B is only because of the mass flux through the boundary @B, here represented by %u  n, where n denotes the outer normal vector to @B (Fig. 2.1): Z

Z

Z

%.t2 ; x/ dx  B

%.t1 ; x/ dx D  B

t2

Z @B

t1

%.t; x/u.t; x/  n.x/ dSx dt:

(2.2)

One should keep in mind that formula (2.2) contains all the relevant piece of information given by physics. In particular, only integrability of % and the “normal trace” of %u on @B are needed for (2.2) to make sense, while the subsequent discussion uses purely mathematical arguments based on the (possibly unjustified) hypothesis of smoothness of all fields in question. First, apply Gauss–Green theorem to rewrite (2.2) in the form: Z

Z

Z

%.t2 ; x/ dx  B

Fig. 2.1 The mass flux %u  n D %.t; x/u.t; x/  n.x/ through a point .t; x/ 2 @B at time t

%.t1 ; x/ dx D  B

t2 t1

Z

  divx %.t; x/u.t; x/ dx dt: B

2.2 Balance of Momentum

27

Furthermore, fixing t1 D t, performing the limit t2 ! t and interchanging the time derivative and integration over B we may use the mean value theorem to obtain Z @t %.t; x/ dx D lim

t2 !t

B

1  t2  t

Z

Z %.t2 ; x/ dx  B

%.t; x/ dx

 (2.3)

B

Z t2 Z   1 divx %.s; x/u.s; x/ dx ds t2 !t t2  t t B Z   D  divx %.t; x/u.t; x/ dx:

D  lim

B

Finally, as relation (2.3) should hold for any volume element B, we may infer that   @t %.t; x/ C divx %.t; x/u.t; x/ D 0:

(2.4)

Relation (2.4) is a first order partial differential equation called equation of continuity. Obviously, both % and u must be at least continuously differentiable for (2.4) to make sense. In the future, we introduce a weak formulation of (2.4) in which, roughly speaking, all derivatives in (2.4) are understood in the sense of the mathematical theory of distributions. Such an approach is in fact closer to the original integral formulation (2.2).

2.2 Balance of Momentum Using arguments similar to the preceding part, we may derive balance of momentum in the form     (2.5) @t %.t; x/u.t; x/ C divx %.t; x/u.t; x/ ˝ u.t; x/ D divx T.t; x/ C %.t; x/f.t; x/; or, equivalently in view of (2.4), h i %.t; x/ @t u.t; x/ C u.t; x/  rx u.t; x/ D divx T.t; x/ C %.t; x/f.t; x/; where the symbol T denotes the Cauchy stress characterizing the internal forces acting on the fluid, and where f is the (specific) external volume force. Equation (2.5) represents a mathematical formulation of Newton’s second law. The force f has its origin in the external objects acting on the fluid (gravitational, magnetic, etc. forces), while the specific form of the Cauchy stress tensor reflects

28

2 Mathematical Model

the material properties of the fluid. These expressed in terms of the constitutive relations are needed to close the system of Eqs. (2.4) and (2.5).

2.3 Constitutive Relations and Navier–Stokes System We adopt the standard mathematical definition of a fluid being a material satisfying Stokes’ law T D S  pI; where S represents viscous stress and p is a scalar function termed pressure. In addition, we suppose that the fluid is barotropic, meaning the pressure p and the density are interrelated through an explicitly given equation of state p D p.%/: While the pressure p can be seen as a static variable, the specific form of the viscous stress S reflects the fluid resistance to motion and as such must be related to the velocity gradient rather than the velocity itself. For the purpose of this book, we focus on a class of fluids for which S D S.rx u/ is a linear function of the velocity gradient, more specifically S obeys Newton’s rheological law   2 S D S.rx u/ D rx u C rxt u  divx uI C divx uI: 3

(2.6)

The scalars and are termed the shear and bulk viscosity coefficient, respectively. Similar to the pressure, the viscosity coefficients may depend on %; however, we will always assume that they are constant, more precisely, > 0,  0. The specific form of the stress tensor (2.6) can be derived from general physical principles accepted in the continuum mechanics, in particular from the fact that the fluid is (assumed to be) isotropic, with material properties independent of the reference frame, see Chorin and Marsden [19]. Remark 1 Of course, relation (2.6) represents a very particular case of a linearly viscous fluid. More complicated rheologies may occur in the real world applications; however, mathematics of the so-called non-Newtonian fluids goes beyond the scope of this book. Since the viscosity coefficients are constant, we may introduce  D  23 and write 2 divx T D x u C . C /rx divx u  rx p.%/; > 0;    ; 3

2.4 Spatial Domain and Boundary Conditions

29

in particular, Eqs. (2.4) and (2.5) can be rewritten in a concise form as

Navier–Stokes System @t % C divx .%u/ D 0;

(2.7)

@t .%u/ C divx .%u ˝ u/ C rx p.%/ D x u C . C /rx divx u C %f: (2.8)

The system of Eqs. (2.7) and (2.8) should be compared with the “more famous” incompressible Navier–Stokes system, where the density is set constant, say % 1, while (2.7) and (2.8) “reduce” to divx u D 0;

(2.9)

@t u C divx .u ˝ u/ C rx p D x u C f:

(2.10)

Unlike in (2.8), the pressure p in (2.10) is an unknown function determined (implicitly) by the fluid motion! The pressure in the incompressible Navier–Stokes system (2.9) and (2.10) has therefore a nonlocal character and may depend on the far field behavior of the fluid system. Remark 2 The fact that the shear viscosity > 0 is strictly positive plays an important role in the mathematical analysis of the Navier–Stokes system. The motion of an inviscid fluid is governed by the Euler system @t % C divx .%u/ D 0;

(2.11)

@t .%u/ C divx .%u ˝ u/ C rx p.%/ D %f;

(2.12)

for which the theory contains so far unsurmountable difficulties due to the inevitable presence of singularities in the form of shock waves that may develop in a finite time, see, e.g., Smoller [85].

2.4 Spatial Domain and Boundary Conditions The system of differential equations (2.7) and (2.8) describes the motion “inside” the fluid itself. In applications, the fluid is usually confined to a bounded physical space represented by a spatial domain ˝  R3 . As is well known, the presence of the physical boundary @˝ and the associated problem of fluid–structure interaction represent a source of substantial difficulties in the mathematical analysis of fluids in motion.

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2 Mathematical Model

In order to avoid technicalities, we adopt the common hypothesis that the viscous fluid adheres completely to the physical boundary expressed through the no-slip boundary condition uj@˝ D 0:

(2.13)

We note that the problem of the choice of correct boundary conditions in the real world applications is rather complex, some parts of the boundaries may allow for in and/or out flux of the fluid, the fluid domain may not be a priori known (free boundary problems), among many other physically relevant situations. The interested reader may consult Priezjev and Troian [83] for further discussion.

2.5 Initial Conditions and Well Posedness Given the initial state at a reference time t0 , say t0 D 0, the time evolution of the fluid should be determined as a solution of the Navier–Stokes system (2.7) and (2.8), supplemented with the boundary conditions (2.13) as the case may be. To this end, we prescribe the initial density %.0; x/ D %0 .x/; x 2 ˝;

(2.14)

together with the initial distribution of the momentum, .%u/.0; x/ D .%u/0 .x/; x 2 ˝;

(2.15)

as, strictly speaking, the momentum balance (2.8) is an evolutionary equation for %u rather than u. Such a difference will become clear in the so-called weak formulation of the problem discussed in the forthcoming section. Ideally, the Navier–Stokes system (2.7) and (2.8) supplemented with the boundary conditions (2.13) and the initial conditions (2.14) and (2.15) should give rise to a mathematically well-posed problem admitting a unique solution on a given time interval .0; T/ for any physically admissible choice of the initial data. Here, “physically admissible” means the density %0 to be strictly positive, the momentum .%u/0 bounded, etc. Moreover, in view of the real world applications, the solution %; u should be computable by means a suitable numerical scheme, whereas the latter should be implementable in order to obtain reliable numerical results in the real time. It is probably needless to say that the present state of the art of the mathematical analysis of nonlinear systems like (2.7) and (2.8) is far from being sufficient in order to perform completely such a programme. On the other hand, however, certain progress has been made during the past few decades and the aim of this book is to make the reader familiar with the principal ideas that have emerged. In particular, the existence theory we develop is based on a convergent numerical scheme presented in Part II of this book.

Chapter 3

Weak Solutions

A vast class of nonlinear evolutionary problems arising in mathematical fluid mechanics including the Navier–Stokes system (2.7) and (2.8) is not known to admit classical (differentiable, smooth) solutions for all choices of data and on an arbitrary time interval. The existence of classical solutions has been established under rather restrictive conditions, notably if • the problem is studied in a simplified geometry, typically in the spatial dimension N D 1, see Antontsev et al. [2]; • the initial data represent a small perturbation of a static state, see Matsumura and Nishida [69, 70]; • the existence time interval .0; T/ is short, see Tani [88], Valli [95, 96], and Valli and Zajaczkowski [97], among others. On the other hand, most of the real world problems call for solutions defined in-the-large approached in the numerical simulations. In order to perform a rigorous analysis, we have to introduce a concept of generalized or weak solution, for which, roughly speaking, derivatives are interpreted in the sense of distributions. Another motivation, at least in the case of the compressible Navier–Stokes system (2.7) and (2.8), is the possibility to study the fluid dynamics emanating from irregular initial state, for instance, the density %0 may not be continuous. As shown by Hoff [52], the singularities incorporated initially will “survive” in the system at any time; thus the weak solutions are necessary in order to describe the dynamics. The main deficiency of the present theory is that the class of weak solutions may be too large to give rise to a well-posed problem. Although the presence of viscosity may and effectively does provide a certain regularizing effect, the weak solutions of the Navier–Stokes system (2.7) and (2.8) are not known to be uniquely determined by the data. Note that the situation is even more delicate for the inviscid fluids governed by the Euler system (2.11) and (2.12), for which the weak solutions

© Springer International Publishing Switzerland 2016 E. Feireisl et al., Mathematical Theory of Compressible Viscous Fluids, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-319-44835-0_3

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3 Weak Solutions

are not unique, even when subjected to various physically relevant admissibility conditions, see Chiodaroli and Kreml [16], Chiodaroli et al. [17], and De Lellis and Székelyhidi [24] for the most recent examples of ill-posedness of problems in fluid dynamics.

3.1 Equation of Continuity: Weak Formulation A weak formulation of the equation of continuity (2.7) is intimately related to its integral form (2.2). We consider a test function ' belonging to the class Cc1 ..0; T/  ˝/ defined on the space-time cylinder .0; T/  ˝. Multiplying (2.7) on ', integrating the resulting expression over .0; T/  ˝, and performing obvious by parts integration, we obtain Z

T 0

Z   %.t; x/@t '.t; x/ C %.t; x/u.t; x/  rx '.t; x/ dx dt D 0:

(3.1)

˝

We say that a pair of functions Œ%; u is a weak solution to Eq. (2.7) in .0; T/  ˝, if the integral identity is satisfied for any ' 2 Cc1 ..0; T/  ˝/. Remark 3 Sometimes, it is more convenient to keep the concise form (2.7) and to say that the derivatives are satisfied in the sense of distributions—a statement equivalent to (3.1). Note that (3.1) does not provide any information on the boundary behavior of the fields %, u, in particular, it does not include the boundary and the initial conditions.

3.1.1 Weak–Strong Compatibility Apparently, the weak formulation (3.1) makes sense as soon as both % and %u are merely locally integrable in the open set .0; T/˝. It is easy to see that any classical (smooth) solution of Eq. (2.7) is also a weak solution. Similarly, any weak solution that is continuously differentiable satisfies (2.7) pointwise. Such a property is called weak–strong compatibility.

3.1.2 Weak Continuity in Time Up to now, we have left apart the problem of satisfaction of the initial condition (2.14). Obviously, some kind of weak continuity is needed for (2.14) to make sense. To this end, we suppose that both the density %.t; / and the momentum

3.1 Equation of Continuity: Weak Formulation

33

%u.t; / are locally integrable in space and globally integrable in time, specifically, % 2 L1 .0; TI L1loc .˝//; %u 2 L1 .0; TI L1loc .˝I R3 //:

(3.2)

Taking '.t; x/ D

2 Cc1 .0; T/;  2 Cc1 .˝/

.t/.x/;

as a test function in (3.1) we may infer, by virtue of (3.2), that the function Z t 7! ˝

%.t; x/.x/ dx is absolutely continuous in Œ0; T

(3.3)

for any  2 Cc1 .˝/ provided % 2 L1loc .Œ0; T  ˝/. In particular, the initial condition (2.14) may be imposed in the sense that Z

Z lim

t!0C

˝

%.t; x/.x/ dx D

˝

%0 .x/.x/ dx for any  2 Cc1 .˝/:

Remark 4 To be more precise, we should say that the density % considered as a function of the time t 2 .0; T/ ranging in the space L1loc .˝/ can be modified on a set of times in Œ0; T of zero Lebesgue measure in such a way that (3.3) holds. We may equivalently say that % 2 Cweak .Œ0; TI L1 .K// for any compact K  ˝: Now, take '" .t; x/ D where

"

" .t/'.t; x/;

' 2 Cc1 .Œ0; T  ˝/;

2 Cc1 .0; /, 0

see Fig. 3.1. Fig. 3.1 Graphs of cutoff functions "1 and "2 with respect to time, "1 > "2

"

 1;

"

% 1Œ0;  as " ! 0;

(3.4)

34

3 Weak Solutions

Taking '" as a test function in (3.1) and letting " ! 0, we conclude, making use of (3.3), that Z Z %.; x/'.; x/ dx  %0 .x/'.0; x/ dx ˝

Z



D 0

˝

Z   %.t; x/@t '.t; x/ C %.t; x/u.t; x/  rx '.t; x/ dx dt ˝

for any  2 Œ0; T and any ' 2 Cc1 .Œ0; T  ˝/. We may go even further and replace the initial time by any 1 2 Œ0; T obtaining Z ˝

Z D

2

Z 

1

˝

%.t; x/'.t; x/ dx

tD2 (3.5) tD1

 %.t; x/@t '.t; x/ C %.t; x/u.t; x/  rx '.t; x/ dx dt

for any 0  1 < 2  T and any ' 2 Cc1 .Œ0; T  ˝/. Formula (3.5) can be alternatively used as a definition of weak solution to the equation of continuity (2.7). Furthermore, anticipating the no-slip boundary condition (2.13), we may ask (3.5) to be satisfied for any test functions ' 2 Cc1 .Œ0; T  R3 / restricted to ˝, meaning ' need not vanish on @˝. Accordingly, the class of admissible weak solutions must be restricted to functions integrable up to the boundary, specifically, we require % 2 L1 .0; TI L1 .˝// and %u 2 L1 ..0; T/  ˝I R3 /. It is interesting to compare (3.5) with the original integral formulation of the principle of mass conservation stated in (2.2). To this end, we take '" .t; x/ D " .x/; with " 2 Cc1 .B/ such that 0  "  1; " % 1B as " ! 0: It is easy to see that Z

Z ˝

%.; x/'" .; x/ dx 

Z ˝

Z

%0 .x/'" .0; x/ dx !

%.; x/ dx  B

%0 .x/ dx B

as " ! 0, which coincides with the expression on the left-hand side of (2.2). Consequently, the right-hand side of (3.5) must posses a limit and we set Z

 0

Z

Z ˝

%.t; x/u.t; x/  rx " .x/ dx dt ! 

 0

Z @B

%.t; x/u.t; x/  n dSx dt;

3.1 Equation of Continuity: Weak Formulation

35

which may be seen as a definition of the integral of the normal trace of the momentum %u over the lateral boundary of the space time cylinder .0; /  ˝. We may therefore conclude that weak solutions possess a kind of “weak” normal trace on the boundary of the cylinder .0; /  B that satisfies (2.2), see Chen and Frid [14] for a more elaborate treatment of the concept of normal traces for solutions to general conservation laws.

3.1.3 Total Mass Conservation, Positive Density, and Vacuum In this section, we adopt the more restrictive definition of the weak solutions based on the integral identity (3.5), with the test functions that do not vanish on the boundary, for which a better integrability of %, %u is required, namely, % 2 L1 .0; TI L1 .˝//, %u 2 L1 ..0; T/  ˝I R3 /. Under these circumstances, it is easy to observe that (3.5) holds with ˝ replaced by R3 provided %, u have been extended to be zero outside ˝. If ˝  R3 is bounded, we may choose ' in (3.5) that coincides with the characteristic function 1Œ0; ˝ on the space time cylinder Œ0;   ˝ to obtain Z

Z ˝

%.; x/ dx D

˝

%0 .x/ dx D M0 for any   0;

(3.6)

meaning, the total mass M0 of the fluid is a constant of motion. A physical mass density must be positive or at least nonnegative at any time. Unfortunately, strict positivity of % cannot be deduced from the weak formulation (3.5) even if we suppose that %0 > 0 in the whole domain ˝. Indeed we need extra regularity of the velocity field u to ensure unique solvability of the streamline equation (2.1). Formally, we have d %.t; X.t// D %.t; X.t//divx u.t; X.t//; X.0/ D X0 dt

(3.7)

as long as (2.1) admits a unique solution for any X0 2 ˝. Relation (3.7) implies positivity of the density on condition that divx u is bounded, or at least divx u 2 L1 .0; TI L1 .˝//:

(3.8)

This condition along with certain regularity of u in turn implies unique solvability of the system of ordinary differential equations (2.1) in a generalized sense, see DiPerna and Lions [26]. Unfortunately, however, the degree of regularity hidden in (3.8) is not available or at least not known for the weak solutions to the Navier– Stokes system (2.7) and (2.8), for which the available energy estimates yield the velocity gradient merely square integrable in .0; T/  ˝, see Chap. 4 below.

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3 Weak Solutions

Fig. 3.2 Density profile in a fixed time t with a vacuum region

Remark 5 As a matter of fact, any weak solution of the compressible Navier–Stokes system emanating from smooth initial data and satisfying (3.8) must be regular as shown by Sun et al. [87] and [39]. In the light of the previous discussion we may therefore only expect that %  0 for a.a. .t; x/ 2 .0; T/  ˝;

(3.9)

meaning a weak solution may hypothetically develop a vacuum region in a finite time even if %0 > 0 (Fig. 3.2). Given the anticipated regularity of u, even (3.9) cannot be deduced directly from the weak formulation (3.5) and must be stated as an integral part of a proper definition of suitable weak solutions to the Navier–Stokes system, see Chap. 5 below.

3.2 Balance of Momentum: Weak Formulation Following the steps detailed in the preceding part, we introduce a weak formulation of the balance of momentum (2.8) in the form of a family of integral identities Z

T 0

Z  ˝

.%u/.t; x/  @t '.t; x/ C .%u ˝ u/.t; x/ W rx '.t; x/  Cp.%/.t; x/divx '.t; x/ dx dt Z

T

D 0

Z  rx u.t; x/ W rx '.t; x/ ˝

 C. C /divx u.t; x/divx '.t; x/  %.t; x/f.t; x/  '.t; x/ dx dt satisfied for any test function ' 2 Cc1 ..0; T/  ˝I R3 /.

(3.10)

3.2 Balance of Momentum: Weak Formulation

37

Of course, we have tacitly assumed that all quantities appearing in (3.10) are at least locally integrable in .0; T/  ˝. In particular, as (3.10) contains explicitly rx u, we have to assume integrability of this term. As we shall see in Chap. 4, one can expect, given the available a priori bounds, rx u to be square integrable, specifically, u 2 L2 .0; TI W 1;2 .˝I R3 //: Moreover, as the functions in the Sobolev space W 1;2 .˝I R3 / possess well-defined traces on @˝, the no-slip boundary conditions (2.13) may be incorporated by the stipulation u 2 L2 .0; TI W01;2 .˝I R3 //; where W01;2 .˝I R3 / is the Sobolev space obtained as the closure of Cc1 .˝I R3 / in the W 1;2 -norm. Remark 6 Strictly speaking, traces for Sobolev functions are available only for domains ˝ with certain smoothness (for instance Lipschitz). On the other hand, the space W01;2 can be meaningfully defined for any domain as the closure of smooth compactly supported functions.

3.2.1 Weak Continuity in Time Using the same arguments as in Sect. 3.1.2, we may show the weak continuity in time of the momentum %u making meaningful the initial condition (2.15). To this end, we suppose that %u 2 L1 .0; TI L1loc .˝I R3 // and that all remaining quantities in the weak formulation (3.10) are at least locally integrable in space and globally in time. Thus, exactly as in Sect. 3.1.2, we may convert (3.10) to the integral identity Z

2 1

Z  .%u/.t; x/  @t '.t; x/ C .%u ˝ u/.t; x/ W rx '.t; x/ ˝

 Cp.%/.t; x/divx '.t; x/ dx dt Z 2 Z  rx u.t; x/ W rx '.t; x/ D 1

˝

 C. C /divx u.t; x/divx '.t; x/  %.t; x/f.t; x/  '.t; x/ dx dt Z tD2 C %u.t; x/  '.t; x/ dx ˝

tD1

(3.11)

38

3 Weak Solutions

for any 0  1 < 2  T, ' 2 Cc1 .Œ0; T  ˝I R3 /, where %u 2 Cweak .Œ0; TI L1 .K; R3 // for any compact K  ˝: Remark 7 It is worth noting that it is the momentum %u rather than the velocity u that is weakly continuous in time. As a matter of fact, one can loose control over the time oscillations of u on the hypothetical vacuum zones where % D 0, cf. Sect. 3.1.3. There appears a slightly ambiguous situation as both % and %u have well-defined instantaneous values for any time t 2 Œ0; T whereas u has not.

3.2.2 Admissibility Criteria A natural question arises, namely, if the weak formulation enlarging substantially the set of relevant solutions is likely to provide a unique solution to the initialboundary value problem (2.7), (2.8), (2.13)–(2.15). As we have already seen in Sect. 3.1.3, certain desired properties of solutions such as nonnegativity of the density must be “appended” as an integral part of the definition. We may also require a weaker statement termed weak–strong uniqueness for the Navier–Stokes system, meaning, the weak and strong solutions emanating from the same initial data should coincide as long as the latter exist. As shown in [39], such property holds true provided the weak solution satisfies in addition the energy inequality discussed in the next chapter.

Chapter 4

A Priori Bounds

A priori bounds are natural constraints imposed on the set of (hypothetical) smooth solutions by the data as well as by the differential equations satisfied. A priori bounds determine the function spaces framework the (weak) solutions are looked for. By definition, they are formal, derived under the principal hypothesis of smoothness of all quantities in question. A priori bounds have their counter part in the stability estimates derived for the associated numerical scheme discussed in Part II. As we will see, the a priori bounds available for the Navier–Stokes system are rather poor and can be derived by means of elementary integration. Unfortunately, they are the best available for our problem although some substantial and mathematically rather delicate improvements have been obtained recently by Plotnikov and Weigant [81] in the simplified 2-D geometry, see also [82] for the stationary case.

4.1 Total Mass Conservation Recalling that a priori estimates are formal and derived under the hypothesis of smoothness of all quantities in question we may go back to formula (3.7) obtaining  inf %.0; y/ exp tkdivx ukL1 ..0;T/˝/

y2˝

(4.1)

 %.t; x/   sup %.0; y/ exp tkdivx ukL1 ..0;T/˝/

y2˝

for any t 2 Œ0; T, x 2 ˝. As already pointed out the bounds established in (4.1) depend on kdivx ukL1 on which we have no information. Thus we may infer only © Springer International Publishing Switzerland 2016 E. Feireisl et al., Mathematical Theory of Compressible Viscous Fluids, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-319-44835-0_4

39

40

4 A Priori Bounds

that %.t; x/  0:

(4.2)

In addition, relation (4.2) combined with the total mass conservation (3.6) yields k%.t; /kL1 .˝/ D k%0 kL1 .˝/ ; %0 D %.0; /:

(4.3)

4.2 Total Energy Balance The kinetic energy balance is obtained by taking the scalar product of the momentum equation (2.5) with u. Making use of the continuity equation (2.4) we deduce  @t

   1 1 %juj2 C divx %juj2 u C divx . p.%/u/  p.%/divx u  divx .Su/ C S W rx u 2 2 (4.4) D %f  u:

Remark 8 In contrast with the preceding part of the book, we will systematically omit writing down the explicit dependence of all quantities on the independent variables .t; x/, unless such notation is necessary to avoid confusion. Keeping in mind that u satisfies the no-slip boundary condition (2.13), we may integrate (4.4) by parts obtaining d dt

Z  ˝

1 %juj2 2



Z ˝

where Z ˝

Z

dx 

p.%/divx u dx C

Z S.rx u/ W rx u dx D

2

˝

˝

Z S.rx u/ W rx u dx D

2

.jrx uj C . C /jdivx uj / dx 

˝

Z ˝

%f  u dx; (4.5)

jrx uj2 dx: (4.6)

Remark 9 Note that, in general, S.rx u/ W rx u ¤ jrx uj2 C . C /jdivx uj2 ; however, (4.6) can be justified by simple by parts integration keeping in mind the u vanishes on @˝.

4.2 Total Energy Balance

41

Next, the term involving the driving force f may be handled as Z

Z ˝



%f  u dx 

1 kfkL1 ..0;T/˝/ 2

˝

p p jfj % %juj dx

Z ˝

Z % dx C

˝

(4.7)

 %juj2 dx I

whence the integral on the right-hand side of (4.5) may be controlled by the lefthand side by means of Gronwall’s argument. Finally, we focus on the integral Z ˝

p.%/divx u dx:

Multiplying Eq. (2.4) by b0 .%/ we obtain the renormalized equation of continuity   @t b.%/ C divx .b.%/u/ C b0 .%/%  b.%/ divx u D 0:

(4.8)

This equation, derived later also in a weak form, plays a crucial role in the analysis of our problem. The idea of renormalization can be traced back to the pioneering works of Kruzhkov [63] and later DiPerna and Lions [26]. In particular, the choice Z b.%/ D P.%/ %

% 1

p.z/ dz z2

(4.9)

gives rise to b0 .%/%  b.%/ D p.%/; and, in accordance with the no-slip boundary condition, Z

d  p.%/divx u dx D dt ˝

Z P.%/ dx: ˝

Going back to (4.5), we obtain the total energy balance d dt

Z  ˝

 Z Z 1 %juj2 C P.%/ dx C S.rx u/ W rx u dx D %f  u dx: 2 ˝ ˝

Remark 10 The quantity Z  E.t/

˝

 1 2 %juj C P.%/ .t; x/ dx 2

(4.10)

42

4 A Priori Bounds

represents the total mechanical energy of the fluid. It follows from (4.6) and (4.10) that E.t/ is (for f D 0) a nonincreasing function of t and strictly decreasing as soon as rx u ¤ 0. As the First law of thermodynamics asserts that the total energy of a closed system is conserved, the quantity E.t/ cannot be the total energy of a viscous fluid. As seen in (4.10), and in accordance with the Second law of thermodynamics, a part of the mechanical energy is irreversibly converted to another form of internal energy usually associated with the production of heat as long as the fluid is in motion. In the framework of our simplified model, the resulting changes of temperature and their influence on the fluid motion are not taken into account. A mathematical theory for the full energetically complete system has been developed in [35]. The function P introduced in (4.9) is called pressure potential. In order to deduce a priori bounds from the energy balance (4.10), it is convenient, although not strictly necessary, P to be a convex function of %. To this end, we adopt a physically grounded thermodynamic stability hypothesis that the pressure is an increasing function of the density, specifically p 2 CŒ0; 1/ \ C1 .0; 1/; p.0/ D 0; p0 .%/ > 0 for any % > 0:

(4.11)

Under hypothesis (4.11), the pressure potential is a convex function of % bounded from below and we may use the energy balance, together with (4.6) and (4.7) and the standard Gronwall argument, to deduce the following a priori bounds: p sup k %u.t; /kL2 .˝IR3 /  c.E0 ; T; kfkL1 ..0;T/˝IR3 / /;

t2Œ0;T

Z

sup t2Œ0;T

Z

T 0

(4.12)

˝

P.%/.t; / dx  c.E0 ; T; kfkL1 ..0;T/˝IR3 / /;

krx u.t; /k2L2 .˝IR33/ dt  c.E0 ; T; kfkL1 ..0;T/˝IR3 / /;

(4.13)

(4.14)

where E0 denotes the initial energy Z  E0 D

˝

1 %0 ju0 j2 C P.%0 / 2

 dx:

Finally, as u satisfies the no-slip condition (2.13), we may use Poincaré’s inequality applied to (4.14) to obtain Z 0

T

ku.t; /k2W 1;2 .˝IR3 / dt  c.E0 ; T; kfkL1 ..0;T/˝IR3 / /: 0

(4.15)

4.3 Pressure Estimates

43

Remark 11 Seeing that %u D

p p % %u

we may use (4.3) and (4.12), together with Hölder’s inequality to obtain sup k%u.t; /kL1 .˝IR3 /  c.E0 ; T; kfkL1 ..0;T/˝IR3 / /:

(4.16)

t2Œ0;T

Remark 12 The hypothesis that f is a bounded and measurable function is not optimal and can be relaxed at the expense of unnecessary technical difficulties.

4.3 Pressure Estimates In order to exploit (4.13), certain growth of the pressure must be assumed in addition to (4.11). Here and hereafter we therefore assume that p0 .%/ D p1 > 0 for some   1: %!1 % 1 lim

(4.17)

The specific value of the exponent  will play a role of a “critical” parameter. We recall that the theory developed in this book yields positive existence results as soon as  > 32 , ˝  R3 . The convergence of the numerical scheme in Part II requires a stronger restriction  > 3. Remark 13 Hypothesis (4.17) is motivated by the standard isentropic state equation p.%/ D a% ; a > 0; where  is the adiabatic exponent (Fig. 4.1). Physically relevant values are 1    5 5 3 , where  D 1 is the so-called isothermal case, while  D 3 describes the monoatomic gas, see Eliezer et al. [29]. Under hypotheses (4.11) and (4.17), the pressure potential satisfies P.%/  c1 %  c2 ; c1 > 0; in particular, bound (4.13) implies Z sup t2Œ0;T

˝

p.%/.t; / dx  c.E0 ; T; kfkL1 ..0;T/˝IR3 / /:

(4.18)

44

4 A Priori Bounds

Fig. 4.1 The isentropic state relation between the density and the pressure

p p Moreover, writing %u D % %u, we may use the estimates (4.12)–(4.15), together with Hölder’s inequality and the Sobolev embedding relation W 1;2 ,! L6 , to deduce 8 9 p p k%uk 2  k %kL2 .˝/ k %ukL2 .˝IR3 // ; ˆ > ˆ > < = L  C1 .˝IR3 // I ˆ > ˆ : k%uk 6 ;  k%kL .˝/ kukL6 .˝IR3 //  k%kL .˝/ kukW 1;2 .˝IR3 // > 0

L  C6 .˝IR3 //

whence k%uk

2

L1 .0;TIL  C1 .˝IR3 //

C k%uk

6

L2 .0;TIL  C6 .˝IR3 //

(4.19)

 c.E0 ; T; kfkL1 ..0;T/˝IR3 / /; and, similarly, k%u ˝ ukL1 .0;TIL1 .˝IR33 // C k%u ˝ uk

3

L1 .0;TIL  C3 .˝IR33 //

(4.20)

 c.E0 ; T; kfkL1 ..0;T/˝IR3 / /: If  > 32 , we have

3  C3

> 1 in (4.20); whence, by interpolation,

k%u˝ukLq ..0;T/˝IR33 /  c.E0 ; T; kfkL1 ..0;T/˝IR3 / / for a certain q > 1:

(4.21)

4.3 Pressure Estimates

45

We point out that bound (4.21), meaning bound on the convective term %u˝u in the reflexive space Lq , q > 1, is the only part of the existence proof where the hypothesis  > 32 is really needed! In view of (4.19)–(4.21), all terms appearing in the weak formulation (3.5) and (3.11) are at least equi-integrable in the Lebesgue space L1 ..0; T/˝/ with the only exception of the pressure p.%/ bounded only by (4.18). A seemingly direct way to improve (4.18) is to “compute” the pressure from the momentum balance (2.8): 1 p.%/ D 1 x divx @t .%u/  x divx divx .%u ˝ u/ 1 C1 x divx divx S.rx u/ C x divx .%f/;

where 1 x is a suitable “inverse” of the Laplacian. Since the pseudo-differential p operator 1 x divx divx is bounded in the L -spaces, the only problematic term is 1 x divx @t .%u/. However, since the pressure is positive, we may multiply the above relation by b.%/, and, after by parts integration, to replace 1 1 x divx @t .%u/b.%/  x divx .%u/@t b.%/;

where @t b.%/ can be computed by means of the renormalized equation (4.8). In order to justify the above programme, we first suppose  >3

(4.22)

and use the quantity rx 1 x Œ%;

'D

2 Cc1 .0; T/;  2 Cc1 .˝/

as a test function in the integral identity (3.11). The operator 1 x is defined as the inverse of the Laplacian on the whole R3 here applied to the cut-off % extended to be zero outside ˝, specifically, 1 x Œ%



1 F!x



 1 Fx! Œ% ; jj2

where F denotes the standard Fourier transform, see Sect. 1.5. The resulting expression reads Z

T 0

Z

T

D 0

Z h ˝

Z ˝

i h  2 p.%/%  . C /%divx u dx dt

i . C /divx u  p.%/ rx   rx .1 x Œ%/ dx dtC

(4.23)

46

4 A Priori Bounds

Z

T 0

Z ˝



rx u W rx rx .1 x Œ%/ dx dt  Z

Z

T

 0

˝

Z

0

Z ˝

%f  rx 1 x Œ% dx dt

 @t .%u/  rx .1 x Œ%/ dx dt:

C 0

T

 .%u ˝ u/ W rx rx .1 x Œ%/ dx dt

Z

T

Z

˝

Since  > 3, we get, by virtue of (4.13) and the standard elliptic estimates, krx 1 x Œ%kL1 ..0;T/˝IR3 /  c.; E0 ; T; kfkL1 ..0;T/˝IR3 / / and krx .rx 1 x Œ%/kL1 .0;TIL .˝IR33 //  c.; E0 ; T; kfkL1 ..0;T/˝IR3 / /: Consequently, by virtue of the energy estimates established in (4.12)–(4.21) combined with Hölder’s inequality, we get ˇZ ˇ ˇ ˇ

T

Z ˝

0

ˇZ ˇ  c. ; ; E0 ; T; kfkL1 / C ˇˇ

T 0

ˇ ˇ  2 p.%/% dx dtˇˇ Z ˝

 @t .%u/ 

(4.24)

rx .1 x Œ%/

Finally, we integrate by parts the rightmost integral Z

T 0

Z Z

˝ T

 @t .%u/  rx .1 x Œ%/ dx dt Z

D Z

T

Z

0

C 0

˝

˝

@t  %u  rx .1 x Œ%/ dx dt

 %u  rx .1 x Œdivx .%u// dx dt;

where the right-hand side is bounded in view of (4.19) and  > 3. Going back to (4.24) we may therefore infer that Z K

p.%/% dx dt  c.K; E0 ; T; kfkL1 ..0;T/˝IR3 / /;

ˇ ˇ dx dtˇˇ :

4.3 Pressure Estimates

47

or, equivalently, k%kL C1 .K/  c.K; E0 ; T; kfkL1 ..0;T/˝IR3 / / for any compact K  .0; T/  ˝: (4.25) Remark 14 Estimate (4.25) can be slightly improved, namely k%kL C1 ..0;T/K/  c.K; E0 ; T; kfkL1 ..0;T/˝IR3 / / for any compact K  ˝: (4.26) Indeed the only “singular” integral in (4.24) containing @t Z

T 0

Z ˝

is

@t  %u  rx .1 x Œ%/ dx dt:

Since 2

%u 2 Cweak .Œ0; TI L  C1 .˝I R3 //; % 2 Cweak .Œ0; TI L .˝//; this integral remains bounded if we let

% 1.

Remark 15 Similar estimates can be obtained in the general case  > 32 , where we take 'D

ˇ rx 1 x Œ% ;

for ˇ > 0 sufficiently small, see Part III.

2 Cc1 .0; T/;  2 Cc1 .˝/

Chapter 5

Weak Formulation Revisited

In Chap. 3, we have introduced the weak formulation of both the equation of continuity (3.5) and the momentum balance (3.11). On the other hand, we have seen in Chap. 4 that regular solutions of the Navier–Stokes system satisfy also the renormalized equation of continuity (4.8), together with the total energy balance (4.10). Under the general hypothesis considered in this book, the piece of information encoded in (4.8) and (4.10) cannot be obtained directly from the weak formulation (3.5) and (3.11). Accordingly, it seems convenient to include both (4.8) and (4.10) in the definition of weak solutions to the problem (2.7), (2.8), (2.13)– (2.15) as a kind of additional admissibility criteria. Similar to the preceding part, we suppose that ˝  R3 is a bounded domain with Lipschitz boundary and the pressure p satisfies hypotheses (4.11) and (4.17). Moreover, in accordance with the energy bounds established in (4.10), we take %0 2 L .˝/; %0  0 a.a. in ˝;

j.%u/0 j2 2 L1 .˝/; %0

f 2 L1 ..0; T/  ˝I R3 /:

(5.1) (5.2)

Remark 16 Note that hypothesis (5.1) requires implicitly the initial momentum .%u/0 to vanish on the vacuum set fx 2 ˝ j %0 .x/ D 0g. In view of the a priori bounds obtained in Chap. 4, hypotheses (5.1) are optimal, while (5.2) could be extended to a slightly less regular class of functions.

© Springer International Publishing Switzerland 2016 E. Feireisl et al., Mathematical Theory of Compressible Viscous Fluids, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-319-44835-0_5

49

50

5 Weak Formulation Revisited

5.1 Regularity Class Motivated by the a priori bounds established in Chap. 4, we look for weak solutions belonging to the following class: • the density % D %.t; x/ is a nonnegative function a.a. in .0; T/  ˝, % 2 Cweak .Œ0; TI L .˝//; p.%/ 2 L˛ ..0; T/  K/ for a certain ˛ > 1 and any compact K  ˝I • the velocity field u D u.t; x/ satisfies u 2 L2 .0; TI W01;2 .˝I R3 //I • the momentum %u D .%u/.t; x/ satisfies 2

%u 2 Cweak .Œ0; TI L  C1 .˝I R3 //;

p %u 2 L1 .0; TI L2 .˝I R3 //:

5.2 Renormalized Equation of Continuity We start by introducing a class of (nonlinear) functions b such that b 2 C1 Œ0; 1/; b.0/ D 0; b0 .r/ D 0 whenever r  Mb ;

(5.3)

where Mb is a constant that may differ for each particular b. A weak formulation of the mass conservation which includes both (3.5) and (4.8) reads

Renormalized Equation of Continuity Z

T 0

Z h ˝

i  .% C b.%// @t ' C.% C b.%// urx ' C b.%/  b0 .%/% divx u' dx dt (5.4)

Z D ˝

.%0 C b.%0 // '.0; / dx

for any ' 2 Cc1 .Œ0; T/R3 / and any b belonging to the class specified in (5.3).

For b 0 we obtain the standard weak formulation of (3.5).

5.3 Momentum Equation

51

Remark 17 One could also write (5.4) in the form Z

2 1

Z h ˝

i  .% C b.%// @t ' C .% C b.%// u  rx ' C b.%/  b0 .%/% divx u' dx dt Z D ˝

.% C b.%// .t; /'.t; / dx

tD2

(5.5)

tD1

for any 0  1 < 2  T. Here, a certain ambiguity appears as the weakly continuous representative of % may not coincide with the weakly continuous representative of b.%/ at any t. Fortunately, such a situation does not occur as it can be shown that the (weak) renormalized solutions are strongly continuous in time, specifically, % 2 C.Œ0; TI L1 .˝//: The interested reader may check the proof in [34]. Remark 18 Since the test functions considered in (5.4) are not required to have compact support in ˝, this relation actually holds on the whole physical space R3 provided %, u were extended to be zero outside ˝. Note also that (5.4) implies that the initial condition %.0; / D %0 ./ is fulfilled.

5.3 Momentum Equation From Chap. 3, we recall

Momentum Balance Z ˝

Z D Z 

2

1 2 1

%u.t; /  '.t; / dx

tD2 (5.6) tD1

Z   %u  @t ' C %u ˝ u W rx ' C p.%/divx ' dx dt ˝

Z   rx u W rx ' C . C /divx udivx '  %f  ' dx dt ˝

for any 0  1  2 and for any test function ' 2 Cc1 .Œ0; T  ˝I R3 /.

52

5 Weak Formulation Revisited

Note that (5.6) already includes the satisfaction of the initial condition (2.15) provided we set %u.0; / D .%u/0 :

5.4 Energy Inequality Unfortunately, the weak solutions are not known to be uniquely determined by the initial data. Therefore it is desirable to introduce as much physically grounded conditions as allowed by the construction of the weak solutions. One of them is

Energy Inequality  Z Z  Z   1 2 jrx uj2 C . C /jdivx uj2 dx dt %juj C P.%/ .; / dx C 2 ˝ 0 ˝ (5.7)  Z  Z Z 1 2  j.%u/0 j C P.%0 / dx C %f  u dx dt 2%0 ˝ ˝ 0 for a.a.  2 .0; T/, with the pressure potential Z P.%/ D %

% 1

p.z/ dz: z2

Obviously, the energy inequality (5.7) is a weak counterpart of the energy equality (4.10). The equality is lost in the passage to weak solutions as the dissipative term Z

 0

Z   jrx uj2 C . C /jdivx uj2 dx dt ˝

is only weakly lower semi-continuous with respect to the topology induced by the energy bounds (4.14). The fact that energy is not being “produced” in our class of weak solutions represents an important admissibility criterion that eliminates the oscillatory weak solutions constructed by De Lellis and Székelyhidi [23] in the context of the Euler system. The weak solutions satisfying the energy inequality (5.7) enjoy the property of weak–strong uniqueness; they coincide with a strong solution emanating from the same initial data as long as the latter exists, see [39].

5.4 Energy Inequality

53

Summarizing the material of this chapter we get Definition 1 We say that Œ%; u is a weak solution to the Navier–Stokes system (2.7) and (2.8) in .0; T/  ˝ satisfying the boundary conditions (2.13) and the initial conditions (2.14) and (2.15) if: • Œ%; u belongs to the regularity class specified in Sect. 5.1; • the renormalized equation of continuity (5.4) and the momentum balance (5.6) are satisfied in .0; T/  ˝; • the energy inequality (5.7) holds for a.a.  2 .0; T/.

Chapter 6

Weak Sequential Stability

The property of weak sequential stability plays a crucial role in the analysis of any nonlinear problem. It states that the solution set of a given problem is (weakly) precompact with respect to the topologies induced by the available a priori estimates. In our context, this property can be stated as follows:

Weak Sequential Stability Given a family fŒ%" ; u" g">0 of weak solutions of the compressible Navier– Stokes system (2.7) and (2.8), with the boundary condition (2.13), emanating from the initial data %" .0; / D %0;" ; .%u/.0; / D .%u/0;" ; we want to show that %" ! %; u" ! u as " ! 0 in a certain sense and at least for suitable subsequences, where %, u is another weak solution of the same problem.

Although showing weak sequential stability does not provide an explicit proof of existence of the weak solutions, its verification represents one of the prominent steps towards a rigorous existence theory and/or convergence of a numerical scheme. In the class of weak solutions, the proof of weak sequential stability amounts to handling two fundamental issues: • Concentrations. The family fŒ%" ; u" g">0 may weakly converge to a measure, not to a function. A typical example is the so-called gravitational collapse, where © Springer International Publishing Switzerland 2016 E. Feireisl et al., Mathematical Theory of Compressible Viscous Fluids, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-319-44835-0_6

55

56

6 Weak Sequential Stability

the density concentrates around certain points becoming a combination of Dirac measures in the limit. Fortunately, in our setting, concentrations are eliminated by a priori bounds, notably by the pressure estimate (4.26). • Oscillations. The family fŒ%" ; u" g">0 may develop uncontrolled oscillations similar to sin "t , " ! 0. As a result, the (weak) limit b.%; u/ of nonlinear compositions fb.%" ; u" /g" may not coincide with the desired b.%; u/. Eliminating the possibility of density oscillations in the compressible Navier–Stokes system is a cornerstone of the mathematical theory developed by Lions [68]. The key quantity is the so-called effective viscous flux p.%/  . C 2 /divx u that is in fact more regular than its two components. As we shall see below, the effective viscous flux satisfies Lions’ identity h i Œ p.%/  . C 2 /divx u b.%/ D p.%/  . C 2 /divx u b.%/

(6.1)

for any b that can be used, together with the renormalized equation of continuity, to rule out possible density oscillations. In the remaining part of this chapter, we establish the property of weak sequential stability for a sequence of weak solutions fŒ%" ; u" g">0 under the same hypotheses as in Chap. 4, notably the pressure p satisfies (4.11) and (4.17), with  > 3: The relevant modifications for the (so far) critical case  > 32 will be discussed in Part III. For the sake of simplicity, we also take f 0 noticing that all results remain valid as long as f satisfies (5.2).

6.1 Uniform Bounds To begin, we need to show uniform bounds on the family of weak solutions fŒ%" ; u" g">0 in terms of the initial data independent of " ! 0. To this end, we suppose that Z  ˝

1 j.%u/0;" j2 C P.%0;" / 2%0;"

 dx  E0 ;

(6.2)

where the constant E0 is independent of ".

6.1.1 Energy Bounds As the weak solutions satisfy the energy inequality (5.7), the desired estimates can be obtained exactly as in Sect. 4.2. Specifically, we get (recall that

6.1 Uniform Bounds

57

% 2 Cweak .Œ0; TI L .˝//) sup k%" .t; /kL .˝/  c.E0 /

(6.3)

t2.0;T/

and p ess sup k %" u" .t; /kL2 .˝IR3 //  c.E0 /;

(6.4)

t2.0;T/

together with Z 0

T

ku" .t; /k2W 1;2 .˝IR3 / dt  c.E0 /:

(6.5)

Here and hereafter, the symbol c.: : : / denotes a generic constant independent of ". Furthermore, interpolating (6.3) and (6.4) we obtain p p p p k%" u" kLq .˝IR3 / D k %" %" u" kLq .˝IR3 /  k %" kL2 .˝/ k %" u" kL2 .˝IR3 / ; where qD

2 > 1 provided  > 1:  C1

Thus we may conclude that (recall that %u 2 Cweak .Œ0; TI Lq .˝I R3 //) supt2Œ0;T k%" u" .t; /kLq .˝IR3 /  c.E0 /; q D

2 :  C1

(6.6)

Next, applying a similar treatment to the convective term in the momentum equation, we have k%" u" ˝ u" kLq .˝IR33 / D k%" u" kL2=. C1/ .˝IR3 / ku" kL6 .˝IR3 / ; with q D

6 : 4 C 3

Using the standard embedding relation W 1;2 .˝/ ,! L6 .˝/;

(6.7)

we may infer that Z 0

T

k%" u" ˝ u" k2Lq .˝IR33/ dt  c.E0 ; T/; q D

6 ; 4 C 3

(6.8)

58

6 Weak Sequential Stability

noticing that 3 6 > 1 as long as  > : 4 C 3 2 Remark 19 Note that the previous considerations apply to any weak solution to the compressible Navier–Stokes system satisfying the energy inequality and yield the desired estimates in the whole range  > 32 . We will use this observation in Part III.

6.1.2 Pressure Estimates Uniform pressure bounds require an analogue of the integral identity (4.23) in the class of weak solutions. To this end, we consider the quantities 'D

rx 1 x ŒŒ%" ı  ;

2 Cc1 .0; T/;  2 Cc1 .˝/

as test functions in the momentum equation (5.6), where Œ%" ı D !ı .1˝ %" / is a regularization of %" by a spatial convolution with a family of regularizing kernels !ı (Fig. 6.1), !ı .y/ D

1 y ; ! 2 Cc1 .R3 /; !  0; !.y/ D !.jyj/; ! ı3 ı

Z R3

!.y/ dy D 1: (6.9)

As we observed in Remark 18, the equation of continuity (5.4) holds on the whole space R3 provided %" , u" were extended to be zero outside ˝. In particular,

Fig. 6.1 A family of regularizing kernels

6.1 Uniform Bounds

59

we deduce that @t Œ%" ı D divx .Œ%" u" ı / in .0; T/  R3 ; where taking convolution of (5.4) with !ı is nothing else than using !ı .x  y/ as a test function. In particular, as the momentum %" u" is weakly continuous in time, the function Œ%" ı is continuously differentiable in both t and x, therefore ' is a legitimate test function for (5.6). Were Œ%" ; u"  sufficiently smooth, we could have tested without mollifying in space. As similar arguments will be used repeatedly in Parts II and III, we find it convenient to explain them in detail here. After a straightforward manipulation we obtain Z 0

Z

Z

T

T

˝

Z h

0

˝

h i  2 p.%" /Œ%" ı  . C /Œ%" ı divx u" dx dt D i . C /divx u"  p.%" / rx   rx .1 x ŒŒ%" ı / dx dt

Z

Z

T

C 0

Z

˝

Z

T

 0

˝

Z

 .%" u" ˝ u" / W rx rx .1 x ŒŒ%" ı / dx dt Z

T



@t

0

Z



rx u" W rx rx .1 x ŒŒ%" ı / dx dt

˝

Z

T

C ˝

0

 %" u"  rx .1 x ŒŒ%" ı / dx dt

 %" u"  rx .1 x Œdivx Œ%" u" ı / dx dt;

where we can perform the limit ı ! 0 to conclude that Z

T

Z ˝

0

Z

h i  2 p.%" /%"  . C /%" divx u" dx dt D

Z h

T

˝

0

i . C /divx u"  p.%" / rx   rx .1 x Œ%" / dx dt

Z

Z

T

C 0

Z

˝

Z

T

 0

˝

Z 0

Z

T

 .%" u" ˝ u" / W rx rx .1 x Œ%" / dx dt Z

T



@t Z

C 0



rx u" W rx rx .1 x Œ%" / dx dt

˝

˝

 %" u"  rx .1 x Œ%" / dx dt

 %" u"  rx 1 x Œdivx .%" u" / dx dt:

(6.10)

60

6 Weak Sequential Stability

Remark 20 The last integrand should be interpreted as  %" u"  rx 1 x Œdivx .%" u" / 1 D  %" u"  rx 1 x divx Œ%" u"    %" u"  rx x Œ%" rx   u" :

From (6.10), we may deduce the local pressure estimates Z

T 0

Z p.%" /%" dx dt  c.E0 ; T; K/ for any compact K  ˝;

(6.11)

K

exactly as in Sect. 4.3.

6.2 Limit Passage Our ultimate goal is to perform the limit " ! 0. In view of the uniform bounds established in the previous section, we may assume that %" ! % weakly-(*) in L1 .0; TI L .˝//;

(6.12)

u" ! u weakly in L2 .0; TI W01;2 .˝I R3 //

(6.13)

passing to suitable subsequences as the case may be. Moreover, since %" satisfies the equation of continuity (5.5), the convergence in (6.12) may be strengthened to %" ! % in Cweak .Œ0; TI L .˝//:

(6.14)

Indeed it is enough to show that the family of scalar t-dependent functions Z t 7! ˝

%" .t; / dx

converges in C.Œ0; T/ to the limit Z t 7! ˝

%.t; / dx

for any  2 Cc1 .˝/. However, this follows easily from Eq. (5.5) and the uniform bounds (6.3) and (6.6). Specifically, taking '.t; x/ D .t/.x/, 2 Cc1 .0; T/, b D 0 in (5.5), we deduce that Z Z d %" .t; / dx D %" u"  rx  dx in .0; T/; dt ˝ ˝

6.2 Limit Passage

61

where the right-hand side is uniformly bounded in L1 .0; T/ for any fixed . The desired conclusion follows by a direct application of the classical Arzelà–Ascoli theorem.

6.2.1 Compactness in Convective Terms Our next goal is to establish convergence of the convective terms. To this end, we first observe that the compact embedding W01;2 .˝/ ,!,! Lq .˝/ for any 1  q < 6 gives rise, by duality, to compactness of the embedding (cf. Theorem 5) Lp .˝/ ,!,! W 1;2 .˝/ for p > Consequently, bounded sets in Lp .˝/, p > particular, relation (6.14) yields,

6 5

6 : 5

are precompact in W 1;2 .˝/, in

%" ! % in C.Œ0; TI W 1;2 .˝//; which, combined with (6.13) and the uniform bound (6.6), gives rise to 2

%" u" ! %u weakly-(*) in L1 .0; TI L  C1 .˝I R3 //:

(6.15)

Now, using the arguments leading to (6.14), the convergence (6.15) may be improved to 2

%" u" ! %u in Cweak .Œ0; TI L  C1 .˝I R3 //:

(6.16)

Finally, seeing that 2 6 3 2 > as long as  > I whence L  C1 .˝/ ,!,! W 1;2 .˝/;  C1 5 2

we may apply the same treatment to the cubic term %" u" ˝ u" to conclude %" u" ˝ u" ! %u ˝ u weakly in Lq ..0; T/  ˝I R33 / for a certain q > 1: Remark 21 Note that the preceding steps are valid for any  > 32 .

(6.17)

62

6 Weak Sequential Stability

6.2.2 Limit in the Equation of Continuity Unfortunately, we cannot pass to the limit directly in the renormalized equation of continuity (5.4) for a nonlinear function b. Instead, we start with b D 0 and let " ! 0 in (5.4). Using (6.14) and (6.16), we deduce, similar to Sect. 3.1.2, Z  Z  %.; /'.; /  %0 '.0; / dx D ˝

 0

Z  ˝

 %@t ' C %u  rx ' dx dt

(6.18)

for any  2 Œ0; T and any ' 2 Cc1 .Œ0; T  R3 /, where %0 is a weak limit of the sequence f%0;" g">0 in L .˝/. In order to derive the renormalized version of (6.18), we use the procedure proposed by DiPerna and Lions [26] based on spatial regularization by means of the convolution with the family of kernels introduced in (6.9). Following the arguments leading to (6.10) we obtain @t Œ%ı C divx .Œ%ı u/ D divx .Œ%ı u/  divx Œ%uı in .0; T/  R3 :

(6.19)

Since all quantities in (6.19) are regular, we can multiply both sides of the equation by b0 .Œ%ı /, where b is as in (5.3), obtaining  @t b .Œ%ı / C divx .b .Œ%ı / u/ C b0 .Œ%ı / Œ%ı  b .Œ%ı / divx u

(6.20)

D b0 .Œ%ı / .divx .Œ%ı u/  divx Œ%uı / : Now, if we show that the expression on the right-hand side of (6.20) vanishes for ı ! 0, the limit of the left-hand side will yield (5.4). To make this argument rigorous, we use the celebrated Friedrichs commutator lemma Lemma 2 Let N  2, 1  ˇ < 1, 1  p; q; ˛  1, Assume for I  R a bounded time interval ˇ

% 2 L˛ .II Lloc .RN //;

1 q

C

1 ˇ

 1,

1 p

C

1 ˛

1;q

u 2 Lp .II Wloc .RN I RN //:

Then divx .Œ%ı u/  divx Œ%uı ! 0 in Ls .II Lrloc .RN // as ı ! 0; where

1 s

D

1 ˛

C

1 p

and

1 r

D

1 q

C ˇ1 .

Proof We have divx .Œ%ı u/  divx Œ%uı

Z D Œ%ı divx u C

RN

%.t; y/.u.t; x/  u.t; y//  rx !ı .x  y/ dy D I1ı C I2ı :

 1.

6.2 Limit Passage

63

Clearly, I1ı ! %divx u strongly in Ls .II Lr .K// for any compact subset of RN . Hence we have to show that I2ı ! %divx u in the same space. First we show that kI2ı kLr .B.0;R//  Ck%kLs .B.0;RC1//krx ukLq .B.0;RC2/IRNN /

(6.21)

for a.a. t 2 .0; T/ independently of ı. To this aim, we apply several times a suitable change of variables and Hölder’s inequality to verify Z

ˇZ ˇ ˇ

kI2ı krLr .B.0;R//

ˇ u.t; x/  u.t; x  ız/ ˇ  rx !.z/ dzˇ dx ı B.0;R/ jzj1 Z Z ˇ u.t; x/  u.t; x  ız/ ˇr ˇ ˇ  C.!/ j%.t; x  ız/jr ˇ ˇ dz dx ı B.0;R/ jzj1 Z  Z ˇ u.t;  C ız/  u.t; / ˇq  qr ˇ ˇ r C j%.t; /j ˇ dz d ˇ ı B.0;RC1/ z1

D

%.t; x  ız/

 Ck%krLˇ .B.0;RC1//krx ukrLq .B.0;RC2/IRNN / : Next, let us show that I2ı ! %divx u strongly in Lr .B.0; R// for a.a. t 2 I. Indeed, due to (6.21) it is enough to show it for % smooth in the spatial variable. We have Z Z u.t; x/  u.t; x  ız/ ı  rx !.z/ dz dx: %.t; x  ız/ I2 D N ı R jzj1 1;q

As u 2 Wloc .RN I RN / for a.a. t 2 I, u.t; x/  u.t; x  ız/ Dz ı

Z

1 0

rx u.t; x  ız/ d ! z  rx u.t; x/

as ı ! 0C , for a.a. t 2 I and .x; z/ 2 RN  B.0; 1/. As % is smooth, %.t; x  ız/ ! %.t; x/ for a.a. t 2 I. Therefore by Vitali’s theorem Z RN

I2ı '

Z

Z

dx !

z ˝ rx ! dz W B.0;1/

for a.a. t 2 I and any ' 2 Cc1 .RN /.

RN

Z %rx u' dx D 

RN

%divx u' dx

64

6 Weak Sequential Stability

Finally, as kI2ı kLs .IILr .B.0;R///  C

Z I

k%ksLˇ .B.0;RC1//krx uksLq .B.0;RC2/IRNN / dt

 Ck%ksL˛ .IILˇ .B.0;RC1///krx uksLp .IILq .B.0;RC2/IRNN // and I2ı ! %divx u strongly in Lrloc .RN / for a.a. t 2 I, the Lebesgue dominated convergence theorem yields the claim of the lemma.  Since b0 is bounded, Lemma 2 implies that the expression on the right-hand side of (6.20) tends to zero in L1 ..0; T/  ˝/ as soon as % 2 L2 ..0; T/  ˝/. We have therefore shown that the limit Œ%; u is a solution of the renormalized equation of continuity (5.4). Remark 22 It is important to note that the above argument is not applicable in the full range of  > 32 . As shown by Lions [68] the critical value for  in the threedimensional case is  D 95 . A more delicate procedure yielding validity of the renormalized equation for the limit Œ%; u will be presented in Part III.

6.2.3 Limit in the Momentum Equation Given the weak compactness of the family fŒ%" ; u" g">0 , it is easy to perform the weak limit in the momentum equation to obtain Z ˝

Z D

2

1

Z 

%u.t; /  '.t; / dx

tD2 (6.22) tD1

Z   %u  @t ' C %u ˝ u W rx ' C p.%/divx ' dx dt ˝

2 1

Z   rx u W rx ' C . C /divx u divx ' dx dt ˝

for any 0  1  2 and for any test function ' 2 Cc1 .Œ0; T  ˝I R3 /, with %u.0; / D .%u/0  a weak limit of f.%u/0;" g">0 : Thus it remains to show the crucial relation p.%/ D p.%/

6.3 Strong Convergence of the Densities

65

or, equivalently, %" ! % a.a. in .0; T/  ˝:

(6.23)

This will be carried over in a series of steps specified in the remaining part of this chapter. Remark 23 Strictly speaking, the strong (pointwise) convergence claimed in (6.23) is equivalent to p.%/ D p.%/ only if p is not linear on any segment in its domain of definition.

6.3 Strong Convergence of the Densities Our ultimate goal to establish the weak sequential stability for the Navier–Stokes system is to show (6.23). To this end, we follow the idea of Lions [68] controlling possible oscillation in the sequence f%" g">0 by means of the renormalized equation of continuity (5.4).

6.3.1 Amplitude of Density Oscillations We take b.%/ D % log.%/ (see also Fig. 6.2) in the renormalized equation of continuity (5.4). Strictly speaking, such a choice of b is not allowed by (5.3); Fig. 6.2 The graph of the function b.%/ D % log.%/

66

6 Weak Sequential Stability

however, we check easily that (5.4) remains valid approximating % log.%/ by a suitable family bı .%/ % % log.%/ and using the Lebesgue dominated convergence theorem to perform the limit ı ! 0. Consequently Z

T 0

Z  %" log .%" / @t

 %" divx u"

˝

for any 2 Cc1 Œ0; T/, Chap. 2, we get

dx dt D  ˝

%0;" log.%0;" / dx

(6.24)

.0/ D 1. Furthermore, repeating the procedure from

Z

Z ˝

Z



%" log.%" /.; / dx C



Z

Z ˝

0

%" divx u" dx dt D

˝

%0;" log.%0;" / dx

(6.25)

for any 0    T. Consequently, we may let " ! 0 to obtain Z

Z ˝

% log %.; / dx C

 0

Z

Z ˝

%divx u dx dt D

˝

%0 log.%0 / dx:

(6.26)

On the other hand, we have shown in Sect. 6.2.2 that the limit Œ%; u also satisfies the renormalized equation of continuity, in particular, Z

Z ˝

% log.%/.; / dx C Z D ˝



0

Z ˝

%divx u dx dt

%0 log.%0 / dx;

which, combined with (6.26), gives rise to Z  ˝

Z  % log.%/  % log.%/ .; / dx C

 0

Z   %divx u  %divx u dx dt

(6.27)

˝

Z   %0 log.%0 /  %0 log.%0 / dx: D ˝

The function % 7! % log.%/ being strictly convex, the quantity Z  ˝

 % log.%/  % log.%/ .; / dx

can be seen as a measure of possible oscillations of the sequence f%" g" at the time . Indeed we recall the following rather standard result. Lemma 3 Suppose that %" ! % weakly in L2 .Q/;

6.3 Strong Convergence of the Densities

67

and % log.%/ D % log.%/: Then %" ! % in L1 .Q/: Proof Suppose that 0 < ı  %: Consequently, because of convexity of z 7! z log.z/, we have a.a. in Q %" log.%" /  % log.%/ D .log.%/ C 1/ .%"  %/ C ˛.ı/j%"  %j2 ; ˛.ı/ > 0; therefore Z fı%g

j%"  %j2 dx dt

Z 1 .log.%/  1/.%"  %/ dx dt D ˛.ı/ fı%g Z 1 .%" log.%" /  % log.%// dx dt: C ˛.ı/ fı%g Thus we conclude that (at least up to a subsequence) %" ! % a.a. in the set f%  ıg for any ı > 0: Now, by seeing that %" ! % a.a. in the set f% D 0g and, by the Lebesgue dominated convergence theorem, jf0 < % < ıgj ! 0 as ı ! 0; we obtain the desired conclusion.  Now assume, for a moment, that we can show Z Z Z Z %divx u dx dt  %divx u dx dt for any  > 0: 0

˝

0

˝

(6.28)

68

6 Weak Sequential Stability

Then relation (6.27) would imply % log.%/ D % log.%/

(6.29)

on condition that %0 log.%0 / D %0 log.%0 / or, equivalently %0;" ! %0 (strongly) in L1 .˝/:

(6.30)

Recall that we may assume without loss of generality that %0;" ! %0 weakly in L .˝/; however, (6.30) requires strong (a.a. pointwise) convergence of the initial densities. Under hypothesis (6.30), the proof of strong convergence of the sequence f%" g">0 therefore reduces to showing (6.28). This will be done in the next section. Remark 24 Hypothesis (6.30) requires strong compactness of the initial densities. Obviously, the strong convergence of f%" g">0 would fail the initial data should develop oscillations. On the other hand, however, the amplitude of oscillations decays uniformly with growing time provided the pressure % 7! p.%/ is a strictly convex function, see [34].

6.3.2 Effective Viscous Flux The desired inequality (6.28) can be seen as a direct consequence of the effective viscous flux equation (6.1). Indeed, for b.%/ D % relation (6.1) reads   %divx u  %divx u  lim inf p.%" /%"  p.%/% "!0

D lim inf ..p.%" /  p.%//.%"  %//  0 "!0

as the pressure is monotone (nondecreasing). This may look like a slightly dubious argument since the quantities p.%" /%" are bounded only in L1loc ..0; T/˝/; however, it is exactly what we will make rigorous below. Our starting point is a slightly modified version of identity (6.10) Z

Z

T

Z

h i  2 p.%" /%"  . C 2 /%" divx u" dx dt D

0

˝

T

Z h ˝

0

Z

T

C 0

i . C 2 /divx u"  p.%" / rx   rx .1 x Œ%" / dx dt Z ˝



curlx u"  curlx rx .1 x Œ%" / dx dt

(6.31)

6.3 Strong Convergence of the Densities

Z

Z

T

 0

˝

Z

Z

@t

0

Z

 .%" u" ˝ u" / W rx rx .1 x Œ%" / dx dt

T



˝

Z

T

69

C 0

˝

 %" u"  rx .1 x Œ%" / dx dt

 %" u"  rx .1 x divx Œ%" u" / dx dt;

where we have used the identity Z

Z ˝

rx v W rx w dx D

Z ˝

curlx v  curlx w dx C

˝

divx v divx w dx:

(6.32)

Note that (6.32) can be verified easily for any couple of functions v; w 2 Cc1 .˝I R3 / and then extended for arbitrary v; w 2 W01;2 .˝I R3 / by density. Next, we may apply the same arguments used in the derivation of (6.10) to the limit momentum equation (6.22) to obtain Z

T

Z

0

Z

˝

Z h

T

˝

0

h i  2 p.%/%  . C 2 /%divx u dx dt D

Z

i . C 2 /divx u  p.%/ rx   rx .1 x Œ%/ dx dt Z

T

C ˝

0

Z

(6.33)



curlx u  curlx rx .1 x Œ%/ dx dt

Z

T

 0

˝

Z

Z

T

 0

Z

 .%u ˝ u/ W rx rx .1 x Œ%/ dx dt

T

@t

˝

Z

C 0

˝

 %u  rx .1 x Œ%/ dx dt

 %u  rx .1 x divx Œ%u/ dx dt:

Obviously, in order to establish the desired relation (6.1), it is enough to show that all integrals on the right-hand side of (6.31) tend to their counterpart in (6.33). To this end, we first observe that, in view of the compact embedding W 1; .˝/ ,!,! C.˝/;  > 3; and the density convergence (6.14), the standard elliptic estimates yield 1 3 rx 1 x Œ%"  ! rx x Œ% in C.Œ0; T  ˝I R /:

70

6 Weak Sequential Stability

Using the weak convergence of fŒ%" ; u" g">0 established in the previous part, we can therefore “eliminate” all terms in (6.31) and (6.33) containing rx 1 x Œ%" , rx 1 Œ%, respectively. Accordingly, we obtain x Z

T

Z

h i  2 p.%" /%"  . C 2 /%" divx u" dx dt

lim

"!0 0

˝

Z

Z

T



˝

0

Z

T

i h  2 p.%/%  . C 2 /%divx u dx dt Z

D lim Z

"!0 T

 0

˝

0

Z

˝

Z

0

Z

T

Z

 0

 %" u"  rx .1 x divx Œ%" u" / dx dt

.%" u" ˝ u" / W .rx ˝ rx /.1 x Œ%" / dx dt T



˝

(6.34)

Z ˝



 %u  rx .1 x divx Œ%u/ dx dt

 .%u ˝ u/ W .rx ˝ rx /.1 Œ%/ dx dt : x

It remains to show that the right-hand side of (6.34) equals zero. To see this, we use the technique commonly known as compensated compactness.

6.3.3 Compactness via Div–Curl Lemma Div–Curl lemma, developed by Murat and Tartar [74, 89], represents an efficient tool for handling compactness in nonlinear problems, where the classical Sobolev embeddings of Rellich–Kondraschev type are not applicable.

Div–Curl Lemma Lemma 4 Let B  RM be an open set. Suppose that vn ! v weakly in Lp .BI RM /; wn ! w weakly in Lq .BI RM / (continued)

6.3 Strong Convergence of the Densities

71

as n ! 1, where 1 1 1 C D < 1: p q r Let, moreover, 1;s fDIVŒvg1 .B/; nD1 be precompact in W 1;s .BI RMM / fCURLŒwg1 nD1 be precompact in W

for a certain s > 1. Then vn  wn ! v  w weakly in Lr .B/:

Remark 25 Here, the operators DIV and CURL are considered on a general M-dimensional space, specifically [cf. (1.1) and (1.2)] DIVŒu D

M X

 @xj uj ; fCURLŒwgi;j D @xi wj  @xj wi :

jD1

Proof We give the proof only for a very special case that will be needed in the future, namely, we assume that Z div vn D 0; wn D rx ˚n ;

RM

˚n dy D 0:

(6.35)

The interested reader may consult, e.g., Tartar [89] or [35, Chap. 10, Theorem 10,21] for the proof in the general setting. Given the local character of the weak convergence, it is enough to show the result for B D RM . By the same token, we may assume that all functions are compactly 1;s supported. We recall that a (scalar) sequence fgn g1 .RM / if nD1 is precompact in W s M M gn D div Œhn ; with fhn g1 nD1 precompact in L .R I R /:

Now, it follows from the standard compactness arguments (compact embeddings for Sobolev spaces) that ˚n ! ˚ (strongly) in Lq .RM /; rx ˚ D w:

72

6 Weak Sequential Stability

Taking ' 2 Cc1 .RM / we have Z

Z vn  wn ' dy D

RM

Z D

RM

vn  rx ˚n ' dy

vn  rx '˚n dy ! 

RM

Z

Z RM

v  rx '˚ dy

v  w' dy;

D RM

which completes the proof under the simplifying hypothesis (6.35).  Our ultimate goal is to apply Lemma 4 to show Z

Z

T

R3

0

Z

Z

T



R3

0

u"  rx .1 x divx Œ%" u" /%" dx dt

.%" u" ˝ u" / W .rx ˝ rx /.1 x Œ%" / dx dt !

Z 0

Z

T



Z

T

R3

Z R3

0

 %u  rx .1 x divx Œ%u/ dx dt

.%u ˝ u/ W .rx ˝ rx /.1 x Œ%/ dx dt;

where all functions have been extended to be zero outside ˝. Consider a bilinear form (recall Sect. 1.5.1) Œv; w D

3  X

 v i Ri;j ŒW j   wi Ri;j Œv j  ; Ri;j D @xi 1 x @xj :

i;jD1

We have 3   X v i Ri;j ŒW j   wi Ri;j Œv j  i;jD1

D

3 X

  .v i  Ri;j Œv j /Ri;j ŒW j   .wi  Ri;j ŒW j /Ri;j Œv j 

i;jD1

D U  V  W  Z;

(6.36)

6.3 Strong Convergence of the Densities

73

where Ui D

3 3 X X .v i  Ri;j Œv j /; W i D .wi  Ri;j ŒW j /; divx U D divx W D 0; jD1

jD1

and 0 V i D @xi @

3 X

1

0 1 3 X 1 @xj W j A ; Z i D @xi @ 1 @xj v j A ; i D 1; 2; 3:

jD1

jD1

Thus a direct application of Div–Curl lemma (Lemma 4) yields Œv" ; w"  ! Œv; w weakly in Ls .R3 / whenever v" ! v weakly in Lp .R3 I R3 /, w" ! w weakly in Lq .R3 I R3 /, and 1 1 1 C D < 1: p q s Seeing that 2

%" ! % in Cweak .Œ0; TI L .R3 //; %" u" ! %u in Cweak .Œ0; TI L  C1 .˝I R3 //; we conclude that %" .t; /rx 1 Œdivx .%" u" /.t; /.%" u" /.t; /.rx ˝rx /1 Œ%" .t; / ! 1

%.t; /rx  Œdivx .%u/.t; /  .%u/.t; /  .rx ˝ rx /1 Œ%.t; / weakly in Ls .˝I R3 / for all t 2 Œ0; T; where sD

2 > 1 since  > 3:  C3

In view of the compact embedding W01;1 .˝/ ,!,! Lq .˝/ for any q


3 : 2

(6.38)

Consequently, the convergence in (6.36) would follow if 0

1;q0

u" ! u weakly in Lr .0; TI W0 .˝I R3 //; where r0 ; q0 are the dual exponents to r; q in (6.38). Unfortunately, this is not the case as q0 could be large and fu" g">0 converges weakly only in the space L2 .0; TI W01;2 .˝I R3 //. To fix this difficulty, we use the fact that the quantity %" rx 1 Œdivx .%" u" /  .%" u" /  .rx ˝ rx /1 Œ%"  enjoys better integrability than (6.38). Indeed it follows from the estimates (6.3), (6.4), Hölder’s inequality, and the standard elliptic regularity estimates that Z

T 0

  %" rx 1 Œdivx .%" u" /  .%" u" /  .rx ˝ rx /1 Œ%" 2

6

L  C12 .˝/

dt (6.39)

 c; 6 > 65 . where, as  > 3, we have  C12 Next, extending u" to be zero outside ˝, we write

u" D u"  Œu" ı C Œu" ı ; where Œı is the spatial regularization via convolution with the family of regularizing kernels (6.9). Clearly, Œu" ı ! Œuı weakly in L2 .0; TI W 1;q .R3 I R3 //I whence, in accordance with (6.37) and (6.38), Z 0

Z

T

 0

Z

T

R3

Z R3

%" Œu" ı  rx .1 x divx Œ%" u" / dx dt

.%" Œu" ı ˝ u" / W .rx ˝ rx /.1 x Œ%" / dx dt !

6.3 Strong Convergence of the Densities

Z 0

Z

T

Z

T

R3

Z

 0

R3

75

 %Œuı  rx .1 x divx Œ%u/ dx dt

.%Œuı ˝ u/ W .rx ˝ rx /.1 x Œ%/ dx dt;

as " ! 0 for any ı > 0. In order to complete the proof of (6.36) we have to show that Œu" ı  u" is “small” in L2 .0; TI Lq .˝// for any fixed 1  q < 6;

(6.40)

uniformly with respect to " cf. (6.39). To this end, we use the following assertion. Lemma 5 Let v 2 L2 .R3 / be such that Z R3

jv.x C y/  v.x/j2 dx  h2 M 2 for all jyj < h:

(6.41)

Then kŒvı  vkL2 .R3 /  ıM provided ı < h. Proof We have Z Œvı .x/  v.x/ D Z D

R3

R3

.!ı .x  y/v.y/  !ı .y/v.x// dy

!ı .y/ .v.x C y/  v.x// dy:

By Jensen’s inequality, jŒvı .x/  v.x/j2 Z Z  !ı .y/jv.x C y/  v.x/j2 dy D !ı .y/jv.x C y/  v.x/j2 dy: R3

fjyj0 be a sequence of weak solutions to the problem (2.7), (2.8), (2.13), emanating from the initial data fŒ%0;" ; .%u/0;" g">0 satisfying (6.2) and such that %0;" ! %0 in L1 .˝/: Then, at least for a suitable subsequence, %" ! % weakly-(*) in L1 .0; TI L .˝// and in L1 ..0; T/  ˝/; u" ! u weakly in L2 .0; TI W01;2 .˝I R3 //; where Œ%; u is a weak solution of the same problem with the initial data Œ%0 , .%u/0 , the latter being a weak limit of the sequence f.%u/0;" g">0 .

6.3 Strong Convergence of the Densities

77

It is worth noting that Theorem 12 does not imply the existence of weak solutions for the Navier–Stokes system. The only conclusion is that the set of weak solutions is (weakly) compact with respect to the natural energy bounds. Nevertheless, as we shall see in Part II, the arguments used in the proof of Theorem 12 may be easily adapted to show convergence of a suitable approximation scheme yielding a constructive existence proof. Remark 26 The hypothesis that ˝ is a Lipschitz domain has been used only for convenience to avoid possible technical difficulties with the embedding relations for Sobolev functions. The same conclusion can be obtained for a general domain ˝  R3 , the reader may consult [37] for details.

Part II

Existence of Weak Solutions via a Numerical Method

Chapter 7

Numerical Method

We show that the weak solutions to the Navier–Stokes system exist, globally in time, for any finite energy initial data. The proof will be constructive in the sense that the desired weak solution is obtained as a suitable limit of a numerical scheme. By a numerical scheme we mean a finite number of algebraic equations yielding an approximate solution of the problem. To this end, we use the method of time discretization in combination with a mixed finite-volume finite-element scheme to solve that resulting “stationary” problems. The scheme is implicit, the numerical approximation at any time level is obtained as a solution of a finite system of nonlinear algebraic equations resulting from the spatial discretization. The proposed numerical method produces approximate or numerical solutions of the Navier–Stokes system (2.7) and (2.8), with the boundary conditions (2.13), and the initial data (2.14) and (2.15). In order to avoid technicalities, we restrict ourselves to the class of initial data %0 2 L .˝/; %0 > 0 a.a. in ˝; .%u/0 D %0 u0 ; u0 2 L1 .˝I R3 /:

(7.1)

Moreover, similar to Part I, we consider the pressure p D p.%/ satisfying hypotheses (4.11) and (4.17), with  > 3: As we shall see in Part III, the validity of the existence theory can be extended to the so far optimal value  > 32 , however, via a “nonconstructive” proof based on a two-level approximation scheme.

© Springer International Publishing Switzerland 2016 E. Feireisl et al., Mathematical Theory of Compressible Viscous Fluids, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-319-44835-0_7

81

82

7 Numerical Method

7.1 Time Discretization To make the presentation as simple as possible, we take an equi-distant time step t > 0 and approximate v.kt; x/  v k .x/; x 2 ˝; k D 0; 1; : : : : Furthermore, we approximate the time derivatives by the difference (Fig. 7.1) Dt v k D

v k  v k1 ; k D 1; 2; : : : : t

Finally, we set v 0 .x/ D v0 .x/  a (prescribed) initial value of v; and extend v k D v 0 for all k D 1; 2; : : : :

Fig. 7.1 Discrete valued function v D .v j /j extended on a piecewise constant function with the corresponding time differences Dt v D .Dt v j /j (the dotted line represents the according slopes)

7.2 Spatial Discretization

83

Accordingly, we take %0 D %0 ; u0 D u0 ;

(7.2)

and replace the Navier–Stokes system by a family of (stationary) partial differential equations: Dt %k C divx .%k uk / D 0; Dt .%k uk / C divx .%k uk ˝ uk / C rx p.%k / D x uk C . C /rx divx uk ;

(7.3) (7.4)

with the boundary condition uk j@˝ D 0;

(7.5)

for k D 1; 2; : : : . Clearly, in order to obtain an approximate solution on the time interval Œ0; T it is enough to solve ŒT=t differential equations. Note that solutions at the level k depend on those at the level k1 through the discretized time derivative Dt . The scheme is implicit as uk is obtained in terms of uk1 by solving a system of nonlinear partial differential equations.

7.2 Spatial Discretization The next step is to replace the system of differential equations (7.3) and (7.4) by a finite system of nonlinear algebraic equations at each level k D 1; 2; : : : To this end, we use a variant of the mixed finite-volume finite element method proposed in [58].

7.2.1 Spatial Domain, Triangulation, and Mesh The physical space ˝  R3 —the spatial domain occupied by the fluid—will be approximated by a family of polygonal domains f˝h gh>0 . To keep the presentation simple, we focus on polyhedral approximations, where ˝h can be expressed as a union of a finite number of compact tetrahedra Ej , ˝ h D [Ej 2Eh Ej ; intŒEi  \ intŒEj  D ; for i ¤ j: The family Eh of all tetrahedra covering ˝h is termed mesh. The parameter h > 0 related to the characteristic diameter of the mesh plays a role of the degree of spatial discretization analogous to the time step t. We recall that the following notation

84

7 Numerical Method

will be systematically used in this chapter:
0 independent of t; hI similarly >

A  B , A  cB; with c > 0 independent of t; h; and
0 such that Ki  ˝h ; Ke  R3 n ˝ h for all 0 < h < h0 :

(7.6)

Remark 27 The situation ˝ ¤ ˝h is termed unfitted mesh in numerical analysis, see, e.g., Babuška and Aziz [4, 5] (Fig. 7.2). Sometimes, it is also convenient to assume ˝  ˝h to facilitate the treatment of more complicated boundary conditions. Each tetrahedron E 2 Eh is determined by its vertices, E D cofx1 ; x2 ; x3 ; x4 g; Fig. 7.2 An example of an unfitted mesh for ˝

7.2 Spatial Discretization

85

Fig. 7.3 An example of a tetrahedral element E D cofx1 ; x2 ; x3 ; x4 g and its outer normal vector nE , for the face D cofx1 ; x2 ; x4 g the normal n might have opposite direction

the boundary @E consists of four faces—intersection of @E with the four hyperplanes containing three distinct vertices of E. Faces in the mesh will be denoted , h is the family of all faces in the mesh. We distinguish internal and external faces, h D int [ ext ; ext D f 2 h j  @˝h g; int D h n ext : To each face in the mesh, we associate its normal vector n . The normals associated with the faces in ext are always taken to be external normals to @˝h . Remark 28 If  @E, we denote nE the external normal vector to @E (Fig. 7.3). There are four distinct external normal vectors for each E 2 Eh . In this book, we shall always assume that the mesh is shape regular, more specifically: • For each face 2 int there exist exactly two tetrahedra EK ; EL 2 Eh such that D @EK \ @EL : • There exists h > 0 such that hE  h; E  h for all E 2 Eh ; where hE denotes the diameter of E, and E the radius of the largest ball contained in E (Fig. 7.4). Remark 29 We may define a reference element EQ as the convex hull of the three basis vectors e1 D Œ1; 0; 0, e2 D Œ0; 1; 0, e3 D Œ0; 0; 1 and the zero vector Œ0; 0; 0.

86

7 Numerical Method

Fig. 7.4 An element E with diameter E and the largest inscribed ball of radius hE

All elements E 2 Eh of a shape regular mesh can be written as E D hAE EQ C aE ; AE 2 R33 ; aE 2 R3 ; q where all eigenvalues of the symmetric matrix below away from zero uniformly for h ! 0.

AE ATE are bounded above and

7.2.2 Finite Elements, Finite Volumes, and Function Spaces Numerical methods based on finite elements use finite-dimensional analogues of the standard function spaces defined piecewise on each element E 2 Eh . Typically, these functions are smooth on each individual element, therefore they possess welldefined traces on each face . We denote v out .x/ D lim v.x C ın /; v in .x/ D lim v.x  ın / for x 2 ; 2 int I ı!0C

ı!0C

and an analogous definition of v in for 2 ext . We simply write v for v in if  @E, cf. Remark 28. Analogously, we define ŒŒv D v out  v in ; fvg D

v out C v in 1 ; and hvi D 2 j j

Z

v dSx :

The variational (weak) formulation of the problem will be replaced by its discrete counterpart consisting in resolution of a finite system of possibly nonlinear

7.2 Spatial Discretization

87

equations. In contrast with the finite element method, the finite volume methods use equations on each face resulting from by parts integration. For problems like the Navier–Stokes system containing equations of mixed type (hyperbolic–parabolic), it is convenient to use a combination of both methods. The density %, susceptible to develop sharp gradients, will be approximated at each time step k by a piece-wise constant function %kh belonging to the space ˇ o n ˇ Qh .˝h / D r 2 L1 .˝h / ˇ rjE D const for each E 2 Eh : As we have seen in the previous part, the velocity field is more regular, at least with respect to the spatial variable. On the other hand, we need the approximate solutions resulting from the numerical scheme to be stable, meaning they should satisfy some form of the energy inequality yielding the necessary uniform estimates. Accordingly, the velocity field u will be approximated by a piece-wise affine function ukh in the space Vh .˝h I R3 /, where ˇ n ˇ Vh .˝h / D v 2 L2 .˝h / ˇ vjE D affine function for each E 2 Eh ;

Z Z in out v dSx D v dSx for any 2 int :



We also introduce

 ˇ Z ˇ V0;h .˝h / D v 2 Vh .˝h / ˇ v in dSx D 0 for any 2 ext :

Remark 30 The spaces Vh are known as Crouzeix–Raviart finite elements. Note that Vh is not a subspace of the Sobolev space W 1;2 , where we expect to recover the velocity u. In terms of the usual terminology, such finite elements are called non-conforming (Fig. 7.5).

7.2.3 Discretization of Convective Terms, Upwind Our goal is to replace the integral Z CD

˝

ru  rx  dx

that appears in the weak formulation of the convective term: divx .ru/ by its discrete counterpart. We suppose that r is approximated by a piece-wise constant function rh 2 Qh .˝h /, and u by a piece-wise affine function uh 2 V0;h .˝h I R3 /. Now, suppose that a smooth test function  is also approximated by a piece-wise

88

7 Numerical Method

Fig. 7.5 Sketch of a part of a function from Vh .˝h / with a sketch of the meaning of the boundary constraints

constant one. Performing formally the corresponding limit passage in C we obtain: Ch D

XZ

2 h

frh g uh  n ŒŒ dSx D

XZ 2 h



frh g huh  ni ŒŒ dSx ;

where ŒŒ, or, more precisely ŒŒ =h can be viewed as a suitable approximation of rx . One is therefore tempted to take Ch as a substitute for the convective term. However, in order to make the scheme stable, we need some kind of “dissipation” added in the approximate problem. In the original system of differential equations, a suitable regularization is standardly obtained R by adding artificial viscosity in the form "x r, or, in the variational setting " ˝ rx r  rx  dx, where " > 0 is a small parameter. At the discrete level, we may consider for ˛ 2 .0; 1/ Ch D

X Z

2 h

frh g huh  n i ŒŒ dSx  h˛

Z

 ŒŒrh  ŒŒ dSx ;

or a more subtle variant  XZ  1 Ch D frh g huh  ni  ŒŒrh  jhuh  ni j ŒŒ dSx : 2 2 h

The expression frh g huh  ni 

1 ŒŒrh  jhuh  ni j D rhout Œhuh  ni  C rhin Œhuh  ni C 2

is called upwind in numerical analysis.

(7.7)

(7.8)

7.2 Spatial Discretization

89

The “dissipative part” 12 ŒŒrh  jhuh  ni j in the upwind (7.8) is active only if huh  ni ¤ 0, while its counterpart can be seen as a finite volume approximation of the artificial viscosity term h˛ r. In this book, we use a modified upwind formula that can be seen as a compromise between (7.7) and (7.8), namely  UpŒrh ; uh  D

rhout 

Crhin

Œhuh  ni C h˛  C Œhuh  ni  h˛  2

Œhuh  ni C h˛ C C Œhuh  ni  h˛ C 2

 (7.9)

 ;

which can be equivalently written as 1 UpŒrh ; uh  D frh g huh  ni  maxfh˛ I j huh  ni jg ŒŒrh  „ ƒ‚ … 2 „ ƒ‚ … convective part

(7.10)

dissipative part

D rhout Œhuh  ni  C rhin Œhuh  ni C  h˛

  ŒŒrh  huh  ni  ; 2 h˛

where we have chosen (Fig. 7.6) 8 0 for z < 1; ˆ ˆ < z C 1 if  1  z  0; .z/ D ˆ 1  z if 0 < z  1; ˆ : 0 for z > 1:

(7.11)

Thus the numerical approximation of the convective term takes the form Z ˝

Fig. 7.6 The graph of function 

ru  rx  dx 

X Z 2 int



UpŒrh ; uh  ŒŒ dSx :

(7.12)

90

7 Numerical Method

Remark 31 Observe that (7.12) is invariant with respect to the choice of the normal vector to the face .

7.3 Numerical Scheme Let ˘hQ W L1 .˝h / ! Qh .˝h / denote the projection onto the space of piece-wise constant functions defined as Z 1 Q ˘h ŒvjE D hviE D v dx for each E 2 Eh : jEj E Occasionally, we will also use a shorter notation ˘hQ Œv D hvi : Since, by virtue of Jensen’s inequality, p

k˘hQ ŒvkLp .˝h / D

X

 jEj

E2Eh

1 jEj

p

Z v dx E



XZ E2Eh

p

E

jvjp dx D kvkLp .˝h / ;

the projection ˘hQ is bounded on Lp .˝h / with the norm one for any 1  p < 1, and, obviously, also for p D 1. For v 2 Vh .˝h / and a differential operator D acting on the space-variables, we denote Dh vjE D Dv for each E 2 Eh : We set %0h D ˘hQ Œ%0  2 Qh .˝h /; u0h D ˘hQ Œu0  2 Qh .˝I R3 /;

(7.13)

and determine recursively %kh 2 Qh .˝h /, ukh 2 V0;h .˝h I R3 /, k D 1; 2; : : : as a solution of the following system of equations:

Continuity Method Z ˝h

Dt %kh  dx 

for any  2 Qh .˝h /;

X Z 2 int



UpŒ%kh ; ukh  ŒŒ dSx D 0

(7.14)

7.4 Discrete Renormalization

91

Momentum Method Z ˝h

X Z  k ˝ k ˛ ˝ ˛ Dt %h uh   dx  UpŒ%kh ukh ; ukh   ŒŒhi dSx 2 int

Z

Z C ˝h

rh ukh

(7.15)



W rh  dx C . C /

Z ˝h

divh ukh divh 

dx D ˝h

p.%kh /divh  dx

for any  2 V0;h .˝h I R3 /. Remark 32 As Qh .˝h /, V0;h .˝h I R3 / are finite-dimensional spaces, system (7.14) and (7.15) consists of a finite number of nonlinear algebraic equations. Moreover, the reader will have noticed a formal similarity of the numerical method to the weak formulation (3.1) and (3.10). This observation will be useful in the analysis of consistency and convergence of the numerical solutions. Our goal is to construct a weak solution %, u of the Navier–Stokes system as a suitable limit for h ! 0 of the approximate solutions %kh , ukh . This amounts to showing: k1 k k • Solvability. Given %k1 h , uh , system (7.14) and (7.15) admits a solution %h , uh for k D 1; 2; : : : . • Stability. The approximate solutions satisfy a discrete version of the renormalized equation of continuity (5.4) and the energy inequality (5.7). In particular, all the bounds established in Sect. 6.1 remain valid for the approximate solutions. • Consistency. The approximate solutions satisfy the weak formulation of the Navier–Stokes system supplemented with extra h-dependent terms that vanish in the limit h ! 0. • Convergence. The approximate solutions %kh , ukh converge for h ! 0 up to a suitable subsequence to a weak solution %, u of the original problem.

7.4 Discrete Renormalization In the following two sections, we derive discrete counterparts of the renormalized equation of continuity (5.4), or rather its integrated version, and the energy inequality (5.7). Both relations will follow from the numerical method (7.14) and (7.15) by means of elementary algebraic manipulations and the mean value theorem.

92

7 Numerical Method

Our first goal is to derive a discrete analogue of the integrated renormalized equation of continuity Z

d dt

b.%/ dx C ˝

Z   b0 .%/%  b.%/ divx u dx D 0: ˝

It follows from the mean value theorem that 1 2   b.%kh /  b.%hk1 / D b0 .%kh / %kh  %hk1  b00 .skh / %kh  %hk1 ; 2

with skh 2 minf%kh I %hk1 g; maxf%kh I %hk1 g ; whence Z ˝h

Dt %kh b0 .%kh /

Z dx D ˝h

Dt b.%kh /

1 dx C 2

Z b ˝h

00

.skh /

2  k %h  %k1 h dx: t

(7.16)

Next, we consider b0 .%kh / as a test function in the upwind operator in the continuity method (7.14). Decomposition (7.10) gives rise to X Z

2 int

UpŒ%kh ; ukh 



b

0

.%kh /



X Z ˚ 

%kh ukh  n b0 .%kh / dSx dSx D 2 int



Z ˇ˝ ˛ ˇ

0 k

1 X b .%h / dSx ;  maxfh˛ I ˇ ukh  n ˇg %kh 2 2 int

where, furthermore, X Z ˚  X X Z 0 k

k k %h uh  n b .%h / dSx D  2 int



E2Eh @E

X X Z

E

˚ k 0 k k %h b .%h /uh  n dSx

Z k

0 k k 1 X X %h b .%h /uh  n dSx D %kh b0 .%kh /ukh  n dSx  2 E2E E E2Eh @E E h @E Z X X Z

1 %kh b0 .%kh /ukh  n dSx : D b0 .%kh /%kh divh ukh dx  2 E2E ˝h E h

@E

To handle the most right integral, write Z ˝h

D

X X Z E2Eh E 2@E

E

b.%kh /divh ukh

dx D

XZ E2Eh

b.%kh /ukh  n dSx D 

E

b.%kh /divh ukh dx

Z k

k 1 X X b.%h / uh  n dSx : 2 E2E E h

E 2@E

7.4 Discrete Renormalization

93

Consequently, summing up the previous estimates we obtain X Z 2 int





UpŒ%kh ; ukh  b0 .%kh / dSx D

Z

 ˝h

b.%kh /  b0 .%kh /%kh divh ukh dx

(7.17)

Z ˇ˝ ˛ ˇ

0 k

1 X b .%h / dSx maxfh˛ I ˇ ukh  n ˇg %kh 2 2 int Z 

 k k

1 X X uh  n dSx : C b.%h /  b0 .%kh / %kh 2 E2E E 

h

E 2@E

Thus, putting together (7.16) with (7.17) we may infer that Z

Z ˝h

Dt b.%kh /



dx C ˝h

1 C 2

b0 .%kh /%kh  b.%kh / divh ukh dx

Z b

00

˝h

.skh /

2  k %h  %k1 h dx t

Z ˇ˝ ˛ ˇ

0 k

1 X C b .%h / dSx maxfh˛ I ˇ ukh  n ˇg %kh 2 2 int Z  k



 k 1 X X  b.%h /  b0 .%kh / %kh uh  n dSx D 0: 2 E2E E h

E 2@E

In addition, we have Z 

 k k

1 X X  b.%h /  b0 .%kh / %kh uh  n dSx 2 E2E E h E 2@E Z  k



 k 1 X b.%h /  b0 .%kh /in %kh juh  nj dSx D 2 2 h Z  k



 k 1 X b.%h /  b0 .%kh /out %kh juh  nj dSx ;  2 2 h

which, combined with (7.18), yields Z

Z ˝h

Dt b.%kh /



dx C

1 C 2

˝h

Z b ˝h

00

b0 .%kh /%kh  b.%kh / divh ukh dx

.skh /

2  k %h  %k1 h dx t

(7.18)

94

7 Numerical Method

˝ k ˛ ! Z uh  n k

0 k

h˛ X %h b .%h / dSx C  2 2 h˛ int XZ      C b .%kh /in  b .%kh /out  b0 .%kh /out .%kh /in  .%kh /out  2 h



ˇ˝ ˛ ˇ  ˇ ukh  n ˇ dSx D 0:

Thus we may use the mean value theorem to obtain a discrete counterpart to the (integrated) renormalized equation of continuity in the form Z

Z Dt

b.%kh /

˝h

C

h˛ 2

dx C ˝h

1 C 2 X Z 2 int





 b0 .%kh /%kh  b.%kh / divh ukh dx

(7.19)

 k 2 %h  %k1 h dx b t ˝h ˝ k ˛ ! k

uh  n 0 k

%h  b .%h / dSx h˛

Z

00

.skh /

Z

2 ˝ ˛ 1 X C b00 .zkh / %kh j ukh  n j dSx D 0; 2 2 h

where

k k1 skh 2 Qh .˝h /; skh 2 minf%kh I %k1 h g; maxf%h I %h g ;

(7.20)

and zkh piece-wise constant on any face ,

zkh 2 minf.%kh /out I .%kh /in g; maxf.%kh /out I .%kh /in g :

(7.21)

Remark 33 Relation (7.19) has been derived on condition that b is twice continuously differentiable on the range of %kh . As we shall see below, the approximate solutions can be found to satisfy %kh > 0; whence b 2 C2 .0; 1/ is admissible in (7.19).

7.5 Energy Inequality An important property of the numerical scheme (7.14) and (7.15) is that the approximate solutions %kh , ukh satisfy an energy inequality. To see this, we take  D ukh as a test function in the momentum method (7.15) and estimate the integrals in the resulting expression as follows.

7.5 Energy Inequality

95

7.5.1 Pressure Potential To begin, we use the renormalized continuity method (7.19) to obtain Z

Z ˝h

p.%kh /divh ukh dx D Dt

˝h

1 2

P.%kh / dx 

Z ˝h

P00 .skh /

 k 2 %h  %hk1 dx t

(7.22)

˝ k ˛ ! Z k

uh  n 0 k

h˛ X  %h  P .%h / dSx 2 2 h˛ int Z

2 ˝ ˛ 1 X  P00 .zkh / %kh j ukh  n j dSx D 0; 2 2 h

where skh and zkh satisfy (7.20) and (7.21), respectively, and the pressure potential P is given by (4.9).

7.5.2 Time Derivative of the Total Energy We have Z ˝h

Dt .%kh

˝ k˛ uh /  ukh dx D

Z D ˝h

Dt %kh

Z ˝h

˝ ˛ ˝ ˛ %kh ukh  %k1 uk1 ˝ k ˛  uh dx t

ˇ˝ k ˛ˇ2 ˇ u ˇ dx C h

Z

Z %

k1

˝h

(7.23)

˝ ˛ ˝ ˛ ˝ k ˛ ukh  uk1 h dx uh  t

Z ˇ˝ k ˛ˇ2 ˇ˝ ˛ˇ2 1 ˇ ˇ uh dx C D %hk1 Dt ˇ ukh ˇ dx 2 ˝h ˝h ˇ˝ ˛ ˝ ˛ ˇ2 Z t k1 ˇˇ ukh  uhk1 ˇˇ %h ˇ C ˇ dx: ˇ ˇ t ˝h 2 Dt %kh

7.5.3 Convective Term The convective term can be handled in a similar fashion as its analogue in the continuity method in Sect. 7.4. We have X Z 2 int



˝ ˛ ˝ ˛

UpŒ%kh ukh ; ukh   ukh dSx

(7.24)

96

7 Numerical Method

D

X Z ˚ 2 int



˝ ˛ ˝ ˛

%kh ukh  ukh ukh  n dSx

Z ˚ ˝ ˛  ˝ ˛

˝ ˛

1 X dSx :  max h˛ I j ukh  n j %kh ukh  ukh 2 2 int

˝ ˛ On the other hand, the choice  D 12 j ukh j2 as a test function in the continuity method (7.14) gives rise to 1 2

Z

Z ˝ ˛ ˝ ˛

1 X Dt %kh j ukh j2 dx D UpŒ%kh ; ukh  j ukh j2 dSx 2 2 ˝h

(7.25)

int

D

X Z ˚  ˚˝ ˛ ˝ ˛

%kh ukh ukh ukh  n dSx 2 int



2 int



Z ˚ ˝ ˛ 

˝ ˛

1 X  max h˛ I j ukh  n j %kh  j ukh j2 dSx 4 2 int X Z ˚  ˚˝ ˛ ˝ ˛

%kh ukh ukh ukh  n dSx D Z ˚ ˝ ˛  ˚˝ ˛ k

˝ k ˛

1 X %h  uh dSx :  max h˛ I j ukh  n j ukh 2 2 int

Next observe that Dt .akh bkh / D bkh Dt akh C ak1 Dt bkh ; and, consequently, a simple combination of (7.23), (7.24), and (7.25) yields Z ˝h

Dt .%kh

X Z ˝ k˛ ˝ ˛ ˝ ˛

k uh /  uh dx  UpŒ%kh ukh ; ukh   ukh dSx

2 int

ˇ˝ ˛ ˝ ˛ ˇ2 t k1 ˇˇ ukh  uhk1 ˇˇ %h ˇ ˇ dx ˇ ˇ t ˝h ˝h 2 X Z  ˚  ˚˝ ˛ ˚ ˝ ˛  ˝ ˛

C %kh ukh  %kh ukh  ukh ukh  n dSx

1 D 2

Z

 ˝ ˛ Dt %kh j ukh j2 dx C

2 int

Z



Z ˚ ˝ ˛  ˝ ˛

˝ ˛

1 X dSx C max h˛ I j ukh  n j %kh ukh  ukh 2 2 int Z ˚ ˝ ˛  ˚˝ ˛ k

˝ k ˛

1 X %h  uh dSx :  max h˛ I j ukh  n j ukh 2 2 int

(7.26)

7.5 Energy Inequality

97

Using a straightforward algebraic identity

˝ ˛

˝ ˛

˚˝ ˛ k

˝ k ˛

˚ k  ˝ k ˛

2 %kh ukh  ukh  ukh %h  uh D %h uh

we get Z ˚ ˝ ˛  ˝ ˛

˝ ˛

1 X max h˛ I j ukh  n j %kh ukh  ukh dSx 2 2 int Z ˚ ˝ ˛  ˚˝ ˛ k

˝ k ˛

1 X %h  uh dSx  max h˛ I j ukh  n j ukh 2 2 int ˝ k ˛ ! Z ˚ k  ˝ k ˛

2 uh  n h˛ X dSx %h uh D  2 2 h˛ int Z ˚ k  ˝ k ˛

2 ˇ˝ k ˛ ˇ 1 X ˇ u  n ˇ dSx : % uh C h 2 2 h int

Similarly, we have ˚ k  ˚˝ k ˛ ˚ k ˝ k ˛ 1 k

˝ k ˛

%h uh I %h uh  %h uh D  4 whence X Z  ˚  ˚˝ ˛ ˚ ˝ ˛  ˝ ˛

%kh ukh  %kh ukh  ukh ukh  n dSx 2 int



D

Z k

˝ k ˛

2 k 1 X uh  n dSx : %h uh 4 2 int

Thus, summing up the previous estimates we obtain Z ˝h

Dt .%kh 1 D 2

X Z ˝ k˛ ˝ ˛ ˝ ˛

k uh /  uh dx  UpŒ%kh ukh ; ukh   ukh dSx 2 int



ˇ˝ ˛ ˝ ˛ ˇ2 t k1 ˇˇ ukh  uhk1 ˇˇ %h ˇ ˇ dx ˇ ˇ t ˝h ˝h 2 ! ˝ ˛ Z ˚ k  ˝ k ˛

2 ukh  n h˛ X %h uh  dSx C 2 2 h˛ int Z ˚ k  ˝ k ˛

2 ˇ˝ k ˛ ˇ 1 X ˇ u  n ˇ dSx % uh C h 2 2 h

Z

 ˝ ˛ Dt %kh j ukh j2 dx C

int

Z

(7.27)

98

7 Numerical Method



Z k

˝ k ˛

2 k 1 X %h uh uh  n dSx ; 4 2 int

where, finally, Z ˚ k  ˝ k ˛

2 ˇ˝ k ˛ ˇ 1 X ˇ u  n ˇ dSx % uh h 2 2 h int Z k

˝ k ˛

2 k 1 X %h uh  uh  n dSx 4 2 int

Z  k in ˝ k ˛ C ˝ k ˛

2 ˝ ˛ 1 X .%h / Œ uh  n   .%kh /out Œ ukh  n  uh D dSx : 2 2 int

7.5.4 Conclusion Taking the sum of (7.22) and (7.27) we obtain a discrete analogue of the total energy balance, which is the main source of the stability estimates needed in the proof of convergence of the numerical method.

Discrete Energy Balance 

Z ˝h

Z



C ˝h

1 D 2

Z

Dt

 1 k ˝ k˛ 2 k % j u j C P.%h / dx 2 h h

(7.28)

jrh ukh j2 C . C /jdivh ukh j2 dx

ˇ˝ ˛ ˝ 2 ˛ ˇ2  k Z %h  %hk1 t k1 ˇˇ ukh  uhk1 ˇˇ dx  %h ˇ P ˇ dx ˇ ˇ t t ˝h ˝h 2 ! ˝ ˛ Z k

0 k

ukh  n h˛ X  %h P .%h /  dSx 2 2 h˛ int Z

2 ˝ ˛ 1 X  P00 .zkh / %kh j ukh  n j dSx 2 2 00

.skh /

h

(continued)

7.6 Well Posedness

99

˝ k ˛ ! Z ˚ k  ˝ k ˛

2 uh  n h˛ X dSx %  uh   2 2 h h˛ int Z ˝ ˛  k in ˝ k ˛ C ˝ k ˛

2 1 X  dSx ; .%h / Πuh  n   .%kh /out Πukh  n  uh 2 2 int

where skh and zkh satisfy (7.20) and (7.21), respectively. Remark 34 The expression on the right-hand side of (7.28) represents the so-called numerical dissipation; it is non-positive, at least for a nondecreasing pressure, meaning for a convex pressure potential P.

7.6 Well Posedness Our ultimate goal in this chapter is to show that the numerical method proposed in (7.14) and (7.15) admits a solution Œ%kh ; ukh  for any k D 0; 1; : : : . This will be done recursively, assuming the solutions exist up to k  1, we show solvability of the system (7.14) and (7.15) at the level k.

7.6.1 Positivity of the Numerical Densities As the constitutive equations contain nonlinearities defined only for positive (nonnegative) values of the density, it is important to observe that this property is preserved by the numerical scheme. As the latter is defined on the family of domains ˝h that vary with h ! 0, it is convenient to have the initial data extended to the whole space R3 . We will therefore suppose that %0 2 L .R3 /; %0 j˝h  % > 0 for all h > 0;

(7.29)

where % is a positive constant. This is obviously consistent with the hypothesis (7.1) as well as with the compactness property of the family f˝h gh>0 specified in (7.6). Consequently, the approximation of the initial density satisfies %0;h D ˘hQ Œ%0  D h%0 i > 0 in ˝h for all h > 0:

100

7 Numerical Method

Lemma 6 Suppose that %kh 2 Qh .˝h / satisfies (7.14), where %hk1 > 0 in ˝, and ukh 2 V0;h .˝h I R3 /. Then %kh > 0 in ˝h : Proof We first show that %kh  0. To see this observe that the renormalized continuity method (7.19) gives rise to Z

Z ˝h

b.%kh / dx 

Z ˝h

b.%hk1 / dx C t

˝h

  b.%kh /  b0 .%kh /%kh divh ukh dx

whenever b is a convex function. This can be seen by choosing a suitable approximation of b by a family of smooth functions with nonnegative second derivative. Taking bm .%/ D maxf%I 0g and using positivity of %hk1 we immediately obtain bm .%kh / D 0 yielding %kh  0 in ˝h : Now assume that there exists E 2 Eh such that %kh jE D 0. Take  D 1E as a test function in the continuity method (7.14) to obtain ˛ ˝ ˛ ˝ t 0 D %kh E D %k1  h E jEj

X E f@En@˝h g

Z E

Up.%kh ; ukh / dSx ;

(7.30)

where, in agreement with (7.10), Z E

Z D E

Up.%kh ; ukh / dSx

˝ ˛  1 k out  ˝ k ˛ uh  n  maxfh˛ ; j ukh  nj g dSx  0 .%h / 2

in contrast with (7.30).



Obviously, Lemma 6 implies that the approximate densities remain strictly positive for k D 1; 2; : : : as long as the initial data are taken strictly positive.

7.6 Well Posedness

101

7.6.2 Solvability of the Numerical Scheme Finding solutions to the numerical scheme (7.14) and (7.15) amounts to solving a finite system of nonlinear algebraic equations. The usual approach is to convert the problem to finding a fixed point of a suitable nonlinear mapping. We have Proposition 1 Let the pressure p D p.%/ satisfy (4.11), and let %hk1 2 Qh .˝h /; %hk1 > 0; uhk1 2 V0;h .˝h I R3 / be given. Then the numerical method (7.14) and (7.15) admits a solution %kh 2 Qh .˝/; %kh > 0; ukh 2 V0;h .˝h I R3 /:

Remark 35 Note that Proposition 1 claims existence and does not address uniqueness of the family of numerical solutions. Proof Our goal is to apply Schaeffer’s fixed point theorem (Theorem 11 from Sect. 1.7). We set Z D V0;h .˝h I R3 / and examine solvability of (7.14) for a fixedukh . Step 1 We claim that (7.14) admits a unique solution %kh and that the mapping ukh 2 V0;h .˝I R3 / 7! %kh 2 Qh .˝h / is continuous. Indeed, given ukh , the solution %kh is determined by a finite system of linear equations

˝h

Z

X Z

Z %kh  dx  t

2 int



UpŒ%kh ; ukh  ŒŒ dSx D

˝h

%hk1  dx

(7.31)

for all  D 1E , E 2 Eh , where the associated homogeneous problem Z ˝h

%kh 

dx  t

X Z 2 int



UpŒ%kh ; ukh  ŒŒ dSx D 0

admits a unique (zero) solution. This can be seen revoking the renormalized formula (7.19) for the convex functions b.r/ D jrj, exactly as in the proof of Lemma 6. Moreover, by virtue of Lemma 6, the unique solution %kh is positive, therefore admitting a uniform bound Z

Z ˝h

j%kh j dx D

Z ˝h

%kh dx D

˝h

%hk1 dx:

102

7 Numerical Method

In particular, the associated mapping ukh 7! %kh Œukh  is continuous on the finite dimensional space V0;h .˝h I R3 /. Step 2 Rewrite (7.15) as Z Z k rh uh W rh  dx C . C / ˝h

˝h

˝ ˛ divh ukh divh  dx D F Œukh I  ;

(7.32)

 2 V0;h .˝h I R3 /, where F is a linear form on the space V0;h .˝h I R3 / defined as ˝

F Œukh I 

˛

Z D ˝h

p.%kh /divh  

Z C ˝h

dx C

X Z

2 int

˝ ˛ UpŒ%kh ukh ; ukh   ŒŒhi dSx

˝ ˛  ˝ ˛ %hk1 uhk1    %kh ukh   dx: t

Here, the density %kh is determined by ukh as the unique solution of (7.14) discussed in Step 1. The bilinear form on the left-hand side of (7.32) may be viewed as a scalar product on the (finite-dimensional) space V0;h .˝h I R3 /; whence solving (7.32) is equivalent to finding a fixed point of the mapping T .ukh / 7! R ı F , where R is given by the Riesz representation formula associated with this scalar product. The existence of a fixed point ukh is guaranteed by Theorem 11 as soon as we check that all possible solutions to the problem Z

Z

˝h

rh ukh W rh  dx C . C /

Z D 

Z C ˝h

˝h

p.%kh /divh 

dx C 

˝h

X Z 2 int



divh ukh divh  dx

(7.33)

˝ ˛ UpŒ%kh ukh ; ukh   ŒŒhi dSx

˝ ˛  ˝ ˛ %hk1 uhk1    %kh ukh   dx for any ; and some  2 Œ0; 1; t

belong to a bounded set in Vh .˝h I R3 /. However, solutions of (7.33) coincide with those of the numerical scheme (7.14) and (7.15) with the viscosity coefficients replaced by =, =. Thus the desired bound follows easily from the energy inequality (7.28), where the pressure potential P is a convex function as p is nondecreasing. We also remark that (7.33) admits only the trivial (zero) solution if  D 0.  Remark 36 Unfortunately, we are not able to show that the numerical scheme admits a unique solution at any time step. Of course, the problem of uniqueness of numerical solutions is somehow related to the same question for the limit problem, where the issue is still largely open in particular in the context of weak solutions.

Chapter 8

Stability of the Numerical Method

Stability in numerical analysis means that the approximate solutions admit the same bounds as indicated by the a priori estimates for the original problem. The fact that the numerical solutions satisfy the energy inequality (7.28) plays a crucial role.

8.1 Function Spaces It is convenient to introduce a topology on the discrete function spaces Qh and Vh mimicking their continuous counterparts. We introduce the (semi)-norms kvk2H 1 .˝ / D h

Q

X Z ŒŒv2 dSx for v 2 Qh .˝h /; h 2 int

and kvk2H 1 .˝ / D V

h

Z ˝h

jrh vj2 dx D

XZ E2Eh

jrx vj2 for v 2 V0;h .˝h /: E

It is easy to check that k  kH 1 .˝h / is a norm on V0;h .˝h /, while k  k2H 1 .˝

is a

© Springer International Publishing Switzerland 2016 E. Feireisl et al., Mathematical Theory of Compressible Viscous Fluids, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-319-44835-0_8

103

V

Q

h/

semi-norm vanishing on constant functions in Qh .˝h /. We show that these norms share certain compactness properties with the Sobolev norm krx vkL2 . The key result is the following assertion.

104

8 Stability of the Numerical Method

Proposition 2 We have Z

X Z ŒŒv2 jv.x/  v.x  /j dx  hjj dSx h fx2˝h jdistŒx;@˝h >jjg 2
jjg

D jj

X

j j j ŒŒv j D jj

2 int

X Z 2 int



(8.2)

j ŒŒv j dSx :

Next, since we deal with a shape regular mesh, observe that each element E possesses a finite number Nh .E/ of neighboring elements En such that distŒEn ; E  h, where Nh .E/  N, with N independent of h and E. Consequently, max jv.x  /  v.x/j  N max j ŒŒv j

(8.3)

2 int

x2˝h

whenever jj < h. Interpolating (8.2), (8.3) we conclude that Z fx2˝h jdistŒx;@˝h >jjg

jv.x  /  v.x/j2 dx  jjN

XZ 2 h



ŒŒv2 dSx

yielding obviously (8.1).  Similar estimates hold also for functions in V0;h .˝h /. Indeed it is enough to realize the Z 2h sup j ŒŒv.x/ j  jrh vj dx for any 2 int and v 2 V0;h .˝h /; jE1 [ E2 j E1 [E2 x2 (8.4) where E1 and E2 are the neighboring elements sharing the face . Consequently, we get, exactly as in the proof of Proposition 2, Z


ı : Consequently, by Jensen’s inequality and (8.1), Z


ıg

jŒvı  vj2 dx  h2 kvk2H 1 .˝ / whenever 0 < ı  h Q

h

(8.6)

for any v 2 Qh .˝h /. Similarly, in accordance with (8.5), Z


ı :

(8.7)

106

8 Stability of the Numerical Method

Thus, revoking Jensen’s inequality and (8.1), we get Z


ıg

jrx Œvı j2 dx 

h kvk2H 1 .˝ / for any v 2 Qh .˝h / h Q ı

(8.8)

provided 0 < ı  h, and, replacing (8.1) by (8.5), Z ˝h


0 for all %  0; lim

Remark 40 In comparison with (4.17) we require the pressure derivative p0 .0/ to be strictly positive. This in turn implies that the pressure potential P is strictly convex, specifically, P00 .%/ D

p0 .%/ > P > 0 for all % > 0: %

This fact will facilitate considerably certain estimates. We remark that the hypothesis p0 .0/ > 0 can be relaxed, see Karper [58], in contrast with the requirement  > 3 that is absolutely necessary for the present numerical scheme to converge.

108

8 Stability of the Numerical Method

8.2.1 Total Mass Conservation We start by a simple observation that the numerical scheme conserves the total mass, namely Z

Z ˝h

%kh dx D

˝h

%0h dx D

Z ˝h

˘hQ Œ%0  dx for k D 0; 1; : : : ; h > 0:

(8.15)

Indeed (8.15) follows by taking  D 1 in the continuity method (7.14).

8.2.2 Energy Estimates In view of hypothesis (8.14), the energy balance (7.28) gives rise to the following estimates: Z ˝ ˛ sup %kh j ukh j2 dx  2Eh0 ; (8.16) ˝h

k1

sup k%kh kL .˝h /  c.Eh0 /;

(8.17)

k1

where Eh0 D



Z ˝h

 1 0 ˝ 0˛ 2 < %h j uh j C P.%0h / dx  1: 2

In addition, we record the “integral” estimates on the dissipative terms t

XZ k1

˝h

jrx ukh j2 dx D t

X

0 . We also have, in accordance with (10.15), 2

%u 2 Cweak .Œ0; TI L  C1 .R3 I R3 //; %u.0; / D %0 u0 :

10.3 Strong Convergence of the Numerical Densities

145

10.3 Strong Convergence of the Numerical Densities In view of (10.21) it remains to resolve the most delicate problem, namely the strong (pointwise a.a.) convergence of the numerical densities: %h ! % a.a. in .0; T/  ˝:

(10.22)

We follow closely the steps performed in the “continuous” case in Sect. 6.3. First observe that the limit functions %, u satisfy relation (6.33), specifically Z Z

T

Z

h i  2 p.%/%  . C 2 /%divx u dx dt D

0

˝

T

Z h

0

˝

Z

i . C 2 /divx u  p.%/ rx   rx .1 x Œ%/ dx dt Z

T



curlx u  curlx rx .1 x Œ%/ dx dt

C 0

˝

Z

Z

T



@t

0

Z

Z

T



(10.23)

˝

0

Z

˝

 .%u ˝ u/ W rx rx .1 x Œ%/ dx dt

Z

T

C 0

 %u  rx .1 x Œ%/ dx dt

˝

 %u  rx .1 x Œdivx .%u// dx dt

that can be verified for any 2 Cc1 .0; T/,  2 Cc1 .˝/ exactly as in Sect. 6.3.2. Clearly, a natural counterpart of (6.31) in the numerical context is (10.3) provided we are able to justify the “by parts integration” leading from Z

T

Z ˝h

0

 rh uh W rx rx 1 x Œ%h  dx dt

to Z

Z

T 0

˝h

Z

T

 curlh uh  curlx   rx 1 x Œ%h  dx dt

Z

C 0

˝h

 divh uh divx   rx 1 x Œ%h  dx dt:

146

Since

10 Convergence

Z

Z ˝h

rh uh W rx G dx D

Z ˝h

Z

rh uh W .rx 

rxT /G Z

curlh uh  curlx G dx C

D ˝h

˝h

Z

Z

rh uh W rxT G dx 

C ˝h

˝h

dx C ˝h

rh uh W rxT G dx

divh uh divx G dx

divh uh divx G dx;

it is enough to control the last two integrals. Lemma 9 Let v 2 V0;h .˝h I R3 /, G 2 Cc1 .˝I R3 /. Then ˇZ ˇ Z ˇ ˇ T ˇ rh v W rx G dx  divh v divx G dxˇˇ ˇ ˝h

˝h


3: Let the initial data %0 and .%u/0 be given such that (continued)

154

10 Convergence

Theorem 14 (continued) %0 2 L1 .˝/; %0 > 0 a.a. in ˝; .%u/0 D %0 u0 ; u0 2 L1 .˝I R3 /: Then the Navier–Stokes system (2.7) and (2.8), supplemented with the noslip boundary condition (2.13), admits a weak solution Œ%; u in .0; T/  ˝ in the sense specified in Definition 1.

Remark 53 There are two rather inconvenient hypotheses imposed on the pressure in Theorem 14, namely the nonphysical value of the adiabatic exponent  > 3 and p0 .0/ > 0. Note that the so-called isentropic pressure-density state equation takes the form p.%/ D a% for a certain  in the range 1 <  

5 : 3

(10.31)

We relax the restrictions on the pressure in the next section allowing for p satisfying (10.31) with  > 32 . Unfortunately, this concerns only the proof of existence of weak solutions, the convergence of the numerical scheme in this case remains an open problem.

Part III

Existence Theory for General Pressure

Chapter 11

Weak Solutions with Artificial Pressure

The last part of this book is devoted to the proof of existence of weak solutions to the compressible Navier–Stokes system for a general pressure law p D p.%/. More specifically, we assume that the pressure satisfies 3 p0 .%/ D p1 > 0;  > : %!1 % 1 2 (11.1)

p 2 C1 Œ0; 1/; p.0/ D 0; p0 .%/ > 0 for % > 0; lim

Note that (11.1) includes the so-called isentropic pressure density state equation % D a% ; a > 0;  >

3 ; 2

and, in particular, the state equation of a monoatomic gas for which  D 53 . Our starting point is the existence result stated in Theorem 14. We consider a modified pressure law in the form (see also Fig. 11.1)  p" .%/ D p.%/ C " % C % ; with " > 0; > 3

(11.2)

for which Theorem 14 from the end of Part II yields the existence of a family of solutions fŒ%" ; u" g">0 . The initial data Œ%0;" ; u0;"  must be chosen to satisfy the hypotheses of Theorem 14, and Z  ˝

1 " % j.%u/0;" j2 C P.%0;" / C "%0;" log.%0;" / C 2%0;"  1 0;"

 dx  E0 ;

© Springer International Publishing Switzerland 2016 E. Feireisl et al., Mathematical Theory of Compressible Viscous Fluids, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-319-44835-0_11

157

158

11 Weak Solutions with Artificial Pressure

Fig. 11.1 The graph of the modified pressure law p"

uniformly for " ! 0, %0;" ! %0 in L .˝/; .%u/0;" ! .%u/0 in L1 .˝I R3 /; where Œ%0 ; .%u/0  comply with (5.1), and Z P.%/ D %

% 1

(11.3)

p.z/ dz: z2

Then we send " ! 0 to obtain the desired solution Œ%; u as a suitable limit of the sequence fŒ%" ; u" g">0 . The analysis is technically more complicated than in Part I; however, some steps are quite similar to those in the proof of Theorem 12. The main stumbling block is the fact that the pressure estimates elaborated in Sect. 4.3 must be modified and give rise to weaker bounds for the density %. Notably, the density is no longer square integrable and the renormalization technique of DiPerna and Lions used in Sect. 6.2.2 cannot be applied. Instead, we introduce a concept of oscillations defect measure to show that the limit Œ%; u represents a renormalized solution of the equation of continuity.

11.1 Uniform Bounds Theorem 14 provides a family of approximate solutions fŒ%" ; u" g">0 in the set .0; T/  ˝ enjoying all properties stated in Chap. 5. In particular, they satisfy the renormalized equation of continuity (5.4) and the energy inequality Z  ˝

  1 1 %" ju" j2 C P.%" / C " %" log.%" / C % " .; / dx 2 1

(11.4)

11.3 Pressure Estimates

159

Z



C Z   ˝

Z   jrx u" j2 C . C /jdivx u" j2 dx dt ˝

0

  1 1 % 0;" dx dt j.%u/0;" j2 C P.%0;" / C " %0;" log.%0;" / C 2%0 1

for a.a.  2 .0; T/.

11.2 Energy Estimates Since the energy of the initial data is bounded uniformly for " ! 0, we can deduce uniform energy bounds exactly as in Sect. 6.1. More specifically, we have: sup k%" .t; /kL .˝/  c.E0 /;

(11.5)

t2.0;T/

Z

T 0

ku" .t; /k2W 1;2 .˝IR3 / dt  c.E0 /;

supt2Œ0;T k%" u" .t; /k

2

L  C1 .˝IR3 /

Z

T 0

k%" u" .t; /k2

6

L  C6 .˝IR3 /

 c.E0 /;

dt  c.E0 /;

(11.6) (11.7) (11.8)

and, finally, Z

T 0

with have

6 4 C3

k%" u" ˝ u" k2Lq .˝IR33 / dt  c.E0 ; T/; q D

6 ; 4 C 3

(11.9)

> 1 as soon as  > 32 , see (6.3)–(6.8) and Remark 19. In addition, we sup k%" .t; /k L 

t2.0;T/

1 c.E0 /: "

(11.10)

11.3 Pressure Estimates The pressure estimates are slightly more elaborate than their counterparts in Sect. 6.1.2. The main idea is to take the quantities 'D

rx 1 x ŒŒb.%" /ı;  ;

2 Cc1 .0; T/;  2 Cc1 .˝/

160

11 Weak Solutions with Artificial Pressure

as test functions in the momentum equation (5.6), where Œb.%" /ı; D  t Œ!ı x b.%" / is the regularization via convolutions in both space and time. Specifically, !ı D !ı .x/ is the family of regularizing kernel introduced in (6.9), while  D  .t/ satisfies Z 1    ./ D ! ; ! 2 Cc1 .R/; !  0; !./ D !./; !.t/ dt D 1:   R The function b is taken to comply with hypothesis (5.3), in particular, the composition b.%" / solves the renormalized equation of continuity (5.5). Accordingly, extending %" .t; / D %0;" ; u" .t; / D 0 for t  0, and %" .t; / D %" .T; /, u" .t; / D 0 for t > T (and, in accordance with our convention, to be zero outside ˝), we observe the (5.5) holds for any 1 < t < 1, x 2 R3 . Thus we have @t Œb.%" /ı; D divx .Œ%" b.%" /u" ı; / C

 i h b.%" /  b0 .%" /%" divx u"

ı;

in R  R3 :

Taking ' as a test function in the momentum equation (5.6) we obtain, exactly as in Sect. 6.1.2, Z 0

T

Z

2

˝

Z

h i   p.%" / C " %" C % " Œb.%" /ı;  . C /Œb.%" /ı; divx u" dx dt Z h

T

D

˝

0

Z

i . C /divx u"  p" .%" / rx   rx .1 x ŒŒb.%" /ı; / dx dt Z

T

C 0

Z

˝

Z

T

 0

˝

Z 0

Z

T

C 0

Z

T

C 0

Z

 .%" u" ˝ u" / W rx rx .1 x ŒŒb.%" /ı; / dx dt Z

T





rx u" W rx rx .1 x ŒŒb.%" /ı; / dx dt

@t Z ˝

˝

 %" u"  rx .1 x ŒŒb.%" /ı; / dx dt

 %" u"  rx .1 x Œdivx Œb.%" /u" ı; / dx dt

 i  h  1 0 dt;  %" u"  rx x  b .%" /%"  b.%" / divx u" ı;

11.3 Pressure Estimates

161

and, letting  ! 0, ı ! 0, Z

T 0

Z

2

˝

Z D

h

i   p.%" / C " %" C % " b.%" /  . C /b.%" /divx u" dx dt Z h

T

0

˝

Z

i . C /divx u"  p" .%" / rx   rx 1 x Œb.%" / dx dt Z

T

C 0

Z

˝

Z

T

 0

Z  0

Z

T

˝

0

T

C 0

Z



rx u" W rx rx 1 x Œb.%" / dx dt

 .%" u" ˝ u" / W rx rx 1 x Œb.%" / dx dt Z

T

@t Z

C Z

(11.11)

˝

˝

 %" u"  rx 1 x Œb.%" / dx dt

 %" u"  rx 1 x Œdivx .b.%" /u" / dx dt

h   i 0  %" u"  rx 1 .% /%  b.% / div u  b dt: " " " x " x

Remark 54 Although relation (11.11) was derived for the nonlinearities b satisfying the rather restrictive hypothesis (5.3), its validity can be extended to a large class of functions by means of a suitable approximation b  bı and the Lebesgue dominated convergence theorem. In accordance with what we have said in Remark 54, we take 8 ˆ < % for 0  %  1; 1 ˇ1 b D bı ; bı .0/ D 0; b0ı .%/ D % nfor 1 < % < ı ; o  ˆ : max 1  %  1 I 0 for %  1 ; ı ı ı ˇ1 with the exponent ˇ > 0 to be chosen below. Obviously, the function bı belongs to class (5.3), and relation (11.11) holds for bı .%" /, ı > 1. In order to deduce uniform pressure estimates, we have to choose ˇ > 0 so small that all the integrals on the right-hand side of (11.11) are controlled by the energy bounds established in (11.5)–(11.10) uniformly for ı > 0. For 0 < ˇ < 13 we have, by virtue of (11.5), bı .%" / bounded in L1 .0; TI L3 .˝// uniformly in both ı and ". Consequently, the same arguments as in Sect. 6.1.2 can be used to handle most of the integrals on the right-hand side of (11.11), more precisely,

162

11 Weak Solutions with Artificial Pressure

we deduce Z TZ 0

   2 p.%" / C ".%" C % " / bı .%" / dx dt  c. ; ; E0 ; T/

˝

ˇZ ˇ C ˇˇ

T

Z

0

(11.12)

h  i ˇˇ  0 ˇ  b  %" u"  rx 1 .% /%  b .% / div u ı " x " dtˇ : x ı " "

To control the rightmost integral in (11.12), we use the elliptic estimates, together with the Sobolev embedding W 1;p .˝/ ,! Lq .˝/; 1  q 

3p ; 1  p < 3; 3p

to obtain  h  i    0  b .% /%  b .% / div u rx 1  " " ı " x " x ı

Lq .˝IR3 /

(11.13)

 h  i    0  b .% /%  b .% / div u  c rx 1 ı " x "  x ı " "       c  b0ı .%" /%"  bı .%" / divx u" 

W 1;p .˝IR3 /

Lp .˝IR3 /

   c b0ı .%" /%"  bı .%" /Lr .˝/ kdivx u" kL2 .˝/ ; where 1 1 1 C D : r 2 p Consequently, in view of estimates (11.5), (11.6), and (11.8), we can adjust the exponents in (11.13) to satisfy 1D

 C6 1  C6 3p  C6 1 1 6 C D C D C C ; rD > 1: 6 q 6 3p 6 r 6 4  6

Thus, in agreement with (11.5), the choice 0 < ˇ    0 b .%" /%"  bı .%" / ı

Lr .˝/

4 6 6

renders the term

bounded in L1 .0; T/;

therefore the right-hand side of (11.13) remains bounded for ı ! 0. Letting ı ! 0 in (11.12) we infer that Z

T 0

Z   p.%" / C ".%" C % " / %ˇ" dx dt  c.K; E0 ; T/ for any compact K  ˝; K

(11.14)

11.4 Weak Limit of the Sequence of Approximate Solutions

163

whenever

1 4  6 I : 0 < ˇ  min 3 6 

11.4 Weak Limit of the Sequence of Approximate Solutions Since the uniform bounds we have at hand are essentially the same as in Sect. 6.1, the first part of the proof of convergence including the weak compactness of convective terms can be performed in the same way as in Sects. 6.2.1–6.2.3. We have

%" ! %

8 < in Cweak .Œ0; TI L .˝//; and :

weakly in L Cˇ ..0; T/  K/ for any compact K  ˝; ˇ > 0; u" ! u weakly in L2 .0; TI W01;2 .˝I R3 //;

passing to suitable subsequences if necessary. Moreover, exactly as in Sect. 6.2.2, 2

6

%" u" ! %u in Cweak .Œ0; TI L  C1 .˝I R3 // and weakly in L2 .0; TI L  C6 .˝I R3 //; where the limit satisfy the equation of continuity Z

2 1

Z

Z ˝

Œ%@t ' C %u  rx ' dx dt D

˝

%.t; /'.t; / dx

tD2

;

(11.15)

tD1

for any 0  1 <   T and any test function ' 2 Cc1 .Œ0; T  R3 /, where %.0; / D %0 : Finally, by virtue of the pressure estimates (11.14), ˇ

k"% " kL1 ..0;T/K/ D " Cˇ

Z 0

T

Z



K

" Cˇ % " dx dt ! 0 as " ! 0;

(11.16)

similarly k"%" kL1 ..0;T/K/ ! 0 as " ! 0: Consequently, letting " ! 0 in the weak formulation of the momentum equation (5.6), and recalling that p.%" / ! p.%/

weakly in L

 Cˇ 

..0; T/  ˝/;

164

we obtain Z

11 Weak Solutions with Artificial Pressure

2 1

Z   %u  @t ' C %u ˝ u W rx ' C p.%/divx ' dx dt

(11.17)

˝

Z 

2 1

Z  ˝

 rx u W rx ' C . C /divx udivx ' dx dt Z

D ˝

%u.t; /  '.t; / dx

tD2 tD1

for any 0  1 < 2  T and any test function ' 2 Cc1 .Œ0; T  ˝I R3 /. Similar to Chap. 6, the ultimate step of the proof is to show that p.%/ D p.%/; or, equivalently, %" ! % a.a. in .0; T/  ˝:

(11.18)

As a byproduct (however, necessary for the proof) we shall also prove that Œ%; u is a renormalized solution to the equation of continuity, cf. (5.4). This is the objective of the next chapter.

Chapter 12

Strong Convergence of the Approximate Densities

In this chapter, we establish weak convergence of the densities for the perturbed Navier–Stokes system endowed with the artificial pressure (11.2). Following the arguments used in Sect. 6.3 we need the renormalized formulation of the continuity equation (5.4). Unfortunately, the regularization technique of DiPerna and Lions [26] is no longer applicable as the limit density % is not (known to be) square integrable. Instead, we exploit another piece of information hidden in the effective viscous flux identity (6.1).

12.1 Effective Viscous Flux The derivation of the effective viscous flux identity (6.1) follows, with obvious modifications, the arguments delineated in Sect. 6.3.2. We start by rewriting (11.11) in the form Z

T 0

Z ˝

2 Z

h

i   p.%" / C " %" C % " b.%" /  . C 2 /b.%" /divx u" dx dt Z h

T

D

˝

0

Z Z

i . C 2 /divx u"  p" .%" / rx   rx 1 x Œb.%" / dx dt Z

T

C 0

Z

T

˝

 0

(12.1)

Z  0

˝



curlx u"  curl rx 1 x Œb.%" / dx dt

 .%" u" ˝ u" / W rx rx 1 x Œb.%" / dx dt

T

@t

Z ˝

 %" u"  rx 1 x Œb.%" / dx dt

© Springer International Publishing Switzerland 2016 E. Feireisl et al., Mathematical Theory of Compressible Viscous Fluids, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-319-44835-0_12

165

166

12 Strong Convergence of the Approximate Densities

Z

T

C Z

0

Z

T

C ˝

0

Z ˝

 %" u"  rx 1 x Œdivx .b.%" /u" / dx dt

h  i   b0 .%" /%"  b.%" / divx u" dx dt  %" u"  rx 1 x

for b satisfying (5.3). The next step is to use, first formally, the quantities '.t; x/ D

.t/.x/rx 1 x Œb.%/

(12.2)

as test functions in the momentum balance (11.17). As Œ%" ; u"  satisfy the renormalized equation of continuity (5.4), we have b.%" / ! b.%/ in Cweak .Œ0; TI Lq .˝// for any 1  q < 1; and, in view of the compact embedding W 1;2 .˝/ ,!,! Lp .˝/ for any 1  p < 6; b.%" /u" ! b.%/u weakly in L2 ..0; T/  ˝I R3 /:

(12.3)

Remark 55 The spaces in (12.3) are not optimal in view of the available bounds. Consequently, the limit for " ! 0 in (5.4) gives rise to Z

T 0

Z h ˝

Z i b.%/@t ' C b.%/u  rx ' dx dt D

T 0

Z  ˝

 b0 .%/%  b.%/divx u ' dx dt (12.4)

for any test function ' 2 Cc1 ..0; T/  R3 /. Now, we may use (11.16) and repeat the arguments leading to (12.1), using the quantities (12.2) as test functions in the limit momentum equation (11.17), to obtain Z

T

Z

0

Z

h i  2 p.%/b.%/  . C 2 /b.%/divx u dx dt D

˝

Z h

T

˝

0

Z

T

i . C 2 /divx u  p.%/ rx   rx 1 x Œb.%/ dx dt Z

C ˝

0

Z

T

Z

 0

˝

h i curlx u  curlx rx 1 x Œb.%/ dx dt   .%u ˝ u/ W rx rx 1 x Œb.%/ dx dt

(12.5)

12.2 Oscillations Defect Measure

Z

T

 0

Z

T

Z @t Z

C Z

T

Z

C 0

˝

0

˝

167

 %u 

˝

 %u  rx 1 x Œb.%/ dx dt

 %u  rx 1 x Œdivx .b.%/u/ dx dt rx 1 x

    0  b .%/%  b.%/ divx u dx dt:

Comparing (12.1) and (12.5) with (6.31) and (6.33) we observe easily that we are in the same situation as in Sect. 6.3.2, with %" replaced by b.%" /, and b.%/ in place of the weak limit %. Thus exactly same arguments as in Sect. 6.3.2, based on Div–Curl lemma, can be applied to obtain the effective viscous flux identity h i p.%/b.%/  p.%/ b.%/ D . C 2 / b.%/divx u  b.%/divx u in .0; T/  K

(12.6)

for any compact K  ˝ and any b as in (5.3), which is exactly (6.1). Remark 56 Relation (12.6) is stated locally in ˝ in accordance with the pressure estimates (11.14).

12.2 Oscillations Defect Measure We introduce a new tool to describe possible oscillations in the weakly converging sequence f%" g">0 . To this end, we use the cut-off functions (see also Fig. 12.1) Tk .r/ D kT

r k

;

(12.7)

where k  1, T 2 C1 Œ0; 1/, 8 < r for 0  r  1; T.r/ D T 00 .r/  0 for r 2 .1; 3/; : 2 for r  3: The oscillations defect measure associated with the sequence f%" g">0 is defined as   Z osc˛ Œ%" ! %.Q/ D sup lim sup jTk .%" /  Tk .%/j˛ dx dt ; k1

for any Q  .0; T/  R3 .

"!0

Q

(12.8)

168

12 Strong Convergence of the Approximate Densities

Fig. 12.1 The graph of the cut-off function Tk

Remark 57 Note that osc “measures” possible oscillations and cuts-off concentrations in the sequence f%" g">0 . In particular, osc.Q/ vanishes whenever %" ! % a.a. in Q.

12.2.1 Oscillations Defect Measure and Renormalization The following assertion is crucial for the existence proof in the absence of square integrability of the density. Lemma 10 Let Q  R  R3 be an open set. Let %" ! % weakly in L1 .Q/; u" ! u weakly in Lr .QI R3 /;

(12.9)

rx u" ! rx u weakly in Lr .QI R33 /; r > 1 be a sequence of weak solutions to the renormalized equation of continuity (4.8) in Q. Suppose that osc˛ Œ%" ! %.Q/ < 1

(12.10) (continued)

12.2 Oscillations Defect Measure

169

Lemma 10 (continued) for 1 1 C < 1: ˛ r

(12.11)

Then also the limit Œ%; u is a weak solution of the renormalized equation of continuity in Q.

Remark 58 For Œ%; u being a weak solution of (4.8) in Q means that the integral identity Z h   i b.%/@t ' C b.%/u  rx ' C b.%/  b0 .%/% divx u' dx dt D 0 Q

holds for any ' 2 Cc1 .Q/, and any b satisfying (5.3). Proof First of all, note that it is enough to show the result on J K with J a bounded time interval, K a ball such that J  K  Q. Recall that we consider functions b.r/ of class C1 .Œ0; 1// which are constant for large values of r. Due to the hypotheses of the lemma and the results and arguments already used in Chap. 3, we know that Tk .%" / ! Tk .%/

in Cweak .JI Lq .K//

Tk .%" /u" ! Tk .%/u

for any 1  q < 1;

weakly in Lr .J  KI R3 /;

where Tk are the cut-off functions introduced in (12.7). Similar to (12.4), we get Z Z h i Tk .%/@t ' C Tk .%/u  rx ' dx dt J

K

Z Z   Tk0 .%/%  Tk .%/ divx u' dx dt D J

K

for any test function ' 2 Cc1 .Q/. Obviously, the functions Tk .%/ are square integrable. Thus we may use the regularization procedure of DiPerna and Lions [26], exactly as in Sect. 6.2.2, to deduce Z Z h     i b Tk .%/ @t ' C b Tk .%/ u  rx ' dx dt (12.12) J

K

Z Z h    i  C b Tk .%/  b0 Tk .%/ Tk .%/ divx u' dx dt J

K

Z Z

D J

K

   b0 Tk .%/ Tk0 .%/%  Tk .%/ divx u' dx dt:

170

12 Strong Convergence of the Approximate Densities

Seeing that k%  Tk .%/kL1 .Q/  lim inf k%"  Tk .%" /kL1 .Q/ ! 0 for k ! 1 "!0

as f%" g">0 is equi-integrable in Q, we may let k ! 1 in (12.12) to obtain the desired expression on the left-hand side of the limit identity. Consequently, it is enough to show that Z Z    b0 Tk .%/ Tk0 .%/%  Tk .%/ divx u' dx dt ! 0 for k ! 1 (12.13) J

K

for any fixed ' and any b such that b0 .r/ D 0 for r  M: Denote Qk;M D f.t; x/ 2 J  KI jTk .%/j  Mg: We have

       Tk .%/ T 0 .%/%  Tk .%/ divx u k  

(12.14)

L1 .JK/

 c sup kdivx u" kLr .JK/ lim inf kTk .%" /  Tk0 .%" /%" kLr0 .Qk;M / ; "!0

">0

1 1 C 0 D 1: r r

Using hypothesis (12.11), we get kTk .%" /  Tk0 .%" /%" kLr0 .Qk;M /

(12.15)

 kTk .%" /  Tk0 .%" /%" kL1 .Qk;M / kTk .%" /  Tk0 .%" /%" kL1 ; ˛ .Q k;M / for some 0 <  < 1. As the family f%" g">0 is equi-integrable, due to a similar argument as above sup kTk .%" /  Tk0 .%" /%" kL1 .JK/ ! 0 ">0

for k ! 1:

(12.16)

Now, recalling that 0  Tk0 .%" /%"  Tk .%" /, we obtain kTk .%" /  Tk0 .%" /%" kL˛ .Qk;M /

   kTk .%" /  Tk .%/kL˛ .Qk;M / C kTk .%/  Tk .%/kL˛ .JK/ C kTk .%/kL˛ .Qk;M /   1  kTk .%" /  Tk .%/kL˛ .Qk;M / C osc˛ Œ%" ! %.Q/ C MjJ  Kj ˛ :

12.2 Oscillations Defect Measure

171

Therefore 1

lim sup kTk .%" /  Tk0 .%" /%" kL˛ .Qk;M /  2osc˛ Œ%" ! %.Q/ C MjJ  Kj q < 1: "!0

(12.17)

Combining (12.14)–(12.17) we obtain (12.13). The lemma is proved.  In the present situation, we intend to use Lemma 10 with r D 2 which requires ˛ > 2. Thus Lemma 10 offers an alternative to the regularization provided we can show boundedness of the oscillations defect measure with ˛ > 2. Although this might seem equivalent or even more difficult than showing square integrability of %, we shall see in the next section that osc˛ Œ%" ! %..0; T/  ˝/ remains bounded with ˛ D  C 1 > 2:

12.2.2 Oscillations Defect Measure and Effective Viscous Flux Boundedness of the oscillations defect measure follows from the effective viscous flux identity (12.6). The pressure p, satisfying hypotheses (4.11) and (4.17), can be written in the form p.%/ D pB .%/ C pM .%/ C a% ; a > 0;

(12.18)

where jpB j  B is bounded and pM a nondecreasing function of %. Consequently, pM .%/Tk .%/  pM .%/ Tk .%/  0; and relation (12.6) can be written as % Tk .%/  % Tk .%/ 

  C 2  Tk .%/.divx u  pB .%//  Tk .%/.divx u  pB .%// : a

Next, observe that  jTk .%" /  Tk .%/j C1  %"  % .Tk .%" /  Tk .%//; and, as % is convex and Tk .%/ concave, %  lim inf %" ; Tk .%/  lim sup Tk .%" /: "!0

"!0

(12.19)

172

12 Strong Convergence of the Approximate Densities

We conclude that lim sup jTk .%" /  Tk .%/j C1  % Tk .%/ % Tk .%/ % Tk .%/C% Tk .%/ "!0

(12.20)

   D % Tk .%/  % Tk .%/ C %  % Tk .%/  Tk .%/  % Tk .%/  % Tk .%/: On the other hand, a short inspection of the right-hand side of (12.19) yields Tk .%/.divx u  pB .%//  Tk .%/.divx u  pB .%//

(12.21)

  D lim Tk .%" /  Tk .%/ .divx u"  pB .%" // : "!0

Combining (12.19)–(12.21) we obtain Z

T

Z

lim sup "!0

Z

T

 lim sup "!0

0

0

jTk .%" /  Tk .%/j C1 dx dt

K

Z ˇ ˇ ˇ ˇ ˇTk .%" /  Tk .%/ˇ jdivx u"  pB .%" /j dx dt K

     sup kdivx u"  pB .%" /kL2 ..0;T/˝/ lim sup Tk .%" /  Tk .%/ ">0

L2 ..0;T/K/

"!0

for any compact K  ˝. Seeing that     lim sup Tk .%" /  Tk .%/

L2 ..0;T/K/

"!0

     lim sup kTk .%" /  Tk .%/kL2 ..0;T/K/ C lim sup Tk .%/  Tk .%/ "!0

"!0

L2 ..0;T/K/

 2 lim sup kTk .%" /  Tk .%/kL2 ..0;T/K/ ; "!0

and  C 1 > 2, we may infer, by virtue of the uniform bounds (11.6), Z

T

Z

jTk .%" /  Tk .%/j C1 dx dt  c;

lim sup "!0

0

K

where the constant is independent of k as well as of the compact K  ˝. Recalling our convention that all functions are extended to be zero outside ˝, we get the desired conclusion osc C1 Œ%" ! %..0; T/  R3 / < 1:

(12.22)

12.3 Existence of Weak Solutions

173

As  C 1 > 2, we may apply Lemma 10 to deduce that the limit functions Œ%; u satisfy the renormalized equation of continuity (5.4) in .0; T/  R3 . Remark 59 Note that the above argument applies whenever  C 1 > 2, meaning for the full “isentropic” range  > 1.

12.3 Existence of Weak Solutions Our ultimate goal is to show the strong convergence of the sequence f%" g">0 claimed in (11.18). To this end, we adapt the arguments of Sect. 6.3.1. To begin, consider the function (see also Fig. 12.2) Z Lk .%/ D %

% 1

Tk .r/ drI r2

note that L0k .%/

Z D

in particular, L00k .%/  0 for % > 0. Fig. 12.2 The graph of the function Lk

1

%

Tk .r/ Tk .%/ ; dz C r2 %

174

12 Strong Convergence of the Approximate Densities

Then %L0k .%/  Lk .%/ D Tk .%/ and we have (it follows by the limit passage " ! 0 in the renormalized equation of continuity) d dt

Z

Z ˝

Lk .%/ dx C

˝

Tk .%/divx u dx D 0 a.a. in .0; T/:

Similarly, by virtue of (12.22) and Lemma 10, d dt

Z

Z ˝

Lk .%/ dx C

˝

Tk .%/divx u dx D 0 a.a. in .0; T/:

Since the initial densities converge strongly, see (11.3), we obtain Z h ˝

Z i Lk .%/  Lk .%/ .; / dxC Z



D 0

Z h ˝

 0

Z h ˝

i Tk .%/divx u  Tk .%/divx u dx dt

(12.23)

i Tk .%/divx u  Tk .%/divx u dx dt;

where, by virtue of (12.6), the second integral on the left-hand side is nonnegative. Now we use (12.22), to control the expression on the right-hand side of (12.23): ˇZ ˇ ˇ ˇ

 0

Z h ˝

ˇ i ˇ Tk .%/divx u  Tk .%/divx u dx dtˇˇ

 kdivx ukL2 ..0;T/˝/ kTk .%/  Tk .%/kL2 ..0;T/˝/  ckTk .%/  Tk .%/kL1 ..0;T/˝/ kTk .%/  Tk .%/kL1 2 ..0;T/˝/ ; with  > 0, where kTk .%/  Tk .%/kL1 ..0;T/˝/ ! 0 as k ! 1; and, in view of (12.22), kTk .%/  Tk .%/kL2 ..0;T/˝/ < 1: Thus, letting k ! 1 in (12.23), we obtain the desired conclusion % log.%/  % log.%/ D 0 a.a. in .0; T/  ˝; yielding, exactly as in Sect. 6.3.1, the strong convergence of the density stated in (11.14). We have proved the following existence result.

12.3 Existence of Weak Solutions

175

Theorem 15 Let ˝  R3 be a bounded Lipschitz domain, and let T > 0 be arbitrary. Suppose that the initial data satisfy %0 2 L .˝/; %0  0 a.a. in ˝;

j.%u/0 j2 2 L1 .˝/: %0

(12.24)

Let the pressure p 2 C1 Œ0; 1/ satisfy p0 .%/ D p1 > 0; %!1 % 1

(12.25)

3 : 2

(12.26)

p.0/ D 0; p0 .%/ > 0 for all % > 0; lim where >

Then the Navier–Stokes system (2.7) and (2.8), endowed with the no-slip boundary condition uj@˝ D 0

(12.27)

admits a weak solution Œ%; u in .0; T/  ˝ in the sense specified in Chap. 5.

Remark 60 Although the proof was done under the simplifying assumption f D 0, it is a straightforward exercise for the reader to adapt it to the case f 2 L1 ..0; T/  ˝I R3 /; or even f 2 Lp .0; TI Lq .˝I R3 //; with suitable exponents p and q. Remark 61 Certain hypotheses concerning the pressure can be considerably relaxed. As a matter of fact, it is enough that the pressure satisfies (12.18) with p.%/  % ,  > 32 for % ! 1. Indeed it is straightforward to check that the whole proof of existence can be repeated replacing the artificial pressure by a more general perturbation

" .%/ C "% C "%

176

12 Strong Convergence of the Approximate Densities

where " 2 Cc1 Œ0; 1/ are supported in a compact set K  Œ0; 1/ and converge to zero uniformly as " ! 0. In particular, the pressure p in the class CŒ0; 1/ \ C1 .0; 1/ [cf. (4.11)] or even more general perturbations can be included as long as they grow at least as % ,  > 3=2, for % ! 1. The reader will have noticed that formula (12.18) allows for pressure-density state equations that need not be monotone, at least on a compact interval, see [33]. More general results in this direction were recently obtained by Bresch and Jabin [7].

Chapter 13

Concluding Remarks and Suggestions for Further Reading

In the book, we deliberately focused on purely mathematical aspects omitting the physical background of the modeling of motion of compressible viscous fluids arising from classical continuum mechanics. The interested reader may consult the standard reference material by Batchelor [6], Lamb [65], Landau and Lifshitz [66], or the more recent treatment by Gallavotti [46, 47]. Mathematical aspects of the theory are accented in Chorin and Marsden [19], Truesdell [91, 92], or Truesdell and Rajagopal [93]. Mathematical theory based on the concept of weak solutions as developed in this book is particularly suitable for studying convergence of numerical schemes as illustrated in Part II. On the other hand, it is legitimate to ask whether or not the weak solutions are really needed in the context of viscous fluids. As shown by Hoff [52], Hoff and Santos [54], singularities persist for compressible viscous fluids at any time provided they were imposed initially. Hoff [53] even proposed a concept of solution that would be weak enough to capture singularities but strong enough to ensure uniqueness. For smooth initial data, existence and uniqueness of regular solutions are known at least on (possibly) short times intervals, see Valli and Zajaczkowski [97], Mucha and Zajaczkowski [73], or the more recent result by Cho et al. [18]. In addition, the smooth solutions with the initial state close to an equilibrium can be extended globally in time, see Matsumura and Nishida [70], Matsumura and Padula [71], or Salvi and Straškraba [84]. Similar results in the context of Besov spaces can be found in the papers by Danchin [20–22] and [15], where in the last three papers also the heat conducting fluids are considered. Whether or not any smooth solution to the barotropic Navier–Stokes system can be extended globally in time is an outstanding open problem. Sun et al. [87] showed that any possible blow-up of smooth solutions must be due to the appearance of density concentrations. In other words, a local smooth solution remains regular provided the density, or rather the pressure, remains bounded. On the other hand, weak and smooth solutions coincide as long as the latter exists, see [38, 39] (also © Springer International Publishing Switzerland 2016 E. Feireisl et al., Mathematical Theory of Compressible Viscous Fluids, Advances in Mathematical Fluid Mechanics, DOI 10.1007/978-3-319-44835-0_13

177

178

13 Concluding Remarks and Suggestions for Further Reading

Germain [48] for other classes of solutions). We therefore conclude that weak solutions starting from regular initial data are in fact regular on the time interval on which the density remains bounded, see also Wen and Zhu [98] for further improvement of this criterion. Another possible singularity is the appearance of a vacuum—a zone of vanishing density. Such a situation is excluded in the mono-dimensional case as shown by Hoff and Smoller [55]. As we have seen above, the appearance of a vacuum from regular data must be necessarily accompanied by density concentrations at the same time or earlier. Many results concerning the weak and strong solutions for the Navier–Stokes equations can be found in the books by Lions [68] and Novotný and Straškraba [79], where also the steady case is considered. The extensions for the evolutionary heat-conducting case are treated in the monographs by Feireisl [34] and Feireisl and Novotný [35]. Let us briefly mention problems which are close to the studied system, however, are not covered in this book. The existence of weak solutions for the steady Navier– Stokes equations was extended with respect to the Lions’ monograph [68] to a larger set of the exponents  by Frehse et al. [44], Jesslé and Novotný [56], and recently using a slightly different technique by Plotnikov and Weigant in [82]. Similar results for the heat-conducting case can be found in the papers by Novotný and Pokorný [77, 78] and Jesslé et al. [57]. The existence of weak solutions for time-periodic solutions, which is in a sense a combination of the technique for the steady and the evolutionary problem, was proved by Feireisl et al. [36, 40] with extensions by Axmann and Pokorný [3]. All the results above considered only the case when there is no inflow into the domain. The problem for nonzero inflow condition in the evolutionary case was studied by Novo in [76] and generalized by Girinon in [49]. This problem for the steady flow is the case of large data (weak solutions) totally open. To conclude we remark that the state of the art of the mathematical theory of compressible viscous fluids is fairly satisfactory in the simplified mono-dimensional case, where smooth data give rise to smooth solutions and discontinuous initial data to (unique) weak solutions, see Kazhikhov [59], Zlotnik and Amosov [100], or the monograph by Antontsev et al. [2]. Last but not least, there is a large piece of work in numerical analysis based on finite element methods to barotropic compressible Navier–Stokes. Kellogg and Liu [61] or Chaps. 4.2 and 4.3 in the monograph of Feistauer et al. [42] as well as the references cited therein could be a good material for further studies. The finite volume methods in combination with upwinding is a technique frequently used to discretize Euler fluxes, see the review paper of Herbin et al. [51] and Gurris et al. [50], among others. A mixed method combining the advantages of finite volume– finite element approach was employed by Feistauer et al. [41], the reader may also consult the references in [42, 20, Chap. 4.4]. Earlier results of similar character were obtained by Bristeau et al. [10, 11], Carlenzoli et al. [12], and others.

References

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Index

1Q – the characteristic function of the set Q, 2 BV.Q/ – the space of functions of bounded variation, 16 C.Q/ – the space of continuous functions on Q, 5 Ck .Q/ – the space of k-times continuously differentiable functions, 6 Cck .Q/ – the space of ktimes continuously differentiable functions with compact support, 7 Ck; .Q/ – the space of functions with Hölder continuous derivatives, 6 Cweak .QI X/ – the space of functions on Q continuous with respect to the weak topology on X, 6 L p .QI X/ – the Lebesgue space of integrable functions ranging in X, 8 P (pressure potential), 42 W k;p .QI RM / – Sobolev space, 11  – Laplace operator, Laplacian, 5 ,! continuous embedding, 13 ,!,! compact embedding, 13 1 F!x – inverse Fourier transform, 17 Fx! – Fourier transform, 17 M.Q/ – the space of Radon measures, 7 I – the identity matrix (tensor), 2 r – gradient, 4 % (density), 25 jQj – the Lebesgue measure of the set Q, 7 a ˝ b – the tensor product of vectors a, b, 2 curl – vorticity, 5 u (velocity), 25 D 0 .QI RM / – the space of distributions, 7 div – divergence, 4 a priori bounds, 39

Arzelà–Ascoli Theorem, 6

balance of momentum, 27, 51 weak formulation, 36

Cauchy stress tensor, 28 concentrations, 55 consistency, 91, 111 formulation of the continuity method, 114 formulation of the momentum method, 122 continuity method, 90 convergence convective terms, 140 fields equations, 144 numerical scheme, 135 time derivatives, 143 Crouzeix–Raviart finite elements, 87

discrete energy balance, 98 discrete Sobolev embedding, 106 Div-Curl lemma, 70 divergence, 4 domain convergence, 84 Dunford–Pettis Theorem, 21

effective viscous flux, 56, 68, 149, 165 embedding relations, 13 energy inequality, 52 discretized, 94

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186 equation of continuity, 27 weak formulation, 32 equi-integrability, 21

finite element, 83, 86 finite volume, 86 First law of thermodynamics, 42 Fourier transform, 17 Friedrichs commutator lemma, 62

gradient, 4 Green’s formula, 15 Gronwall’s lemma, 10

Hölder’s inequality, 9 Hörmander–Mikhlin Theorem, 20 Helly’s Theorem, 17

interpolation inequality, 9 inverse divergence, 19 isentropic state equation, 157

Jensen’s inequality, 10

kinetic energy balance, 40

Lebesgue point, 8 Lebesgue space, 8 Lions’ identity, 56 Lipschitz domain, 10

mesh, 83 shape regular, 85 unfitted, 84 momentum method, 91 monoatomic gas state equation, 157

Navier–Stokes system, 29 weak solution, 53, 153 Newton’s rheological law, 28

Index no-slip boundary condition, 30 numerical dissipation, 109

oscillations, 56 oscillations defect measure, 167

Poincaré’s inequality, 16 discrete, 112, 114 polyhedral approximation, 83 pressure potential, 42 pseudodifferential operator, 18

reference element, 85, 112 regularizing kernels, 58 Rellich–Kondrachov embedding Theorem, 13 renormalized equation of continuity, 41, 50 discretization, 92 weak formulation, 50 Riesz representation Theorem, 9 Riesz transform, 19

Schaeffer’s fixed point Theorem, 22 Second law of thermodynamics, 42 Sobolev space, 10 dual space, 12 Sobolev–Slobodeckii space, 14 spatial discretization, 83 stability, 91, 103 Stokes’ law, 28

time discretization, 82 total energy balance, 41 total mass conservation, 39 Trace Theorem, 15 triangulation, 83

upwind, 88 weak L1 compactness, 21 weak sequential stability, 55 weak solution, 31, 53 weak–strong compatibility, 32 weak–strong uniqueness, 38, 52