Compressible Navier-Stokes equations: theory and shape optimization

Monografi e M a t e ma t y c z n e Instytut Matematyczny Polskiej Akademii Nauk (IMPAN) Volume 73 (New Series) Found...

27 downloads 654 Views 3MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

Monografi e

M a t e ma t y c z n e

Instytut Matematyczny Polskiej Akademii Nauk (IMPAN)

Volume 73 (New Series) Founded in 1932 by S. Banach, B. Knaster, K. Kuratowski, S. Mazurkiewicz, W. Sierpinski, H. Steinhaus

Managing Editor: Marek Bozejko (Wroclaw University, Poland) Editorial Board: Jean Bourgain (IAS, Princeton, USA) Joachim Cuntz (University of Mnü ster, Germany) Ursula Hamenstdä t (University of Bonn, Germany) Gilles Pisier (Texas A&M University, USA) Piotr Pragacz (IMPAN, Poland) Andrew Ranicki (University of Edinburgh, UK) Slawomir Solecki (University of Illinois, Urbana-Champaign, USA) Przemyslaw Wojtaszczyk (IMPAN and Warsaw University, Poland) Jerzy Zabczyk (IMPAN, Poland) Henryk Zoladek (Warsaw University, Poland)

Volumes 31–62 of the series Monografie Matematyczne were published by PWN – Polish Scientific Publishers, Warsaw

Pavel Plotnikov • Jan Sokołowski

Compressible Navier–Stokes Equations Theory and Shape Optimization

Pavel Plotnikov Siberian Branch of RAS Lavrentyev Institute of Hydrodynamics Novosibirsk Russia

Jan Sokołowski Institut Élie Cartan de Nancy, UMR 7502 Université de Lorraine, CNRS Vandœuvre-lès-Nancy France

ISBN 978-3-0348-0366-3 ISBN 978-3-0348-0367-0 (eBook) DOI 10.1007/978-3-0348-0367-0 Springer Basel Heidelberg New York Dordrecht London Library of Congress Control Number: 2012945409 Mathematics Subject Classification (2010): 35Q30, 35Q35, 35Q93, 49Q10, 49Q12, 49K20, 49K40, 76N10, 76N25 © Springer Basel 2012 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. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. 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. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media (www.birkhauser-science.com)

Contents Preface

ix

Introduction

xi

1

2

Preliminaries 1.1 Functional analysis . . . . . . . . . . . . . . . . . . 1.1.1 Banach spaces . . . . . . . . . . . . . . . . 1.1.2 Interpolation of Banach spaces . . . . . . . 1.1.3 Invertibility of linear operators . . . . . . . 1.1.4 Fixed point theorems . . . . . . . . . . . . 1.2 Function spaces . . . . . . . . . . . . . . . . . . . . 1.2.1 Hölder spaces . . . . . . . . . . . . . . . . . 1.2.2 Measure and integral . . . . . . . . . . . . . 1.2.3 Lebesgue measure in Rd . Lebesgue spaces . 1.3 Compact sets in Lp spaces . . . . . . . . . . . . . . 1.3.1 Functions of bounded variation . . . . . . . 1.3.2 Lp spaces of Banach space valued functions 1.4 Young measures . . . . . . . . . . . . . . . . . . . . 1.5 Sobolev spaces . . . . . . . . . . . . . . . . . . . . 1.6 Molliﬁers and DiPerna & Lions lemma . . . . . . . 1.7 Partial diﬀerential equations . . . . . . . . . . . . . 1.7.1 Elliptic equations . . . . . . . . . . . . . . . 1.7.2 Stokes problem . . . . . . . . . . . . . . . . 1.7.3 Parabolic equations . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

1 2 2 6 8 8 9 9 12 16 20 20 23 29 32 35 39 39 42 43

Physical background 2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Isentropic ﬂows. Compressible Navier-Stokes equations 2.2 Boundary and initial conditions . . . . . . . . . . . . . . . . . 2.3 Power and work of hydrodynamic forces . . . . . . . . . . . . 2.4 Navier-Stokes equations in a moving frame . . . . . . . . . . 2.5 Flow around a moving body . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

45 45 47 48 50 51 53

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

v

vi

Contents

3

Problem formulation 3.1 Weak solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Renormalized solutions . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Work functional. Optimization problem . . . . . . . . . . . . . . .

57 59 60 61

4

Basic statements 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . 4.2 Bounded energy functions . . . . . . . . . . . . . 4.3 Functions of bounded mass dissipation rate . . . 4.4 Compactness properties . . . . . . . . . . . . . . 4.4.1 Two lemmas on compensated compactness 4.4.2 Proof of Theorem 4.4.2 . . . . . . . . . . 4.5 Basic integral identity . . . . . . . . . . . . . . . 4.6 Continuity of viscous ﬂux . . . . . . . . . . . . . 4.7 Viscous ﬂux. Localization . . . . . . . . . . . . .

5

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

63 63 63 68 69 71 76 78 87 95

Nonstationary case. Existence theory 5.1 Problem formulation. Results . . . . . . . . . . . . . . . . . . 5.2 Regularized equations . . . . . . . . . . . . . . . . . . . . . . 5.3 Passage to the limit. The ﬁrst level . . . . . . . . . . . . . . . 5.3.1 Step 1. Convergence of densities and momenta . . . . 5.3.2 Step 2. Weak convergence of the kinetic energy tensor 5.3.3 Step 3. Weak limits of ε∇n ⊗ un . . . . . . . . . . . 5.3.4 Step 4. Proof of estimates (5.3.8–5.3.9) . . . . . . . . . 5.3.5 Step 5. Proof of Theorem 5.3.1 . . . . . . . . . . . . . 5.3.6 Proof of Theorem 5.3.2 . . . . . . . . . . . . . . . . . 5.3.7 Local pressure estimate . . . . . . . . . . . . . . . . . 5.3.8 Normal derivative of the density . . . . . . . . . . . . 5.4 Passage to the limit. The second level . . . . . . . . . . . . . 5.5 Passage to the limit. The third level . . . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

99 99 105 118 121 122 124 125 129 130 132 136 139 157

6

Pressure estimate

7

Kinetic theory. Fast density oscillations 7.1 Problem formulation. Main results 7.2 Proof of Theorem 7.1.9 . . . . . . . 7.3 Proof of Theorem 7.1.12 . . . . . . 7.4 Adiabatic exponent γ > 3/2 . . . .

8

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

167

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

175 175 185 204 223

Domain convergence 8.1 Hausdorﬀ and Kuratowski-Mosco convergences 8.2 Capacity, quasicontinuity and ﬁne topology . . 8.3 Applications to ﬂow around an obstacle . . . . 8.4 S-compact classes of admissible obstacles . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

225 225 229 235 244

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

Contents 9

Flow around an obstacle. Domain 9.1 Preliminaries . . . . . . . . 9.2 Existence theory . . . . . . 9.3 Main stability theorem . . .

vii dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

249 249 250 255

10 Existence theory in nonsmooth domains 281 10.1 Existence theory in nonsmooth domains . . . . . . . . . . . . . . . 281 10.2 Continuity of the work functional . . . . . . . . . . . . . . . . . . . 282 10.3 Applications of the stability theorem to optimization problems . . 296 11 Sensitivity analysis. Shape gradient of the drag functional 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Change of variables in Navier-Stokes equations . . . . . . . 11.1.2 Extended problem. Perturbation theory . . . . . . . . . . . 11.2 Existence of solutions . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Function spaces. Results . . . . . . . . . . . . . . . . . . . . 11.2.2 Step 1. Estimates of composite functions . . . . . . . . . . . 11.2.3 Step 2. Stokes equations . . . . . . . . . . . . . . . . . . . . 11.2.4 Step 3. Transport equations . . . . . . . . . . . . . . . . . . 11.2.5 Step 4. Fixed point scheme. Estimates . . . . . . . . . . . . 11.2.6 Step 5. Proof of Theorem 11.2.6 . . . . . . . . . . . . . . . 11.3 Dependence of solutions on N . . . . . . . . . . . . . . . . . . . . . 11.3.1 Problem formulation. Basic equations. Transposed problem 11.3.2 Transposed problem . . . . . . . . . . . . . . . . . . . . . . 11.3.3 Special dual space . . . . . . . . . . . . . . . . . . . . . . . 11.3.4 Very weak solutions . . . . . . . . . . . . . . . . . . . . . . 11.3.5 Uniqueness and existence. Main theorem . . . . . . . . . . . 11.4 Shape derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Preliminaries. Results . . . . . . . . . . . . . . . . . . . . . 11.4.2 Proof of Theorem 11.4.6 . . . . . . . . . . . . . . . . . . . . 11.4.3 Conclusion. Material and shape derivatives . . . . . . . . . 11.5 Shape derivative of the drag functional. Adjoint state . . . . . . . .

299 299 300 304 308 308 312 314 315 319 328 330 330 333 347 348 354 355 355 359 370 371

12 Transport equations 12.1 Introduction . . . . . . . . . . . . . . . . . . 12.2 Existence theory in Sobolev spaces . . . . . 12.3 Proof of Theorem 12.2.3 . . . . . . . . . . . 12.3.1 Normal coordinates . . . . . . . . . 12.3.2 Model equation . . . . . . . . . . . . 12.3.3 Local estimates . . . . . . . . . . . . 12.3.4 Estimates near the inlet . . . . . . . 12.3.5 Partition of unity . . . . . . . . . . . 12.3.6 Proofs of Lemmas 12.3.2 and 12.3.3 12.3.7 Proof of Lemma 12.3.4 . . . . . . . .

377 377 380 382 383 384 385 390 392 395 398

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

viii 13 Appendix 13.1 Proof of Lemma 2.4.1 . . . . . . . . . . . 13.2 Normal coordinates . . . . . . . . . . . . . 13.3 Geometric results. Approximation of unity 13.4 Singular limits of normal derivatives . . .

Contents

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

407 407 411 413 418

Bibliography

441

Notation

451

Index

453

Preface This book is a result of scientiﬁc collaboration, for more than ten years, between three countries, France, Russia and Poland. The main support for the joint research comes from Université de Lorraine, Institut Élie Cartan and from Centre Nationale de la Recherche Scientiﬁque (CNRS) for the visits of Pavel I. Plotnikov in France. Also the support of the Systems Research Institute of the Polish Academy of Sciences for the visits in Poland is acknowledged. The research of Pavel I. Plotnikov was partially supported by Contract 02.740.11.0617 with the Ministry of Education and Science of the Russian Federation and by grant 10-01-00447-a from the Russian Foundation of Basic Research, while the research of Jan Sokołowski was partially supported by the project ANR-09-BLAN-0037 Geometric analysis of optimal shapes (GAOS) ﬁnanced by the French Agence Nationale de la Recherche (ANR) and by grant N51402132/3135 of the Polish Ministerstwo Nauki i Szkolnictwa Wyższego “Optymalizacja z wykorzystaniem pochodnej topologicznej dla przepływów w ośrodkach ściśliwych” in the Systems Research Institute of the Polish Academy of Sciences. The monograph is published in the Polish mathematical series by Birkhäuser Basel. The authors are greatly indebted to Jerzy Trzeciak for his careful work on editing the monograph, and his patience. Pavel Plotnikov Lavrentyev Institute of Hydrodynamics and Novosibirsk State University Jan Sokołowski Université de Lorraine, Institut Élie Cartan and Systems Research Institute of the Polish Academy of Sciences Nancy, Novosibirsk and Warsaw, March 2012

ix

Introduction This monograph is devoted to the study of boundary value problems for equations of viscous gas dynamics, named compressible Navier-Stokes equations. The mathematical theory of Navier -Stokes equations is very interesting in its own right, but its principal signiﬁcance lies in the central role Navier-Stokes equations now play in ﬂuid dynamics. Most of this book concentrates on those aspects of the theory that have proven useful in applications. The mathematical study of compressible Navier-Stokes equations dates back to the late 1950s. It seems that Serrin [121] and Nash [92] were the ﬁrst to consider the mathematical questions of compressible viscous ﬂuid dynamics. An intensive treatment of compressible Navier-Stokes equations starts with pioneering papers by Itaya [60], Matsumura & Nishida [87], Kazhikhov & Solonnikov [63], and Hoﬀ [56] on the local theory for nonstationary problems, and by Beirão da Veiga [11, 12], Padula [103], and Novotný & Padula [98, 99] on the theory of stationary problems for small data. A global theory of weak solutions to compressible Navier-Stokes equations was developed by P.-L. Lions in 1998. These results were essentially improved, sharpened and generalized by E. Feireisl. We refer the reader to the books by Lions [80], Feireisl [34], Novotný & Straškraba [101], and Feireisl & Novotný [37] for the state of the art in the domain. Although the theory is satisfactory in what concerns local time behavior and small data, many issues of global behavior of solutions for large data are far from being understood. There are a vast range of unsolved problems concerning questions such as regularity of solutions to compressible Navier-Stokes equations, the theory of weak solutions for small adiabatic exponents, existence theory for heat conducting ﬂuids. However, these problems will not be our primary concern here. We are mainly interested in three problems that we describe brieﬂy below. Existence theory. This issue is important since no progress in the mathematical theory of Navier-Stokes equations can be made without answering the basic questions on their well-posedness. We focus on existence results for the inhomogeneous in/out ﬂow problem, in particular the problem of the ﬂow around a body placed in a ﬁnite domain. Notice that the majority of known results are related to viscous gas ﬂows in domains bounded by impermeable walls. In/out ﬂow problems are still poorly investigated. We refer to the paper by Novo [93], where an existence

xi

xii

Introduction

theorem was proved for constant boundary data, and recent work by Girinon [52], where the existence of a weak solution was established for convex ﬂow domains with inlet independent of the time variable. We give an existence result in the general nonstationary case without imposing restrictions on the geometry of the ﬂow domain and the behavior of boundary data. In contrast, the question of existence of global weak solutions to the stationary in/out ﬂow problem remains essentially unsolved. Local strong solutions close to the uniform ﬂow have been studied by Farwig [29] and Kweon & Kellogg [69, 72, 73]. With applications to shape optimization theory in mind, we consider the problem of the ﬂow around a body placed in a bounded domain for small Mach and Reynolds numbers. Stability of solutions with respect to nonsmooth data and domain perturbations. Propagation of rapid oscillations in compressible ﬂuids. In compressible viscous ﬂows, any irregularities in the initial and boundary data are transferred inside the ﬂow domain along ﬂuid particle trajectories. The transport of singularities in viscous compressible ﬂows was studied by Hoﬀ [56, 57]. In this book we discuss the propagation of rapid oscillations of the density, which can be regarded as acoustic waves. The main idea is that any rapidly oscillating sequence is associated with some stochastic ﬁeld named the Young measure (see Tartar [128] and Perthame [106] for basic ideas). We establish that the distribution function of this stochastic ﬁeld satisﬁes a kinetic equation of a special form, which leads to a rigorous model for propagation of nonlinear acoustic waves. Notice that oscillations can be induced not only by oscillations of initial and boundary data, but also by irregularities of the boundary of the ﬂow domain. Domain dependence of solutions to compressible Navier-Stokes equations. This issue is important because of applications to shape optimization theory. The latter is a branch of the general calculus of variations which deals with the shapes of geometric and physical objects instead of parameters and functions. The classic examples of shape optimization problems are the isoperimetric problem and Newton’s problem of the body of minimal resistance. We refer to [126], [18], [21], [46], [54], [62], [91] for a general account of the theory and the relevant references. The ﬁrst global result on domain dependence of solutions to compressible NavierStokes equations is due to Feireisl [33], who proved that the set of solutions to compressible Navier-Stokes equations is compact provided the set of ﬂow domains is compact in the Kuratowski-Mosco topology and their boundaries have “uniformly small” volumes. We prove that the compactness result holds true if the set of ﬂow domains is compact in the Kuratowski-Mosco topology, and also that some cost functionals, such as the drag and the work of hydrodynamical forces, are continuous in this topology. With applications to shape optimization in mind, we consider the shape diﬀerentiability of strong solutions and give formulae for the shape derivative of the drag functional. Let us also mention that in the incompressible case the shape diﬀerentiability of the drag functional was considered in [14], [15], [123]. Finally, we refer to Mohammadi & Pironneau [90] for the relevant references in applied shape optimization for ﬂuids.

Introduction

xiii

We also discuss the mathematical questions which are not related directly to Navier-Stokes equations. Among them are the theory of boundary value problems for transport equations and the problem of vanishing viscosity for diﬀusion equations with convective terms. The basic idea of the book is to give as more details as possible and to avoid using complicated mathematical tools. In particular, we do not use compensated compactness results, the Bogovski lemma, or semigroup theory. Only the undegraduate background in mathematical analysis and elementary facts from functional analysis are assumed of the reader. The material is organized in twelve chapters and an appendix. We start in Chapter 1 with a review of standard topics from real and functional analysis. The chapter includes, mainly without proofs, basic facts on measure and integral, functional analysis, elliptic and parabolic equations. We focus on measure theory, since the notion of Young measure is widely used throughout the book. Most of the material will be familiar to the reader and can be omitted. Possible exceptions are Section 1.3.1 containing a general formula for integration by parts in the LebesgueStieltjes integral, Section 1.4 where the notion of Young measure is introduced, and Sections 1.1.2 and 1.5 devoted to interpolation theory and Sobolev spaces. In Chapter 2 we collect the basic physical facts concerning compressible Navier-Stokes equations including the formulation of equations in a moving coordinate frame and formulae for the hydrodynamical forces and the work of these forces. In Chapter 3 we give the mathematical formulation of the main boundary value problem for compressible Navier-Stokes equations and discuss the notions of weak and renormalized solutions. Chapters 4–11 can be considered as the core of the book. Our considerations are based on the approach developed by P.-L. Lions and E. Feireisl. The main ingredients of their method are Lions’s result on weak continuity of the eﬀective viscous ﬂux, Lp -estimates of the pressure function, and the theory of the oscillation defect measure developed by E. Feireisl. Some of these results are of general character and hold true for any system of mass and impulse-momentum laws. We assemble all such results in Chapter 4, where we consider the system of balance laws which is formulated as follows. Assume that a medium occupies a domain Ω ⊂ Rd , d = 2, 3. We want to ﬁnd a velocity ﬁeld u : Ω × (0, T ) → Rd and a density function : Ω × (0, T ) → R+ satisfying the momentum and mass balance equations ∂t (u) + div( u ⊗ u) = div T + f in Ω × (0, T ), ∂t + div(u − g) = 0 in Ω × (0, T ),

(0.0.1)

where f , g are given vector ﬁelds, and T is the stress tensor. The class of such systems includes compressible Navier-Stokes equations and their numerous modiﬁcations. At this stage we do not specify the form of the stress tensor and do not impose boundary and initial conditions on the density and the velocity. Instead we assume that they satisfy some integrability conditions. In Chapter 4

xiv

Introduction

we consider the basic properties of solutions to equations (0.0.1). The results we present include elementary facts on integrability of functions with ﬁnite energy (Section 4.2), and the standard material concerning weak compactness properties of the impulse momentum u and the kinetic energy tensor u ⊗ u (Section 4.4). Section 4.6 is more important: we prove there the general form of P.-L. Lions’s result on weak continuity of the viscous ﬂux, which is the most important result of the mathematical theory of viscous compressible ﬂows. Chapter 5 deals with existence theory for the nonstationary in/out ﬂow boundary value problem for compressible Navier-Stokes equations, which are a particular case of system (0.0.1) with the stress tensor of the form T = ∇u + (∇u) + (λ − 1) div u I − p() I,

(0.0.2)

where λ is some constant, and p() is a monotone function such that p() ∼ γ at inﬁnity. The in/out ﬂow problem for equations (0.0.1)–(0.0.2) can be formulated as follows. Let a vector ﬁeld U : Ω × (0, T ) → Rd and a nonnegative function ∞ : Ω × (0, T ) → R be given. We want to ﬁnd a solution of (0.0.1)–(0.0.2) satisfying the initial and boundary conditions u = U, = ∞ on ∂Ω × {t = 0}, u = U on ∂Ω × (0, T ), = ∞ on Σin ,

(0.0.3)

where the inlet Σin is the open subset of ∂Ω × (0, T ) which consists of all points (x, t) such that the vector U(x, t) points to the inside of Ω×(0, T ). The peculiarity of this problem is that we deal with the boundary value problem for the mass balance equations. In Chapter 5 we prove that for the adiabatic exponent γ > 2d and smooth initial and boundary data satisfying the compatibility conditions, the problem has a renormalized solution. We follow the multilevel regularization scheme proposed by E. Feireisl, but with a diﬀerent regularization technique. The main ingredient of our method is the estimates of the normal derivatives of solutions to singularly perturbed transport equations (Section 5.3.8). These estimates are nontrivial and their derivation is based on Aronson-type inequalities for the heat kernels of diﬀusion equations with convective terms. Another essential ingredient of our method is the systematic use of Young measure theory. Chapter 6 is of technical character. There we prove that for the solution to problem (0.0.1)–(0.0.3) constructed in Chapter 5, the pressure p() is locally integrable with some exponent greater than 1. In Chapter 7 the results obtained are extended to the range of adiabatic exponents (3/2, ∞) common for homogeneous boundary value problems. In this chapter we propose a new approach to the boundary value problems with fast oscillating boundary data and develop a theory of such problems based on the kinetic formulation of the governing equation. We deal with the sequence of solutions (u , ), > 0, to problem (0.0.1)–(0.0.3) with regularized pressure functions of the form p = p() + εn , and the initial and boundary data ∞ . We assume that the sequence ∞ is only bounded, but need not converge to any limit in the

Introduction

xv

strong sense. In particular this class of data includes rapidly oscillating functions of the form x t , ∞ = F x, t, , where F (x, t, y, τ ) is a bounded function, periodic in y and τ . Under these assumptions, the sequence ( , u , p( )) converges only weakly to some limit (, u, p) as → 0. Following [128] we conclude that this limit admits a representation (x, t) = s dμx,t (s), p(x, t) = p(s) dμx,t (s), (0.0.4) R

R

where μx,t is a probability measure on the real line named the Young measure. It is completely characterized by the distribution function f (x, t, s) = μx,t (−∞, s]. Notice that the sequence converges strongly if and only if the distribution function is deterministic, i.e., f (1−f ) = 0. The basic idea underlying the method of kinetic equations (see [106]) is that the distribution function satisﬁes a diﬀerential relation named a kinetic equation. Usually the kinetic equation contains some undeﬁned terms and cannot be considered as an equation in the strict sense of this word. A remarkable property of compressible Navier-Stokes equations is that in this particular case the kinetic equation can be written in closed form as s (p(τ ) − p) dτ f (x, t, τ ) = 0. ∂t f + div(f u) − ∂s sf div u + λ + 1 (−∞,s] In Chapter 7 we derive the kinetic equation and show that, when combined with relations (0.0.4) and the momentum balance equations, it gives a closed system of integro-diﬀerential equations which describes the propagation of rapid oscillations in a compressible viscous ﬂow. We also prove that if the data are deterministic and the function f satisﬁes some integrability condition, then any solution to the kinetic equation satisfying some integrability conditions is deterministic. This fact is a general property of the kinetic equation and has no connection with the theory of Navier-Stokes equations. It follows that if ∞ converges strongly, then so does . In the next chapters we apply the kinetic equation method to the analysis of the domain dependence of solutions to compressible Navier-Stokes equations. We restrict our considerations to the problem of the ﬂow around an obstacle placed in a ﬁxed domain. In this problem Ω = B \ S is a condenser type domain, B is a ﬁxed hold-all domain and S is a compact obstacle. It is assumed that U vanishes on S × (0, T ). In Chapter 8 we collect the basic facts concerning domain convergence and related questions from capacity theory. The most important is Hedberg’s theorem (Theorem 8.2.22) on approximation of Sobolev functions. In this chapter we also introduce the notion of S-convergence, which plays a key role in the next chapters. Denote by CS∞ (B) the set of all smooth functions deﬁned in B and vanishing on S B. Let WS1,2 (B) be the closure of CS∞ (B) in the W 1,2 (B)-norm. It is clear

xvi

Introduction

that WS1,2 (B) is a closed subspace of W 1,2 (B). A sequence of compact sets Sn B is said to S-converge to S if • there is a compact set B B such that Sn , S ⊂ B and Sn converges to S in the standard Hausdorﬀ metric; • for any sequence un u weakly convergent in W 1,2 (B) with un ∈ WS1,2 (B), n the limit element u belongs to WS1,2 (B); (B) with un → u • whenever u ∈ WS1,2 (B), there is a sequence un ∈ WS1,2 n strongly in W 1,2 (B). We investigate in great detail the properties of S-convergence and give examples of classes of obstacles which are compact with respect to this convergence. In Chapter 9 we prove the central result on the domain stability of solutions to compressible Navier-Stokes equations. We show that if a sequence Sn of compact obstacles S-converges to a compact obstacle S then the sequence of corresponding solutions to the in/out ﬂow problem contains a subsequence which converges to a solution to the in/out ﬂow problem in the limiting domain. In Chapter 10 we sharpen this result by proving that the typical cost functionals, such as the work of hydrodynamical forces, are continuous with respect to S-convergence. As a conclusion we establish the solvability of the problem of minimization of the work of hydrodynamical forces in the class of obstacles with a given ﬁxed volume. Chapter 11 is devoted to the shape sensitivity analysis of the stationary boundary problem for compressible Navier-Stokes equations. Here we prove the local existence and uniqueness results for the in/out ﬂow problem for compressible Navier-Stokes equations under the assumption that the Reynolds and Mach numbers are suﬃciently small. We show the weak diﬀerentiability of solutions with respect to the shape of the ﬂow domain and derive formulae for the derivatives and corresponding system of adjoint equations, which are of a practical interest. The results obtained are based on the theory of strong solutions to boundary value problems for transport equations which is presented in Chapter 12.

Chapter 1

Preliminaries We use the following notation throughout the monograph. A vector in n-dimensional Euclidean space is denoted by v = (v1 , . . . , vn ), whether it is a column vector or a row vector. For a matrix A = (Aij ), i, j = 1, . . . , n, with real entries Aij , i is the row index and j the column index, and A = (Aji ) stands for the transposed matrix. The product of a matrix by a column vector is denoted by Av, and of a row vector by a matrix by vA. The product of two matrices C = AB is a matrix with the entries Cij = Aik Bkj with the summation convention over repeated indices. In particular, the scalar product of two vectors is v · u = vi ui , and (Av)i = Aij vj , while vA = A v with (vA)i = (A v)i = Aji vj . The tensor product of two vectors u, v ∈ Rn is the matrix A := u ⊗ v with the entries Aij := ui vj , i, j = 1, . . . , n. For the product of this matrix with a vector we have (u ⊗ v)w = u(v · w) and w(u ⊗ v) = (w · u)v. The derivatives of a scalar or a vector function with respect to the time variable are denoted by ∂t v = ∂v ∂t , and similarly for the spatial variables. There is a diﬀerence between the Jacobian of a vector function and its gradient: the Jacobian is denoted by ⎡ ⎢ ⎢ ⎢ ∂v ∂v ∂v Dv = (∂xj vi ) = , , =⎢ ⎢ ∂x1 ∂x2 ∂x3 ⎢ ⎣

∂v1 ∂x1 ∂v2 ∂x1 ∂v3 ∂x1

∂v1 ∂x2 ∂v2 ∂x2 ∂v3 ∂x2

∂v1 ∂x3 ∂v2 ∂x3 ∂v3 ∂x3

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

P. Plotnikov and J. Sokoáowski, Compressible Navier–Stokes Equations: Theory and Shape Optimization, Monografie Matematyczne 73, DOI 10.1007/978-3-0348-0367-0_1, © Springer Basel 2012

1

2

Chapter 1. Preliminaries

and the gradient is its transpose

⎡

⎢ ⎢ ⎢ ∇v = Dv = (∂xi vj ) = [∇v1 , ∇v2 , ∇v3 ] = ⎢ ⎢ ⎢ ⎣

∂v1 ∂x1 ∂v1 ∂x2 ∂v1 ∂x3

∂v2 ∂x1 ∂v2 ∂x2 ∂v2 ∂x3

∂v3 ∂x1 ∂v3 ∂x2 ∂v3 ∂x3

⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

Therefore, the nonlinear term in the Navier-Stokes equations is a vector denoted by v∇v = ((vj ∂xj vi )) = [v · ∇v1 , v · ∇v2 , v · ∇v3 ], where we sum over the repeated indices j = 1, 2, 3, and v · ∇v1 , v · ∇v2 , v · ∇v3 stand for column vectors according to our convention. In general, for a function u : Rd → R we denote by ∂ |α| u ∂αu = α1 d ∂x1 . . . ∂xα d the partial derivative of order |α| = α1 + · · · + αd with the multiindex α = (α1 , . . . , αd ), for integers αi , i = 1, . . . , d. We also use the simpliﬁed notation e.g., ∂x2 for the collection of all the ∂2 second order derivatives ∂x2i xj = , i, j = 1, . . . , d, of a scalar function ∂xi ∂xj Rd x → (x) ∈ R, and ∂xk u, k = 1, 2, for the collections of the ﬁrst order and of the second order derivatives of a vector function Rd x → u(x) ∈ Rd . For simplicity we write, e.g., u ∈ Lp (Ω) to mean that all components uj of a vector function u = (u1 , . . . , ud ) belong to the space Lp (Ω), in other words Lp (Ω) stands here for Lp (Ω; Rd ). The same convention is used for other spaces, e.g., in our notation (v, ϕ, ζ) ∈ C 1+γ (Ω) × C γ (Ω)2 means that all components of the vector function v belong to the space C 1+γ (Ω), and the scalar functions ϕ, ζ belong to C γ (Ω). The notation ∂x2 u ∈ Lp (Ω) means that all the second order derivatives belong to Lp (Ω). In this way we avoid the notation with multiindices unless strictly necessary. For a given symmetric tensor S = (Sij ), its divergence is the vector denoted by div S with the components div Si = ∂xj Sij , summed over j = 1, 2, 3. The product of two tensors is the scalar A : B = Ai,j Bi,j . On the other hand, points in Rd are denoted by x, y with coordinates x = (xi ), y = (yi ); this is an exception from the vector notation.

1.1 Functional analysis 1.1.1 Banach spaces We recall some well known facts; the main sources are [24] and [131]. A normed space A is a linear space over the ﬁeld of real numbers equipped with a norm

1.1. Functional analysis

3

· A : A → R such that uA ≥ 0, uA = 0 if and only if u = 0, λuA = |λ| uA for all λ ∈ R and u ∈ A, u + vA ≤ uA + vA for all u, v ∈ A. A sequence un ∈ A converges (converges strongly) to u ∈ A if un − uA → 0 as n → ∞. In this case we write un → u or limn→∞ un = u. A normed A space is complete if um − un A → 0 as m, n → ∞ implies the existence of u ∈ A such that un → u. Complete normed spaces are named Banach spaces. A set G ⊂ A is open if for any a ∈ A there is ε > 0 such that the ball {x ∈ A : x − aA < ε} is contained in A. A set F ⊂ A is closed if for any sequence F un → u the limit u belongs to F . Obviously F is closed if and only if A \ F is open. The closure of D ⊂ A is denoted by cl D or D. We say that a set D ⊂ A is dense in a set E ⊂ A if E ⊂ cl D. Embedding. We say that a Banach space A is continuously embedded in a Banach space B or that the embedding of A in B is bounded if A ⊂ B and there exists c > 0 such that uB ≤ cuA for all u ∈ A. In this case we write A → B. Product, sum and intersection. The Cartesian product A × B of Banach spaces A, B consists of all pairs (u, v), where u ∈ A, v ∈ B, and is equipped with the norm uA + vA . Let A, B be Banach spaces, both subsets of an ambient Banach space Z. Then the intersection A ∩ B equipped with the norm uA∩B = uA +uB and the algebraic sum A+B := {w = u+v : u ∈ A, v ∈ B} equipped with the norm wA+B = inf{uA + vB : u + v = w} are Banach spaces. Linear operators. Let A, B be Banach spaces. Linear mappings T : A → B are called linear operators. A linear operator is bounded if it has a ﬁnite norm T L(A,B) := sup T uB = inf{c : T uB ≤ cuA for all u ∈ A}. uA ≤1

Equipped with this norm, the set L(A, B) of bounded linear operators becomes a Banach space. Duality. The dual space A of a Banach space A consists of all continuous linear functionals u : A → R. The duality pairing between A and A is deﬁned by u , u := u (u). Equipped with the norm u A := supuA ≤1 |u , u|, A becomes a Banach space. We have (see [49, Thm. 5.13]) Theorem 1.1.1. Let Banach spaces A and B be subsets of a Banach space Z and suppose A ∩ B is dense in Z. Then (A ∩ B) = A + B ,

(A + B) = A ∩ B .

4

Chapter 1. Preliminaries

The structure of the dual of a Cartesian product is simpler. For every element w of the dual space (A×B) , there are u ∈ A and v ∈ B such that w , (u, v) = u , u + v , v and w (A×B) = u A + v B . Hence (A × B) can be identiﬁed with A × B . Hahn-Banach theorem. We will use the following result which is a particular case of the famous Hahn-Banach theorem ([24, Ch. II.3, Thm. 11] and [24, Ch. V.1, Thm. 12]). Theorem 1.1.2. (i) Let B be a closed linear subspace of a Banach space A and v ∈ B . Then there is w ∈ A whose restriction to B coincides with v and w A = v B . (ii) Let K be a closed convex subset of a Banach space A and e ∈ A \ K. Then there is u ∈ A such that u , e > supu∈K u , u. Weak convergence. Weak topology. Let A be a Banach space. A sequence un ∈ A is said to converge weakly to u ∈ A (denoted un u) if u , un → u , u as n → ∞ for all u ∈ A . A sequence un ∈ A is said to converge weakly to u ∈ A (denoted un u ) if un , u → u , u as n → ∞ for all u ∈ A. A set G ⊂ A is said to be open in the weak topology if for every u ∈ G there are ﬁnite collections of ui ∈ A and εi > 0, i = 1, . . . , n, such that {v ∈ A : |ui , v − u| < εi , i = 1, . . . , n} ⊂ G. Weakly closed sets are complements of weakly open sets. For every F, E ⊂ A, a mapping ϕ : F → E is continuous in the weak topology (weakly continuous) if for every v = ϕ(u) with u ∈ F , and every weakly open set G v, there is a weakly open set D u such that ϕ(w) ∈ G ∩ E for all w ∈ F ∩ D. A mapping ϕ : F → E is sequentially weakly continuous if for any sequence F un u ∈ F , we have ϕ(un ) ϕ(u). A Banach space A is weakly complete if for any sequence un ∈ A, n ≥ 1, with the property u , um − un → 0 as m, n → ∞ for all u ∈ A , there is u ∈ A such that un u. Reﬂexivity. Let A be a Banach space. For every u ∈ A, the mapping κu : A → R deﬁned by κu (u ) = u , u is continuous and so can be regarded as an element of A . Thus we get the so-called natural isometry A u → κu ∈ A . The space A is reﬂexive if the natural isometry takes A onto A , i.e., for every u ∈ A there exists a unique u ∈ A such that u = κu . In this case we can identify A and A . The Eberlein-Šmulian theorem ([131, Ch. 5. 4]) shows that a reﬂexive Banach space is weakly complete: Theorem 1.1.3. A Banach space is weakly complete if and only if it is reﬂexive.

1.1. Functional analysis

5

Weak compactness. Let A be a Banach space. A set K ⊂ A is weakly compact if any family {Gα } of weakly open sets such that K ⊂ α Gα contains a ﬁnite

subfamily Gαi , i = 1, . . . , n, such that K ⊂ i Gαi . A set K is sequentially weakly compact if any sequence {un } ⊂ K contains a subsequence {unk } which converges weakly to some element u ∈ K. In Banach spaces bounded sets usually have some weak compactness properties. The following theorem ([67, Ch. 4.3, Thm. 4]) is a simplest result of this sort. Theorem 1.1.4. Let a Banach space A be separable, i.e. A contains a dense countable subset. Then every bounded sequence {un } ⊂ A contains a subsequence {unk } which converges weakly to some element u ∈ A . In the nonseparable case we have the following Alaoglu theorem [24, Ch. V.4, Thm. 2]. Theorem 1.1.5. Let A be a reﬂexive Banach space. Then every closed ball in A is weakly compact. The connection between weak compactness and sequential weak compactness is established by the second Eberlein-Šmulian theorem ([24, Ch. V.6, Thm. 1]): Theorem 1.1.6. Let A be a Banach space and K ⊂ A. Then every sequence un ∈ K contains a subsequence unk u ∈ A if and only the weak closure of K is weakly compact. On the other hand, in view of [24, Ch. V.3, Thm. 13], every convex closed set of a Banach space is weakly closed. This result along with Theorems 1.1.5 and 1.1.6 leads to Theorem 1.1.7. Every closed ball in a reﬂexive Banach space A is sequentially weakly compact. In particular, every bounded sequence {un } ⊂ A contains a subsequence {unk } which converges weakly to some element u ∈ A. Metrizability of the weak topology. Let A be a Banach space and K be a closed ball in A. The weak topology on K (induced weak topology) is said to be metrizable if there is a symmetric distance function ρ : K × K → R with the properties • ρ(u, v) ≥ 0, ρ(u, v) = 0 if and only if u = v, • ρ(u, w) ≤ ρ(u, v) + ρ(v, w), • a set E ⊂ K is the intersection K ∩ G, where G is weakly open, if and only if for every u ∈ E, the set E contains a ball {v ∈ K : ρ(u, v) < }, ε > 0. The following theorem ([24, Ch. V.5, Thm. 2]) connects metrizability and separability. Theorem 1.1.8. If A is a Banach space and A is separable, then the weak topology on every closed ball of A is metrizable.

6

Chapter 1. Preliminaries As a consequence we get

Lemma 1.1.9. Let A be a Banach space such that A is separable and let K be a closed ball in A. If a mapping ϕ : K → K is sequentially weakly continuous, then it is weakly continuous. Proof. Let ϕ : K → K be sequentially weakly continuous. Assume, contrary to our claim, that it is not weakly continuous. Then there is u ∈ K and a weakly open set V ϕ(u) with the property that every weakly open set G u contains w ∈ G ∩ K such that ϕ(w) does not belong to V . Since V is weakly open and ϕ(u) is in V , there are ui ∈ A and εi > 0, i = 1, . . . , m, such that {v : |ui , v − ϕ(u)| < εi , i = 1, . . . , m} ⊂ V, hence

max |ui , ϕ(w) − ϕ(u)| ≥ min εi =: δ > 0. i

Since the weak topology on K is metrizable, there are weakly open sets Gn such that Gn ∩ K = {v : ρ(u, v) < 1/n}. Hence there are wn ∈ K, n ≥ 1, such that ρ(wn , u) < 1/n and

max |ui , ϕ(wn ) − ϕ(u)| ≥ δ > 0.

1≤i≤m

(1.1.1)

Let us prove that wn u. Choose u ∈ A and ε > 0. Then the set Uε = {v : |ui , v − u| < ε} is weakly open and contains u. Hence Uε ∩ K contains some ball {v ∈ K : ρ(v, u) < r}. It follows that for n ≥ 1/r, the elements wn belong to Uε , which leads to lim supn→∞ |ui , wn −u| ≤ ε. Since ε is arbitrary, we conclude that ui , wn − u → 0 and hence wn u. By the sequential continuity of ϕ it follows that ϕ(wn ) ϕ(u), which contradicts (1.1.1).

1.1.2 Interpolation of Banach spaces In this section we recall some results from interpolation theory (see [16] for the proofs). We only consider the so-called real interpolation method, which is diﬀerent from the widely used complex interpolation method (see [16], [129]). Let A0 and A1 be Banach spaces, contained in some ambient normed space. For t > 0 introduce nonnegative functions K : A0 + A1 → R and J : A0 ∩ A1 → R deﬁned by K(t, u, A0 , A1 ) =

inf

u=u0 +u1 ui ∈Ai

(u0 A0 + tu1 A1 ),

J(t, u, A0 , A1 ) = max{uA0 , tuA1 }. The function K is positive, increasing and concave in t. For each s ∈ (0, 1), 1 < r < ∞, the K-interpolation space [A0 , A1 ]s,r,K consists of all elements u ∈ A0 +A1

1.1. Functional analysis

7

having the ﬁnite norm u[A0 ,A1 ]s,r,K =

∞

t−1−sr K(t, u, A0 , A1 )r dt

1/r .

(1.1.2)

0

On the other hand, the J-interpolation space [A0 , A1 ]s,r,J consists of all elements u ∈ A0 + A1 which admit a representation ∞ v(t) (1.1.3) u= dt, v(t) ∈ A1 ∩ A0 for t ∈ (0, ∞), t 0 where v is Bochner integrable, and have the ﬁnite norm u[A0 ,A1 ]s,r,J = inf

v(t)

∞

t

−1−sr

1/r r

J(t, v(t), A0 , A1 ) dt

< ∞,

(1.1.4)

0

where the inﬁmum is taken over the set of all v(t) satisfying (1.1.3). The ﬁrst main result of interpolation theory states that for all s ∈ (0, 1) and r ∈ (1, ∞) the Banach spaces [A0 , A1 ]s,r,K and [A0 , A1 ]s,r,J are isomorphic topologically and algebraically. Hence the two norms introduced are equivalent, and we can omit the indices J and K. The following properties of interpolation spaces will be used throughout this book. The ﬁrst concerns the interpolation of subspaces and directly follows from the deﬁnitions. Lemma 1.1.10. If A˜i are closed subspaces of Ai for i = 0, 1, then [A˜0 , A˜1 ]s,r ⊂ [A0 , A1 ]s,r and u[A0 ,A1 ]s,r ≤ u[A˜0 ,A˜1 ]s,r . The second lemma, which is a particular case of [16, Thm. 3.4.2], establishes the density properties of A1 ∩ A0 . Lemma 1.1.11. Let 0 < s < 1 and 1 < r < ∞. Then the linear space A1 ∩ A0 is dense in [A0 , A1 ]s,r . In particular, if A1 ⊂ A0 , then A1 is dense in [A0 , A1 ]s,r (even if A1 is not dense in A0 ). The following representation for the interpolation of dual spaces ([16, Thm. 3.7.1]) is important. Lemma 1.1.12. Let Ai , i = 0, 1, be Banach spaces such that Ai are linear subspaces of some normed space A and A1 ∩ A0 is dense in A0 and A1 in the topology of A. Then the Banach spaces [(A0 ) , (A1 ) ]s,r and ([A0 , A1 ]s,r ) are isomorphic topologically and algebraically. Hence the spaces can be identiﬁed with equivalent norms. It follows from this and Lemma 1.1.11 that if A1 ⊂ A0 is dense in A0 , then, since A0 ⊂ A1 , the Banach space A0 is dense in ([A0 , A1 ]s,r ) = [(A0 ) , (A1 ) ]s,r for 0 < s < 1 < r < ∞. The following lemma is a central result of interpolation theory.

8

Chapter 1. Preliminaries

Lemma 1.1.13. Let Ai , Bi , i = 0, 1, be Banach spaces, such that Ai are linear subspaces of some Banach space A, and Bi are linear subspaces of some normed space B, and let T : Ai → Bi be a bounded linear operator. Then, for all s ∈ (0, 1) and r ∈ (1, ∞), the operator T : [A0 , A1 ]s,r → [B0 , B1 ]s,r is bounded and T L([A0 ,A1 ]s,r ,[B0 ,B1 ]s,r ) ≤ c max T L(A0 ,B0 ) , T L(A1 ,B1 ) .

1.1.3 Invertibility of linear operators Lemma 1.1.14. Let V be a reﬂexive Banach space, V its dual, and A : V → V a linear operator satisfying, for some σ > 0, A[ϕ], ϕ ≥ σϕ2V

for all ϕ ∈ V.

Then A has an inverse A−1 : V → V , and A−1 [ψ]V ≤ σ −1 ψV

for all ψ ∈ V .

(1.1.5)

Proof. Clearly, there is an inverse A−1 : Im A → V . Moreover, for ϕ = A−1 [ψ], A−1 [ψ]V ψV ≥ ψ, A−1 [ψ] = A[ϕ], ϕ ≥ σϕ2V = σA−1 [ψ]2V , which gives (1.1.5). It remains to prove that Im A = V . Since A−1 is bounded, Im A is a closed subspace of V . If Im A = V then by the Hahn-Banach theorem 1.1.2, there is an element 0 = ω ∈ V such that ψ, ω = 0 for all ψ ∈ Im A. This gives A[ω], ω = 0, a contradiction.

1.1.4 Fixed point theorems The following three famous theorems on solvability of nonlinear functional equations (see [68] for instance) are exploited in this book. The ﬁrst is the Schauder theorem: Theorem 1.1.15. Let K be a convex, bounded, closed subset of a Banach space X, and Ξ : K → K a continuous mapping such that Ξ(K) is a relatively compact subset of X. Then there is a point x ∈ K such that Ξ(x) = x. The continuity and compactness conditions can be replaced by the following hypothesis: The mapping Ξ takes each weakly convergent sequence {xn } ⊂ K to the strongly convergent sequence {Ξ(xn )} ⊂ K. The second result is the more general Leray-Schauder theorem: Theorem 1.1.16. Let G be an open bounded subset of a Banach space X, and let Ξ(τ, ·) : cl G → X, τ ∈ [0, 1], be a one-parameter family of mappings that satisﬁes:

1.2. Function spaces

9

• The mapping Ξ : [0, 1] × cl G → X is continuous. • For τ ∈ [0, 1] and any bounded set A ⊂ G, the image Ξ(τ, A) is a relatively compact subset of X. • Ξ(τ, x) = x for all τ ∈ [0, 1] and x ∈ ∂G, i.e., the mapping Ξ(τ, ·) has no ﬁxed points at the boundary of G. • The mapping Ξ(0, ·) has a unique ﬁxed point x0 ∈ G. Moreover, the mapping I − Ξ(0, ·) is a homeomorphism of some neighborhood of x0 , where I stands for the identity mapping. Then the mapping Ξ(1, ·) has a ﬁxed point x1 ∈ G, that is, x1 = Ξ(1, x1 ). In many applications it is useful to replace the Schauder ﬁxed point theorem by the following Tikhonov ﬁxed point theorem (see [24, Ch. V.10, Thm. 5]): Theorem 1.1.17. Let A be a locally convex linear topological space and K be a convex compact set in A. Assume furthermore that a mapping ϕ : K → K is continuous in the topology of A. Then there exists w ∈ K such that w = ϕ(w). For our purposes the Tikhonov theorem is applied in the following less abstract form. Theorem 1.1.18. Let A be a reﬂexive Banach space such that A is separable. Let K ⊂ A be a closed ball and ϕ : K → K be a sequentially continuous mapping, i.e., ϕ(un ) ϕ(u) for any K un u ∈ K. Then there exists w ∈ K such that w = ϕ(w). Proof. The space A equipped with the weak topology (the system of weakly open sets) becomes a locally convex linear topological space. In view of Theorem 1.1.5, the closed ball K is a compact convex subset of this space. It remains to note that, by Lemma 1.1.9, every sequentially continuous mapping ϕ : K → K is weakly continuous, and apply Theorem 1.1.17.

1.2 Function spaces 1.2.1 Hölder spaces Deﬁnition 1.2.1. Let Ω be a domain in Rd . We denote by C(Ω) the Banach space of all bounded continuous functions u : Ω → R with the norm uC(Ω) = sup |u(x)|. x∈Ω

All such functions have unique continuous extensions to the closure cl Ω; the extensions will be denoted by the same letter, We say that a function u : Ω → R is compactly supported in Ω if supp u = cl {x ∈ Ω : u(x) = 0} Ω,

10

Chapter 1. Preliminaries

where supp u Ω means that supp u is a compact subset of the open set Ω. We denote by Cc (Ω) the linear space of all continuous functions u : Ω → R compactly supported in Ω. Deﬁnition 1.2.2. For every integer m ≥ 0, we denote by C m (Ω) the Banach space which consists of all functions u on Ω such that ∂ α u ∈ C(Ω) for all |α| ≤ m and

uC m (Ω) = max ∂ α uC(Ω) < ∞. |α|≤m

For every integer m ≥ 0 and real β ∈ (0, 1), we denote by C m+β (Ω) the Banach space which consists of all functions u ∈ C m (Ω) with the ﬁnite norm uC m+β (Ω) = uC m (Ω) +

|α|=m

sup x,y∈Ω,x =y

|∂ α u(x) − ∂ α u(y)| . |x − y|β

We will identify C 0 (Ω) with C(Ω) and C m+0 (Ω) with C m (Ω). We denote by C (Ω) the linear space of functions u : Ω → R such that all derivatives of u belong to the class C(Ω). The symbol C0∞ (Ω) denotes the subspace of C ∞ (Ω) which consists of compactly supported functions. ∞

Domains Deﬁnition 1.2.3. Let Ω ⊂ Rd be a bounded domain, m ≥ 0 be an integer and β ∈ [0, 1). We say that ∂Ω (or equivalently Ω) belongs to the class C m+β , denoted ∂Ω ∈ C m+β , if it has the following property. For every point x0 ∈ ∂Ω, there exist a neighborhood Ux0 of x0 in Rd , a Cartesian coordinate system y = (y1 , . . . , yd−1 , yd ) := (y, yd ), centered at x0 , and a function η ∈ C m+β (Bd−1 ), Bd−1 = {|y| < 1}, such that in the new coordinates Ux0 ∩ ∂Ω = {y : yd = η(y), y ∈ Bd−1 }.

(1.2.1)

In other words, ∂Ω is the graph of a C m+β function in a neighborhood of each of its points. Let ∂Ω ∈ C m+β . Assume that an integer l ≥ 0 and σ ∈ [0, 1) satisfy the inequality l + σ ≤ m + β. We say that a function u : ∂Ω → R belongs to the class C l+σ (∂Ω) if for every x0 ∈ ∂Ω there exists ϕ ∈ C l+σ (Bd−1 ) such that u|Ux0 ∩∂Ω = ϕ(y). Deﬁnition 1.2.4. Let Ω ⊂ Rd be a bounded domain. We say that ∂Ω (or equivalently Ω) is Lipschitz if it admits a representation (1.2.1) with a Lipschitz function η in a neighborhood of every point x0 ∈ ∂Ω.

1.2. Function spaces

11

Hölder spaces of Banach space valued functions Deﬁnition 1.2.5. Let X be a Banach space and T ∈ (0, ∞). We will denote by C(0, T ; X) the Banach space of all bounded continuous functions u : [0, T ] → X with the norm uC(0,T ;X) = sup u(t)X . t∈[0,T ]

We say that the mapping u : [0, T ] → X is diﬀerentiable at t ∈ (0, T ) if there is ∂t u(t) ∈ X such that u(s) − u(t) = 0. ∂ lim u(t) − t s→t s − t X Deﬁnition 1.2.6. Let X be a Banach space, T ∈ [0, ∞), and m ≥ 0 be an integer. We denote by C m (0, T ; X) the linear space of all mappings u : [0, T ] → X such that for every integer k ∈ [0, m], the mapping ∂tk u : (0, T ) → X is continuous and has a continuous extension onto [0, T ]. Endowed with the norm uC m (0,T ;X) = max ∂tk uC(0,T ;X) , 0≤k≤m

C m (0, T ; X) becomes a Banach space. For β ∈ [0, 1), we denote by C m+β (0, T ; X) the Banach space which consists of all functions u ∈ C m (0, T ; X) with the ﬁnite norm uC m+β (0,T ;X) = uC m (0,T ;X) +

∂tm u(t) − ∂sm u(s)X . |t − s|β s,t∈[0,T ], t =s sup

Arzelà-Ascoli theorem Theorem 1.2.7. Let Ω ⊂ Rd be a bounded domain. Let un ∈ C(Ω), n ≥ 1, be a bounded and equicontinuous sequence, i.e., for every ε > 0 there is δ > 0 such that |un (x) − un (y)| ≤ ε

for all x, y ∈ Ω with |x − y| ≤ δ, and all n ≥ 1.

Then there exists a subsequence {unk } ⊂ {un } and u ∈ C(Ω) such that unk → u in C(Ω) as k → ∞. For Banach space valued functions, the Arzelà-Ascoli theorem reads: Theorem 1.2.8. Let X be a Banach space, K ⊂ X be compact and T ∈ (0, ∞). Let un : [0, T ] → K, n ≥ 1, be an equicontinuous sequence, i.e., for every ε > 0 there is δ > 0 such that un (t) − un (s)X ≤ ε

for all s, t ∈ [0, T ] with |s − t| ≤ δ and all n ≥ 1.

Then there exists a subsequence {unk } ⊂ {un } and u ∈ C(0, T ; X) such that unk → u in C(0, T ; X) as k → ∞.

12

Chapter 1. Preliminaries

Corollary 1.2.9. Let X and Y be Banach spaces and m + β > 0. Assume that the embedding Y → X is compact. Then every bounded sequence un ∈ C m+β (0, T ; Y ), n ≥ 1, contains a subsequence which converges in C(0, T ; X). Lemma 1.2.10. Let Ω be a bounded domain in Rd with ∂Ω ∈ C k . Assume that nonnegative integers l, m and real μ, ν ∈ [0, 1) satisfy l + σ < m + β ≤ k. Then for every u ∈ C m+β (Ω), λ uC l+σ (Ω) ≤ cu1−λ C(Ω) uC m+β (Ω) ,

λ = (l + σ)/(m + β),

where c is independent of u. Corollary 1.2.11. Under the assumptions of the previous lemma, the embedding C m+β (Ω) → C l+σ (Ω) is compact.

1.2.2 Measure and integral General theory. Let X be a nonempty set. A collection A of subsets of X is said to be a σ-algebra if the following conditions are fulﬁlled: • ∅ ∈ A; if E ∈ A then X \ E ∈ A.

• If En ∈ A, n ≥ 1, then n En ∈ A. Notice that if En ∈ A, n ≥ 1, then n En ∈ A. Example 1.2.12. The most important example of a σ-algebra is the algebra of Borel sets which is deﬁned as follows. Let Ω be an arbitrary open subset of Rd (most often, Ω = Rd ). The Borel σ-algebra or the σ-algebra of Borel sets B(Ω) is the minimal σ-algebra in Ω containing all open subsets of Ω. Remark 1.2.13. For any countable collections of open sets Gi ⊂ Rd and closed sets Fi ⊂ Rd , the sets Gi ∩ Ω, Gi ∩ Ω, Fi ∩ Ω, Fi ∩ Ω, i

i

i

i

belong to B(Ω). There is not any constructive way to recognize Borel sets among other subsets of Ω. Measures. A map μ : A → [0, ∞] is called a measure (or a nonnegative measure) if μ(∅) = 0, μ En = μ(En ) n

n

for every countable collection of pairwise disjoint sets En ∈ A. A measure μ is said to be σ-ﬁnite if X can be represented as a countable union of ﬁnite measure sets; μ is said to be ﬁnite if μ(X) < ∞. A measure μ is a probability measure if μ(X) = 1. The triple (X, A, μ) is called a measure space.

1.2. Function spaces

13

Integral. A function u : X → R is said to be measurable with respect to A if u−1 (E) ∈ A for every Borel set E ⊂ R. A simple function is a measurable function u : X → R whose range consists of ﬁnitely many points, i.e., uk χk (x), uk ∈ R, χk (x) = 1 for x ∈ Ek , χk (x) = 1 otherwise, u(x) = k

and {Ek } ⊂ A is a ﬁnite collection of pairwise disjoint measurable sets covering X. Notice that for any measurable function u : X → R there is an increasing sequence of simple functions u(n) such that u(n) (x) ≤ u(n+1) (x),

lim u(n) (x) = u(x) in X.

n→∞

(1.2.2)

A simple function u ≥ 0 is integrable if u dμ := uk μ(Ek ) < ∞. X

k

Here, we adopt the standard convention 0 · ∞ = 0 for μ(Ek ) = ∞. Deﬁnition 1.2.14. A nonnegative measurable function u : X → R is integrable (or μ-integrable) if there is a sequence of nonnegative simple functions u(n) , n ≥ 1, with the following properties: u(n) (x) u(x) as n → ∞ in X,

the limit

u(n) dμ =:

lim

n→∞

u dμ

X

X

exists. An arbitrary measurable function u : X → R is deﬁned to be integrable if there are nonnegative integrable functions u± such that u = u+ − u− . The following three theorems play an important role in integration theory. The ﬁrst is the famous Lebesgue dominated convergence theorem. Theorem 1.2.15. Let (X, A, μ) be a measure space, and let un : X → [−∞, ∞] be a sequence of integrable functions such that the limit lim un (x) = u(x)

n→∞

exists for μ-a.e. x ∈ X, i.e., for all x except a set of zero measure. If there is an integrable function v such that |un (x)| ≤ v(x) then u is integrable and

for μ-a.e. x ∈ X,

lim

n→∞

un dμ = X

u dμ. X

14

Chapter 1. Preliminaries The second is the monotone convergence Fatou theorem [42]:

Theorem 1.2.16. Let (X, A, μ) be a measure space, and let un : X → [−∞, ∞] be a sequence of integrable functions such that un dμ < ∞. un ≤ un+1 a.e. in X and sup n

X

Then u(x) = limn→∞ un (x) is integrable and un dμ = u dμ. lim n→∞

X

X

The third is the Levi theorem: Theorem 1.2.17. Let (X, A, μ) be a measure space, and let un : X → [0, ∞] be a sequence of integrable nonnegative functions such that the limit lim un (x) = u(x)

n→∞

exists for μ-a.e. x ∈ X. Then u is integrable and u dμ ≤ lim inf un dμ. X

n→∞

X

Radon measures. Let Ω be an open subset of Rd , and let B(Ω) be the σ-algebra of Borel subsets of Ω. A function u : Ω → R is called a Borel function if it is measurable with respect to B(Ω). A measure μ : B(Ω) → [0, ∞] is called a Borel measure (or a nonnegative Borel measure). Deﬁnition 1.2.18. A Borel measure μ : B(Ω) → [0, ∞] is a Radon measure if μ(K) < ∞ for every compact set K ⊂ Ω. Since Rd is locally compact and every open set in Rd is a union of a countable collection of its compact subsets, every Radon measure in Ω has the following regularity properties: • For every open set D ⊂ Ω, μ(D) = sup {μ(K) : K ⊂ D, K compact}. • For every Borel E ⊂ Ω, μ(E) = inf {μ(A) : E ⊂ A, A open}. Remark 1.2.19. Any bounded Borel function is integrable over a Borel set E of ﬁnite measure. If μ(Ω) < ∞, then any bounded Borel function is integrable over Ω.

1.2. Function spaces

15

Remark 1.2.20. If E ⊂ Ω is a Borel set and u : Ω → R is a Borel function, then χE (x) dμ, χE (x)u(x) dμ, u(x) dx := μ(E) := E

Ω

Ω

provided that the integrals on the right hand sides exist. Here, χE is the characteristic function of the set E, i.e., χE (1 − χE ) = 0 a.e. Signed measures. Let Ω be an open set in Rd . A signed Radon measure on Ω is a function μ : B(Ω) → [−∞, ∞] such that • μ(∅) = 0 and μ takes at most one of the values ±∞, so that we have either μ : B(Ω) → (−∞, ∞] or μ : B(Ω) → [−∞, ∞). • For every countable collection En , n ≥ 1, of pairwise disjoint sets En ∈ B(Ω) we have En = μ(En ). μ n

n

• |μ(K)| < ∞ for every compact K ⊂ Ω. Notice that the series on the right hand side of the last equality converges absolutely if the left hand side is ﬁnite. The following Jordan decomposition theorem (see [42]) shows that the theory of signed measures is the same as the theory of nonnegative measures. Theorem 1.2.21. Let μ : B(Ω) → [−∞, ∞] be a signed Borel measure. Then there is a unique pair of nonnegative mutually singular Borel measures μ± , one of which is ﬁnite, such that μ = μ+ − μ− . Recall that measures μ± are mutually singular if there exists a Borel set F ⊂ Ω such that for any Borel E ⊂ Ω, μ+ (E) = μ+ (F ∩ E), μ− (E) = μ− (Ω \ F ) ∩ E . The nonnegative Radon measure |μ| := μ+ + μ− is called the absolute variation of the signed measure μ. Riesz theorem (see [42]). Let Ω be an open subset of Rd . Recall that Cc (Ω) denotes the linear space of continuous functions u : Ω → R compactly supported in Ω. Deﬁnition 1.2.22. A linear functional l : Cc (Ω) → R is locally bounded if for every compact set K ⊂ Ω, there exists a constant CK such that |l(u)| ≤ CK uC(Ω) for all continuous functions u with supp u ⊂ K.

16

Chapter 1. Preliminaries

Theorem 1.2.23. Let l : Cc (Ω) → R be a locally bounded linear functional. Then there exists a unique signed Radon measure μ : B(Ω) → [−∞, ∞] such that l(u) = u dμ for every u ∈ Cc (Ω). (1.2.3) Ω

Moreover if l(u) ≥ 0 for every nonnegative u ∈ Cc (Ω), then μ is a nonnegative Radon measure. Banach space of Radon measures. Denote by C0 (Ω) the closed subspace of C(Ω) which consists of all continuous functions u ∈ C(Ω) vanishing at ∂Ω such that u(x) → 0 as |x| → ∞ (if Ω is unbounded). It is clear that Cc (Ω) ⊂ C0 (Ω), hence any continuous functional l : C0 (Ω) → R is locally bounded in Cc (Ω). In particular, l meets all requirements of Theorem 1.2.23 and admits representation (1.2.3) with a Radon measure μ. The following result is a consequence of the Riesz theorem. Theorem 1.2.24. Let l : C0 (Ω) → R be a bounded linear functional. Then there exists a unique signed Radon measure μ : B(Ω) → (−∞, ∞) such that l(u) = u dμ for every u ∈ C0 (Ω). (1.2.4) Ω

Moreover, |μ|(Ω) = μ+ (Ω) + μ− (Ω) = lC0 (Ω) :=

sup {u∈C0 (Ω):uC(Ω) =1}

|l(u)|,

where μ± is the Jordan decomposition of μ deﬁned by Theorem 1.2.21. Conversely, for any signed Radon measure μ in Ω with |μ(Ω)| < ∞, relation (1.2.4) deﬁnes a continuous functional l ∈ C0 (Ω) such that l = |μ|(Ω). The totality of all signed Radon measures μ with μM(Ω) := |μ|(Ω) < ∞ forms a Banach space M(Ω) which is isomorphic to the dual space C0 (Ω) .

1.2.3 Lebesgue measure in Rd . Lebesgue spaces There is a unique Radon measure μL in Rd with the property μL (Q) =

d

(bi − ai ) for any parallelepiped Q =

i=1

d

(ai , bi ).

i=1

The following construction is due to Lebesgue. We say that E ⊂ Rd has Lebesgue measure zero if for any > 0, there is a Borel set B such that E⊂B

and

μL (B) < .

1.2. Function spaces

17

Next, we say that a set A ⊂ Rd is Lebesgue measurable if A = B ∪ E, where B is a Borel set and E is a zero measure set. In this case we set m(A) = μL (B). Denote by Ld the totality of all Lebesgue measurable sets. It is known that Ld is a σ-algebra in Rd , and m : Ld → R is a nonnegative measure in Rd . The measure m is called the Lebesgue measure in Rd or the Lebesgue extension of the measure μL . The integral with respect to the Lebesgue measure is named the Lebesgue integral. Remark 1.2.25. Obviously this procedure may be applied to all Borel measures μ on Rd . Thus we can get the Lebesgue extension denoted by μ of any Borel measure μ, but the classes of μ-measurable sets and μ-integrable functions strongly depend on μ. Further, when no measure is explicitly indicated, we say simply measure, measurable function, integrable function for Lebesgue measure and Lebesgue measurable or integrable function. We also use the standard notation dx := dm and write meas A for m(A). Let Ω ⊂ Rd be a measurable set. We say that two measurable functions u and v are equivalent in Ω if u(x) = v(x) for a.e. x ∈ Ω (i.e. for all x ∈ Ω, possibly except a set of zero measure). From now on we identify a measurable function with its equivalence class. Deﬁnition 1.2.26. Let Ω ⊂ Rd be a measurable set and 1 ≤ p ≤ ∞. Then we deﬁne the Lebesgue space Lp (Ω) := {u : Ω → R : u is measurable and uLp (Ω) < ∞}, where for 1 ≤ p < ∞, uLp (Ω) :=

1/p

|u(x)|p dx

,

Ω

and if p = ∞, then uL∞ (Ω) := ess sup |u| = inf{k ∈ R : |u(x)| ≤ k for a.e. x ∈ Ω}. Ω

It follows from the Minkowski inequality u1 + · · · + un Lp (Ω) ≤ u1 Lp (Ω) + · · · + un Lp (Ω) that Lp (Ω) is a normed space. Moreover, it is a Banach space. We say that u ∈ Lploc (Ω) if u ∈ Lp (K) for every compact set K ⊂ Ω. Note that Lploc Ω) is just a linear space.

18

Chapter 1. Preliminaries

Hölder inequality. Duality. The main tool for studying Lebesgue spaces is the Hölder inequality n n ui 1 ≤ ui p L (Ω)

L

i=1

i (Ω)

i=1

which holds, provided that the right hand side is ﬁnite, for all exponents 1 1 + ··· + = 1. p1 pn

pi ∈ [1, ∞] with If meas Ω is ﬁnite and

1 1 1 ≥ + ··· + , p p1 pn

p, pi ∈ [1, ∞],

then there is a constant c, depending on Ω and pi , such that for all ui ∈ Lpi (Ω), n ui

≤c

Lp (Ω)

1=1

n

ui Lpi (Ω) .

(1.2.5)

1=1

The Hölder inequality leads to the following description of the dual spaces. Theorem 1.2.27. Let Ω ⊂ Rd be a measurable set and 1 ≤ p < ∞. Then for any continuous linear functional l ∈ Lp (Ω) there is a unique function v ∈ Lq (Ω), with q = p/(p − 1) for p > 1, and q = ∞ for p = 1, such that for all u ∈ Lp (Ω), l(u) = u(x)v(x) dx. Ω

Moreover lLp (Ω) :=

sup {u∈Lp (Ω):uLp (Ω) =1}

|l(u)| = vLq (Ω) .

In other words, the dual space Lp (Ω) is isomorphic to Lq (Ω). Weak convergence in Lebesgue spaces. Let Ω be a measurable subset of Rd . Assume that 1 ≤ q < ∞, q = q/(q − 1), i.e., 1 1 + = 1. q q Recall that a sequence (fk )k≥1 ⊂ Lq (Ω) converges weakly to f ∈ Lq (Ω) in Lq (Ω), written fk f weakly in Lq (Ω), provided that

fk g dx → Ω

f g dx as k → ∞ Ω

(1.2.6)

1.2. Function spaces

19

for each g ∈ Lq (E). A sequence (fk )k≥1 ⊂ Lq (Ω) is weakly convergent if the limit lim fk g dx (1.2.7) k→∞

Ω

exists for all g ∈ L (Ω). If 1 < q < ∞, then each weakly convergent sequence converges weakly to some element f ∈ Lq (Ω). This is not true for q = 1. The theory of weak convergence in L∞ (Ω) is complicated. For this reason we replace it with the weak convergence. Recall that a sequence (fk )k≥1 ⊂ L∞ (Ω) converges weakly to f ∈ L∞ (Ω) in L∞ (Ω) provided that fk g dx → f g dx as k → ∞ q

Ω

Ω

for all g ∈ L (Ω). Now we recall some boundedness properties of weakly convergent sequences. 1

Lemma 1.2.28. Assume that fk f weakly in Lq (Ω). Then (fk )k≥1 is bounded in Lq (Ω) and f Lq (Ω) ≤ lim inf k→∞ fk Lq (Ω) . Moreover, if 1 < q < ∞ and f Lq (Ω) = limk→∞ fk Lq (Ω) then fk → f converges strongly (i.e., in norm) in Lq (Ω) as k → ∞. For 1 < q < ∞, the Banach space Lq (Ω) is reﬂexive, hence bounded sequences are relatively weakly compact: Lemma 1.2.29. Assume 1 < q < ∞ and the sequence (fk )k≥1 is bounded in Lq (Ω). Then there is a subsequence (fkj )j≥1 and an element f ∈ Lq (Ω) such fkj f weakly in Lq (Ω) as j → ∞. For q = 1 the boundedness is necessary but not suﬃcient for the weak compactness. In order to formulate the corresponding result it is convenient to give the following deﬁnition. Deﬁnition 1.2.30. We say that a set M ⊂ L1 (Ω) is equi-integrable if for any ε > 0 there exists δ(ε) > 0, depending only on M and ε, such that |f | dx < ε A

for each A ⊂ Ω with meas A < δ(ε) and for all f ∈ M. This deﬁnition is equivalent to the following property: M ⊂ L1 (Ω) is equiintegrable if and only if there is a continuous function Φ : [0, ∞) → [0, ∞) with t−1 Φ(t) → ∞ as t → ∞

and

Φ(|f |) dx < ∞.

sup f ∈M

Ω

Lemma 1.2.31. Assume the sequence (fk )k≥1 is bounded in L1 (Ω) and equi-integrable. Then there exists a subsequence fkj and a function f ∈ L1 (Ω) with fkj f weakly in L1 (Ω).

20

Chapter 1. Preliminaries

1.3 Compact sets in Lp spaces Throughout this book we use the following simple compactness criterion in Lp spaces (see [42]). Proposition 1.3.1. Let Ω ⊂ Rd be a bounded Lebesgue measurable set. A set K ⊂ Lp (Ω), 1 ≤ p < ∞, is relatively compact in Lp (Ω) if and only if the following two conditions are satisﬁed: (i) Any sequence of functions fn ∈ K contains a subsequence fnm which converges in measure, i.e., lim

m,k→∞

meas {x ∈ Ω : |fnm (x) − fnk (x)| ≥ } = 0

(ii) For any > 0 there is δ > 0 such that |f |p dx < for all f ∈ K and

A⊂Ω

for all > 0.

with meas A < δ.

A

The condition (ii) is equivalent to the following: There is a continuous function Φ : R+ → R+ with the properties Φ(|f |p ) dx < ∞. lim t−1 Φ(t) = ∞, sup t→∞

f ∈K

Ω

In particular, we have the following condition. Lemma 1.3.2. Let Ω ⊂ Rd be a bounded Lebesgue measurable set. If a sequence fn ∈ Lr , 1 < r ≤ ∞, converges in measure and is bounded in Lr (Ω), then fn converges strongly in any Ls (Ω) with 1 ≤ s < r.

1.3.1 Functions of bounded variation Let I be an arbitrary interval (open, closed, half-open, or inﬁnite). The total variation of a function F : I → R over I is deﬁned by I

F := sup

q

|F (tk+1 ) − F (tk )|,

(1.3.1)

k=0

where the supremum is taken over all t0 < · · · < tq such that ti ∈ I. A function F is said to be of bounded variation on I if the total variation of F over I is ﬁnite. Every monotone bounded function is a function of bounded variation. Conversely, every function of bounded variation on I can be represented as a diﬀerence of two monotone bounded functions: F = F + − F − , F + (t) = F, F − = F + − F. (−∞,t]∩I

1.3. Compact sets in Lp spaces

21

In particular, every function F : I → R of bounded variation is continuous everywhere on I except on a countable set of jump points t at which the limits F (t + 0) = lim F (z), z t

F (t − 0) = lim F (z) z t

exist. Further we use the following simple lemma. Lemma 1.3.3. If F ∈ L1 (I) and f ∈ L1 (I) satisfy η (t)F (t) dt = − η(t)f (t) dt for all η ∈ C0∞ (I), I

I

then F can be changed on a set of zero measure in such a way that F ∈ C[0, T ], F ≤ |f (t)|dt, F C[0,T ] ≤ cF L1 (I) + f L1 (I) . I

I

The following theorem due to Helly (see [67]) is the most important result in the theory of bounded variation functions. Theorem 1.3.4. Let a sequence of functions Fn : [0, T ] → R, T < ∞, satisfy the condition Fn < ∞. sup n

[0,T ]

Then there is a subsequence, still denoted by Fn , and a function F of bounded variation such that F ≤ lim inf Fn . Fn (t) → F (t) as n → ∞ for all t ∈ [0, T ], n

[0,T ]

[0,T ]

Lebesgue-Stieltjes integral. Let I be an arbitrary interval and U ⊃ I be an open interval. Let F : U → R be a function of bounded variation. Then there exists a unique signed Borel measure μF on U with the properties μF [a, b] = F (b + 0) − F (a − 0), μF [a, b) = F (b − 0) − F (a − 0),

μF (a, b) = F (b − 0) − F (a + 0), μF (a, b] = F (b + 0) − F (a + 0).

The Borel measure μF is called the Lebesgue-Stieltjes measure associated with F . Every ﬁnite Borel measure μ on an open interval U can be regarded as the Lebesgue-Stieltjes measure associated with the function F (s) = μ (−∞, s] ∩ U ). If functions of bounded variation F, F˜ : U → R coincide everywhere except on a countable set, then μF = μF˜ . The integral with respect to the Borel measure μF is called the LebesgueStieltjes integral and is denoted by f (s) dF (s) := f (s) dμF for all Borel functions f. I

I

22

Chapter 1. Preliminaries

Example 1.3.5. Let g ∈ C[a, b] and suppose f : [a, b] → R admits the representation t f (t) = fac (t) + fj (t), fac (t) = f (s) ds, fj (t) = ck θ(t − ak ), 0

k

where θ(t) = 0 for t < 0 and

θ(t) = 1 for t ≥ 1,

f ∈ L (a, b) and {ak } ⊂ [a, b] is a ﬁnite or inﬁnite sequence. Then b f= |f (s)| ds + |ck | 1

a

[a,b]

and

k:ak >a

b

g(s) df (s) =

g(s)f (s) ds +

a

[a,b]

ck g(ak ).

(1.3.2)

k

Example 1.3.6. Let μ ∈ M(R) = C0 (R) be a nonnegative ﬁnite Radon measure on R. Then the distribution function f of the measure μ, (1.3.3)

f (s) = μ(−∞, s],

has bounded variation and μf = μ. The function f is nondecreasing and satisﬁes f (s) = lim f (s + h), h 0

lim f (s) = 0.

s→−∞

(1.3.4)

The most important fact in the theory of the Lebesgue-Stieltjes integral is the following integration by parts formula: Lemma 1.3.7. Let I be an arbitrary interval and U ⊃ I an open interval. Let f, g : U → R be functions of bounded variation, and let S denote the set of points s ∈ I at which both f and g are discontinuous. Then f dg + g df = μf g (I) + A(s), I

where

I

s∈S

f (s + 0) + f (s − 0) ]μg ({s}) + [g(s) − 12 g(s + 0) + g(s − 0) ]μf ({s}),

A(s) = [f (s) −

1 2

μf g [α, β] = f (β + 0)g(β + 0) − f (α − 0)g(α − 0), μf g (α, β) = f (β − 0)g(β − 0) − f (α + 0)g(α + 0), μf g [α, β) = f (β − 0)g(β − 0) − f (α − 0)g(α − 0), μf g (α, β] = f (β + 0)g(β + 0) − f (α + 0)g(α + 0), f (β − 0)g(β − 0) = lim f (s)g(s) s→∞

f (α + 0)g(α + 0) = lim f (s)g(s) s→−∞

In particular:

for β = ∞, for α = −∞.

1.3. Compact sets in Lp spaces

23

(i) If S is empty or if f (s) = 12 f (s+0)+f (s−0) and g(s) = 12 g(s+0)+g(s−0) for all s ∈ S, then f dg + I

g df = μf g (I). I

(ii) If f and g are continuous from the right at all points of S, then 1 f dg + g df = μf g (I) + μf ({s})μg ({s}). 4 I I s∈S

1.3.2 Lp spaces of Banach space valued functions Deﬁnition 1.3.8. Let Ω ⊂ Rd be a (Lebesgue) measurable set, and let Y be a Banach space. We say that a function s : Ω → Y is simple if s admits a representation s(x) =

N

ci χEi (x),

(1.3.5)

i=1

where the sets Ei ⊂ Ω are measurable and mutually disjoint, ci ∈ Y are distinct elements, and χEi is the characteristic function of the set Ei . Deﬁnition 1.3.9. (i) A function u : Ω → Y is strongly measurable (or simply measurable) if there is a sequence {sn } of simple measurable functions such that lim sn (x) − u(x)Y = 0 for a.e. x ∈ Ω. n→∞

(ii) A function u : Ω → Y is weakly measurable if for any u ∈ Y the function x ∈ Ω → u , u(x) ∈ R

is measurable.

(iii) A function u : Ω → Y is weakly measurable if for any v ∈ Y the function x ∈ Ω → u(x), v ∈ R is measurable. If a function u is strongly measurable, then the scalar function u(x)Y is measurable. It is diﬃcult to check whether a given function is strongly measurable or not. The relation between strong measurability and the standard notion of measurability is given by the following theorem ([24, Ch. III.6, Lemma 9]). Theorem 1.3.10. Let Ω ⊂ Rd be a measurable set and Y be a Banach space. A function u : Ω → Y is strongly measurable if and only if the following conditions hold: (I) There is a set E of zero Lebesgue measure such that the set u(Ω \ E) is separable, i.e., contained in the closure of some countable set. (II) u−1 (G) is measurable for every open set G ⊂ Y , or equivalently (II’) u−1 (G) is measurable for every Borel set G ⊂ Y .

24

Chapter 1. Preliminaries

Condition (I) is obviously fulﬁlled if Y is separable. The following theorem due to Pettis shows that the separability of Y guarantees the equivalence of strong and weak measurability (see [42, Thm. 2.104], [24, Ch. III.6, Thm. 11]). Theorem 1.3.11. Let Ω ⊂ Rd be a measurable set of ﬁnite Lebesgue measure and let Y be a Banach space. A function u : Ω → Y is strongly measurable if and only if it is weakly measurable and there is a set E ⊂ Ω of zero measure such that the set u(Ω \ E) ⊂ Y is separable. In particular, if Y is separable, then any weakly measurable function is strongly measurable. The Egoroﬀ and Lusin theorems remain true for strongly measurable functions: Theorem 1.3.12. Let Ω ⊂ Rd be a measurable set of ﬁnite Lebesgue measure and let Y be a Banach space. Assume that strongly measurable functions un converge a.e. in Ω to a measurable function u, i.e., for almost every x ∈ Ω, the sequence un (x) converges to u(x) strongly in Y . Then for every ε > 0 there is a measurable set E ⊂ Rd with meas E < ε such that un (x) − u(x)Y → 0 uniformly on Ω \ E. Theorem 1.3.13. Let Ω ⊂ Rd be a measurable set of ﬁnite Lebesgue measure, Y be a Banach space, and u : Ω → Y be a strongly measurable function. Then for every ε > 0 there is an open set G ⊂ Rd with meas G < ε such u is continuous on Ω \ G. Bochner integral. Let Ω ⊂ Rd be a Lebesgue measurable set. A simple measurable function s of the form (1.3.5) is integrable if ci = 0 whenever meas Ei = ∞. The Bochner integral of s with respect to the Lebesgue measure is deﬁned as follows: N N s(x) dx = ci meas Ei and s(x) dx = ci meas(Ei ∩ E) Ω

E

i=1

i=1

for any measurable set E ⊂ Ω. Deﬁnition 1.3.14. Let Ω ⊂ Rd be a measurable set. A strongly measurable function u : Ω → Y is Bochner integrable if there is a sequence {sn } of simple integrable functions such that lim sn (x) − u(x)Y = 0 for a.e. x ∈ Ω, lim sn (x) − u(x)Y dx = 0.

n→∞

n→∞

Ω

The Bochner integral of u over Ω is deﬁned to be the limit u(x) dx = lim sn (x) dx. Ω

n→∞

Ω

This limit exists and it is independent of the choice of the sequence sn convergent to u.

1.3. Compact sets in Lp spaces

25

The following important theorem due to Bochner shows that measurability and integrability in norm are equivalent to Bochner integrability ([42, Thm. 2.108]). Theorem 1.3.15. A strongly measurable function u is Bochner integrable over Ω if and only if the norm u(x)Y is Lebesgue integrable over Ω. Moreover, ≤ u dx u(x)Y dx. Ω

Ω

Y

Many general results concerning the properties of real-valued integrable functions remain true for Bochner integrals. In particular we have the following version of the Lebesgue dominated convergence theorem ([24, Ch. III.6, Cor. 16]). Theorem 1.3.16. Let Ω ⊂ Rd be a Lebesgue measurable set, Y be a Banach space, and suppose strongly measurable functions un : Ω → Y converge a.e. in Ω to a function u. Furthermore assume that there is a nonnegative function g ∈ L1 (Ω) such that un (x)Y ≤ g(x) a.e. in Ω. Then u is Bochner integrable and un (x) − u(x)Y dx = 0. lim n→∞

Ω

The next theorem due to Bochner justiﬁes the “diﬀerentiation” of the Bochner integral ([131, Ch. 5,5, Thm. 2]). Theorem 1.3.17. Let Ω ⊂ Rd be an open set, Y be a Banach space, and u : Ω → Y be a Bochner integrable function. Then for a → 0 and for a.e. x0 ∈ Ω, a−d u(x) dx → u(x0 ) strongly in Y. Qa (x0 )

Here Qa (x0 ) denotes the cube centered at x0 with side length a. Lp spaces. We are now in a position to deﬁne Lp spaces of functions which take values in Banach spaces. We distinguish between strongly and weakly measurable functions. Deﬁnition 1.3.18. Let Ω be a measurable subset of Rd and Lp (Ω; Y ) be the linear space of all strongly measurable functions u : Ω → Y with the ﬁnite norm 1/p uLp (Ω;Y ) = u(x)pY dx , Ω

with 1 ≤ p < ∞. If we identify all functions in the same equivalence class, then Lp (Ω; Y ) becomes a Banach space. L∞ (Ω; Y ) is the normed space of all (equivalence classes of) strongly measurable functions u : Ω → Y with the ﬁnite norm uL∞ (Ω;Y ) = ess sup u(x)Y := inf{c : u(x) < c for a.e. x ∈ Ω}. x∈Ω

26

Chapter 1. Preliminaries

Remark 1.3.19. It follows from Theorem 1.3.15 that all functions u ∈ Lp (Ω; Y ) are Bochner integrable. The following theorem describes the basic properties of the spaces Lp (Ω; Y ) ([42, Thm. 2.110]). Theorem 1.3.20. Let Y be a Banach space and Ω be a measurable subset of Rd . Then (i) Lp (Ω; Y ) is a Banach space for 1 ≤ p ≤ ∞; (ii) the family of all integrable simple functions is dense in Lp (Ω; Y ) for 1 ≤ p < ∞, i.e., for any u ∈ Lp (Ω; Y ) and ε > 0 there is a simple function s of the form (1.3.5) such that s − uY ≤ ε. Deﬁnition 1.3.21. Let Ω be a measurable subset of Rd , let Y be a Banach space, Y the dual space to Y , and 1 ≤ p < ∞. Then Lpw (Ω; Y ) is the space of all (equivalence classes of) weakly measurable functions u : Ω → Y with the ﬁnite norm uLpw (Ω;Y ) =

u(x)Y dx

1/p

.

Ω

If we identify functions with their equivalence classes, then Lpw (Ω; Y ) becomes a Banach space. L∞ w (Ω; Y ) is the Banach space of all (equivalence classes of) weakly measurable functions u : Ω → Y with the ﬁnite norm = ess sup u(x)Y . uL∞ w (Ω;Y )

x∈Ω

The properties of the spaces Lp (Ω; Y ) are similar to the properties of Lp (Ω); in particular we have the following extension of Theorem 1.3.16 ([24, Ch. III.6, Cor. 16]). Theorem 1.3.22. Let Ω ⊂ Rd be a Lebesgue measurable set, Y be a Banach space, p ∈ [1, ∞), and suppose that strongly measurable functions un : Ω → Y converge a.e. in Ω to a function u. Furthermore, assume that there is a nonnegative function g ∈ L1 (Ω) such that un (x)pY ≤ g(x) a.e. in Ω. Then u ∈ Lp (Ω; Y ) and lim un − uLp (Ω;Y ) = 0.

n→∞

Subspaces of Lp (Ω; Y ). In applications, various subspaces Z of some “hold-all” Banach spaces Y are considered. A natural question is whether Lp (Ω; Z) is then a subspace of Lp (Ω; Y ). The answer is given by the following simple lemma. Lemma 1.3.23. Let Z be a closed subspace of a Banach space Y , and suppose a function u : Ω → Z belongs to Lp (Ω; Y ) for some p ∈ [1, ∞]. Then u ∈ Lp (Ω; Z). Proof. Since u(x)Z = u(x)Y , it suﬃces to show that u : Ω → Z is strongly measurable. By Theorem 1.3.10 there are a set E ⊂ Rd of zero measure and a

1.3. Compact sets in Lp spaces

27

countable subset Y = {yi } of Y with the property that for every x ∈ Ω \ E and ε > 0 there is yi ∈ Y such that u(x) − yi ≤ ε. Next for every i, choose zi ∈ Z such that zi − yi Y ≤ 2 dist(yi , Z). and set Z = {zi } ⊂ Z. Note that if u(x) − yi Y ≤ ε, then dist(yi , Z) ≤ ε and so u(x) − zi Y ≤ u(x) − yi Y + zi − yi Y ≤ 3ε. In other words, for every x ∈ Ω \ E and ε > 0, there exists zi ∈ Z satisfying u(x) − zi Y ≤ 3ε. This means that u(X \ E) ⊂ cl Z ⊂ Z.

(1.3.6)

Let us prove that for every open G ⊂ Z, the set u−1 (G) is measurable. Notice that the set F = Z \ G is closed in Y . Since the mapping u : Ω → Y is measurable, it follows from assertion (II’) in Theorem 1.3.10 that u−1 (F ) is measurable and hence the set u−1 (G) = Ω \ u−1 (F ) is measurable. From this and (1.3.6) we conclude that u : Ω → Z meets all requirements of assertions (I)–(II) in Theorem 1.3.10 and hence is strongly measurable. Duality. The following result (see [42, Thm. 2.112]) is important for us since it leads to the notion of Young measure which is widely used throughout this book. Theorem 1.3.24. Let Ω be a measurable subset of Rd , let Y be a Banach space and let 1 ≤ p < ∞, q = p/(p − 1). • Assume that Y is separable. Then a functional V : Lp (Ω; Y ) → R belongs to (Lp (Ω; Y )) if and only if there is v ∈ Lqw (Ω; Y ) such that V, u =

v(x), u(x) dx. Ω

Moreover v is unique and the norm of the functional V equals vLqw (Ω,Y ) . • Assume that Y is reﬂexive. Then a functional V : Lp (Ω; Y ) → R belongs to (Lp (Ω; Y )) if and only if there is v ∈ Lq (Ω; Y ) such that V, u =

v(x), u(x) dx. Ω

Moreover, v is unique and the norm of V is equal to vLq (Ω,Y ) . Corollary 1.3.25. If Y is reﬂexive and 1 < p < ∞, then Lp (Ω; Y ) is reﬂexive.

28

Chapter 1. Preliminaries

Anisotropic spaces. The anisotropic Lebesgue spaces are the most important class of Lp spaces. In the analysis of nonstationary problems we often need functions with diﬀerent properties with respect to time and the spatial variables. In this case it is convenient to replace the standard Lebesgue space by an anisotropic space which takes into account the features of the problem. Let Ω ⊂ Rd be a measurable set, T > 0 and let Q be the cylinder Ω × (0, T ). For any measurable function u : Q → R we write u(t) instead u(·, t), i.e., u(x, t) denotes the value of u at the point (x, t), but u(t) denotes a function of the spatial variable x which is deﬁned in Ω and takes the value u(x, t) at x. Deﬁnition 1.3.26. Let p, q ∈ [1, ∞] be given. Then Lq (0, T ; Lp (Ω)) := u : Q → R : u measurable and uLq (0,T ;Lp (Ω)) < ∞ , where for 1 ≤ q < ∞,

uLq (0,T ;Lp (Ω)) :=

T

u(t)Lp (Ω) dt

1/q

,

0

while if q = ∞, then uL∞ (0,T ;Lp (Ω)) := ess sup u(t)Lp (Ω) . t∈(0,T )

The theory of anisotropic Lebesgue spaces is the same as the theory of standard Lebesgue spaces. In particular, we have the following generalization of the Hölder inequality. Assume that the domain Ω is of ﬁnite measure, and the exponents p, q, pi , qi , 1 ≤ i ≤ n, satisfy the conditions 1 1 , ≥ p p i=1 i n

p, q, pi , qi ∈ [1, ∞],

1 1 . ≥ q q i=1 i n

(1.3.7)

Then there is a constant c, depending on Ω and p, q, pi , qi , such that for all ui ∈ Lqi (0, T ; Lpi (Ω)), n ui

Lq (0,T ;Lp (Ω))

i=1

≤c

n

ui Lqi (0,T ;Lpi (Ω)) .

(1.3.8)

i=1

Remark 1.3.27. If in (1.3.7) there are equalities instead of inequalities, then inequality (1.3.8) is fulﬁlled for all measurable sets Ω. Moreover, in this case the constant c is 1. Next, we have the following representation for dual spaces. For any exponent p ∈ [1, ∞] set p = p/(p − 1) for 1 < p < ∞,

p = ∞ for p = 1,

p = 1 for p = ∞.

1.4. Young measures

29

Theorem 1.3.28. Let Ω ⊂ Rd be a measurable set and 1 ≤ p, q < ∞. Then for any continuous functional l ∈ Lq (0, T ; Lp (Ω)) there is a unique function v ∈ Lq (0, T ; Lp (Ω)) such that for all u ∈ Lp (Ω), l(u) = u(x, t)v(x, t) dxdt. Q

Moreover lLq (0,T ;Lp (Ω)) :=

sup uLq (0,T ;Lp (Ω)) =1

|l(u)| = vLq (0,T ;Lp (Ω)) .

In other words, Lq (0, T ; Lp (Ω)) is isomorphic to Lq (0, T ; Lp (Ω)). In accordance with Theorem 1.3.28 we have the following deﬁnitions for weak convergence in Lq (0, T ; Lp (Ω)): Deﬁnition 1.3.29. Let 1 < p, q ≤ ∞. A sequence fn ∈ Lq (0, T ; Lp (Ω)), n ≥ 1, converges to f weakly in Lq (0, T ; Lp (Ω)) if T T fn ξ dxdt = f ξ dxdt for any ξ ∈ Lq (0, T ; Lp (Ω)). (1.3.9) lim n→∞

0

Ω

0

Ω

If (1.3.9) holds and 1 < p, q < ∞ then the sequence fn ∈ Lq (0, T ; Lp (Ω)), n ≥ 1, converges to f weakly in Lq (0, T ; Lp (Ω)). The following result establishes the weak compactness of bounded sets. Theorem 1.3.30. Let fn ∈ Lq (0, T ; Lp (Ω)), n ≥ 1, be a bounded sequence in Lq (0, T ; Lp (Ω)), with 1 < p, q ≤ ∞. Then there are a subsequence, still denoted by fn , and a function f ∈ Lq (0, T ; Lp (Ω)) such that fn converges to f weakly (weakly for p, q < ∞) in Lq (0, T ; Lp (Ω)).

1.4 Young measures Let Ω ⊂ Rd be a measurable set. A Young measure (sliced measure, local probability distribution) is a family of Borel probability measures μx in R, depending on a parameter x ∈ Ω, such that the mapping x → μx is weakly measurable with respect to the Lebesgue measure in Rd . Recall that C0 (R) ⊂ C(R) denotes the closed subspace of all continuous functions u : R → R such that u(x) → 0 as |x| → ∞, and M(R) denotes the dual space C0 (R) of Radon measures deﬁned by Theorem 1.2.24. It is clear that the Banach space Y := C0 (R) is separable and it meets all requirements of Theorem 1.3.24. Applying that theorem to the space L1 (Ω; C0 (R)) we conclude that (L1 (Ω; C0 (R))) = L∞ w (Ω; M(R)). Thus we introduce the following deﬁnition.

30

Chapter 1. Preliminaries

Deﬁnition 1.4.1. Let Ω ⊂ Rd be a measurable set. A Young measure μ is an element of L∞ w (Ω; M(R)) such that μx ≥ 0,

μx , 1 = 1 for a.e. x ∈ Ω,

i.e. μx is a probability measure on R for a.e. x ∈ Ω. It is clear that the norm in L∞ w (Ω; M(R)) of every Young measure is equal to 1. Hence a Young measure belongs to the unit sphere in this space and the set of Young measures is not compact in the weak topology. It is of interest when the weak limit of Young measures is also a Young measure. The following lemma gives an answer to this question. Lemma 1.4.2. Let Ω ⊂ Rd be a bounded measurable set and {μn }n≥1 be a sequence of Young measures. Furthermore, assume that for any ε > 0 there is N such that for all n, |λ|≥N

dμnx (λ) < ε.

Then there is a subsequence, still denoted by {μn }, and a Young measure μ ∈ L∞ w (Ω; M(R)) such that weakly in L∞ w (Ω; M(R)).

μn μ

Proof. Since L1 (Ω; C0 (R)) is separable, any bounded subset of the dual space is weak sequentially compact. Hence the sequence {μn } contains a subsequence, still denoted by μn , which converges weakly in L∞ w (Ω; M(R)) to an element μ. By Theorem 1.3.24, μ can be represented as a weak measurable family of Radon measures μx . It remains to show that μx is a probability measure for a.e. x ∈ Ω. To this end choose an arbitrary nonnegative function ψ ∈ C0 (Ω), and select η ∈ C0 (R) with 0 ≤ η ≤ 1, and η(x) = 1 for |x| ≤ N . We have (1 − ε) ψ(x) dx ≤ ψ(x) dx − ψ(x) (1 − η(x/N )) dμnx dx Ω Ω R Ω n ψ(x) η(x/N ) dμx dx ≤ ψ(x) dx. = R

Ω

Ω

Letting n → ∞ we obtain (1 − ε) ψ(x) dx ≤ ψ(x) η(x/N ) dμx dx ≤ ψ(x) dx. Ω

R

Ω

Ω

Letting N → ∞ and next ε → 0 we arrive at ψ(x) dμx dx = ψ(x) dx for all nonnegative ψ ∈ C0 (Ω). Ω

R

Ω

Hence μx is a probability measure for a.e. x ∈ Ω.

1.4. Young measures

31

Further applications require two important deﬁnitions. Deﬁnition 1.4.3. Let Ω ⊂ Rd be a measurable set and B ⊂ Rm be a Borel set. A function f : Ω × B → R is said to be a normal integrand if for a.e. x ∈ Ω the function f (x, ·) is lower semicontinuous and there exists a Borel function g : Ω × B → R such that f (x, ·) = g(x, ·) for a.e. x ∈ Ω. Deﬁnition 1.4.4. Let Ω ⊂ Rd be a measurable set and B ⊂ Rm be a Borel set. A function f : Ω × B → R is said to be a Carathéodory integrand if for a.e. x ∈ Ω the function f (x, ·) is continuous and for every y ∈ B the function f (·, y) is measurable. Every Carathéodory integrand is a normal integrand. The following fundamental theorem on Young measures is the most important result of the theory. Theorem 1.4.5. Let Ω be a bounded subset of Rd and let {vn : Ω → R} be a sequence of measurable functions with the following property: lim lim sup meas{x : |vn (x)| > t} = 0.

t→∞ n→∞

(1.4.1)

Then there is a subsequence, still denoted by vn , and a Young measure μ ∈ L∞ w (Ω; M(R)) such that for any ϕ ∈ C0 (R): • ϕ(vn ) ϕ weakly in L∞ (Ω), where ϕ(x) = μx , ϕ. • For any normal integrand F : Ω × R → R bounded from below, F (x, vn (x)) dx ≥ F (x) dx, where F (x) = μx , F (x, ·). lim inf n→∞

Ω

Ω

• For any Carathéodory integrand F : Ω × R → R such that the functions {F (x, vn (x))} are equi-integrable, lim F (x, vn (x)) dx = F (x) dx, where F (x) = μx , F (x, ·). n→∞

Ω

Ω

Remark 1.4.6. Condition (1.4.1) is equivalent to the following: There is a continuous function g : R+ → R such that lims→∞ g(s) = ∞ and g(|vn |) dx < ∞. sup n

Ω

As a corollary we obtain the following result on weak convergence in Lr (Ω). Theorem 1.4.7. Let Ω ⊂ Rd be a bounded measurable set. Assume that a sequence vn ∈ Lr (Ω), 1 < r < ∞, converges weakly in Lr (Ω) to a function v. Then after passing to a subsequence we can assume that there is a Young measure μ ∈ L∞ ω (Ω; M(R)) such that for all continuous functions g : R → R satisfying lim g(s)/|s|r → 0

as |s| → ∞,

(1.4.2)

32

Chapter 1. Preliminaries

g(vn ) converges weakly in L1 (Ω) to g(x) = μx , g. The Young measure μ has the properties μx , |s|r ∈ L1 (Ω), g(x) = μx , g

a.e. in Ω

for all continuous g satisfying (1.4.2), and |v(x)|r = | μx , s |r ≤ μx , |s|r

a.e. in Ω.

Moreover, vn converges to v strongly in any Ls (Ω), 1 ≤ s < r, if and only if μx is the Dirac measure at v(x), i.e. μx , g = g(v(x)).

1.5 Sobolev spaces First we recall some basic facts from the theory of Sobolev-Slobodetsky spaces which can be found in [2] and [129]. Let Ω be an open subset of Rn . For every multiindex α = (α1 , . . . , αd ) with nonnegative integers αi and f ∈ L1loc (Ω), a function g ∈ L1loc (Ω) is the α-th generalized (or distributional ) derivative of f (denoted g = ∂ α f ) if gϕ dx = (−1)α1 +···+αd Ω

f ∂ α ϕ dx Ω

for all ϕ ∈ C0∞ (Ω). The number |α| = α1 + · · · + αd is the order of the derivative g. For an integer l ≥ 0 and for an exponent r ∈ [1, ∞), we denote by W l,r (Ω) the Sobolev space of functions having all generalized derivatives up to order l in Lr (Ω). Endowed with the norm uW l,r (Ω) = sup|α|≤l ∂ α uLr (Ω) , it becomes a Banach space. For real 0 < s < 1 < r < ∞, the fractional Sobolev space W s,r (Ω) is obtained by the real interpolation method between Lr (Ω) and W 1,r (Ω), i.e., W s,r (Ω) = [Lr (Ω), W 1,r (Ω)]s,r , and consists of all measurable functions with the ﬁnite norm uW s,r (Ω) = uLr (Ω) + |u|s,r,Ω ,

where |u|rs,r,Ω

|x − y|−d−rs |u(x) − u(y)|r dxdy.

=

(1.5.1)

Ω×Ω

In view of Lemma 1.1.11, the space W 1,r (Ω) is dense in W s,r (Ω). In particular, if Ω is a bounded domain of class C 1 , then C ∞ (Ω) is dense in W s,r (Ω). In general, the Sobolev space W l+s,r (Ω), 0 < s < 1 < r < ∞, l ≥ 0 an integer, is deﬁned as the space of measurable functions with the ﬁnite norm uW l+s,r (Ω) = uW l,r (Ω) + sup|α|=l ∂ α uW s,r (Ω) . Furthermore, the notation W0s,r (Ω), 0 ≤ s ≤ 1, stands for the closed subspace s,r of W (Rd ) which consists of all u ∈ W s,r (Rd ) vanishing outside of Ω. We identify functions of W0s,r (Ω) with their restrictions to Ω. We have W s,r (Ω) = W0s,r (Ω) for sr < 1.

1.5. Sobolev spaces

33

For 1 < r < ∞, set W00,r (Ω) = W00,r (Ω) and W01,r (Ω) = W01,r (Ω). For all 0 < s < 1 and 1 < r < ∞, we denote by W0s,r (Ω) the interpolation space [W00,r (Ω), W01,r (Ω)]s,r endowed with one of the equivalent norms deﬁned by the real interpolation method. In other words, W0s,r (Ω) is obtained by real interpolation of the subspaces W01,r (Ω) = W01,r (Ω) ⊂ W 1,r (Ω) and W00,r (Ω) = Lr (Ω). Notice that in view of Lemma 1.1.11, the space W01,r (Ω) is dense in W0s,r (Ω) for 0 < s < 1 < r < ∞. It follows that C0∞ (Ω) is dense in W0s,r (Ω) and hence W0s,r (Ω) is the completion of C0∞ (Ω) in the W0s,r -norm. But in general W0s,r (Ω) does not coincide with the completion of C0∞ (Ω) in the W s,r -norm. Generally speaking, W0s,r (Ω) is not a subspace of W s,r (Ω), and its norm is stronger than the norm of W s,r (Ω). Embedding theorems. Let Ω be the whole Rn or a bounded domain with C 1 boundary. Let sr > d and 0 ≤ α < s − d/r be real numbers. Then the embedding W s,r (Ω) → C α (Ω) is continuous. If Ω is a bounded domain of class C 1 , then the embedding is compact, i.e., every bounded subset of W s,r (Ω) is relatively compact in C α (Ω). Moreover, for sr > d, and all u, v ∈ W s,r (Ω), uvW s,r (Ω) ≤ c(r, s)uW s,r (Ω) vW s,r (Ω) .

(1.5.2)

Let Ω be a bounded domain of class C 1 . If sr < d and t−1 = r−1 − d−1 s, then the embedding W s,r (Ω) → Lt (Ω) is continuous. If Ω is an arbitrary domain of class C 1 , then the embedding is continuous under the additional assumption r < d [2, Thm. 7.57]. We also have (see [2, Thm. 7.58]), for s > 0, 1 < r < β < ∞ and α = s − dr−1 + dβ −1 , uW α,β (Ω) ≤ c(r, s, α, β, Ω)uW s,r (Ω) .

(1.5.3)

Duality. Let Ω be the whole Rn or a bounded domain with C 1 boundary. We ﬁx s ∈ [0, 1]. We deﬁne uv dx (1.5.4) u, v = Ω

for all functions such that the right hand side makes sense. For r ∈ (1, ∞), each element v ∈ Lr (Ω) determines a functional Lv in (W0s,r (Ω)) , r = r/(r − 1), s ≥ 0, by the identity Lv (u) = u, v. We introduce the (−s, r)-norm of an element v ∈ Lr (Ω) to be, by deﬁnition, the norm of the functional Lv , that is, vW −s,r (Ω) =

sup

|v, u|.

(1.5.5)

u∈W0s,r (Ω) u s,r =1 W0

(Ω)

Let W −s,r (Ω) be the completion of Lr (Ω) with respect to the (−s, r)-norm. For an integer s, the space W −s,r (Ω) is topologically and algebraically isomorphic to

34

Chapter 1. Preliminaries

(W0s,r (Ω)) and can be identiﬁed with it. Notice that for 0 < s < 1, the Banach space W −s,r (Ω) is not an interpolation space between Lr (Ω) and W01,r (Ω). Similarly, we can deﬁne W −s,r (Ω) to be the completion of Lr (Ω) in the norm vW −s,r (Ω) =

|v, u|.

sup

(1.5.6)

u∈W0s,r (Ω) u s,r =1 W0

(Ω)

We intensively exploit the spaces W −s,r (Ω) in Chapter 11. The main properties of these spaces are listed in the following lemma. Lemma 1.5.1. Let real s, r satisfy 0 < s < 1 < r < ∞. Then • W −s,r (Ω) is topologically and algebraically isomorphic to the Banach space (W0s,r (Ω)) and can be identiﬁed with it; • W −s,r (Ω) is topologically and algebraically isomorphic to the interpolation space [Lr (Ω), W0−1,r (Ω)]s,r = ([Lr (Ω), W01,r (Ω)]s,r ) and can be identiﬁed with it;

• (Lr (Ω)) , which can be identiﬁed with Lr (Ω), is dense in W −s,r (Ω) and hence C0∞ (Ω) is dense in W −s,r (Ω). Proof. Observe that for v ∈ Lr (Ω), Lv : u → v, u is a continuous functional on Lr (Ω), i.e., Lv ∈ (Lr (Ω)) . Since W0s,r (Ω) is continuously embedded in Lr (Ω), Lv is a continuous functional on W0s,r (Ω), i.e., Lv ∈ (W0s,r (Ω)) . In view of (1.5.6), the norm vW −s,r (Ω) is exactly the norm Lv (W s,r (Ω)) . From this and the def0

inition of W −s,r (Ω) we conclude that W −s,r (Ω) is the completion of (Lr (Ω)) in the (W0s,r (Ω)) -norm. Notice that W01,r (Ω) is dense in Lr (Ω). Setting A = A0 = Lr (Ω), A1 = W01,r (Ω) and applying Lemma 1.1.12 we conclude that

[Lr (Ω), W01,r (Ω)]s,r

= [(Lr (Ω)) , (W01,r (Ω)) ]s,r

By the deﬁnition of W0s,r (Ω), we have (W0s,r (Ω)) ≡ [Lr (Ω), W01,r (Ω)]s,r . Thus we get

(W0s,r (Ω)) = [(Lr (Ω)) , (W01,r (Ω)) ]s,r .

(1.5.7)

Next, the inclusion W01,r (Ω) ⊂ Lr (Ω) implies (Lr (Ω)) ⊂ (W01,r (Ω)) . Applying Lemma 1.1.11 we ﬁnd that (Lr (Ω)) is dense in [Lr (Ω) , (W01,r (Ω)) ]s,r . From this and (1.5.7) we conclude that Lr (Ω) is dense in (W0s,r (Ω)) . Hence (W0s,r (Ω))

1.6. Molliﬁers and DiPerna & Lions lemma

35

is the completion of (Lr (Ω)) in the (W0s,r (Ω)) -norm. But so is W −s,r (Ω). Thus we get

W −s,r (Ω) = (W0s,r (Ω)) = [Lr (Ω), W0−1,r (Ω)]s,r = ([Lr (Ω), W01,r (Ω)]s,r ) .

It remains to note that C0∞ (Ω) is dense in Lr (Ω) which is identiﬁed with (Lr (Ω)) . Hence C0∞ (Ω) is dense in W −s,r (Ω). It is well known that the operator ∇ : W s,r (Ω) → W s−1,r (Ω) is continuous for s = 0, 1. It follows that ∇ : W s,r (Ω) → W s−1,r (Ω) is continuous for all s ∈ (0, 1), and hence ∇uW s−1,r (Ω) ≤ c(s, r)uW s,r (Ω)

for all s ∈ (0, 1), 1 < r < ∞.

(1.5.8)

Extension. Let k ≥ 1 be an integer and Ω ⊂ Rd be a bounded domain of class C k . Then for every r ∈ [1, ∞] there exists a bounded extension operator Ξ : W k,r (Ω) → W k,r (Rd ) such that Ξu = u in Ω and ΞuW k,r (Rd ) ≤ cuW k,r (Ω) , where c depends on k and Ω.

1.6 Molliﬁers and DiPerna & Lions lemma A nonnegative function θ : R → R+ is said to be a mollifying kernel in R if ∞ θ(s) ds = 1, supp θ ⊂ (−1, 1), θ(−s) = θ(s). (1.6.1) θ ∈ C0 (R), R

For any f ∈ L1loc (Rd × R) deﬁne the molliﬁers [f ],k (x, s) := k θ(k(τ − s))f (x, τ ) dτ, [f ]m, (x, s) := md

Rd

(1.6.2)

R

Θ(m(y − x))f (y, s) dy,

Θ(x) =

d

θ(xi ).

(1.6.3)

i=1

We write simply [f ]m,k instead of [[f ]m, ],k . Notice that the composition of molliﬁers is also a molliﬁer. More precisely, we have [[f ],k ],k (x, s) = k θ2 (k(τ − s))f (x, τ ) dτ, where θ2 (s) = θ(s − ξ)θ(ξ) dξ. R

R

(1.6.4)

Recall some simple properties of mollifying operators. For any function f locally integrable in Rd , the molliﬁed function [f ]m, has continuous derivatives

36

Chapter 1. Preliminaries

of all orders in Rd . Let Ω be a bounded domain in Rd and extend f ∈ Lr (Ω), 1 ≤ r < ∞, by zero outside of Ω over the whole Rd . Then f − [f ]m, Lr (Ω) → 0 as m → ∞. If, in addition, f depends on the time variable t ∈ (0, T ) and belongs to the space Lq (0, T ; Lr (Ω)), 1 ≤ q ≤ ∞, 1 ≤ r < ∞, f − [f ]m, Lq (0,T ;Lr (Ω)) → 0 as m → ∞ when q < ∞ and f − [f ]m, Ls (0,T ;Lr (Ω)) → 0 as m → ∞ for all s < ∞. Next if f ∈ Lq (0, T ; W s,r (Rd )), s ∈ R, 1 < q, r < ∞, then f − [f ]m, Lq (0,T ;W s,r (Rd )) → 0 as m → ∞. The commutator of a mollifying operator and the operator of multiplication by a function from a Sobolev space is bounded in a suitable Lebesgue space, and its norm vanishes as the parameter m tends to zero. The following lemma is a particular case of a general result due to DiPerna and P.-L. Lions [23]. 1,2 (Rd )). Lemma 1.6.1. Suppose that f ∈ L∞ (0, T ; Lrloc (Rd )) and u ∈ L1 (0, T, Wloc ∞ r 1 1,2 This means that f ∈ L (0, T ; L (B)) and u ∈ L (0, T, W (B)) for every ball B ⊂ Rd . Then for q −1 = 2−1 + r−1 and a compact set E ⊂ Rd ,

div([f u]m, − [f ]m, u) → 0

in L1 (0, T ; Lq (E))

as m → ∞.

(1.6.5)

The main application of this lemma is to justify the renormalization procedure for weak solutions to the linear transport equation ∂t + div(u) = F

in Rd × (0, T ),

(x, 0) = 0 (x) in Rd .

(1.6.6)

If and u are suﬃciently smooth, then it is easy to check that for any C 1 function ϕ : R → R, the composite function ϕ() satisﬁes ∂t ϕ() + div(ϕ()u) + (ϕ () − ϕ()) div u = ϕ ()F ϕ()(x, 0) = ϕ(0 )(x) in Rd .

in Rd × (0, T ),

(1.6.7)

(1.6.7) is called the renormalized equation and derivation of (1.6.7) from the original equation (1.6.6) is called the renormalization procedure. The justiﬁcation of the renormalization procedure for weak nonsmooth solutions is nontrivial. Throughout this book we use the following lemma which is a particular case of a general result established in [23]:

1.6. Molliﬁers and DiPerna & Lions lemma

37

Lemma 1.6.2. Assume that functions ∈ L∞ (0, T ; Lr (Rd )), 0 ∈ Lr (Rd ) with r−1 ≤ 2−1 − d−1

2 < r ≤ ∞,

vanish for |x| > R, and a function F ∈ L1 (0, T ; L1 (Ω)) also vanishes for |x| > R, with large R. Furthermore, assume that a vector ﬁeld u ∈ Lα (0, T ; W 1,2 (Rd )), α > 1, and the functions , F satisfy the integral identity

T

Rd

0

((∂t ψ + ∇ψ · u) + ψF ) dxdt +

Rd

ψ(x, 0)0 dx = 0

(1.6.8)

for all ψ ∈ C ∞ (Rd × (0, T )) vanishing for all suﬃciently large |x| and for t = T . Then the integral identity

T

0

Rd

T

ϕ()(∂t ψ + ∇ψ · u) dxdt + − 0

T

Rd

ϕ ()ψF dxdt

Rd

0

(ϕ () − ϕ ()) div u dxdt +

Rd

ψ(x, 0)0 dx = 0 (1.6.9)

holds for any C 1 function ϕ : R → R such that |ϕ(s)| + |(1 + |s|)ϕ (s)| ≤ M < ∞. Molliﬁers and functions of bounded variation. The following lemma is widely used when applying the kinetic equation method to Navier-Stokes equations. Lemma 1.6.3. Let f : R → R be a monotone function of bounded variation and let H be a bounded function on R such that the limits limτ →s±0 H(τ ) exist at each s ∈ R. Then for any ψ ∈ C(R), ˜ lim [ψ[H],k ],k = lim [ψH],k = ψ H,

k→∞

where

k→∞

(1.6.10)

1 1 ˜ H(s) = lim H(s − h) + lim H(s + h). 2 h 0 2 h 0

˜ is Borel and The function H ˜ df. lim ψ[H],k ∂s [f ],k ds ≡ lim ψ[H],k d[f ],k = ψH k→∞

k→∞

R

In particular,

R

2 R

(1.6.11)

R

ψ f˜ ds f (s) =

R

ψ ds f 2 (s).

(1.6.12)

38

Chapter 1. Preliminaries

Proof. We have

[ψ[H],k ],k (s) = ψ[[H],k ],k (s) + k

θ(ks − kτ )(ψ(τ ) − ψ(s))[H],k (τ ) dτ.

Since ψ is continuous and H is bounded, we have k θ(kτ − ks)(ψ(τ ) − ψ(s))[H],k (τ ) dτ ≤ ck θ(kτ − ks)|ψ(τ ) − ψ(s)| dτ R R = c θ(τ )|ψ(s + τ /k) − ψ(s)| dτ → 0 as k → ∞. (1.6.13) R

Notice that the kernel θ2 is even, which yields s − + ˜ H(s) = H (s)k θ2 (k(τ − s)) dτ + H (s)k −∞

∞

θ2 (k(τ − s)) dτ,

s

where H ± (s) = limτ →s±0 H(τ ). From this and (1.6.4) we get s ˜ [[H],k ],k − H(s) = θ2 (τ )(H(s + τ /k) − H − (s)) dτ −∞ ∞ θ2 (τ )(H(s + τ /k) − H + (s)) dτ → 0 +

(1.6.14)

s

˜ as k → ∞. Combining (1.6.13) and (1.6.14) we obtain limk→∞ [ψ[H],k ],k = ψ H. ˜ ˜ Repeating these arguments we obtain limk→∞ [ψH],k = ψ H. It follows that H is ˜ and hence it is Borel. Next, a pointwise limit of continuous functions [H],k = ψ H ψ[H],k d[f ],k = ψ[H],k (s)∂s k θ(k(τ − s))f (τ ) dτ ds. (1.6.15) R

R

R

Integrating by parts in the Stieltjes integral gives ∂s k θ(k(τ − s))f (τ ) dτ = −k 2 θ (k(τ − s))f (τ ) dτ R R = k θ(k(τ − s)) dτ f. R

Since ψ[H],k is continuous and bounded, the Fubini theorem yields ψ[H],k (s) k θ(k(τ − s)) dτ f R R k ψ[H],k (s)θ(k(τ − s)) ds dτ f (τ ) = [ψ[H],k ],k dτ f (τ ). = R

Thus we get

R

R

R

ψ[H],k d[f ],k =

R

[ψ[H],k ],k dτ f (τ ).

1.7. Partial diﬀerential equations

39

Letting k → ∞, noting that the continuous functions [ψ[H],k ],k converge pointwise ˜ and applying the Lebesgue dominated convergence theorem we arrive at to ψ H (1.6.11). It remains to prove (1.6.12). Assume for a moment that ψ ∈ C 1 (R). Notice that ψ d[f ]2,k = − ψ [f ]2,k ds → − ψ f˜2 ds. (1.6.16) 2 ψ[f ],k d[f ],k = R

R

R

R

A function of bounded variation is continuous everywhere except on some countable set. Hence f˜2 = f 2 a.e. in R. From this and the integration by parts formula in Lemma 1.3.7 we obtain ψ f˜2 ds = ψ f 2 ds = − ψ ds f 2 . (1.6.17) R

R

R

On the other hand relation (1.6.11) with H replaced by f gives ψ[f ],k d[f ],k = 2 ψ f˜ df. 2 lim k→∞

R

R

Combining this with (1.6.16) and (1.6.17) we obtain (1.6.12) for ψ ∈ C 1 (R). It remains to note that every function ψ ∈ C(R) is a pointwise limit of a sequence of uniformly bounded functions ψn ∈ C 1 (R), say ψn = [ψ],n .

1.7 Partial diﬀerential equations—selected facts 1.7.1 Elliptic equations Poisson equation.

The simplest elliptic equation is the Poisson equation in Rd , Δu = f

in Rd ,

d≥2

(1.7.1)

which is understood in the sense of the theory of distributions. This means that the functions u and f are integrable on every compact subset of Rd and uΔϕ dx = f ϕ dx for all ϕ ∈ C0∞ (Rd ). Rd

Rd

Its solution is given by explicit formulae, which along with the classical estimates of the Newton potential lead to the following result: Lemma 1.7.1. For any bounded set Ω and f ∈ W s−2,r (Rd ),

s ∈ R,

1 < r < ∞,

with supp f ⊂ Ω, equation (1.7.1) has a solution with the properties:

40

Chapter 1. Preliminaries • This solution is analytic outside of Ω. • It satisﬁes lim sup |x|d−2 |u(x)| < ∞

for d > 2,

|x|→∞

lim sup (log |x|)−1 |u(x)| < ∞

for d = 2,

|x|→∞

uW s,r (BR ) ≤ c(R, s, r, Ω)f W s−2,r (Rd ) . Here, BR is the ball {x ∈ Rd : |x| < R} of an arbitrary radius R < ∞, and the constant c depends only on R, s, r, and Ω. The relation f → u determines a linear operator u = Δ−1 [f ]. In this framework we can deﬁne the linear operators Aj = ∂xj Δ−1 ,

Rkj = ∂xk ∂xj Δ−1 ,

1 ≤ k, j ≤ d.

The Riesz operator Rkj is a singular integral operator and by the ZygmundCalderón theorem (see [127]) it is bounded in any space Lp (Rd ) with 1 < p < ∞. The Zygmund-Calderón theorem and Lemma 1.7.1 imply Corollary 1.7.2. Under the assumptions of Lemma 1.7.1 for every s ∈ R, r ∈ (1, ∞), and f ∈ W s,r (Rd ), Aj [f ]W s+1,r (BR ) ≤ c0 f W s,r (Rd ) , Rkj [f ]W s,r (Rd ) ≤ c1 f W s,r (Rd ) . where c0 depends only on R > 0, s, r, and c1 depends only on s and r.

Corollary 1.7.3. For every r ∈ (1, ∞) and f ∈ Lr (Rd ), g ∈ Lr (Rd ), r = r/(r −1), gRij [f ] dx = f Rij [g] dx. Rd

Rd

Boundary value problems for elliptic equations. In Chapter 5 we use existence and uniqueness results for solutions to elliptic equations. Denote by α a vector with nonnegative integer components α = (α1 , . . . , αd ),

αi ∈ N ∪ {0},

|α| = α1 + · · · + αd ,

and set ξ α = ξ1α1 . . . ξdαd

for ξ ∈ Rd ,

∂ α = ∂xα11 · · · ∂xαdd .

Remark 1.7.4. We also use the notation ∂xk u for the totality of derivatives of order k of a function u, and the notation ∂nk u for the normal derivative of order k of u at the boundary of the domain of deﬁnition of u.

1.7. Partial diﬀerential equations

bαβ

41

Let Ω be a bounded domain in Rd with C ∞ boundary. Assume that functions ∈ C ∞ (Ω), |α| = |β| = m, satisfy the ellipticity condition 2m ≤ bαβ ξ α ξ β ≤ cb |ξ|2m c−1 b |ξ|

for all ξ ∈ Rd ,

cb > 0

(recall the summation convention). Let us consider the boundary value problem ∂ β (bαβ ∂ α v) = f in Ω, ∂ κ v = 0 on ∂Ω, 0 ≤ |κ| ≤ m − 1. (1.7.2) |α|=|β|=m

The following theorem (see [3], [78]) is a particular case of the general theory of boundary value problems for elliptic equations. Theorem 1.7.5. Let s ≥ −m be an integer. Then for any f ∈ W s,2 (Ω) problem (1.7.2) has a unique solution v ∈ W 2m+s,2 (Ω),

with

v ∈ W0m,2 (Ω),

which satisﬁes the estimate vW 2m+s,2 (Ω) ≤ c f W s,2 (Ω) . Here, the constant c depends only on bαβ , Ω and s. Remark 1.7.6. For s ∈ [−m, 0] equation (1.7.2) is understood in the sense of distributions, which means that we have the integral identity |β| (−1) bαβ ∂ α v∂ β ϕ dx = f, ϕ for all ϕ ∈ C0∞ (Ω). Ω

We also have the following result: Lemma 1.7.7. Let Ω ⊂ Rd , d = 2, 3, be a bounded domain with C ∞ boundary. Let b ∈ C ∞ (Ω) with b > σ > 0. Furthermore, assume that ∈ L1 (Ω) is a nonnegative function and the operator A : W04,2 (Ω) → W −4,2 (Ω) is deﬁned by A[ϕ], ψ = (bΔ2 ϕΔ2 ψ + bϕψ + ϕψ) dx for all ϕ, ψ ∈ W04,2 (Ω). (1.7.3) Ω

Then the inverse A−1 : W −4,2 (Ω) → W04,2 (Ω) exists and satisﬁes A−1 [ψ]W 4,2 (Ω) ≤ σ −1 ψW −4,2 (Ω) 0

for all ψ ∈ W −4,2 (Ω).

(1.7.4)

Proof. Note that W04,2 (Ω) → C 2 (Ω) and 1/2 (|Δϕ|2 + |ϕ|2 ) dx . ϕW 4,2 (Ω) ∼ 0

Ω

Hence the space V = W04,2 (Ω) and the operator A meet all requirements of Lemma 1.1.14, and the proof is complete.

42

Chapter 1. Preliminaries

As a consequence of Theorem 1.7.5 and Corollary 1.7.7 we obtain the following lemma for systems of elliptic equations. Lemma 1.7.8. Let d = 2, 3, τ ∈ (0, 1), and 0 ≤ ∈ L2 (Ω),

b ∈ C ∞ (Ω),

b > σ > 0.

Then for all vector ﬁelds f ∈ L2 (Ω), the boundary value problem 1 1 2 Δ (bΔ2 v) + bv + v = f τ τ

in Ω,

v ∈ W04,2 (Ω),

(1.7.5)

has a unique solution; moreover, this solution satisﬁes vW 8,2 (Ω) ≤ c τ f L2 (Ω) . Here, the constant c depends only on b, , and Ω. Proof. Since all requirements of Lemma 1.7.7 are satisﬁed, it follows that problem (1.7.5) has a unique solution, and vW 4,2 (Ω) ≤ cτ f L2 (Ω) . Noting that the embedding W 4,2 (Ω) → C(Ω) is compact we conclude that the vector ﬁeld F = τ f − τ v − bv satisﬁes the estimate FL2 (Ω) ≤ τ f L2 (Ω) + τ L2 (Ω) vW 4,2 (Ω) + bC(Ω) vW 4,2 (Ω) ≤ cτ f L2 (Ω) . On the other hand, the elliptic equation (1.7.5) with the homogeneous boundary conditions can be rewritten as Δ2 (bΔ2 v) + bv = F in Ω,

v ∈ W04,2 (Ω).

Since the operator z → Δ2 (bΔ2 z) satisﬁes all hypotheses of Theorem 1.7.5 we have vW 8,2 (Ω) ≤ cFL2 (Ω) , completing the proof.

1.7.2 Stokes problem We need an auxiliary result for the Stokes problem which we give with the proof. Stokes equations. The following lemma is a straightforward consequence of classical results on solvability of the ﬁrst boundary value problem for Stokes equations (see [50]) and interpolation theory. For Ω bounded, consider the projection of L1 (Ω) onto the subspace of codimension one which consists of the mean value zero functions, 1 q dx, q ∈ L1 (Ω). (1.7.6) Πq := q − meas Ω Ω

1.7. Partial diﬀerential equations

43

Lemma 1.7.9. Let Ω ⊂ R3 be a bounded domain with ∂Ω ∈ C 2 and (F, G) ∈ W s−1,r (Ω) × W s,r (Ω) for 0 < s < 1, 1 < r < ∞, (F, G) ∈ W s−1,r (Ω) × W s,r (Ω) for s = 0, 1, 1 < r < ∞. Then the boundary value problem Δu − ∇π = F, u = 0 on ∂Ω,

in Ω,

div u = ΠG Ππ = π,

(1.7.7)

has a unique solution (u, π) ∈ W s+1,r (Ω) × W s,r (Ω) such that. uW s+1,r (Ω) + πW s,r (Ω) ≤ c(Ω, r, s)(F W s−1,r (Ω) + GW s,r (Ω) )

(1.7.8)

for 0 < s < 1 < r < ∞ and uW s+1,r (Ω) + πW s,r (Ω) ≤ c(Ω, r, s)(F W s−1,r (Ω) + GW s,r (Ω) )

(1.7.9)

for s = 0, 1 and 1 < r < ∞. Proof. Note that, by Theorem 6.1 in [50], for any F ∈ W s−1,r (Ω) and G ∈ W s,r (Ω) with s = 0, 1, problem (1.7.7) has a unique solution u, π satisfying (1.7.9). Thus the relation (F, G) → (u, π) determines a linear operator T : W s−1,r (Ω) × W s,r (Ω) → W s+1,r (Ω) × W s,r (Ω). Therefore, Lemma 1.7.9 is a consequence of Lemma 1.1.13. Remark 1.7.10. We stress that for every s ∈ [0, 1], the solution to problem (1.7.7) given by the lemma belongs to the class W01,r (Ω) × Lr (Ω) and is a weak solution in the sense of the deﬁnition given in [50, Ch. 4]. Notice also that the statement of the lemma does not hold for 0 < s < 1 if we replace W s−1,r by W s−1,r since the latter is not an interpolation space for s ∈ (0, 1).

1.7.3 Parabolic equations Let Ω be a bounded domain in Rd with C 2+α boundary and Q = Ω × (0, T ). For any α ∈ (0, 1) denote by C α/2,α (Q) and C 1+α/2,2+α (Q) the Banach spaces of all functions u : cl Q → R with the ﬁnite norms uC α/2,α (Q) = uC(Q) +

|u(x, t) − u(x, τ )| |t − τ |α/2 (t,τ,x)∈[0,T ]2 ×Ω sup

+

sup (t,x,y)∈[0,T ]×Ω

2

|u(x, t) − u(y, t)| |x − y|α

(where Ω = cl Ω) and uC 1+α/2,2+α (Q) = uC 1,2 (Q) + +

|∂t u(x, t) − ∂t u(x, τ )| |t − τ |α/2 (t,τ,x)∈[0,T ]2 ×Ω sup

sup

i,j (t,x,y)∈[0,T ]×Ω

2

|∂xi ∂xj u(x, t) − ∂xi ∂xj u(y, t)| . |x − y|α

44

Chapter 1. Preliminaries

Remark 1.7.11. By the Arzelà-Ascoli compactness theorem, the embedding C κ (0, T ; C 1 (Ω)) → C α/2,α (Q) is compact for all 0 < α < κ ≤ 1. The following results are due to Ladyzhenskaya, Solonnikov and Ural’tseva [74]. Theorem 1.7.12. Let Ω ⊂ Rd be a bounded domain with C 2+α boundary and let b ∈ C α/2,α (Q)

and

k ∈ C α/2,α (Q).

Furthermore, assume that ∞ ∈ C 1+α/2,2+α (Q) and f ∈ C α/2,α (Q) satisfy the compatibility condition ∂t ∞ − εΔ∞ + b · ∇∞ + k∞ = f

for (x, t) ∈ ∂Ω × {t = 0}.

Then the boundary value problem ∂t − εΔ + b · ∇ + k = f = ∞

on ∂Ω × (0, T ),

in Q, = ∞ for t = 0,

(1.7.10a) (1.7.10b)

has a unique solution ∈ C 1+α/2,2+α (Q), and C 1+α/2,2+α (Q) ≤ c(∞ C 1+α/2,2+α (Q) + f C α/2,α (Q) ). Here, the constant c depends only on T , Ω, ε and bC α/2,α (Q) , kC α/2,α (Q) . Moreover, if ∞ > c∞ > 0 and f ≥ 0, then inf Q (x, t) > c > 0, where the constant c depends only on Ω, T and bC α/2,α (Q) , kC α/2,α (Q) , c∞ . Theorem 1.7.13. Let Ω ⊂ Rd be a bounded domain with C 2+α boundary and suppose that functions ∞ , f satisfy the condition ∂t ∞ L2 (Q) + ∞ L2 (0,T ;W 2,2 (Ω)) + ∞ C(0,T ;W 1,2 (Ω)) + f L2 (Q) < ∞. Then the boundary value problem ∂t − εΔ = f in Q, = ∞ on ∂Ω × (0, T ),

(1.7.11a) = ∞

for t = 0,

(1.7.11b)

has a unique solution , and the solution satisﬁes ∂t L2 (Q) + L2 (0,T ;W 2,2 (Ω)) + C(0,T ;W 1,2 (Ω)) ≤ c ∂t ∞ L2 (Q) + ∞ L2 (0,T ;W 2,2 (Ω)) + ∞ C(0,T ;W 1,2 (Ω)) + f L2 (Q) , where the constant c depends only on T , Ω and ε.

Chapter 2

Physical background 2.1 Governing equations Suppose a compressible ﬂuid (which we may also call a gas) occupies a domain Ω ⊂ R3 named the ﬂow domain. The ﬂow domain can vary in time and its position and even shape can depend on the time variable t. In this case we write Ωt to stress the dependence on t. The state of the ﬂuid is characterized completely by the macroscopic quantities: the density (x, t), the velocity u(x, t), and the temperature ϑ(x, t). These quantities are called state variables in the following. The governing equations represent three basic principles of ﬂuid mechanics: the mass balance ∂t + div(u) = 0 in Ω,

(2.1.1a)

the balance of momentum ∂t (u) + div(u ⊗ u) + ∇p = f + div S(u)

in Ω,

(2.1.1b)

and the energy conservation law ∂t E + div((E + p)u) = div(S(u)u) + div(κ∇ϑ) + (f) · u + Q.

(2.1.1c)

Here, the vector ﬁeld f denotes the density of external mass force, the heat conduction coeﬃcient κ is a positive constant, the given function Q is the intensity of the external energy ﬂux, the viscous stress tensor S has the form 2 S(u) = ν1 ∇u + ∇u − div u I + ν2 div u I, (2.1.1d) 3 in which the viscosity coeﬃcients νi , i = 1, 2, satisfy the inequality 43 ν1 + ν2 > 0, the energy density E is given by E=

1 |u|2 + e, 2

P. Plotnikov and J. Sokoáowski, Compressible Navier–Stokes Equations: Theory and Shape Optimization, Monografie Matematyczne 73, DOI 10.1007/978-3-0348-0367-0_2, © Springer Basel 2012

45

46

Chapter 2. Physical background

where e is the density of internal energy. The physical properties of the ﬂuid are reﬂected through constitutive equations relating the state variables to the pressure p and the internal energy density e. The common point of view is that p and e can be represented as functions of and ϑ. The functions p(, ϑ) and e(, ϑ) are not arbitrary but should satisfy the Gibbs equation 1 p(, ϑ) de(, ϑ) − d = ds(, ϑ), ϑ ϑ2 which means that the left hand side is the exact diﬀerential of some function s named entropy density. In the classical case of perfect polytropic gases, the pressure and the internal energy density are deﬁned by the following formulae, called the constitutive law : (2.1.1e) p = Rm ϑ, e = cv ϑ. Here, Rm is a positive constant inversely proportional to the molecular weight of the gas that is Rm = cp − cv , with γ := cp /cv > 1, where cv is the speciﬁc heat capacity at constant volume and cp is the speciﬁc heat capacity at constant pressure; cv , cp are positive constants. In this notation the entropy density s takes the form s = log e + (γ − 1) log .

(2.1.2)

The system of diﬀerential equations (2.1.1) is called the compressible NavierStokes-Fourier equations. It is useful to rewrite the governing equations in dimensionless form which is widely used in applications. To this end, we denote by uc , c , pc , ϑc the typical values of velocity, density, pressure, and temperature, and by lc and Tc the typical values of length scale and time intervals. For simplicity we assume that uc = lc /Tc . Under this assumption, the characteristic values form four dimensionless combinations which are named: the Reynolds number, the Pecle number, the Mach number, and the viscosity ratio (see [119]), Re =

c u c lc , ν1

Pe =

p c lc uc , κc ϑ c

Ma2 =

c u2c , pc

λ=

1 ν2 + . 3 ν1

Denote also by fc and Qc the characteristic values of mass force and heat inﬂux. They form two dimensionless combinations Fr2m =

u2c , fc lc

Θ=

c Qc lc . pc uc

Note that here the characteristic quantities c , ϑc , and pc should be compatible with the constitutive law. For instance if the pressure is deﬁned by (2.1.1e), then pc = Rm c ϑc . Observe that the speciﬁc values of the constants γ, λ, and

2.1. Governing equations

47

Pe depend only on the physical properties of the ﬂuid. For example, for the air under standard conditions, we have γ = 7/5 and Pe = 7/10. The passage to the dimensionless variables is deﬁned as follows: x → lc x, t → Tc t, u → uc u, → c , ϑ → ϑc ϑ, f → c fc f , κ → κc κ,

(2.1.3)

and when performed in (2.1.1), leads to the following system of diﬀerential equations for dimensionless quantities in the scaled domain lc−1 Ω, still denoted by Ω: 1 1 1 div S(u) + 2 f in Ω, ∇p = Re Ma2 Frm ∂t + div(u) = 0 in Ω, 1 κ div ∇ϑ ∂t (s) + div(su) − Pe ϑ 2 1 Ma κ Q 2 = S(u) : ∇u + |∇ϑ| + Θ , ϑ Re Peϑ ϑ ∂t (u) + div(u ⊗ u) +

(2.1.4a) (2.1.4b)

(2.1.4c)

where u ⊗ u stands for the tensor product of two vectors and the dimensionless viscous stress tensor is deﬁned by S(u) = ∇u + ∇u + (λ − 1) div u I,

div S(u) = Δu + λ∇ div u.

(2.1.5)

If the ﬂow domain Ωt varies in time, the above equations should be considered in the moving scaled domain.

2.1.1 Isentropic ﬂows. Compressible Navier-Stokes equations The ﬂow is barotropic if the pressure depends only on the density. The most important example of such ﬂows are isentropic ﬂows. In order to deduce the governing equations for isentropic ﬂows we note that for a perfect ﬂuid with νi = κ = 0, i = 1, 2, the entropy takes a constant value at each material point. Hence in this case the governing equations have a family of explicit solutions with the entropy s = const. By (2.1.1e) and (2.1.2) in this case we have p() = (γ − 1) exp(sc )γ , where the positive constant sc is a characteristic value of the entropy (without loss of generality we can take (γ − 1) exp(sc ) = 1). Assuming that this relation holds for νi = 0, i = 1, 2, we arrive at the system of compressible Navier-Stokes equations

48

Chapter 2. Physical background

for isentropic ﬂows of a viscous compressible ﬂuid in dimensionless form: 1 ∇p() Ma2 1 1 div S(u) + 2 f = Re Frm ∂t + div(u) = 0 in Ω. ∂t (u) + div(u ⊗ u) +

in Ω,

(2.1.6a) (2.1.6b)

Recall that the exponent γ depends on the physical properties of the ﬂuid. In particular [25], γ = 5/3 for mono-atomic, γ = 7/5 for diatomic and γ = 4/3 for polyatomic gases. It is worth noting that the entropy production for a viscous gas is proportional to S(u) : ∇u. Generally speaking, this means that the total entropy of a viscous gas increases in time and in contrast to the nonviscous case the existence of isentropic solutions for viscous gas dynamics equations is unlikely. In this sense, the compressible Navier-Stokes equations are thermodynamically inconsistent. The equations can be considered as an approximation of the real physical problem. However, this situation is not unusual in mathematical physics. We recall that the standard heat equation is also thermodynamically inconsistent. Nevertheless, the compressible Navier-Stokes equations play an important role in the theory of compressible ﬂuid dynamics as the only example of physically relevant equations for which we have nonlocal existence results.

2.2 Boundary and initial conditions Boundary conditions. The governing equations should be supplemented with boundary conditions. The typical boundary conditions for the velocity ﬁeld are: the ﬁrst boundary condition (Dirichlet-type condition) u = U on ∂Ω,

(2.2.1)

the second boundary condition (Neumann-type condition) (S(u) − p I)n = Sn

on ∂Ω,

(2.2.2)

where n is the outward normal vector to ∂Ω, and U and Sn are given vector ﬁelds. Important particular cases are the no-slip boundary condition with U = 0, and the zero normal stress condition with Sn = 0. A third type of physically and mathematically reasonable condition is the no-stick boundary condition u · n = 0,

((S(u) − p I)n) × n = 0 on ∂Ω,

which corresponds to the case of frictionless boundary. The typical boundary conditions for the temperature are the Dirichlet and Neumann boundary conditions.

2.2. Boundary and initial conditions

49

The formulation of boundary conditions for the density is a more delicate task. Assume that the velocity u satisﬁes the ﬁrst boundary condition (2.2.7), and split the boundary of the ﬂow region into three disjoint sets called the inlet Σin , the outgoing set Σout , and the characteristic set Σ0 ; these sets are deﬁned by Σin = {x ∈ ∂Ω : U · n < 0}, Σ0 = {x ∈ ∂Ω : U · n = 0}.

Σout = {x ∈ Σ : U · n > 0},

(2.2.3)

The density should be given on the inlet: = ∞

on Σin .

(2.2.4)

The boundary conditions for the density are not needed in the case of Σin = ∅. In particular, there are no boundary conditions for the density if the velocity satisﬁes the no-slip or no-stick conditions, i.e., whenever Σin = Σout = ∅. If the ﬂow domain depends on t, i.e. Ω = Ωt , and material points of ∂Ωt are moving with a velocity V(x, t) then the boundary conditions become u = U on ∂Ωt ,

= ∞

on Σtin ,

(2.2.5)

where the time dependent inlet is deﬁned by Σtin = {x ∈ ∂Ωt : (U(x, t) − V(x, t)) · n < 0}. Initial conditions. At the initial time t = 0 the distributions of the velocity ﬁeld, density and temperature should be prescribed in Ω := Ω0 for solutions of the Navier-Stokes-Fourier equations. The velocity ﬁeld and density should be prescribed for solutions of the compressible Navier-Stokes equations u(x, 0) = u0 (x),

(x, 0) = 0 (x) in Ω.

(2.2.6)

Flows around moving bodies. The problem of a gas ﬂow around a moving body is of practical importance. It presents an example of an exterior boundary value problem and is formulated as follows: Suppose that at time t a body occupies a compact set St ⊂ Rd . Its position and shape can vary in time, but we assume that there is no mass ﬂux through its boundary. Let material points of ∂St be moving with the velocity V(x, t). Then the problem is to ﬁnd the velocity ﬁeld u and the density satisfying the compressible Navier-Stokes equations (2.1.6) along with the boundary and initial conditions u = V on ∂St , u(x, 0) = u0 (x), u(x, t) → 0,

(x, 0) = 0 (x) in Ω0 , → ∞ as |x| → ∞.

(2.2.7)

50

Chapter 2. Physical background

2.3 Power and work of hydrodynamic forces The stress tensor in a viscous compressible ﬂow is deﬁned by T = S(u) − p I,

(2.3.1)

where the viscous stress tensor S(u) is given by (2.1.1d), and the force acting from the side of the ﬂow at the boundary point x ∈ ∂Ω is equal to Rf = −T n = (−S(u) + p I) n.

(2.3.2)

Recall that n is the outward normal vector to ∂Ω. If the ﬂow domain Ω = Ωt varies in time, and material points on its boundary are moving with a given velocity Vs (x, t), the surface density of the power developed by the hydrodynamic force Rf is given by Jdens = −T n · Vs = (−S(u) + p I) n · Vs . (2.3.3) The total hydrodynamic force RΩ and the power JΩ developed by the hydrodynamic forces are equal to RΩ = (−S(u) + p I) n dS, JΩ = (−S(u) + p I) n · Vs dS. (2.3.4) ∂Ωt

∂Ωt

Therefore, the total work of the hydrodynamic forces over the time period [0, T ] is T (2.3.5) (−S(u) + p I) n · Vs dS dt. WΩ = ∂Ωt

0

After scaling (2.1.3) of independent and state variables and the following scaling of the hydrodynamic force, power, and work: RΩ → c u2c lc2 RΩ ,

JΩ → c u3c lc2 JΩ ,

WΩ → Tc c u3c lc2 WΩ ,

expressions (2.3.4)–(2.3.5) can be rewritten in dimensionless form as 1 (∇u + (∇u) + (λ − 1) div u − σp I) n dS, RΩ = − Re ∂Ωt 1 JΩ = − (∇u + (∇u) + (λ − 1) div u − σp I) n · Vs dS, Re ∂Ωt and 1 WΩ = − Re

0

T

(∇u + (∇u) + (λ − 1) div u − σp I) n · Vs dSdt.

(2.3.6)

(2.3.7)

∂Ωt

Here, we use the notation λ=

1 ν2 + , 3 ν1

σ=

Re . Ma2

(2.3.8)

2.4. Navier-Stokes equations in a moving frame

51

2.4 Navier-Stokes equations in a moving frame If the ﬂow region varies in time, then it is convenient, for technical and practical reasons, to reduce the corresponding boundary value problem for ﬂuid dynamics equations to a problem in a ﬁxed domain by a change of independent variables. In this section we describe such a change of variables in the case when Ωt evolves like an absolutely rigid body. Assume that x and t are dimensionless variables. Recall that a one-parameter family of mappings y → x(y, t) represents a rigid body motion in Euclidean space Rd if and only if x = U(t)y + a(t),

(2.4.1)

where a(t) is an arbitrary vector function and U(t) is an arbitrary one-parameter family of special orthogonal matrices, i.e., U U = I and det U = 1. For simplicity, we assume that a and U are twice continuously diﬀerentiable in t and U(0) = I,

a(0) = 0.

The velocity V of the rigid body motion (2.4.1) is deﬁned by ˙ ˙ V(y, t) = U(t)y + a(t), ˙ where a(t) =

d dt a(t).

(2.4.2)

Introduce also the vector ﬁeld

˙ ˙ W(y, t) = U (t) V(y, t) = U (t)U(t)y + U (t)a(t).

(2.4.3)

Next set v(y, t) = U (t) u(x(y, t), t) − W(y, t),

ρ(y, t) = (x(y, t), t).

(2.4.4)

Lemma 2.4.1. Let the ﬂow domain be of the form Ωt = U(t) Ω0 +a(t). Then smooth functions (u, ) satisfy equations (2.1.6) in Ωt if and only if the functions (v, ρ), deﬁned by (2.4.4), satisfy the equations 1 div S(v) Re 1 + ∇p(ρ) + C v = ρ f ∗ Ma2 in Ω0 ,

∂t (ρ v) + div(ρ v ⊗ v) −

∂t (ρ) + div(ρv) = 0

in Ω0 ,

(2.4.5a) (2.4.5b)

where the viscous stress tensor takes the form S(v) = ∇v + ∇v + (λ − 1) div v, and the skew-symmetric matrix C = (Cij )d×d and the vector ﬁeld f ∗ are deﬁned by ∂Wi ∂Wj − , Cij = ∂yj ∂yi

52

Chapter 2. Physical background

and

1 1 2 2 U f (x(y, t), t) − ∂t W + 2 ∇|W| . Frm Here, Wi are the components of the vector ﬁeld W given by (2.4.3). The expressions (2.3.6) and (2.3.7) for the power developed by the hydrodynamic forces and the work produced by these forces read in the new variables 1 JΩ = − (∇v + (∇v) + (λ − 1) div v I − σp(ρ) I) n · W dS, (2.4.6) Re ∂Ω0 f∗ =

and WΩ = −

1 Re

T 0

∇v + (∇v) + (λ − 1) div v I − σp(ρ) I n · W dSdt.

∂Ω0

(2.4.7) It is worth noting that from the mathematical standpoint, the only diﬀerence between equations (2.1.6) and (2.4.5) is the presence of the Coriolis force Cv in (2.4.5). Remark 2.4.2. In three dimensions the expression for the Coriolis force can be written in the traditional form of the cross product Cv = 2ω × v,

where the angular velocity is 2ω = rot W.

(2.4.8)

A similar representation holds in the two-dimensional case if we assume that the angular velocity is orthogonal to the plane of motion. Stationary ﬂows. In general, the matrix C and the force f ∗ depend on the time variable t and equations (2.4.5) are nonautonomous. The equations become autonomous and lead to stationary solutions only if ∂t W = ∂t {U (t)f (x(y, t), t)} = 0.

(2.4.9)

By (2.4.3), the equality ∂t W = 0 is equivalent to the relations U(t) = eBt ,

˙ a(t) = e−Bt b,

in which B is a skew-symmetric matrix and b is a constant vector. If this is the case, then the matrix C = B and the corresponding angular velocity ω are constants. In particular, condition (2.4.9) is fulﬁlled if the vector ﬁeld f and the angular velocity ω are constants such that f × ω = 0. Example 2.4.3. The most important case is the Galileo transformation such that U = I,

a(t) = v∞ t,

In this case C = 0,

f∗ =

v∞ = const.

1 f (y − v∞ t, t). Fr2m

2.5. Flow around a moving body

53

Example 2.4.4. Another important example is the motion of gas on rotating Earth. Assume that the origin of the coordinate system is at the center of Earth and the x1 axis is directed to the north pole. In this case the dimensionless angular velocity is given by ω = ωe1 , e1 = (1, 0, 0), ω = ωe /Tc , where ωe is the angular velocity of Earth. If (y, t) is a moving frame connected with Earth, then x = eCω t y,

i.e. U(t) = eCω t ,

Here, the matrix Cω deﬁned by Cω y ⎛ ⎞ 0 0 0 Cω = ω ⎝ 0 0 −1 ⎠ , 0 1 0 ⎛ cos ωt 0 0 1 eCω t = cos ωt ⎝ 0 0

a(t) = 0.

= ω × y has the form

⎞ ⎛ sin ωt 0 0 ⎠ + sin ωt ⎝ 0 1 0

0 0 1

⎞ 0 −1 ⎠ . 0

The gravity force is determined from the Newton law and, after scaling, can be taken in the form x f = − 3, |x| and W = Cω y. It follows from Lemma 2.4.1 that the functions v and ρ satisfy (2.4.5) with 1 y ω2 + f∗ = − 2 ∇(y22 + y32 ). 2 Frm |y|3 This means that in the rotating system of coordinates we have to take into account the Coriolis and centrifugal forces, and the system of governing equations is autonomous, i.e., no terms in the equations are explicitly time dependent, but the solutions are functions of the time and the spatial variables.

2.5 Flow around a moving body Let us return to the ﬂow of a viscous gas around a moving body. Assume that the body is rigid and its conﬁguration St at time t is determined in dimensionless variables by the relations St = U(t)S + a(t),

U U = I,

U(0) = I,

a(0) = 0,

(2.5.1)

where the initial conﬁguration S is a ﬁxed compact subset of Rd . In particular, a ˙ ˙ point of S evolves with the dimensionless velocity V(y, t) = U(t) y + a(t). Then the modiﬁed velocity ﬁeld v(y, t) = U (t)u(x(y, t), t) − W(y, t) with W = U (t)V(y, t)

54

Chapter 2. Physical background

and the density ρ(y, t) = (x(y, t), t) satisfy equations (2.4.5) in the ﬂow domain Ω0 = Rd \ S. It follows from (2.2.7), (2.4.2), and (2.4.3) that in addition ρ and v satisfy the following boundary and initial conditions: v = 0 on ∂S,

(2.5.2)

v(y, 0) = v0 (y), ρ(y, 0) = 0 (y) in Rd \ S, v(y, t) + W(y, t) → 0, ρ → ∞ as |y| → ∞, ˙ ˙ − a(0)). where the initial vector ﬁeld is v0 (y) = u0 (y) − (U(0)y For the purposes of numerical simulation, the boundary value problems in unbounded domains should be replaced by modiﬁed problems in bounded domains. To this end, we choose an arbitrary hold-all domain B ⊂ R3 , for instance, a suﬃciently large ball, such that S B. Next, we transfer the boundary conditions from inﬁnity to ∂B and arrive at the following boundary value problem for v and ρ, for a given T > 0. Problem 2.5.1. Find functions (v, ρ) satisfying 1 div S(v) Re 1 ∇p(ρ) + C v = ρ f ∗ + Ma2

∂t (ρ v) + div(ρ v ⊗ v) −

in Ω × (0, T ),

(2.5.3a)

in Ω × (0, T ),

(2.5.3b)

v = 0 on ∂S × (0, T ), v = −W on ∂B × (0, T ), ρ = ∞ on Σin , v(y, 0) = v0 (y) in Ω, ρ(y, 0) = 0 (y) in Ω,

(2.5.3c)

∂t ρ + div(ρv) = 0

where Ω = B \ S,

Σin = {(y, t) ∈ ∂B × (0, T ) : W(y, t) · n(y) > 0}.

Now we recalculate the work WS of the hydrodynamic forces acting on the moving body. To this end we can use (2.4.7), which gives T 1 WS = − ∇v + (∇v) + (λ − 1) div v I − σp(ρ) I n · W dSdt. Re 0 ∂S (2.5.4) But this formula is inconvenient since it involves the surface integral of ∇v and p which is not deﬁned for weak solutions. We replace (2.5.4) by a more complicated but robust formula which only contains volume integrals. Lemma 2.5.2. Let (v, ρ) be a classical solution to problem (2.5.3) in a cylinder (B \ S) × (0, T ) and suppose a function η ∈ C ∞ (B \ S) satisﬁes the condition η(x) = 0 in a neighborhood of ∂B,

η(x) = 1 in a neighborhood of ∂S. (2.5.5)

2.5. Flow around a moving body

55

Then

η ρ(·, T )v(·, T ) · W(·, T ) − ρ0 v0 · W(·, 0) dx+

WS = −

T

B\S

# 1 T : ∇(ηW) + η(ρf ∗ − Cv) · W dxdt, ρηv · ∂t W + ρ(v ⊗ v) − Re (2.5.6)

"

B\S

0

where T = ∇v + (∇v) + (λ − 1) div v I − σp(ρ) I. Proof. Since η vanishes on ∂B and equals 1 on ∂S, we can integrate by parts to obtain − Tn · W dS = − ηTW · n dS ∂S ∂Ω div T · (ηW) dx − T : ∇(ηW) dx. =− B\S

B\S

From this, the identity 1 1 1 p(ρ)I = S(v) − T, 2 Re Re Ma and (2.5.3a) we conclude that

T

"

WS = −

η∂t (ρv) · W + div(ρv ⊗ v) · (ηW) + ηCv · W B\S

0

# 1 T : ∇(η∇W) dxdt. − ηρf · W + Re ∗

(2.5.7)

Since ηv ⊗ v = 0 on ∂B ∪ ∂S, we can integrate by parts to obtain

T

− 0

η∂t (ρv) · W + div(ρv ⊗ v) · (ηW) dxdt

B\S

T

= 0

ρηv · ∂t W + ρ(v ⊗ v) : ∇(ηW) dxdt

B\S

−

η ρ(·, T )v(·, T ) · W(·, T ) − ρ0 v0 · W(·, 0) dx.

B\S

Inserting this into (2.5.7) we arrive at (2.5.6).

Chapter 3

Problem formulation In this chapter we give the mathematical formulation of the problem which we intend to consider in this monograph. Recall that our main objective is to develop a mathematical theory for the drag optimization problem and investigation of the related mathematical questions in the theory of compressible Navier-Stokes equations. We focus on the problem of the compressible viscous gas ﬂow around a moving rigid body (2.5.3) and on the question of optimal choice of the shape of this body in order to minimize the drag functional in the stationary case or the work in the nonstationary case. By abuse of notation we return to the original notation and we will write x, u, instead of y, v, ρ, and f instead of f ∗ . Next notice that explicit values of the Reynolds and Mach numbers are not essential for the general mathematical theory of Navier-Stokes equations, provided that these numbers are separated from 0 and bounded from above. The Coriolis force is also immaterial for the mathematical theory. Hence without loss of generality we may assume that Ma2 = Re = 1 and

lc = uc = Tc = 1,

C = 0.

(3.0.1)

Next, in problem (2.5.3) of the ﬂow around an obstacle, the velocity ﬁeld vanishes on the obstacle S, equals −W on the boundary of the hold-all domain B, and coincides with a given vector ﬁeld v0 at the initial time. It is convenient to introduce a vector ﬁeld U : Rd × [0, T ) → Rd such that U = −W on ∂B × (0, T ),

U = 0 on S × (0, T ),

U = v0 on (B \ S) × {t = 0}. (3.0.2) When v0 is smooth and satisﬁes an obvious compatibility condition we can assume that U is also a smooth vector ﬁeld. Thus we come to the following boundary value problem in the ﬂow domain Ω = B \ S ⊂ Rd ,

d = 2, 3.

P. Plotnikov and J. Sokoáowski, Compressible Navier–Stokes Equations: Theory and Shape Optimization, Monografie Matematyczne 73, DOI 10.1007/978-3-0348-0367-0_3, © Springer Basel 2012

57

58

Chapter 3. Problem formulation

Problem 3.0.3. For given T > 0 and for given functions ∞ : Ω × [0, T ) → R+ , U : Ω × [0, T ) → Rd ,

(3.0.3)

f : Ω × (0, T ) → Rd , ﬁnd a velocity ﬁeld u : Ω×[0, T ) → Rd and a density : Ω×[0, T ) → R+ satisfying ∂t ( u) + div( u ⊗ u) + ∇p(ρ) = div S(u) + f ∂t + div(u) = 0

in Ω × (0, T ),

in Ω × (0, T ),

(3.0.4b)

u = 0 on ∂S × (0, T ), u = U on ∂B × (0, T ), = ∞ on Σin , u(x, 0) = U(x, 0) in Ω, (x, 0) = ∞ (x, 0)

(3.0.4a)

(3.0.4c)

in Ω,

where S(u) = ∇u + ∇u + (λ − 1) div u,

div S(u) = Δu + λ∇ div u,

and the inlet Σin is deﬁned by Σin = {(x, t) ∈ ∂B × (0, T ) : U(x, t) · n(x) < 0}. Recall that n is the unit outward normal to the boundary ∂Ω. The problem in the stationary case can be formulated as follows: Problem 3.0.4. For given U : ∂Ω → Rd ,

∞ : ∂Ω → Rd ,

f : Ω → Rd ,

(3.0.5)

ﬁnd a velocity ﬁeld u : Ω → Rd and a density : Ω → R+ satisfying div( u ⊗ u) + ∇p(ρ) = div S(u) + f div(u) = 0 u=0

in Ω,

in Ω,

(3.0.6a) (3.0.6b)

on ∂S,

u = U on ∂B, = ∞ on Σin , where the inlet Σin is deﬁned by Σin = {x ∈ ∂B : U(x) · n(x) < 0}.

(3.0.6c)

3.1. Weak solutions

59

Remark 3.0.5. For stationary ﬂows, if U · n = 0 on ∂Ω, the inlet is an empty set. In this particular case the total mass M of the gas should be prescribed: (x) dx = M. Ω

3.1 Weak solutions All known results in the global existence theory for compressible Navier-Stokes equations concern the so-called weak solutions. The notion of a weak solution arises in a natural way from the formulation of the governing equations as a system of conservation laws. In our particular case weak solutions are deﬁned as follows: Deﬁnition 3.1.1. Let Ω ⊂ Rd be a bounded domain with C 3 boundary ∂Ω and Q be the cylinder Ω × (0, T ). A couple (, u) : Q → R+ × Rd is a weak solution to Problem (3.0.3) if the following conditions are satisﬁed: • The state variables satisfy , p, |u|2 ∈ L∞ (0, T ; L1 (Ω)), u ∈ L2 (0, T ; W 1,2 (Ω)), u = U on ∂Ω × [0, T ].

(3.1.1) (3.1.2)

• The integral identity u · ∂t ψ + u ⊗ u : ∇ψ + p div ψ − S(u) : ∇ψ dxdt Q + f · ψ dxdt + ∞ U · ψ(·, 0) dx = 0 (3.1.3) Q

Ω ∞

holds for all vector ﬁelds ψ ∈ C (Q) satisfying ψ = 0 in a neighborhood of {∂Ω × [0, T ]} ∪ {Ω × {T }}, which means that the test vector ﬁelds vanish at the lateral side and the lid of the cylinder Q. • The integral identity ( ∂t ζ + u · ∇ζ) dxdt = Q

ζ∞ U · n dSdt −

∂Ω×[0,T ]

∞ ζ(·, 0) dx (3.1.4) Ω

holds for all functions ζ ∈ C ∞ (Q) satisfying ζ = 0 in a neighborhood of {(∂Ω × [0, T ]) \ Σin } ∪ {Ω × {T }}, which means that the support of the test function ζ meets the boundary of the cylinder Q at the inlet Σin and the bottom Ω × {0}.

60

Chapter 3. Problem formulation

3.2 Renormalized solutions The mass balance equation (3.0.4b) is a ﬁrst order linear partial diﬀerential equation for the density . Given a smooth solution to this equation, the composite function ϕ() satisﬁes the ﬁrst order diﬀerential equation ∂t ϕ() + div(ϕ()u) + (ϕ () − ϕ()) div u = 0.

(3.2.1)

Thus, a smooth solution to the mass balance equation generates a solution in the form ϕ() to the ﬁrst order quasi-linear diﬀerential equation (3.2.1); the latter solution is associated with the prescribed function ϕ. Equation (3.2.1) derived for ϕ() is called a renormalized equation. The derivation of a renormalized equation is called renormalization. The renormalization procedure is simple for smooth solutions to the mass balance equation, but becomes nontrivial for weak solutions. It turns out that the renormalized equation is convenient for our purposes. A natural way to avoid the renormalization procedure for weak solutions to the mass balance equation is simple replacement of the mass balance equation by its renormalized version. This means that it is required that the density and the velocity satisfy (3.2.1) for a speciﬁc class of functions ϕ. Thus we come to the following deﬁnition: Deﬁnition 3.2.1. Let (, u) : Q → R+ × Rd be a weak solution to Problem 3.0.3 satisfying all hypotheses of Deﬁnition 3.1.1. We say that (, u) is a renormalized solution to Problem 3.0.3 if the integral identity Q

ϕ() ∂t ζ + ϕ()u · ∇ζ − ϕ () − ϕ() ζ div u dxdt = ζϕ(∞ )U · n dSdt − ϕ(∞ )ζ(·, 0) dx (3.2.2) ∂Ω×[0,T ]

Ω

holds for all functions ζ ∈ C ∞ (Q) satisfying ζ = 0 in a neighborhood of {(∂Ω) \ Σin } ∪ Ω × {T } , and for all functions ϕ ∈ C01 (R). Remark 3.2.2. Obviously, any C 1 function ϕ : R → R+ can be approximated by a sequence of compactly supported C 1 functions ϕn such that ϕn (s) → ϕ(s),

ϕn (s) → ϕ (s) as n → ∞ uniformly on compact subsets of R,

Hence the integral identity (3.2.2) is satisﬁed for all C 1 functions ϕ such that ϕ(), ϕ () ∈ L2 (Q).

3.3. Work functional. Optimization problem

61

3.3 Work functional. Optimization problem If we adopt the conventions formulated at the beginning of this chapter, in particular if we assume that the dimensionless parameters satisfy condition (3.0.1), then Lemma 2.5.2 leads to the following expression for the work WS of the hydrodynamic forces acting on the moving obstacle S: WS = −

T

B\S

+ 0

where

η (·, T )u(·, T ) · W(·, T ) − 0 U · W(·, 0) dx

ηu · ∂t W + (u ⊗ u) − T : ∇(ηW) + η(f ) · W dxdt,

(3.3.1)

B\S

T = ∇u + (∇u) + (λ − 1) div u I − p() I.

Recall that if the obstacle S moves in atmosphere like a solid body then its physical position St at time t is deﬁned by (2.5.1) with Tc = 1 (because of (3.0.1)), i.e., St = U(t)S + a(t).

(3.3.2)

Here the unitary matrix U and the vector ﬁeld a(t) can be deﬁned by an appropriate ﬂight planning scenario. In this case the vector ﬁeld W is given by (2.4.3), which in accordance with (3.0.1) reads ˙ ˙ W(x, t) = U (t)U(t)x + U (t)a(t). (3.3.3) Remark 3.3.1. Notice that both vector ﬁelds U and W are deﬁned in the strip Rd × [0, T ], but the equality U = −W holds only on the surface ∂B × (0, T ). From the mathematical point of view this connection and representation (3.3.3) are not essential and we cam consider U and W as independent smooth vector ﬁelds. When a solid body moves in atmosphere along the trajectory prescribed by (3.3.2), the corresponding value WS is nothing other than the work done by the engine which moves the body. A natural question is to minimize this work by choosing an appropriate shape of S from some suitable class S.

Chapter 4

Basic statements 4.1 Introduction The mathematical theory of compressible Navier-Stokes equations is based on a number of statements which are not related directly to the speciﬁc structure of the Navier-Stokes equations themselves but are of a general character. In this chapter we assemble the basic facts of general character derived for solutions to the mass and momentum conservation equations ∂t + div(u − g) = 0, ∂t (u) + div(u ⊗ u) = div T + ρf .

(4.1.1)

The majority of the statements formulated and proved here are independent of the form of the constitutive laws and pertain to both original and regularized equations of the motion of viscous ﬂuids. We specify neither the form of the stress tensor T nor the mass source g. In fact, system (4.1.1) is not complete and it can be regarded as a diﬀerential relation between the state variables , u and T. Moreover, we do not impose boundary and initial conditions for the state variables at this stage. Instead we assume that they satisfy some integrability conditions. The results we present include elementary facts on integrability of functions with ﬁnite energy (Section 4.2), the standard material concerning weak compactness properties of solutions to the mass and momentum balance laws (Section 4.4), and the famous Lions result on the weak continuity of the so-called viscous ﬂux, which is the most important result of the mathematical theory of viscous compressible ﬂows (Section 4.6).

4.2 Bounded energy functions In the mathematical theory of ﬂuid mechanics, the stress tensor T is related to the density and to the velocity vector ﬁeld u by some constitutive relations. For

P. Plotnikov and J. Sokoáowski, Compressible Navier–Stokes Equations: Theory and Shape Optimization, Monografie Matematyczne 73, DOI 10.1007/978-3-0348-0367-0_4, © Springer Basel 2012

63

64

Chapter 4. Basic statements

example, for adiabatic ﬂows of the viscous gas, the constitutive law reads T = S(u) − p()I,

p() = cγ ,

where γ is the adiabatic exponent, and S(u) is the viscous stress tensor. The constitutive relations should be in agreement with the second principle of thermodynamics, which yields the boundedness of the total energy and of the integral of energy dissipation rate. Thus L∞ (0,T ;Lγ (Ω)) + |u|2 L∞ (0,T ;L1 (Ω)) ≤ E, uL2 (0,T ;W 1,2 (Ω)) ≤ E,

(4.2.1)

where the constant E depends on initial and boundary data. These inequalities are called the energy estimates or the ﬁrst estimates for solutions of (4.1.1). Functions (, u) satisfying (4.2.1) are said to be bounded energy functions. For the sake of generality we do not specify the constitutive relations and simply postulate estimates (4.2.1). In this section we derive the integrability properties of the density and of the velocity ﬁelds satisfying the energy estimates. This means that we obtain a collection of Lp -estimates for the momentum u and the kinetic energy tensor u ⊗ u in terms of the constant E and the domain Ω. The results are of general character and are used throughout the monograph. Let Ω ⊂ Rd , d = 2, 3, be a bounded domain with Lipschitz boundary and Q = Ω × (0, T ). Assume that a density : Q → R+ and a velocity ﬁeld u : Q → Rd satisfy the energy inequalities (4.2.1). This raises the question on the existence of robust estimates for the momentum u and the kinetic energy tensor u ⊗ u. The following proposition gives estimates which are suﬃcient for our purposes. We introduce the deviation parameter a and the critical exponent b by the equalities a=

2 1 − , d γ

b=

2γ . γ+1

(4.2.2)

Proposition 4.2.1. Assume that and u satisfy inequalities (4.2.1) with γ > d/2, and that α ∈ [2, ∞] and β ∈ [1, ∞) satisfy the conditions 1 1 a + ≥ α β b

for α < ∞, d = 3

a 1 1 + > α β b

and

α = ∞, d ≥ 2,

for α < ∞, d = 2.

(4.2.3) (4.2.4)

Then there is a constant cE , depending only on E, Ω, γ, α, and β, such that uLα (0,T ;Lβ (Ω)) ≤ cE .

(4.2.5)

Proof. First, we consider the most complicated case α < ∞. Represent β |u|β in the form √ β |u|β = λ ( |u|)μ |u|ω

4.2. Bounded energy functions

65

with nonnegative λ, μ, ω satisfying (4.2.6)

λ + μ/2 = μ + ω = β. Next, for all nonnegative exponents m, n, l such that 1 1 1 + + = 1, n m l

λn = γ,

(4.2.7)

μm = 2,

the Hölder inequality implies

1/n √

1/m

1/l |u| dx ≤ dx |u| dx |u|ωl dx Ω Ω Ω Ω 1/n 1/m 1/l 1/l γ 2 ωl = dx |u| dx |u| dx ≤ cE |u|ωl dx , β

β

λn

Ω

Ω

μm

μm

Ω

Ω

which leads to

ω/β

uLβ (Ω) ≤ cE uLωl (Ω) . Now set k = ωl,

(4.2.8)

α = 2β/ω.

Thus we get uLα (0,T ;Lβ (Ω)) =

≤ cE

0

T

(t)u(t)α Lβ (Ω) dt

T 0

u(t)2Lk (Ω) dt

1/α

1/α

2/α

= cE uL2 (0,T ;Lk (Ω)) .

(4.2.9)

Notice that estimate (4.2.9) holds if and only if the exponents m, n, l and λ, μ, ω are nonnegative and satisfy relations (4.2.6) and (4.2.7). This leads to some restrictions on the range of the parameters β and α = 2β/ω. To analyze such restrictions we notice that relations (4.2.6) and (4.2.7) give the following expressions for n and m in terms of ω, l, and γ: 1 1 1 1 γ 1 = = γ−ω− . (4.2.10) 1+ω− , n 1+γ l m 1+γ l It follows that 1/m ≤ γ/n. Hence m ≥ 0 yields n ≥ 0. Moreover if m, n, l and ω are nonnegative, then by (4.2.6) and (4.2.7), so are λ, μ and β. Hence the restrictions on the set of parameters are m ≥ 0,

l ≥ 0,

ω ≥ 0.

Next, relations (4.2.6), (4.2.7), and (4.2.10) give 2 γ β= γ−ω− + ω, 1+γ l

(4.2.11)

66

Chapter 4. Basic statements

which yields, in terms of γ and β, 1 γ−1 γ+1 1 1 γ+1 1 =1+ ω− β=β + 1− − . l 2γ 2γ β γ α 2γ Substituting this into (4.2.10) we can rewrite (4.2.11) in the form β ≥ 0,

α ≥ 2,

γ−1 γ+1 1 + ≥ . β γα 2γ

(4.2.12)

This system of inequalities is incomplete, because they do not guarantee the existence of the integral on the right hand side of (4.2.9). In order to cope with this diﬃculty notice that the embedding W 1,2 (Ω) → Lk (Ω) is bounded for 1 ≤ k ≤ 2d/(d − 2) if d ≥ 3,

and

k < ∞ if d = 2.

(4.2.13)

Recall that k = 2lβ/α and α < ∞. It follows from this and (4.2.13) that the embedding W 1,2 (Ω) → Lk (Ω) is bounded for d−2 β 1 ≥ l d α

and

1/l > 0.

(4.2.14)

Combining this with (4.2.9) and (4.2.1) we deduce that uLα (0,T ;Lβ (Ω)) ≤ cE for α, β satisfying (4.2.14). Let us consider inequalities (4.2.14) in more detail. By using the formula for 1/l we rewrite (4.2.14) as 2 1 β γ+1 1+ − − β ≥ 0, d γ α 2γ (4.2.15) 1 β γ+1 1+ 1− − β > 0. γ α 2γ Since we are considering the case α < ∞, these inequalities can be replaced by 1 β γ+1 2 − −β ≥ 0 for d ≥ 3 (4.2.16) 1+ d γ α 2γ and 1+

2 1 − d γ

γ+1 β −β > 0 for d = 2. α 2γ

(4.2.17)

Recalling the notations a and b we can rewrite (4.2.16)–(4.2.17) in the equivalent form a 1 1 a 1 1 + ≥ for d ≥ 3, + > for d = 2. (4.2.18) α β b α β b This leads to the statement of Proposition 4.2.1 in the case of α < ∞.

4.2. Bounded energy functions

67

It remains to consider the case α = ∞. By the Hölder inequality, 1/(γ+1) γ/(γ+1) (|u|)2γ/(γ+1) dx ≤ γ dx |u|2 dx , Ω

Ω

Ω

which gives uL∞ (0,T ;L2γ/(γ+1) (Ω)) ≤ L∞ (0,T ;Lγ (Ω)) |u|2 L∞ (0,T ;L1 (Ω)) ≤ cE .

Later we use only the particular cases of Proposition 4.2.1 with α = ∞, 2. The corresponding result is given by the following corollary: Corollary 4.2.2. Assume that and u satisfy inequalities (4.2.1) with γ > d/2. Then there is a constant cE , depending only on E, β, α, and Ω, such that uL∞ (0,T ;L2γ/(γ+1) (Ω)) ≤ cE , uL2 (0,T ;Lβ (Ω)) ≤ cE

for all β ∈ [1, ∞]

uLα (0,T ;L2 (Ω)) ≤ cE

for all α ∈ [1, 4γ/d − 2).

with

1 1 1 1 > + − , β 2 γ d

Proof. Take α = ∞, 2 and next β = 2 in (4.2.3) and (4.2.4).

Proposition 4.2.1 leads to a useful estimate for the kinetic energy density. Corollary 4.2.3. Suppose that γ > d/2, the functions , u satisfy (4.2.1), and a ≥ 1, b ≥ 1 satisfy the inequalities 1 1 1 d−2 1 − . (4.2.19) 1 ≤ a ≤ 2, a + > + a 2 b b 2d Then there is a constant cE , depending only on E, a, b, γ and Ω, such that |u|2 La (0,T ;Lb (Ω)) ≤ cE .

(4.2.20)

In particular |u|2 L2 (0,T ;Lτ (Ω)) ≤ cE |u| L1 (0,T ;Lτ (Ω)) ≤ cE 2

$ for all τ ∈ 1, (1 − 2−1 a)−1 , $ for all τ ∈ 1, (1 − a)−1 .

(4.2.21) (4.2.22)

Proof. Proposition 4.2.1 and the Hölder inequality yield |u|2 La (0,T ;Lb (Ω)) ≤ uLα (0,T ;Lβ (Ω)) uL2 (0,T ;Lq (Ω)) ≤ cuLα (0,T ;Lβ (Ω)) uL2 (0,T ;W 1,2 (Ω)) ≤ C(E) for all nonnegative exponents α, β, q satisfying 1 1 1 = + , a α 2

1 1 1 = + , b β q

68

Chapter 4. Basic statements

and

a 1 1 + > , α β b

q<

2d . d−2

It is easy to check that such exponents exist for all a and b satisfying (4.2.19). It remains to note that estimates (4.2.21) and (4.2.22) are particular cases of (4.2.19)–(4.2.20).

4.3 Functions of bounded mass dissipation rate Along with bounded energy functions (, u) that satisfy the energy estimates (4.2.1), we consider functions with the ﬁnite mass dissipation rate, ε (1 + )γ−2 |∇(x, t)|2 dxdt ≤ E < ∞, (4.3.1) Q

where the mass diﬀusion coeﬃcient ε is a small parameter. Because such functions have some additional regularity compared to bounded energy functions, the results of Propositions 4.2.1 and Corollaries 4.2.2–4.2.3 can be improved. In this section we obtain an estimate for the convection term ∇⊗u and the “internal energy” γ . Proposition 4.3.1. Let Ω be a bounded domain with Lipschitz boundary. Assume that and u satisfy inequalities (4.2.1) and (4.3.1) with γ > 2d. Then there is a constant cE , depending only on E, Ω, γ, and q, such that |∇| |u|L1 (0,T ;Lq (Ω)) ≤ ε−1/2 cE

for all 1 ≤ q ≤

3γ/2 L1 (Q) ≤ cE ε−1 .

d , d−1

(4.3.2) (4.3.3)

Proof. Inequalities (4.2.1) and (4.3.1) yield εγ/2 L2 (0,T ;W 1,2 (Ω)) = ε

γ + Q

1/2 γ 2 γ−2 |∇|2 dxdt ≤ ε1/2 cE , (4.3.4) 4

where the constant cE depends only on Ω, E, and γ. The same arguments give 1/2 εL2 (0,T ;W 1,2 (Ω)) = ε (2 + |∇|2 ) dxdt ≤ ε1/2 cE . (4.3.5) Q

On the other hand, the Hölder inequality implies the estimate ε|∇(t)| |u(t)|Lq (Ω) ≤ cε∇(t)L2 (Ω) u(t)Lm (Ω) whenever

1 1 1 ≥ + . q 2 m

4.4. Compactness properties

69

Since the embedding Lm (Ω) → W 1,2 (Ω) is bounded for m < 2d/(d − 2), we conclude that ε∇(t)u(t)Lq (Ω) ≤ cε1/2 ε1/2 ∇(t)L2 (Ω) u(t)W 1,2 (Ω) and hence ε∇(t)u(t)L1 (0,T ;Lq (Ω)) ≤ cε1/2 cE whenever

1 1 d−2 d > + ⇔ q< , q 2 2d d−1 which yields (4.3.2). Next, 1/2 1/2 1/2 3γ/2 dx ≤ γ dx 2γ dx ≤ cE 2γ dx ≤ cE γ/2 2L4 (Ω) . Ω

Ω

Ω

Ω

Since 4 < 2d/(d − 2), we have γ/2 (t)2L4 (Ω) ≤ cγ/2 (t)2W 1,2 (Ω) . Thus we get

3γ/2 dxdt ≤ cE Q

0

T

γ/2 (t)2W 1,2 (Ω) dt ≤ cE ε−1 .

4.4 Compactness properties The compactness properties of solutions to the mass and momentum balance equations are crucial for the theory of weak solutions to Navier-Stokes equations. There are two reasons for this. The ﬁrst is existence theory. Notice that the standard proof of existence of weak solutions to boundary value problems for Navier-Stokes equations consists of two steps. First, we construct a sequence of approximate solutions with uniformly bounded energy. Next, we prove that any weak limit of such a sequence is a solution to the governing equations. The latter step is almost trivial for incompressible ﬂuids, but leads to diﬃcult problems for compressible Navier-Stokes equations. The second reason for the necessity of a detailed study of weak compactness properties of the solution set to the Navier-Stokes equations is that the compactness of this set implies its stability with respect to perturbations of given data. In this framework, the term compactness is not appropriate and should be replaced by weak compactness or, more precisely, weak closedness. In this section we consider elementary weak compactness properties of bounded energy functions; integrability of such functions was studied in Section 4.2. Assume that Ω is a bounded domain with Lipschitz boundary and set Q = Ω × (0, T ),

0 < T < ∞.

Let us consider sequences of vector ﬁelds un : Q → Rd and of functions n , ϕn : Q → R, n ≥ 1, such that n ≥ 0. Assume that the following condition is satisﬁed:

70

Chapter 4. Basic statements

Condition 4.4.1. • There are exponents d/2 < γ < ∞, d/2 < s ≤ ∞ and a constant E independent of n such that the functions n , ϕn , and the vector ﬁelds un satisfy the “energy inequalities” γn L1 (Q) + un L2 (0,T ;W 1,2 (Ω)) ≤ E,

(4.4.1)

n |un | L∞ (0,T ;L1 (Ω)) ≤ E, ϕn L∞ (0,T ;Ls (Ω)) ≤ E.

(4.4.2) (4.4.3)

2

• There are integrable vector ﬁelds un , vn , w : Q → Rd , functions gn , hn : Q → R and matrix-valued functions Vn : Q → Rd×d such that un L1 (Q) + vn L1 (Q) + wn L1 (Q) + gn L1 (Q) + hn L1 (Q) + Vn L1 (Q) ≤ E

(4.4.4)

and ∂t n = div un + gn , ∂t ϕn = div vn + hn , ∂t (n un ) = div Vn + wn .

(4.4.5)

The following theorem on weak compactness properties of the momentum u and of the kinetic energy tensor u ⊗ u is the main result of this section. Theorem 4.4.2. Let a sequence (un , n , ϕn ) satisfy Conditions 4.4.1. Then there is a subsequence, still denoted by (un , n , ϕn ), a vector ﬁeld u ∈ L2 (0, T ; W 1,2 (Ω)), and functions ∈ L∞ (0, T ; Lγ (Ω)), ϕ ∈ L∞ (0, T ; Ls (Ω)) such that weakly in L∞ (0, T ; Lγ (Ω)),

n ϕn ϕ

weakly in L∞ (0, T ; Ls (Ω)),

un u

2

weakly in L (0, T ; W

1,2

(4.4.6)

(Ω));

for m−1 > 2−1 + γ −1 − d−1 , n un u ϕn un ϕu

weakly in L2 (0, T ; Lm (Ω)), 2

m

weakly in L (0, T ; L (Ω));

(4.4.7) (4.4.8)

and for 1 < b < (1 − 2−1 a)−1 , n un ⊗ un u ⊗ u

weakly in L2 (0, T ; Lb (Ω)).

(4.4.9)

Moreover, for a.e. t ∈ (0, T ), n (t) (t) ϕn (t) ϕ(t) n (t)un (t) (t)u(t)

weakly in Lγ (Ω), weakly (weakly for s = ∞) in Ls (Ω),

(4.4.10a) (4.4.10b)

weakly in L2γ/(γ+1) (Ω).

(4.4.10c)

4.4. Compactness properties

71

The remaining part of this section is devoted to the proof of Theorem 4.4.2. It is based on two auxiliary compactness lemmas which describe the pointwise convergence of weakly convergent sequences and the weak continuity of products of two weakly convergent sequences. These results can be regarded as a simplest version, appropriate for our purposes, of the compensated compactness principle.

4.4.1 Two lemmas on compensated compactness First, we investigate pointwise convergence of integrable functions depending on the time variable t ∈ (0, T ). Let us consider a bounded sequence of functions Hn ∈ L∞ (0, T ; Lr (Ω)), n ≥ 1, r ∈ (1, ∞]. We prove a simple criterion for the weak convergence of the functions Hn (t, ·). For every ξ ∈ C0∞ (Ω) set Hn (t)ξ dx. Fn [ξ](t) = Ω ∞

Since the sequence Hn is bounded in L (0, T ; Lr (Ω)), the functions Fn [ξ] belong to L∞ (0, T ). Lemma 4.4.3. Let Hn be a bounded sequence in L∞ (0, T ; Lr (Ω)) for r ∈ (1, ∞]. Furthermore, assume that for every ξ ∈ C0∞ (Ω) and for n ≥ 1, the function Fn [ξ] can be modiﬁed on a set of zero measure in such a way that Fn [ξ] ≤ Cξ , (4.4.11) [0,T ]

where Cξ is independent of n. Then there is a function H ∈ L∞ (0, T ; Lr (Ω)) and a subsequence, still denoted by Hn , such that for a.e. t ∈ [0, T ], Hn (t) H(t) weakly (weakly for r = ∞) in Lr (Ω). % For the deﬁnition of the variation and elementary properties of functions of bounded variation we refer to Section 1.3.1. Proof. For every n, there is a set Tn of full measure in [0, T ] such that

Hn (t)Lr (Ω) ≤ Hn L∞ (0,T ;Lr (Ω)) ,

t ∈ Tn .

Set T = n Tn . Next, choose a countable set of functions ξm ∈ C0∞ (Ω), m ≥ 1, which is dense in Lr (Ω), where r = r/(r − 1) for r < ∞ and r = 1 for r = ∞. It suﬃces to prove that for all m and t ∈ T , the limit lim Hn (t)ξm dx (4.4.12) n→∞

Ω r

exists. Indeed, ﬁx an arbitrary ξ ∈ L (Ω). For every δ > 0, there is m such that ξ − ξm Lr (Ω) < δ. In view of lim sup (Hp (t) − Hq (t))ξm dx = 0, p,q→∞

Ω

72

Chapter 4. Basic statements

we have for t ∈ T , lim sup (Hp (t) − Hq (t))ξ dx ≤ lim sup (Hp (t) − Hq (t))ξm dx p,q→∞ p,q→∞ Ω Ω + lim sup (|Hp (t)| + |Hq (t)|)|ξm − ξ| dx p,q→∞ Ω ≤ c lim sup |Hp (t)| + |Hq (t)|Lr (Ω) ξ − ξm Lr (Ω) ≤ cδ. p,q→∞

Since δ > 0 is arbitrary, we conclude that for all t ∈ T and ξ ∈ Lr (Ω), the limit Hn (t)ξ dx lim n→∞

Ω

exists. Hence the sequence Hn (t) converges weakly (weakly for r = ∞) in Lr (Ω). Recall that the Lebesgue space Lr (Ω) with r ∈ (1, ∞) is weakly complete and L∞ (Ω) is weak complete. Therefore, there exists H(t) such that Hn (t) converges to H(t) weakly in Lr (Ω) (weakly for r = ∞). Let us prove the existence of the limit (4.4.12). It follows from (4.4.11) that for every ﬁxed m the functions Fn(m) = Fn [ξm ],

n ≥ 1,

are uniformly bounded in [0, T ] and have uniformly bounded variations. Now we apply a diagonal process. By the Helly compactness theorem 1.3.4, there is a (1) subsequence {H1,n } ⊂ {Hn } and a bounded function F1,∞ : [0, T ] → R such that the functions (1) H1,n (t)ξ1 dx F1,n (t) := Ω

(1) F1,∞

converge to pointwise on [0, T ]. In its turn, there is a subsequence {H2,n } ⊂ {H1,n } ⊂ {Hn } such that the functions (m) F2,n (t) := H2,n (t)ξm dx, m = 1, 2, Ω

(m)

converge to some bounded functions F2,∞ pointwise on [0, T ] as n → ∞. Repeating these arguments we conclude that there are sequences · · · ⊂ {Hk,n } ⊂ {Hk−1,n } ⊂ · · · ⊂ {Hn } (m)

and bounded functions Fk,∞ , m ≥ 1, such that for every 1 ≤ m ≤ k, and for n → ∞, (m) (m) Fk,n (t) := Hk,n (t)ξm dx → Fk,∞ (t) pointwise on [0, T ]. Ω

4.4. Compactness properties

73

Hence for every m ≥ 1, the diagonal sequence (m) {Fn,n (t)} := Hn,n (t)ξm dx Ω

converges pointwise on [0, T ] to a function F∞ ∈ L∞ (0, T ). Notice that {Hn,n } is a subsequence of {Hn }. By abuse of notation we write simply Hn instead of Hn,n . In other words, there is a subsequence of Hn , still denoted by Hn , such that for every m ≥ 1, (m) Hn (t)ξm dx → F∞ (t) pointwise on [0, T ], (4.4.13) (m)

Ω

which yields the existence of the limit (4.4.12).

The second auxiliary lemma establishes additional suﬃcient conditions for pointwise weak convergence of sequences of time dependent integrable functions. Lemma 4.4.4. Let r ∈ (1, ∞], {Hn }n≥1 a bounded sequence in L∞ (0, T ; Lr (Ω)), and {Sn }n≥1 a bounded sequence in L∞ (0, T ; W −k,h (Ω)), with k ≥ 0 and 1 < h< ∞, such that Sn (t)W −k,h (Ω) → 0

as n → ∞

for a.e. t ∈ (0, T ).

(4.4.14)

Furthermore, assume that ∂t (Sn + Hn ) = div Qn + qn ,

(4.4.15)

where the sequences {Qn }n≥1 and {qn }n≥1 are bounded in L1 (Q). Then there is a function H ∈ L∞ (0, T ; Lr (Ω)) and a subsequence, still denoted by Hn , such that Hn (t) H(t)

weakly (weakly for r = ∞) in Lr (Ω)

for a.e. t ∈ [0, T ].

Proof. Choose a positive constant M and a set T of full measure in (0, T ) such that Sn L∞ (0,T ;W −k,h (Ω)) + Hn L∞ (0,T ;Lr (Ω)) ≤ M, Sn (t)W −k,h (Ω) + Hn (t)Lr (Ω) ≤ M For any ξ ∈ C0∞ (Ω) set

for all t ∈ T .

Fn [ξ](t) = Sn (t), ξ +

Hn (x, t)ξ(x) dx, Ω

where ·, · stands for the duality pairing between W −k,h (Ω) and W0k,h (Ω). We have T T η (t)Fn [ξ](t) dt = η(t)Nn [ξ](t) dt for all η ∈ C0∞ (0, T ), (4.4.16) 0

0

74

Chapter 4. Basic statements

where

(Qn · ∇ξ − qn ξ) dx.

Nn [ξ](t) = Ω

The linear mapping ξ → Nn [ξ](t) admits the estimate |Nn [ξ](t)| ≤ (Qn (t)L1 (Ω) + qn L1 (Ω) )ξC 1 (Ω) . Since the sequences {Qn }n≥1 and {qn }n≥1 are bounded in L1 (Q), we have

T

|Nn [ξ](t)| dt ≤ cξC 1 (Ω) . 0

Next we have, a.e. on (0, T ), |Fn [ξ](t)| ≤ Sn (t)W −k,h (Ω) ξC k (Ω) + Hn Lr (Ω) ξC(Ω) ≤ cM ξC k (Ω) . It follows that the function t → Fn [ξ](t) meets all requirements of Lemma 1.3.3, so after a possible change of its values on a set of zero measure we obtain Fn [ξ] ≤ cξC k (Ω) . [0,T ]

It now follows from the Helly Theorem 1.3.4 that for every ξ ∈ C0k (Ω), there exists a subsequence, still denoted by Fn [ξ], which converges everywhere in [0, T ]. Choosing a countable set {ξm } dense in C0k (Ω), arguing as in the proof of Lemma 4.4.3, and applying the diagonal process, we conclude that there is a subsequence, still denoted by Fn , such that for all t ∈ (0, T ) and ξ ∈ C0k (Ω), the limits limn→∞ Fn [ξ](t) exist. In particular, for all t ∈ T and ξ ∈ C0k (Ω), the limits Hn (x, t)ξ(x) dx lim Sn (t), ξ + n→∞

Ω

exist. By (4.4.14) we can choose T in such a way that lim Sn (t), ξ = 0 for all t ∈ T and ξ ∈ C0k (Ω).

n→∞

Therefore, as n → ∞, Hn (x, t)ξ(x) dx converges for all t ∈ T and ξ ∈ C0k (Ω). Ω

Since the sequence {Hn (t)} is bounded in Lr (Ω), it follows that it converges weakly in Lr (Ω) (weakly for r = ∞) to some H(t), obviously belonging to L∞ (0, T ; Lr (Ω)). The next lemma concerns the continuity properties of a product of weakly convergent sequences.

4.4. Compactness properties

75

Lemma 4.4.5. Let 2d/(d + 2) < r ≤ ∞ and suppose sequences Hn and un satisfy the conditions Hn L∞ (0,T ;Lr (Ω)) + un L2 (0,T ;W 1,2 (Ω)) ≤ c, Hn (t) H(t)

un u weakly in L2 (0, T ; W 1,2 (Ω)), weakly (weakly for r = ∞) in Lr (Ω) for a.e. t ∈ (0, T ).

Then Hn un H u weakly in L2 (0, T ; Lq (Ω)) for all q ∈ (1, ∞) with q −1 > 2−1 + r−1 − d−1 . Proof. The Hölder inequality implies Hn un L2 (0,T ;Lq (Ω)) ≤ Hn L∞ (0,T ;Lr (Ω)) uL2 (0,T ;Lσ (Ω)) ≤ cuL2 (0,T ;Lσ (Ω)) for all σ ∈ [1, ∞] such that q −1 ≥ r−1 + σ −1

or equivalently σ −1 ≤ q −1 − r−1 .

(4.4.17)

It follows from the assumptions of the lemma that q −1 − r−1 > (d − 2)/(2d). Hence there is a σ < 2d(d − 2) satisfying (4.4.17). For such a σ the embedding W 1,2 (Ω) → Lσ (Ω) is bounded, which leads to the estimate Hn un L2 (0,T ;Lq (Ω)) ≤ cuL2 (0,T ;Lσ (Ω)) ≤ cuL2 (0,T ;W 1,2 (Ω)) ≤ c. Hence, it suﬃces to show that for all vector ﬁelds ω ∈ C0∞ (Q)3 , lim (Hn un − Hu) · ω dxdt = 0. n→∞

(4.4.18)

Q

Now set r = r/(r − 1) for 1 < r < ∞ and r = 1 for r = ∞. Notice that r < 2d(d − 2) for r > 2d/(d + 2) and

W01,2 (Ω) → Lr (Ω) is compact for r < 2d/(d − 2).

(4.4.19)

The continuity of L2 (0, T ; W 1,2 (Ω)) → L2 (0, T ; Lr (Ω)) implies that

un − u 0 weakly in L2 (0, T ; Lr (Ω)). Since Hω ∈ L2 (0, T ; Lr (Ω)) it follows that lim Hω · (un − u) dxdt = 0. n→∞

Q

Thus we get lim (Hn un − Hu) · ω dxdt ≤ lim (Hn − H)un · ω dxdt.

n→∞

Q

n→∞

Q

76

Chapter 4. Basic statements

By (4.4.19) the embedding of the dual spaces Lr (Ω) → W −1,2 (Ω) is compact, which yields Hn (t) → H(t) strongly in W −1,2 (Ω) for a.e. t ∈ (0, T ).

(4.4.20)

Moreover, the sequence Hn is bounded in L∞ (0, T ; W −1,2 (Ω)). This leads to lim (Hn − H)un · ω dxdt n→∞

Q

T

Hn (t) − H(t)W −1,2 (Ω) ω · un (t)W 1,2 (Ω) dt

≤ lim

n→∞

≤ lim

0

n→∞

≤ c lim

n→∞

T

ω ·

0

0

un (t)2W 1,2 (Ω)

dt

1/2 0

T

Hn (t) − H(t)2W −1,2 (Ω) dt

T

Hn (t) − H(t)2W −1,2 (Ω) dt

1/2

1/2

.

It follows from (4.4.20) and the Lebesgue dominated convergence theorem that the last integral vanishes as n → ∞.

4.4.2 Proof of Theorem 4.4.2 Notice that by Theorem 1.3.30 on weak compactness of bounded sets there is (, ϕ, u) such that, after passing to a subsequence of (n , ϕn , un ) if necessary, the convergences (4.4.6) hold. Next we derive the limit relations (4.4.10). We begin by proving (4.4.10a). Set Hn = n . It follows from (4.4.5) that for any ξ ∈ C0∞ (Ω) the functions of the time variable t ∈ (0, T ), n (t)ξ dx, Fn [ξ](t) := Ω

satisfy the integral identity T η (t)Fn [ξ](t) dt = 0

T

η(t)Gn [ξ](t) dt

0

for all η ∈ C0∞ (0, T ),

where

(un · ∇ξ − gn ξ) dx.

Gn [ξ](t) = Ω

We have the inequality |Gn [ξ](t)| ≤ (un (t)L1 (Ω) + gn L1 (Ω) )ξC 1 (Ω) , which combined with condition (4.4.4) leads to T |Gn [ξ](t)| dt ≤ EξC 1 (Ω) =: Cξ . 0

(4.4.21)

4.4. Compactness properties

77

Applying Lemma 1.3.3 we conclude that, after possible changes of values on a set of zero measure, Fn [ξ] ≤ Cξ . (4.4.22) [0,T ]

It follows that n meets all requirements of Lemma 4.4.3 with Hn and r replaced by n and γ, respectively. This yields (4.4.10a). The same arguments give (4.4.10b). To prove (4.4.10c), we observe that by (4.4.1)–(4.4.2), the sequences n , un meet all requirements of Corollary 4.2.2. It follows that the functions n un are uniformly bounded in L∞ (0, T ; L2γ/(γ+1) (Ω)). Next, set Hn (t) = n (t)un (t). It follows from (4.4.5) that the functions n (t)un · ξ dx, ξ ∈ C0∞ (Ω), Fn [ξ](t) = Ω

satisfy the integral identity T η (t)Fn [ξ](t) dt = 0

where the function

T

η(t)Gn [ξ](t) dt, 0

Gn [ξ](t) =

(Vn (t) : ∇ξ − wn · ξ) dx Ω

satisﬁes |Gn [ξ](t)| ≤ c(Vn (t)L1 (Ω) + wn (t)L1 (Ω) )ξC 1 (Ω) . From this and inequalities (4.4.4) in Condition 4.4.1 we obtain T |Gn [ξ](t)| dt ≤ C(E)ξC 1 (Ω) =: Cξ . 0

Hence the sequence Fn [ξ] satisﬁes (4.4.22) and Hn = n un meets all requirements of Lemma 4.4.3, which yields (4.4.10c). To prove (4.4.7), it suﬃces to notice that by (4.4.10a) and (4.4.6), the functions n and un meet all requirements of Lemma 4.4.5 with Hn replaced by n and r replaced by γ. Repeating the same arguments we obtain (4.4.8). It remains to prove (4.4.9). By (4.4.10c), the sequence Hn (t) = n (t)un (t) converges weakly in L2γ/(γ+1) (Ω) to H(t) = (t)u(t) for a.e. t ∈ (0, T ). Moreover, the functions Hn (t) are uniformly bounded in L∞ (0, T ; L2γ/(γ+1) (Ω)). Since γ > d/2, we have 2γ/(γ + 1) > 2d/(d + 2). Hence the functions Hn and un meet all requirements of Lemma 4.4.5 with r = 2γ/(γ + 1), so n un ⊗ un converges to u ⊗ u weakly in L2 (0, T ; Lb (Ω)) for all b ∈ (1, ∞) satisfying b−1 > 2−1 − d−1 + r−1 = 2−1 − d−1 + (γ + 1)(2γ)−1 = 1 − 2−1 a. This completes the proof of Theorem 4.4.2.

78

Chapter 4. Basic statements

4.5 Basic integral identity The compactness results obtained in Section 4.4.2 follow from the general fact that the governing equations represent a system of conservation laws in divergence form. But the momentum u and the kinetic energy tensor u ⊗ u are not the only nonlinear terms in compressible Navier-Stokes equations. In particular, the stress tensor usually includes the pressure p, which is a nonlinear function of the density . The question of weak compactness properties for the composite functions p(n ), n ≥ 1, is diﬃcult, and its resolution requires a sophisticated mathematical technique. The key observation is that the quantity p−(λ+1) div u, called the eﬀective viscous ﬂux or eﬀective viscous pressure, enjoys better regularity properties compared to p and div u. This results from a detailed examination of the projection of the momentum balance equation onto the space of potential vector ﬁelds, as is proved in [34], [80]. Fortunately, it is not necessary to investigate this projection in detail. All needed information can be obtained from a special integral identity which is derived in this section. Before we formulate the corresponding results let us introduce some notation. Recall the operators (see Section 1.7.1 for details) Ai = ∂xi Δ−1 ,

Rij = ∂xi ∂xj Δ−1 ,

1 ≤ i, j ≤ d.

Introduce also the vector-valued operator A and the matrix-valued operator R deﬁned by A = (Ai )1≤i≤3 , R = (Rij )3×3 . The symbol R[·] will stand for the action of the linear operator R on a function: • for a scalar function η it is a matrix with the entries (R[η])ij := Rij η; • for a vector function v it is a vector with the components (R[v])i := Rij vj ; • for a tensor function T it is a matrix with the entries (R[T])ij := Rik Tkj . Let Ω be a bounded domain in Rd with Lipschitz boundary, and denote Q = Ω × (0, T ). Suppose that the functions (, ϕ, ), the vector ﬁelds (u, g, gϕ ), and the matrix-valued functions (T, G) satisfy the following equations: The ﬁrst is the generalized mass balance equation with a source ∂t + div(u − g) = 0 in Q,

(4.5.1)

the second is the generalized momentum balance equation ∂t (u) + div(u ⊗ u) = div(T + G) + f

in Q,

(4.5.2)

and the third is the “renormalized” mass balance equation ∂t ϕ + div(ϕu − gϕ ) + = 0 in Q. Furthermore, we assume that the following conditions are satisﬁed.

(4.5.3)

4.5. Basic integral identity

79

Condition 4.5.1. • There are exponents γ > 1, 1 < r < ∞, 1 < s ≤ ∞, and a constant E such that γ L1 (Q) + uL2 (0,T ;W 1,2 (Ω)) ≤ E,

(4.5.4)

|u| L∞ (0,T ;L1 (Ω)) ≤ E, ϕL∞ (0,T ;Ls (Ω)) ≤ E.

(4.5.5) (4.5.6)

2

• For r = r/(r − 1),

T ∈ L1 (Q), T ∈ Lr (Q ), ϕ ∈ Lr (Q ) for every compact Q Q. • We have

g, gϕ , ∈ L2 (Q),

(4.5.7)

G ∈ L1 (0, T ; Lq (Ω)),

where q = s/(s − 1) for s < ∞,

and

1 < q ≤ 2 for s = ∞.

Throughout this section we assume that all functions considered are extended by zero over the whole space Rd . The extended functions are denoted by the same symbol. Theorem 4.5.2. Under Condition 4.5.1, if the exponents s, γ and functions gϕ , G satisfy either 2d < γ ≤ s < ∞, (4.5.8) or s = ∞,

γ > d/2,

gϕ = 0,

G = 0,

(4.5.9)

then the integral identity H · u + ζ(N + P + Q) · u + S dxdt (4.5.10) ξϕR : [ζ(T + G)] dxdt = Q

Q

holds for all ξ ∈

C0∞ (Q)

and ζ ∈ C0∞ (Ω). Here,

H = R[ξϕ] (ζu) − (ξϕ) R[ζu],

(4.5.11)

S = (∇ζ ⊗ A[ξϕ]) : (u ⊗ u − T − G) + ζA[ξϕ] · (f ),

(4.5.12)

N = R[ξ gϕ ],

(4.5.13)

P = A[ϕ∂t ξ + ϕu · ∇ξ],

Q = −A[gϕ · ∇ξ + ξ].

Furthermore, all functions in (4.5.10) are integrable in Q and HL∞ (0,T ;Lι (Ω)) + PL2 (0,T ;W 1,2 (Ω)) + A[ξϕ]L∞ (0,T ;W 1,2d (Ω)) ≤ c, QL2 (0,T ;W 1,2 (Ω)) + NL2 (Q) ≤ c(gL2 (Q) + gϕ L2 (Q) + L2 (Q) ). (4.5.14) Here, the constant c depends only on the functions ξ, ζ, and on E, and the exponent ι ≥ 1 is arbitrary satisfying the inequalities ι<

2γ γ+1

for d/2 < γ ≤ 2d,

ι<

2γ γ+3

for γ > 2d.

80

Chapter 4. Basic statements

The proof is based on three auxiliary lemmas which give the construction and describe the properties of a special vector ﬁeld ω. Lemma 4.5.3. Assume that solutions to equations (4.5.1)–(4.5.3) meet all requirements of Theorem 4.5.2 and ξ ∈ C0∞ (Q), ζ ∈ C0∞ (Ω). Then the time derivative of the vector ﬁeld ω = ζA[ξϕ] has the representation ∂t ω = ζ N + P + Q − R[ξϕ u] , (4.5.15) where the remainders N, P, and Q are deﬁned by (4.5.13) and satisfy estimates (4.5.14). The momentum ϕu satisﬁes the estimate ϕuL2 (0,T ;Lρ (Ω)) + R[ξϕuL2 (0,T ;Lρ (Ω)) ≤ c,

(4.5.16)

where ρ is an arbitrary number such that 2 ≤ ρ < ∞,

ρ−1 > 2−1 + s−1 − d−1 ,

(4.5.17)

and the constant c depends only on ξ, the constant E in (4.5.4)–(4.5.7), the domain Ω, and the exponents γ, s, ρ. Proof. Observe that (4.5.3) is equivalent to the integral identity ∂t ς ϕ + (ϕu − gϕ ) · ∇ς − ς dxdt = 0

(4.5.18)

Q

for every ς ∈ C0∞ (Q). Choose any η ∈ C0∞ (Q) and set ς = ξAi [ηζ] ∈ C0∞ (Q). Substituting ς into (4.5.18) and noting that all functions in (4.5.18) are compactly supported in Q we arrive at ξϕAi [ζ∂t η] dxdt + ∂t ξϕ Ai [ηζ] dxdt Rd ×[0,T ] Rd ×[0,T ] ξ Ai [η∇ζ] · (ϕu − gϕ ) dxdt + ξ Ai [ζ∇η] · (ϕu − gϕ ) dxdt + Rd ×[0,T ] Rd ×[0,T ] + Ai [ηζ]∇ξ · (ϕu − gϕ ) dxdt − ξAi [ηζ] dxdt = 0. Rd ×[0,T ]

Rd ×[0,T ]

Using the skew-symmetry of Ai we can rewrite this identity in the form ∂t η{ζAi [ξϕ]} dxdt + ζAi [ξ(ϕu − gϕ )] · ∇η dxdt Rd ×[0,T ] Rd ×[0,T ] + η {ζAi [∂t ξϕ]} dxdt + η{∇ζ · Ai [ξ(ϕu − gϕ )]} dxdt R3 ×[0,T ] Rd ×[0,T ] η{ζ Ai [∇ξ · (ϕu − gϕ )]} dxdt − η{ζAi [ξ]} dxdt = 0. + Rd ×[0,T ]

Rd ×[0,T ]

(4.5.19)

4.5. Basic integral identity

81

In order to derive expressions for partial derivatives of ω, we have to integrate by parts in the integrals of this identity. The justiﬁcation of this procedure requires estimates of the integrands. To derive such estimates ﬁx ρ satisfying (4.5.17). By the Hölder inequality, we have ϕ(t)u(t)Lρ (Ω) ≤ ϕ(t)Ls (Ω) u(t)Lυ (Ω) , where υ −1 = ρ−1 − s−1 . By (4.5.17) we have υ < 2d/(d − 2), hence the embedding W 1,2 (Ω) → Lυ (Ω) is bounded. Thus we get ϕ(t)u(t)Lρ (Ω) ≤ cϕ(t)Ls (Ω) u(t)W 1,2 (Ω) and ϕuL2 (0,T ;Lρ (Ω)) ≤ cϕL∞ (0,T ;Ls (Ω)) uL2 (0,T ;W 1,2 (Ω)) ≤ c(E).

(4.5.20)

Since ζ ∈ C0∞ (Rd ) is compactly supported in Ω and ξ ∈ C0∞ (Rd+1 ) is compactly supported in Q, the vector ﬁeld ζAi [ξϕu] is compactly supported in Q. From this and Corollary 1.7.2 we obtain Ai [ξϕu] L2 (0,T ;W 1,2 (Ω)) ≤ Ai [ξϕu]L2 (0,T ;W 1,ρ (Ω)) ≤ ϕuL2 (0,T ;Lρ (Ω)) ≤ c(ξ, E).

(4.5.21)

The same arguments give Ai [ξgϕ ]L2 (0,T ;W 1,2 (Ω)) + Ai [ξ]L2 (0,T ;W 1,2 (Ω)) ≤ c(ξ)(gϕ L2 (Q) + L2 (Q) ). (4.5.22) Since s ≥ 2, we have Ai [∂t ξϕ]L2 (0,T ;W 1,2 (Ω)) ≤ c(ξ)ϕL2 (0,T ;L2 (Ω)) ≤ c(ξ, E).

(4.5.23)

We can now integrate by parts to obtain ϕ ζAi [ξ(ϕu − g )] · ∇η dxdt = − ηζ div Ai [ξ(ϕu − gϕ )] dxdt Rd ×[0,T ] Rd ×[0,T ] η∇ζ · Ai [ξ(ϕu − gϕ )] dxdt. − Rd ×[0,T ]

Substituting this result into (4.5.19) we arrive at

∂t η{ζAi [ξϕ]} dxdt − Rd ×[0,T ] +

Rd ×[0,T ]

Rd ×[0,T ]

ηζ div Ai [ξ(ϕu − gϕ )] dxdt

ηζ Ai [∂t ξϕ + ∇ξ · (ϕu − gϕ ) − ξ] dxdt = 0

82

Chapter 4. Basic statements

for all i = 1, . . . , d. We can rewrite this equality in vector form

Rd ×[0,T ]

∂t η ω dxdt −

Rd ×[0,T ]

ηζR [ξ(ϕu − gϕ )] dxdt

+

Rd ×[0,T ]

or equivalently

ηζ A [∂t ξϕ + ∇ξ · (ϕu − gϕ ) − ξ] dxdt = 0,

∂t η ω dxdt +

Q

ηζ(N + P + Q − R[ξϕu]) dxdt = 0. Q

Since η ∈ C0∞ (Q) is arbitrary, this yields (4.5.15). It remains to note that inequality (4.5.14) for N follows from the boundedness of the operator R in L2 (Rd ). Inequalities (4.5.14) for P, Q easily follow from (4.5.21)–(4.5.23). Finally, inequality (4.5.20) and the boundedness of R in Lebesgue spaces imply (4.5.16). General properties of ω are listed in the second lemma. Lemma 4.5.4. Under the assumptions of Theorem 4.5.2 and Lemma 4.5.3, if the exponents s and γ satisfy condition (4.5.8), then ∂t ω ∈ L2 (Q),

∇ω ∈ L∞ (0, T ; Ls (Ω)) ∩ Lr (Q).

(4.5.24)

If s, γ, Gn and gnϕ satisfy (4.5.9), then ∂t ω ∈ L2 (0, T ; Lτ (Ω)),

∇ω ∈ L∞ (0, T ; Lσ (Ω)) ∩ Lr (Q)

(4.5.25)

for all τ , σ such that 1 < τ < 2d/(d − 2) and 1 < σ < ∞. Proof. If s and γ satisfy (4.5.8), then (4.5.24) for ∂t ω follows from estimates (4.5.14), (4.5.16), and equality (4.5.15) (since ρ > 2 in (4.5.16)). Recall that estimates (4.5.14), (4.5.16), and equality (4.5.15) are established by Lemma 4.5.3. It remains to consider the case of γ, s, G, and gϕ which satisfy (4.5.9). Then N = 0 and ∂t ω = ζ(P + Q − R[ξϕu]). It follows from estimates (4.5.14) and the Sobolev embedding theorem that for all τ < 2d/(d − 2), PL2 (0,T ;Lτ (Ω)) + QL2 (0,T ;Lτ (Ω)) ≤ cPL2 (0,T ;W 1,2 (Ω)) + cQL2 (0,T ;W 1,2 (Ω)) < ∞. On the other hand, since s = ∞ by (4.5.9), estimate (4.5.16) implies R[ξϕu]L2 (0,T ;Lτ (Ω)) ≤ c for all τ < 2d/(d − 2).

4.5. Basic integral identity

83

Hence ∂t ω satisﬁes (4.5.25). To estimate ∇ω notice that ∇ω = ζR[ξϕ] + ∇ζ ⊗ A[ξϕ]. Since ϕ ∈ L∞ (0, T ; Ls (Ω)) ∩ Lrloc (Q) and ξ is compactly supported in Q, we have ξϕ ∈ L∞ (0, T ; Ls (Ω)) ∩ Lr (Q). Hence inclusions (4.5.24)–(4.5.25) for ∇ω follow from the estimates of the operators R and A given by Corollary 1.7.2. Lemma 4.5.5. The integral identity ∂t ω · (u) + ∇ω : (u ⊗ u − T − G) + ω · (f ) dxdt = 0

(4.5.26)

Q

holds for all vector ﬁelds ω compactly supported in Q with the properties

∂t ω ∈ L2 (0, T ; Lβ (Ω)),

ω ∈ L1 (0, T ; Lγ/(γ−1) (Ω)),

∇ω ∈ L∞ (0, T ; Lb (Ω)) ∩ L∞ (0, T ; Lq (Q)) ∩ Lr (Q), where q = q/(q − 1) and β , b are some numbers satisfying (1 − 2−1 a)−1 = 2a−1 < b < ∞, κ = κ/(κ − 1) < β < ∞,

(4.5.27)

(4.5.28)

where κ−1 = 2−1 + γ −1 − d−1 . Proof. We ﬁrst observe that by (4.5.2), the integral identity (4.5.26) holds for all ω ∈ C0∞ (Q). Let us consider in detail all the terms involved in this identity. It follows from (4.5.4)–(4.5.5) that (, u) are functions of bounded energy and meet all requirements of Corollaries 4.2.2 and 4.2.3. Applying Corollary 4.2.2 we obtain uL2 (0,T ;Lβ (Ω)) ≤ c(E) for all β ∈ [1, κ).

(4.5.29)

On the other hand, Corollary 4.2.3 implies u ⊗ u ∈ L2 (0, T ; Lb (Ω)) for all b ∈ [ 1, (1 − 2−1 a)−1 ).

(4.5.30)

By Condition 4.5.1, we have ∈ L∞ (0, T ; Lγ (Ω)),

T ∈ Lrloc (Q),

G ∈ L1 (0, T ; Lq (Ω)).

(4.5.31)

Let now ω be an arbitrary vector ﬁeld satisfying (4.5.27) and compactly supported in Q. To give a rigorous proof of (4.5.26) for ω we consider the molliﬁcation [ω]m,m of ω, deﬁned by (1.6.1), (1.6.3). Since ω is compactly supported in Q and satisﬁes (4.5.27), the smooth function [ω]m,m is compactly supported in Q for suﬃciently large m. Inserting [ω]m,m in (4.5.26) we obtain ∂t [ω]m,m · (u) + ∇[ω]m,m : (u ⊗ u − T − G) + [ω]m,m · (f ) dxdt = 0. Q

(4.5.32)

84

Chapter 4. Basic statements

By general properties of molliﬁers we have

∂t [ω]m,m = [∂t ω]m,m → ∂t ω

in L2 (0, T ; Lβ (Ω)),

∇[ω]m,m = [∇ω]m,m ∇ω

weakly in L∞ (0, T ; Lq (Ω)),

∇[ω]m,m = [∇ω]m,m → ∇ω

in L2 (0, T ; Lb (Ω)) ∩ Lr (Q),

[ω]m,m → ω

in L1 (0, T ; Lγ/(γ−1) (Ω)).

Letting m → ∞ in (4.5.32) and recalling (4.5.29)–(4.5.31) we conclude that ω satisﬁes the integral identity (4.5.26). We are now in a position to complete the proof of Theorem 4.5.2. Our ﬁrst task is to check that the special vector ﬁeld ω = ζA[ξϕ], deﬁned by Lemma 4.5.3, satisﬁes the assumptions of Lemma 4.5.5. This follows directly from (4.5.25) if γ and s satisfy (4.5.9). Otherwise if γ and s satisfy (4.5.8), then s = q ,

2d < γ ≤ s < ∞ and

(4.5.33)

which along with (4.5.24) implies that ω ∈ L∞ (0, T ; W 1,2d (Ω)) ⊂ L∞ (Q), thus ω obviously satisﬁes (4.5.27). Next, for γ > d/2, the quantity κ satisﬁes the inequalities κ > 2 ⇒ κ/(κ − 1) < 2. On the other hand, inclusion (4.5.24) implies ∂t ω ∈ L2 (0, T ; L2 (Ω)). Combining these results we conclude that ∂t ω also satisﬁes (4.5.27) with an arbitrary exponent β ∈ (κ/(κ − 1), 2]. Let us prove that ∇ω satisﬁes (4.5.27). Inequalities (4.5.33) imply 1 2 a = − > 3/(2d), 2a−1 < 4d/3. d γ In particular, we have s > 2a−1 and we can take b = s ≡ q . On the other hand, inclusion (4.5.24) implies ∇ω ∈ L∞ (0, T ; Ls (Ω)) ∩ Lr (Q). Hence, for this choice of q and b we conclude that

∇ω ∈ L∞ (0, T ; Lb (Ω)) ∩ L∞ (0, T ; Lq (Ω)) ∩ Lr (Q).

(4.5.34)

Thus ∇ω also satisﬁes condition (4.5.27). Hence the vector ﬁeld ω satisﬁes (4.5.27), and we can substitute ω into the integral identity (4.5.26) and next apply Lemma 4.5.3 to obtain ζ (N + P + Q − R[ξϕ u]) · u + ∇ω : (u ⊗ u − T − G) + ω · (f ) dxdt = 0. Q

Recalling the relation ∇ω = ζR[ξϕ] + ∇ζ ⊗ A[ξϕ],

4.5. Basic integral identity

85

and formulae (4.5.11)–(4.5.13) we can rewrite the above identity in the form ζR[ξϕ] : (u ⊗ u − T − G) Q + ζ (N + P + Q − R[ξϕ u]) · u + S dxdt = 0. (4.5.35) Since ξ is compactly supported in Q and ζ is compactly supported in Ω, it follows from Corollary 4.2.2 and Lemma 4.5.3 that ζu ∈ L2 (0, T ; Lβ (Rd )),

ξϕu ∈ L2 (0, T ; Lρ (Rd ))

for all β and ρ satisfying 1 > β −1 > 2−1 + γ −1 − d−1 ,

2−1 > ρ−1 > 2−1 + s−1 − d−1 .

(4.5.36)

On the other hand, inequalities (4.5.8) and (4.5.9) yield 2−1 + γ −1 − d−1 + 2−1 + s−1 − d−1 = 1 + γ −1 + s−1 − 2d−1 < 1. Hence there exist β and ρ satisfying (4.5.36) such that β −1 + ρ−1 = 1. Applying Corollary 1.7.3 we obtain R[ξϕu] · (ζu) dxdt = R[ξϕu] · (ζu) dxdt Q Rd ×(0,T ) ξϕR[ζu] · u dxdt = ξϕR[ζu] · u dxdt. (4.5.37) = Rd ×(0,T )

Q

By Condition 4.5.1 and (4.5.8), (4.5.9), we have ϕ ∈ L∞ (0, T ; Ls (Ω)) and s ≥ q . Since ξ is compactly supported in Q we infer that ξϕ ∈ L∞ (0, T ; Lq (Rd )). Since ζ is compactly supported in Ω, Condition 4.5.1 yields ζG ∈ L1 (0, T ; Lq (Rd )). From this and the symmetry properties of the operator R given by Corollary 1.7.3 we conclude that R[ξϕ] : ζG dxdt = R[ξϕ] : ζG dxdt Q Rd ×(0,T ) = ξϕR : [ζG] dxdt = ξϕR : [ζG] dxdt. (4.5.38) Rd ×(0,T )

Q

Since ξ is compactly supported in Q we By Condition 4.5.1, we have ϕ ∈ infer that ξϕ ∈ ∩Lr (Rd × (0, T )). Moreover, there is h > 0 such that ξϕ and R[ξϕ] are supported in Rd × [h, T − h]. On the other hand, since T ∈ Lrloc (Q) and ζ is compactly supported in Ω we have ζT ∈ Lr (Rd × (h, T − h)). Applying Corollary 1.7.3 we obtain R[ξϕ] : ζT dxdt = R[ξϕ] : ζT dxdt Q Rd ×[h,T −h] ξϕR : [ζT] dxdt = ξϕR : [ζT] dxdt. (4.5.39) = Lrloc (Q).

Rd ×[h,T −h]

Q

86

Chapter 4. Basic statements

Inserting (4.5.37)–(4.5.39) into (4.5.35) we arrive at ζR[ξϕ] : (u ⊗ u) − ξϕ R[ζu] · u ξϕ R : [ζ(T + G)] dxdt = Q Q + ζ (N + P + Q) · u + S dxdt,

(4.5.40)

which combined with the obvious identity ζR[ξϕ] : (u ⊗ u) − ξϕR[ζu] · u ≡ H · u leads to (4.5.10). It remains to prove inequalities (4.5.14). For N, P, and Q, they follow directly from Lemma 4.5.3. Next, the hypotheses of Theorem 4.5.2 imply s > 2d, hence ϕL∞ (0,T ;L2d (Ω)) ≤ cϕL∞ (0,T ;Ls (Ω)) ≤ c, which along with Corollary 1.7.2 yields (4.5.14) for A[ξϕ]. It remains to estimate H. The Hölder inequality and boundedness of R in Lebesgue spaces (see Corollary 1.7.2) gives HL∞ (0,T ;Lι (Ω)) ≤ cϕL∞ (0,T ;Lτ (Ω)) uL∞ (0,T ;Lσ (Ω)) for all τ , σ such that 1 < σ, τ < ∞,

1 1 1 + ≤ < 1. τ σ ι

By (4.5.4)–(4.5.5) the functions (, u) have bounded energy and meet all requirements of Proposition 4.2.1. Hence we can apply Corollary 4.2.2 of that proposition to obtain uL∞ (0,T ;Lσ (Ω)) ≤ c(E) for all σ ∈ (1, 2γ/(γ + 1)]. By (4.5.6) we have ϕL∞ (0,T ;Lτ (Ω)) ≤ c(E) for all τ ∈ (1, s]. Hence HL∞ (0,T ;Lι (Ω)) ≤ c(ζ, E)

(4.5.41)

for all ι satisfying 1/s + (γ + 1)/(2γ) < 1/ι < 1. If γ and s satisfy (4.5.8), then ∞ > s ≥ γ > 2d and 1/s + (γ + 1)/(2γ) ≤ (γ + 3)/(2γ). Hence inequality (4.5.41) holds for all ι ∈ (1, 2γ/(γ+3)). If γ and s satisfy condition (4.5.9), then s = ∞ and γ > d/2. In this case (γ+1)/(2γ) < 1/ι < 1, and inequality (4.5.41) holds for all ι ∈ (1, 2γ/(γ + 1)). This leads to estimate (4.5.14) for H.

4.6. Continuity of viscous ﬂux

87

4.6 Proof of continuity of viscous ﬂux for general stress tensors This section is devoted to the famous P.-L. Lions result on weak continuity of “viscous ﬂux”. This result occupies an important place in the theory of compressible ﬂows and plays a crucial role in the proof of existence of solutions to compressible Navier-Stokes equations. The result is formulated in a most general setting for the dynamics of a compressible medium with a general stress tensor T. Therefore, a general system of mass balance (4.6.1) and moment balance equations (4.6.2) is considered. Such a general result can be applied to diﬀerent modiﬁcations of compressible Navier-Stokes equations including diﬀerent regularization schemes deﬁned for these equations. First we formulate the explicit assumptions on classes of functions under consideration and on the governing equations. Let Ω be a bounded domain in Rd , d = 2, 3, with a Lipschitz boundary, and Q = Ω × [0, T ]. Furthermore, assume that functions n : Q → R+ , vector ﬁelds un : Q → Rd , and matrix-valued functions Tn : Q → Rd×d , n ≥ 1, satisfy the diﬀerential equations ∂t n + div(n un − gn ) = 0 in Q, ∂t (n un ) + div(n un ⊗ un ) = div(Tn + Gn ) + ρn f

(4.6.1) in Q,

(4.6.2)

which are understood in the sense of distributions. Here, f ∈ C(Q) and gn ∈ L2 (Q) are given vector ﬁelds. Along with the “density” and “velocity” ﬁeld (n , un ) we consider the “renormalized density” ϕn , n ≥ 1, satisfying ∂t ϕn + div(ϕn un − gnϕ ) + n = 0 in Q,

(4.6.3)

for a given vector ﬁeld gnϕ ∈ L2 (Q) and function n ∈ L2 (Q). We impose neither boundary nor initial conditions for solutions to (4.6.1)–(4.6.3). Moreover, no relation is imposed between the “stress tensor” Tn and the “state variables” (n , un ). Instead it is assumed that the state variables satisfy the following conditions: Condition 4.6.1. • There are exponents 1 < γ < ∞, 1 < s ≤ ∞, 1 < r < ∞ and a constant E independent of n such that γn L1 (Q) + un L2 (0,T ;W 1,2 (Ω)) ≤ E,

(4.6.4)

n |un | L∞ (0,T ;L1 (Ω)) ≤ E, ϕn L∞ (0,T ;Ls (Ω) ≤ E.

(4.6.5)

2

(4.6.6)

The stress tensors Tn are uniformly integrable, Tn L1 (Q) ≤ E.

(4.6.7)

Moreover, for every compact subset Q Q there is a constant C(Q ) such that ϕn Lr (Q ) + Tn Lr (Q ) ≤ C(Q ).

(4.6.8)

88

Chapter 4. Basic statements • The given elements in (4.6.1)–(4.6.3) satisfy Gn L1 (0,T ;Lq (Ω)) → 0 as n → ∞, gn L2 (Q) +

gnϕ L2 (Q) n

L2 (Q)

(4.6.9)

→ 0 as n → ∞, ≤ E,

(4.6.10) (4.6.11)

where q = s/(s − 1) for s < ∞ and 1 < q ≤ 2 for s = ∞. • There are functions ∈ L∞ (0, T ; Lγ (Ω)),

u ∈ L2 (0, T ; W 1,2 (Ω)), ϕ ∈ L∞ (0, T ; Ls (Ω)) ∩ Lrloc (Q),

T ∈ L1 (Q) ∩ Lrloc (Q) and

such that the following convergences hold: n weakly in L∞ (0, T ; Lγ (Ω)), ϕn ϕ weakly in L∞ (0, T ; Ls (Ω)),

(4.6.12)

un u weakly in L2 (0, T ; W 1,2 (Ω)), ϕn ϕ weakly in Lr (Q ),

Tn T weakly in Lr (Q )

for any Q Q. Here r = r/(r − 1). Deﬁnition 4.6.2. The generalized viscous ﬂux Vn , associated with the system of conservation laws (4.6.1)–(4.6.2), is deﬁned as follows: Vn = R : [ζTn ] := Rij [ζTn,ij ],

(4.6.13)

where ζ ∈ C0∞ (Ω) is an arbitrary function of the spatial variable x. In our framework, the expression for the viscous ﬂux involves a nonlocal operator and an arbitrary cut-oﬀ function. The problem of localization of the viscous ﬂux is discussed in Section 4.7. Notice that ζ is compactly supported in Ω and T belongs to Lrloc (Ω × (0, T )). It follows that the matrix-valued functions ζT and R[ζT] belong to Lr (Rd × [h, T − h]) for every h ∈ (0, T ). From this and condition (4.6.12) we conclude that Vn V = R : [ζT] weakly in Lq (Rd × [h, T − h])) for h ∈ (0, T ).

(4.6.14)

The following theorem, which is the main result of this section, gives the weak continuity of the product Vn ϕn . Theorem 4.6.3. Assume that n , un , Tn are solutions to equations (4.6.1)–(4.6.3) satisfying Condition 4.6.1. Furthermore, assume that the exponents s, γ, q and the elements gnϕ , Gn satisfy either 2d < γ ≤ s < ∞,

(4.6.15)

4.6. Continuity of viscous ﬂux

89

or γ > d/2, Then for all ξ ∈

C0∞ (Q)

s = ∞,

gnϕ = 0,

Gn = 0.

(4.6.16)

C0∞ (Ω),

and ζ ∈ ξ Vn ϕn dxdt = ξ V ϕ dxdt. lim

n→∞

Q

(4.6.17)

Q

Since ξϕn , ξϕ ∈ Lr (Q) are supported in Ω×(h, T −h) for some h > 0, the integrals in (4.6.17) are well deﬁned. Proof. Fix ξ ∈ C0∞ (Q) and ζ ∈ C0∞ (Ω). If n , un , Tn are solutions to equations (4.6.1)–(4.6.3) satisfying Condition 4.6.1 and either (4.6.15) or (4.6.16), then they obviously satisfy (4.5.1)–(4.5.7) and meet all requirements of Theorem 4.5.2. Hence they satisfy the integral identity (4.5.10), which in this case can be rewritten in the form ξ ϕn Vn dxdt = − ξϕn R : [ζGn ] dxdt Q Q + Hn · un + ζn (Nn + Pn + Qn ) · un + Sn dxdt. (4.6.18) Q

Here, we denote Hn = R[ξϕn ](ζn un ) − (ξϕn ) R[ζn un ], Sn = (∇ζ ⊗ A[ξϕn ]) : (n un ⊗ un − Tn − Gn ) + ζA[ξϕn ] · (n f ), Nn = R[ξ gnϕ ], Pn = A[ϕn ∂t ξ + ϕn un · ∇ξ], Qn =

−A[gnϕ

(4.6.19) (4.6.20) (4.6.21)

· ∇ξ + n ξ].

Our task is to pass to the limit on the right hand side of identity (4.6.18) as n → ∞. The considerations are based on the following Lemma 4.6.4. Under the assumptions of Theorem 4.6.3, the sequence (n , ϕn , un ) meets all requirements of Theorem 4.4.2 and satisﬁes (4.4.6)–(4.4.10c). Proof. It suﬃces to show that (n , ϕn , un ) and γ, s satisfy Condition 4.4.1. Observe that s, γ > d/2 by (4.6.15)–(4.6.16). Hence s, γ satisfy Condition 4.4.1. Next, estimates (4.4.6) in this condition coincide with (4.6.4)–(4.6.6) in Condition 4.6.1 of Theorem 4.6.3. It follows from (4.6.1)–(4.6.3) that (n , ϕn , un ) satisfy equations (4.4.5) in Condition 4.4.1 with un = −n un + gn ,

gn = 0,

Vn = Tn + Gn − n un ⊗ un , wn = n f , vn = −ϕn un + gnϕ , hn = −n . Estimates (4.6.4)–(4.6.7) and relations (4.6.10) in Condition 4.6.1 obviously imply the boundedness of the sequences un , vn , hn , wn , gn and Vn in L1 (Q). Hence (n , ϕn , un ) satisfy Condition 4.4.1.

90

Chapter 4. Basic statements

The remaining part of the proof of Theorem 4.6.3 naturally falls into three steps. Step 1. Weak continuity of remainders. assumptions of the theorem, Pn P = A[ϕ∂t ξ + ϕu · ∇ξ],

First of all we prove that under the Qn Q = −A[ξ]

(4.6.22)

weakly in L2 (0, T ; W 1,2 (Ω)). Since Pn and Qn are bounded in L2 (0, T ; W 1,2 (Ω)), it suﬃces to show that for all η ∈ C0∞ (Q), (Pn − P)η dxdt = 0, lim (Qn − Q)η dxdt = 0. (4.6.23) lim n→∞

n→∞

Q

Q

As η and ξ are compactly supported in Q, and the operator A is skew-symmetric on the space of compactly supported square integrable functions, we have (Pn − P)η dxdt = (ϕ − ϕn )ξt A[η] dxdt + (ϕu − ϕn un ) · ∇ξA[η] dxdt. Q

Q

Q

Notice that ∇ξA[η] ∈ C0∞ (Q). By Lemma 4.6.4 and Theorem 4.4.2 the sequence ϕn (t) converges weakly (weakly for s = ∞) in Ls (Ω) to ϕ(t) for almost every t ∈ [0, T ]. Moreover, un converges to u weakly in L2 (0, T ; W 1,2 (Ω)). Since s > 2d, we can apply Lemma 4.4.5 with Hn = ϕn and r = s to infer that ϕn un converges to ϕu weakly in L2 (0, T ; Lm (Ω)) for some m ∈ (1, ∞). This yields the ﬁrst equality in (4.6.23). Next, we have (Qn − Q)η dxdt = (n − )ξt A[η] dxdt + gnϕ · ∇ξA[η] dxdt. Q

Q

Q

Notice that gnϕ → 0 in L2 (0, T ; L2 (Ω)) and n weakly in L2 (0, T ; L2 (Ω)) which leads to the second equality in (4.6.23). Next, by Lemma 4.6.4 and Theorem 4.4.2, we have n (t)un (t) (t)u(t) weakly in Lr (Ω), r = 2γ/(γ + 1), for almost every t ∈ [0, T ]. Moreover, n un is uniformly bounded in L∞ (0, T ; Lr (Ω)). By the hypotheses of Theorem 4.6.3 we have γ > d/2, hence r = 2γ/(γ + 1) > 2d/(d + 2). Therefore, the functions Hn = n un meet all requirements of Lemma 4.4.5. Applying that lemma to Hn · Pn and Hn · Qn and using (4.6.22) we conclude that n un · Pn u · P,

n un · Qn u · Q weakly in L2 (0, T ; Lm (Ω)) (4.6.24)

for some m ∈ (1, ∞). Now our task is to prove that n un · Nn vanishes as n → ∞. If condition (4.6.16) is fulﬁlled, then Nn ≡ 0. If (4.6.15) holds, then γ > 2d. By

4.6. Continuity of viscous ﬂux

91

Condition 4.6.1, the bounded energy functions (n , un ) meet all requirements of Proposition 4.2.1. Applying Corollary 4.2.2 of that proposition we conclude that n un L2 (0,T ;Lβ (Ω)) ≤ c whenever

β −1 > 2−1 + γ −1 − d−1 ,

hence for β = 2.

On the other hand, since the operator R is bounded in L2 (Rd ), we have Nn L2 (Q) ≤ cgnϕ L2 (Q) , which along with (4.6.10) gives n un · Nn L1 (Q) ≤ cn un L2 (Q) gnϕ L2 (Q) ≤ cgnϕ L2 (Q) → 0. Combining (4.6.22)–(4.6.25) we ﬁnally obtain ζn Nn + Pn + Qn · un dxdt → ζ P + Q · u dxdt Q

(4.6.25)

as n → ∞. (4.6.26)

Q

Step 2. Weak continuity of Sn . Now our task is to prove that the functions Sn deﬁned by (4.6.20) satisfy the limit relation Sn dxdt = S dxdt, (4.6.27) lim n→∞

Q

Q

where S = (∇ζ ⊗ A[ξϕ]) : (u ⊗ u − T) + ζA[ξϕ] · (f ). Observe that, since s ≥ 2d, Condition 4.6.1 implies ϕn L∞ (0,T ;L2d (Ω)) ≤ ϕn L∞ (0,T ;Ls (Ω)) ≤ c. Recall that by Lemma 4.6.4 the functions ϕn meet all requirements of Theorem 4.4.2. It now follows from (4.4.10b) that there is a set T of full measure in [0, T ] with the properties ϕn (t)L2d (Ω) ≤ c for all t ∈ T and ϕn (t) ϕ(t) weakly in L2d (Ω) for all t ∈ T . Since ξ is compactly supported in Ω, it follows that A[ξϕn ](t)W 1,2d (Ω) ≤ c and A[ξϕn ](t) A[ξϕ](t) weakly in W 1,2d (Ω) as n → ∞ for all t ∈ T . Recalling the compactness of the embedding W 1,2d (Ω) → C(Ω) we conclude that for all t ∈ T , A[ξϕn ](t)C(Ω) ≤ c

92

Chapter 4. Basic statements

and A[ξϕn ](t) → A[ξϕ](t) in C(Ω) as n → ∞. Therefore, the functions A[ξϕn ] are uniformly bounded in Q and the sequence A[ξϕn ] converges a.e. to A[ξϕ]. In particular A[ξϕn ] converges strongly in any Lm (Q) with 1 ≤ m < ∞. On the other hand, (n , un ) satisfy all hypotheses of Theorem 4.4.2. It follows that n and the matrix-valued functions n un ⊗ un converge weakly to and u ⊗ u, respectively, in Lp (Q) for some p > 1. This leads to the following convergence: Ω

(∇ζ ⊗ A[ξϕn ]) : (n un ⊗ un ) + ζA[ξϕn ] · (n f ) dxdt (∇ζ ⊗ A[ξϕ]) : (u ⊗ u) + ζA[ξϕ] · f ) dxdt. →

(4.6.28)

Ω

Next, note that ξ ∈ C0∞ (Q) and hence there is h > 0 such that ξ(x, t) = 0 for t ∈ (0, h) ∪ (T − h, T ). Hence there is a function η ∈ C0∞ (0, T ) such that ξη = ξ in Q. We have A[ξϕn ] = ηA[ξϕn ] and

∇ζ ⊗ A[ξϕn ] = η∇ζ ⊗ A[ξϕn ].

Hence the function on the left hand side of the latter identity is compactly supported on the set Q = supp ηζ Q. Recall that A[ξϕn ] converges in Lm (Q) for all m > 1 and, in particular, it con verges in Lr (Q). Since Tn converges to T weakly in Lr (Q ), we have (∇ζ ⊗ A[ξϕn ]) : Tn dxdt = (∇ζ ⊗ A[ξϕn ]) : Tn dxdt Q Q → (∇ζ ⊗ A[ξϕ]) : T dxdt = (∇ζ ⊗ A[ξϕ]) : T dxdt.

Q

On the other hand, relation (4.6.9) yields |Gn | dxdt → 0 as n → ∞. (∇ζ ⊗ A[ξϕn ]) : Gn dxdt ≤ c Q

(4.6.29)

Q

(4.6.30)

Q

Combining (4.6.28)–(4.6.30) we arrive at the desired relation (4.6.27). Step 3. Weak continuity of Hn . This is the most important step. We prove that the sequence of vector functions Hn (t) converges weakly for almost every t ∈ [0, T ]. This results from the following lemma on the commutator of the operator R and multiplication by a smooth function.

4.6. Continuity of viscous ﬂux

93

Lemma 4.6.5. Let η ∈ C0∞ (Ω) and deﬁne a matrix-valued operator Cη : Lp (Rd ) → Lp (Rd ) with entries Cη,ij by Cη [·] = ηR[·] − R[η ·]. Then for any p ∈ (1, ∞), there is a constant c depending only on p, d and η such that Cη vW 1,p (Ω) ≤ cvLp (Rd ) for all v ∈ Lp (Rd ). Proof. It suﬃces to assume that v ∈ C0∞ (Ω). Then Cη v ∈ C ∞ (Rd ) and we can apply the Laplace operator to Cη to obtain ΔCη,ij v ≡ ∂xi ∂xj η v − ΔηRij v + 2∂xk (∂xk η Rij v) − ∂xi (∂xj η v) − ∂xj (∂xi η v), which leads to the identity Cη,ij v ≡ Δ−1 ∂xi ∂xj η v − ΔηRij v + 2Ak (∂xk η Rij v) − Ai (∂xj η v) − Aj (∂xi η v). The desired estimate obviously follows from the boundedness of R in Lebesgue spaces and the smoothing properties of the operators A and Δ−1 established in Corollary 1.7.2. Let us show that this lemma implies the weak continuity of Hn (t). Recall that the functions (n , ϕn , un ) satisfy all conditions of Theorem 4.5.2. By estimate (4.5.14) in that theorem, there is a full measure set T in [0, T ] such that Hn (t)Lι (Ω) ≤ c for all t ∈ T and n ≥ 1,

(4.6.31)

where the exponent ι is greater than 2d/(d + 2). Hence it suﬃces to prove that for a suitable choice of T , the equality ηHn (t) dx = ηH(t) dx lim (4.6.32) n→∞

Ω

Ω

holds for all η ∈ C0∞ (Ω). Notice that (4.6.19) combined with the symmetry properties of R yields the identity R[ξϕn ](ζn un ) − (ξϕn )R[ζn un ] η dx ηHn (t) dx = Ω Ω Cη [ξ(t)ϕn (t)] (ζn (t)un (t)) dx. = Ω

By Lemma 4.6.4, the functions (n , ϕn , un ) meet all requirements of Theorem 4.4.2. It follows from relation (4.4.10b) in that theorem and the inequality s > 2d that for a suitable choice of T , ϕn (t) ϕ(t) weakly in L2d (Ω) for all t ∈ T . Now Lemma 4.6.5 implies that Cη [ξϕn (t)] Cη [ξϕ(t)] weakly in W 1,2d (Ω) for

94

Chapter 4. Basic statements

all t ∈ T . Since W 1,2d (Ω) is compactly embedded into C(Ω), we ﬁnally infer that for all t ∈ T , Cη [ξϕn (t)] → Cη [ξϕ(t)] uniformly in Ω

as n → ∞.

On the other hand, by (4.4.10c), we have n (t)un (t) (t)u(t) weakly in Lβ (Ω), β = 2γ/(γ + 1), for a.e. t ∈ [0, T ]. Hence there is a set T of full measure in [0, T ] such that n Cη [ξϕn (t)]un Cη [ξϕ(t)]u(t) weakly in Lβ (Ω) for all t ∈ T . Thus we get ηHn (t) dx = Cη [ξ(t)ϕn (t)] (ζn (t)un (t)) dx lim n→∞ Ω Ω Ω Cη [ξ(t)ϕ(t)] ζ(t)u(t)) dx = ηH(t) dx = Ω

Ω

for all t ∈ T . This leads to (4.6.32). Thus, the sequence Hn is bounded in L∞ (0, T ; Lι (Ω)) and satisﬁes (4.6.31)–(4.6.32). Since ι > 2d/(d + 2), the sequences Hn = Hn and un meet all requirements of Lemma 4.4.5, hence there is m ∈ (1, ∞) such that Hn · un H · u weakly in L2 (0, T ; Lm (Ω)) as n → ∞. In particular,

Hn · un dxdt =

lim

n→∞

Q

H · u dxdt.

(4.6.33)

Q

Step 4. Proof of Theorem 4.6.3. We are now in a position to complete the proof of the theorem. Combining (4.6.26), (4.6.27), (4.6.33) and recalling the integral identity (4.6.18) we obtain lim

n→∞

ξϕn Vn dxdt = lim ξϕn R : [ζTn ] dxdt n→∞ Q Hn · un + ζn (Nn + Pn + Qn ) · un + Sn dxdt = lim n→∞ Q − lim ξϕn R : [ζGn ] dxdt n→∞ Q = H · u + ζ(P + Q) · u + S dxdt − lim ξϕn R : [ζGn ] dxdt.

Q

Q

n→∞

Q

4.7. Viscous ﬂux. Localization

95

Since s ≥ γ and q ≥ γ/(γ − 1), it follows from (4.6.9) that ξϕn R : [ζGn ] dxdt ≤ cϕn L∞ (0,T ;Lγ (Ω)) R : [ζGn ]L1 (0,T ;Lγ/(γ−1) (Ω)) Q

≤ cζGn L1 (0,T ;Lq (Ω)) → 0 as n → ∞, which leads to the convergence lim H · u + ζ(P + Q) · u + S dxdt. ξϕn Vn dxdt = n→∞

Q

(4.6.34)

Q

Recall that by Lemma 4.6.4 the functions (n , ϕn , un ) satisfy the hypotheses of Theorem 4.4.2. It follows from (4.4.6)–(4.4.9) and relations (4.6.9)–(4.6.12) that the limit functions (, ϕ, u) satisfy equations (4.6.1)–(4.6.3) with gn = gϕ n = 0, Gn = 0, n = , and Tn = T. Obviously the limit functions satisfy estimates (4.6.4)–(4.6.6). Therefore, (, ϕ, u) meet all requirements of Theorem 4.5.2, so H · u + ζ(P + Q) · un + S dxdt. ξϕR : [ζT] dxdt = Q

Q

Since R : [ζT] = V, this identity along with (4.6.34) implies (4.6.17), which completes the proof of Theorem 4.6.3.

4.7 Viscous ﬂux. Localization Theorem 4.6.3 has a serious drawback that expression (4.6.13) for the viscous ﬂux involves a nonlocal singular operator and an arbitrary cut-oﬀ function. In this section we show that in the practically important case of compressible NavierStokes equations this result can be formulated in a local form without singular integrals. Recall that in the speciﬁc case of Navier-Stokes equations, the stress tensor has the representation Tn = S(un ) − pn I,

(4.7.1)

where the viscous stress tensor S is given by S(u) = ∇u + ∇u + (λ − 1) div u I. Deﬁne the “genuine viscous ﬂux” Vn by Vn = pn − (1 + λ) div un .

(4.7.2)

The following theorem, a straightforward consequence of Theorem 4.6.3, is the main result of this section. This is a generalization of known results, in the sense that the stress tensor T plays no special role, and the result is given for an arbitrary solution ϕ of the mass balance transport equation instead of the density .

96

Chapter 4. Basic statements

Theorem 4.7.1. Suppose that, under the assumptions of Theorem 4.6.3, the sequence Tn , n ≥ 1, admits representation (4.7.1) and r ≥ 2. Then there is a subsequence of Vn , still denoted by Vn , such that Vn V = p − (1 + λ) div u

weakly in Lr (Q )

for every Q Q.

Moreover, for every η ∈ C0∞ (Q), we have lim η Vn ϕn dxdt = η V ϕ dxdt. n→∞

Q

(4.7.3)

(4.7.4)

Q

Proof. By (4.6.4), the sequence S(un ) is bounded in L2 (Q). Moreover, estimates (4.6.7)–(4.6.8) imply the boundedness of Tn in L1 (Q) and Lrloc (Q). Notice that r = r/(r − 1) ≤ 2. Now formula (4.7.1) implies that pn is bounded in L1 (Q) and Lrloc (Q). Passing to a subsequence we can assume that pn converges weakly to some p ∈ L1 (Q) ∩ Lrloc (Q) in Lr (Q ) for every Q Q. Furthermore, div un → div u weakly in L2 (Q). These results along with (4.7.2) imply (4.7.3). Now choose ζ ∈ C0∞ (Ω) such that η(x, t)ζ(x) = η(x, t) for all (x, t) ∈ Rd+1 . It is easy to see that ζTn = ∇(ζun ) + ∇(ζun ) + ζ (λ − 1) div un − pn I − ∇ζ ⊗ un − un ⊗ ∇ζ. (4.7.5) Notice that the operator R : W 1,2 (Rd ) → W 1,2 (Rd ) is bounded and commutes with diﬀerentiation. In particular, R : [∇w + ∇(w) ] = Rij [ ∂xi (wj ) + ∂xj (wi ) ] = ∂xi ∂xj Δ−1 [ ∂xi (wj ) + ∂xj (wi ) ] = ∂xj ∂x2i Δ−1 wj + ∂xi ∂x2j Δ−1 wi = ∂xj wj + ∂xi wi = 2 div w for all w ∈ W 1,2 (Rd ). Thus we get R : [∇(ζun (t)) + ∇(ζun (t)) ] = 2 div(ζun (t)).

(4.7.6)

Next, for every v ∈ Lr (Ω) with 1 < r < ∞, we have R : [v I] = Rij [vδij ] = Rii [v] =

d

∂x2i Δ−1 v = v,

i=1

hence for a.e. t ∈ [0, T ], R : [ζ((λ − 1) div un (t) − pn (t)) I] = ζ(λ − 1) div un (t) − ζpn (t). Combining (4.7.5)–(4.7.7) we obtain R : [ζTn ] = 2 div(ζun (t)) + ζ(λ − 1) div un (t) − ζpn (t) − R : [∇ζ ⊗ un + un ⊗ ∇ζ],

(4.7.7)

4.7. Viscous ﬂux. Localization

97

which leads to R : [ζTn (t)] = ζ (λ + 1) div un (t) − pn (t) − Wn (t), where Wn (t) = R : [∇ζ ⊗ un (t) + un (t) ⊗ ∇ζ] − 2∇ζ · un (t). Recalling the deﬁnition (4.7.2) of Vn we can rewrite this identity in the form ζVn (t) = −R : [ζTn (t)] − Wn (t) = −Vn (t) − Wn (t).

(4.7.8)

Repeating the same arguments we arrive at the following representation of the limit V : (4.7.9) ζV (t) = −R : [ζT(t)] − W (t) = −V − W (t), where W (t) = R : [∇ζ ⊗ u(t) + u(t) ⊗ ∇ζ] − 2∇ζ · u(t).

(4.7.10)

It follows from (4.7.8) and the equality ηζ = η that lim η Vn ϕn dxdt = − lim η Vn ϕn dxdt − lim η Wn ϕn dxdt. (4.7.11) n→∞

n→∞

Q

n→∞

Q

Q

Since Tn and ϕn meet all requirements of Theorem 4.6.3, we have lim η Vn ϕn dxdt = η V ϕ dxdt. n→∞

Q

Q

Combining this with (4.7.9) and (4.7.11) we obtain η Vn ϕn dxdt = η V ϕ dxdt − lim η Wn ϕn − W ϕ) dxdt. lim n→∞

Q

Q

n→∞

Q

It remains to prove that the limit on the right hand side is simply zero. Since the operator R : W 1,2 (Rd ) → W 1,2 (Rd ) is bounded and un u weakly in L2 (0, T ; W 1,2 (Ω)), we have Wn W

weakly in L2 (0, T ; W 1,2 (Ω)).

On the other hand, by Lemma 4.6.4, the sequence ϕn meets all requirements of Theorem 4.4.2, so ϕn (t) ϕ(t) weakly (weakly for s = ∞) in Ls (Ω). Moreover, s > 2d. Hence the couple (ϕn , Wn ) meets all requirements of Lemma 4.4.5, so Wn ϕn W ϕ weakly in L2 (0, T ; Lm (Ω)) for all m−1 > s−1 + (d − 2)/(2d). This completes the proof of Theorem 4.7.1.

Chapter 5

Nonstationary case. Existence theory 5.1 Problem formulation. Results In this chapter we prove the existence of weak solutions to the general initialboundary value problem for compressible Navier-Stokes equations, which includes problem (3.0.3) as a particular case. The results obtained are only of preliminary character: we assume that the pressure p() is a fast growing function of the density, and the initial and boundary data are suﬃciently smooth and satisfy some compatibility conditions. These restrictions are removed in Section 10.1. We use a three-level regularization method similar to the method used in monographs [34] and [37], but our regularization scheme is diﬀerent because of diﬃculties caused by inhomogeneous boundary conditions. In order to make the presentation clearer and avoid unnecessary technical difﬁculties, we assume that the ﬂow domain Ω, the function p in the constitutive law p = p(), and the given functions U, ∞ , and f satisfy the following hypotheses: Condition 5.1.1. • The ﬂow domain Ω ⊂ Rd , d = 2, 3, is a bounded domain ∞ with C boundary. For given T > 0 we denote by Q the cylinder with lateral surface ST and parabolic boundary T deﬁned by Q = Ω × (0, T ),

ST = ∂Ω × (0, T ),

T

= ST ∪ (cl Ω × {t = 0}). (5.1.1)

• We consider the general constitutive law p = p() and the internal energy density P () = s−2 p(s) ds (5.1.2) 0

with the properties P () − P () = p(),

P () = −1 p ().

(5.1.3)

P. Plotnikov and J. Sokoáowski, Compressible Navier–Stokes Equations: Theory and Shape Optimization, Monografie Matematyczne 73, DOI 10.1007/978-3-0348-0367-0_5, © Springer Basel 2012

99

100

Chapter 5. Nonstationary case. Existence theory

• The function p : [0, ∞) → R+ has continuous ﬁrst derivative with p () ≥ 0,

p(0) = p (0) = 0.

(5.1.4)

There are an exponent γ > 1 and a constant cE > 0 such that γ ≤ cE (P () + 1),

γ ≤ cE (p() + 1),

p() ≤ cE (P () + 1).

(5.1.5)

The function P is twice continuously diﬀerentiable on R+ and for all ∈ [1, ∞), γ γ c−1 E ≤ p() ≤ cE ,

γ−1 c−1 ≤ p () ≤ cE γ−1 . E

(5.1.6)

Moreover, P () ≥ c−1 E > 0 for all ∈ [0, ∞),

|∇P (∞ )| ≤ cE .

(5.1.7)

• There is an exponent υ ∈ (1, γ] and a constant cp > 0 such that υ ≤ cp (P () + 1),

υ ≤ cp (p() + 1),

p() ≤ cp (P () + 1).

(5.1.8)

The constant cE in (5.1.5)-(5.1.7) is greater than cp . • Let Γ = (ST \Σin )∩cl Σin be the interface between the inlet and the outgoing subset of ST . We assume that lim σ −d meas Oσ < ∞,

σ→0

(5.1.9)

where Oσ = {(x, t) ∈ Rd+1 : dist((x, t), Γ) ≤ σ} is a tubular neighborhood of Γ. • The initial and boundary data satisfy ∞ , U ∈ C ∞ (Q), f ∈ C(Q), and UC 1 (Rd ×(0,T )) + ∞ L∞ (Q) + f C(Q) ≤ ce ,

∞ C 1 (Rd ×(0,T )) ≤ cE (5.1.10) where ce is some positive constant. Moreover, the data satisfy the compatibility condition ∞ = 0 in the vicinity of the edge ∂Ω × {t = 0}.

(5.1.11)

Remark 5.1.2. Throughout this chapter we consider various modiﬁcations of the governing equations and their regularizations depending on some parameters. The main ingredients of the theory are a priori estimates of solutions to the governing equations. The a priori estimates are bounds for norms of the solutions in appropriate Banach spaces. These bounds depend on the ﬂow domain, initial and boundary data, the constitutive law, and parameters in the equations. With an application to the shape optimization problem in mind, we distinguish a priori

5.1. Problem formulation. Results

101

estimates which depend only on the diameter of the ﬂow domain and on “rough” characteristics of the data, and more precise estimates, depending on ﬁne geometric properties of the ﬂow domain and on the smoothness properties of the functions involved in the governing equations. Further, we denote by ce generic constants depending only on λ,

γ,

diam Ω,

T,

∞ L∞ (

T)

,

UC 1 (Q) ,

f C(Q) ,

cp .

The notation cE,p stands for a generic constant depending on ce and on Ω, cE , γ and the norms ∞ C 3 (Q) , UC 4 (Q) . In other words, ce depends only on rough characteristics of the data and on the constitutive law, while cE,p depends on the ﬁne structure of the ﬂow domain, the pressure function, and the given data. Now we formulate the evolution problem under consideration. Problem 5.1.3. Find a velocity ﬁeld u and a density function satisfying ∂t (u) + div( u ⊗ u) + ∇p(ρ) = div S(u) + f ∂t + div(u) = 0 in Q, u=U = ∞

in Q,

= ST ∪ (cl Ω × {t = 0}), on Σin , |t=0 = ∞ in Ω,

on

T

(5.1.12a) (5.1.12b) (5.1.12c)

where the inlet is of the form Σin = {(x, t) ∈ ST : U(x, t) · n(x) < 0}, and S(u) = ∇u + ∇u + (λ − 1) div u,

div S = Δu + λ∇ div u.

We claim that system (5.1.12) admits a weak renormalized solution which is deﬁned as follows: Deﬁnition 5.1.4. A couple u ∈ L2 (0, T ; W 1,2 (Ω)),

∈ L∞ (0, T ; Lγ (Ω))

is said to be a weak renormalized solution to problem (5.1.12) if (u, ) satisﬁes • The kinetic energy is bounded, i.e., |u|2 ∈ L∞ (0, T ; L1 (Ω)). • The velocity satisﬁes the nonhomogeneous Dirichlet boundary condition u = U on ST . This condition makes sense since u ∈ L2 (0, T ; W 1,2 (Ω)) has a trace on ST = ∂Ω × (0, T ). • The integral identity u · ∂t ξ + u ⊗ u : ∇ξ + p div ξ − S(u) : ∇ξ dxdt Q + f · ξ dxdt + (∞ U · ξ)(x, 0) dx = 0 (5.1.13) Q

Ω ∞

holds for all vector ﬁelds ξ ∈ C (Q) equal to 0 in a neighborhood of the lateral side ST and of the top Ω × {t = T }.

102

Chapter 5. Nonstationary case. Existence theory

• The integral identity Q

ϕ()∂t ψ + ϕ()u · ∇ψ + ψ(ϕ() − ϕ ()) div u dxdt = ψϕ(∞ )U · n dΣ − (ϕ(∞ )ψ)(x, 0) dx (5.1.14) Σin

Ω

holds for all ψ ∈ C ∞ (Q) vanishing in a neighborhood of the surface ST \ Σin and in a neighborhood of the top Ω × {t = T }, and for all smooth functions ϕ : [0, ∞) → R such that lim sup(|ϕ()| + |ϕ ()|) < ∞.

(5.1.15)

→∞

This means that ϕ has minimal admissible smoothness and ϕ is bounded at inﬁnity. Remark 5.1.5. Since u ∈ L2 (0, T ; W 1,2 (Ω)), the integral identity (5.1.14) also holds for functions ϕ satisfying the growth conditions lim sup(−γ/2 |ϕ()| + −γ/2+1 |ϕ ()|) < ∞.

(5.1.16)

→∞

The existence of weak renormalized solutions to problem (5.1.12) is the main result of this chapter. Theorem 5.1.6. Assume that Condition 5.1.1 is fulﬁlled, and that γ > 2d. Then problem (5.1.12) has a weak renormalized solution which meets all requirements of Deﬁnition 5.1.4 and satisﬁes the estimate uL2 (0,T ;W 1,2 (Ω)) + |u |2 L∞ (0,T ;L1 (Ω)) + P ()L∞ (0,T ;L1 (Ω)) ≤ ce , (5.1.17) where the constant ce is as in Remark 5.1.2. Estimate (5.1.17) depends only on the pressure estimate (5.1.8) from below. The remaining part of this chapter is devoted to the proof of this theorem. Proceeding from a recent construction [34, 37], it contains three contributions to the existence theory for problem (5.1.12). The ﬁrst is solvability of regularized equations. The second states a priori estimates for solutions of the regularized equations; the estimates are independent of the regularization parameters. The third establishes compactness properties for these solutions. We begin with the construction of regularized equations. There are several ways of regularizing the governing equations and deﬁning approximate solutions: the ﬁctitious domain method with mollifying the velocity vector ﬁeld in the mass balance equation; penalization of governing equations (see [80]); time discretization combined with existence results for the stationary problem (see [80]); parabolic or elliptic regularization of the mass balance equations

5.1. Problem formulation. Results

103

combined with correction of the momentum balance equations (see [80], [34]). Notice that justiﬁcation of the penalization method is known at present only for a time independent convex surface Σin (see [93], [52]). We cannot use the time discretization method since the stationary nonhomogeneous boundary value problem is more diﬃcult to analyze compared to nonstationary problems. In fact, the stationary problem remains essentially unsolved. We are going to use a three-level approximation scheme to regularize the governing equations. The basic element of the scheme is the standard parabolic regularization proposed by P.-L. Lions and E. Feireisl (see [80], [34]), ∂t (ε uε ) + div((ε uε − ε∇ε ) ⊗ uε ) + ∇pε = div S + ε f , ∂t ε + div(ε u) = εΔε ,

(5.1.18) (5.1.19)

with the boundary and initial conditions uε = U,

ε = ∞

on

T.

We emphasize that, in contrast to [34, 37], the density ε should satisfy the Dirichlet condition at least on the inlet Σin in view of the boundary conditions (5.1.12c) for the density. Equations (5.1.18)–(5.1.19) can be considered as a mathematical model of compressible ﬂows with mass diﬀusion and such equations are of independent interest. The peculiarity of this model is that the momentum balance equation degenerates for ε = 0, therefore, the model needs additional corrections. To make such a correction we notice that the main diﬀerence between the homogeneous and nonhomogeneous boundary value problems is that the test function ψ in the integral identity (5.1.14) is nonzero on Σin . The corresponding integral identity for the regularized mass balance equation reads Q

ε ∂t ψ + (ε uε − ε∇ε ) · ∇ψ dxdt = ψ(∞ U + ε∇ε ) · n dΣ − ∞ ψ(x, 0) dx. Σin

Ω

In order to obtain, in the limit ε → 0, the integral identity (5.1.14) for , i.e., with ϕ() = , we have to show that solutions to the regularized mass balance equations (5.1.19) satisfy ε ∇ε ∇ψ dxdt → 0, ε ψ ∂n ε dΣ → 0 as ε → 0. Q

Σin

The ﬁrst relation easily follows from the energy estimate, but the second is less trivial to justify. Here, we are dealing with the vanishing viscosity limit for the normal derivative of solutions to an equation (5.1.19) with a convective term. This question arises in a natural way in the theory of elliptic-parabolic equations with nonnegative characteristic form. In the elliptic case it was shown in [102] that

104

Chapter 5. Nonstationary case. Existence theory

ε∂n ε tends to 0 on Σin provided that the vector ﬁelds uε have uniformly bounded derivatives in a neighborhood of Σin and the functions ε are uniformly bounded in this neighborhood. In Section 13.4 we extend this result to the parabolic case with integrable ε , but the restriction on the smoothness of uε cannot be weakened. To make solutions smooth near the boundary as ε → 0, we take the regularized equations in the form Δ2 (bΔ2 ∂t (u − U)) + b∂t (u − U) + ∂t (u) + div((u − ε∇) ⊗ u) + ∇p(ρ) = div S(u) + f ∂t + div(u) = εΔ in Q,

in Q,

(5.1.20a) (5.1.20b)

along with the boundary and initial conditions = ∞

on

T,

u − U ∈ C(0, T ; W04,2 (Ω)),

u|t=0 = U in Ω,

(5.1.20c)

where b ∈ C ∞ (Ω) is an arbitrary positive function. In Section 5.2 we apply the Leray-Schauder Theorem 1.1.16 to show that problem (5.1.20) has a strong solution and derive estimates for this solution. The results obtained are collected in Theorem 5.2.1. Then we start a long way from the regularized equations to compressible Navier-Stokes equations. First, we take b(x) = a(x) + σ, σ > 0, where a is a smooth cut-oﬀ function such that a = 1 in the δ/2-neighborhood of ∂Ω and a = 0 outside the δ-neighborhood of ∂Ω. We consider σ, δ and ε as the parameters of regularization. Thus we have a three-level regularization, and the proof of Theorem 5.1.6 involves three successive passages to the limit: σ → 0, then ε → 0, and ﬁnally δ → 0. The ﬁrst step is made in Section 5.3. Actually, we show that there is a sequence σn → 0 such that the corresponding solutions to problem (5.1.20) converge weakly in Q to some functions (ε , uε ). We also prove that ε satisﬁes the mass diﬀusion equation (5.1.19) in Q, but the momentum balance equation (5.1.18) holds true only in a smaller cylinder. However, the functions (ε , uε ) remain smooth in a neighborhood of Σin . The results obtained in Section 5.3 are collected in Theorem 5.3.1. By Theorem 5.3.1 the velocity ﬁeld u and the density ε still remain smooth in a neighborhood of the lateral surface ST . We use this fact in Section 5.4, where we exploit Proposition 5.3.11 to prove that ε∂n ε → 0 on the inlet as ε → 0. Using that proposition we show that the weak limit points of ε serve as renormalized solutions to the mass balance equation and satisfy the integral identity (5.1.14). However, we are unable to prove the strong convergence of ε at this stage. Instead we obtain a representation of a weak limit p of p(ε ) in terms of a Young measure associated with the sequence ε . Letting δ → 0 in Section 5.5 completes the proof of the three-level convergence of the regularizing scheme.

5.2. Regularized equations

105

5.2 Regularized equations This section is devoted to existence theory for the regularized problem (5.1.20). Here we prove that for the adiabatic exponent γ > 2d, the regularized problem has a solution in C 1,2 (Q), and derive estimates for this solution independent of the regularization parameter b. The result is given by the following theorem. Theorem 5.2.1. Let the data U, ∞ , f and the pressure function p() satisfy Conditions 5.1.1 with γ > 2d. Let b ∈ C ∞ (Ω) satisfy b > σ > 0. Then there exists ε0 > 0, depending only on cE,p , such that for all ε ∈ (0, ε0 ), problem (5.1.20) has a solution u ∈ C 1 (0, T ; W 8,2 (Ω)),

∈ C 1+α/2,2+α (Q),

0 < α < 1,

≥ 0.

(5.2.1)

The couple (, u) satisﬁes the energy estimates b|Δ2 u|2 + b|u|2 + |u|2 + P (x, t) dx sup t∈(0,T )

Ω

+

|∇u|2 + εP ()|∇|2 dxdt ≤ cE,p ,

(5.2.2)

Q

uL2 (0,T ;W 1,2 (Ω)) ≤ cE,p ,

|u|2 L∞ (0,T ;L1 (Ω)) ≤ cE,p ,

(5.2.3)

L∞ (0,T ;L1 (Ω)) + ε(1 + ) |∇| L1 (Q) ≤ cE,p , uL2 (Q) + uL∞ (0,T ;Lβ (Ω)) ≤ cE,p ,

(5.2.4) (5.2.5)

|u|2 L2 (0,T ;Lz (Ω)) ≤ cE,p ,

(5.2.6)

γ

ε

γ−2

1/2

|∇| |u|L1 (0,T ;Ls (Ω)) ≤ cE,p ,

3γ/2

ε

2

(5.2.7)

L1 (Q) ≤ cE,p ,

(5.2.8)

where β = 2γ/(γ + 1), s = d/(d − 1), and z is an arbitrary number satisfying 1 ≤ z < (1 − 2−1 a)−1 ,

a = 2d−1 − γ −1 .

For every t ∈ [0, T ], and u satisfy b|Δ2 u|2 + |bu|2 + |u|2 + P (x, t)dx Ω t + |∇u|2 + ε(P () − ε1/3 )|∇|2 dxdt Ω 0 t ≤ ce + M (t) + ce ece (t−s) M (s) ds,

(5.2.9)

(5.2.10)

0

where the constant ce is speciﬁed by Condition 5.1.1 and t M (t) = 2 b|Δ2 U|2 (x, t) dx + 8ε2/3 |∇P (∞ )|2 dxdt 0 Ω Ω t ∂t P (∞ ) + ∇P (∞ ) · u dxdt. −4 0

Ω

(5.2.11)

106

Chapter 5. Nonstationary case. Existence theory

In order to apply a ﬁxed point argument we reduce problem (5.1.20) to an operator equation in a space of smooth functions. Observe that the governing equations can be rewritten in an equivalent form Δ2 (bΔ2 v) + bv + v = Ψ[u, ] in Q, v ∈ C(0, T ; W04,2 (Ω)), t u(x, t) = U(x, t) + v(x, s) ds in Q,

(5.2.12a) (5.2.12b)

0

∂t + div(u) = εΔ in Q, = ∞ on T = ST ∪ (cl Ω × {t = 0}),

(5.2.12c) (5.2.12d)

where the nonlinear operator Ψ is deﬁned by Ψ[u, ] = div S(u) + f − (u − ε∇)∇u − ∇p(ρ) − ∂t U.

(5.2.13)

Let us consider the family of mappings Ξτ : (˜ u, ˜) → (u, ) deﬁned by u, ˜] in Q, Δ2 (bΔ2 v) + bv + τ ˜v = τ Ψ[˜ t u(x, t) = U(x, t) + v(x, s) ds in Q,

v ∈ C(0, T ; W04,2 (Ω)),

(5.2.14a) (5.2.14b)

0

u) − εΔ = 0 in Q, ∂t + div(˜ = ∞ on T .

(5.2.14c) (5.2.14d)

Problem (5.2.12) can be written as a ﬁxed point problem (u, ) = Ξ1 (u, ),

(u, ) ∈ X,

(5.2.15)

in an appropriate Banach space X. We set X = C κ (0, T ; C02 (Ω)) × C 1+α/2,2+α (Q) with 0 < α < κ < 1.

(5.2.16)

Theorem 5.2.1 will be proved if we show that the one-parameter family of operators Ξτ meets all requirements of the Leray-Schauder theorem 1.1.16 in X. We check these in a sequence of lemmas. The ﬁrst yields the continuity properties of Ξτ . Lemma 5.2.2. The mapping X × [0, 1] (u, , τ ) → Ξτ (u, ) ∈ X is continuous. Moreover, for any τ ∈ [0, 1] the mapping Ξτ : X → X is compact, i.e., it takes bounded subsets of X into relatively compact sets. Proof. First we show that Ξτ : X → C 1 (0, T ; W 8,2 (Ω)) × C 1+ι/2,2+ι (Q) is continu, ˜) strongly in X. Since uous for all τ ∈ [0, 1] and any ι ∈ (0, κ]. Let (˜ un , ˜n ) → (˜ the embedding X → C(0, T ; C 2 (Ω)) × C 1 (Q) is bounded, we have ˜n → u ˜ in C(0, T ; C 2 (Ω)), u

˜n → ˜ in C 1 (Q) as n → ∞.

u, ˜) in C(0, T ; C(Ω)). By Lemma It now follows from (5.2.13) that Ψ(˜ un , ˜n ) → Ψ(˜ 1.7.8, the unique solution vn (t) to the equation Δ2 (bΔ2 vn (t)) + bvn (t) + τ ˜n (t)vn (t) = τ Ψ(˜ un , ˜n )(t),

vn (t) ∈ W04,2 (Ω),

5.2. Regularized equations

107

satisﬁes the inequality vn (t)W 8,2 (Ω) ≤ cτ Ψ(˜ un (t), ˜n (t))L2 (Ω) ≤ cτ, where c is independent of n. Thus, the sequence vn is bounded in C(0, T ; W 8,2 (Ω)). Next, we have Δ2 (bΔ2 (v − vn )) + b(v − vn ) + τ ˜(v − vn ) = τ Ψ(˜ u, ˜) − Ψ(˜ un , ˜n ) + τ (˜ n − ˜)vn , which along with Lemma 1.7.8 leads to u(t), ˜(t)) − Ψ(˜ un (t), ˜n (t))C(Ω) vn − vC(0,T ;W 8,2 (Ω)) ≤ c max Ψ(˜ (0,T )

+ c max ˜ (t) − ˜n (t)C(Ω) max ˜ vn (t)C(Ω) → 0 (0,T )

(0,T )

as n → ∞. Hence un − uC 1 (0,T ;W 8,2 (Ω)) → 0 as n → ∞. Therefore, equations (5.2.14a) and (5.2.14b) determine a continuous mapping {(˜ u, ˜) → u} : X → C 1 (0, T ; W 8,2 (Ω)). Let us now return to the parabolic boundary value problem (5.2.14c)-(5.2.14d). We rewrite (5.2.14c) in the form ˜ · ∇ + (div u ˜ ) = 0. ∂t − εΔ + u

(5.2.17)

˜ and div u ˜ of equation (5.2.17) Notice that for any (˜ u, ˜) ∈ X, the coeﬃcients u belong to C κ/2,κ (Q). Therefore, (5.2.14c) meets all requirements of Theorem 1.7.12 and problem (5.2.14c)–(5.2.14d) has a unique solution ∈ C 1+κ/2,2+κ (Q). Thus equations (5.2.14c)–(5.2.14d) deﬁne a mapping {(˜ u, ˜) → } : X → C 1+κ/2,1+κ (Q). Let us prove that the mapping is continuous. Assume that (˜ un , ˜n ) converges to (˜ u, ˜) in X. We have ˜, ˜n → u u

˜ n → div u ˜ in C κ/2,κ (Q). div u

By Theorem 1.7.12, the corresponding solutions n to problem (5.2.14c)–(5.2.14d) are uniformly bounded in C 1+κ/2,2+κ (Q). The diﬀerence − n satisﬁes ˜ · ∇( − n ) + (div u ˜ )( − n ) ∂t ( − n ) − εΔ( − n ) + u ˜ )∇n + div(˜ ˜ )n , = (˜ un − u un − u − n = 0 on

T ST

∪ (cl Ω × {t = 0}).

(5.2.18) (5.2.19)

108

Chapter 5. Nonstationary case. Existence theory

Since = n = ∞ on T and ∞ vanishes near the edge ∂Ω × {t = 0}, the initial and boundary data in equation (5.2.18) satisfy the compatibility conditions of Theorem 1.7.12. It follows that − n C 1+κ/2,2+κ (Q) ≤ c(un − uC κ/2,κ (Q) + div(un − u)C κ/2,κ (Q) )n C 1+κ/2,2+κ (Q) ≤ cun − uC κ (0,T ;C 2 (Ω)) → 0 as n → ∞, and the mapping {(˜ u, ˜) → } : X → C 1+κ/2,2+κ (Q) is continuous. Combining the results obtained we conclude that the mapping Ξτ : X → C 1 (0, T ; W 8,2 (Ω)) × C 1+κ/2,2+κ (Q) is continuous. Since the embedding C 1 (0, T ; W 8,2 (Ω)) × C 1+κ/2,2+κ (Q) → X is compact, it follows that the mapping Ξτ : X → X is continuous and compact. The continuity of this mapping with respect to τ is obvious. Next, we have to prove that all ﬁxed points of the operators Ξτ belong to some bounded set in X. This means that the set of all solutions to equations (5.2.12) in X (under the assumption that they exist) is uniformly bounded in X. In this way we also establish the desired estimates (5.2.2)–(5.2.10). In fact all these estimates can be derived from the ﬁrst energy estimate which is a consequence of the energy conservation law. We begin by proving (5.2.10) and split the proof into three lemmas. Lemma 5.2.3. Under the assumptions of Theorem 5.2.1, let (u, ) ∈ X be a ﬁxed point of the operator Ξτ with 0 < τ ≤ 1. Then there is a constant ce as in Theorem 5.2.1 and ε0 > 0, depending only on ce , such that for all ε < ε0 , 1 2

1 t b|Δ (u − U)| + b|u − U| + |u − U| (x, t) dx + |∇u|2 dxds 2 0 Ω Ω t t 1 4/3 2 2 ε |∇| + ce P () + ce |u| dxds. ≤ p() div u dxds + ce + 2 0 0 Ω Ω (5.2.20)

2

2

2

2

Proof. We have already proved that each ﬁxed point (u, ) ∈ X of Ξτ belongs to C 1,2 (Q) and ∂t u ∈ C(0, T ; W 8,2 (Ω)). Hence the ﬁxed point is a strong solution to the equation b 1 2 Δ (bΔ2 ∂t (u − U)) + ∂t (u − U) + ∂t (u − U) = Ψ[u, ] in Q. τ τ Multiplying this equation by u − U and integrating the result over the cylinder

5.2. Regularized equations

109

Ω × [0, t] we arrive at the identity t b 1 2 2 Δ bΔ ∂t (u − U) + ∂t (u − U) · (u − U) dxdt τ τ Ω 0 t 1 ∂t |u − U|2 + (u − ε∇)∇|u − U|2 dxdt + 2 0 Ω t + ∇p − (Δu + λ∇ div u) · (u − U) dxdt 0 Ω t t + ∂t U + (u − ε∇)∇U · (u − U) dxdt = f · (u − U) dxdt. 0

0

Ω

Ω

(5.2.21)

Since ∂nk (u − U) = 0 on the lateral surface ST for all integers 0 ≤ k ≤ 3, and u − U = 0 at the bottom Ω × {t = 0}, and ∂t + div(u − ε∇) = 0, we can integrate by parts in (5.2.21) to obtain 1 b (|Δ2 (u − U)|2 + |u − U|2 ) + |u − U|2 (x, t) dx 2 Ω τ t t 2 |u| + λ| div u|2 dxdt + I1 + I2 + I3 = p div u dxdt, + 0

Ω

where

0

(5.2.22)

Ω

t (∂t U · u + u∇U · u − ε∇∇U · u − f · u) dxdt, Ω t I2 = − (∂t U · U + u∇U · U − ε∇∇U · U − f · U) dxdt, 0 Ω t (∇u : ∇U + λ div u div U − p div U) dxdt. I3 = −

I1 =

0

0

Ω

Recalling that the absolute values of U, f , and the ﬁrst derivatives of U are bounded by the constant ce we get the estimate t |I1 + I2 + I3 | ≤ ce (p + + |u| + |u|2 + ε|∇| |u| + ε|∇|) dxdt 0 Ω t + ce |∇u| dxdt. (5.2.23) 0

Ω

The Cauchy inequality implies |u| ≤ + |u|2 , |∇u| ≤ δ|∇u|2 + δ −1 , 1 1 ce ε|∇| |u| ≤ ε4/3 |∇|2 + 4c2e ε2/3 |u|2 , εce |∇| ≤ ε2 |∇|2 + 4c2e , 4 4

110

Chapter 5. Nonstationary case. Existence theory

for every δ ∈ (0, 1). Substituting these estimates into (5.2.23) we obtain t ce + ce |I1 + I2 + I3 | ≤ p + + |u|2 + (ε2/3 + δ)|u|2 dxdt δ 0 Ω t t 1 + ε4/3 |∇|2 dxdt + δce |∇u|2 dxdt. (5.2.24) 2 Ω Ω 0 0 Since u − U vanishes at ST , the Poincaré inequality yields t t |u|2 dxdt ≤ |u − U|2 dxdt + |U|2 dxdt Ω Ω Ω 0 0 0 t t ≤ ce + |u − U|2 dxdt ≤ ce + ce |∇u − ∇U|2 dxdt Ω Ω 0 0 t t t 2 2 |∇u| dxdt + ce |∇U| dxdt ≤ ce + ce |∇u|2 dxdt. ≤ ce + ce t

0

Ω

0

Ω

0

Ω

Inserting this into (5.2.24) we obtain |I1 + I2 + I3 | t ce 1 + ce p + ce + ce |u|2 + ε4/3 |∇|2 + ce (2ε2/3 + δ)|∇u|2 dxdt. ≤ δ 2 Ω 0 In turn, substituting this into (5.2.22) and recalling that τ ≤ 1 we arrive at 1 b|Δ2 (u − U)|2 + b|(u − U)|2 + |u − U|2 (x, t) dx 2 Ω t + (1 − ce δ − ce ε2/3 ) |∇u|2 dxdt Ω 0 t t 1 ce 2 4/3 2 + ce p + ce + ce |u| + ε |∇| dxdt + p div u dxdt. ≤ δ 2 Ω Ω 0 0 (5.2.25) Next, by (5.1.8) we have υ −2 −1 −2 −1 ≥ c−2 P () ≥ c−1 p p − cp p − cp − cp ≥ cp − ce .

Inserting this into (5.2.25) we obtain 1 b|Δ2 (u − U)|2 + b|(u − U)|2 + |u − U|2 (x, t) dx 2 Ω t 2/3 + (1 − ce δ − ce ε ) |∇u|2 dxdt 0 Ω t t ce 1 4/3 2 2 ce P () + ce |u| + ε |∇| dxdt + ≤ + p div u dxdt. δ 2 0 0 Ω Ω

5.2. Regularized equations

111 2/3

Finally choosing ε0 and δ, depending only on ce , such that ce δ + ce ε0 arrive at the desired inequality (5.2.20).

≤ 1/2 we

Lemma 5.2.4. There is a constant ce as in Remark 5.1.2 such that for all t ∈ (0, T ) and τ ∈ (0, 1], any ﬁxed point (u, ) of the operator Ξτ satisﬁes t

1 1/3 P () − ε |∇|2 dxdt P ()(x, t) dx + ε 2 0 Ω Ω t ≤ ce + 2ε2/3 |∇P (∞ )|2 dxdt Ω 0 t t − ∂t P (∞ ) + u · ∇P (∞ ) dxdt − p div u dxdt.

1 2

0

Ω

0

(5.2.26)

Ω

Proof. If (u, ) is a ﬁxed point of Ξτ , then satisﬁes ∂t − εΔ + u · ∇ + div u = 0 = ∞

on

in Q,

(5.2.27)

T.

Since ∞ ∈ C ∞ (Q) satisﬁes the compatibility conditions (5.1.11), the boundary value problem for meets all requirements of Theorem 1.7.12 and hence has a unique solution ∈ C 1+α/2,2+α (Q) which is strictly positive in Q. Thus we can multiply both sides of the diﬀusion equation in (5.2.27) by P () − P (∞ ) and use the identity P () − P () = p() to obtain 0 = (∂t − εΔ + u · ∇ + div u )(P () − P (∞ )) = ∂t (P () − P (∞ )) + div (P () − P (∞ ))u − ε div (P () − P (∞ ))∇ + εP ()|∇|2 − ε∇ · ∇P (∞ ) + ∂t P (∞ ) + ∇P (∞ ) · u + p() div u. Integrating this equality by parts over the cylinder Ω×(0, t) and using the identity P (∞ )∞ − P (∞ ) = p(∞ ) we arrive at t P () − P (∞ ) (x, t) dx + ε P ()|∇|2 dxdt 0 Ω Ω t t + ∂t P (∞ ) + u · ∇P (∞ ) dxdt − ε ∇ · ∇P (∞ ) dxdt 0 0 Ω Ω t t p(∞ )(x, 0) dx + p(∞ )U · n dSdt = − p div u dxdt, +

Ω

0

∂Ω

0

Ω

112

Chapter 5. Nonstationary case. Existence theory

which leads to the inequality (P () − ce )(x, t) dx Ω t + εP ()|∇|2 + ∂t P (∞ ) + u · ∇P (∞ ) dxdt Ω 0 t t −ε ∇ · ∇P (∞ ) dxdt ≤ ce − p div u dxdt. 0

Ω

0

Ω

Next notice that ε|∇P (∞ )| |∇| ≤

1 4/3 ε |∇|2 + 2ε2/3 |∇P (∞ )|2 , 2

which leads to t 1 (P () − ce )(x, t) dx + ε P () − ε1/3 |∇|2 dxdt 2 Ω Ω 0 t + ∂t P (∞ ) + u · ∇P (∞ ) dxdt 0 Ω t t 2/3 ≤ ce + 2ε |∇P (∞ )|2 dxdt − p div u dxdt. 0

0

Ω

(5.2.28)

Ω

Next, condition (5.1.8) and the Young inequality imply υ ≤ cp P () + cp ,

ce ≤ δυ + δ −υ/(υ−1) ceυ/(υ−1) ,

where δ > 0 is an arbitrary number. In particular, for suﬃciently small δ > 0 and a suitable choice of the constant ce , we have ce ≤ P ()/2 + ce . Inserting (5.2.29) into (5.2.28) we obtain the desired estimate (5.2.26).

(5.2.29)

Lemma 5.2.5. There is a constant ce as in Remark 5.1.2 and ε0 > 0, depending only on cE,p , such that for all ε ∈ (0, ε0 ), t ∈ (0, T ), and τ ∈ (0, 1], any ﬁxed point (u, ) of the operator Ξτ satisﬁes Ω

b|Δ2 u|2 + b|u|2 + |u|2 + P () (x, t) dx t |∇u|2 + ε(P () − ε1/3 )|∇|2 dxdt + 0 Ω t ≤ ce + ce (P () + |u|2 ) dxdt + M (t), (5.2.30) 0

Ω

5.2. Regularized equations

Ω

113

b|Δ2 u|2 + b|u|2 + |u|2 + P () (x, t) dx t + |∇u|2 + εP ()|∇|2 dxdt 0 Ω t ≤ cE,p + cE,p (P () + |u|2 ) dxdt, 0

(5.2.31)

Ω

where M (t) is deﬁned by (5.2.11). Proof. Adding (5.2.20) and (5.2.26) we arrive at 1 2

Ω

b|Δ2 (u − U)|2 + b|u − U|2 + |u − U|2 + P () (x, t) dx 1 t |∇u|2 + εP ()|∇|2 − ε4/3 |∇|2 dxdt + 2 0 Ω t ≤ ce + ce (P () + |u|2 ) dxdt 0 Ω t t + 2ε2/3 |∇P (∞ )|2 dxdt − ∂t P (∞ ) + u · ∇P (∞ ) dxdt. 0

Ω

0

Ω

(5.2.32)

Next, using the inequality (a − b)2 ≥ 2−1 a2 − b2 and noting that |U|, b ≤ ce we obtain

b 2 b |Δ (u − U)|2 + |u − U|2 + |u − U|2 (x, t) dx 2 2 2 Ω 1 1 ≥ b|Δ2 u|2 + b|u|2 + |u|2 (x, t) dx − b|Δ2 U|2 + ce + ce (x, t) dx. 4 Ω 2 Ω

From this and inequality (5.2.29) we obtain 1 2

Ω

b|Δ2 (u − U)|2 + b|u − U|2 + |u − U|2 + P (x, t) dx 1 b|Δ2 u|2 + b|u|2 + |u|2 + 2P − ce (x, t) dx ≥ 4 Ω 1 (b|Δ2 U|2 )(x, t) dx − ce − 2 Ω 1 b|Δ2 u|2 + b|u|2 + |u|2 + P (x, t) dx ≥ 4 Ω 1 (b|Δ2 U|2 )(x, t) dx − ce . (5.2.33) − 2 Ω

114

Chapter 5. Nonstationary case. Existence theory

Next, substituting (5.2.33) into (5.2.32) we ﬁnally obtain b|Δ2 u|2 + b|u|2 + |u|2 + P () (x, t) dx Ω t + |∇u|2 + ε(P () − ε1/3 )|∇|2 dxdt 0 Ω t ≤ ce + c e (P () + |u|2 ) dxdt + 2 (b|Δ2 U|2 )(x, t) dx 0 Ω Ω t 2/3 +4 2ε |∇P (∞ )|2 dxdt − (∂t P (∞ ) + u · ∇P (∞ )) dxdt. 0

(5.2.34)

Ω

Recalling formula (5.2.11) for M (t) we conclude that (5.2.34) coincides with claim (5.2.30). To prove (5.2.31) notice that in view of (5.1.7) there is ε0 , depending only on cE and hence only on cE,p , such that for all ε ∈ (0, ε0 ) we have 1 P (). (5.2.35) 2 On the other hand, in view of Remark 5.1.2 we have 2 |M | ≤ cE,p + cE,p ( + |u| ) dx ≤ cE,p + cE,p (P () + |u|2 ) dx. P () − ε1/3 ≥

Ω

Ω

Inserting this into (5.2.34) leads to estimate (5.2.31).

Lemma 5.2.6. There is a constant ce as in Remark 5.1.2 and ε0 > 0, depending only on cE,p , such that for all ε ∈ (0, ε0 ), t ∈ (0, T ), and τ ∈ (0, 1], any ﬁxed point (u, ) of the operator Ξτ satisﬁes inequalities (5.2.2) and (5.2.10). Proof. Choose ε0 > 0 that meets all requirements of Lemma 5.2.5. We begin by proving (5.2.10). It follows from (5.2.30) that for all t ∈ (0, T ), t 2 (|u| + P ())(x, t) dx − ce (P () + |u|2 ) dxdt ≤ ce + M (t). 0

Ω

Ω

Multiplying by exp(−ce t) we can rewrite this inequality in the equivalent form d −ce t t e (P () + |u|2 ) dxdt ≤ e−ce t (ce + M (t)). dt 0 Ω Integrating over (0, t) and multiplying the result by exp(ce t) we arrive at t t 2 (P () + |u| ) dxdt ≤ ce + ece (t−s) M (s)ds. 0

Ω

0

Inserting this into the right hand side of (5.2.30) we obtain (5.2.10). It remains to prove (5.2.2). To this end notice that (5.2.31) implies, for all t ∈ (0, T ), t (|u|2 + P ())(x, t) dx − ce (P () + |u|2 ) dxdt ≤ cE,p . Ω

0

Ω

5.2. Regularized equations

115

Arguing as before we obtain t t (P () + |u|2 ) dxdt ≤ ece (t−s) cE,p ds ≤ cE,p . 0

0

Ω

Inserting this into the right hand side of (5.2.31) we obtain (5.2.2).

Lemma 5.2.7. Let ε0 be deﬁned by Lemma 5.2.5. Then for any ε ∈ (0, ε0 ) and τ ∈ (0, 1], every ﬁxed point (u, ) ∈ X of the operator Ξτ satisﬁes inequalities (5.2.3)–(5.2.8). Proof. We begin by proving (5.2.3). The Poincaré inequality yields u(t)L2 (Ω) ≤ u(t) − U(t)L2 (Ω) + U(t)L2 (Ω) ≤ ce ∇(u(t) − U(t))L2 (Ω) + U(t)L2 (Ω) ≤ ce ∇u(t)L2 (Ω) + ce ∇U(t))L2 (Ω) + U(t)L2 (Ω) ≤ ce ∇u(t)L2 (Ω) + ce . Thus we get u2L2 (0,T ;W 1,2 (Ω)) =

0

T

(u(t)2L2 (Ω) + ∇u(t)2L2 (Ω) ) dt

≤ ce 0

T

∇u(t)2L2 (Ω) dt + ce ,

which along with (5.2.2) gives the ﬁrst inequality in (5.2.3); the second is a straightforward consequence of (5.2.2). Notice that by conditions (5.1.5)–(5.1.7) and formula (5.1.3) we have −1 γ P () ≥ c−1 E − cE ,

P () ≥ (2cE )−1 (1 + )γ−2 .

(5.2.36)

On the other hand, (5.2.2) implies sup P (x, t) dx + ε P ()|∇|2 dxdt ≤ cE,p . t∈(0,T )

Q

Ω

Combining this with (5.2.36) we arrive at (5.2.4). It remains to prove (5.2.5)– (5.2.8). It follows from (5.2.3) that the ﬁxed point (u, ) is a bounded energy function. Moreover, its energy is uniformly bounded by a constant cE,p . Therefore, the ﬁxed point (u, ) satisﬁes all hypotheses of Proposition 4.2.1 with E replaced by cE,p . Hence we can apply Corollaries 4.2.2 and 4.2.3 to obtain estimates (5.2.5) and (5.2.6). Since γ > 2d, it follows from estimates (5.2.3)–(5.2.4) that the functions (u, ) have the dissipation rate ε (1 + )γ−2 |∇(x, t)|2 dxdt ≤ cE,p . Q

116

Chapter 5. Nonstationary case. Existence theory

In particular, these functions meet all requirements of Proposition 4.3.1 with E replaced by cE,p . It remains to notice that inequalities (5.2.7)–(5.2.8) coincide with (4.3.2)–(4.3.3) in that proposition. Lemmas 5.2.6 and 5.2.7 give uniform estimates for the ﬁxed points of the operator Ξτ in the energy space norm. The next lemma shows that the boundedness of the ﬁxed points of Ξτ in the Banach space X is a straightforward consequence of the energy estimates. Lemma 5.2.8. There is a constant c independent of τ ∈ (0, 1] such that (u, )X < c for each ﬁxed point (u, ) ∈ X of the operator Ξτ . Proof. We begin by analysing the nonlinear operator Ψ which is part of the operator Ξτ . It follows from the deﬁnition (5.2.13) of Ψ that for every ζ ∈ C02 (Ω) and t ∈ (0, T ), Ψ[u, ](x, t)ζ(x) dx = − S(u) : ∇ζ dx + p() div ζ dx Ω Ω Ω ( u − ε∇)∇u − f + ∂t U ζ dx. − Ω

We have |S(u)| + p + ( u − ε∇)∇u − f + ∂t U ≤ cE,p (|∇u|2 + |u|2 + γ + |u|2 + ε2 |∇|2 + 1), which gives Ψ[u, ](x, t)ζ(x) dx ≤ cE,p ζC 1 (Ω) u(t)2W 1,2 (Ω) + γ (t)L1 (Ω) Ω

+(t)u(t)2L2 (Ω) + ε2 ∇(t)2L2 (Ω)

≤ cE,p ζC 1 (Ω) 1 + ε2 ∇(t)2L2 (Ω) + u(t)2W 1,2 (Ω) + (t)u(t)2L2 (Ω) . Here, we use the estimate γ (t)L1 (Ω) ≤ cE,p that follows from (5.2.4). Since the embedding W 4,2 (Ω) → C 1 (Ω) is bounded, we conclude that Ψ[u, ](·, t)W −4,2 (Ω) ≤ cE,p 1 + ε2 ∇(t)2L2 (Ω) + u(t)2W 1,2 (Ω) + (t)u(t)2L2 (Ω) . Applying Lemma 1.7.7 to problem (5.2.14a) we obtain v(t)W 4,2 (Ω) ≤ cE,p 1 + ε2 ∇(t)2L2 (Ω) + u(t)2W 1,2 (Ω) + (t)u(t)2L2 (Ω) . (5.2.37) Noting that t

v(x, s) ds

u(x, t) = U(x, t) + 0

5.2. Regularized equations

117

we arrive at uC(0,T ;W 4,2 (Ω)) ≤ cE,p 1+ε2 ∇2L2 (Q) +u2L2 (0,T ;W 1,2 (Ω)) +u2L2 (0,T ;L2 (Ω)) . Now estimates (5.2.3)–(5.2.5) yield uC(0,T ;W 4,2 (Ω)) ≤ cE,p . Thus we get uC(0,T ;C 2 (Ω)) ≤ uC(0,T ;W 4,2 (Ω)) ≤ c,

(5.2.38)

where c is independent of , u and τ . From this and (5.2.4) we obtain div(u)L2 (Q) ≤ c. Next, we apply Theorem 1.7.13 to the mass diﬀusion equation (5.2.14c) to obtain C(0,T ;W 1,2 (Ω)) ≤ c div(u)L2 (Q) ≤ c, where c is independent of u, and τ . Combining this with (5.2.38) we conclude that for a.e. t ∈ (0, T ), ∇(·, t)L2 (Ω) + u(·, t)W 1,2 (Ω) ≤ c, which along with (5.2.37) yields v(·, t)W 4,2 (Ω) ≤ c. Hence uC 1 (0,T ;C 2 (Ω)) ≤ uC 1 (0,T ;W 4,2 (Ω)) ≤ c + vC(0,T ;W 4,2 (Ω)) ≤ c.

(5.2.39)

In particular, we have uC 1 (Q) + div uC 1 (Q) ≤ c. Thus we can apply Theorem 1.7.12 to the diﬀusion equation (5.2.12c) to obtain for all α ∈ (0, 1) the estimate C 1+α/2,2+α (Q) ≤ c. Combining this with (5.2.39) we conclude that the norms of all ﬁxed points of the operator Ξτ in X are bounded by a constant c independent of the ﬁxed points and of τ . Proof of Theorem 5.2.1. We are now in a position to complete the proof of Theorem 5.2.1. First of all we notice that by Lemma 5.2.2 the mapping X × [0, 1] (u, , τ ) → Ξτ (u, ) ∈ X is continuous. Moreover, for any τ ∈ [0, 1] the mapping Ξτ : X → X is compact. Set G = {(u, ) ∈ X : (u, )X < c}, where c is the constant of Lemma 5.2.8. By that lemma, Ξτ for τ ∈ (0, 1] has no ﬁxed points on ∂G = {(u, )X = c}. Next, for τ = 0 we have Ξ0 (u, ) = (U, 0 ), where 0 ∈ C 1+α/2,2+α (Q) is a solution to the boundary value problem ∂t 0 + div(0 U) = εΔ0

in Q,

0 = ∞

on

T.

118

Chapter 5. Nonstationary case. Existence theory

It is clear that the homeomorphism I − Ξ0 : X → X has the only zero (U, 0 ) and we can assume that c is so large that the ball G includes the point (U, 0 ). It follows that the mappings Ξτ and the set G meet all requirements of the LeraySchauder theorem 1.1.16, so Ξ1 has a ﬁxed point (u, ) in the ball G. It follows from the deﬁnition (5.2.16) of the Banach space X that (u, ) ∈ C κ (0, T ; C 2 (Ω)) × C 1+κ/2,2+κ (Q) for any κ ∈ (0, 1). Moreover, as is proved in Lemma 5.2.2, the mapping X (u, ) → Ξτ (u, ) ∈ C 1 (0, T ; W 4,2 (Ω)) × C 1+κ/2,2+κ (Q) is continuous. Therefore, for any ﬁxed point (u, ), u belongs to C 1 (0, T ; W 4,2 (Ω)). Since the embedding W 4,2 (Ω) → C 1 (Ω) is continuous, it follows from (5.2.13) that Ψ[u, ] ∈ C 1 (0, T ; C(Ω)). We also have v ≡ ∂t u ∈ C(0, T ; C 1 (Ω)) and hence τ Ψ[u, ] − bv − τ v ∈ C(0, T ; L2 (Ω)).

(5.2.40)

In view of (5.2.14a) the ﬁxed point (u, ) satisﬁes the equation Δ2 (bΔ2 v) = τ Ψ[u, ] − bv − τ v. Applying Theorem 1.7.5 to this equation and recalling (5.2.40) we conclude that u ∈ C 1 (0, T ; W 8,2 (Ω)). It remains to note that the energy estimate (5.2.2) in Theorem 5.2.1 is given by Lemma 5.2.6. Estimates (5.2.3)–(5.2.8) are given by Lemma 5.2.7, and (5.2.10) is established in Lemma 5.2.6. The proof of Theorem 5.2.1 is complete.

5.3 Passage to the limit. The ﬁrst level In this section we perform the ﬁrst step on the way from the regularized equations to the compressible Navier-Stokes equations. We investigate the behavior of solutions when the coeﬃcient b tends to zero. At this stage we let the coeﬃcient b tend to zero inside the cylinder Q while the diﬀusion coeﬃcient ε remains strictly positive. More precisely, we assume that b(x) = a(x) + σ

in Ω,

(5.3.1)

where σ is a positive parameter and a ∈ C ∞ (Ω) is a nonnegative function with a(x) = 1 for dist(x, ∂Ω) ≤ δ/2,

a(x) = 0

for dist(x, ∂Ω) ≥ δ.

(5.3.2)

Here, δ is a ﬁxed small positive number. For given δ we denote by Ωδ a subdomain in Ω and by Qδ the corresponding subcylinder in Q, deﬁned by Ωδ = {x ∈ Ω : dist(x, ∂Ω) > δ},

Qδ = Ωδ × (0, T ).

(5.3.3)

5.3. Passage to the limit. The ﬁrst level

119

With this notation the function a takes the value 0 in Qδ and 1 in Q \ Qδ/2 . It is clear, in view of (5.3.2), that for this choice of b equations (5.1.20) meet all requirements of Theorem 5.2.1 and so have a solution satisfying inequalities (5.2.2)–(5.2.10). Fix a and choose an arbitrary sequence σn → 0 as n → ∞. Denote by (un , n ) solutions to problem (5.1.20) corresponding to σn . It follows from estimates (5.2.2)–(5.2.8) that there is a subsequence, still denoted by (un , n ), and functions (u, ), χ, with the following properties: n weakly in L3γ/2 (Q), ∇n ∇ weakly in L2 (Q), un u weakly in L2 (0, T ; W 1,2 (Ω)), √ a + σn Δ2 un χ weakly in L∞ (0, T ; L2 (Ω)).

(5.3.4)

Our task is to pass to the limit as σn → 0 in the regularized equations (5.1.20) and to obtain a system of equations and boundary conditions for the limits and u. The corresponding result is given by the following theorems. The ﬁrst gives estimates for the couple (u, ) and shows that (u, ) is a weak solution to the momentum balance equation. Theorem 5.3.1. Let the pressure function p(), domain Ω, and functions U, ∞ satisfy Condition 5.1.1 with the adiabatic exponent γ > 2d. Let a be a given function satisfying condition (5.3.2) and σn a sequence with σn → 0 as n → ∞. Furthermore assume that (un , n ) are solutions to problem (5.1.20) with b = a + σn that meet all requirements of Theorem 5.2.1 and satisfy conditions (5.3.4). Then there exists a subsequence, still denoted by (un , n ), and a matrix-valued function K with the following properties: (i) n → strongly in Ls (Q) for 1 < s < 3γ/2 and n un ⊗ un K

weakly in L2 (0, T ; Lz (Ω)) for 1 < z < (1 − 2−1 a)−1 (5.3.5)

with a = 2d−1 − γ −1 . Moreover, we can characterize the limits as follows: K = u ⊗ u in Qδ , √ χ = a Δ2 u in Q \ Qδ .

(5.3.6) (5.3.7)

(ii) The functions , u satisfy the energy inequality

ess sup t∈(0,T )

a|Δ2 u|2 + a|u|2 + |u|2 + P () (x, t) dx

Ω

T

+ 0

Ω

|∇u|2 + εP ()|∇|2 dxdt ≤ cE,p . (5.3.8)

120

Chapter 5. Nonstationary case. Existence theory Moreover, for almost every t ∈ (0, T ), a|Δ2 u|2 + a|u|2 + |u|2 + P () (x, t) dx Ω t + |∇u|2 + ε(P () − ε1/3 )|∇|2 dxdt Ω 0 t ≤ ce + M (t) + ce ece (t−s) M (s) ds,

(5.3.9)

0

where M (t) is given by (5.2.11) with b replaced by a. The constants ce and cE,p are as in Remark 5.1.2. (iii) The functions , u satisfy inequalities (5.2.3)–(5.2.8). (iv) The integral identity 2 aΔ (u − U) · ∂t Δ2 ζ + a(u − U) · ∂t ζ + u · ∂t ζ dxdt Q f · ζ dxdt + (u − ε∇) ⊗ u + p() I − S(u) : ∇ζ dxdt + Q Q + ∞ (x, 0)U(x, 0) · ζ(x, 0) dx = 0 (5.3.10) Ω ∞

holds for all vector ﬁelds ζ ∈ C (Q) satisfying ζ(x, T ) = 0

in Ω,

ζ(x, t) = 0

in Q \ Qδ .

(5.3.11)

The second theorem describes the behavior of (u, ) near ST and shows that is a weak renormalized solution to the mass diﬀusion equation. Theorem 5.3.2. Under the assumptions of Theorem 5.3.1, set Aδ := Ω\Ωδ/3 . Then uL∞ (0,T ;C 2 (Ω\Ωδ/2 )) ≤ c(δ)cE,p .

(5.3.12)

There is a constant c(δ, ε), depending only on δ and ε, such that ∂t L2 (0,T ;L2 (Aδ )) + L2 (0,T ;W 2,2 (Aδ )) + C(0,T ;W 1,2 (Aδ )) ≤ c(δ, ε) cE,p . (5.3.13) In particular, the normal derivative ∂n is well deﬁned and square-integrable on the lateral boundary ST . The integral identity (∂t ψ − ε∇ · ∇ψ + u · ∇ψ) dxdt Q ψ(ε∇ − ∞ U) · n dSdt + (ψ∞ )(x, 0) dx = 0 (5.3.14) + ST

Ω

5.3. Passage to the limit. The ﬁrst level

121

holds for all ψ ∈ C ∞ (Q) vanishing at t = T . Moreover,

ϕ()∂t ψ + ϕ()u − ε∇ϕ() · ∇ψ − ψ ϕ () − ϕ() div u dxdt Q + ψ εϕ (∞ )∇ − ϕ(∞ )U · n dSdt + (ψϕ(∞ ))(x, 0) dx ≥ 0 ST

(5.3.15)

Ω

for all smooth nonnegative functions ψ, vanishing in a neighborhood of ST \Σin and in a neighborhood of the top Ω×{t = T }, and for all convex functions ϕ ∈ C 2 [0, ∞) satisfying the growth condition |ϕ()| + |ϕ ()| + |ϕ ()2 | ≤ C2 . The proof of Theorem 5.3.1 naturally falls into four steps given in Sections 5.3.1–5.3.5. The proof of Theorem 5.3.2 appears in Section 5.3.6.

5.3.1 Step 1. Convergence of densities and momenta In this section we show that the sequence n converges almost everywhere in the cylinder Q. This fact results from the following lemma due to Yu. Dubinski and J.-L. Lions [77]. Lemma 5.3.3. Let X → Y → Z be continuously embedded Banach spaces such that Y is reﬂexive and the embedding X → Y is compact. Furthermore, assume that n is a bounded sequence in Lr (0, T ; X) for some 1 < r ≤ ∞, such that the sequence ∂t n is bounded in L1 (0, T ; Z). Then there is a subsequence, still denoted by n , and a function ∈ Lr (0, T ; Y ) such that for all s ∈ (1, r), n →

in Ls (0, T ; Y )

as n → ∞.

This lemma implies the following proposition. Proposition 5.3.4. Under the assumptions of Theorem 5.3.1, as n → ∞, n → p(n ) → p() n un u

in Ls (Q) for all s ∈ [1, 3γ/2), in Ls (Q) for all s ∈ [1, 3/2),

(5.3.16) (5.3.17)

weakly in L2 (Q).

(5.3.18)

Proof. Let us prove that n converges a.e. in Q. Consider the triple of Banach spaces X = W 1,2 (Ω) ⊂ Y = L2 (Ω) ⊂ Z = W −4,2 (Ω). Obviously, it meets the requirements of Lemma 5.3.3. The sequence n is bounded in L2 (0, T ; X) by (5.2.4).To estimate the time derivative of n in Z, choose ζ ∈ W04,2 (Ω) ⊂ C 2 (Ω). Recall that by Theorem 5.2.1 the function n ∈ C 1,2 (Q) is a

122

Chapter 5. Nonstationary case. Existence theory

solution to the diﬀusion equation (5.1.20b) with u replaced by un . Multiplying this equation by ζ and integrating over Ω we obtain |∂t n , ζ| = (εn Δζ + n un · ∇ζ) dx. Ω

By the energy estimate (5.2.2) and inequality (5.1.5) we conclude that |∂t n , ζ| ≤ cζC 2 (Ω) (n + n |un |) dx Ω ≤ cζW 2,4 (Ω) (γn + n |un |2 + 1) dx ≤ c, Ω

where the constant c is independent of n. Hence the sequence ∂t n is bounded in L∞ (0, T ; Z), so Lemma 5.3.3 yields the compactness of the sequence n in any Ls (Q) with 1 ≤ s < 2. On the other hand, by (5.3.4) this sequence converges weakly to in L3γ/2 (Q). Hence it converges to strongly in Ls (Q). On the other hand, in view of estimate (5.2.8), the sequence n is bounded in L3γ/2 (Q). Thus Lemma 1.3.2 shows that n converges strongly to in all spaces Lr (Q) with r < 3γ/2, which yields (5.3.16). Relation (5.3.17) obviously follows from (5.3.16) and the restrictions (5.1.6) imposed on p(). Next, un converges weakly in L2 (Q) in view of (5.3.4). By the strong convergence of n in L2 (Q) we conclude that lim n un · ξ dxdt = u · ξ dxdt for all ξ ∈ C0∞ (Q). n→∞

Q

Q

Since by (5.2.5) the sequence n un is bounded in L2 (Q), we see that the momentum n un converges to u weakly in L2 (Q).

5.3.2 Step 2. Weak convergence of the kinetic energy tensor In this section the weak convergence of the matrix-valued functions n un ⊗ un is investigated. Proposition 5.3.5. Under the assumptions of Theorem 5.3.1, n un ⊗ un K ∈ L2 (0, T ; Lz (Ω))

weakly in L2 (0, T ; Lz (Ω))as n → ∞

for all z satisfying 1 < z < (1 − 2−1 a)−1 , where a = 2d−1 − γ −1 . Moreover, K = u ⊗ u in Qδ . Proof. We begin with the observation that (un , n ) is a solution to the regularized problem (5.1.20), meets all requirements of Theorem 5.2.1, and admits the energy estimate (5.2.2). Notice that in view of condition (5.1.5) the estimate (5.2.2) coincides with (4.2.1), and hence (un , n ) are functions of bounded energy in the sense of Section 4.2. Moreover, their energy is bounded by a constant cE,p independent of n. Hence the conclusion of Proposition 4.2.1 holds for (un , n ). In

5.3. Passage to the limit. The ﬁrst level

123

particular, we can apply estimate (4.2.21) in Corollary 4.2.3 to deduce that the sequence n un ⊗ un is bounded in L2 (0, T ; Lz (Ω)). Hence, after passing to a subsequence, we may assume that this sequence converges weakly in L2 (0, T ; Lz (Ω)) to a matrix-valued function K. It remains to prove K coincides with u ⊗ u in the smaller cylinder Qδ . It suﬃces to show that lim n un ⊗ un : H dxdt = u ⊗ u : H dxdt (5.3.19) n→∞

Qδ

Qδ

for any matrix-valued function H ∈ C0∞ (Qδ ). To do this we prove that the momentum n un (x, t) converges weakly for almost every t ∈ (0, T ), and next apply Lemma 4.4.5. Let us consider the vector functions Hn = n un and Sn = σn Δ4 un + σn un . It follows from representations (5.3.1)–(5.3.2) with σ = σn that the coeﬃcient b(x) is equal to σn in the subdomain Ωδ . From this and the energy estimate (5.2.2) we conclude that √ σn (Δ2 un + un )L∞ (0,T ;L2 (Ωδ )) ≤ cE,p , where the constant cE,p is independent of n. This leads to √ Sn L∞ (0,T ;W −4,2 (Ω)) ≤ cE,p σn .

(5.3.20)

Since σn → 0 as n → ∞, we can assume, passing to a subsequence if necessary, that Sn (t)W −4,2 (Ωδ ) → 0 as n → ∞ for a.e. t ∈ (0, T ). (5.3.21) On the other hand, inequality (5.2.5) yields the estimate n un L∞ (0,T ;Lβ (Ω)) ≤ cE,p

for β = 2γ/(γ + 1),

which gives Hn L∞ (0,T ;Lβ (Ω)) ≤ cE,p .

(5.3.22)

As b = σn in Qδ , the ﬁrst equation (5.1.20a) in the system of regularized equations (5.1.20) reads ∂ (Sn + Hn ) = div Qn + qn in Qδ , ∂t where Qn = S(un ) − p(n )I − n un ⊗ un + ε∇n ⊗ un ,

qn = n f .

Since (un , n ) are solutions to the regularized equations (5.1.20) and meet all requirements of Theorem 5.2.1, they satisfy estimates (5.2.3)–(5.2.8), which combined with the constraints on the pressure growth rate imposed by (5.1.6) leads to Qn L1 (Q) + qn L1 (Q) ≤ cE,p

124

Chapter 5. Nonstationary case. Existence theory

with a constant cE,p independent of n. It now follows from (5.3.20)–(5.3.21) that the sequence Hn meets all requirements of Lemma 4.4.4 with r = β, k = 4, h = 2 BK’, and with Ω replaced by Ωδ , so Hn (t) = n un H(t) weakly in Lβ (Ωδ ) for a.e. t ∈ [0, T ]. In view of (5.3.18) we have H = u. Since β > 2d/(d + 2), it now follows from (5.3.4) that the sequences Hn = n un and un satisfy all hypotheses of Lemma 4.4.5, and hence n un ⊗ un = Hn ⊗ un H ⊗ u = u ⊗ u weakly in L2 (0, T ; Lm (Ωδ )) for m−1 > β −1 + (d − 2)(2d)−1 = 1 − 2−1 a, which yields the desired relation (5.3.19).

5.3.3 Step 3. Weak limits of ε∇n ⊗ un Lemma 5.3.6. Under the assumptions of Theorem 5.3.1, the couple (, u) satisﬁes estimate (5.2.7), and the equality lim ε (∇n ⊗ un ) : ∇ζ dxdt = ε (∇ ⊗ u) : ∇ζ dxdt (5.3.23) n→∞

Q

Q ∞

holds for any vector ﬁeld ζ ∈ C (Q) vanishing in a neighborhood of ST . Proof. First observe that by (5.2.4), we have ε1/2 ∇n L2 (Q) ≤ cE,p . Hence ∇n ∇ weakly in L2 (Q) and so ε1/2 ∇L2 (Q) ≤ cE,p . The Hölder inequality implies |∇| |u|L1 (0,T ;Lγ/(γ−1) (Ω)) ≤ ∇L2 (0,T ;L2 (Ω)) uL2 (0,T ;Lr (Ω)) ≤ ε−1/2 cE,p uL2 (0,T ;Lr (Ω)) for r−1 = (γ − 1)γ −1 − 2−1 or equivalently r−1 = 2−1 − γ −1 . Since γ > d, we have r < 2d/(d − 2). Hence the embedding W 1,2 (Ω) → Lr (Ω) is bounded. Thus we get |∇| |u|L1 (0,T ;Lγ/(γ−1) (Ω)) ≤ ε−1/2 cE,p uL2 (0,T ;W 1,2 (Ω)) ≤ cE,p ε−1/2 , which is the desired estimate (5.2.7). It remains to prove (5.3.23). To this end notice that un and n are continuously diﬀerentiable in x. Hence we can integrate by parts to obtain ε ∇n ⊗ un : ∇ζ dxdt = −ε n (∇un : ∇ζ + un · Δζ) dxdt. (5.3.24) Q

Q

Since 2 < 3γ/2, it follows from Proposition 5.3.4 that n → strongly in L2 (Q). On the other hand, (5.3.4) implies ∇un ∇u and un u weakly in L2 (Q). Letting n → ∞ in (5.3.24) leads to lim ε ∇n ⊗ un : ∇ζ dxdt = −ε (∇u : ∇ζ + u · Δζ) dxdt. (5.3.25) n→∞

Q

Q

Integrating by parts on the right hand side we obtain (5.3.23).

5.3. Passage to the limit. The ﬁrst level

125

5.3.4 Step 4. Proof of estimates (5.3.8–5.3.9) Now we are in a position to establish the basic energy estimates (5.3.8)–(5.3.9). Our considerations are based on the energy estimates (5.2.2), (5.2.10) and the following auxiliary lemmas. Lemma 5.3.7. Let χ be deﬁned by the limit relation (5.3.4). Then √ χ = aΔ2 u in Q \ Qδ , χ = 0 in Qδ . (5.3.26) Proof. Set χn = (a+σn )1/2 Δ2 un . It follows from Theorem 5.2.1 that the sequence χn is bounded in L∞ (0, T ; L2 (Ω)). Hence, passing to a subsequence if necessary, we can assume that χn converges weakly in L∞ (0, T ; L2 (Ω)) to a function χ ∈ L∞ (0, T ; L2 (Ω)). Since σn → 0, we conclude that (a + σn )Δ2 un a1/2 χ weakly in L∞ (0, T ; L2 (Ω)). Next, notice that for any ζ ∈ C0∞ (Rd+1 ) compactly supported in Q \ Qδ , we have (a + σn )−1/2 χn ζ dxdt = un Δ2 ζ dxdt → uΔ2 ζ dxdt. Q

Q

Q

On the other hand, since a is strictly positive on the support of ζ, (a + σn )−1/2 χn ζ dxdt → a−1/2 χζ dxdt. Q

Q

Thus we obtain the integral identity −1/2 χ a uΔ2 ζ dxdt ζ dxdt = Q\Qδ

Q\Qδ

for all ζ ∈ C0∞ (Q \ Qδ ),

−1/2 χ

which yields Δ u = a in Q \ Qδ . Next, since Δ2 un converges to Δ2 u in the 1/2 sense of distributions in Q, the sequence σn Δ2 un converges to zero in the sense of distributions in the cylinder Qδ . Hence χ = 0 in Qδ . 2

Lemma 5.3.8. Under the assumption of Theorem 5.3.1, for any 0 ≤ T < T ≤ T ,

T

a|Δ u| dxdt ≤ lim inf 2

T

Ω T

T T

n→∞

P () dxdt ≤ lim inf

T

2

n→∞

Ω

|u| dxdt ≤ lim inf

T T T T

Ω

Ω

Ω

2

T

T

T T

n→∞

Ω

|∇u|2 dxdt ≤ lim inf

T

n→∞

Ω

P ()|∇| dxdt ≤ lim inf Ω

T T

T T

T

(5.3.27a)

P (n ) dxdt,

(5.3.27b)

n |un |2 dxdt,

(5.3.27c)

|∇un |2 dxdt,

(5.3.27d)

P (n )|∇n |2 dxdt.

(5.3.27e)

Ω

2

n→∞

(a + σn )|Δ2 un |2 dxdt,

Ω

126

Chapter 5. Nonstationary case. Existence theory

Proof. Since the sequence χn =

T T

√

a + σn Δ2 un converges to χ in L2 (Q), we have

|χ|2 dxdt ≤ lim inf n→∞

Ω

T

T

(a + σn )|Δ2 un |2 dxdt, Ω

which along with Lemma 5.3.7 yields (5.3.27a). Next notice that the function P is nonnegative and continuous. Moreover, in view of (5.3.16), n converges to in L1 (Q). Passing to a subsequence we can assume that n converges to a.e. in Q. Hence the sequence of nonnegative functions P (n ) converges a.e. to P (), and the Levi Theorem yields (5.3.27b). Let us prove (5.3.27c). Since (un , n ) satisfy the energy inequality (5.2.2), √ the sequence n un is bounded in L2 (Q). On the other hand, un u weakly in √ √ 2 L (Q), n → strongly in L2γ (Q), and 2γ > 2. Thus we get √ √ n un ξ dxdt → uξ dxdt as n → ∞ for all ξ ∈ C(Q). Q

√

Q

√

Hence n un u weakly in L2 (Q). Recalling that the norm in a Banach space is lower semicontinuous in the weak convergence, we arrive at (5.3.27c). Inequality (5.3.27d) obviously follows from the weak convergence of ∇un to ∇u in L2 (Q). It remains to prove (5.3.27e). To this end introduce the function s& Φ(s) = P (τ ) dτ. 0

It follows from the constitutive relations (5.1.3) and (5.1.6) that Φ is continuously diﬀerentiable and Φ() ≤ cE (1 + )γ/2 . By (5.2.8), the sequence n is bounded in L3γ/2 (Q) and hence Φ(n ) is bounded in L3γ/(γ+1) (Q). Noting that n → a.e. in Q and 3γ/(γ + 1) > 2, we conclude that Φ(n ) → Φ() in L2 (Q). On the other hand, inequality (5.2.2) yields the boundedness of the sequence & P (n )∇n = ∇Φ(n ) in L2 (Q). Moreover, for any vector ﬁeld ξ ∈ C0∞ (Q), we have & P (n )∇n · ξ dxdt = ∇Φ(n ) · ξ dxdt = − Φ(n ) div ξ dxdt Q Q Q & →− Φ() div ξ dxdt = ∇Φ() · ξ dxdt = P ()∇ · ξ dxdt. Q

It follows that (5.3.27e).

&

Q

P (n )∇n

&

Q

P ()∇ weakly in L2 (Q), which leads to

The next lemma ensures the validity of inequalities (5.3.8)–(5.3.9). Lemma 5.3.9. Under the assumptions of Theorem 5.3.1 the couple (u, ) satisﬁes estimates (5.3.8)–(5.3.9).

5.3. Passage to the limit. The ﬁrst level

127

Proof. We begin by proving (5.3.8). Choose t0 ∈ (0, T ), and h > 0 so small that (t0 − h, t0 + h) ⊂ (0, T ). It follows from (5.2.2) with (u, ) replaced by (un , n ) that t0 +h 1 (a|Δ2 un |2 + a|u|2 + |un |2 + P (n ))(x, t) dxdt ≤ cE,p . 2h t0 −h Ω Letting n → ∞ and applying (5.3.27a)–(5.3.27c) with T = t0 − h and T = t0 + h leads to t0 +h 1 (a|Δ2 u|2 + a|u|2 + |u|2 + P ())(x, t) dxdt ≤ cE,p . 2h t0 −h Ω Letting h → 0 gives (a|Δ2 u|2 + a|u|2 + |u|2 + P ())(x, t0 ) dx ≤ cE,p

for a.e. t0 ∈ (0, T ). (5.3.28)

Ω

Next, (5.2.2) implies

(|∇un |2 + εP ()|∇|2 ) dxdt ≤ cE,p .

Q

Letting n → ∞ and applying (5.3.27a)–(5.3.27c) with T = 0 and T = T gives (|∇u|2 + εP ()|∇|2 ) dxdt ≤ cE,p . (5.3.29) Q

Combining (5.3.28) and (5.3.29) leads to the desired estimate (5.3.8). To prove (5.3.9) integrate both sides of (5.2.10), with (u, ) replaced by (un , n ) and b replaced by a + σn , to obtain

t0 +h

t0 −h

(a + σn )|Δ2 un |2 + (a + σn )|un |2 + n |un |2 + P (n ) (x, t) dxdt

Ω t0 +h

+ t0 −h

≤

t

|∇un |2 + ε(P (n ) − ε1/3 )|∇n |2 (x, s) dxds dt

0 Ω t0 +h

ce + Mn (t) + ce

t0 −h

t

ece (t−s) Mn (s) ds dt,

(5.3.30)

0

where

t

(a + σn )|Δ U| (x, t) dx + 8ε |∇P (∞ )|2 dxdt 0 Ω t (5.3.31) n ∂t P (∞ ) + ∇P (∞ ) · un dxdt. −4 2

Mn (t) = 2

Ω

0

Ω

2

2/3

128

Chapter 5. Nonstationary case. Existence theory

Letting n → ∞ in (5.3.30) and applying Lemma 5.3.8 with T = t0 − h and T = t0 + h we arrive at t0 +h a|Δ2 u|2 + a|u|2 + |u|2 + P () (x, t) dxdt t0 −h

Ω

t0 +h

+

t

t0 −h

0

≤ lim

n→∞

|∇u|2 + ε(P () − ε1/2 )|∇|2 (x, s) dxds dt

Ω t0 +h

t0 −h

t ce + Mn (t) + ce ece (t−s) Mn (s) ds dt.

(5.3.32)

0

It follows from estimates (5.2.3)–(5.2.4) that the sequences n and n un are bounded in L∞ (0, T ; L1 (Ω)). Hence Mn (t) is uniformly bounded on (0, T ). In fact the functions Mn have uniformly bounded derivatives on (0, T ). On the other hand, Proposition 5.3.4 implies that n → in L2 (Q) and n un u weakly in L2 (Q), which yields for all t ∈ (0, T ), t t n ∂t P (∞ )+∇P (∞ )·un dxdt → ∂t P (∞ )+∇P (∞ )·u dxdt. 0

Ω

0

Ω

It now follows from (5.3.31) that t 2 2 2/3 a|Δ U| (x, t) dx + 8ε |∇P (∞ )|2 dxdt Mn (t) → 2 Ω Ω 0 t −4 ∂t P (∞ ) + ∇P (∞ ) · u dxdt =: M (t) Ω

0

for all t ∈ (0, T ). Recall that the functions Mn (t) are uniformly bounded on (0, T ). Applying the Lebesgue dominated convergence theorem we arrive at

t Mn (t) + ce ece (t−s) Mn (s) ds dt

t0 +h

lim

n→∞

t0 −h

0

t0 +h

= t0 −h

t M (t) + ce ece (t−s) M (s) ds dt. 0

Substituting this into (5.3.32) we ﬁnally obtain 1 2h

t0 +h

t0 −h

1 + 2h

a|Δ2 u|2 + a|u|2 + |u|2 + P () dxdt

Ω

t0 +h

t0 −h

t 0

1 ≤ 2h

|∇u|2 + ε(P () − ε1/3 )|∇|2 (x, s) dxds dt

Ω t0 +h

t0 −h

t ce + M (t) + ce ece (t−s) M (s) ds dt.

(5.3.33)

0

Letting h → 0 we arrive at the desired inequality (5.3.9).

5.3. Passage to the limit. The ﬁrst level

129

5.3.5 Step 5. Proof of Theorem 5.3.1 We are now in a position to prove Theorem 5.3.1. Proof of (i)–(ii). First note that the strong convergence of n is exactly the statement of Proposition 5.3.4. Relation (5.3.5) is proved in Proposition 5.3.5. Next, representation (5.3.6) for the matrix-valued function K is a direct consequence of Proposition 5.3.5, and Lemma 5.3.7 gives (5.3.7). This proves (i). To prove (ii), notice that the energy estimates (5.3.8) and (5.3.9) are established in Lemma 5.3.9. Proof of (iii). The energy estimate (5.3.8) gives T 2 |∇u|2 + εP ()|∇|2 dxdt ≤ cE,p . ess sup (|u| + P ())(x, t) dx + t∈(0,T )

0

Ω

Ω

On the other hand, the constitutive relations (5.1.3)–(5.1.6) imply P () ≥ cE,p (γ − 1),

P () ≥ cE,p (1 + γ−2 ).

It follows that uL2 (0,T ;W 1,2 (Ω)) ≤ cE,p ,

|u|2 L∞ (0,T ;L1 (Ω)) ≤ cE,p ,

L∞ (0,T ;L1 (Ω)) + ε(1 + ) γ

γ−2

|∇| L1 (Q) ≤ cE,p , 2

(5.3.34) (5.3.35)

which yields (5.2.3)–(5.2.4). Moreover, by (5.3.34)–(5.3.35), (u, ) are bounded energy functions and meet all requirements of Proposition 4.2.1 with E replaced by cE,p . Hence, Corollary 4.2.2 implies uL∞ (0,T ;L2γ/(γ+1) (Ω)) ≤ cE,p , which immediately yields (5.2.5). In its turn, applying Corollary 4.2.3 we get the estimate |u|2 L2 (0,T ;Lz (Ωδ )) ≤ cE,p , which gives (5.2.6). Since γ > 2d, it follows from (5.3.34) and (5.3.35) that (u, ) are bounded dissipation rate functions and satisfy all hypotheses of Proposition 4.3.1. Estimate (4.3.3) in that proposition yields 3γ/2 L1 (Q) ≤ cE,p ε−1 , which is exactly the required estimate (5.2.8). It remains to note that estimate (5.2.7) is given by Lemma 5.3.6. Proof of (iv). Now choose ζ ∈ C ∞ (Q) which vanishes at the top of Q and outside of Qδ . Multiplying by ζ the regularized momentum equation (5.1.20a) with b, u, replaced by a + σn , un , and n , respectively, and integrating the result by parts we arrive at σn Δ2 (un − U) · ∂t Δ2 ζ + σn (un − U) · ∂t ζ + u · ∂t ζ dxdt Q + n un ⊗ un − ε∇n ⊗ un + p(n )I − S(un ) : ∇ζ dxdt Q + n f · ζ dxdt + (∞ U)(x, 0) dx = 0. Q

Ω

130

Chapter 5. Nonstationary case. Existence theory

Letting n → ∞ and using (5.3.4) and Propositions 5.3.4, 5.3.5 and Lemma 5.3.6 we obtain the integral identity (5.3.10). The proof of Theorem 5.3.1 is complete.

5.3.6 Proof of Theorem 5.3.2 Since the coeﬃcient a(x) equals 1 in the annulus Ω \ Ωδ/2 , estimate (5.3.8) implies 2 ess sup |Δ u(x, t)|2 + |u(x, t)|2 dx ≤ cE,p . t∈(0,T )

Ω\Ωδ/2

Since the boundary of Ω \ Ωδ/2 is a surface of class C 3 depending on δ only, we conclude that uL∞ (0,T,W 4,2 (Ω\Ωδ/2 )) ≤ c(δ)cE,p . Next, since the embedding W 4,2 (Ω \ Ωδ/2 ) → C 2+α (Ω \ Ωδ/2 ) is bounded for 0 < α < 1/2 with a bound depending on δ, it follows that uL∞ (0,T ;C 2+α (Ω\Ωδ/2 )) ≤ c(δ)cE,p

for all α ∈ (0, 1/2),

(5.3.36)

which leads to (5.3.12). Let us derive an estimate for . Choose η ∈ C ∞ (Ω) with η = 1 in Ω \ Ωδ/3 ,

η = 0 outside of Ω \ Ωδ/2 .

Next, set ρn = ηn . Recall that n satisﬁes the diﬀusion equation (5.2.12c) with (u, ) replaced by (un , n ). It follows that ∂t ρn − εΔρn = fn

in Q,

ρn = η∞

on

T,

(5.3.37)

where fn = −ηun · ∇n − η div un n − εΔηn − 2ε∇η∇n . Notice that by (5.3.36) we have |fn | ≤ c(η, δ)ce,p (|n | + |∇n | + 1). Since η depends only on δ, it now follows from the energy estimate (5.2.4) that fn L2 (Q) ≤ c(ε, δ)cE,p . Thus we can apply Theorem 1.7.13 to the initial-value problem (5.3.37) to obtain the estimate ∂t ρn L2 (0,T ;L2 (Ω)) + ρn L2 (0,T ;W 2,2 (Ω)) + ρn C(0,T ;W 1,2 (Ω)) ≤ c(ε, δ)cE,p . Recall the notation Aδ for the annulus Ω \ Ωδ/3 . Since n = ρn in Aδ × (0, T ), we obtain ∂t n L2 (0,T ;L2 (Aδ )) + n L2 (0,T ;W 2,2 (Aδ )) + n C(0,T ;W 1,2 (Aδ )) ≤ c(ε, δ)cE,p . (5.3.38) As n weakly in L2 (0, T ; L2 (Aδ )) we can assume by passing to a subsequence that n weakly in L2 (0, T ; W 2,2 (Aδ )) and weakly in L∞ (0, T ; W 1,2 (Aδ )).

5.3. Passage to the limit. The ﬁrst level

131

Moreover we can assume that ∂t n ∂t weakly in L2 (0, T ; L2 (Aδ )). Since the norm is lower semicontinuous in the weak convergence, the limit also satisﬁes estimate (5.3.38), which leads to the desired estimate (5.3.13). As the embedding W 1,2 (Ω \ Ωδ/3 ) → L2 (∂Ω) is continuous (see [2, Thm. 5.22]) we have ∇L2 (∂Ω) ≤ c(δ)W 2,2 (Aδ ) . From this and estimates (5.3.38), (5.3.13) we obtain ∇n L2 (ST ) ≤ c(δ)n L2 (0,T ;W 2,2 (Aδ )) ≤ c(ε, δ)cE,p , ∇L2 (ST ) ≤ c(δ, ε)L2 (0,T ;W 2,2 (Aδ )) ≤ c(ε, δ)cE,p .

(5.3.39) (5.3.40)

Moreover, the weak convergence n in L2 (0, T ; W 2,2 (Aδ )) yields ∇n ∇ weakly in L2 (ST ).

(5.3.41)

Now, ﬁx ψ ∈ C ∞ (Q) vanishing in a neighborhood of Σin and in a neighborhood of Ω × {t = T }. Multiplying (5.1.20b) by ψ and integrating the result by parts in Q we arrive at the integral identity Q

(n ∂t ψ − ε∇n · ∇ψ + n un · ∇ψ) dxdt ψ(ε∇n − ∞ U) · n dSdt + + ST

(ψ∞ )(x, 0) dx = 0. (5.3.42)

∂Ω

Recall that by Theorem 5.3.1, n → strongly in L2 (Q), ∇n → ∇ weakly in L2 (Q), and un u weakly in L2 (0, T ; W 1,2 (Ω). It follows that Q

(n ∂t ψ − ε∇n · ∇ψ + n un · ∇ψ) dxdt (∂t ψ − ε∇ · ∇ψ + u · ∇ψ) dxdt. → Q

On the other hand, relation (5.3.41) yields ψ∇n · n dSdt → ST

ψ∇ · n dSdt.

(5.3.43)

ST

Letting n → ∞ in (5.3.42) we obtain the desired integral identity (5.3.14). It remains to prove (5.3.15). To this end ﬁx a convex function ϕ satisfying ϕ ∈ C 2 [0, ∞),

|ϕ()| + |ϕ ()| + |ϕ ()2 | ≤ C2 ,

ϕ () ≥ 0.

(5.3.44)

Next, choose a smooth nonnegative function ψ vanishing in a neighborhood of Σin and in a neighborhood of Ω × {t = T }. Multiplying (5.1.20b) by ϕ (n )ψ and

132

Chapter 5. Nonstationary case. Existence theory

integrating the result by parts over Q we obtain

ϕ(n )(∂t ψ + ∇ψ · un ) dxdt − ε ϕ (n )∇n ∇ψ dxdt Q Q − ϕ (n )n − ϕ(n ) div un ψ dxdt + (ϕ(∞ )ψ)(x, 0) dx Q Ω − ϕ(∞ )U·n dSdt+ε ϕ (∞ )∇n ·n ψ dSdt = ε ψϕ (n )|∇n |2 dxdt ≥ 0. ST

ST

Q

(5.3.45) It follows from the growth conditions (5.3.44) and estimate (5.2.8) that the sequences ϕ(n ) and ϕ (n )n are bounded in L3γ/4 (Q). Moreover, we have already shown that n → a.e. in Q. Since 3γ/4 > 2 these sequences converge strongly in L2 (Q) to ϕ() and ϕ () respectively. On the other hand, div un converges weakly in L2 (Q) to div u. Hence we can pass to the limit to obtain Q

ϕ(n )(∂t ψ + ∇ψ · un ) dxdt − ε ϕ (n )∇n ∇ψ dxdt Q − ϕ (n )n − ϕ(n ) div un ψ dxdt Q → ϕ()(∂t ψ + ∇ψ · u) dxdt − ε ϕ ()∇∇ψ dxdt Q Q − ϕ (n ) − ϕ() div uψ dxdt.

(5.3.46)

Q

Letting n → ∞ in (5.3.45) and applying (5.3.43), (5.3.46) we obtain inequality (5.3.15). The proof of Theorem 5.3.2 is complete.

5.3.7 Local pressure estimate Theorem 5.3.1 guarantees the existence of the limits (u, ) of a sequence of solutions (un n ) to the regularized equations (5.1.20) with b = a + σn , where σn → 0 and a is concentrated near the lateral boundary ST of the cylinder Q. Recall that Theorem 5.3.1 provides the estimate p()L∞ (0,T ;L1 (Ω)) ≤ cE,p , where a constant cE,p as in Remark 5.1.2 depends only on the diameter of the ﬂow domain Ω, length T of the time interval, and the given data and elementary properties of the constitutive law (5.1.4), (5.1.10). Estimate (5.2.8) for established in Theorem 5.3.1 gives an estimate for the pressure function p() in L3/2 (Q), but this estimate depends on the small parameter ε and it is useless. Here, we show that the pressure function satisﬁes L(1+γ)/γ estimates on compact subsets of the cylinder Q. The corresponding result is given in the following proposition.

5.3. Passage to the limit. The ﬁrst level

133

Proposition 5.3.10. Under the assumptions of Theorem 5.3.1 there exists δ0 > 0 independent of ε such that for for all δ ∈ (0, δ0 ] and all compact subsets Q of the cylinder Qδ , γ+1 dxdt + p()(γ+1)/γ dxdt ≤ cE,p (Q ), (5.3.47) Q

Q

where the constant cE,p (Q ) depends only on the constant cE,p and Q , δ0 . In particular, cE,p (Q ) is independent of δ ≤ δ0 . Proof. The proof is based on the basic integral identity (4.5.10) of Theorem 4.5.2. First, by the integral identities (5.3.10) and (5.3.14), the couple (u, ) satisﬁes the mass and momentum balance equations ∂t + div(u − g) = 0 in Qδ , ∂t (u) + div(u ⊗ u) = div(T + G) + ρf

in Qδ ,

which are understood in the sense of distributions. Here, the matrix-valued functions T, G and the vector-valued function g are deﬁned by G = ε∇ ⊗ u,

T = S(u) − p() I,

g = ε∇,

S(u) = ∇u + ∇u + (λ − 1) div u I.

(5.3.48)

Next, set r = 3,

s = γ,

q = s/(s − 1),

ϕ = ,

gϕ = g = ε∇,

= 0.

(5.3.49)

Let us show that the functions , ϕ, the vector ﬁeld u, and the matrix-valued functions T, G meet all requirements of Theorem 4.5.2 in the cylinder Qδ ⊂ Q. To this end we have to check that , ϕ, u, and T satisfy conditions (4.5.4)–(4.5.7), and then to prove that g and G belong to L2 (Q) and L1 (0, T ; Lq (ΩT )), respectively. By Theorem 5.3.1, the functions , u, and ϕ = satisfy estimates (5.2.3) and (5.2.4). It is easily seen that these imply (4.5.4)–(4.5.6) with E replaced by cE,p . It follows from (5.2.3) and (5.2.4) that TL1 (Q) + gL2 (Q) ≤ cE,p .

(5.3.50)

It also follows from Theorem 5.3.1 that and u satisfy inequalities (5.2.7)–(5.2.8), which leads to the estimate ε3γ/2 L1 (Q) + ε1/2 |∇||u|L1 (0,T ;Ld/(d−1) (Ω)) ≤ cE,p . Since q < d/(d − 1) we deduce that G ∈ L1 (0, T ; Lq (Ω)). As p() ≤ cE,p (γ + 1), we ﬁnd from (5.2.3) and (5.2.8) that TL3/2 (Q) ≤ ∇uL3/2 (Q) + pL3/2 (Q) ≤ cE,p /ε.

134

Chapter 5. Nonstationary case. Existence theory

Since r = 3/2, it follows that the matrix-valued function T ∈ Lr (Q) satisﬁes (4.5.7). Hence the functions , ϕ, , the vector ﬁelds u, g, gϕ , and the tensors T, G satisfy Condition 4.5.1 of Theorem 4.5.2. Finally, the exponents γ and s obviously satisfy condition (4.5.8) of that theorem. Thus H · u + ζ(N + P + Q) · u + S dxdt (5.3.51) ξR : [ζ(T + G)] dxdt = Qδ

Qδ

for all ξ ∈ C0∞ (Qδ ) and ζ ∈ C0∞ (Ωδ ). Here, H = R[ξ] (ζu) − (ξ) R[ζu], S = (∇ζ ⊗ A[ξ]) : (u ⊗ u − T − G) + ζA[ξ] · (f ), N = R[ξ g], P = A[∂t ξ + u · ∇ξ], Q = −A[g · ∇ξ]. Using the relation R : [ζpI] = ζp, we can rewrite equality (5.3.51) in the form ξζp() dxdt = ξR : [ζ(S(u) + G)] dxdt Qδ Qδ − H · u + ζ(N + P + Q) · u + S dxdt. (5.3.52) Qδ

Now our task is to estimate all terms on the right hand side of (5.3.52). Let us estimate N, P, and Q. By (5.2.3) and (5.2.4), (u, ) are ﬁnite energy functions, and the energy is bounded by a constant cE,p . Hence, (u, ) satisﬁes all hypotheses of Proposition 4.2.1 with E replaced by cE,p . We apply Corollary 4.2.2 of this proposition to obtain uL2 (0,T ;Lκ (Ω)) ≤ cE,p

for all κ−1 > 2−1 + γ −1 − d−1 .

(5.3.53)

Since γ > 2d, we can take κ = 2 to obtain uL2 (Q) ≤ cE,p .

(5.3.54)

Notice that ξ, ζ ∈ C0∞ (Rd ) are compactly supported in Ωδ . It follows that the vector ﬁeld ζA[ξu] is compactly supported in Qδ and satisﬁes ζA[ξu]L2 (0,T ;W 1,2 (Ω)) ≤ c(ξ)cE,p .

(5.3.55)

Next, since is in L∞ (0, T ; Lγ (Ω)) with γ > 2d > 2, we have A[∂t ξ]L∞ (0,T ;W 1,2 (Ω)) ≤ c(ξ)cE,p .

(5.3.56)

Combining (5.3.55) and (5.3.56) we obtain PL2 (0,T ;W 1,2 (Ω)) ≤ c(ξ)cE,p .

(5.3.57)

5.3. Passage to the limit. The ﬁrst level

135

Since the vector ﬁeld g = ε∇ ∈ L2 (0, T ; L2 (Ω)) satisﬁes inequality (5.2.4), the same arguments as before lead to A[ξg]L2 (0,T ;W 1,2 (Ω)) + R[ξg]L2 (0,T ;L2 (Ω)) ≤ c(ξ)cE,p .

(5.3.58)

This gives QL2 (0,T ;W 1,2 (Ω)) ≤ c(ξ)cE,p ,

NL2 (0,T ;L2 (Ω)) ≤ c(ξ)cE,p .

(5.3.59)

Combining estimates (5.3.57)–(5.3.59) we get N + P + QL2 (Q) ≤ c(ξ, ζ)cE,p , which along with (5.3.54) leads to ζ(N + P + Q) · uε L1 (Q) ≤ c(ξ, ζ)cE,p .

(5.3.60)

Recall that for γ > d the embedding W 1,γ (Ω) → L∞ (Ω) is compact. Thus we get A[ξ]L∞ (Q) ≤ cA[ξ]L∞ (0,T ;W 1,γ (Ω)) ≤ cξL∞ (0,T ;Lγ (Ω)) ≤ c(ξ, ζ)cE,p . Moreover, by (5.2.7) we have GL1 (Q) ≤ GL1 (0,T ;Lγ/(γ−1) (Ω)) ≤ εcE,p . This along with (5.3.50) leads to SL1 (Q) ≤ c(ξ, ζ)cE,p |u|2 L1 (Q) + T + GL1 (Q) + f L1 (Q) ≤ c(ξ, ζ)cE,p .

(5.3.61)

(5.3.62)

Let us estimate the vector ﬁeld H. Notice that the operator R is bounded in Lκ (Rd ) for all κ ∈ (1, ∞). From this and the Hölder inequality we obtain HL2 (Q) ≤ R[ξ]L∞ (0,T ;Lγ (Ω)) ζuL2 (0,T ;Lσ (Ω)) + ξL∞ (0,T ;Lγ (Ω)) R[ζu] L2 (0,T ;Lσ (Ω)) ≤ cξL∞ (0,T ;Lγ (Ω)) ζuL2 (0,T ;Lσ (Ω)) ≤ cE,p ζuL2 (0,T ;Lσ (Ω)) ,

where

σ −1 = 2−1 − γ −1 .

It is easy to check that with γ > 2d, we have σ −1 = 2−1 − γ −1 > 2−1 + γ −1 − d−1 = κ, which along with (5.3.53) leads to ζuL2 (0,T ;Lσ (Ω)) ≤ cE,p c(ζ).

136

Chapter 5. Nonstationary case. Existence theory

We thus get HL2 (Q) ≤ c(ξ, ζ)cE,p . This inequality along with the estimate uL2 (0,T ;W 1,2 (Ω)) ≤ cE,p implies H · uL1 (Q) ≤ c(ξ, ζ)cE,p .

(5.3.63)

Next, the boundedness of R in Lebesgue spaces yields R : [ζ(S(u) + G)]L1 (Q) ≤ c(ζ)L∞ (0,T ;Lγ (Ω)) R[ζS]L1 (0,T ;Lγ/(γ−1) (Ω)) + R[ζG]L1 (0,T ;Lγ/(γ−1) (Ω)) ≤ c(ζ)cE,p SL1 (0,T ;Lγ/(γ−1) (Ω)) + GL1 (0,T ;Lγ/(γ−1) (Ω)) . Since γ/(γ − 1) < 2 we have S(u)L1 (0,T ;Lγ/(γ−1) (Ω)) ≤ cuL2 (0,T ;W 1,2 (Ω)) ≤ cE,p . Combining this result with (5.3.61) we get R : [ζ(S(u) + G)]L1 (Q) ≤ c(ζ)cE,p .

(5.3.64)

Finally, combining estimates (5.3.60), (5.3.62), (5.3.63), and (5.3.64) with identity (5.3.52) we arrive at ξζp() dxdt ≤ c(ξ, ζ)cE,p . Qδ

It remains to note that for any compact Q Qδ there are nonnegative functions ζ ∈ C0∞ (Ω) and ξ ∈ C0∞ (Q) such that ζξ = 1 on Q .

5.3.8 Normal derivative of the density In this section we derive an estimate for the normal derivative of the density on the boundary. The result is given by the following proposition. Proposition 5.3.11. Under the assumptions of Theorem 5.3.1, for any compact set Σ Σin ⊂ ST , there is a constant C independent of ε such that with κ = 1/2 − d/(2γ), ∂ ε ≤ Cεκ (Lγ (Q) + ∞ C 2 (Q) ) ∂n

on Σ .

(5.3.65)

Before the proof of Proposition 5.3.11, we provide a general result on the normal derivative of the solution to the diﬀusion equation with vanishing viscosity. This result is of independent interest, and it is necessary to justify our regularization procedure.

5.3. Passage to the limit. The ﬁrst level

137

Normal derivative of a singularly perturbed transport equation. Let (z, t∗ ) ∈ Σin and let c0 , ι, l be positive constants. We introduce the following condition on the velocity ﬁeld u: Condition 5.3.12.

• Denote by Qι the cylinder Qι := B(z, ι) × (t∗ − ι, t∗ + ι),

(5.3.66)

where B(z, ι) ⊂ Rd is the ball of radius ι centered at z. The velocity ﬁeld u admits the representation u = ∇H + v H = 0,

in Q ∩ Qι ,

v · n = 0 on ST ∩ Qι ,

(5.3.67)

where the function H and the vector ﬁeld v satisfy HC 2 (Q∩Qι ) ≤ c0 , ess sup v(·, t)C 2 (B(z,ι)∩Ω) ≤ c0 .

(5.3.68)

t∈(t∗ −ι,t∗ +ι)

• The “potential” H satisﬁes 1 1 ∂t H(x, t) + |∇H(x, t)|2 > l > 0 in Qι ∩ Q, 2 4 −∇H(x, t) · n(x) > l > 0 on Qι ∩ ST .

(5.3.69)

Consider the following boundary value problem in the cylinder Qι ∩ Q: ∂t + div(u) − εΔ = 0 in Qι ∩ Q, = ∞ on ST ∩ Qι .

(5.3.70a) (5.3.70b)

The following theorem whose proof is given in Section 13.4 plays a key role in our further considerations. Theorem 5.3.13. Let Ω ⊂ Rd be a bounded domain with C 3 boundary and suppose there are (z, t∗ ) ∈ ST and ι > 0 such that u satisﬁes Condition 5.3.12 in a cylinder Qι . Let ∈ Lγ (Qι ∩ Q), γ > d + 2, be a strong solution to problem (5.3.70) with ∂t , ∂x2 ∈ L2 (Qι ). Then there are constants C and a0 < ι, depending only on d, γ, Ω, ι, and on the constants c0 , l in Condition 5.3.12, such that ∂ ε ≤ Cεκ (Lγ (Qι ∩Q) + ∞ C 2 (Qι ∩Q) ) on ST ∩ Qa0 , ∂n where κ = 2−1 − d(2γ)−1 , Qa0 is deﬁned by (5.3.66) for ι := a0 .

138

Chapter 5. Nonstationary case. Existence theory

Proof of Proposition 5.3.11. In order to apply Theorem 5.3.13 we must check that , u satisfy its hypotheses. The proof of this fact is based on the following lemma which gives the construction of H. Lemma 5.3.14. Let Σ Σin be a compact subset of Σin in the topology of ST = ∂Ω × (0, T ) and (z, t∗ ) ∈ Σ . Furthermore, assume that and u satisfy all conditions of Theorem 5.3.1. Then there are positive constants ι, l, c0 , depending only on Σ , Ω, δ and cE,p , such that Condition 5.3.12 is fulﬁlled for all (z, t∗ ) ∈ Σ . We stress that the constants ι, l, and c0 do not depend on the choice of (z, t∗ ) ∈ Σ . Proof. Choose ι < δ/3 so small that the closure of the ι-neighborhood of Σ in the topology of ST is contained in Σin . This means that there is a compact set Σ ⊂ Σin such that Qι ∩ ST Σ for all (z, t∗ ) ∈ Σ . For such a choice there is l ∈ (0, 1) depending only on ι and ∂Ω, U such that −U · n > 4l1/2

on Qι ⊂ ST

for all (z, t∗ ) ∈ Σ .

More precisely, the constant l > 0 depends on the second order derivatives of the normal coordinates, i.e., on the curvatures of ∂Ω and their derivatives. Next choose ι > 0 depending on ∂Ω such that for any z ∈ ∂Ω the intersection B(z, ι) ∩ Ω is a normal neighborhood of z (see Section 13.2 for deﬁnitions and details), i.e. every point x ∈ B(z, ι) has a representation x = ω(x) + τ (x)n(ω(x)),

ω, τ ∈ C 2 (B(z, ι)),

ω(x) ∈ ∂Ω.

Notice that τ (x) = dist(x, ∂Ω) and ∇τ = n on ∂Ω. It follows from Theorem 5.3.2 that ess sup(0,T ) uC 2+α (Ω\Ωδ/3 ) ≤ c(δ)cE,p . By the choice of ι we have Qι ∩ Q ⊂ (Ω \ Ωδ/3 ) × (0, T ), which leads to ess sup uC 2+α (B(z,ι)∩Ω) ≤ c(ε, δ)cE,p .

(5.3.71)

(t∗ −ι,t∗ +ι)

Set H(x, t) = −U(x, t)·n(ω(x))τ (x). Since τ n(ω) ∈ C 2 B(z, ι) for all z ∈ ∂Ω, the function H belongs to C 2 (B(z, ι) × (0, T )). Hence H ∈ C 2 (Qι ) for all cylinders Qι centered at points (z, t∗ ) ∈ Σ . Moreover, the norm HC 2 (Qι ) is bounded by a constant depending only on ι, U and the curvature of ∂Ω. It is clear that |∂t H(x, t)| ≤ cE,p dist(x, ∂Ω) ≤ cE,p ι in Qι

for all (z, t∗ ) ∈ Σ ,

and −∇H · n = −U · n > 4l1/2 > l > 0 on ST ∩ Qι

for all (z, t∗ ) ∈ Σ .

Choosing ι, depending on U, ∂Ω, and l, suﬃciently small we deduce that |∇H| > 3l1/2 > 0 on Q ∩ Qι

for all (z, t∗ ) ∈ Σ .

In particular, we have 1 1 ∂t H(x, t) + |∇H(x, t)|2 ≥ 2l − cE,p ι. 2 4 ∗ Choosing ι < c−1 E,p l we see that H satisﬁes Condition 5.3.12 at each (z, t ) ∈ Σ .

5.4. Passage to the limit. The second level

139

Proof of Proposition 5.3.11. In view of Lemma 5.3.14 there are positive constants ι, c0 , l and a function H, depending only on Σ , Ω, δ, such that u and H satisfy Condition 5.3.12 for all (z, t∗ ) ∈ Σ . Then Qι ∩ Q ⊂ (Ω \ Ωδ/3 ) × (0, T ). By Theorem 5.3.2, is a strong solution to problem (5.3.70) in the cylindrical annulus (Ω \ Ωδ/3 ) × (0, T ) and hence in Qι ∩ Q. Hence and u satisfy Condition 5.3.12 with constants ι, l, c0 depending only on δ, cE,p , Ω and Σ . Thus the couple (u, ) meets all requirements of Theorem 5.3.13, which completes the proof.

5.4 Passage to the limit. The second level Theorem 5.3.1 guarantees the existence of the limits (u, ) of a sequence of solutions (un n , ) to the regularized equations (5.1.20) with b = a + σn , where σn → 0 and a is concentrated in the δ-neighborhood of the lateral boundary ST = ∂Ω × (0, T ). These limits depend of course on a small parameter ε. It is convenient to manifest this dependence explicitly and give the following deﬁnition: Deﬁnition 5.4.1. If (, u) are weak limits of solutions to the regularized equations deﬁned by Theorem 5.3.1, K is a weak limit of the kinetic energy tensors n un ⊗un , and M (t) is the function in the energy estimate (5.3.9), then ε := ,

uε := u,

Kε := K,

Mε := M.

Now for ﬁxed a and δ our task is to pass to the limit ε → 0 in the governing equations and boundary conditions. We begin with the observation that by Theorem 5.3.1, the functions ε and uε satisfy the estimates, independent of ε, uε L2 (0,T ;W 1,2 (Ω)) ≤ cE,p ,

ε |uε |2 L∞ (0,T ;L1 (Ω)) ≤ cE,p ,

(5.4.1)

γε L∞ (0,T ;L1 (Ω)) + ε(1 + ε )γ−2 |∇ε |2 L1 (Q) ≤ cE,p ,

(5.4.2)

ε uε L∞ (0,T ;Lβ (Ω)) ≤ cE,p ,

(5.4.3)

ε |uε |2 L2 (0,T ;Lz (Ωδ )) ≤ cE,p ,

(5.4.4)

|∇ε |uε L1 (0,T ;Ls (Ω)) ≤ ε−1/2 cE,p ,

(5.4.5)

L1 (Q) ε3γ/2 ε

≤ cE,p ,

(5.4.6) −1

−1

where β = 2γ/(γ + 1), s = d/(d − 1), and z ∈ (1, (1 − 2 a) ), a = 2d − γ −1 . Recall that the constant cE,p is independent of δ, a and ε. It also follows from Proposition 5.3.5 that Kε L2 (0,T ;Lz (Ω)) + Kε L∞ (0,T ;L1 (Ω)) ≤ cE,p .

−1

(5.4.7)

The representation Kε = ε uε ⊗ uε holds in the subcylinder Qδ . Next, by Proposition 5.3.10, for all Q Qδ and δ ∈ (0, δ0 ), with δ0 independent of ε and a, ε Lγ+1 (Q ) ≤ c(Q )E,p

(5.4.8)

140

Chapter 5. Nonstationary case. Existence theory

where the constant c(Q )E,p is independent of ε and δ ∈ (0, δ0 ). Moreover, for almost every t ∈ (0, T ), a|Δ2 uε |2 + a|uε |2 + ε |uε |2 + P (ε ) (x, t) dx Ω t + |∇uε |2 + ε(P (ε ) − ε1/3 )|∇ε |2 dxdt 0 Ω t ≤ ce + Mε (t) + ce ece (t−s) Mε (s) ds, (5.4.9) 0

where

t (a|Δ2 U|2 ) dxdt + 8ε2/3 |∇P (∞ )|2 dxdt Ω Ω 0 t −4 ε ∂t P (∞ ) + ∇P (∞ ) · uε dxdt.

Mε (t) = 2

0

(5.4.10)

Ω

Recall that the constants ce and cE,p are as in Remark 5.1.2. Hence, passing to a subsequence if necessary, we can assume that the sequences ε , uε tend to δ , uδ as ε → 0, and ε δ

weakly in L∞ (0, T ; Lγ (Ω)),

uε uδ

weakly in L2 (0, T ; W 1,2 (Ω)),

ε δ

weakly in L

γ+1

(Q ),

(5.4.11)

for every compact set Q Qδ .

The natural question arises on convergence of the sequences of kinetic energy tensors, momenta, and pressures as ε → 0. The weak convergence of the momenta and the kinetic energy tensors easily follows from Theorem 4.4.2: Lemma 5.4.2. Let δ and uδ be deﬁned by (5.4.11). Then there exist subsequences, still denoted by ε , uε , such that for ε → 0, ε uε δ uδ

weakly in L∞ (0, T ; L2γ/(γ+1) (Ω)),

ε uε ⊗ uε δ uδ ⊗ uδ

weakly in L2 (0, T ; Lz (Ω)),

(5.4.12)

for every z ∈ (1, (1 − 2−1 a)−1 ), where a is given by (4.2.2). Proof. By Theorem 5.3.1 the functions ε and uε satisfy the mass and momentum balance equations ∂t ε = div uε in Q, ∂t (ε uε ) = div Vε + wε

in Qδ ,

where uε = −ε uε + ε∇ε , wε = ε f , Vε = S(uε ) − p(ε ) I − ε uε ⊗ uε + ε∇ε ⊗ uε .

5.4. Passage to the limit. The second level

141

It follows from estimates (5.4.1) and (5.4.2) that the sequences uε and ε are bounded in L2 (0, T ; W 1,2 (Ω)) and L∞ (0, T ; Lγ (Ω)) respectively. On the other hand, in view of estimates (5.4.2) and (5.4.3), the sequence uε is bounded in L1 (Q). Hence the functions ε and the vector ﬁelds uε satisfy Condition 4.4.1 of Theorem 4.4.2 with n replaced by ε. Therefore ε uε δ uδ

weakly in L2 (0, T ; Lm (Ω)) for m−1 > 2−1 + γ −1 − d−1 .

Since the sequence ε uε is bounded in L∞ (0, T ; L2γ/(γ+1) (Ω)), it also converges to δ uδ weakly in L∞ (0, T ; L2γ/(γ+1) (Ω)). This leads to the ﬁrst relation in (5.4.12). To prove the second, notice that by (5.4.1), (5.4.2), and (5.4.5) the sequence Vε is bounded in L1 (Q). On the other hand, since f is bounded, it follows from (5.4.2) that the sequence wε is bounded in L1 (Qδ ). Therefore the sequences ε , uε , Vε , and wε satisfy all hypotheses of Theorem 4.4.2 with n replaced by ε and Q replaced by Qδ . This yields the second relation in (5.4.12). However, in contrast to the ﬁrst level limit, we are unable to prove the strong convergence of ε and pε at this stage. Instead we show the weak convergence of the pressure and obtain the representation of the weak limits of p(ε ), and, more generally, composite functions ϕ(ε ), in terms of the Young measure associated with the sequence ε (see Section 1.4 for deﬁnitions and basic statements of Young measure theory). In addition, we prove that the limit function δ serves as a renormalized solution to the mass transport equation. To obtain the representation for the weak limits of ε we recall that, in view of (5.4.2) and (5.4.8), the sequence ε is bounded in Lγ (Q) and in Lγ+1 (Q ) for every subcylinder Q ⊂ Q. Hence we can apply the fundamental theorem on Young measures (Theorem 1.4.5) to obtain Theorem 5.4.3. There is a subsequence, still denoted by ε , and a Young measure μδ ∈ L∞ w (Q; M(R)) such that for any ϕ ∈ C0 (R): • As ε → 0, ϕ(ε ) ϕ

weakly in L∞ (Q),

where

ϕ(x) = μδxt , ϕ.

• For a.e. (x, t) ∈ Q the measures μδxt are supported on [0, ∞). • For any Carathéodory function f : Q × R → R bounded from below, lim inf f (x, t, ε (x, t)) dx ≥ f (x, t) dx, where f (x, t) = μxt , f (x, t, ·). ε→0

Q

Q

Moreover, if in addition |f (x, t, λ)|λ−γ → 0

as λ → ∞,

uniformly in (x, t) ∈ Q,

then

f (x, t, ε (x, t)) dx =

lim

ε→0

Q

f (x, t) dx. Q

142

Chapter 5. Nonstationary case. Existence theory If Q Q and |f (x, t, λ)|λ−γ−1 → 0

uniformly in (x, t) ∈ Q ,

as λ → ∞,

then lim

ε→0

Q

f (x, t, ε (x, t)) dx =

f (x, t) dx. Q

Recall that a function f : Q × R → R is said to be a Carathéodory function if for a.e. (x, t) ∈ Q the function f (x, t, ·) is continuous and for every y ∈ R the function f (·, ·, y) is measurable. It is worth noting that the weak limits δ of the sequence ε have the representation δ λ dμδxt (λ). (5.4.13) δ (x, t) = μxt , λ := [0,∞)

Since p(λ), P (λ) are continuous functions with asymptotic behavior p(λ), P (λ) ∼ λγ at inﬁnity, the application of Theorem 5.4.3 to the pressure function p(ε ) and the internal energy function P (ε ) leads to the following result: Lemma 5.4.4. The functions pδ , P δ : Q → R+ deﬁned by pδ (x, t) = μδxt , p(λ),

P δ (x, t) = μδxt , P (λ)

belong to L∞ (0, T ; L1 (Ω)). For any measurable function ψ : Q → R+ , we have ψpδ (x, t) dxdt ≤ lim inf ψp( ) dxdt, →0 Q Q (5.4.14) ψP δ (x, t) dxdt ≤ lim inf ψP ( ) dxdt. →0

Q

Q

For every Q Q, the function pδ : Q → R

+

given by

pδ (x, t) = μδxt , p(λ)λ belongs to L1 (Q ) and for any continuous function ψ : Q → R+ , ψpδ (x, t) dxdt ≤ lim inf ψp( ) dxdt. →0

Q

For all ψ ∈ L∞ (Q ),

Q

Q

(5.4.15)

Q

ψpδ (x, t) dxdt = lim

→0

ψP δ (x, t) dxdt = lim

→0

Q

Q

ψp( ) dxdt, (5.4.16) ψP ( ) dxdt.

Moreover, pδ (x, t) − pδ (x, t)δ (x, t) ≥ 0 μδxt

a.e. in Q.

(5.4.17)

Equality holds if and only if is the Dirac measure concentrated at δ (x, t), i.e., μδxt , ϕ = ϕ(δ (x, t)) for every ϕ ∈ C0 (R).

5.4. Passage to the limit. The second level

143

Proof. The lemma is a straightforward consequence of Theorem 5.4.3, applied to the Carathéodory functions ψ(x, t)p(λ) and ψ(x, t)P (λ). To prove (5.4.17) notice that '

( μδxt , p(λ) − p(δ (x, t) (λ − δ (x, t)) = μδxt , p(λ)λ − μxt , p(λ)δ (x, t) − p(δ (x, t))μδxt , λ + p(δ (x, t))δ (x, t)μδxt , 1 = μδxt , p(λ)λ − μδxt , p(λ)δ (x, t) − p(δ (x, t))δ (x, t) + p(δ (x, t))δ (x, t) = μδxt , p(λ)λ − μδxt , p(λ)δ (x, t) = pδ (x, t) − pδ (x, t)δ (x, t).

Since p is strictly monotone, the left hand side of this identity is positive and vanishes if and only if μδxt is the Dirac measure concentrated at δ (x, t). The following Theorem 5.4.5 is the main result of this section. It gives estimates, independent of δ, for the weak limits (δ , uδ ). It also ensures that δ and uδ serve as a renormalized solution to the mass balance equation. Next, the theorem shows that in the subcylinder Q2δ the functions (δ , uδ ) satisfy the modiﬁed moment balance equation with the pressure function p() replaced with the function pδ deﬁned in Lemma 5.4.4. We do not claim the equality p(δ ) = pδ at this stage. However, we make a step forward towards proving this equality by showing that the diﬀerence of pδ and pδ vanishes as δ → 0. Theorem 5.4.5. Under the above assumptions, the couple (δ , uδ ) and the Young measure μδ have the following properties: (i) The functions (δ , uδ ) satisfy the estimates uδ L2 (0,T ;W 1,2 (Ω)) ≤ cE,p , γδ L∞ (0,T ;L1 (Ω))

δ |uδ |2 L∞ (0,T ;L1 (Ωδ )) ≤ cE,p ,

+ pδ L∞ (0,T ;L1 (Ω)) ≤ cE,p ,

(5.4.18) (5.4.19)

δ uδ L∞ (0,T ;Lβ (Ω)) ≤ cE,p ,

(5.4.20)

δ |uδ |2 L2 (0,T ;Lz (Ωδ )) ≤ cE,p ,

(5.4.21)

where β = 2γ/(γ + 1), s = d/(d − 1), and z is an arbitrary number in (1, (1 − 2−1 a)−1 ). Recall that the constant cE,p is as in Remark 5.1.2 and it is independent of δ. For any compact set Q Qδ0 and for all δ ∈ (0, δ0 ], where δ0 is deﬁned in Proposition 5.1.18, there is a constant c(Q ) independent of δ such that δ Lγ+1 (Q ) + pδ L1 (Q ) ≤ c(Q ).

(5.4.22)

144

Chapter 5. Nonstationary case. Existence theory

(ii) The integral identity δ uδ ⊗ uδ + pδ I − S(uδ ) : ∇ζ dxdt δ uδ · ∂t ζ dxdt + Q Q + δ f · ζ dxdt + (∞ U · ζ)(x, 0) dx = 0 (5.4.23) Q

Ω

holds for any vector ﬁeld ζ ∈ C ∞ (Q) such that ζ(x, t) = 0

if x ∈ Ω \ Ω2δ or t = T.

(iii) The integral identity ϕ(δ )∂t ψ + ϕ(δ )uδ · ∇ψ − (ϕ (δ )δ − ϕ(δ )) div uδ dxdt Q − ψϕ(∞ )U · n dΣ + (ψϕ(∞ ))(x, 0) dx = 0 (5.4.24) Σin

Ω

holds for all smooth functions ψ vanishing in a neighborhood of Ω × {t = T } and in a neighborhood of ST \ Σin , and for all ϕ ∈ C 2 [0, ∞) satisfying the growth condition |ϕ()| + |ϕ ()| + |ϕ ()2 | ≤ C2 . (iv) For any compact set Q Qδ , (pδ − pδ δ ) dxdt ≤ cE,p meas(Q \ Q )(γ−2)/(2γ) .

(5.4.25)

(5.4.26)

Q

(v) For almost every t ∈ (0, T ), t

{δ |uδ |2 + P δ }(x, t) dx + Ω

0

∇uδ |2 dxdt

Ω

≤ ce + Mδ (t) + ce

t

ece (t−s) Mδ (s) ds,

(5.4.27)

0

where

(a|Δ2 U|2 )(x, t) dx.

Mδ (t) = 2

(5.4.28)

Ω

The remainder of this section is devoted to the proof of Theorem 5.4.5. By abuse of notation the subscript δ is omitted, and we write simply , u and μ instead of δ , uδ and μδ . We split the proof of each claim (i)–(v) into a sequence of lemmas.

5.4. Passage to the limit. The second level

145

Proof of claim (i). In view of (5.4.11) the functions ε converge to δ weakly in L∞ (0, T ; Lγ (Ω)) and weakly in Lγ+1 (Q ) for every Q Q. Next, the functions uε converge weakly in L2 (0, T ; W 1,2 (Ω)). Finally, by Lemma 5.4.2 the momenta ε uε converge weakly in L∞ (0, T ; Lβ (Ω)) and the kinetic energy tensors ε uε ⊗uε converge to δ uδ ⊗ uδ weakly in L2 (0, T ; Lz (Ωδ )). Therefore, (5.4.18)–(5.4.22) follow from (5.4.1)–(5.4.8) and from the lower semicontinuity of the norm in the weak topology. Next, (5.4.22) is a straightforward consequence of (5.4.8) and (5.4.15). It remains to prove (5.4.19) for p := pδ . To this end choose t0 ∈ (0, T ) and h > 0. Consider the nonnegative Carathéodory function f (x, t, λ) = p(λ) for |t − t0 | < h,

and

f (x, t) = 0

otherwise.

From Lemma 5.4.4, (5.4.14), (5.4.2) and the inequality p() ≤ ce (1+γ ) we obtain 1 2h

t0 +h

1 ε→0 2h

pδ dxdt ≤ lim

t0 −h

Ω

t0 +h

t0 −h

p(ε ) dxdt ≤ cE,p

for all h > 0,

Ω

which obviously implies (5.4.19). Proof of claim (ii). Our next task is to prove (5.4.23). Choose a vector ﬁeld ζ that vanishes in Q \ Q2δ and in a neighborhood of the top Ω × {t = T }. Since a(x) = 0 in Qδ , the integral identity (5.3.10) with and u replaced by ε and uε implies Q

ε uε · ∂t ζ + (ε uε − ε∇ε ) ⊗ uε + p(ε ) I − S(uε ) : ∇ζ dxdt + ε f · ζ dxdt + (∞ U · ζ)(x, 0) dx = 0. (5.4.29) Q

Ω

It follows from (5.4.12) that lim

ε→0

Q

ε uε · ∂t ζ + ε uε ⊗ uε − S(uε ) : ∇ζ + ε f · ζ dxdt δ uδ · ∂t ζ + δ uδ ⊗ uδ − S(uδ ) : ∇ζ + δ f · ζ dxdt. =

(5.4.30)

Q

Next, estimate (5.4.5) yields ε∇ε ⊗ uε : ∇ζ dxdt = 0.

lim

ε→0

(5.4.31)

Q

Now choose h > 0 and notice that Ω2δ × (h, T − h) Q. Then (5.4.16) yields

T −h

(p(ε ) − pδ ) div ζ dxdt = 0.

lim

ε→0

h

Ω2δ

146

Chapter 5. Nonstationary case. Existence theory

From this, (5.4.1), the inequality c(1+γε ) ≥ p(ε ), and estimate (5.4.19) we obtain lim sup ε→0

(p(ε ) − pδ ) div ζ dxdt ≤ lim sup

Q2δ

+ c(ζ) lim sup

ε→0

T

+

ε→0

≤ c(ζ)cE,p

h

T −h

0

h

T

+ T −h

0

T −h h

(p(ε ) − pδ ) div ζ dxdt Ω2δ

(p(ε ) + pδ )) dxdt Ω2δ

dt = 2c(ζ)cE,p h → 0 as h → 0. (5.4.32)

Letting ε → 0 in (5.4.29) and using (5.4.30)–(5.4.32) we obtain (5.4.23). Proof of claim (iii). Now our task is to prove that the limits := δ and u := uδ determine a renormalized solution to the limit transport equation and satisfy the integral identity (5.4.24). The following lemma shows that is a weak solution to the limit transport equation. Lemma 5.4.6. Under the assumptions of Theorem 5.4.5, the integral identity (∂t ψ + ∇ψ · u) dxdt + (∞ ψ)(x, 0) dx = ψ∞ U · n dSdt (5.4.33) Q

ST

Ω

holds for any ψ ∈ C 1 (Q) which vanishes in the vicinity of ST \ Σin and of the top Ω × {T }. Proof. Choose an arbitrary smooth function ψ which is zero in the vicinity of ST \ Σin and of Ω × {t = T }. This means that supp ψ ∩ ST Σin . It follows from the integral identity (5.3.14) with and u replaced by ε and uε that (ε ∂t ψ − ε∇ε · ∇ψ + ε uε · ∇ψ) dxdt Q + ψ(ε∇ε − ∞ U) · n dSdt + (ψ∞ )(x, 0) dx = 0. (5.4.34) ST

Ω κ

Proposition 5.3.11 implies that ε|∂n ε | ≤ cε at all (z, t) ∈ supp ψ ∩ ST Σin , where κ > 0 and c are independent of (z, t) and ε. Hence ψ∇ε · n dSdt ≤ Cεκ → 0 as ε → 0. (5.4.35) ε ST

On the other hand, estimate (5.4.2) implies ∇ε · ∇ψ dxdt ≤ c(ψ)cE,p ε1/2 → 0 as ε 0. ε

(5.4.36)

Q

Letting ε tend to 0 in the integral identity (5.4.34), and using (5.4.12), (5.4.35) and (5.4.36) we obtain (5.4.33).

5.4. Passage to the limit. The second level

147

It follows from Lemma 5.4.6 that := δ is a weak solution to the boundary value problem ∂t + div(u) = 0

in Q,

on Σin ,

= ∞

= ∞

on Ω × {0}.

The next lemma justiﬁes the renormalization procedure (see Section 1.6) and shows that and u determine a renormalized solution to the mass balance equation. Lemma 5.4.7. Under the assumptions of Theorem 5.4.5, the integral identity Q

ϕ()(∂t ψ + ∇ψ · u) dxdt − (ϕ () − ϕ()) div u dxdt Q + ϕ(∞ (·, 0))ψ(·, 0) dx = ψ∞ U · n dSdt

(5.4.37)

ST

Ω

holds for all functions ψ ∈ C 1 (Q) which vanish in the vicinity of ST \ Σin and of the top Ω × {T }, and for any C 2 function ϕ : R → R such that |ϕ(s)| + |(1 + |s|)ϕ (s)| + |(1 + s2 )ϕ (s)| ≤ c(1 + |s|)2 ,

s ∈ R.

(5.4.38)

Proof. First we assume that ϕ satisﬁes the stronger condition |ϕ(s)| + |(1 + |s|)ϕ (s)| + |(1 + s2 )ϕ (s)| ≤ c,

s ∈ R.

(5.4.39)

Choose a test function ψ as in the lemma. Without loss of generality assume that (ψ, ∞ ) and U are extended to C 1 (Rd × R+ ) and C ∞ (Rd × R+ ) functions respectively. Moreover, we can assume that the extensions vanish for large |x| and t. Next, ﬁx an open set D ⊂ Rd × R+ such that cl D ∩ (cl ST \ Σin ) = ∅. Notice that ST ∩ cl D is a compact subset of Σin . Finally, choose cut-oﬀ functions η, ζ ∈ C ∞ (Rd × R+ ) such that 0 ≤ η, ζ ≤ 1,

ζ = η = 0 in the vicinity of ST \ Σin ,

η = 1 in D,

ζ = 1 on supp η.

Now set u∗ = ζu in Q and ω∞ = η∞

u∗ = ζU in Rd × (0, T ) \ Q,

in Rd × (0, T ),

ω = η in Q,

ω = ω∞

in Rd × (0, T ) \ Q,

F + = (ζ)(∂t η + ∇η · u) in Q, F − = ∂t ω∞ + div(ω∞ u∗ ) in Rd × (0, T ) \ Q, and ﬁnally F = F+

in Q,

F = F−

in Rd × (0, T ) \ Q.

148

Chapter 5. Nonstationary case. Existence theory

With this notation and ψ replaced by ηψ, the integral identity (5.4.33) reads ω(∂t ψ + ∇ψ · u∗ ) + ψF + dxdt + (ω∞ ψ)(x, 0) dx = ω∞ u∗ · nψ dSdt. Q

ST

Ω

(5.4.40) Notice that this holds for all ψ ∈ C 1 (Rd × (0, T )) vanishing for t = T , since ω vanishes in a neighborhood of ST \ Σin . Direct calculations show that for every ψ ∈ C 1 (Rd × (0, T )) vanishing at {t = T },

(Rd ×(0,T ))\Q

∗

ω∞ (∂t ψ + ∇ψ · u ) + ψF

−

dxdt + Rd \Ω

=−

(ω∞ ψ)(x, 0) dx

ω∞ u∗ · nψ dSdt.

(5.4.41)

ST

It is important to note that the sign of the right hand side is opposite to the sign of the right hand side in (5.4.40), since the direction of the outward normal to Rd \ Ω is opposite to the direction of n. Combining (5.4.40) and (5.4.41) yields ω(∂t ψ + ∇ψ · u∗ ) + ψF dxdt + (ω∞ ψ)(x, 0) dx = 0 Rd ×(0,T )

Rd

for any ψ ∈ C 1 (Rd ×(0, T )) vanishing for t = T . It follows that ω is a weak solution to the following Cauchy problem in the strip Rd × (0, T ): ∂t ω + div(ωu∗ ) = F

in Rd × (0, T ),

ω = ω∞

for t = 0.

By (5.4.18)–(5.4.20) with δ := and uδ := u we have ω ∈ L∞ (0, T ; Lγ (Rd )), u∗ ∈ L2 (0, T ; W 1,2 (Rd )),

F ∈ L∞ (0, T ; L2γ/(γ+2) (Rd )).

Moreover, these functions vanish for large |x|. Since γ > 2d, we can apply the renormalization Lemma 1.6.2 to conclude that the composite function ϕ(ω) is a weak solution to the Cauchy problem in the strip Rd × (0, T ), ∂t ϕ(ω) + div(ϕ(ω)u∗ ) + (ϕ (ω)ω − ϕ(ω)) div u∗ = ϕ (ω)F, ϕ(ω) = ϕ(η∞ ) ≡ ϕ(ω∞ ) for t = 0. This means that for any C 1 function ψ with ψ(·, T ) = 0,

ϕ(ω)(∂t ψ + ∇ψ · u∗ ) dxdt − ψ(ϕ (ω)ω − ϕ(ω)) div u∗ dxdt d d R ×(0,T ) R ×(0,T ) (ϕ(η∞ )ψ)(x, 0) dx + ψϕ (ω)F dxdt = 0. + Rd

Rd ×(0,T )

5.4. Passage to the limit. The second level

149

On the other hand, straightforward calculations lead to the identity

ϕ(ω)(∂t ψ + ∇ψ · u∗ ) dxdt − (ϕ (ω)ω − ϕ(ω)) div u∗ dxdt Rd ×(0,T )\Q (ϕ(η∞ )ψ)(x, 0) dx + ψϕ (ω)F − dxdt + Rd \Ω Rd ×(0,T )\Q =− ψϕ(ω∞ )u∗ · n dSdt,

Rd ×(0,T )\Q

ST

which gives ϕ(ω)(∂t ψ + ∇ψ · u∗ ) dxdt − ψ(ϕ (ω)ω − ϕ(ω)) div u∗ dxdt Q Q + (ϕ(η∞ )ψ)(x, 0) dx + ψϕ (ω)F + dxdt = ψϕ(ω∞ )u∗ · n dSdt,

Q

Ω

ST

(5.4.42) for every C 1 function ψ in Q vanishing for t = T . Now, choose ψ ∈ C 1 (Rd × (0, T )) equal to 0 in a neighborhood of ST \Σin . The intersection supp ψ ∩ST is a compact subset of Σin . Hence there is an open set D ⊂ Rd × (0, T ) with supp ψ D,

D ∩ (cl ST \ Σin ) = ∅.

For such a D and the corresponding function η, we have ω = ,

ω∞ := η∞ = ∞ ,

u∗ = u on supp ψ ∩ cl Q.

Inserting these into (5.4.42) we obtain (5.4.37). To complete the proof we approximate an arbitrary function ϕ satisfying (5.4.38) by the functions ⎧ ⎪ if |ϕ(s)| ≤ n, ⎨ϕ(s) ϕn (s) = n + 1 if ϕ(s) ≥ n + 1, ⎪ ⎩ −n − 1 if ϕ(s) ≤ −n − 1, which are C 2 on R and satisfy (5.4.39). It is clear that ϕn () → ϕ(),

ϕ () → ϕ ()

everywhere in Q. Moreover, it follows from (5.4.38) and 2 ∈ Lγ/2 (Q) ⊂ L2 (Q) that ϕn ()Lγ/2 (Q) + ϕn ()Lγ/2 (Q) ≤ c.

150

Chapter 5. Nonstationary case. Existence theory

Since γ > 4, it now follows from Lemma 1.3.2 that ϕn () → ϕ(),

ϕn () → ϕ () in L2 (Q) as n → ∞.

(5.4.43)

Substituting ϕn () for ϕ() into (5.4.33), letting n → ∞ and recalling that u, div u ∈ L2 (Q)) we conclude that the integral identity (5.4.37) holds for all ϕ satisfying (5.4.38). It remains to note that claim (iii) is a straightforward consequence of Lemmas 5.4.6 and 5.4.7. Proof of claim (iv). We obtain the desired inequality (5.4.26) in a few steps. Lemma 5.4.8. Let the assumptions of Theorem 5.4.5 be satisﬁed and ϕ be an arbitrary C 2 function on R satisfying condition (5.4.38). Then with the notation := δ and u := uδ for the limits, we obtain ϕ(ε (t)) ϕ(t) ϕ(ε ) uε ϕ u

weakly in Lγ/2 (Ω),

weakly in Lm (Q)

with

(ϕ (ε )ε − ϕ(ε )) div uε (ϕ − ϕ) div u

(5.4.44)

m = 2γ/(γ + 4), m

weakly in L (Q).

(5.4.45) (5.4.46)

Here (ϕ − ϕ) div u is just a notation for a weak limit, and ϕ is deﬁned by the Young measure μδxt , ϕ(x, t) = μδxt , ϕ(λ). (5.4.47) Proof. First, by (5.4.38) and (5.4.2) we have ϕ (ε )ε − ϕ(ε )L∞ (0,T ;Lγ/2 (Ω)) + ϕ(ε )L∞ (0,T ;Lγ/2 (Ω)) ≤ c,

(5.4.48)

where c is independent of ε. Hence, passing to a subsequence, we can assume that ϕ(ε ) converges to some ϕ weakly in L∞ (0, T ; Lγ/2 (Ω)). Moreover, since ϕ()−γ → 0 as → 0, we can apply Theorem 5.4.3 to obtain (5.4.47). Estimate (5.4.48) combined with the Hölder inequality and (5.4.1) gives (ϕ (ε )ε − ϕ(ε )) div uε Lm (Q) ≤ ϕ (ε )ε − ϕ(ε )Lγ/2 (Q) div uε L2 (Q) ≤ c. (5.4.49) Here m = 2γ/(4 + γ) > 1 satisﬁes m−1 = 2−1 + 2γ −1 . Hence, passing to a subsequence, we can assume that (ϕ (ε )ε − ϕ(ε )) div uε converges weakly in Lm (Q) as ε → 0, which leads to (5.4.46). Next, notice that ϕ(ε ) is a solution to the equation ∂t ϕ(ε ) = div vε + hε , where vε = εϕ (ε )∇ε − ϕ(ε )uε , hε = −(ϕ (ε )ε − ϕ(ε )) div uε − εϕ (ε )|∇ε |2 = 0.

5.4. Passage to the limit. The second level

151

By (5.4.48), ϕ(ε ) and ϕ (ε )ε are bounded in L2 (Q). Moreover, in view of (5.4.1), uε and div uε are bounded in L2 (Q). Therefore, (ϕ (ε )ε − ϕ(ε )) div uε and ϕ(ε )uε are bounded in L1 (Q). √ It also follows from (5.4.2) that ε∇ε is bounded in L2 (Q). Since ϕ is bounded it follows that εϕ (ε )∇ε and εϕ |∇ε |2 are bounded in L1 (Q). Therefore the sequences vε and gε are bounded in L1 (Q). Since ϕ(ε ) converges to ϕ weakly in L∞ (0, T ; Lγ/2 ), we can apply Theorem 4.4.2 with n replaced by ε, ϕn replaced by ϕ(ε ), and s replaced by γ/2 to obtain (5.4.44) and (5.4.45). Here we use the inequality m−1 > 2/γ + 2−1 − d−1 . Lemma 5.4.9. Let Dϕ = (ϕ () − ϕ()) div u − (ϕ − ϕ) div u,

Φ = ϕ − ϕ().

Furthermore, assume that ϕ is a C 2 convex function satisfying the growth condition (5.4.38), and ψ ∈ C 1 (Q) is a nonnegative function vanishing in a neighborhood of ST \ Σin and in a neighborhood of Ω × {t = T }. Then Φ(∂t ψ + ∇ψ · u) dxdt + Dϕ ψ dxdt ≥ 0. (5.4.50) Q

Q

Notice that Φ ≥ 0 due the convexity of ϕ. Proof. Recall that by Theorem 5.3.2 the functions ε and uε satisfy inequality (5.3.15), which reads

ϕ(ε )∂t ψ + (ϕ(ε )uε − εϕ (ε )∇ε ) · ∇ψ − ψ(ϕ (ε )ε − ϕ(ε )) div uε dxdt Q + ψ εϕ (∞ )∇ − ϕ(∞ )U · n dSdt + (ψϕ(∞ ))(x, 0) dx ≥ 0. (5.4.51) ST

Ω

By Proposition 5.3.11, ε|∂n ε | ≤ cεκ for all (z, t) ∈ supp ψ ∩ ST Σin , where κ > 0 and c are independent of (z, t) and ε. Hence ψϕ (∞ )∇ε · n dSdt ≤ cεκ → 0 as ε → 0. (5.4.52) ε ST

On the other hand, we have (γ − 2)/2 > 1, which along with the growth condition (5.4.25) and estimate (5.4.2) implies ε Q

|ϕ (ε )∇ε | dxdt ≤ εc |(1 + ε )∇ε | dxdt Q ≤ εc |(1 + ε )(γ−2)/2 ∇ε | dxdt ≤ cε1/2 → 0 as ε → 0. (5.4.53) Q

152

Chapter 5. Nonstationary case. Existence theory

Letting ε → 0 in (5.4.51) and using (5.4.45), (5.4.46), (5.4.52), and (5.4.53) we conclude that ϕ∂t ψ + ϕu · ∇ψ − (ϕ − ϕ) div u dxdt Q − ψϕ(∞ )U · n dΣ + (ψϕ(∞ ))(x, 0) dx ≥ 0 (5.4.54) Σin

Ω

for all smooth nonnegative ψ, vanishing in a neighborhood of Ω × {t = T } and of ST \ Σin . Next, we subtract (5.4.37) from (5.4.54) to obtain (5.4.50). Our further considerations are based on the following geometric lemma: Lemma 5.4.10. Let ∂Ω be a C 1+α surface, 0 < α < 1, U ∈ C 1 (Rd × (0, T )), and u − U ∈ L2 (0, T ; W01,2 (Ω)). Furthermore, assume that a compact “interface” Γ = (ST \ Σin ) ∩ cl Σin satisﬁes condition (5.1.9): lim σ −d meas Oσ < ∞,

(5.4.55)

σ→0

where Oσ is a tubular neighborhood of Γ, Oσ = {(x, t) ∈ Rd+1 : dist((x, t), Γ) ≤ σ}. Then there is a sequence of Lipschitz functions ψn : Q → [0, 1], n ≥ 1, with the following properties: ψn vanishes in some neighborhood of ST \Σin and of {t = T }, ψn 1 everywhere in Q, and lim sup Φ(∂t ψn + ∇ψn · u) dxdt ≤ 0 for all nonnegative Φ ∈ Lq (Q), q > 2. n→∞

Q

Proof. This is a particular case of Theorem 13.3.3.

Lemma 5.4.11. The functions := δ and u := uδ satisfy the inequality ( div u − div u) dxdt ≤ 0. (5.4.56) Q

Proof. Set 3 1 ϕk = − + log k + 2k2 − k 2 3 2 2

for ≤ k −1 ,

ϕk = log for > k−1 .

Obviously ϕk ∈ C 2 [0, ∞) satisﬁes the growth condition (5.4.25). Substituting ϕk and the functions ψn from Lemma 5.4.10 into (5.4.50), letting n → ∞ and using Lemma 5.4.10 we obtain Dϕk dxdt ≥ 0. (5.4.57) Q

5.4. Passage to the limit. The second level

153

Next note that ϕk ()−ϕk ()− = 2k2 −k 2 3

for < k −1 ,

ϕk ()−ϕk () = for ≥ k −1 .

Hence the sequence ϕk − ϕk converges to uniformly on R. Letting k → ∞ in (5.4.57) we obtain (5.4.56). Lemma 5.4.12. For any subcylinder Q Qδ and ψ ∈ C0 (Q ), ψ div u − div u dxdt = lim ψ(p(ε )ε − p) dxdt. (1 + λ) ε→0

Q

Q

Proof. Notice that ε and uε satisfy the diﬀerential equations ∂t ε + div(ε uε − gε ) = 0 in Qδ , ∂t (ε uε ) + div(ε uε ⊗ uε ) = div(Tε + Gε ) + ρε f

in Qδ ,

where Tε = S(uε ) − p(ε ) I,

gε = ε∇ε ,

Gε = ε∇ε ⊗ uε .

Set r = γ + 1,

s = γ, ϕε = ε ,

r = (γ + 1)/γ, ε = 0,

q = s/(s − 1), gεϕ

= gε ,

Let us prove that ε , uε , ϕε , and Tε , Gε , gε meet all requirements of Theorem 4.7.1 with n replaced by ε and Q replaced by Qδ . It follows from estimates (5.4.1)– (5.4.2) that the functions (ε , uε , ϕε ) satisfy inequalities (4.6.4)–(4.6.6). Estimate (5.4.8) implies that ϕε = ε satisﬁes (4.6.8). Next, estimates (5.4.1)–(5.4.2) yield the L1 -estimate (4.6.7) for Tn . On the other hand, the inequalities p ≤ c(γ + 1) and (γ + 1)/γ ≤ 2 along with (5.4.2) and (5.4.8) give, for every Q Qδ , Tε L(γ+1)/γ (Q ) ≤ S(uε )L(γ+1)/γ (Q ) + p(ε )L(γ+1)/γ (Q ) ≤ c(Q ). It follows that Tε satisﬁes (4.6.8). Next (5.4.2) and (5.4.5) imply gε L2 (Q) = ε∇ε L2 (Q) ≤ ε1/2 c → 0 Gε L1 (0,T ;Lq (Ω)) ≤ ε∇ε |uε |L1 (0,T ;Ld/(d−1) (Ω)) ≤ ε

as ε → 0, 1/2

c→0

as ε → 0,

since q < d/(d − 1). Hence gε and Gε satisfy (4.6.9)–(4.6.10) with n → ∞ replaced by ε → 0. Therefore, ε , uε , ϕε , and Tε , Gε , gε satisfy Condition 4.6.1. Applying Theorem 4.7.1 and recalling that ϕε = ε we obtain ψ(p(ε ) − (λ + 1) div uε )ε dxdt = ψ(p − (λ + 1) div u) dxdt. lim ε→0

Qδ

This completes the proof.

Qδ

154

Chapter 5. Nonstationary case. Existence theory

Lemma 5.4.13. Under the assumptions of Theorem 5.4.5 inequality (5.4.26) holds for any subcylinder Q Qδ . Proof. Choose Q Qδ and a nonnegative function ψ ∈ C0∞ (Qδ ) such that ψ = 1 on Q . It follows from Lemmas 5.4.4 with and 5.4.12 and (5.4.56) that with p := pδ ,

=:= δ ,

, p := pδ ,

u := uδ ,

we have (p − p) dxdt ≤ ψ(p − p) dxdt ≤ lim ψ(p(ε )ε − p) dxdt ε→0 Q Q Qδ = (1 + λ) ψ( div u − div u) dxdt = (1 + λ) ( div u − div u) dxdt Qδ Q + (1 + λ) (ψ − 1)( div u − div u) dxdt Q ≤ (1 + λ) (ψ − 1)( div u − div u) dxdt Q ≤ (1 + λ) | div u − div u| dxdt. (5.4.58) Q\Q

Next, estimates (5.4.1)–(5.4.2) and the Hölder inequality imply that for m = 2γ/(γ + 2), div uLm (Q) ≤ lim inf ε div uε Lm (Q) ε→0

≤ lim inf ε Lγ (Q) div uε L2 (Q) ≤ cE,p . ε→0

The same arguments combined with (5.4.18)–(5.4.19) give div uLm (Q) ≤ cE,p , hence | div u − div u| dxdt Q\Q

≤ meas(Q \ Q )1/m div u − div uLm (Q) ≤ cE,p meas(Q \ Q )1/m , where 1/m = (γ − 2)/(2γ). Inserting this into (5.4.58) we obtain (5.4.26).

It remains to note that the statement of Lemma 5.4.13 is exactly claim (iv) of Theorem 5.4.5. Proof of claim (v). The proof is based on the following lemma analogous to Lemma 5.3.8. This lemma is valid for any weak limit, however it is formulated for the weak limit under considerations.

5.4. Passage to the limit. The second level

155

Lemma 5.4.14. For any 0 ≤ T < T ≤ T , the weak limit (u, ) of the sequence (uε , ε ) satisﬁes

T

T T

P dxdt ≤ lim inf ε→0

Ω

T

T T

|u| dxdt = lim inf

T

ε→0

Ω

T T

|∇u|2 dxdt ≤ lim inf ε→0

Ω

T

(5.4.59a)

ε |uε |2 dxdt,

(5.4.59b)

|∇uε |2 dxdt.

(5.4.59c)

2

T T

P (ε ) dxdt, Ω

Ω

Ω

Proof. First observe that if ψ(x, t) is the characteristic function of Ω × [T , T ], then (5.4.59a) turns into (5.4.14). Let us prove (5.4.59b). By Lemma 5.4.2 the sequence ε uε ⊗ uε converges weakly in L2 (0, T ; Lz (Ω)). Hence it also converges weakly in Lz (Q), whence in Lz (Ω × (T , T )). Recalling that |u|2 = tr u ⊗ u we obtain (5.4.59b). Inequality (5.4.59c) obviously follows from the weak convergence of ∇uε to ∇u in L2 (Q). Lemma 5.4.15. The functions Mε : [0, T ] → R deﬁned by (5.4.10) are uniformly bounded and lim Mε (t) ≤ Mδ (t) + ce = 2 (a|Δ2 U|2 )(x, t) dx + ce (5.4.60) ε→0

Ω

where ce is a constant as in Remark 5.1.2. Proof. By (5.4.2) and (5.4.3) the sequences ε and ε uε are uniformly bounded in L∞ (0, T ; L1 (Ω)). Hence the boundedness of Mε (t) is a straightforward consequence of (5.4.10): t Mε (t) = Mδ (t) + 4 0

ε2/3 |∇P (∞ )|2 − ε ∂t P (∞ ) + ∇P (∞ ) · uε dxds.

Ω

Thus we get limε→0 Mε = Mδ + 4 limε→0 Iε , where t ε ∂t P (∞ ) + ∇P (∞ )uε dxds. Iε (t) = − 0

Ω

Next, by relation (5.4.11) and Lemma 5.4.2 we have ε δ =: weakly in L∞ (0, T ; Lγ (Ω)) and ε uε δ uδ =: u weakly in L∞ (0, T ; L2γ/(γ+1) (Ω)). Hence letting ε → 0 we obtain t I := lim Iε (t) = − (∂t P (∞ ) + ∇P (∞ )u) dxds. ε→0

0

Ω

It follows from Theorem 13.3.3 (see Appendix) that there is a sequence of Lipschitz functions ψn : Ω × (0, t) → R such that

156

Chapter 5. Nonstationary case. Existence theory

• 0 ≤ ψn ≤ 1, ψn 1 in Ω × (0, t) as n → ∞; • ψn vanishes in a neighborhood of ∂Ω × (0, t) \ Σin and of Ω × {t}; • for any nonnegative function Φ ∈ L2 (Ω × (0, t)), lim inf Φ(∂t ψn + ∇ψn · u) dxds ≤ 0. n→∞

(5.4.61)

Ω×(0,t)

We have I = limn→∞ In , where t In = − ψn (∂t P (∞ ) + ∇P (∞ )u) dxds Ω 0 t = P (∞ )(∂t ψn + ∇ψn · u) dxds Ω 0 t ∂t (P (∞ )ψn ) + ∇(P (∞ )ψn ) · u dxds. − Ω

0

Notice that ψn P (∞ ) is a Lipschitz function vanishing in a neighborhood of Ω×{t} and in a neighborhood of ∂Ω × (0, t) \ Σin . Extended by zero over Q, ψn P (∞ ) becomes a Lipschitz function vanishing in a neighborhood of ST \ Σin and in a neighborhood of Ω × {t = T }. Hence ψn P (∞ ) can be regarded as a test function in the mass balance equation. In particular, the integral identity (5.4.33) with ψ replaced by ψn P (∞ ) implies t ∂t (P (∞ )ψn ) + ∇(P (∞ )ψn ) · u dxds Ω 0 t = ψn ∞ P (∞ )U · n dS(x)ds − (ψn ∞ P (∞ ))(x, 0) dx ≤ ce . 0

∂Ω

Ω

Thus we get t In (t) ≤ 0

δ P (∞ )(∂t ψn + ∇ψn · u) dxds + ce .

Ω

Since Φ := P (∞ ) is nonnegative and belongs to L2 (Q) we can apply (5.4.61) to obtain t δ P (∞ )(∂t ψn + ∇ψn · u) dxds ≤ 0, lim n→∞

0

Ω

which leads to I = limn→∞ In ≤ ce . Combining this result with the identity limε→0 Mε = Mδ + 4I we obtain (5.4.60). Lemma 5.4.16. Let u = uδ , = δ and P = P δ be deﬁned by (5.4.11) and Lemma 5.4.4, respectively. Then for almost every t ∈ (0, T ) the functions (u, ) and P satisfy inequality (5.4.27).

5.5. Passage to the limit. The third level

157

Proof. Choose t0 ∈ (0, T ) and h > 0 such that (t0 − h, t0 + h) ⊂ (0, T ). Recall that P ≥ 0 in view of (5.1.7). Integrate both sides of (5.4.9) over (t0 − h, t0 + h) to obtain

t0 +h

(ε |uε |2 + P (ε ))(x, t) dxdt +

t0 −h

Ω

≤

t0 +h

t

t0 −h t0 +h

+ t0 −h

0

t0 +h

t0 −h

t 0

(|∇uε |2 )(x, s) dxds dt

Ω

ε4/3 |∇ε |2 dxds dt

Ω

t ece (t−s) Mε (s) ds dt. ce + Mε (t) + ce 0

It follows from (5.2.4) that ε4/3 |∇ε |2 L1 (Q) ≤ cE,p ε1/3 → 0 as ε → 0. Letting ε → 0 and applying Lemma 5.4.14 with T = t0 − h and T = t0 + h and Lemma 5.4.15 we obtain 1 2h

t0 +h

t0 −h

t0 +h t 1 (|u| + P )(x, t) dxdt + (|∇u|2 )(x, s) dxds dt 2h t0 −h 0 Ω Ω t0 +h t 1 ≤ ce + Mδ (t) + ce ece (t−s) Mδ (s) ds dt. 2h t0 −h 0

2

Letting h → 0 yields (5.4.27).

It remains to notice that the last claim (v) coincides with the statement of Lemma 5.4.16. This completes the proof of Theorem 5.4.5.

5.5 Passage to the limit. The third level In this section we pass to the limit as δ → 0 and thus we complete the proof of the main Theorem 5.1.6. We begin with the observation that the functions δ and uδ satisfy estimates (5.4.18)–(5.4.22) of Theorem 5.4.5. Hence, passing to a subsequence, we can assume that there exist ∈ L∞ (0, T ; Lγ (Ω)) ∩ Lγ+1 loc (Q) and u ∈ L∞ (0, T ; W 1,2 (Ω)) with the properties δ weakly in L∞ (0, T ; Lγ (Ω)), uδ u weakly in L2 (0, T ; W 1,2 (Ω)), γ+1

δ weakly in L

(5.5.1)

(Q ) for any compact Q Q.

Finally arguing as in the proof of Lemma 5.4.2 we can assume that δ uδ u weakly in L∞ (0, T ; L2γ/(γ+1) (Ω)), δ uδ ⊗ uδ u ⊗ u weakly in L2 (0, T ; Lz (Ω)).

(5.5.2)

158

Chapter 5. Nonstationary case. Existence theory

In the previous section we have proved the existence of a Young measure μδ associated with the sequence ε . Recall that μδ is an element of the Banach space L∞ w (Q; M(R)), where M(R) is the space of Borel measures on R. By Theorem 1.2.24, the latter space is the dual to the Banach space C0 (R) of all continuous functions on the real axis vanishing at inﬁnity. Since C0 (R) is separable, we can apply the generalized Riesz Theorem 1.3.24 to deduce that L∞ w (Q; M(R)) is the dual of L1 (Q; C0 (R)), and for every functional V ∈ L1 (Q; C0 (R)) , there exists a unique parameterized measure ν ∈ L∞ (Q; M(R)) such that νxt , v(x, t, ·) dxdt for all v ∈ L1 (Q; C0 (R)). V, v = Q

The converse is true in the sense that every ν ∈ L∞ (Q; M(R)) determines an element of the dual space L1 (Q; C0 (R)) . Since L1 (Q; C0 (R)) is separable, it follows from the Alaoglu theorem that every bounded sequence in L1 (Q; C0 (R)) contains a weakly convergent subsequence. By deﬁnition, the Young measures δ μδ , δ > 0, belong to the space L∞ w (Q; M(R)) and the norm of μ in this space is equal to 1. Hence, passing to a subsequence, we can assume that μδ converges ∞ weakly in L∞ w (Q; M(R)) to some μ ∈ Lw (Q; M(R)). This means that for all 1 v ∈ L (Q; C0 (R)), δ μxt , v(x, t, ·) dxdt → μxt , v(x, t, ·) dxdt as δ → 0. (5.5.3) Q

Q

It is worth noting that this convergence alone does not imply that μ is a Young measure. Now our task is to show that the limit functions (u, ) determine a renormalized solution to problem (5.1.12) in the sense of Deﬁnition 5.1.4 and meet all requirements of Theorem 5.1.6. We split the proof into several lemmas. The ﬁrst shows that the parameterized measure μ deﬁned by (5.5.3) is a Young measure. Lemma 5.5.1. (i) The parameterized measure μ deﬁned by (5.5.3) is a Young measure. The limit function deﬁned by (5.5.1) has the representation (x, t) = μxt , λ. (ii) Let p() be a pressure function and deﬁne the internal energy s−2 p(s) ds. P () = 0

Then the functions p(x, t) := μxt , p(λ),

P (x, t) := μxt , P (λ)

belong to L∞ (0, T ; L1 (Ω)), and p(x, t) = μxt , p(λ)λ belongs to L1 (Q ) for any compact set Q Q.

5.5. Passage to the limit. The third level

159

(iii) For any nonnegative ψ ∈ L∞ (Q) and compact Q Q, ψp dxdt ≤ lim inf ψpδ dxdt, δ→0 Q Q ψP dxdt ≤ lim inf ψP δ dxdt, δ→0

Q

Q

δ→0

Q

ψp dxdt ≤ lim inf

(5.5.4)

Q

ψpδ dxdt,

ψp dxdt = lim

Q

δ→0

Q

ψpδ dxdt, (5.5.5)

where pδ and pδ are deﬁned in Lemma 5.4.4. Proof. To avoid repetitions we give the proof only for the pressure function p. The same proof works for P . Proof of claim (i). First observe that by Lemma 5.4.3 and (5.4.19), μδ , p(λ) dxdt = pδ dxdt ≤ cE,p , Q

Q

and for any ﬁxed compact set Q Q, μδ , p(λ)λ dxdt = Q

Q

pδ dxdt ≤ cE,p (Q ).

It now follows from the inequality γ ≤ ce (p() + 1) that ∞ ∞ δ −γ+1 λ dμxt (λ) dxdt ≤ cE,p N , p(λ) dμδxt (λ) dxdt ≤ cE,p N −1 . Q

Q

N

N

(5.5.6)

Now ﬁx η ∈ C0∞ (R) with the properties 0 ≤ η ≤ 1,

η() = 1 if || ≤ 1,

η() = 0 if || ≥ 2.

The limit element μ deﬁnes a weakly measurable mapping Q (x, t) → μxt ∈ M(R). By the deﬁnition of weak limit, the Radon measure μxt is nonnegative. Since all μδxt are supported on the convex set [0, ∞), so is μxt . Next, for any ψ ∈ L∞ (Q) we have ψμxt , η(λ/N )λ dxdt = lim ψμδxt , η(λ/N )λ dxdt. (5.5.7) δ→0

Q

Q

Assume now that ψ ∈ L∞ (Q) is nonnegative. We have δ ψμxt , η(λ/N )λ dxdt = ψδ (x, t) dxdt + ON , Q

Q

160

Chapter 5. Nonstationary case. Existence theory

where the remainder is of the form ψμδxt , (η(λ/N ) − 1)λ dxdt. ON = Q

By the ﬁrst inequality in (5.5.6), |ON | ≤ ψL∞ (Q) Q

∞

λ dμδxt dxdt ≤ cE,p ψL∞ (Q) N −γ+1 .

N

Thus we get ψμδxt , η(λ/N )λ dxdt − ψδ dxdt ≤ cE,p ψL∞ (Q) N −γ+1 . lim sup δ→0

Q

Q

Since δ converges to weakly in Lγ (Q), we now deduce from (5.5.7) that ψμxt , η(λ/N )λ dxdt − ψ dxdt ≤ cE,p ψL∞ (Q) N −γ+1 . (5.5.8) Q

Q

Notice that η(λ/N )λ λ as N → ∞ on [0, ∞). Hence by the Fatou Theorem 1.2.16, μxt , η(λ/N )λ μxt , λ as N → ∞. Letting N → ∞ in (5.5.8) we obtain ψ(μxt , λ − ) dxdt = 0 for all nonnegative ψ ∈ L∞ (Q).

(5.5.9)

Q

Therefore μxt , λ = (x, t) for a.e. (x, t) ∈ Q. Now replace μxt , η(λ/N )λ by μxt , η(λ/N ) in (5.5.7) and repeat the previous arguments. Thus we arrive at μxt , 1 = 1 for a.e. (x, t) ∈ Q. Hence μ is a Young measure. This proves claim (i). Proof of claim (ii). Relation (5.5.3) with v = η(λ/N )p(λ) implies μxt , η(λ/N )p(λ) dxdt = lim μδxt , η(λ/N )p(λ) dxdt δ→0 Q Q ≤ lim inf μδxt , p(λ) dxdt = lim inf pδ dxdt. δ→0

Q

δ→0

It now follows from (5.4.19) that μxt , η(λ/N )p(λ) dxdt ≤ cE,p .

Q

(5.5.10)

Q

Since η(λ/N )p(λ) p(λ) as N → ∞, by the Fatou Theorem 1.2.16 we have μxt , η(λ/N )p(λ) μxt , p(λ) = p(x, t) as N → ∞.

5.5. Passage to the limit. The third level

161

Applying once more Theorem 1.2.16 we arrive at cE,p ≥ μxt , η(λ/N )p(λ) dxdt μxt , p(λ) dxdt Q

as N → ∞.

Q

Hence p ∈ L1 (Q). Next choose Q Q and deﬁne v(x, t, λ) = p(λ)λη(λ/N ) if (x, t) ∈ Q and v(x, t, λ) = 0 otherwise. By (5.5.3) we have μxt , η(λ/N )p(λ)λ dxdt = lim μδxt , η(λ/N )p(λ)λ dxdt δ→0 Q Q δ ≤ lim inf μxt , p(λ)λ dxdt = lim inf pδ dxdt. δ→0

δ→0

Q

Q

From this and (5.4.22) we get μxt , η(λ/N )p(λ)λ dxdt ≤ cE,p (Q ). Q

Since η(λ/N )p(λ)λ p(λ)λ as N → ∞, we apply the Fatou Theorem 1.2.16 twice to obtain μxt , η(λ/N )p(λ)λ μxt , p(λ)λ = p(x, t) as N → ∞ and

cE,p ≥

Q

μxt , η(λ/N )p(λ)λdxdt

Q

μxt , p(λ)λ dxdt

as N → ∞.

Hence p belongs to L1 (Q). This completes the proof of claim (ii). Proof of claim (iii). Let ψ ∈ L∞ (Q) be nonnegative. By the convergence (5.5.3) with v = ψ(x, t)η(λ/N )p(λ), we have

Q

ψμxt , η(λ/N )p(λ) dxdt = lim ψμδxt , η(λ/N )p(λ) dxdt δ→0 Q δ ≤ lim inf ψμxt , p(λ) dxdt = lim inf ψpδ dxdt. δ→0

δ→0

Q

Q

Since η(λ/N )p(λ) p(λ) as N → ∞, the Fatou Theorem 1.2.16 yields μxt , η(λ/N )p(λ) μxt , p(λ) as N → ∞. Applying once more Theorem 1.2.16 we arrive at ψμxt , η(λ/N )p(λ) dxdt ψμxt , p(λ) dxdt Q

Q

as N → ∞.

(5.5.11)

162

Chapter 5. Nonstationary case. Existence theory

Therefore, we can let N → ∞ in (5.5.11) to obtain (5.5.4). Now we repeat these arguments to derive (5.5.5). Fix Q Q. Notice that Q Qδ for all suﬃciently small δ. In view of (5.5.11), ψμxt , η(λ/N )p(λ) dxdt = lim ψμδxt , η(λ/N )p(λ) dxdt (5.5.12) δ→0

Q

Q

for any nonnegative ψ ∈ L∞ (Q). On the other hand, ψμδxt , η(λ/N )p(λ) dxdt = ψpδ (x, t) + ON , Q

Q

where ON

= Q

ψμδxt , (η(λ/N ) − 1)p(λ) dxdt.

By the second inequality in (5.5.6), ∞ |ON | ≤ ψL∞ (Q ) p(λ) dμδxt dxdt ≤ c(Q )ψL∞ (Q ) N −1 . Q

Thus we get lim sup δ→0

N

Q

ψμδxt , η(λ/N )p(λ) dxdt

−

Q

ψpδ dxdt ≤ c(Q )ψL∞ (Q ) N −1 .

We now ﬁnd from (5.5.12) that − cN −1 + ψμxt , η(λ/N )p(λ) dxdt ≤ lim inf ψpδ dxdt δ→0 Q Q ψpδ dxdt ≤ cN −1 + ψμxt , η(λ/N )p(λ) dxdt. ≤ lim sup δ→0

Q

(5.5.13)

Q

On the other hand, applying the Fatou theorem twice we obtain lim ψμxt , η(λ/N )p(λ) dxdt = ψμxt , p(λ) dxdt = ψp dxdt. N →∞

Q

Q

Letting N → ∞ in (5.5.13) we obtain ψμxt , p(λ) = lim δ→0

Q

Q

Q

ψpδ dxdt,

which yields the equality in (5.5.5). It remains to prove the inequality in (5.5.5). For any nonnegative ψ ∈ L∞ (Q) we have ψμxt , η(λ/N )p(λ)λ dxdt = lim ψμδxt , η(λ/N )p(λ)λ dxdt δ→0 Q Q δ ≤ lim inf ψμxt , p(λ)λ dxdt = lim inf ψpδ dxdt. (5.5.14) δ→0

Q

δ→0

Q

5.5. Passage to the limit. The third level

163

Since η(λ/N )p(λ)λ p(λ)λ as N → ∞, arguing as before, we apply the Fatou theorem and let N → ∞ to obtain the inequality in (5.5.5). Lemma 5.5.1 implies the strong convergence of the sequence δ . Lemma 5.5.2. The limits , p, and P satisfy p = p() and P = P (). Moreover, δ → a.e. in Q. Proof. First we show that for a.e. (x, t) ∈ Q, the Young measure μxt is the Dirac measure concentrated at (x, t). We begin with the observation that the conclusion of Lemma 5.4.4 obviously holds for μxt in place of μδxt . By that lemma it suﬃces to prove that p − p = 0 a.e. in Q. Notice that p − p ≥ 0 by (5.4.17). Next, ﬁx a compact subset Q Q and let ψ be the characteristic function of Q . Then (5.5.5) implies p dxdt ≤ lim inf pδ dxdt and p dxdt = lim pδ dxdt, δ→0

Q

which leads to

Q

Q

Q

Q

δ→0

(p − p) dxdt ≤ lim inf δ→0

Q

(pδ − pδ ) dxdt.

Combining this with (5.4.26) we obtain (p − p) dxdt ≤ cE,p meas(Q \ Q )(γ−2)/(2γ) . Q

Choosing an increasing sequence of sets Q whose union covers Q and recalling that p − p ≥ 0 we arrive at (p − p) dxdt = 0, and so p − p = 0 a.e. in Q. Q

Applying Lemma 5.4.4 we conclude that μxt is the Dirac measure concentrated at (x, t) for a.e. (x, t) ∈ Q. In particular, p(x, t) = μxt , p(λ) = p((x, t)),

P (x, t) = μxt , P (λ) = P ((x, t)).

This leads to the desired equalities in Lemma 5.5.2. Next notice that the sequence 2δ is bounded in Lγ/2 (Q) and converges weakly in this space to 2 . We have 2 (x, t) = μxt , λ2 = (x, t)2 , since μxt is the Dirac measure concentrated at (x, t). This yields 2 lim δ (x, t) dxdt = (x, t)2 dxdt. δ→0

Q

Q

Therefore, δ converges to weakly in L (Q) and δ L2 (Q) → L2 (Q) . This is possible if and only if δ → strongly in L2 (Q). Passing to a subsequence we can assume that δ converges to a.e. in Q. 2

164

Chapter 5. Nonstationary case. Existence theory Our next task is to derive an energy estimate for (u, ).

Lemma 5.5.3. Let (u, ) be deﬁned by (5.5.1). Then t (P () + |u|2 )(x, t) dx + |∇u(x, s)|2 dxds ≤ ce 0

Ω

(5.5.15)

Ω

for a.e. t ∈ (0, T ), where the constant ce is as in Remark 5.1.2. Proof. Choose t0 ∈ (0, T ) and h > 0 such that (t0 − h, t0 + h) ⊂ (0, T ). Let ψ(x, t) be the characteristic function of Ω × (t0 − h, t0 + h). By the choice of ψ and (5.5.4),

t0 +h

P (x, t) dxdt ≤ lim inf

t0 −h

δ→0

Ω

t0 +h

P δ (x, t) dxdt.

t0 −h

Ω

By (5.5.2), the sequence δ uδ ⊗ uδ converges to u ⊗ u weakly in L2 (0, T ; Lz (Ω)). Hence it also converges weakly in L1 (Q), whence in L1 (Ω × (t0 − h, t0 + h)). Recalling that |u|2 = tr u ⊗ u we obtain

t0 +h t0 −h

|u|2 dxdt = lim inf δ→0

Ω

t0 +h

t0 −h

δ |uδ |2 dxdt. Ω

Since ∇uδ converges to ∇u weakly in L2 (Q), we also have t t |∇u|2 dxds ≤ lim inf |∇uδ |2 dxds. Ω

0

δ→0

0

t

|∇u|2 dxds dt

Ω

It follows that 1 2h

t0 +h

t0 −h

(P + |u|2 )(x, t) dx +

Ω

1 ≤ lim inf δ→0 2h

t0 +h

t0 −h

0

Ω

t

(P δ + δ |uδ | )(x, t) dx + 2

0

Ω

|∇uδ |2 dxds dt.

Ω

(5.5.16) Inequality (5.4.27) implies 1 2h ≤

t0 +h

t

Ω t0 −h t0 +h

1 2h

t0 −h

|∇uδ |2 dxds dt

(P δ + δ |uδ | )(x, t) dx + 2

0

t

ce + Mδ (t) + ce 0

Ω

ece (t−s) Mδ (s) ds dt ≤ ce + ce sup Mδ (t), t∈[0,T ]

(5.5.17) √

where Mδ (t) = 2 aΔ2 U(t)2L2 (Ω) . Notice that U ∈ C ∞ (Q). The nonnegative function a does not exceed 1 and is supported in the δ-neighborhood of ∂Ω. This

5.5. Passage to the limit. The third level

165

leads to the estimate Mδ ≤ cE,p δ. Inserting this in (5.5.17), and applying (5.5.16), we obtain t0 +h t 1 (P + |u|2 )(x, t) dx + |∇u|2 dxdt dt ≤ ce . 2h t0 −h 0 Ω Ω

Letting h → 0 and noting that P = P () we arrive at (5.5.15).

The next lemma shows that the limit functions (u, ) satisfy the mass balance equation. Lemma 5.5.4. Let (u, ) be deﬁned by (5.5.1). Then the integral identity

Q

u · ∂t ξ + u ⊗ u : ∇ξ + p() div ξ − S(u) : ∇ξ dxdt + f · ξ dxdt + (∞ U · ξ)(x, 0) dx = 0 (5.5.18) Q

Ω

holds for all vector ﬁelds ξ ∈ C ∞ (Q) equal to 0 in a neighborhood of the lateral side ST = ∂Ω × (0, T ) and of the top Ω × {T }. Proof. Choose a vector ﬁeld ζ which vanishes in a neighborhood of ST and of Ω × {T }. There is a δ0 > 0 such that supp ζ ⊂ Q2δ0 . It follows from (5.5.1)–(5.5.2) that δ uδ · ∂t ζ + δ uδ ⊗ uδ − S(uδ ) : ∇ζ + δ f · ζ dxdt lim δ→0 Q = u · ∂t ζ + u ⊗ u − S(u) : ∇ζ + f · ζ dxdt. (5.5.19) Q

Since for any h > 0 the set Q = Ω2δ0 × [h, T − h] is a compact subset of Qδ0 , the equality in (5.5.5) with ψ = div ζ implies lim sup δ→0

div ζ (p − pδ ) dxdt ≤ lim sup

Q2δ0

+c lim sup δ→0

h

T

+ 0

T −h

Ω2δ

δ→0

T −h

h

(p + pδ ) dxdt ≤ c 0

div ζ(p − pδ ) dxdt Ω2δ0

h

T

+ T −h

dt = 2ch → 0 (5.5.20)

as h → 0. Letting δ → 0 in (5.4.23), using (5.5.19)–(5.5.20) and recalling the equality p = p() in Lemma 5.5.2 we obtain the integral identity (5.5.18). Hence the limit functions and u satisfy the momentum balance equation. Finally we show that is a renormalized solution to the mass balance equation.

166

Chapter 5. Nonstationary case. Existence theory

Lemma 5.5.5. Let (u, ) be deﬁned by (5.5.1). Then the integral identity Q

ϕ()∂t ψ + ϕ()u · ∇ψ + ψ ϕ() − ϕ () div u dxdt = ψϕ(∞ )U · n dΣ − (ϕ(∞ )ψ)(x, 0) dx (5.5.21) Σin

Ω

holds for all ψ ∈ C ∞ (Q) vanishing in a neighborhood of the surface ST \ Σin and of the top Ω × {T }, and for all smooth functions ϕ : [0, ∞) → R such that lim sup(−1 |ϕ()| + |ϕ ()|) < ∞.

(5.5.22)

→∞

Proof. Fix ψ and ϕ satisfying the conditions of this lemma. Recall that by the integral identity (5.4.24) we have Q

ϕ(δ )∂t ψ + ϕ(δ )uδ · ∇ψ − ϕ (δ )δ − ϕ(δ ) div uδ dxdt ψϕ(∞ )U · n dΣ + (ψϕ(∞ ))(x, 0) dx = 0. (5.5.23) − Σin

Ω

In view of Lemma 5.5.2, the sequence δ converges to a.e. in Q. Since ϕ and ϕ are continuous, the sequences ϕ(δ ) and ϕ (δ )δ converge to ϕ() and ϕ (), respectively, a.e. in Q. On the other hand, it follows from (5.4.19) and the growth condition (5.5.22) imposed on ϕ that the sequences ϕ(δ ) and ϕ (δ )δ are bounded in Lγ (Q). By the Lebesgue dominated convergence theorem, these sequences converge to ϕ() and ϕ (), respectively, in L1 (Q). Letting δ → 0 in (5.5.23) we obtain the integral identity (5.5.21). We are now in a position to complete the proof of the main Theorem 5.1.6. It follows from Lemmas 5.5.4 and 5.5.5 that the functions (u, ) deﬁned by (5.5.1) satisfy the integral identities (5.1.13), (5.1.14) and meet all requirements of Deﬁnition 5.1.4 of a renormalized solution to problem (5.1.12). On the other hand, in view of Lemma 5.5.3, (u, ) satisfy the energy estimate (5.1.17). This completes the proof of Theorem 5.1.6.

Chapter 6

Pressure estimate In Chapter 5 we have proved the existence of renormalized solutions of problem (5.1.12) under the assumptions that the ﬂow domain, the given data U, ∞ , and the constitutive relation p = p() satisfy Condition 5.1.1. Moreover, it was also assumed that the adiabatic exponent γ in Condition 5.1.1 satisﬁes γ > 2d. In this case Theorem 5.1.6 gives the existence of a renormalized solution (u, ) satisfying the energy estimate (5.1.17). It follows from that estimate that the energy of (u, ) is bounded by a constant ce as in Remark 5.1.2, that is, depending only on diam Ω, UC 1 (Q) , ∞ L∞ ( T ) , and the constant cp which depends only on p(). In addition, Proposition 5.3.10 ensures that the pressure function p() is integrable with exponent (γ + 1)/γ > 1 on every compact subset of the cylinder Q. This local estimate plays a crucial role in the proof of the main Theorem 5.1.6. Unfortunately, this estimate involves the constant cE,p , which depends on higher order derivatives of the boundary data. In this chapter we show that for a suitable l = l(γ) > 1 and γ > d/2 the Ll -norm of the pressure function p() is bounded on every compact subset Q Q by a constant depending only on ce and Q . In Chapter 7 this result is applied in order to establish existence theory for the adiabatic exponent γ > d/2 and for fast oscillating boundary data. Problem 6.0.1. For given f ∈ C(Q), ﬁnd functions (u, ) satisfying the system of equations ∂t (u) + div( u ⊗ u) + ∇p(ρ) = div S(u) + f ∂t + div(u) = 0 in Ω.

in Q,

(6.0.1a) (6.0.1b)

We introduce the following condition on the pressure function p(). Condition 6.0.2. For the general constitutive law p = p() and the internal energy density

P () =

s−2 p(s) ds,

0

P. Plotnikov and J. Sokoáowski, Compressible Navier–Stokes Equations: Theory and Shape Optimization, Monografie Matematyczne 73, DOI 10.1007/978-3-0348-0367-0_6, © Springer Basel 2012

167

168

Chapter 6. Pressure estimate

the functions p, P : [0, ∞) → R+ are continuous and there are an exponent γ > 1 and a constant cp > 0 such that γ ≤ cp (P () + 1),

γ ≤ cp (p() + 1),

p() ≤ cp (P () + 1).

(6.0.2)

Proposition 6.0.3. Let Ω be a bounded domain with boundary of class C ∞ in Rd , d = 2, 3, and suppose the pressure function p satisﬁes Condition 6.0.2 with γ ∈ (d/2, ∞). Let ∈ L∞ (0, T ; Lγ (Ω)), u ∈ L2 (0, T ; W 1,2 (Ω)) be a renormalized solution to equations (6.0.1) such that p ∈ L∞ (0, T ; L1 (Ω)) ∩ Lrloc (Q), 1 < r < 2, and uL2 (0,T ;W 1,2 (Ω)) + |u|2 L∞ (0,T ;L1 (Ω)) + P ()L∞ (0,T ;L1 (Ω)) ≤ ce .

(6.0.3)

Then for any exponent θ from the interval 0 ≤ θ < min{2γ/d − 1, γ/2} and for any cylinder Q Q, there is a constant c, depending only on cp , ce , Q , γ and θ, such that p()θ dxdt ≤ c. (6.0.4) Q

Proof. First, ﬁx θ ∈ (0, min{2γ/d − 1, γ/2}) and K > 1. Set ⎧ θ ⎪ for 0 ≤ < K, ⎨ θ θ−1 2 2 ϕ() = K − θK { /2 − (1 + K) + K + K /2} for K ≤ ≤ K + 1, ⎪ ⎩ ϕ(K + 1) for > K + 1. The function ϕ is continuously diﬀerentiable, with the second order derivative piecewise continuous on (0, ∞). It is easily seen that ϕ() ≤ θ on this interval. Set ϕ := ϕ(). Since K > 1 is arbitrary, it suﬃces to prove that p()ϕ() dxdt ≤ c(ce , Q ), (6.0.5) Q

where c(ce , Q ) is independent of K. It follows from Deﬁnition 5.1.4 of renormalized solutions that the functions (u, , ϕ) satisfy the integral identities u · ∂t ξ + u ⊗ u : ∇ξ + p div ξ − S(u) : ∇ξ + f · ξ dxdt = 0, Q (∂t ψ + u · ∇ψ) dxdt = 0, Q ϕ∂t ψ + ϕu · ∇ψ + ψ ϕ() − ϕ () div u dxdt = 0, Q

Chapter 6. Pressure estimate

169

which hold for all vector ﬁelds ξ ∈ C0∞ (Q) and functions ψ ∈ C0∞ (Q). This means that (u, , ϕ) satisfy in the cylinder Q the equations ∂t (u) + div(u ⊗ u) = div T + f , ∂t + div(u) = 0,

(6.0.6)

∂t ϕ + div(ϕu) + = 0, where T = S(u) − p() I,

= ϕ() − ϕ () div u.

(6.0.7)

Next, it follows from (6.0.3) and (6.0.2) that TL1 (Q) ≤ ce ,

T ∈ Lr (Q ).

(6.0.8)

Moreover, the boundedness of ϕ and estimate (6.0.3) imply ϕ ∈ L∞ (Q),

∈ L2 (Q).

It follows that the functions (u, , ϕ), and tensors T satisfy Condition 4.5.1 with g = gϕ = 0, G = 0. Hence they meet all requirements of Theorem 4.5.2 with s = ∞, γ. Now ﬁx a cylinder Q Q and two nonnegative functions ξ ∈ C0∞ (Q) and ζ ∈ C0∞ (Ω) such that ξ = ζ = 1 on Q . Now, it is convenient to consider ζ ∈ C0∞ (Ω) as a function in C ∞ (Ω × R) which is constant with respect to the time variable. By Theorem 4.5.2 the functions (u, , ϕ) satisfy the integral identity (4.5.10), which in our case reads H · u + ζ(P + Q) · u + S dxdt. (6.0.9) ξϕR : [ζT] dxdt = Q

Q

Here, we denote H = R[ξϕ](ζu) − (ξϕ) R[ζu], S = (∇ζ ⊗ A[ξϕ]) : (u ⊗ u − T) + ζA[ξϕ] · (f ),

(6.0.10) (6.0.11)

P = A[ϕ∂t ξ + ϕu · ∇ξ],

(6.0.12)

Q = −A[ξ].

Using the identity R : [ζp() I] = ζp()I we obtain from (6.0.9) ξζϕp() dxdt = ξϕR : [ζS(u)] dxdt Q Q H · u + ζ(P + Q) · u + S dxdt. − Q

170

Chapter 6. Pressure estimate

Since ζ = ξ = 1 on Q , we have ϕp() dxdt ≤ |ξϕR : [ζS(u)]| dxdt Q Q + (|H · u| + |ζ|(|P| + |Q|)|u| + |S| dxdt.

(6.0.13)

Q

Let us estimate all terms on the right hand side. We have R : [ζS(u)]L2 (0,T ;L2 (Rd )) ≤ ζS(u)L2 (0,T ;L2 (Rd )) ≤ cS(u)L2 (0,T ;L2 (Ω)) ≤ c∇uL2 (0,T ;L2 (Ω)) ≤ c(ce , ζ). On the other hand, the inequality ϕ ≤ θ along with (6.0.2) and (6.0.3) implies ξϕL∞ (0,T ;Lγ/θ (Rd )) ≤ θ L∞ (0,T ;Lγ/θ (Ω)) ≤ θL∞ (0,T ;Lγ (Ω)) ≤ c(ce , ξ). Recalling that 1 ≥ 1/2 + θ/γ and applying the Hölder inequality we arrive at ξϕR : [ζS(u)]L1 (Q) ≤ cR : [ζS(u)]L2 (0,T ;L2 (Rd )) ξϕL∞ (0,T ;Lγ/θ (Rd )) ≤ c(ce , ξ, ζ). (6.0.14) Next, it follows from the Hölder inequality that u · HL1 (Q) ≤ cuL2 (0,T ;Lm (Ω)) ζuL2 (0,T ;Ln (Ω)) R[ξϕ]L∞ (0,T ;Lκ (Ω)) + cuL2 (0,T ;Lm (Ω)) R[ζu]L2 (0,T ;Ln (Ω)) ξϕL∞ (0,T ;Lκ (Ω)) for all exponents satisfying n, m, κ ∈ (1, ∞),

n−1 + m−1 + κ−1 ≤ 1.

By the boundedness of the operator R in Lebesgue spaces on Rd , we have R[ξϕ]L∞ (0,T ;Lκ (Rd )) ≤ cξϕL∞ (0,T ;Lκ (Rd )) ≤ cϕL∞ (0,T ;Lκ (Ω)) and R[ζu]L2 (0,T ;Ln (Rd )) ≤ cζuL2 (0,T ;Ln (Rd )) ≤ cuL2 (0,T ;Ln (Ω)) . Thus we get u · HL1 (Q) ≤ cuL2 (0,T ;Lm (Ω)) uL2 (0,T ;Ln (Ω)) ϕL∞ (0,T ;Lκ (Ω)) . Now set κ = γ/θ, which gives ϕκ ≤ γ , hence (by (6.0.2) and (6.0.3)) ϕL∞ (0,T ;Lκ (Ω)) ≤ ce .

(6.0.15)

Chapter 6. Pressure estimate

171

It follows that u · HL1 (Q) ≤ cuL2 (0,T ;Lm (Ω)) uL2 (0,T ;Ln (Ω)) for

n−1 + m−1 ≤ 1 − θ/γ.

n, m ∈ (1, ∞),

(6.0.16) (6.0.17)

By the energy estimate (6.0.3) the functions (u, ) meet all requirements of Proposition 4.2.1. Applying Corollary 4.2.2 of that proposition we conclude that uL2 (0,T ;Lm (Ω)) ≤ ce

if m−1 > 2−1 + γ −1 − d−1 .

(6.0.18)

On the other hand, the embedding theorem implies uL2 (0,T ;Ln (Ω)) ≤ cuL2 (0,T ;W 1,2 (Ω)) ≤ ce

if n−1 > 2−1 − d−1 .

(6.0.19)

Since θ/γ < 2d−1 − γ −1 , we have (2−1 + γ −1 − d−1 ) + (2−1 − d−1 ) = 1 + γ −1 − 2d−1 < 1 − θ/γ. Hence there exist m, n ∈ (1, ∞) satisfying m−1 > 2−1 + γ −1 − d−1 ,

n−1 > 2−1 − d−1 ,

and

n−1 + m−1 ≤ 1 − θ/γ.

For such m and n substituting (6.0.18)–(6.0.19) into (6.0.16) leads to u · HL1 (Q) ≤ c(ce , ζ, ξ).

(6.0.20)

Let us estimate the vector ﬁeld P. For any σ ∈ [1, ∞) we have ϕ∂t ξ + ϕu · ∇ξL2 (0,T ;Lσ (Ω)) ≤ c(ξ)ϕL∞ (0,T ;Lσ (Ω)) + c(ξ)ϕuL2 (0,T ;Lσ (Ω)) . Moreover, by the Hölder inequality we have ϕuL2 (0,T ;Lσ (Ω)) ≤ c(Ω)ϕL∞ (0,T ;Ln (Ω)) uL2 (0,T ;Lm (Ω))

(6.0.21)

for all exponents m and n satisfying n−1 + m−1 ≤ σ −1 ,

m, n ∈ [1, ∞].

Set n = γ/θ and note that for such n we have ϕn ≤ γ . Thus we get ϕL∞ (0,T ;Ln (Ω)) ≤ γ θL∞ (0,T ;L1 (Ω)) ≤ ce , hence ϕuL2 (0,T ;Lσ (Ω)) ≤ ce uL2 (0,T ;Lm (Ω)) −1

−1

for any exponent m ∈ (1, ∞) satisfying m ≤ σ − θγ for m−1 > 2−1 − d−1 the embedding theorem implies

−1

(6.0.22) . On the other hand,

uL2 (0,T ;Lm (Ω)) ≤ cuL2 (0,T ;W 1,2 (Ω)) ≤ ce .

(6.0.23)

172

Chapter 6. Pressure estimate

Combining (6.0.21)–(6.0.23) yields ϕuL2 (0,T ;Lσ (Ω)) ≤ ce

(6.0.24)

for all σ such that there exists m with 2−1 − d−1 < m−1 < σ −1 − θγ −1 . In other words, inequality (6.0.24) holds for all σ such that σ ∈ (1, ∞),

σ −1 > θγ −1 + 2−1 − d−1 .

(6.0.25)

Notice that σ < γθ−1 , which yields θ/γ

ϕL∞ (0,T ;Lσ (Ω)) ≤ cϕL∞ (0,T ;Lγ/θ (Ω)) ≤ ce γ L∞ (0,T ;L1 (Ω)) ≤ ce .

(6.0.26)

In view of (6.0.24) and (6.0.26) we get ϕ∂t ξ + ϕu · ∇ξL2 (0,T ;Lσ (Ω)) ≤ c(ξ)ce

when σ −1 > θγ −1 + 2−1 − d−1 . (6.0.27)

Since ξ is compactly supported in Q, we have A[ϕ∂t ξ + ϕu · ∇ξ]L2 (0,T ;W 1,σ (Ω)) ≤ c(ξ)ce . Recalling that the embedding W 1,σ (Ω) → Lι (Ω) is bounded if ι−1 > σ −1 − d−1 , we deduce that PL2 (0,T ;Lι (Ω)) = A[ϕ∂t ξ + ϕu · ∇ξ]L2 (0,T ;Lι (Ω)) ≤ c(ξ)ce

(6.0.28)

for all ι with σ −1 − d−1 < ι−1 ≤ 1, where σ is an arbitrary number satisfying (6.0.25). From this and (6.0.25) we conclude that (6.0.28) holds for all exponents ι satisfying (6.0.29) 2−1 + θγ −1 − 2d−1 < ι−1 ≤ 1. Next, the inequality θγ −1 < 2d−1 − γ −1 yields 2−1 + γ −1 − d−1 < 1 − (2−1 + θγ −1 − 2d−1 ).

(6.0.30)

It follows that there are m, ι ∈ (1, ∞) such that 2−1 + γ −1 − d−1 < m−1 < 1 − ι−1

and

ι > 2−1 + θγ −1 − 2d−1 .

For such m and ι, the Hölder inequality along with (6.0.18) and (6.0.28) implies the desired estimate for P: ζu · PL1 (Q) ≤ c(ξ, ζ)uL2 (0,T ;Lm (Ω)) PL2 (0,T ;Lι (Ω)) ≤ c(ξ, ζ)ce uL2 (0,T ;Lm (Ω)) ≤ c(ξ, ζ)ce .

(6.0.31)

Chapter 6. Pressure estimate

173

We use similar arguments to estimate Q. Notice that || ≤ cθ | div u|. By the Hölder inequality, we have ξL2 (0,T ;Lr (Ω)) ≤ c(ξ)∇uL2 (0,T ;L2 (Ω)) θ L∞ (0,T ;Lγ/θ (Ω)) = c(ξ)∇uL2 (0,T ;L2 (Ω)) θL∞ (0,T ;Lγ (Ω)) ≤ c(ξ)ce for τ −1 = θγ −1 + 2−1 . Notice that τ ∈ (1, ∞) since θγ −1 < 2−1 . Since ξ is compactly supported in Q, we get QL2 (0,T ;W 1,τ (Ω)) = A[ξ]L2 (0,T ;W 1,τ (Ω)) ≤ c(ξ)ce . Recall that the embedding W 1,τ (Ω) → Lω (Ω) is bounded if ω −1 > τ −1 − d−1 . Thus we get QL2 (0,T ;Lω (Ω)) ≤ c(ξ)ce

if ω −1 > θγ −1 + 2−1 − d−1 , ω ≥ 1.

(6.0.32)

Next, it follows from (6.0.30) that 2−1 + γ −1 − d−1 < 1 − (θγ −1 + 2−1 − d−1 ). Hence there are m, ω ∈ (1, ∞) with 2−1 + γ −1 − d−1 < m−1 = 1 − ω −1 ,

ω −1 > θγ −1 + 2−1 − d−1 .

For such m and ω, the Hölder inequality along with (6.0.18) and (6.0.32) implies the estimate for Q: ζu · QL1 (Q) ≤ c(Ω, ξ, ζ)uL2 (0,T ;Lm (Ω)) QL2 (0,T ;Lω (Ω)) ≤ c(ξ, ζ)ce uL2 (0,T ;Lm (Ω)) ≤ c(ξ, ζ)ce .

(6.0.33)

It remains to estimate S. First, we observe that ξϕ is compactly supported in Q and satisﬁes θ/γ ξϕL∞ (0,T ;Lγ/θ (Ω)) ≤ cγ L∞ (0,T ;L1 (Ω)) ≤ ce , hence A[ξϕ]L∞ (0,T ;W 1,γ/θ (Ω)) ≤ c(ξ)ce . Since γ/θ > d, the embedding W 1,γ/θ (Ω) → C(Ω) is bounded, whence A[ξϕ]L∞ (Q) ≤ c(ξ)ce . It follows that SL1 (Q) ≤ c(ξ, ζ)ce (|u|2 L1 (Q) + TL1 (Q) + L1 (Q) + 1) ≤ c(ξ, ζ)ce . Inserting this estimate along with (6.0.20), (6.0.31), and (6.0.33) into (6.0.13) and noting that ζ, ξ depend only on Q we obtain (6.0.5), and the proposition follows.

Chapter 7

Kinetic theory. Fast density oscillations 7.1 Problem formulation. Main results In Chapter 5 the existence of renormalized solutions to problem (5.1.12) is proved under restrictive regularity assumptions on the ﬂow domain, the given data U, ∞ , and the constitutive relation p = p(). In particular, the adiabatic exponent γ satisﬁes γ > 2d. The restrictions are used mainly in the proof of the existence of strong solutions to the regularized equations and to justify the limit passages when the regularization parameters go to zero. As the existence of a renormalized solution is proved, the restrictions on the data and the constitutive relations should be weakened. It is most important to relax the restriction γ > 2d and extend the existence theory to the range γ > d/2, common for homogeneous boundary value problems. Another problem is to extend the solvability of the nonhomogeneous problem to nonsmooth densities ∞ . These problems are closely related to the question of compactness of the set of solutions to compressible Navier-Stokes equations corresponding to various pressure functions and boundary data. Indeed, we can approximate the pressure function p() with a fast growing artiﬁcial pressure pε and next approximate the bounded function ∞ with a sequence of smooth functions ∞ , ∈ (0, 1]. Then we can apply Theorem 5.1.6 to obtain a sequence of solutions (u , ) to problem (5.1.12) with p replaced by p and ∞ replaced by ∞ . By estimate (5.1.17) in Theorem 5.1.6 the solutions (u , ) have uniformly bounded energies. Hence, passing to a subsequence if necessary, we can assume that (u , ) converge weakly in the energy space to some limit functions (u, ). Our goal now is to derive equations for the limit (u, ). We can expect that the equations depend on the approximation methods. Strongly convergent approximation should lead to Navier-Stokes equations, but weakly convergent approximation

P. Plotnikov and J. Sokoáowski, Compressible Navier–Stokes Equations: Theory and Shape Optimization, Monografie Matematyczne 73, DOI 10.1007/978-3-0348-0367-0_7, © Springer Basel 2012

175

176

Chapter 7. Kinetic theory. Fast density oscillations

may lead to some other equations. In the case of weakly convergent approximation we encounter the problem of fast oscillating data arising in nonlinear acoustics. In the most general case such data appear as weak limits of sequences of smooth data. To give a rigorous meaning to the above discussion we formulate explicit assumptions on the artiﬁcial pressure function p(), its approximations p , and the boundary data: the densities ∞ and the velocity U. Condition 7.1.1. The function p : [0, ∞) → R+ has continuous ﬁrst derivative and p () ≥ 0,

p(0) = p (0) = 0.

(7.1.1)

There are an exponent γ > d/2 and a constant cp > 0 such that for ∈ [0, ∞), γ ≤ cp (P () + 1),

γ ≤ cp (p() + 1),

p() ≤ cp (P () + 1),

(7.1.2)

and for ∈ [1, ∞), γ γ c−1 p ≤ p() ≤ cp ,

γ−1 c−1 ≤ p () ≤ cp γ−1 , p

p(1) ≤ cp .

(7.1.3)

In particular, (7.1.4) p() ≤ cp (γ + 1). - −1 Therefore, the energy density P = 0 s p(s) ds is twice continuously diﬀerentiable on (0, ∞), but P () can develop a singularity at = 0. Condition 7.1.2. form

• We take the approximation of the pressure function in the p () = χ p() + (2 + n ), ∈ (0, 1], (7.1.5)

where n > max{γ, 4d} and χ ∈ C ∞ (R) is a nondecreasing function such that χ(s) = 0 for s ≤ 1/2,

χ(s) = 1

for s ≥ 1.

(7.1.6)

• The function U ∈ C ∞ (Rd × (0, T )) and the C ∞ boundary ∂Ω satisfy the geometric condition (5.1.9) in Condition 5.1.1. ∞ • The nonnegative function ∞ ∈ C (Q), ∈ (0, 1], vanishes in a neighborhood of the edge ∂Ω × {0}.

• The functions ∞ are uniformly bounded on the parabolic boundary ST ∪ (cl Ω × {t = 0}) of the cylinder Q and ∞ ∞ ∈ L∞ (

T)

weakly in L∞ (

T)

as → 0.

T

=

(7.1.7)

The weak convergence in (7.1.7) implies fast oscillations of solutions in the model. We denote by ce , as in Remark 5.1.2, generic constants depending only on λ,

γ,

diam Ω,

T,

∞ L∞ (

T)

,

UC 1 (Q) ,

f C(Q) ,

cp .

7.1. Problem formulation. Main results

177

Regularized problem. Let us consider the regularized boundary value problem ∂t ( u ) + div( u ⊗ u ) + ∇p ( ) = div S(u ) + f ∂t + div( u ) = 0 in Q, u = U on ST , = ∞ on Σin , u |t=0 = U, |t=0 = ∞ in Ω.

in Q,

(7.1.8a) (7.1.8b) (7.1.8c)

The following result is a direct consequence of Theorem 5.1.6: Theorem 7.1.3. Let Condition 7.1.2 be satisﬁed. Then problem (7.1.8) has a renormalized solution which satisﬁes the estimate u L2 (0,T ;W 1,2 (Ω)) + |u |2 L∞ (0,T ;L1 (Ω)) + P ( )L∞ (0,T ;L1 (Ω)) ≤ ce , (7.1.9) where ce is independent of . For any θ, 0 < θ < min{2γd−1 − 1, 2−1 γ} and a compact set Q Q, there is a constant ce (Q ) independent of such that p ( )θ dxdt ≤ ce (Q ), (γ+θ + εn+θ ) dxdt ≤ ce (Q ). (7.1.10) Q

Q

Proof. It suﬃces to check that Ω, p , U and ∞ satisfy Condition 5.1.1 with (υ, γ) replaced by (γ, n). It follows from Condition 7.1.2 that Ω, U, and ∞ obviously satisfy Condition 5.1.1. It remains to check that the pressure function p( ) and the internal energy density 1 2 1 n s −1 −1 p(s) ds + + . (7.1.11) s p(s) ds = s χ P := 2 n 0 0 satisfy (5.1.4)–(5.1.8). First we show that p and P satisfy (5.1.4)–(5.1.7). Since p is continuously diﬀerentiable and χ(s/) vanishes on [0, /2], the function P is twice continuously diﬀerentiable on [0, ∞). Since all functions in representations (7.1.5) and (7.1.11) are nonnegative and monotone, we have n ≤ −1 p (),

n ≤ n−1 P ().

(7.1.12)

Notice that p are uniformly bounded on [0, 1]. Next, (7.1.3) yields, for ≥ 1, p () ≤ cp γ−1 + (2 + nn−1 ) ≤ c()n−1 . (7.1.13) This leads to the inequality p () ≤ c()(n + 1), which along with (7.1.12) implies p () ≤ cp γ + (2 + n ) ≤ c()n ,

p () ≤ c()(P () + 1).

(7.1.14)

Finally note that P () = −1 p ≥ 2. It follows from (7.1.12)–(7.1.14) that pε and P satisfy the desired relations (5.1.5) with γ replaced by n and CE replaced by a constant c(), depending on cp and .

178

Chapter 7. Kinetic theory. Fast density oscillations

It remains to show that p and P satisfy (5.1.8). We ﬁrst observe that (7.1.11) can be rewritten in the form 1 1 s s−1 1 − χ P () = P () − P∗ () + 2 + n , P∗ () = p(s) ds. 2 n 0 (7.1.15) It is easy to check that 0 ≤ P∗ () ≤ s−1 p(s) ds = η(), where η() → 0 as → 0. (7.1.16) 0

Now (7.1.2) yields γ −1 −1 −1 γ cp − C, P () ≥ P () − P∗ () ≥ c−1 p − cp − η(1) ≥ 2

(7.1.17)

where C depends only on cp and γ. Hence γ ≤ ce (P + 1) for some constant ce depending only on cp and γ. Next, in view of (7.1.4) we have p () ≤ p() + (2 + n ) ≤ p() + nP () ≤ cp γ + cp + nPε (). Combining this with (7.1.17) we arrive at p () ≤ (2c2p + n)P () + C, where C depends only on cp and γ. Hence p ≤ ce (P + 1)

(7.1.18)

for some constant ce . Recall that χ(/) − 1 vanishes for ≥ 1, which yields (1 − χ(/))p() ≤ cp . Applying (7.1.2) we obtain γ p () ≥ p() − (1 − χ(/))p() ≥ c−1 p − 1 − cp .

(7.1.19)

Hence γ ≤ ce (p + 1) for a suitable constant ce . Now we assemble the results obtained and arrive at γ ≤ ce (P + 1),

γ ≤ ce (p + 1),

p ≤ ce (P + 1) for ≥ 0.

(7.1.20)

Therefore, p and P satisfy (5.1.8) with υ and cp replaced by γ and ce . It follows that Ω, U, ∞ , and p satisfy Condition 5.1.1 with exponents (υ, γ) with υ ∈ (1, γ) replaced by (γ, n) with γ ∈ (1, n). Thus we can apply Theorem 5.1.6 to conclude that problem (7.1.8) has a renormalized solution (u , ) which satisﬁes estimate (7.1.9) with a constant ce depending only on diam Ω, UC 1 (Q) , the constant cp in Condition 7.1.1, and ∞ L∞ ( T ) . Next, notice that in view of (7.1.20), the function p satisﬁes Condition 6.0.2. It then follows from estimate (7.1.9) that the solution (u , ) meets all requirements of Proposition 6.0.3 with p replaced by p and cp replaced by ce . Hence p ( )θ dxdt ≤ ce (Q ), (7.1.21) Q

7.1. Problem formulation. Main results

179

which is exactly the ﬁrst inequality in (7.1.10). In order to prove the second, notice that (7.1.19) and the inequality p () ≥ n imply cp γ + n ≤ 2p ( ) + cp . Since θ ≤ γ, estimate (7.1.9) implies θ γ dxdt ≤ ce (Q ) ( + 1) dxdt ≤ ce (Q ) (P + 1) dxdt ≤ ce (Q ), Q

Q

Q

which gives γ+θ n+θ (cp + ) dxdt ≤ 2 Q

Q

p θ dxdt

+ cp

θ dxdt ≤ ce (Q ).

Q

Together with (7.1.21), this leads to the second estimate in (7.1.10).

Let us consider the sequence of solutions to problem (7.1.8) deﬁned by Theorem 7.1.3. Passing to a subsequence we can assume that there exist u ∈ L2 (0, T ; W 1,2 (Ω)) and ∈ L∞ (0, T ; Lγ (Ω)) such that u u weakly in L2 (0, T ; W 1,2 (Ω)), weakly in L∞ (0, T ; Lγ (Ω)), γ+θ

weakly in L

(7.1.22)

(Q ) for all Q Q.

Notice that in Condition 7.1.2 it is assumed that the sequence of boundary and initial data ε∞ is only weakly convergent, and it may be rapidly oscillating as → 0. The well known example of a rapidly oscillating sequence is x t , (7.1.23) ∞ = R x, t, , where R(x, t, y, τ ) is a smooth function periodic in y and τ . The theory of weak convergence for such sequences is called two-scale convergence theory. In view of the mass transport equation (7.1.8c), oscillations in boundary and initial data are transferred inside the ﬂow domain along ﬂuid particle trajectories. Hence we can expect that these oscillations induce rapid oscillations of the density in the ﬂow domain, and the propagation of such oscillations will now be under discussion. It is worth noting that density oscillations can be regarded as sound waves, studied in acoustics. Rapid oscillations appear if the wavelength of the sound is small compared to the diameter of the ﬂow domain. In that case, the sound waves can be considered in the same way as light rays and wave fronts in optics. For example, in classical linear acoustics, sound propagation is described by the wave equation ∂t2 − c2 Δ = 0.

(7.1.24)

180

Chapter 7. Kinetic theory. Fast density oscillations

The passage from the wave equation to geometric acoustics can be performed if approximate solutions of the wave equation (7.1.24) in the form = A(x, t)e s(t,x,) i

(7.1.25)

are considered. Substitution of this formula into (7.1.24) and further formal asymptotic analysis leads to the Hamilton-Jacobi equation for the phase function s which describes the position of the wave front (see e.g. [20] for details). This discussion shows that rapid oscillations appear in a natural way in modeling a viscous gas motion in an acoustic ﬁeld. However, the method of asymptotic expansions in the linear theory of partial diﬀerential equations is diﬃcult to apply to nonlinear problems of viscous gas dynamics. The diﬃculty is the lack of information on smoothness properties of solutions to compressible Navier-Stokes equations. It seems that Bakhvalov and Eglit have been the ﬁrst to consider the problem of averaged equations of the one-dimensional motion of a viscous compressible medium with rapidly oscillating boundary and initial data. In [8] new equations of motion are formally derived. It is discovered that the averaging description for this problem cannot be given in terms of the averaging characteristics and the resulting new equations have a substantially diﬀerent form compared to the original equations. These results have been rigorously justiﬁed and generalized by Amosov and Zlotnik [5], [6]. Notice that all those results only apply to the one-dimensional case and with data functions of the form (7.1.23). In this chapter another approach based on Young measure theory is developed. It follows from relations (7.1.22) and the fundamental theorem on Young measures (Theorem 1.4.5) that there exist subsequences, still denoted by ε , ∞ , ∞ ∈ L∞ and Young measures μ ∈ L∞ w (Q; M(R)), μ w ( T ; M(R)), with the following properties: For any continuous function ϕ ∈ C0 (R) we have ϕ( ) ϕ ϕ(∞ ) ϕ∞

weakly in L∞ (Q), weakly in L∞ ( T ),

where ϕ(x, t) = μxt , ϕ,

ϕ∞ (x, t) = μ∞ xt , ϕ.

(7.1.26a) (7.1.26b)

(7.1.26c)

In this framework rapidly oscillating sequences are associated with some Young measures. At this point it is worth noting that a Young measure determines a random function (depending on the spatial variables and the time variable), according to the following deﬁnition ([40]): Deﬁnition 7.1.4. Let (E, A, π) be a probability space, i.e., A is a σ-algebra on the set E and π : A → R is a probability measure, π(E) = 1. A random variable is a Borel map ς : E → R. Recall that ς is Borel if ς −1 (B) ∈ A for any Borel set B ⊂ R. The probability distribution of the random variable ς is the probability measure μ on the real line deﬁned by μ(B) = π(ς −1 (B)) for all Borel sets B ⊂ R.

7.1. Problem formulation. Main results

181

The distribution function (cumulative distribution function) of ς is deﬁned by f (λ) = μ(−∞, λ], λ ∈ R. A family of random variables ςxt labeled by points (x, t) ∈ Q is called a random function (random ﬁeld) on Q. Therefore, we can consider rapidly oscillating sequences and ∞ as random functions on the cylinder Q and on the surface T with the associated probability distributions μxt and μ∞ xt , respectively. Introduce the corresponding cumulative distribution functions f (x, t, s) = μxt (−∞, s],

f∞ (x, t, s) = μ∞ xt (−∞, s].

(7.1.27)

For a.e. (x, t) ∈ Q (resp. (x, t) ∈ T ), the functions f (x, t, s) and f∞ (x, t, s) are monotone and right continuous in s, tend to 1 as s → ∞ and vanish for s < 0. In accordance with Example 1.3.6, for any ϕ ∈ C0 (R) integrals with respect to Young measures can be written in the form of Stieltjes integrals (see Section 1.3.1 for deﬁnition): ∞ μxt , ϕ = ϕ(s) ds f (x, t, s), μxt , ϕ = ϕ(s) ds f∞ (x, t, s). (7.1.28) R

R

In particular, the weak limits and p of the densities and pressure functions p( ) are determined as the expectation values = s ds f (x, t, s), p = p(s) ds f (x, t, s). (7.1.29) R

R

In other words the density and the pressure can be considered as random functions on Q with the common probability measure μxt . Now, the main problem is to derive the so-called kinetic equation for the cumulative probability. In this way the system of governing equations along with boundary conditions for the limit velocity ﬁeld u and the distribution function f can be obtained. To make the presentation clear, ﬁrst, the kinetic equation is postulated, and then rigorously derived. Problem 7.1.5 (Kinetic problem). For given U ∈ C ∞ (Q), f ∈ C(Q) and a distribution function f∞ : T × R → [0, 1], ﬁnd a velocity ﬁeld u ∈ L2 (0, T ; W 1,2 (Ω)) and a distribution function f : Q × R → [0, 1] such that: (i) f is measurable and satisﬁes f (x, t, s) = lim f (x, t, s + h), h 0

f (x, t, s ) ≤ f (x, t, s )

for s ≤ s , (7.1.30)

f (x, t, s) = 0 for s < 0, lim f (x, t, s) = 1, s→∞ γ |s| ds f (x, t, s) dxdt < ∞. Q

R

(7.1.31) (7.1.32)

182

Chapter 7. Kinetic theory. Fast density oscillations

(ii) u and f satisfy the following equations and boundary conditions: ∂t (u) + div(u ⊗ u) + ∇p = div S(u) + f in Q, ∂t f + div(f u) − ∂s (sf div u + s C[f ]) = 0 in Q × R, u = U on T , f = f∞ on Σin × R,

f (x, 0, s) = f∞ (x, 0, s)

on Ω × R.

Here the nonlinear operator C[f ] is deﬁned by 1 (p(τ ) − p) dτ f (x, t, τ ), C[f ] = λ + 1 (−∞,s]

(7.1.33a) (7.1.33b) (7.1.33c) (7.1.33d)

(7.1.33e)

and the functions and p are given by (7.1.29). Since 0 ≤ p ≤ c(1 + γ ), it follows from (7.1.32) that p ∈ L1 (Q). Remark 7.1.6. The equality f (x, t, s) = 0 on (−∞, 0) simply means that the corresponding probability measure vanishes on this interval, i.e. μxt (−∞, 0) = 0. Therefore if a function ϕ is integrable with respect to the measure μxt = ds f (x, t, s), then ϕ ds f (x, t, s) = ϕ ds f (x, t, s). (7.1.34) R

[0,∞)

In particular, =

s ds f (x, t, s), p = p(s) ds f (x, t, s), [0,∞) [0,∞) 1 C[f ](x, t, s) = (p(τ ) − p(x, t)) dτ f (x, t, τ ) for s ≥ 0, λ + 1 [0,s]

C[f ](x, t, s) = 0

(7.1.35)

for s < 0.

Remark 7.1.7. Since 0 ≤ ∞ ≤ ce , it follows from Theorem 1.4.5 that supp μ∞ xt ⊂ [0, ce ]. Therefore, the distribution function f∞ has the properties f∞ (x, t, s) = 0 for s < 0, f∞ (x, t, s) = 1 for s > ce , f∞ (x, t, s) is monotone and right continuous in s.

(7.1.36) (7.1.37)

Further we will consider solutions to Problem 7.1.5 satisfying additional regularity requirements concerning the behavior of the distribution function at inﬁnity. To this end, the notion of regular solution to the kinetic problem is introduced. Deﬁnition 7.1.8. A solution (u, f ) to Problem 7.1.5 is said to be regular if HL1+γ (Q) + sup Vv L1 (Q) < ∞

(7.1.38)

v∈R+

where

min{s, v}(p(s) − p) ds f (x, t, s),

Vv (x, t) = [0,∞)

(7.1.39) f (x, t, s)(1 − f (x, t, s)) ds.

H(x, t) = [0,∞)

7.1. Problem formulation. Main results

183

Results: Convergence of fast oscillating ﬂow to a solution of the kinetic problem. The following theorem which is the ﬁrst main result of this section shows that the vector ﬁeld u and the distribution function f deﬁned by (7.1.26) and (7.1.27) are a regular weak solution to Problem 7.1.5. Theorem 7.1.9. Assume that the pressure function p, the functions U, ∞ , and the domain Ω satisfy Conditions 7.1.1 and 7.1.2. Furthermore assume that (u , ) are solutions to problem (7.1.8) deﬁned by Theorem 7.1.3 and u, f , f∞ are deﬁned by (7.1.22), (7.1.26), and (7.1.27). Then: (i) The distribution function f satisﬁes conditions (7.1.30)–(7.1.31). There is a constant ce , depending only on diam Ω, UC 1 (Q) , the function p(), the time period T , and ϕ∞ L∞ ( T ) , such that sγ ds f (x, t, s) dx ≤ ce . (7.1.40) ess sup t∈(0,T )

Ω

[0,∞)

Moreover, for any Q Q and 0 < θ < min{2γd−1 − 1, γ/2}, there is a constant c depending only on ce , Q and θ such that sγ+θ ds f (x, t, s) dx ≤ c. (7.1.41) Q

[0,∞)

(ii) The integral identity for momentum balance u · ∂t ξ + u ⊗ u : ∇ξ + p div ξ − S(u) : ∇ξ dxdt Q + f · ξ dxdt + (∞ U · ξ)(x, 0) dx = 0 (7.1.42) Q

Ω

∞

holds for all vector ﬁelds ξ ∈ C (Q) equal to 0 in a neighborhood of the lateral boundary ST and of the top Ω × {t = T }. Here the functions and p are deﬁned by (7.1.29). (iii) The integral identity for the distribution function f ∂t ψ + f ∇x ψ · u − f s∂s ψ div u − s∂s ψ C[f ] dsdxdt Q R + f∞ (x, 0, s)ψ(x, 0, s) dsdx − f∞ (x, t, s)ψ(x, t, s)U · n dsdΣ = 0 Ω

R

Σin

R

(7.1.43) holds for all ψ ∈ C ∞ (Q × R) which satisfy the conditions ψ = 0 in a neighborhood of (ST \ Σin ) × R and of Ω × {t = T } × R, ψ(x, t, s) = 0 for all large |s| uniformly in (x, t).

(7.1.44)

The functions C[f ] and f∞ are deﬁned by (7.1.33e), (7.1.27), respectively.

184

Chapter 7. Kinetic theory. Fast density oscillations

(iv) Assume, in addition, that the pressure function admits the representation p() = pc () + pb () where pc : [0, ∞) → R is convex and pb is uniformly bounded. Then the function f satisﬁes condition (7.1.38), i.e., f is a regular solution to the kinetic equation. For the proof, see Section 7.2. Results: Main theorem on the kinetic equation. Deterministic data. The theory of the kinetic equation (7.1.33b) is of independent interest aside from the theory of Navier-Stokes equations itself. The following theorem on kinetic equations with deterministic data, which is the second main result of this chapter, makes it possible, among other things, to prove compactness properties of solutions to compressible Navier-Stokes equations and to investigate the domain dependence of solutions to these equations. Before formulating the result we give the following Deﬁnition 7.1.10. The random function μxt is deterministic if for a.e. (x, t) the probability measure μxt is concentrated at a single point. This means that there is a measurable function : Q → R such that the distribution function f (x, t, s) = 0 for s < (x, t) and f (x, t, s) = 1 for s ≥ (x, t). Let us consider the boundary value problem for the kinetic equation ∂t f + div(f u) − ∂s (sf div u + s C[f ]) = 0 in Q × R, f = f∞ on Σin × R, f (x, 0, s) = f∞ (x, 0, s) on Ω × R . Here the nonlinear operator C is deﬁned by 1 C[f ] = (p(τ ) − p) dτ f (x, t, τ ), λ + 1 (−∞,s]

(7.1.45a) (7.1.45b)

p= R

p(s) ds f (x, t, s).

(7.1.45c)

We emphasize that here u is a given vector ﬁeld that has nothing to do with Navier-Stokes equations. The next question concerns the deterministic case. If the boundary data f∞ is deterministic, is a solution to Problem 7.1.5 deterministic? This question is important because if f is deterministic, then obviously p = p() and a solution of the kinetic equation becomes a weak renormalized solution of the mass balance equation. The following theorem which is the main result of this chapter gives a positive answer to this question. Our goal is to derive conditions under which any solution to problem (7.1.45) with deterministic boundary and initial distribution function f∞ is deterministic. Throughout this chapter we assume that the domain Ω, the given vector ﬁeld u, and f , f∞ satisfy the following conditions: Condition 7.1.11. • Ω ⊂ Rd is a bounded domain with C ∞ boundary. The vector ﬁeld u ∈ L2 (0, T ; W 1,2 (Ω)) satisﬁes the following boundary condition on the parabolic boundary T = ST ∪ (cl Ω × {t = 0}): u = U on

T.

(7.1.46)

7.2. Proof of Theorem 7.1.9

185

• The lateral surface ST = ∂Ω × (0, T ) and the function U ∈ C ∞ (Rd × (0, T )) satisfy the geometric condition (5.1.9). • The distribution functions f ∈ L∞ (Q × R) and f∞ ∈ L∞ ( T × R) are monotone and right continuous in s for a.e. (x, t) in Q and T , respectively. This means that f (x, t, s ) ≤ f (x, t, s ) for s ≤ s ,

lim f (x, t, s + h) = f (x, t, s),

h 0

and the same condition holds for f∞ . Moreover, f (x, t, s) → 1, f∞ (x, t, s) → 1 as s → ∞, f (x, t, s) = 0, f∞ (x, t, s) = 0 for s < 0. • p : R+ → R+ satisﬁes Condition 7.1.1 with γ > d/2, and there is c > 0 such that sγ ds f (x, t, s) dx < c < ∞, (7.1.47) ess sup t∈(0,T )

Ω

[0,∞)

• There is θ > 0 such that for every Q Q, sγ+θ ds f (x, t, s) dx < c(Q ) < ∞. Q

[0,∞)

• There is a positive ∞ ∈ C 1 (Rd × (0, T )) such that for a.e. (x, t) ∈ f∞ (x, t, s) = 0 for s < ∞ (x, t), f∞ (x, t, s) = 1 for s ≥ ∞ (x, t).

T,

(7.1.48)

Theorem 7.1.12. Let Condition 7.1.11 be satisﬁed. Furthermore assume that f satisﬁes equations (7.1.45) and inequality (7.1.38), i.e., f is a regular solution to problem (7.1.45). Then there is ∈ L∞ (0, T ; Lγ (Ω)) such that for a.e. (x, t) ∈ Q, f (x, t, s) = 0

for s < (x, t),

f (x, t, s) = 1

for s ≥ (x, t).

(7.1.49)

7.2 Proof of Theorem 7.1.9 Proof of claim (i). We split the proof into a sequence of lemmas. Lemma 7.2.1. Under the assumptions of Theorem 7.1.9, ψ(x, t) sγ ds f (x, t, s) dxdt ≤ lim sup ψ(x, t)( )γ dxdt, (7.2.1) →0 Q [0,∞) Q ψ(x, t) sθ+γ ds f (x, t, s) dxdt ≤ lim sup ψ(x, t)( )θ+γ dxdt, Q

[0,∞)

→0

Q

(7.2.2)

186

Chapter 7. Kinetic theory. Fast density oscillations

for all nonnegative ψ ∈ L∞ (Q) and all Q Q. Moreover for all ψ ∈ L∞ (Q ), pψ dxdt = ψ(x, t) p(s) ds f (x, t, s) dxdt Q Q [0,∞) ψ(x, t)p( ) dxdt. (7.2.3) = lim →0

Q

Proof. Consider the nonnegative Carathéodory function F (x, t, s) = ψ(x, t)|s|γ . It follows from estimate (7.1.9) and the inequality sγ ≤ cp (1 + P (s)) that F (x, t, ) dxdt ≤ ce ψL∞ (Q) (1 + P ( )) dxdt ≤ ce ψL∞ (Q) . Q

Q

Hence the integrals of F (x, t, ) are uniformly bounded. Recall that f is the distribution function of the Young measure μxt associated with the sequence , i.e., dμxt (s) = ds f (x, t, s). Hence we can apply Theorem 1.4.5 with Ω replaced by Q and vn replaced by to obtain (7.2.1). Next consider the integrand F (x, t, s) = ψ(x, t)|s|θ+γ on a compact set Q Q. It follows from estimate (7.1.10) that F (x, t, ) dxdt ≤ ce (Q )ψL∞ (Q) θ+γ dxdt ≤ ce (Q )ψL∞ (Q) . Q

Q

Applying now Theorem 1.4.5 with Ω replaced by Q and vn replaced by we obtain (7.2.2). Finally choose an arbitrary Q Q and ψ ∈ L∞ (Q ). Consider the integrand F (x, t, s) = ψ(x, t)p(s). Since p(s) ≤ c(1 + sγ ) for s ≥ 0, we have 1+θ/γ

|F (x, t, )|1+θ/γ ≤ ψL∞ (Q ) (1 + γ+θ ). It now follows from estimate (7.1.10) that 1+θ/γ |F (x, t, )|1+θ/γ dxdt ≤ ce ψL∞ (Q ) 1 + Q

Q

γ+θ dxdt ≤ c,

where c is independent of . Since θ/γ > 0, it follows that the sequence F (x, t, ) is equi-integrable in Q . Applying Theorem 1.4.5 we conclude that μxt , F (x, t, ·) dxdt = lim F (x, t, ) dxdt, Q

→0

Q

which along with the identity μxt , F (x, t, ·) = ψ(x, t)

p(s) ds f (x, t, s) [0,∞)

yields (7.2.3).

7.2. Proof of Theorem 7.1.9

187

Lemma 7.2.2. Under the assumptions of Theorem 7.1.9, the distribution function f satisﬁes inequalities (7.1.40) and (7.1.41). Proof. Choose t0 ∈ (0, T ) and h > 0 such that (t0 − h, t0 + h) ⊂ (0, T ). Next set ψ(x, t) = 1 if t ∈ (t0 − h, t0 + h) and ψ(x, t) = 0 otherwise. It follows from (7.2.1), (7.1.9), and the inequality sγ ≤ ce (1 + P (s)) that 1 2h

t0 +h

t0 −h

t0 +h 1 sγ ds f (x, t, s) dxdt ≤ lim sup γ dxdt 2h t0 −h Ω →0 [0,∞) t0 +h 1 (1 + P ( )) dxdt ≤ ce . ≤ ce lim sup 2h t0 −h Ω →0

Ω

Letting h → 0 we obtain (7.1.40). Next choose Q Q and set ψ(x, t) = 1 in Q and ψ = 0 elsewhere. Combining (7.1.10) and (7.2.2) we obtain (7.1.41). Proof of claim (ii) Lemma 7.2.3. There exist subsequences, still denoted by , u , such that as → 0, u u

weakly in L∞ (0, T ; L2γ/(γ+1) (Ω)),

u ⊗ u u ⊗ u

weakly in L2 (0, T ; Lz (Ω)),

(7.2.4)

where z is an arbitrary number from the interval (1, (1−2−1 a)−1 ), a = 2d−1 −γ −1 . Proof. By (7.1.8) the functions and u satisfy the mass and momentum balance equations ∂t = div u

in Q,

∂t ( u ) = div V + w

in Q,

which are understood in the sense of distributions. Here, u = − u ,

w = f ,

V = S(u ) − p ( ) I − u ⊗ u .

It follows from estimate (7.1.9) that the sequences u and are bounded in L2 (0, T ; W 1,2 (Ω)) and L∞ (0, T ; Lγ (Ω)), respectively. Moreover, that estimate implies that (u , ) are bounded energy functions in the sense of Section 4.2, and their energies are bounded by the constant ce in (7.1.9) independent of . Hence we can apply Corollary 4.2.2 to conclude that u L∞ (0,T ;L2γ/(γ+1) (Ω)) ≤ c. Hence the sequence u is bounded in L1 (Q) and the functions , u satisfy Condition 4.4.1 of Theorem 4.4.2 with n replaced by . Therefore u u weakly in L2 (0, T ; Lm (Ω)) for m−1 > 2−1 + γ −1 − d−1 .

188

Chapter 7. Kinetic theory. Fast density oscillations

Since the sequence u is bounded in L∞ (0, T ; L2γ/(γ+1) (Ω)), it also converges weakly in that space. This leads to the ﬁrst relation in (7.2.4). To prove the second, notice that by (7.1.9) and the inequalities 0 ≤ p ≤ c(1 + P ), the sequence V is bounded in L1 (Q). On the other hand, since f is bounded, it follows from (7.1.9) that w is bounded in L1 (Q). Therefore the sequences , u , V , and w satisfy all hypotheses of Theorem 4.4.2 with n replaced by . This yields the second relation in (7.2.4). Lemma 7.2.4. Under the assumptions of Theorem 7.1.3 for any Q Q there is a constant c(Q ) independent of such that p ( )Lr (Q ) ≤ c(r, Q )

for 1 ≤ r < 1 + θ(nγ)−1

(7.2.5)

and lim

→0

Q

|p( ) − p ( )| dxdt = 0.

(7.2.6)

Proof. Since p() ≤ c(1 + γ ) and 2 ≤ 1 + n , we have p ( )r dxdt ≤ c (1 + γ + n )r dxdt Q Q nr (rγ + ) dxdt ≤ c + c (θ+γ + n+θ ) dxdt, ≤c+c Q

Q

which along with (7.1.10) yields (7.2.5). Next, we have p () = p() + p∗ () + (2 + n ),

where

p∗ () = (1 − χ(/))p().

Since 1 − χ(/) vanishes for > , we have 0 ≤ p∗ () ≤ max p() = η() → 0 as → 0. ∈[0,]

From this and the inequality 2 ≤ 1 + n we obtain |p () − p()| ≤ η() + + 2n .

(7.2.7)

Next, the Young inequality implies n/(n+θ) n ≤ n+θ + 1. Thus from (7.1.10) we get n θ/(n+θ) dxdt = Q

Q

n/(n+θ) n

dxdt ≤

θ/(n+θ) Q

(n+θ + 1) dxdt

≤ ce (Q )θ/(n+θ) , Combining this with (7.2.7) we obtain |p( ) − p ( )| dxdt ≤ ce + ce η() + ce (Q )θ/(n+θ) → 0 as → 0. Q

7.2. Proof of Theorem 7.1.9

189

Lemma 7.2.5. Under the assumptions of Theorem 7.1.9 the identity (7.1.42) holds for all vector ﬁelds ξ ∈ C ∞ (Q) equal to 0 in a neighborhood of ST = ∂Ω × (0, T ) and of the top Ω × {t = T }. Proof. Choose ξ ∈ C ∞ (Q) as in the statement. Since ( , u ) is a weak solution to the momentum balance equation (7.1.8a), we have

u ⊗ u + p ( ) I − S(u ) : ∇ζ dxdt Q + f · ζ dxdt + ∞ (·, 0)U(·, 0) · ζ(·, 0) dx = 0. (7.2.8)

u · ∂t ζ dxdt + Q

Q

Ω

It follows from Lemma 7.2.3 that u · ∂t ζ + u ⊗ u − S(u ) : ∇ζ + f · ζ dxdt lim →0 Q = u · ∂t ζ + u ⊗ u − S(u) : ∇ζ + f · ζ dxdt.

(7.2.9)

Q

Next notice that ζ vanishes in a neighborhood of ST = ∂Ω × (0, T ). Hence there is a cylinder Qδ = Ωδ × (0, T ),

where

Ωδ = {x ∈ Ω : dist(x, ∂Ω) > δ},

such that supp ζ ⊂ Qδ . Now choose h > 0 and notice that Ωδ × (h, T − h) Q. It follows from (7.2.3) with ψ = div ζ that T −h T −h lim p( ) div ζ dxdt = p div ζ dxdt. (7.2.10) →0

h

h

Ωδ

Ωδ

On the other hand, (7.2.6) with Q = Ωδ × (h, T − h) implies T −h |p( ) − p ( )| dxdt = 0. lim →0

h

Ωδ

Combining this with (7.2.10) we obtain T −h p ( ) div ζ dxdt = lim →0

h

Ωδ

T −h

h

p div ζ dxdt.

(7.2.11)

Ωδ

Next, in view of (7.1.18) we have 0 ≤ p ≤ c(1 + P ). Along with (7.1.9) this implies h T h T + p( ) div ζ dxdt ≤ c + (1 + P ) dxdt ≤ ch. 0

T −h

Ωδ

0

T −h

Ω

(7.2.12)

190

Chapter 7. Kinetic theory. Fast density oscillations

Notice that p(s) ≤ c(1 + sγ ) for s ≥ 0 in view of (7.1.4). Thus p(x, t) := p(s) ds f (x, t, s) ≤ c 1 + sγ ds f (x, t, s) . [0,∞)

[0,∞)

From (7.1.40) we conclude that p(x, t) dx ≤ c,

ess sup t∈(0,T )

which leads to

h

0

T

Ω

+ T −h

p div ζ dxdt ≤ ch.

(7.2.13)

Ωδ

Combining (7.2.11)–(7.2.13), and noting that ζ vanishes outside Ωδ × (0, T ), we arrive at lim sup p ( ) div ζ dxdt − p div ζ dxdt ≤ ch. →0

Q

Q

Finally, letting h → 0 we obtain lim p ( ) div ζ dxdt = p div ζ dxdt. →0

Q

(7.2.14)

Q

Letting → 0 in (7.2.8) and recalling (7.2.9), (7.2.14) we obtain the integral identity (7.1.42). Proof of claim (iii) Lemma 7.2.6. Let the hypotheses of Theorem 7.1.9 be satisﬁed and Φ ∈ C(R+ ) vanish near +∞. Then passing to a subsequence we can assume that Φ div u Φ div u

weakly in L2 (Q),

(7.2.15)

V Φ −(λ + 1)Φ div u + Φp weakly in L (Q), 1 Φ div u = Φ div u + Φp − Φp , λ+1 2

(7.2.16) (7.2.17)

where Φ = Φ( ), V = −(1 + λ) div u + p ( ). Notice that Φ div u is just a notation for the weak limit, whereas p, Φ, and Φp are represented in terms of the Young measure μxt . Proof. Assume that Φ(s) is extended to a function in Cc (R). Moreover, it suﬃces to prove the lemma for Φ ∈ C0∞ (R) since C0∞ (R) is dense in C0 (R). Observe that the governing equations (7.1.8) can be rewritten in the form ∂t ( u ) + div( u ⊗ u ) = div T + f ∂t + div( u ) = 0 in Q.

in Q,

(7.2.18a) (7.2.18b)

7.2. Proof of Theorem 7.1.9

191

Here T = S(u ) − p ( ) I.

(7.2.18c)

Moreover, since (u , ) is a renormalized solution to problem (7.1.8) we have the following equation for Φ = Φ( ): ∂Φ + div(Φ u ) + = 0.

(7.2.18d)

= (Φ ( ) − Φ( ))u .

(7.2.19)

Here

Now choose the exponents 1 < r ≤ 1 +

θ , nγ

r = r /(r − 1),

1 < q ≤ 2,

s = ∞,

(7.2.20)

and set g = gϕ = 0,

G = 0.

(7.2.21)

Let us check that the functions , ϕ := Φ , the vector ﬁelds u , g , gϕ , the matrix-valued functions T , G , and the exponents γ, s, r, q meet all requirements of Theorem 4.7.1 with n → ∞ replaced by → 0. Since the stress tensor T admits representation (7.2.18c) it suﬃces to prove that all these quantities satisfy Condition 4.6.1 and (4.6.16). First, by (7.1.9) and the inequality γ ≤ c(1 + P ) in Condition 7.1.1, the functions , ϕ = Φ , and the vector ﬁelds u satisfy (4.6.4)–(4.6.6) with n and E replaced by and a constant c independent of , respectively. Next, since Φ − Φ is bounded, it follows from (7.1.9) and (7.2.5) that for any Q Q, T L1 (Q) ≤ c,

L2 (Q) ≤ c,

T Lr (Q ) ≤ c(Q ),

(7.2.22)

where c is independent of . Hence T and Φ satisfy (4.6.7)–(4.6.8) with n, E, and ϕ replaced by , c, and Φ , respectively. Next, (7.2.21) and (7.2.22) obviously imply that , g , gϕ , and G satisfy (4.6.9)–(4.6.11). On the other hand, by (7.1.22) and (7.1.26a), u u weakly in L2 (0, T ; W 1,2 (Ω)), weakly in L∞ (0, T ; Lγ (Ω)), Φ Φ

weakly in L∞ (Q).

Hence , u , and Φ satisfy (4.6.12) with n → ∞ replaced by → 0.

192

Chapter 7. Kinetic theory. Fast density oscillations

Next, by (7.2.5) the functions p ( ) are bounded in Lr (Q ) for every Q Q. Hence passing to a subsequence we can assume that p ( ) converges weakly in Lr (Q ) to some p∗ ∈ L1 (Q) ∩ Lrloc (Q). Since r ≤ 2, we have T = S(u ) + p ( ) S(u) + p∗ = T, V = −(λ + 1) div u + p ( ) −(λ + 1) div u + p∗ = V

(7.2.23)

weakly in Lr (Q ). Hence the sequence T also satisﬁes (4.6.12). Therefore, the functions , ϕ = Φ , the vector ﬁelds u , g , gϕ , and the matrix-valued functions T , G satisfy Condition 4.6.1 of Theorem 4.6.3 with exponents γ, s, r, q given by (7.2.20). Obviously the exponents and the functions G , gϕ satisfy (4.6.16). In summary, all these quantities meet all requirements of Theorems 4.6.3 and 4.7.1. Applying Theorem 4.7.1 we obtain lim V Φ η dxdt = V Φ η dxdt for all η ∈ C0∞ (Q), (7.2.24) →0

Q

Q

where V and V are deﬁned by (7.2.23). Now we specify the integrands in (7.2.24). By (7.2.3) and (7.2.6) we have lim ψ(x, t)p ( ) dxdt = ψ(x, t)p dxdt for all ψ ∈ L∞ (Q ). →0

Q

Q

∗

It follows that p = p and hence V = −(λ + 1) div u + p.

(7.2.25)

Since Φ ∈ C0∞ (R), the sequence Φ div u is bounded in L2 (Q). Passing to a subsequence we can assume that there is a function Φ div u ∈ L2 (Q) such that Φ div u Φ div u weakly in L2 (Q).

(7.2.26)

This is (7.2.15). Notice that Φ div u is just a notation for the weak limit. Next, since Φ ∈ C0∞ (R), the sequences Φ( )p ( ) and Φ( )p( ) are bounded in L∞ (Q). Hence there are functions (Φp)∗ and Φp such that for any ψ ∈ L1 (Q), ψp ( )Φ( ) dxdt = ψ(x, t)(Φp)∗ dxdt, (7.2.27) lim →0 Q Q lim ψp( )Φ( ) dxdt = ψ(x, t)Φp dxdt. (7.2.28) →0

Q

Q

Moreover, since Φ is compactly supported, from (7.1.5) we have (p () − p())Φ() → 0 as → 0 uniformly in . Hence (Φp)∗ = Φp. From (7.2.24), (7.2.27), and (7.2.26) we now obtain −(λ + 1)Φ div u + Φp η dxdt, V Φ η dxdt = lim →0

Q

Q

(7.2.29)

7.2. Proof of Theorem 7.1.9

193

which yields (7.2.16). Finally combining (7.2.24), (7.2.29), and (7.2.25) we get −(λ + 1)Φ div u + Φp = −(λ + 1)Φ div u + Φp,

which yields (7.2.17).

Lemma 7.2.7. Let the hypotheses of Theorem 7.1.9 be satisﬁed. Furthermore assume that ϕ ∈ C0∞ (R) and Φ(s) = ϕ (s)s − ϕ(s). Then

1 ϕ(∂t ψ + ∇ψ · u) dxdt − ψ Φp − Φp dxdt λ+1 Q Q − ψΦ div u dxdt + (ψϕ∞ )(x, 0) dx − ψϕ∞ U · n dSdt = 0 (7.2.30) Q

ST

Ω

for all ψ ∈ C ∞ (Q) vanishing in a neighborhood of ST \ Σin and of Ω × {t = T }. Here ϕ= ϕ(s) ds f (x, t, s), p = p(s) ds f (x, t, s), ϕ∞ = ϕ(s) ds f∞ (x, t, s), R R R Φp = Φ(s)p(s) ds f (x, t, s), Φ = Φ(s) ds f (x, t, s). R

R

Proof. Choose ϕ and ψ as in the statement. Set ϕ = ϕ( ), ϕ∞ = ϕ(∞ ), and Φ = Φ( ). Since is a renormalized solution to the boundary value problem ∂t + div( u ) = 0 in Q, = ∞ on Σin , |t=0 = ∞

in Ω,

we have ϕ (∂t ψ + ∇ψ · u ) dxdt − ψΦ div u dxdt Q Q + (ψϕ∞ )(x, 0) dx − Ω

ST

ψϕ∞ U · n dSdt = 0.

Letting → 0 and using (7.1.26a)–(7.1.26b) and (7.2.26) we obtain

ϕ(∂t ψ + ∇ψ · u) dxdt −

Q

ψΦ div u dxdt + (ψϕ∞ )(x, 0) dx −

Q

Ω

Inserting (7.2.17) we obtain (7.2.30).

ST

ψϕ∞ U · n dSdt = 0.

Lemma 7.2.8. Under the assumptions of Theorem 7.1.9 the distribution function f deﬁned by (7.1.26) and (7.1.27) satisﬁes the integral identity (7.1.43).

194

Chapter 7. Kinetic theory. Fast density oscillations

Proof. Choose η ∈ C0∞ (R) and set ∞ ϕ(s) = η(τ ) dτ, Φ(s) = sϕ (s) − ϕ(s) = −sη(s) − s

∞

η(τ ) dτ.

(7.2.31)

s

First, we derive formulae for ϕ, Φ, and Φp. We have ∞ ϕ(x, t) = ϕ(s) ds f (x, t, s) = η(τ ) dτ ds f (x, t, s). R

R

s

Since the distribution function f vanishes for s < 0 and the smooth function ϕ vanishes for all large s and dϕ = −η ds, we can use Lemma 1.3.7 and integrate by parts in the Stieltjes integral to obtain ϕ(x, t) = η(s)f (x, t, s) ds. (7.2.32) R

The same arguments give

ϕ∞ (x, t) =

R

(7.2.33)

η(s)f∞ (x, t, s) ds.

As the smooth function sϕ (s) is compactly supported and d(sϕ (s)) = −(sη) ds, integration by parts gives ϕ = sϕ (s) ds f (x, t, s) = (sη) f (x, t, s) ds. R

R

Combining this with (7.2.32) we obtain Φ(x, t) =

ϕ (x, t)

− ϕ(x, t) =

sη (s)f (x, t, s) ds.

(7.2.34)

R

Next, for a.e. (x, t) ∈ Q denote by F the distribution function of the Borel measure dνxt (s) = p(s)dμxt (s) on R, F (x, t, s) = p(τ ) dτ f (x, t, τ ). (7.2.35) (−∞,s]

Since p(s) is integrable with respect to μxt for a.e. (x, t) ∈ Q, the function F (x, t, s) is monotone, right continuous and F (x, t, s) = 0 for s < 0,

F (x, t, s) → p(x, t) as s → ∞.

Repeating the previous arguments with f replaced by F we obtain the following analogues of formulae (7.2.32)–(7.2.34): ϕ(s)p(s) ds f (x, t, s) = ϕ(s) ds F (x, t, s) = η(s)F (x, t, s) ds, R R R ϕ (s)sp(s) ds f (x, t, s) = ϕ (s)s ds F (x, t, s) = (sη) F (x, t, s) ds, R

R

R

7.2. Proof of Theorem 7.1.9 which gives

195

(ϕ (s)s − ϕ(s))p(s) ds f (x, t, s) = sη F (x, t, s) ds R R η (s)s p(τ ) dτ f (x, t, τ ) ds. =

Φp =

R

(−∞,s]

Combining this with (7.2.34) and recalling formula (7.1.33e) for C[f ] we obtain, for a.e. (x, t) ∈ Q, Φp − Φp = η (s)s (p(τ ) − p) dτ f (x, t, τ ) ds R (−∞,s] = (λ + 1) sη (s)C[f ]) ds. (7.2.36) R ∞

Let ς ∈ C (Q) vanish in a neighborhood of ST \ Σin and of Ω × {t = T }. Then ς and ϕ meet all requirements of Lemma 7.2.7 and hence satisfy the integral identity 1 ϕ(∂t ς + ∇ς · u) dxdt − ς Φp − Φp dxdt λ+1 Q Q ςΦ div u dxdt + (ςϕ∞ )(x, 0) dx − ςϕ∞ U · n dSdt = 0. (7.2.37) − Q

ST

Ω

It now follows from representations (7.2.32) and (7.2.33) that ϕ(∂t ς + ∇ς · u) dxdt = η(s)(∂t ς + ∇ς · u)f (x, t, s) ds dxdt Q

and

Q

(ςϕ∞ )(x, 0) dx − ςϕ∞ U · n dSdt Ω ST = ς(x, 0)η(s)f∞ (x, 0, s) dsdx − Ω

R

ST

R

ςη(s)U · n f∞ (x, t, s) ds dSdt.

Next, (7.2.34) and (7.2.36) imply 1 ς Φp − Φp dxdt = ςsη (s)C[f ](x, t, s) ds dxdt, λ+1 Q Q R ςΦ div u dxdt = ςsη (s)f (x, t, s) div u dsdxdt. Q

(7.2.38)

R

Q

(7.2.39)

R

Inserting (7.2.38)–(7.2.39) into (7.2.37) we ﬁnally obtain f ∂t (ςη) + ∇x (ςη) · u − s∂s (ςη) f div u + C[f ] dxdtds Q×R (ςη)(x, 0, s)f∞ (x, 0, s) dxds − (ςη)f∞ U · n dSdt ds = 0. (7.2.40) + Ω×R

ST ×R

196

Chapter 7. Kinetic theory. Fast density oscillations

Thus we have proved the desired identity (7.1.43) for ψ(x, t, s) = ς(x, t)η(s). It remains to extend this result to all smooth functions ψ(x, t, s) satisfying (7.1.44). Fix such a ψ ∈ C ∞ (Q × R). Denote by ψˆ the Fourier transform of ψ with respect to s, ˆ t, ξ) = √1 ψ(x, ψ(x, t, s)e−iξs ds. 2π R Next choose ζ ∈ C0∞ (R) nonnegative with ζ(0) = 1 and for ξ ∈ R set ˆ t, ξ), ς(x, t, ξ) = ψ(x,

η(s, ξ) = ζ(s/k)eiξs .

Since ψ is C ∞ with respect to all arguments and compactly supported in s, it follows that ψˆ is smooth in x, t, ξ and ˆ t, ξ)| + |∇x ψ(x, ˆ t, ξ)| + |∂t ψ(x, ˆ t, ξ)| ≤ c(m)(1 + |ξ|)−m |ψ(x,

(7.2.41)

for any m ≥ 0. Inserting the deﬁnitions of ς and η in (7.2.40) we arrive at

eiξs (∂t ψˆ + ∇x ψˆ · u)ζ(s/k)f Q×R

ˆ ζ(s/k) + iξζ(s/k) (f div u(x, t) + C[f ]) dxdtds − ψs ˆ 0, ξ)ζ(s/k)f∞ (x, 0, s) dxds eiξs ψ(x, + Ω×R ˆ t, ξ)ζ(s/k)f∞ U · n dSdtds. = eiξs ψ(x,

(7.2.42)

ST ×R

ˆ t, ξ) and the integrals depend on ξ. For every positive N deﬁne Here ψˆ = ψ(x, 1 ψN (x, t, s) = √ 2π

N

ˆ t, ξ) dξ. eiξs ψ(x, −N

Integrating (7.2.42) with respect to ξ over [−N, N ] we obtain Q×R

(∂t ψN + ∇x ψN · u)ζ(s/k)f − s ψN ζ(s/k) + ∂s ψN ζ(s/k) (f div u(x, t) + C[f ]) dxdtds ψN (x, 0, s)ζ(s/k)f∞ (x, 0, s) dxds + Ω×R ψN (x, t, s)ζ(s/k)f∞ U · n dSdtds. =

(7.2.43)

ST ×R

Notice that the nonnegative functions f, f∞ are bounded by 1. Obviously we have |C[f ]| ≤ 2p, a majorant independent of s and integrable on Q. Next, div u ∈ L2 (Q).

7.2. Proof of Theorem 7.1.9

197

Since ζ is compactly supported, it follows that s(f div u + C[f ])ζ(s/k), s(f div u + C[f ])(ζ(s/k)) ∈ L1 (Q × R), ζ(s/k)f ∈ L1 (Q × R),

ζ(s/k)f∞ ∈ L1 (

T

× R).

(7.2.44)

The formula for the inverse Fourier transform and estimates (7.2.41) imply that ψN (x, t, s) → ψ(x, t, s) uniformly in x, t, s as N → ∞. The same holds for ∂t ψN , ∇x ψN and ∂s ψN . Letting N → ∞ in (7.2.43), recalling (7.2.44), and applying the Lebesgue dominated convergence theorem we obtain, for every k > 0,

Q×R

(∂t ψ + ∇x ψ · u)ζ(s/k)f − s ζ(s/k) ψ + ζ(s/k)∂s ψ (f div u(x, t) + C[f ]) dxdtds ψ(x, 0, s)ζ(s/k)f∞ (x, 0, s) dxds + Ω×R ψ(x, t, s)ζ(s/k)f∞ U · n dSdtds. =

(7.2.45)

ST ×R

Finally notice that the smooth function ψ(x, t, s) is compactly supported in s and ζ(s/k) → 1, ζ(s/k) = k −1 ζ (s/k) → 0 as k → ∞. Letting k → ∞ in (7.2.45) we obtain (7.1.43). Proof of claim (iv). We have already proved claims (i)–(iii) of Theorem 7.1.9 which ensure that (u, f ) is a solution to Problem 7.1.5. It remains to prove that this solution is regular in the sense of Deﬁnition 7.1.8, i.e. satisﬁes estimate (7.1.38). We split the proof into a sequence of lemmas. For any ϑ : Q → R and for any ∈ R set φϑ (x, t, ) = min{, ϑ(x, t)}. (7.2.46) It follows from the fundamental theorem on Young measures (Theorem 1.4.5) and the convergences (7.1.26) that for any ϑ ∈ C(Q), φϑ (·, ) φϑ φϑ (x, t) =

weakly in L∞ (Q),

where

min{s, ϑ(x, t)} ds f (x, t, s).

(7.2.47)

[0,∞)

Lemma 7.2.9. Let ϑ ∈ C(Q). Then for any h ∈ Cc (Q), lim

→0

h φϑ ( ) div u − φϑ div u dxdt Q 1 = lim h φϑ ( )p( ) − φϑ p dxdt. λ + 1 →0 Q

(7.2.48)

198

Chapter 7. Kinetic theory. Fast density oscillations

Proof. Choose N > supQ ϑ(x, t) and δ > 0. Since φϑ (x, t, ) is uniformly continuous in Q × [0, N ], there are smooth functions ϕk : [0, N ] → R and ϑk : Q → R, 1 ≤ k ≤ n, such that the function φδ (x, t, ) =

n

ϑk (x, t)ϕk ()

(7.2.49)

k=1

approximates φϑ with accuracy δ, i.e., sup

|φϑ (x, t, ) − φδ (x, t, )| ≤ δ.

(x,t,)∈Q×[0,N ]

Extend ϕk to [0, ∞) by setting ϕk (s) = ϕk (N ) for s > N . This yields an extension of φδ to Q × [0, ∞). We keep the same notation for the extended functions. It follows from (7.2.46) that sup

|φϑ (x, t, ) − φδ (x, t, )| ≤ δ.

(7.2.50)

(x,t,)∈Q×[0,∞)

Let h ∈ Cc (Q). Fix a compact set Q Q such that supp h ⊂ Q . Let us prove that the desired equality (7.2.48) holds for φδ in place of φϑ . By linearity it suﬃces to prove that for every k, lim

→0

hϑk ϕk ( ) div u − ϕk div u dxdt Q 1 = h ϕk ( )p( ) − ϕk p dxdt. lim λ + 1 →0 Q

(7.2.51)

To this end notice that the Lipschitz function ϕk () − ϕk (N ) vanishes for ≥ N and hence, by (7.2.17),

hϑk (ϕk ( ) − ϕk (N )) div u − ϕk − ϕk (N ) div u dxdt →0 Q 1 lim = hϑk (ϕk ( ) − ϕk (N ))p( ) − ϕk − ϕk (N ) p dxdt. λ + 1 →0 Q lim

(7.2.52)

Recalling that div u converges weakly to div u in L2 (Q) and using (7.2.3) we obtain 0 = lim hϑk ϕk (N ) div u − ϕk (N ) div u dxdt, →0 Q hϑk ϕk (N )p( ) − ϕk (N )p dxdt. 0 = lim →0

Q

7.2. Proof of Theorem 7.1.9

199

Combining this with (7.2.52) we arrive at (7.2.51). Summing (7.2.51) with respect to k we obtain lim h φδ ( ) div u − φδ div u dxdt →0 Q 1 lim = h φδ ( )p( ) − φδ p dxdt. (7.2.53) λ + 1 →0 Q Next, by (7.1.9) we have div u L2 (Q) + p( )L1 (Q) ≤ c, which gives h φϑ ( ) − φδ ( ) div u dxdt + h φϑ ( ) − φδ ( ) p( ) dxdt ≤ cδ. Q

Q

(7.2.54) It follows from the obvious inequalities |φϑ − φδ | = |φϑ − φδ | ≤ δ and the inclusion p ∈ L1 (Q) that h φϑ − φδ div u dxdt + h φϑ − φδ p dxdt ≤ cδ. (7.2.55) Q

Q

Combining (7.2.53)–(7.2.55) we obtain # " 1 φϑ ( )p( ) − φϑ p h dxdt φϑ ( ) div u − φϑ div u − −cδ ≤ lim inf →0 λ+1 Q # " 1 ≤ lim sup φϑ ( )p( ) − φϑ p h dxdt ≤ cδ. φϑ ( ) div u − φϑ div u − λ+1 →0 Q Letting δ → 0 we obtain equality (7.2.48).

Now deﬁne the function Tϑ (x, t) = φϑ (x, t, ) − φϑ (x, t). Recall

φϑ (x, t, ) = min φϑ (x, t) =

(7.2.56)

s ds f (x, t, s), ϑ(x, t) ,

[0,∞)

(7.2.57)

min{s, ϑ(x, t)} ds f (x, t, s).

[0,∞)

Lemma 7.2.10. Under the assumptions of Theorem 7.1.9, for every ϑ ∈ C(Q) and every compact set Q Q there is a constant c independent of ϑ and Q such that γ+1 Tϑ Lγ+1 (Q ) ≤ lim sup | min{ , ϑ} − min{, ϑ}|γ+1 dxdt ≤ c. (7.2.58) →0

Q

200

Chapter 7. Kinetic theory. Fast density oscillations

Proof. As the function s → φϑ (x, t, s) deﬁned by (7.2.46) is monotone and its derivative does not exceed 1, it follows that for any s ≤ s , |φϑ (x, t, s ) − φϑ (x, t, s )|1+γ ≤ (φϑ (x, t, s ) − φϑ (x, t, s ))(s − s ). γ

γ

(7.2.59)

Next, (7.1.3) implies s − s ≤ c(p(s ) − p(s )) for 1 ≤ s ≤ s . γ

γ

(7.2.60)

By (5.1.4), the pressure function p satisﬁes s ≤ c(p(s ) + 1) and is bounded on [0, 1], which leads to γ

s − s ≤ c(p(s ) − p(s )) + c for s ≤ 1 ≤ s . γ

γ

(7.2.61)

Combining (7.2.60) and (7.2.61) yields s − s ≤ c(p(s ) − p(s )) + c for 0 ≤ s ≤ s . γ

γ

(7.2.62)

From this and (7.2.59) we ﬁnally obtain |φϑ (x, t, s ) − φϑ (x, t, s )|1+γ ≤ cp (φϑ (x, t, s ) − φϑ (x, t, s ))(p(s ) − p(s ) + c). (7.2.63) Now choose Q Q and h ∈ Cc (Q) such that 0 ≤ h ≤ 1, and h(x, t) = 1 in Q . Notice that Tϑ is the weak limit of the sequence φϑ (x, t, ) − φϑ (x, t, ). It follows that |Tϑ |γ+1 dxdt ≤ h|Tϑ |γ+1 dxdt Q Q ≤ lim h|φϑ (x, t, ) − φϑ (x, t, )|1+γ dxdt. (7.2.64) →0

Q

Now our task is to estimate the right hand side of this inequality. First we use (7.2.63) to obtain lim

→0

h|φϑ (x, t, ) − φϑ (x, t, )|1+γ dxdt Q h φϑ (x, t, ) − φϑ (x, t, ) (p( ) − p()) dxdt ≤ cp lim →0 Q + c lim h φϑ (x, t, ) − φϑ (x, t, ) dxdt →0 Q = cp h I(x, t) dxdt + c h(φϑ − φϑ (x, t, )) dxdt, Q

Q

where I = φϑ p − φϑ p() − φϑ (·, ) p + φϑ (·, ) p().

(7.2.65)

7.2. Proof of Theorem 7.1.9

201

Now rewrite the expression for I in the form I = φϑ p − φϑ p + φϑ − φϑ (·, ) p − p() .

(7.2.66)

Recall that, by the assumptions of Theorem 7.1.9 (claim (iv)), the pressure function has a representation p() = pc ()+pb () where pc is convex and pb is bounded, i.e. there exists c such |pb | ≤ c. Since pc is convex, we have pc ≥ pc (). It follows that p ≥ pc − c ≥ pc () − c ≥ p() − 2c, so p − p() ≥ −2c. On the other hand, since φϑ (x, t, ·) is concave, we have φϑ ≤ φϑ (). Thus φϑ − φϑ () p − p() ≤ 2c φϑ () − φϑ , which along with (7.2.66) leads to I ≤ φϑ p − φϑ p + c φϑ () − φϑ . Combining this with (7.2.65) we arrive at lim

→0

Q

h|φϑ (x, t, ) − φϑ (x, t, )|1+γ dxdt ≤c h φϑ p − φϑ p dxdt + c h φϑ (x, t, ) − φϑ dxdt. Q

(7.2.67)

Q

Next, the obvious inequality 0 ≤ φϑ (x, t, ) ≤ implies 0 ≤ φϑ ≤ and 0 ≤ φϑ (x, t, ) ≤ . From this, (7.1.29) and (7.1.40), we obtain h φϑ (x, t, ) − φϑ dxdt ≤ 2 h dxdt ≤ c. Q

Q

Combining this with (7.2.67) we arrive at lim h|φϑ (x, t, ) − φϑ (x, t, )|1+γ dxdt ≤ c h φϑ p − φϑ p dxdt + c. →0

Q

Q

From this and (7.2.48) we obtain lim

→0

h|φϑ (x, t, ) − φϑ (x, t, )|1+γ dxdt Q ≤ c(λ + 1) h φϑ div u − φϑ div u dxdt + c. (7.2.68) Q

By the deﬁnition of weak convergence, we have h φϑ div u − φϑ div u dxdt = lim hφϑ (x, t, )(div u − div u) dxdt. Q

→0

Q

202

Chapter 7. Kinetic theory. Fast density oscillations

Since φ(x, t, ) is bounded and ∇u ∇u weakly in L2 (Q), it follows that lim hφϑ (x, t, )(div u − div u) dxdt = 0. →0

Q

Combining the above results we arrive at

h φϑ div u − φϑ div u dxdt Q h φϑ (x, t, ) − φϑ (x, t, ) div(u − u) dxdt. = lim →0

(7.2.69)

Q

Next, the Young inequality implies c(λ + 1)h φϑ (x, t, ) − φϑ (x, t, ) div(u − u)

γ+1 1 ≤ 2−1 h|φϑ (x, t, ) − φϑ (x, t, )|γ+1 + 2 γ h c(λ + 1)| div(u − u)| γ .

Since u satisﬁes (7.1.9), (7.1.22), the sequence div(u − u) is bounded in L2 (Q). From this and the inequality (γ + 1)/γ ≤ 2 we obtain γ+1 lim sup | div(u − u)| γ dxdt ≤ c, →0

Q

which gives c(λ + 1) lim

→0

h φϑ (x, t, ) − φϑ (x, t, ) div(u − u) dxdt Q ≤ 2−1 lim h|φϑ (x, t, ) − φϑ (x, t, )|γ+1 dxdt + c. →0

Q

Recalling identity (7.2.69) we arrive at c(λ + 1)

h φϑ div u − φϑ div u dxdt Q ≤ 2−1 lim h|φϑ (x, t, ) − φϑ (x, t, )|γ+1 dxdt + c. →0

Q

Inserting this into (7.2.68) we get h|φϑ (x, t, ) − φϑ (x, t, )|1+γ dxdt ≤ c, 2−1 lim →0

Q

which along with (7.2.64) leads to (7.2.58).

7.2. Proof of Theorem 7.1.9

203

Lemma 7.2.11. Under the assumptions of Theorem 7.1.9, there is a constant c such that (7.2.70) HL1+γ (Q) + sup Vυ L1 (Q) < c, υ∈R+

where

min{s, υ}(p(s) − p) ds f (x, t, s),

Vυ (x, t) =

[0,∞)

(7.2.71) f (x, t, s)(1 − f (x, t, s)) ds.

H(x, t) = [0,∞)

Proof. Pick ϑ ∈ C(Q) nonnegative. Recall the deﬁnition of Tϑ : Tϑ (x, t) = min (x, t), ϑ(x, t) − min{s, ϑ(x, t)} ds f (x, t, s). [0,∞)

By integration by parts (Lemma 1.3.7) we obtain min{s, ϑ(x, t)} ds f (x, t, s) [0,∞) s ds f (x, t, s) + ϑ(x, t) = [0,ϑ(x,t)]

ϑ

=ϑ−

ϑ

(1 − f (x, t, s)) ds.

f (x, t, s) ds = 0

ds f (x, t, s)

(ϑ(x,t),∞)

0

Next, for a.e. (x, t) ∈ Q and every N > 0 we have N γ (1 − f (x, t, N )) = N γ ds f (x, t) ≤ [N,∞)

sγ ds f (x, t) < ∞,

[0,∞)

which leads to limN →∞ N (1 − f (x, t, N )) = 0. Integrating by parts we obtain sds f (x, t, s)) = − sds (1 − f (x, t, s)) [0,N ] [0,N ] (1 − f (x, t, s)) ds − N (1 − f (x, t, N )). = [0,N ]

Letting N → ∞ we arrive at (x, t) = s ds f (x, t, s) = [0,∞)

∞

(1 − f (x, t, s)) ds

for a.e. (x, t) ∈ Q. It follows that ϑ(x,t) f (x, t, s) ds for (x, t) ≥ ϑ(x, t), Tϑ (x, t) = 0 ∞ Tϑ (x, t) = (1 − f (x, t, s)) ds for (x, t) ≤ ϑ(x, t). ϑ(x,t)

(7.2.72)

0

(7.2.73)

204

Chapter 7. Kinetic theory. Fast density oscillations

Next choose a sequence of nonnegative continuous functions ϑn such that ϑn → a.e. in Q. Substituting ϑn into (7.2.73) and letting n → ∞ we conclude that for a.e. (x, t) ∈ Q, ∞ (x,t) Tϑn (x, t) → T = f (x, t, s) ds = (1 − f (x, t, s)) ds. (7.2.74) (x,t)

0

It follows from (7.2.58) that there is a constant c independent of Q and n such that Tϑn Lγ+1 (Q ) ≤ c and hence Tϑn Lγ+1 (Q) ≤ c . Thus T Lγ+1 (Q) ≤ c. Noting that H ≤ T we arrive at the desired estimate for H in (7.2.70). To estimate Vυ notice that by (7.2.48), for any Q Q, 1 Vυ = w-lim(φυ ( ) div u ) − φυ div u, →0 λ+1 where w-lim denotes the weak limit in L2 (Q ). Thus we get 1 Vυ = w-lim(φυ ( ) − φυ ()) div u − w-lim φυ ( ) − φυ () div u. →0 →0 λ+1 From this and (7.2.58) we obtain, for every Q Q, Vυ L1 (Q ) ≤ c lim sup ( div u L2 (Q ) + div uL2 (Q ) )φυ ( ) − φυ ()L2 (Q ) →0

≤ c lim sup φυ ( ) − φυ ()L2 (Q ) ≤ c, →0

where c is independent of Q . Thus we arrive at the estimate for Vυ in (7.2.70). It remains to note that claim (iv) is a direct consequence of Lemma 7.2.11. This completes the proof of Theorem 7.1.9.

7.3 Proof of Theorem 7.1.12 We split the proof into a sequence of lemmas. The ﬁrst lemma determines the extension of a solution to problem (7.1.8) outside of the domain Ω. Lemma 7.3.1. Let f∞ be deﬁned by (7.1.48). Then f∞ ∂t ψ + ∇x ψ · U − s∂s ψ div U + ∂s ψF∞ dsdxdt (Rd \Ω)×(0,T ) R + f∞ (x, 0, s)ψ(x, 0, s) dsdx + f∞ (x, t, s)ψ(x, t, s)U · n ds dSdt = 0 Rd \Ω

R

ST

R

(7.3.1) for all ψ ∈ C ∞ (Rd × (0, T ) × R) vanishing for all suﬃciently large |x|, s, and for t = T . Here F∞ = ∂t ∞ + div(∞ U), (7.3.2)

7.3. Proof of Theorem 7.1.12

205

and n is the outward normal vector to ∂Ω. Notice that (7.3.1) is simply an identity which holds for any deterministic distribution f∞ . Proof. As this is a simpler version of Lemma 7.2.8, we only give some comments. Choose η ∈ C0∞ (R) and set

∞

ϕ(s) =

η(τ ) dτ,

Φ(s) = sϕ (s) − ϕ(s) = −sη(s) −

s

∞

η(τ ) dτ. s

It easily follows from (7.1.48) that ϕ(∞ )(x, t) =

R

(7.3.3)

η(s)f∞ (x, t, s) ds

and Φ(∞ )(x, t) =

R

sη (s)f∞ (x, t, s) ds,

ϕ (∞ )(x, t) =

R

η (s)f∞ (x, t, s) ds.

Multiplying the identity ∂t ∞ + div(∞ U) = F∞ by ϕ (∞ ) we obtain

(7.3.4)

∂t ϕ(∞ ) + div(ϕ(∞ )U) + Φ(∞ ) div U = ϕ (∞ )F∞ . Pick ς ∈ C ∞ (Rd ×(0, T )×R)) vanishing for large |x|+|s| and for t = T . Multiplying the last identity by ς and integrating by parts we arrive at

ϕ(∞ )(∂t ς + ∇ς · U) − ςΦ(∞ ) div U dxdt (Rd \Ω)×(0,T ) + ϕ (∞ )F∞ dxdt + (ςϕ(∞ ))(x, 0) dx (Rd \Ω)×(0,T ) Rd \Ω ςϕ(∞ )U · n dSdt = 0. (7.3.5) + ST

Substituting (7.3.3) and (7.3.4) into (7.3.5) yields (7.3.1) with ψ = ς(x, t)η(s). Arguing as in the proof of Lemma 7.2.8 we conclude that (7.3.1) holds for all test functions as in the hypotheses of Lemma 7.3.1. We now conclude from the integral identity (7.3.1) that f∞ is a weak solution to the equation ∂t f∞ + div(f∞ U) − ∂s (sf∞ div U) + ∂s (F∞ f∞ ) = 0

(7.3.6)

in (Rd × (0, T )) \ Q × R. Now our task is to combine this equation with (7.1.33b) in Q × R and next using cut-oﬀ functions to reduce the boundary value problem (7.1.45) to the Cauchy problem in the strip Rd × (0, T ) × R. To this end introduce

206

Chapter 7. Kinetic theory. Fast density oscillations

a cut-oﬀ function η ∈ C ∞ (Rd × (0, T )) with 0 ≤ η ≤ 1 and vanishing in some neighborhood of ST \ Σin and for all suﬃciently large |x|. Next we set v = u in Q, v = U in (Rd × (0, T )) \ Q, f ∗ (x, t, s) = η(x, t)f (x, t, s) in Q × R, f ∗ (x, t, s) = η(x, t)f∞ (x, t, s) in (Rd × (0, T )) \ Q × R, H = −sf ∗ div u(x, t) − sη(x, t)C[f ] in Q × R, H = −sf ∗ div U(x, t) + f ∗ F∞ (x, t) in (Rd × (0, T )) \ Q × R,

(7.3.7)

Lemma 7.3.2. The integral identity

Rd ×(0,T )×R

f ∗ ∂t ψ + v∇ψ + H∂s ψ + P ψ dxdtds + ηf∞ (x, 0, s)ψ(x, 0, s) dxds = 0 (7.3.8) Rd ×R

holds for all ψ ∈ C ∞ (Rd × (0, T ) × R) vanishing for t = T and for suﬃciently large |x|, s. Here P = (∂t η + ∇η · u)f in Q × R,

P = (∂t η + ∇η · U)f∞ in (Rd × (0, T )) \ Q × R. (7.3.9)

Proof. The integral identity (7.3.1) with ψ replaced by ψη implies

((Rd ×(0,T ))\Q)×R

(ηf∞ ) ∂t ψ + ∇x ψ · U − s∂s ψ div U + ∂s ψF∞ + P ψ dxdtds + ηf∞ (x, 0, s)ψ(x, 0, s) dsdx Rd \Ω R + ηf∞ (x, t, s)ψ(x, t, s)U · n ds dSdt = 0. (7.3.10) ST

R

On the other hand substituting ψη in (7.1.43) we obtain

(f η) ∂t ψ + u∇ψ − ∂s ψs div u + P ψ dxdtds Q×R ∂s ψsη C[f ] dxdtds − ηf∞ (x, t, s)ψ(x, t, s)U · n ds dSdt − Q×R S R T + ηf∞ (x, 0, s)ψ(x, 0, s) dxds = 0. (7.3.11) Ω×R

Summing (7.3.10) and (7.3.11) gives (7.3.8).

7.3. Proof of Theorem 7.1.12

207

The integral identity (7.3.8) means that f ∗ is a weak solution to the Cauchy problem ∂t f ∗ + div(f ∗ v) + ∂s H = P

in Rd × (0, T ) × R,

f ∗ (x, 0, s) = η(x, 0, s)f∞ (x, 0, s) in Rd × R.

(7.3.12)

Recall that f∞ is given by Condition 7.1.11. The next step is the renormalization of this problem. In other words we intend to derive an equation for a composite function Ψ(f ∗ ). We choose Ψ in such a way that it is concave and Ψ(f ) vanishes for every deterministic distribution function f , i.e., for f which takes only two values 0 and 1. The simplest choice is Ψ(f ) = f (1 − f ),

Ψ (f ) = 1 − 2f.

(7.3.13)

The justiﬁcation of the renormalization procedure is based on the following lemma on the properties of the mollifying operator. Recall the deﬁnition (1.6.1)–(1.6.3) of the molliﬁers in Section 1.6: (7.3.14) [f ],k (x, t, τ ) := k θ(k(s − τ ))f (x, t, s) ds, R

[f ]m, (y, t, s) := md

Rd

Θ(m(x − y))f (x, t, s) dx,

Θ(x) =

d

θ(xi ),

(7.3.15)

i=1

and [f ]m,k = [[f ],k ]m, . Here the mollifying kernel θ has the properties θ ∈ C0∞ (R),

θ(s) ds = 1, R

supp θ ⊂ (−1, 1),

θ(−s) = θ(s).

(7.3.16)

Lemma 7.3.3. Suppose g : Rd × (0, T ) × R → R admits a representation g(x, t, s) = sg1 (x, t, s) + g0 (x, t, s),

where

gi (·, t, s)L1 (Rd ) ≤ Mi (t). (7.3.17)

Furthermore assume that χ(s) is continuous, satisﬁes 0 ≤ χ ≤ 1 and vanishes for |s| ≥ N . Then for any m > 0 and k ≥ 1, and for all s ∈ R, [χg],k (·, t, s)L1 (Rd ) ≤ N M1 (t) + M0 (t), [χg]m,k (·, t, s)L1 (Rd ) ≤ N M1 (t) + cM0 (t).

(7.3.18)

Moreover, for every m > 0, N > 0 and k ≥ 1, there is a constant c(m, k, N ) such that for all x ∈ Rd , t ∈ [0, T ], and s ∈ [−N, N ], |∂x [g]m, (x, t, s)| + |∂s [g],k (x, t, s)| ≤ c(N + 2)M1 (t) + cM0 (t).

(7.3.19)

208

Chapter 7. Kinetic theory. Fast density oscillations

Proof. The inequalities k ≥ 1 and χ(τ )|τ | ≤ N imply χ [ g],k (·, t, s)L1 (Rd ) = kθ(k(τ − s))χ(τ )g(x, t, τ ) dτ dx Rd R χ kθ(k(τ − s)) (τ )|g|(x, t, τ ) dxdτ = kθ(k(τ − s))χ(τ )g(·, t, τ )L1 (Rd ) dτ ≤ Rd+1 R ≤ M1 (t) kθ(k(τ −s))χ(τ )|τ | dτ +M0 (t) kθ(k(τ −s)) dτ ≤ N M1 (t)+M0 (t), R

R

which yields the ﬁrst estimate in (7.3.18). To prove the second, we note that in view of the properties of molliﬁers we have [χg]m,k (·, t, s)L1 (Rd ) = [[χg],k ]m, (·, t, s)L1 (Rd ) ≤ [χg],k (·, t, s)L1 (Rd ) . Hence the second inequality in (7.3.18) follows from the ﬁrst. It remains to prove (7.3.19). To this end choose a continuous function χ(s) such that χ(s) = 1 for |s| ≤ N + 1 and χ(s) = 0 for |s| ≥ N + 2. Obviously [g]m,k = [χg]m,k for |s| ≤ N . Hence it suﬃces to prove (7.3.19) for χg. It follows from (7.3.17) that χg(·, t, ·)L1 (Rd+1 ) ≤ (N + 2)(M1 (t) + M0 (t)). By the general properties of molliﬁers, [χg(·, t, ·)]m,k C 1 (Rd+1 ) ≤ cχg(·, t, ·)L1 (Rd+1 ) ≤ c(N + 2)(M1 (t) + M0 (t)),

which gives (7.3.19).

The next proposition justiﬁes the renormalization procedure of the Cauchy problem (7.3.12). Proposition 7.3.4. Let Ψ be given by (7.3.13) and suppose the function f ∗ deﬁned by (7.3.7) satisﬁes (7.3.8). Then for any ψ ∈ C ∞ (Rd × (0, T ) × R) vanishing for t = T and for suﬃciently large |x| + |s|, we have Ψ(f ∗ )(∂t ψ + ∇ψ · u − s∂s ψ div u) − sη∂s ψ Ψ (f ∗ ) C[f ] dxdtds Q×R +2 η sψM ds f ∗ dxdt + N = 0. (7.3.20) Q

R

Here 1 1 M(x, t, s) = lim C[f ](x, t, s − h) + lim C[f ](x, t, s + h), (7.3.21) 2 h 0 2 h 0 Ψ(ηf∞ ) ∂t ψ + ∇ψ · U − ∂s ψ (s div U − F∞ ) dxdtds N= ((Rd ×(0,T ))\Q)×R + ψΨ (f ∗ )P dxdtds + Ψ(ηf∞ (x, 0, s))ψ(x, 0, s) dxds. Rd ×(0,T )×R

Rd ×R

(7.3.22) Proof. The proof naturally falls into ﬁve steps.

7.3. Proof of Theorem 7.1.12

209

Step 1. Integral identity for molliﬁers. Set K(y, τ ; x, s) = kmd θ(k(s − τ ))Θ(m(x − y)) for integers m, k ≥ 1. We have ∗

[f ]m,k (y, t, τ ) ≡

K(y, τ ; x, s)f ∗ (x, t, s) dxds.

(7.3.23)

Rd+1

For a ﬁxed (y, τ ) the function K(y, τ ; ·, ·) belongs to the class C0∞ (Rd+1 ). Now choose h ∈ C ∞ [0, T ] such that h(T ) = 0. Fix (y, τ ) ∈ Rd+1 and set ψ(x, t, s) = h(t)K(y, τ ; x, s). Substituting ψ in the integral identity (7.3.8) gives

T

0

K(y, τ ; x, s)f ∗ (x, t, s) dxds dt

h (t)

Rd+1

T

+

h(t)

Rd+1

0

∇x K(y, τ ; x, s) · v(x, t, s)f ∗ (x, t, s) + ∂s K(y, τ ; x, s)H(x, t, s) + K(y, τ ; x, s)P (x, t, s) dxds dt h(0)ηK(y, τ ; x, s)f∞ (x, 0, s)η(x, 0, s) dxds = 0. + Rd+1

From this, the identities ∇x K(y, τ ; x, s) = −∇y K(y, τ ; x, s), ∂s K(y, τ ; x, s) = −∂τ K(y, τ ; x, s), and (7.3.23) we obtain

T

h (t)[f ∗ ]m,k (y, t, τ ) dt

0

=

T

h(t) divy [vf ∗ ]m,k (y, t, τ ) + ∂τ [H]m,k (y, t, τ ) − [P ]m,k (y, t, τ ) dt

0

− h(0)[ηf∞ ]m,k (y, 0, τ ). (7.3.24) Let us return to notation (x, s) instead of (y, τ ) in this integral identity. It follows from (7.3.24) that the function [f ∗ ]m,k (x, t, s) satisﬁes the diﬀerential equation ∂t [f ∗ ]m,k = div[g(0) ]m,k + ∂s [g (1) ]m,k + [g (2) ]m,k

(7.3.25)

in the sense of distributions. Here g(0) = −f ∗ v,

g (1) = −H,

g (2) = P.

Recall that v = u in the cylinder Q and v = U outside of Q. The function U ∈ C 1 (Rd × [0, T ]) vanishes for all large |x| and u ∈ L2 (0, T ; W 1,2 (Ω)), which gives vL2 (0,T ;W 1,2 (Rd ) ≤ c. Since v vanishes for all large |x|, we have v(·, t)L1 (Rd ) + ∇vL1 (R2 ) ≤ c + cM (t),

(7.3.26)

210

Chapter 7. Kinetic theory. Fast density oscillations

where M (t) = u(·, t)W 1,2 (Ω) ,

M (t)L2 (0,T ) ≤ c.

(7.3.27)

Let us prove that the functions g(0) and g (2) satisfy the inequalities g(0) (·, t, s)L1 (Rd ) + g (2) (·, t, s)L1 (Rd ) ≤ cM (t) + c for a.e. (t, s) ∈ [0, T ] × R. (7.3.28) Since f ∗ does not exceed 1, estimate (7.3.28) for g(0) obviously follows from (7.3.26). Next, since the smooth function η vanishes for all suﬃciently large |x|, estimate (7.3.28) for g (2) follows from formula (7.3.9) for P and estimate (7.3.26) for v. It remains to estimate g (1) . The expression (7.3.7) for H implies |g (1) | = |H| ≤ |s| |C[f ]| + |s| | div v| in Q × R, |g (1) | = |H| ≤ |s| | div v| + |F∞ | in (Rd × (0, T )) \ Q × R.

(7.3.29)

Since ∞ ∈ C 1 (Rd × (0, T )) vanishes for large |x|, the expression (7.3.2) for F∞ yields the estimate F∞ (·, t, s)L1 (Rd \Ω) ≤ c for a.e. (t, s) ∈ [0, T ] × R. From (7.3.26) we obtain H(·, t, s)L1 (Ω) ≤ |s| C[f ](·, t, s) L1 (Ω) + |s|M (t), H(·, t, s)L1 (Rd \Ω) ≤ |s|M (t) + c,

(7.3.30)

for a.e. (t, s) ∈ [0, T ] × R. Next, the function C[f ](x, t, s), deﬁned by (7.1.33e), vanishes for s < 0 and satisﬁes p(s) ds f (x, t, s) + p(x, t) ds f (x, t, s) |C[f ](x, t, s)| ≤ R R (7.3.31) = 2 p(s) ds f (x, t, s) ≡ 2p(x, t). R

Inequality (7.1.3) in Condition 7.1.1 and inequality (7.1.47) in Condition 7.1.11 imply pL∞ (0,T ;L1 (Ω)) ≡ ess sup p(s) ds f (x, t, s) dx t∈(0,T )

≤ c ess sup t∈(0,T )

Ω

[0,∞)

Ω

(1 + sγ ) ds f (x, t, s) dx ≤ c.

(7.3.32)

[0,∞)

Combining this with (7.3.30) and (7.3.31) gives the estimate H(·, t, s)L1 (Rd ) ≡ g (1) (·, t, s)L1 (Rd ) ≤ c|s|M (t) + c|s| + c

(7.3.33)

for a.e. (t, s) ∈ [0, T ] × R. In view of (7.3.28) and (7.3.33) the functions g(0) , g (i) in (7.3.25) meet all requirements of Lemma 7.3.3. Applying inequality (7.3.19) of

7.3. Proof of Theorem 7.1.12

211

that lemma with g replaced by g(0) and g (i) we conclude that the right hand side of (7.3.25) admits the estimate div[g(0) ]m,k + ∂s [g (1) ]m,k + [g (2) ]m,k ≤ c(m, k)(N + 2)(|s| + 1)(M (t) + 1)

(7.3.34)

for all x ∈ Rd , s ∈ [−N, N ], and all N ≥ 1. It now follows from (7.3.27) and (7.3.25) that ∂t [f ∗ ]m,k (x, ·, s)L2 (0,T ) is uniformly bounded on every compact subset of Rd+1 . Since the embedding W 1,2 (0, T ) → C(0, T ) is compact, the functions [f ∗ ]m,k (·, t, ·) converge to some limit as t 0 uniformly on every compact subset of Rd+1 . By (7.3.24) this limit coincides with [ηf∞ ]m,k (x, 0, s). Thus we get [f ∗ ]m,k (x, t, s) → [ηf∞ ]m,k (x, 0, s)

as t 0

(7.3.35)

uniformly on every compact set. Now rewrite equation (7.3.25) in the form ∂t [f ∗ ]m,k + div([f ∗ ]m,k v) + ∂s [H]m,k = [P ]m,k + div Im,k ,

(7.3.36)

where Im,k = div([f ∗ ]m,k v − [f ∗ v]m,k ).

(7.3.37)

Recall that the function [f ∗ ]m,k is inﬁnitely diﬀerentiable with respect to x, s and has the time derivative locally square integrable. Hence it is a strong solution to (7.3.36). Multiplying this equation by the smooth bounded function Ψ ([f ∗ ]k,m ) and noting that Ψ (s)s − Ψ(s) = −s2 we arrive at ∂t Ψ([f ∗ ]k,m ) + div(Ψ([f ∗ ]k,m )v) − [f ∗ ]2k,m div v + Ψ ([f ∗ ]k,m )∂s [H]k,m = Ψ ([f ∗ ]k,m )[P ]k,m + Ψ ([f ∗ ]k,m )Im,k . (7.3.38) Choose ψ ∈ C ∞ (Rd × (0, T ) × R) such that ψ vanishes for t = T and for large |x| and |s|, say ψ(x, t, s) = 0 for |x| ≥ N − 1,

ψ(x, t, s) = 0 for |s| ≥ N − 1.

(7.3.39)

Next choose χ ∈ C ∞ (R) such that χ(s) = 1 for |s| ≤ N,

χ(s) = 0 for |s| ≥ N + 1.

(7.3.40)

It is clear that for k, m ≥ 1, χH = H,

[χH],k = [H],k ,

[χH]m,k = [H]m,k

on the support of ψ. (7.3.41)

212

Chapter 7. Kinetic theory. Fast density oscillations

The same conclusion can be drawn for P . Multiplying (7.3.38) by ψ, integrating the result over Rd × (0, T ) × R, and using (7.3.35) and (7.3.41) we arrive at

Rd ×(0,T )×R

Ψ([f ∗ ]m,k )(∂t ψ + ∇ψ · v) + ψ[f ∗ ]2m,k div v dxdtds

+

Rd ×(0,T )×R

+

Rd ×(0,T )×R

+ Rd ×R

∂s ψ Ψ ([f ∗ ]k,m ) [χH]k,m dxdtds ψΨ ([f ∗ ]k,m )([χP ]m,k + Im,k ) dxdtds

ψ(x, 0, s)Ψ [ηf∞ ]m,k (x, 0, s) dxds − 2Jm,k = 0, (7.3.42)

where

Jm,k =

Rd ×(0,T )×R

ψ[χH]m,k ∂s [f ∗ ]m,k dxdtds.

Here we use the equality Ψ = −2.

Step 2. First level limit. It follows from (7.3.33), (7.3.27) and (7.3.40) that χHL1 (Rd ×(0,T )×R) ≤ c

χ(s)(|s| + 1) (M (t) + 1) dtds (0,T )×R

≤ c(N + 1)2

(M (t) + 1) dt (0,T )

≤ c. Next estimate (7.3.28) for g (2) = P gives χP L1 (Rd ×(0,T )×R) ≤ c

χ(s)(M (t) + 1) dtds (0,T )×R

≤ c(N + 1)

(M (t) + 1) dt (0,T )

≤ c. By the general properties of molliﬁers, it follows that [χH]m,k → χH,

[χP ]m,k → χP

in L1 (Rd × (0, T ) × R) as (m, k) → ∞.

7.3. Proof of Theorem 7.1.12

213

On the other hand, the functions [f ∗ ]m,k are uniformly bounded and converge to f ∗ a.e. in Rd × (0, T ) × R as (m, k) → ∞. This leads to

Rd ×(0,T )×R

Ψ([f ∗ ]m,k )(∂t ψ + ∇ψ · v) + ψ[f ∗ ]2m,k div v dxdtds

+ Rd ×(0,T )×R

∂s ψ Ψ ([f ∗ ]k,m ) [χH]k,m + ψΨ ([f ∗ ]k,m )[χP ]m,k dxdtds

→

Rd ×(0,T )×R

Ψ(f ∗ )(∂t ψ + ∇ψ · v) + ψ(f ∗ )2 div v dxdtds

+ Rd ×(0,T )×R

{∂s ψ Ψ (f ∗ )H + ψΨ (f ∗ )P dxdtds

(7.3.43)

as (m, k) → ∞. Here we use the equalities ψ χ = ψ and ∂s ψ χ = ∂s ψ, which are direct consequences of (7.3.39) and (7.3.40). Repeating this argument we obtain ψ(x, 0, s)Ψ [ηf∞ ]m,k (x, 0, s) dxds → ψ(x, 0, s)Ψ ηf∞ (x, 0, s) dxds Rd ×R

Rd ×R

(7.3.44) as (m, k) → ∞. Next, applying the Lions-Di Perna commutator Lemma 1.6.1 and noting that f ∗ is bounded and ψ is bounded and compactly supported we obtain ψΨ ([f ∗ ]k,m )Im,k dxdtds = 0. (7.3.45) lim lim k→∞ m→∞

Rd ×(0,T )×R

Step 3. Second level limit. Now our task is to pass to the limit in the integral Jm,k = ψ[χH]m,k ∂s [f ∗ ]m,k dxdtds Rd ×(0,T )×R

By abuse of notation we will write simply Hk instead of [χH],k and fk instead of [f ∗ ],k . Notice that the functions Hk and their derivatives with respect to s of any order belong to L1 (Rd × (0, T ) × R). In turn, the functions fk and their derivatives with respect to s are bounded in Rd ×(0, T )×R. It now follows from the symmetry of the mollifying operator that Jm,k ≡ ψ[Hk ]m, ∂s [fk ]m, dxdtds = [ψ[Hk ]m, ]m, ∂s fk dxdtds. Rd ×(0,T )×R

Rd ×(0,T )×R

First we pass to the limit in the latter integral as m → ∞. To this end consider the function Φm,k (t, s) = [Hk ]m, (·, t, s) − Hk (·, t, s)L1 (Rd ) . In view of (7.3.40), Φm,k (t, s) = 0 for |s| ≥ N + 2. Since H satisﬁes (7.3.33), it meets all requirements of Lemma 7.3.3 with g replaced by H, and M1 , M2 replaced

214

Chapter 7. Kinetic theory. Fast density oscillations

by M , 1. It follows from estimate (7.3.18) in that lemma that Φm,k (t, s) ≤ [Hk ]m, (·, t, s)L1 (Rd ) + Hk (·, t, s)L1 (Rd ) = [χH]m,k (·, t, s)L1 (Rd ) + [χH],k (·, t, s)L1 (Rd ) ≤ c(N + 2)(M (t) + 1)

(7.3.46)

for a.e. (t, s) ∈ [0, T ] × R. By the general properties of the mollifying operators, Φm,k (t, s) → 0 as m → ∞ for a.e. (t, s) ∈ [0, T ] × R. On the other hand, in view of (7.3.27), the function M (t) + 1 is integrable on [0, T ]. Hence Φm,k (t, s) has an integrable majorant on the rectangle [0, T ] × [−N − 2, N + 2]. Applying the Lebesgue dominated convergence theorem we obtain 0

T

N +2

−N −2

Φm,k dsdt0 as m → ∞.

Since Φm,k vanishes for |s| ≥ N + 2, it follows that Φm,k → 0 in L1 ((0, T ) × R), which gives [Hk ]m, − Hk → 0 in L1 (Rd × (0, T ) × R) as m → ∞.

(7.3.47)

It follows that $ . [Hk ]m, − Hk 1 d ≤ [Hk ]m, − Hk L1 (Rd ×(0,T )×R) → 0 (7.3.48) m, L (R ×(0,T )×R) as m → ∞. Combining (7.3.47)–(7.3.48) with the identity . . $ $ [Hk ]m, m, − Hk = [Hk ]m, − Hk m, + [Hk ]m, − Hk implies . $ [Hk ]m, m, − Hk → 0 in L1 (Rd × (0, T ) × R) as m → ∞. Letting m → ∞ in the expression for Jm,k with ﬁxed k ≥ 1 and noting that the functions ∂s fk , ψ are bounded we arrive at Jm,k =

Rd ×(0,T )×R

$ . ψ[Hk ]m, m, ∂s fk dxdtds →

Rd ×(0,T )×R

ψHk ∂s fk dxdtds

as m → ∞. Recalling fk = [f ∗ ],k and Hk = [χH],k we ﬁnally obtain lim Jm,k =

m→∞

Rd ×(0,T )×R

ψ[χH],k ∂s [f ∗ ],k dxdtds.

(7.3.49)

7.3. Proof of Theorem 7.1.12

215

Step 4. Third level limit. Now our task is to pass to the limit in (7.3.49) as k → ∞. Let us consider the sequence of functions ψ[χH],k (x, t, s)∂s [f ∗ ],k (x, t, s) ds, (x, t) ∈ Rd × (0, T ). Φk (x, t) = R

Let us show that they have an integrable majorant independent of k. Inequalities (7.3.29) and (7.3.31) imply |χH| ≤ 2χ(s)|s| p(x, t) + χ(s)|s| | div v| in Q × R, |χH| ≤ χ(s)|s| | div v| + χ(s)|F∞ | in (Rd × (0, T )) \ Q × R Now set

p∗ (x, t) = p(x, t) in Q,

p∗ (x, t) = 0

elsewhere.

Since the given vector ﬁeld U belongs to C (R × (0, T )), formula (7.3.2) for F∞ yields |F∞ | ≤ c(|∂t ∞ | + |∇∞ | + ∞ ). (7.3.50) 1

d

It follows that |χH| ≤ c(N + 1) p∗ + | div v| + |∂t ∞ | + |∇∞ | + ∞ =: M(x, t).

(7.3.51)

Since the right hand side is independent of s, the same holds for [χH(x, t, s)],k . Thus we get |Φk (x, t)| ≤ |ψ[χH],k |(x, t, s)∂s [f ∗ ],k (x, t, s) ds ≤ M(x, t) ∂s [f ∗ ],k (x, t, s) ds R

= M(x, t) lim [f ∗ ],k (x, t, s) ≤ M(x, t). s→∞

R

(7.3.52)

Here we use the relation [f ∗ ],k (x, t, s) η(x, t) ≤ 1 as s → ∞, which follows from the similar relation for f ∗ . To estimate M, notice that ∞ ∈ C 1 (Rd × (0, T )) vanishes for all large |x|. Inequalities (7.3.32) imply p∗ L∞ (0,T ;L1 (Ω)) ≤ c. From this and estimates (7.3.26)–(7.3.27) for v we obtain ML1 (Rd ×(0,T )) ≤ c.

(7.3.53)

Our next task is to calculate limk→∞ Φk (x, t). To this end notice that f ∗ is monotone in s, and C[f ] is a diﬀerence of two functions monotone in s. Hence χ(s)H(x, t, s) as a function of s has the right and left limits at each s ∈ R. Therefore, χH and f ∗ , as functions of s, meet all requirements of Lemma 1.6.3. Applying relation (1.6.11) in that lemma and noting that χ, ψ are continuous we obtain ˜ ψ(x, t, s)χ(s)H(x, t, s) ds f ∗ (x, t, s), (7.3.54) lim Φk (x, t) = k→∞

R

216

Chapter 7. Kinetic theory. Fast density oscillations

˜ where H(x, t, s) = 2−1 limh 0 H(x, t, s − h) + 2−1 limh 0 H(x, t, s + h). The expression in (7.3.7) for H implies ˜ H(x, t, s) = −sηM(x, t, s) − s div u(x, t)f˜∗ (x, t, s) in Q × R, where M is given by (7.3.21), and ˜ H(x, t, s) = F∞ (x, t) − s div U(x, t) f˜∗ (x, t, s) in (Rd × (0, T )) \ Q × R. Next, identity (1.6.12) yields 1 1 ∗ ∗ 2 ∗ ˜ ψsf ds f = ψs ds (f ) = − ∂s (sψ)(f ∗ )2 ds, 2 R 2 R R 1 1 ψ f˜∗ ds f ∗ = ψ ds (f ∗ )2 = − ∂s ψ(f ∗ )2 ds. 2 2 R R R ˜ we obtain the existence of the pointwise limit From this and the expression for H lim Φk (x, t) = Φ(x, t)

k→∞

(7.3.55)

given by 1 ∗ 2 Φ = div u(x, t) ∂s (ψs)(f ) ds − ψsη M ds f ∗ in Q, 2 R R 1 1 ∗ 2 Φ = div U ∂s (ψs)(f ) ds − F∞ ∂s ψ (f ∗ )2 ds in (Rd × (0, T )) \ Q. 2 2 R R (7.3.56) Here we use the fact that χ = 1 on the support of ψ. Since by (7.3.52) and (7.3.53) the sequence Φk has an integrable majorant, Lebesgue’s dominated convergence theorem yields lim lim Jm,k = Φ(x, t) dxdt. (7.3.57) k→∞ m→∞

Rd ×(0,T )

Step 5. Renormalization. Letting ﬁrst m → ∞ and then k → ∞ in the integral identity (7.3.42) and using (7.3.43)–(7.3.45) and (7.3.57) we arrive at

Rd ×(0,T )×R

+

Ψ(f ∗ )(∂t ψ + ∇ψ · v) + ψ(f ∗ )2 div v dxdtds + ∂s ψ Ψ (f ∗ )H dxdtds − 2 Φ dxdt Rd ×(0,T )×R Rd ×(0,T ) ψΨ (f ∗ )P dxdtds + ψ(x, 0, s)Ψ(ηf∞ (x, 0, s)) dxds = 0.

Rd ×(0,T )×R

Rd ×R

(7.3.58)

7.3. Proof of Theorem 7.1.12

217

It follows from formulae (7.3.56) for Φ and (7.3.7) for H that

∂s ψ Ψ (f ∗ )Hdxdtds − 2

Q×R

Φ dxdt = 2

Q

η R

Q

sψ M ds f ∗ dxdt

s∂s ψ Ψ (f ∗ )f ∗ + (f ∗ )2 + (f ∗ )2 ψ div u + sη∂s ψ Ψ (f ∗ ) C[f ] dxdtds

− Q×R

(7.3.59) and

((Rd ×(0,T ))\Q)×R

∂s ψ Ψ (f ) H dxdtds − 2

−

∗

((Rd ×(0,T ))\Q)×R

Φ dxdt = (Rd ×(0,T ))\Q

∂s ψ Ψ (f ∗ )f ∗ +(f ∗ )2 s div U−F∞ +(f ∗ )2 ψ div U dxdtds. (7.3.60)

Next notice that Ψ (f ∗ )f ∗ + (f ∗ )2 ≡ Ψ(f ∗ ) and v = u in Q,

v = U and f ∗ = ηf∞

in ((Rd × (0, T )) \ Q.

Inserting (7.3.59)–(7.3.60) into (7.3.58) we obtain the integral identity (7.3.20) which is our claim. The next step is to eliminate the dependence of ψ on x and to replace the integral identity (7.3.20) by an integral inequality containing an arbitrary nonnegative test function depending only on t and s. Lemma 7.3.5. The inequality

Q×R

Ψ(f )(∂t Φ − s∂s Φ div u) − s∂s Φ Ψ (f ) C[f ] dxdtds +2 sΦM ds f dxdt ≥ 0 Q

(7.3.61)

R

holds for any nonnegative function Φ ∈ C ∞ ((0, T ) × R) vanishing for t = T and for all suﬃciently large |s|. Proof. First observe that for every Lipschitz function ψ : Rd × (0, T ) × R → R which vanishes in a neighborhood of the plane {t = T } and for all suﬃciently large |x|, |s|, there is a sequence of functions ψn such that ψn ∈ C ∞ (Rd × (0, T ) × R),

ψn = 0 for t = T and large |x|, |s|,

ψn → ψ uniformly in Rd × (0, T ) × R, and the ﬁrst derivatives of ψn are uniformly bounded and converge to the derivatives of ψ a.e. in Rd × (0, T ) × R. Such a sequence can be obtained by molliﬁcation of ψ. Substituting ψn into (7.3.20) and

218

Chapter 7. Kinetic theory. Fast density oscillations

letting n → ∞ we conclude that the integral identity (7.3.20) holds for all Lipschitz functions ψ deﬁned in Rd × (0, T ) × R such that ψ vanishes in a neighborhood of the plane {t = T } and for all suﬃciently large |x|, |s|. Let us consider the following construction. Introduce the set K = (ST \Σin )∪ {t = T } ⊂ Rd+1 . For every h > 0 denote by O(h) the h-neighborhood of K, O(h) = {(x, t) : dist((x, t), K) < h} Next, choose a Lipschitz function φ : Q → R vanishing in a neighborhood of K. By the sharp form of Whitney’s extension theorem [31], the function φ has a Lipschitz extension over Rd+1 , still denoted by φ. In particular, φ vanishes in O(h1 ) for some h1 > 0. We can choose 0 < h3 < h2 < h1 so small that Bi = {(x, t) : |x|+|t| < h−1 i } contain the cylinder Q. Thus we get B1 \ O(h1 ) B2 \ O(h2 ) B3 \ O(h3 ). Now choose ξ, η ∈ C0∞ (Rd+1 ) with the properties ξ = 1 in B1 \ O(h1 ), η = 1 in B2 \ O(h2 ),

ξ = 0 outside of B2 \ O(h2 ), η = 0 outside of B3 \ O(h3 ).

Finally, choose Φ ∈ C ∞ ((0, T ) × R) satisfying the hypotheses of the lemma and set ψ(x, t, s) = Φ(t, s)φ(x, t)ξ(x, t). Then ψ is Lipschitz in Rd × (0, T ) × Rd and vanishes for t = T and for suﬃciently large |x|, |s|. Moreover, by the choice of ξ, ψ = Φφ in Q × R. On the other hand, η = 1 and f ∗ = f on the support of ψ. Substituting ψ into the integral identity (7.3.20) we obtain

φ Ψ(f )(∂t Φ − s∂s Φ div u) − s ∂s Φ Ψ (f ) C[f ] dxdtds Q×R + ΦΨ(f )(∂t φ + ∇φ · u) dxdtds + 2 φ sΦM ds f dxdt + N = 0, Q×R

R

Q

where N is given by (7.3.22). Since η = 1 on the support of ψ, it follows from the 2 = f∞ and formula (7.3.9) for P that identity f∞ Ψ(ηf∞ ) = Ψ(f∞ ) = 0,

P = 0 on the support of ψ.

Hence N = 0 and φ Ψ(f )(∂t Φ − s∂s Φ div u) − s∂s Φ Ψ (f ) C[f ] dxdtds Q×R + ΦΨ(f )(∂t φ + ∇φ · u) dxdtds + 2 φ sΦM ds f dxdt = 0. (7.3.62) Q×R

Q

R

Observe that by Condition 7.1.11, the vector ﬁeld U and the surface ST = ∂Ω × (0, T ) meet the hypotheses of the geometric Theorem 13.3.3. Hence there exists a

7.3. Proof of Theorem 7.1.12

219

sequence of Lipschitz functions φn : Q → [0, 1] such that each φn vanishes in some neighborhood of ST \ Σin and of {t = T }, φn → 1 everywhere in Q and G(∂t φn + ∇φn · u) dxdt ≤ 0 Q

for all bounded nonnegative functions G. Substituting φ = φn in (7.3.62) and letting n → ∞ we arrive at (7.3.61). Our next task is to obtain an analog of inequality (7.3.61) with the test function Φ independent of s. The derivation of such an inequality is based on the following lemma concerning the properties of C and M: Lemma 7.3.6. For a.e. (x, t) ∈ Q, the function C[f ](x, t, ·) is nonpositive. Moreover, for all g ∈ C0∞ (0, ∞), (λ + 1) g(s)C[f ](x, t, s) ds = g (τ )Vτ (x, t) dτ (7.3.63) R

[0,∞)

where Vτ is deﬁned by (7.1.39). Proof. Since p(s) is monotone, for almost every (x, t) ∈ Q there exists s ∈ [0, ∞), depending on (x, t), such that p(s) ≤ p(x, t) for s ≤ s and p(s) ≥ p(x, t) otherwise. Since f (x, t, s) = 0 for s < 0 we have for any s ≤ s ≤ p(x, t), 1 (p(s) − p) ds f (x, t, s) ≤ 0. C[f ](x, t, s ) − C[f ](x, t, s ) = λ + 1 [s ,s ) The same arguments give C[f ](x, t, s ) ≥ C[f ](x, t, s ) for s ≤ s ≤ s . Hence C[f ](x, t, ·) does not increase on (−∞, s) and does not decrease on (s, ∞). Since C[f ](x, t, ·) vanishes at ±∞, it is nonpositive. The same conclusion can be drawn for M. Next, it follows from the identity (p(z) − p(x, t)) dz f (x, t, z) = − (p(z) − p(x, t)) dz f (x, t, z) [0,s)

[s,∞)

that (λ + 1) R

g(s) C[f ](x, t, s) ds g (τ ) dτ (p(z) − p) dz f (x, t, z) ds = [0,∞) [s,∞) [s,∞) = g (τ ) ds (p(z) − p) dz f (x, t, z) dτ [0,∞) [0,τ ) [s,∞) g (τ ) F (s) dG(s) dτ = [0,∞)

[0,∞)

220

Chapter 7. Kinetic theory. Fast density oscillations

where G(s) = min{s, τ },

(p(z) − p) dz f (x, t, z).

F (s) = [s,∞)

It remains to note that in view of Lemma 1.3.7, F (s) dG(s) = − lim F (s)G(s) − G(s) dF (s) s→−0 [0,∞) [0,∞) = G(s)(p(s) − p) ds f = min{s, τ }(p(s) − p) ds f ≡ Vτ (x, t). [0,∞)

[0,∞)

Now we are in a position to eliminate the variable s from inequality (7.3.61). Lemma 7.3.7. Under the assumption of Theorem 7.1.12, Ψ(f ) dxdtds ≤ 0.

(7.3.64)

Q×R

Proof. Introduce a function ω ∈ C0∞ (R) with the properties ω ≥ 0, supp ω ⊂ (0, 1), ω(s) ds = 1. R

Fix ξ ≥ 1 and set

∞

υ(s) =

ω(τ − ξ) dτ,

h(t) = T − t.

s

Substituting Φ(t, s) = h(t)υ(s) into the integral inequality (7.3.61) we obtain Ψ(f )υ(s) dxdtds + 2 h(t)υ(s)(sM) ds f (x, t, s) dxdt − Q×R Q R ≥ sυ (s)h(t)Ψ(f ) div u dxdtds + sυ (s)h(t)Ψ (f ) C[f ] dxdtds. Q×R

Q×R

(7.3.65) Recall that f , Ψ(f ) and C[f ], M vanish for s < 0. In view of Lemma 7.3.6, M is nonpositive, which leads to h(t)υ(s)(sM) ds f (x, t, s) dxdt ≤ 0. (7.3.66) Q

R

Since υ ≤ 0, we have sυ (s)Ψ(f )h(t) div u dxdtds ≥ T Q×R

sυ (s)Ψ(f )| div u| dxdtds

Q×R

υ (s)Ψ(f )| div u| dxdtds.

≥ (ξ + 1)T Q×R

(7.3.67)

7.3. Proof of Theorem 7.1.12

221

Let us introduce the function | div u| ℘1 (s) = T Q

and note that

Ψ(f )(x, t, τ ) dτ dxdt

υ (s)℘1 (s) ds.

υ (s)Ψ(f )| div u| dxdtds = −

T

(7.3.68)

[0,s]

Q×R

[0,∞)

Then we can write inequality (7.3.67) in the form Ψ(f )h(t)υ (s) div u dxdtds ≥ −(ξ + 1) Q×R

υ (s)℘1 (s) ds.

(7.3.69)

[0,∞)

Next, by Lemma 7.3.6 we have υ (s) C[f ] ≥ 0 and υ (s) C[f ] = 00 for s ≥ ξ + 1, which along with the inequality Ψ (f ) = 1 − 2f ≥ −1 gives sυ (s)h(t)Ψ (f ) C[f ] dxdtds ≥ −(ξ + 1)T υ (s) C[f ] dxdtds. Q×R

Q×R

On the other hand, in view of (7.3.63), we have 1 υ (s) C[f ] dxdtds = υ (s) Vs (x, t) dxdtds. λ + 1 Q×R Q×R Thus we get sυ (s)h(t)Ψ (f ) C[f ] dxdtds ≥ −c(ξ + 1) Q×R

υ (s) Vs (x, t) dxdtds,

Q×R

(7.3.70) where c = T /(λ + 1). Let us introduce the function ℘2 (s) = c Vs dxdt

(7.3.71)

Q

and note that

υ (s) Vs (x, t) dxdtds =

c Q×R

υ (s)℘2 (s) ds; [0,∞)

hence we can rewrite inequality (7.3.70) in the form sυ (s)h(t)Ψ (f ) C[f ] dxdtds ≥ −(ξ + 1) Q×R

υ (s)℘2 (s) ds.

(7.3.72)

[0,∞)

Inserting inequalities (7.3.66), (7.3.69) and (7.3.72) into (7.3.65) we obtain − Ψ(f )υ(s) dxdtds ≥ −(ξ + 1) υ (s)(℘1 (s) + ℘2 (s)) ds. (7.3.73) Q×R

[0,∞)

222

Chapter 7. Kinetic theory. Fast density oscillations

Next recalling the deﬁnition of υ, we get

d υ (s)℘i (s) ds = − ω (s − ξ)℘i (s) ds = dξ [0,∞) [0,∞)

ω(s − ξ)℘i (s) ds, [0,∞)

which along with (7.3.73) yields the inequality

d Ψ(f )υ(s) dxdtds ≤ (1 + ξ) dξ Q×R

ω(s − ξ)(℘1 (s) + ℘2 (s)) ds. [0,∞)

Since υ(s) = 1 for s ≤ ξ and Ψ ≥ 0 we have Ψ(f ) dxdtds ≤ (1 + ξ) Q×[0,ξ]

d dξ

ω(s − ξ)(℘1 (s) + ℘2 (s)) ds.

(7.3.74)

[0,∞)

Let us prove that the functions ℘i , i = 1, 2, are bounded on the positive semiaxis. At this point we use Condition 7.1.11 which says that f satisﬁes inequality (7.1.38), i.e. f is a regular solution to the kinetic equation (7.1.45). Notice that the obvious inequality

s

Ψ(f (x, t, s)) ds ≤ 0

∞

Ψ(f (x, t, s)) ds ≡ H(x, t)

0

along with (7.1.38) implies |℘1 (s)| ≤ c div uL2 (Q) HL2 (Q) ≤ c < ∞ for all s ≥ 0,

(7.3.75)

which yields the boundedness of ℘1 . Using estimate (7.1.38) for Vs we conclude that |℘2 | ≤ c with a constant c independent of ξ. Therefore, the smooth function ξ →

ω(s − ξ)(℘1 (s) + ℘2 (s)) ds [0,∞)

is uniformly bounded on the positive semi-axis. Hence there exists a sequence ξk → ∞ such that (1 + ξk )

d dξ

[0,∞)

[ω(s − ξ)(℘1 (s) + ℘2 (s))]

ξ=ξk

ds → 0 as k → ∞.

Substituting ξ = ξk into (7.3.74) and letting k → ∞, we arrive at (7.3.64).

It follows from Lemma 7.3.7 and Ψ(f ) = f (1 − f ) ≥ 0 that f (1 − f ) = 0 a.e. in Q × R. Hence the distribution function f is deterministic in the sense of Deﬁnition 7.1.10. This completes the proof of Theorem 7.1.12.

7.4. Adiabatic exponent γ > 3/2

223

7.4 Compactness and existence of solutions for adiabatic exponent γ > 3/2 Now we are in a position to answer the questions formulated at the beginning of this chapter. In this section the restriction γ > 2d on the adiabatic exponent γ and the compatibility conditions imposed in Conditions 5.1.1 on the data and the constitutive relations, obeying in Theorem 5.1.6, are relaxed. This is important since the range γ > 2d, assumed in Theorem 5.1.6, in applications corresponds only to liquids and metals, while γ > d/2 covers all gases in the case of space dimension d = 2 and mono-atomic gases in the case d = 3. The compatibility conditions also cannot be accepted from the physical standpoint, since they require the existence of a vacuum zone near ∂Ω ∩ {t = 0}. The main result of this section is the following theorem: Theorem 7.4.1. Suppose the pressure function p() satisﬁes Condition 7.1.1 and has the representation p = pc () + pb (), where pc is convex and pb is bounded. Furthermore, assume that the given functions ∞ ∈ C 1 (Rd × [0, T ]),

U ∈ C ∞ (Rd × [0, T ]),

(7.4.1)

and the bounded domain Ω ⊂ Rd with C ∞ boundary satisfy the geometric Condition 7.1.2. Then problem (5.1.12) has a weak renormalized solution (u, ) which meets all requirements of Deﬁnition 5.1.4 and satisﬁes the estimate uL2 (0,T ;W 1,2 (Ω)) + |u|2 L∞ (0,T ;L1 (Ω)) + L∞ (0,T ;Lγ (Ω)) ≤ ce ,

(7.4.2)

where the constant ce depends only on γ > d/2, UC 1 (Rd ×[0,T ]) , ∞ L∞ (Rd ×[0,T ]) , d, diam Ω, T , and the constant cp in Condition 7.1.1. Moreover, for any cylinder Q Q, and any θ with 0 < θ < min{2γd−1 − 1, 2−1 γ}, there is a constant c, depending only on θ, ce , and Q , such that γ+θ dxdt ≤ c. (7.4.3) Q

Theorem 7.4.1 is a starting point for further investigations of properties of solutions to in/out ﬂow problems for compressible Navier-Stokes equations, important for applications. In particular, as a result of such investigations, the existence of solutions in nonsmooth domains can be shown, and the continuous domain dependence of solutions can be established. Proof. Denote by Υ the bottom edge of the cylinder Q, Υ = {(x, 0) : x ∈ ∂Ω}. Introduce η ∈ C ∞ (Rd × [0, T ]) such that

0 ≤ η ≤ 1, η (x, t) = 1 for dist (x, t), Υ > , η = 0 in a neighborhood of Υ.

(7.4.4) (7.4.5)

224

Chapter 7. Kinetic theory. Fast density oscillations

Next choose a sequence of functions ρ∞ ∈ C ∞ (Rd × [0, T ]) such that ρ∞ → ∞ in C(Q) as → 0. Set ∞ = η ρ∞ . The functions ∞ belong to C ∞ (Rd × (0, T )), and vanish in a neighborhood of the edge Υ. Therefore, the sequence ∞ satisﬁes Condition 7.1.2 and hence meets the requirements of Theorem 7.1.9. Notice that under the hypotheses of Theorem 7.4.1 the artiﬁcial pressure function p given by (7.1.5) and the vector ﬁeld U also satisfy Condition 7.1.2. Applying Theorem 7.1.3 we conclude that the modiﬁed problem (7.1.8) admits a family of renormalized solutions (u , ), > 0, which meets all requirements of Theorem 7.1.3 and, in particular, satisﬁes estimates (7.1.9), (7.1.10). After passing to a subsequence we can assume that there exist u ∈ L2 (0, T ; W 1,2 (Ω)) and ∈ L∞ (0, T ; Lγ (Ω)) such that u u weakly in L2 (0, T ; W 1,2 (Ω)), weakly in L∞ (0, T ; Lγ (Ω)), γ+θ

weakly in L

(7.4.6)

(Q ) for all Q Q.

From this and the fundamental Young measure theorem 1.4.5 we conclude that there exist subsequences, still denoted by and ∞ , and Young measures μ ∈ ∞ L∞ ∈ L∞ w (Q; M(R)) and μ w ( T ; M(R)), with the following properties: For any ϕ ∈ C0 (R) we have ϕ( ) ϕ ϕ(∞ ) ϕ∞

weakly in L∞ (Q),

∞

weakly in L (

where ϕ(x, t) = μxt , ϕ,

(7.4.7) (7.4.8)

T ),

ϕ∞ (x, t) = μ∞ xt , ϕ.

(7.4.9)

μ∞ xt

= δ(· − ∞ (x, t)) Moreover, since ϕ(∞ ) converges to ∞ a.e. in T , we have and the corresponding distribution function f∞ is deterministic in the sense of Deﬁnition 7.1.10. Applying Theorem 7.1.9 we conclude that the limit vector ﬁeld u and the distribution function f of the measure μxt serve as a solution to the kinetic Problem 7.1.5 and this solution is regular in the sense of Deﬁnition 7.1.8. Since the boundary distribution function f∞ is deterministic, we can apply Theorem 7.1.12 to deduce that so is f , and the Young measure μxt , associated with the sequence , is the Dirac measure concentrated at some point (x, t). This means that = ,

p = p(),

ϕ = ϕ().

The integral identity (7.1.42) along with the equality p = p() implies that the couple (u, ) satisﬁes the momentum balance equation. Since in the deterministic case pϕ = p()ϕ() = p ϕ, it follows from the integral identity (7.2.30) and from ϕ = ϕ(),

Φ = Φ() ≡ ϕ () − ϕ()

that is a renormalized solution to the mass balance equations. It remains to note that estimates (7.4.2)–(7.4.3) are consequences of (7.1.9)–(7.1.10).

Chapter 8

Domain convergence The question of domain stability of solutions to compressible Navier-Stokes equations is important for many applications. It can be formulated as follows. Assume that domains Ωn ⊂ Rd converge in some sense to a domain Ω. The question is whether a sequence of solutions to a boundary value problem for compressible Navier-Stokes equation in Ωn contains a subsequence which converges weakly to a solution of the same boundary value problem for compressible Navier-Stokes equations in the limiting domain Ω. In this auxiliary chapter we investigate in detail the question of convergence of subsets of Euclidean space Rd . We discuss two types of convergence, the Hausdorﬀ convergence of compact sets in Rd and the Kuratowski-Mosco convergence of subspaces of Banach spaces. For the purposes of shape optimization we introduce the S-convergence of compact sets, which combines the Hausdorﬀ convergence with the Kuratowski-Mosco convergence. We formulate suﬃcient geometrical conditions which provide the compactness of classes of compact sets in the S topology.

8.1 Hausdorﬀ and Kuratowski-Mosco convergences Hausdorﬀ convergence. If K and S are compact subsets of Rd , then the Hausdorﬀ distance dH between them is deﬁned as follows: dH (K, S) = max sup dist(x, K), sup dist(y, S) . (8.1.1) x∈S

y∈K

Equipped with this metric, the totality of compact subsets of Rd becomes a complete metric space. A remarkable property of this space is that it is locally compact. In particular, we have the following lemma: Lemma 8.1.1. Let B be a bounded domain in Rd . Then any sequence of compact sets Sn ⊂ B contains a subsequence which converges to some compact set S ⊂ B in the Hausdorﬀ metric.

P. Plotnikov and J. Sokoáowski, Compressible Navier–Stokes Equations: Theory and Shape Optimization, Monografie Matematyczne 73, DOI 10.1007/978-3-0348-0367-0_8, © Springer Basel 2012

225

226

Chapter 8. Domain convergence

The notion of the Hausdorﬀ metric is not as useful as it seems at ﬁrst glance, since the topological and metric properties of the Hausdorﬀ limit are not related to the properties of the members of the convergent sequence. Kuratowski-Mosco convergence. Kuratowski-Mosco convergence is widely applied in the analysis of domain dependence of solutions to elliptic equations. It deals with the properties of sequences of functions un from Sobolev spaces W 1,p (Ωn ) on a family of domains Ωn . However, this convergence is related to sequences of Banach spaces rather than of domains. For simplicity of presentation we restrict our considerations to Kuratowski-Mosco convergence of subspaces of a “hold-all” Banach space. Deﬁnition 8.1.2. Let Y be a Banach space and let Xn , n ≥ 1, X, be closed subspaces of Y . The sequence Xn converges to X in the sense of KuratowskiMosco if • for any sequence un u weakly convergent in Y with un ∈ Xn , the limit element u belongs to X; • whenever u ∈ X, there is a sequence un ∈ Xn with un → u strongly in Y . With applications to nonstationary problems in mind, we investigate the properties of Kuratowski-Mosco convergence for spaces of functions depending on the time variable. The following proposition is used in the proof of the domain stability of solutions to initial-boundary value problems for compressible NavierStokes equations. Proposition 8.1.3. Let Y be a separable reﬂexive Banach space, and suppose that a sequence of closed subspaces Xn ⊂ Y converges to a closed subspace X ⊂ Y in the sense of Kuratowski-Mosco. Furthermore, assume that a sequence un ∈ Lr (0, T ; Xn ), n ≥ 1, 1 < r < ∞, converges weakly in Lr (0, T ; Y ) to some u ∈ Lr (0, T ; Y ). Then u ∈ Lr (0, T ; X). Proof. First we prove the result under the assumption that the mappings t → un (t) are uniformly bounded, i.e., un (t)Y ≤ M < ∞ for all t ∈ (0, T ), n ≥ 1.

(8.1.2)

For any interval I ⊂ (0, T ), denote by LI the set of all weak limit points in Y of sequences un (t), t ∈ I. A point u ∈ Y belongs to LI if and only if there exists t ∈ I and a subsequence unm (t) such that unm (t) y weakly in Y as m → ∞. Denote by CI the closure in Y of the convex hull of LI . Since un (t) ∈ Xn , Deﬁnition 8.1.2 implies that LI ⊂ X. Hence CI is a closed convex subset of X. Let us show that there exists a set E of full measure in (0, T ) such that for every t0 ∈ E, u(t) − u(t0 )Y dt = 0, where In = (t0 − 1/2n, t0 + 1/2n). (8.1.3) lim n n→∞

In

8.1. Hausdorﬀ and Kuratowski-Mosco convergences

227

By the Lusin Theorem 1.3.13, for any > 0, there is a closed set T ⊂ (0, T ) such that the mapping u : T → Y is continuous and meas((0, T ) \ T ) < . Denote by E the set of all Lebesgue points of T . Recall that E is of full measure in T . By the deﬁnition of the Lebesgue point, for any t0 ∈ E , lim n meas (In \ E ) = 0.

n→∞

Notice that by (8.1.2) the mapping u : t → u(t) is bounded. Thus we get n In

u(t) − u(t0 )Y dt ≤n u(t) − u(t0 )Y dt + n In ∩E

In \E

u(t) − u(t0 )Y dt

≤ sup u(t) − u(t0 )Y + 2M n meas(In \ E ) → 0 as n → ∞. In ∩E

Therefore, (8.1.3) holds for all t0 ∈ E and hence for all t0 ∈ E = E . Since meas((0, T ) \ E ) < , the set E is of full measure in (0, T ). This proves (8.1.3). Next we show that for all t0 ∈ E, u(t0 ) ∈ CIn , where In = (t0 − 1/2n, t0 + 1/2n). (8.1.4) n

Assume that, contrary to our claim, u(t0 ) ∈ / n CIn . Since CIn decreases, there is / CIn for all n ≥ N . Since CIn is a closed convex an integer N such that u(t0 ) ∈ subset of the Banach space Y , it follows from the Hahn-Banach theorem that there exist a functional f ∈ Y and a constant e such that f (u(t0 )) > e > sup f (u).

(8.1.5)

u∈CIN

In order to obtain a contradiction we will prove that lim sup f (uk (t)) ≤ e for any t ∈ IN .

(8.1.6)

k→∞

To this end ﬁx t ∈ IN and a subsequence uki (t) with lim f (uki (t)) = lim sup f (uk (t)).

i→∞

(8.1.7)

k→∞

By (8.1.2), the sequence uki (t) is bounded in Y . Passing to a subsequence we may assume that uki (t) converges weakly to some v ∈ Y . Thus v ∈ LIN ⊂ CIN . It now follows from (8.1.5) that lim f (uki (t)) = f (v) ≤ e,

i→∞

228

Chapter 8. Domain convergence

which proves (8.1.6). Next, since uk (t) converges to u(t) weakly in Lr (0, T ; Y ), it follows from (8.1.6) that for any n ≥ N , n f (u(t)) dt = lim n f (uk (t)) dt ≤ n lim sup f (uk (t)) dt ≤ e. (8.1.8) k→∞

In

In

In

k→∞

On the other hand, (8.1.3) and (8.1.5) imply lim inf n f (u(t)) dt = lim n f (u(t0 )) dt + lim inf n f (u(t) − u(t0 )) dt n→∞ n→∞ n→∞ In In In = f (u(t0 )) + lim inf n f (u(t) − u(t0 )) dt n→∞ In ≥ f (u(t0 )) − f Y lim u(t) − u(t0 )Y dt n→∞

In

(8.1.9)

= f (x(t0 )) > e,

which contradicts (8.1.8). This proves the inclusion (8.1.4). Since E is a set of full measure in (0, T ), it follows from (8.1.4) that CIn ⊂ X for a.e. t0 ∈ (0, T ), (8.1.10) u(t0 ) ∈ n

which implies the inclusion u ∈ Lr (0, T ; X). Hence the assertion of Proposition 8.1.3 holds for all bounded sequences un (t) satisfying (8.1.2). Let us consider the general case. Choose an arbitrary sequence un (t) with the properties un u

un (t) ∈ Xn

weakly in Lr (0, T ; Y ),

for a.e. t ∈ (0, T )

and set uM n (t) = un (t) if un (t)Y ≤ M

uM n (t) = 0 otherwise,

and

where M is an arbitrary ﬁxed positive number. It is clear that the elements uM n satisfy (8.1.2). By Corollary 1.3.25 the space Lr (0, T ; Y ) is reﬂexive. Hence passing r to a subsequence we can assume that uM n , n ≥ 1, converges weakly in L (0, T ; Y ) M r M r to some u ∈ L (0, T ; Y ). Moreover, as proved above, u ∈ L (0, T ; X). Since 1 M r > 1, the sequences un and uM n converge weakly in L (0, T ; Y ) to u and u , respectively. Thus we get T T M u (t) − u(t)Y dt ≤ lim inf uM n (t) − un (t)Y dt n→∞ 0 0 un (t)Y dt = lim inf n→∞

≤ lim inf

{t:un (t)Y ≥M }

dt

n→∞

1/r

{t:un (t)Y ≥M }

≤ c lim inf meas {t : un (t)Y ≥ M } n→∞

{t:un (t)Y ≥M }

1/r

.

un (t)rY dt

1/r

8.2. Capacity, quasicontinuity and ﬁne topology On the other hand, we have 1 meas {t : un (t)Y ≥ M } ≤ M ≤

1 M

229

{t:un (t)Y ≥M }

T

un Y dt ≤ 0

un (t)Y dt c . M

Combining the results obtained we arrive at T uM (t) − u(t)Y dt ≤ cM −1/r . 0

Therefore, u converges to u in L1 (0, T ; Y ) as M → ∞. Passing to a subsequence we conclude that uM (t) − u(t)Y → 0 as M → ∞ for a.e. t. Since uM (t) ∈ X, the values of the limit function u(t) also belong to X for a.e. t ∈ (0, T ), which proves Proposition 8.1.3. M

8.2 Capacity, quasicontinuity and ﬁne topology In this section we collect basic facts on ﬁne properties of sets and Sobolev functions. The main relevant sources are [1], [28], [76]. Our presentation is mainly based on the monograph [1]. Capacity. Sobolev functions when compared with Lebesgue integrable functions enjoy many additional useful properties. In particular, Sobolev functions are well deﬁned on some sets of zero Lebesgue measure, for instance, on smooth manifolds of codimension one. The properties that characterize pointwise behavior of Sobolev functions are called ﬁne properties. Capacity is a very useful technical tool in the ﬁne theory of Sobolev functions. It can be regarded as a nonnegative function deﬁned on subsets of Euclidean space Rd , and hence capacity theory has many common features with measure theory. But there is one important diﬀerence: capacity is not an additive function of sets but only subadditive. The explicit deﬁnition of capacity (Bessel capacity) is the following (cf. Deﬁnitions 2.2.1, 2.2.2, 2.2.4 in [1]): Deﬁnition 8.2.1. Let K ⊂ Rd be compact. Then Cap(K) = inf {ϕ2W 1,2 (Rd ) : ϕ ∈ C0∞ (Rd ), ϕ ≥ 1 on K}. Deﬁnition 8.2.2. Let G ⊂ Rd be open. Then Cap(G) = sup {Cap(K) : K ⊂ G, K compact}. Deﬁnition 8.2.3. Let A ⊂ Rd be arbitrary. Then Cap(A) = inf {Cap(G) : G ⊃ A, G open}.

230

Chapter 8. Domain convergence

An equivalent deﬁnition can be given in terms of the Bessel kernel GB which /B (ξ) = (1 + |ξ|2 )−1/2 , ξ ∈ Rd . is the inverse Fourier transform of the function G It is well known (see [133]) that a function ϕ is in W 1,2 (Rd ) if and only if it is a convolution ϕ = GB ∗ g, g ∈ L2 (Rd ), and c−1 gL2 (Rd ) ≤ ϕW 1,2 (Rd ) ≤ cgL2 (Rd ) , where c depends only on d. This leads to Cap(A) = inf {gL2 (Rd ) : g ≥ 0, GB ∗ g ≥ 1 on A}. Capacity enjoys many properties similar to properties of measures. It is nonnegative, monotone, and subadditive, i.e. A ⊂ B ⇒ 0 ≤ Cap(A) ≤ Cap(B), Cap(A ∪ B) ≤ Cap(A) + Cap(B), For any compact set K ⊂ Rd , Cap(K) = inf {Cap(G) : G ⊃ K, G is open}. Capacity also enjoys nice metric properties. In particular, it is invariant with respect to translations and rotations and more generally, with respect to all aﬃne isometries. The following lemma given in [1] shows that capacity has some invariance properties with respect to bi-Lipschitz transformations of the ambient Euclidean space. Lemma 8.2.4. Let χ : Rd → Rd be a bijection that is bi-Lipschitz: k −1 |x − x | ≤ |χ(x ) − χ(x )| ≤ k|x − x |

for all x , x ∈ Rd .

(8.2.1)

Then there is a constant c > 0, depending only on k and d, such that for any compact set K ⊂ Rd , c−1 Cap(K) ≤ Cap(χ(K)) ≤ c Cap(K). In many important cases, capacity admits robust estimates. In particular, in the three dimensional case we have c−1 a ≤ Cap(Ba ) ≤ ca

(8.2.2a)

for the ball Ba = {|x| ≤ a} ⊂ R3 , and c−1 a ≤ Cap(Da ) ≤ ca

(8.2.2b)

for the disk Da = {x = (x1 , x2 , 0) : |x| ≤ a} ⊂ R3 , where c is an absolute constant.

8.2. Capacity, quasicontinuity and ﬁne topology

231

In the two-dimensional case, calculating the capacity is more complicated because of the peculiarities of the logarithmic potential. For a ∈ (0, 1), we have (see [76]) c−1 / log(2/a)−1 ≤ Cap(Ba ) ≤ c/ log(2/a)−1 (8.2.2c) for Ba = {|x| ≤ a} ⊂ R2 , and c−1 / log(2/a)−1 ≤ Cap(Ia ) ≤ c/ log(2/a)−1

(8.2.2d)

for Ia = {x = (x1 , 0) : |x1 | ≤ a} ⊂ R2 . Here, the constant c is independent of a. If K is a locally Lipschitz manifold in Rd then Cap(K) = 0 if dim K ≤ d − 2, Cap(K) > 0 if dim K ≥ d − 1. Thickness. The notion of thickness of a set at some point arises in potential theory in connection with the continuity properties of harmonic functions. Recall the classical result on solvability of the Dirichlet problem for the Laplace equation (see [76]). Theorem 8.2.5. Let Ω ⊂ Rd be a bounded domain and f : Rd → R be a continuous function. Assume that for every x ∈ A := Rd \ Ω, ∞

2n(d−2) Cap(B2−n (x) ∩ A) = ∞,

(8.2.3)

n=0

where Ba (x) is the ball of radius a centered at x. Then the Dirichlet problem Δv = 0

in Ω,

v=f

on ∂Ω,

(8.2.4)

has a unique solution v ∈ C(Ω) such that v(y) → f (x)

as Ω y → x

for all x ∈ ∂Ω.

(8.2.5)

Condition (8.2.3) is known as the Wiener regularity condition and it ensures that the closed set Rd \ Ω is in some sense “thick” at the point x. Thus we come to the following deﬁnition of thickness ([1, Def. 6.3.7]): Deﬁnition 8.2.6. We say that a set E ⊂ Rd is thin at a point x if 1 ds Cap(Bs (x) ∩ E) d−1 < ∞, s 0 or equivalently,

∞

2n(d−2) Cap(B2−n (x) ∩ E) < ∞,

(8.2.6)

(8.2.7)

n=0

where Bs (x) = {y ∈ Rd : |y − x| ≤ s}. If a set is not thin at x it is thick at x. A set E is thick if it is thick at every point x ∈ E.

232

Chapter 8. Domain convergence Recall some simple properties of thick and thin sets.

Lemma 8.2.7. If F ⊂ Rd is thick at x and E ⊂ Rd is thin at x then F \ E is thick at x. Moreover, x is a limit point of F . Proof. As capacity is subadditive, we have Cap(Bs (x)∩(F \E)) ≥ Cap(Bs (x)∩F ) − Cap(Bs (x) ∩ E) and

1

Cap(Bs (x) ∩ (F \ E))

0

≥ lim

→0

1

ds sd−1

Cap(Bs (x) ∩ F )

ds sd−1

1

− lim

→0

Cap(Bs (x) ∩ E)

ds sd−1

.

It remains to note that the ﬁrst limit on the right hand side is inﬁnite while the second is ﬁnite. To prove that x is a limit point of F it suﬃces to note that in the opposite case, Cap(Bs (x) ∩ F ) = 0 for all suﬃciently small s, and the integral 0

1

Cap(Bs (x) ∩ F )

ds sd−1

converges at 0. This contradicts the thickness of F at x.

Lemma 8.2.8. Suppose a bi-Lipschitz bijection χ : Rd → Rd satisﬁes condition (8.2.1) and a compact set T is thick at 0. Then for any x ∈ Rd , the set x + χ(T ) is thick at x. Proof. If y ∈ B(0, s/k) then, by (8.2.1), χ(y) ∈ B(0, s). This leads to the inclusion B(0, s) ∩ χ(T ) ⊃ χ(B(0, s/k) ∩ T ). Since capacity is invariant with respect to translations of Rd , we obtain Cap B(x, s) ∩ (x + χ(T )) = Cap B(0, s) ∩ χ(T ) ≥ Cap χ(B(0, s/k) ∩ T ) . On the 8.2.4 with K replaced by χ(B(0, s/k) ∩ T ) implies other hand, Lemma χ Cap (B(0, s/k) ∩ T ) ≥ c(k)Cap(B(0, s/k) ∩ T ). Thus we get Cap B(x, s) ∩ (x + χ(T )) ≥ c(k) Cap(B(0, s/k) ∩ T ). From this we conclude that 1 1 ds ds Cap B(x, s) ∩ (x + χ(T )) d−1 ≥ c(k) Cap(B(0, s/k) ∩ T ) d−1 s s 0 0 1/k ds = c(k) Cap(B(0, s) ∩ T ) d−1 = ∞. s 0 The following examples illustrate the properties of thick sets; the examples are also used in the next sections.

8.2. Capacity, quasicontinuity and ﬁne topology

233

Example 8.2.9 (Flat spearhead). For ω ∈ (0, π/2] consider the plane sector (ω) = {x ∈ R3 : x = (r cos α, r sin α, 0), |α| ≤ ω, r ∈ (0, 1)} ⊂ R3 .

(8.2.8)

Lemma 8.2.10. The set (ω) is thick at 0. Proof. Choose an integer N ≥ [π/ω]+1. Note that the disk Ds = {x ∈ R3 : x3 = 0, x21 + x22 ≤ s2 } can be covered by the N sectors obtained from B(0, s) ∩ (ω) by rotation through iπ/ω, 1 ≤ i ≤ N , about the x3 axis. Since capacity is invariant under rotations, it follows that N Cap(B(0, s) ∩ (ω)) ≥ Cap Ds . Combining this with (8.2.2b) we arrive at Cap(B(0, s) ∩ (ω)) ≥ cs/N , which yields 1 1 ds c ds = ∞. Cap(B(x, s) ∩ (ω)) 2 ≥ s N 0 s 0 Example 8.2.11 (Interval). Let us consider the one-dimensional interval I = {x ∈ R2 : x = (r, 0), r ∈ [0, 1]} ⊂ R2 .

(8.2.9)

Lemma 8.2.12. The set I is thick at 0. Proof. Notice that B(0, s) ∩ I = [0, s] and by (8.2.2d) we have Cap(B(0, s) ∩ I) ≥ c| log s|−1 . Hence 1 1 ds ds = ∞. Cap(B(x, s) ∩ I) 1 ≥ c s 0 0 s log |s| Fine topology, ﬁne continuity. For given x ∈ Rd , we can regard the sets which are thin at x as “small”. This leads to the concepts of ﬁne topology and ﬁne continuity, based on the following deﬁnition (see [1, Defs. 6.4.1–6.4.2]): Deﬁnition 8.2.13. Let x ∈ Rd . Then a set U is called a ﬁne neighborhood of x if x ∈ U and Rd \ U is thin at x. A set G ⊂ Rd is ﬁnely open if it is a ﬁne neighborhood of each of its points. A set F ⊂ Rd is ﬁnely closed if Rd \ F is ﬁnely open. Finely open sets deﬁne the ﬁne topology in Rd , which leads to the notion of a ﬁnely continuous function. Deﬁnition 8.2.14. A real-valued function f deﬁned on a set F ⊂ Rd is ﬁnely continuous at x ∈ F if the set {y ∈ F : |f (y) − f (x)| ≥ } is thin at x for all > 0. As a consequence, we obtain the following proposition (see [1, Prop. 6.4.3]): Proposition 8.2.15. Let f be a real-valued function deﬁned on a set F ⊂ Rd and ﬁnely continuous at a point x ∈ F such that F is thick at x. Then there is a set E ⊂ F such that E is thin at x and lim

F \Ey→x

f (y) = f (x).

234

Chapter 8. Domain convergence

Quasicontinuity. One of the most important ﬁne properties is quasicontinuity, which is deﬁned as follows ([1, Def. 6.1.1]): Deﬁnition 8.2.16. Let B ⊂ Rd be a bounded set. A function f : B → R is said to be quasicontinuous on B if for any > 0 there exists an open set G with Cap(G) < such that f is continuous on B \ G. A function f : Rd → R is quasicontinuous if it is quasicontinuous on every bounded subset of Rd . Remark 8.2.17. Further if a statement is true for all x except for a set E with Cap(E) = 0, then we say that it is true quasieverywhere, written q.e. The following result (see [1, Thm. 6.4.5, Thm. 6.4.6]) establishes the relation between quasicontinuity and ﬁne continuity. Theorem 8.2.18. A function deﬁned on Rd is quasicontinuous if and only if it is ﬁnely continuous q.e. in Rd . Quasicontinuity of elements of Sobolev spaces. Every function f ∈ W 1,2 (Rd ) is, in fact, an equivalence class of measurable functions which coincide oﬀ a set of zero measure. Each element of such a class is called a representative of f . The following statement ([1, Prop. 6.1.2]) states that every Sobolev function becomes quasicontinuous after possible changes of its values on a set of zero measure. Proposition 8.2.19. Every f ∈ W 1,2 (Rd ) has a quasicontinuous representative. It follows from Theorem 8.2.18 that the quasicontinuous representative is ﬁnely continuous q.e. in Rd . The next statement ([1, Thm. 6.1.4]) ensures that the quasicontinuous representative is uniquely deﬁned. Proposition 8.2.20. Let f1 and f2 be quasicontinuous functions, and suppose that f1 = f2 almost everywhere. Then f1 = f2 quasieverywhere. This leads to a new notion of the trace of a Sobolev function on an arbitrary subset of Rd (see [1, Def. 6.1.6]). Deﬁnition 8.2.21. Let f ∈ W 1,2 (Rd ) and let E ⊂ Rd be an arbitrary set. Then the trace of f on E, denoted f |E , is the restriction to E of any quasicontinuous representative of f . Thus the trace is uniquely deﬁned q.e. on E. The following approximation theorem due to Hedberg (see [1, Thm. 9.1.3], [53]) plays an important role in our theory. Theorem 8.2.22. Let f ∈ W 1,2 (Rd ). Let K ⊂ Rd be a closed set. Then the following statements are equivalent: (a) f |K = 0; (b) f ∈ W01,2 (Rd \ K); (c) for any > 0 and any compact F ⊂ Rd \ K, there is an η ∈ C0∞ (Rd \ K) such that η = 1 on F , 0 ≤ η ≤ 1 and f − ηf W 1,2 (Rd ) < .

8.3. Applications to ﬂow around an obstacle

235

8.3 Applications to ﬂow around an obstacle In this section we apply the general results to convergence of condenser type domains. This question is important for our purposes since such a geometry is typical for ﬂow modeling around an obstacle. Let B ⊂ Rd be a bounded holdall domain with C 3 boundary. Assume that the ﬂow occupies a condenser type domain Ω = B \ S, with a compact obstacle S B inside of Ω. Deﬁnition 8.3.1. Let S be a compact subset of B. Denote by CS∞ (B) the linear space of all C ∞ (B) functions u that vanish in some neighborhood of S (depending on u). Denote by WS1,r (B) the completion of CS∞ (B) in the norm of W 1,r (B). In particular, WS1,r (B) is a closed subspace of W 1,r (B). The Banach spaces WS1,2 (B), S B, form a collection of closed subspaces of the Banach space W 1,2 (B). Thus applying Deﬁnition 8.1.2 of Kuratowski-Mosco convergence to this particular case we arrive at the following deﬁnition: Deﬁnition 8.3.2. A sequence of compact sets Sn B, n ≥ 1, converges in the sense of Kuratowski-Mosco to a compact set S B if WS1,2 (B) converges in the n sense of Kuratowski-Mosco to WS1,2 (B), i.e., • for every sequence un u weakly convergent in W 1,2 (B) with un ∈ WS1,2 (B), n 1,2 the limit function u is in WS (B); • conversely, whenever u ∈ WS1,2 (B), there exists a sequence un ∈ WS1,2 (B) n such that un → u strongly in W 1,2 (B). Now Proposition 8.1.3 implies the following corollary: Corollary 8.3.3. Let compact sets Sn , n ≥ 1, converge to a compact set S in the sense of Kuratowski-Mosco and let un ∈ L2 (0, T ; WS1,2 (B)) converge to u weakly n 1,2 2 1,2 2 in L (0, T ; W (B)). Then u ∈ L (0, T ; WS (B)). Theorem 8.2.22 implies the following property of elements of WS1,2 (B): Corollary 8.3.4. Let f ∈ W 1,2 (B). Let S B be a compact set. Then the following statements are equivalent: (a) the quasicontinuous representative of f is equal to 0 q.e. on S; (b) f ∈ WS1,2 (B). Proof. It suﬃces to note that f has an extension F ∈ W 1,2 (Rd ) and apply Theorem 8.2.22. Combined S-convergence. In order to use the advantages of both Hausdorﬀ and Kuratowski-Mosco convergences we introduce the notion of combined convergence of compact subsets of the hold-all domain B.

236

Chapter 8. Domain convergence

Deﬁnition 8.3.5. We say that compact sets Sn B, n ≥ 1, S-converge to a compact set S B if • there is a compact set B B such that Sn , S B ; • the sets Sn converge to S in the Hausdorﬀ metric; (B) converge to WS1,2 (B) in the sense of Kuratowski-Mosco. • the spaces WS1,2 n S

In this case we write Sn −→ S and we say that the compact set S is the S-limit of the sequence Sn . The following proposition shows that every compact obstacle S is the S-limit of a sequence of smooth obstacles Sn ⊃ S. In Chapter 9 this result is applied to existence theory for compressible Navier-Stokes equations in nonsmooth domains. Proposition 8.3.6. Let B ⊂ Rd be a bounded domain with C 1 boundary and S B be a compact set. Then there is a sequence of compact sets Sn ⊃ S with C ∞ S boundaries such that Sn −→ S. The proof of Proposition 8.3.6 is based on two simple but useful lemmas. Lemma 8.3.7. Let B ⊂ Rd be a bounded domain with C 1 boundary and S B be a compact set. Let u ∈ W 1,2 (B) have the property that there is a sequence un ∈ W 1,2 (B) such that un → u strongly in W 1,2 (B) and each un vanishes in some neighborhood of S. Then there exists a sequence ϕn ∈ C ∞ (B) such that each ϕn vanishes in a neighborhood of S and ϕn → u strongly in W 1,2 (B), therefore, u ∈ WS1,2 (B). Proof. Since B is bounded there exists a ball BN = {|x| ≤ N } such that B BN . Since ∂B is a C 1 closed surface, there is an extension un ∈ W 1,2 (Rd ) of un to Rd such that un W 1,2 (Rd ) ≤ cun W 1,2 (B)

and un = 0 in Rd \ BN .

Here the constant c depends only on B. Next choose a sequence of integers mn with the properties 1/mn < 2−d−1 dist(S, supp un ) and

[un ]mn , − un W 1,2 (Rd ) ≤ 1/n.

Here [·]mn , is a molliﬁer deﬁned by (1.6.3), [un ]mn , (x) = mdn Θ(mn (x − y))un (y) dy. Rd

Since Θ(z) is supported in the cube [−1, 1]d , it follows from the choice of mn that the functions ϕn := [un ]mn , ∈ C0∞ (Rd ) vanish in a neighborhood of S and ϕn − uW 1,2 (B) ≤ cun − uW 1,2 (B) + ϕn − un W 1,2 (Rd ) ≤ cun − uW 1,2 (B) + 1/n → 0 as n → ∞.

8.3. Applications to ﬂow around an obstacle

237

Lemma 8.3.8. Let S ⊂ Rd be an arbitrary compact set. Then there is a decreasing sequence of compact sets Sn ⊂ Rd with C ∞ boundaries such that Sn ⊃ S,

dH (S, Sn ) 0

as n → ∞,

where the Hausdorﬀ distance dH is deﬁned by (8.1.1). Proof. Choose η ∈ C ∞ (R) satisfying η(s) = 0 for s ≤ 1, and set

η (s) ≥ 0,

η(s) = 1 for s ≥ 2,

ϕn (x) = η 23n dist(x, S) ,

n ≥ 1.

It is clear that 0 ≤ ϕn ≤ 1 and ϕn (x) = 0 for dist(x, S) ≤ 2−3n , Finally set

fn (x) = 23n+1

Rd

ϕn (x) = 1

for dist(x, S) ≥ 2−3n+1 .

Θ 23n+1 (x − y) ϕn (y) dy

with a nonnegative mollifying kernel Θ such that Θ∈

C0∞ (Rd ),

Θ(x) = 0 for |x| ≥ 1,

Θ(x) dx = 1. Rd

The function fn belongs to C ∞ (Rd ) and satisﬁes fn (x) = 0 for dist(x, S) ≤ 2−3n − 2−3n−1 = 2−3n−1 , fn (x) = 1 for dist(x, S) ≥ 2−3n+1 + 2−3n−1 = 5 · 2−3n−1 . By the Sard Theorem [118], the level set {fn = c} does not contain critical points of fn for almost every c ∈ [0, 1]. Choose cn ∈ (0, 1) with this property and set Sn := {x : fn (x) ≤ cn }. Then Sn is contained in the 5 · 2−3n−1 -neighborhood of S, and it contains the 2−3n−1 -neighborhood of S. It follows that dH (S, Sn ) ≤ 5 · 2−3n−1 → 0 as n → ∞. Since 5 · 2−3(n+1)−1 < ·2−3n−1 , we also have Sn+1 ⊂ Sn . By the choice of cn , ∇fn = 0 on ∂Sn = {fn = cn }, so ∂Sn is a C ∞ surface. Proof of Proposition 8.3.6. It suﬃces to show that the sequence of compact sets Sn deﬁned in Lemma 8.3.8 S-converges to S. By that lemma, Sn → S in the Hausdorﬀ metric. Hence it remains to show that WS1,2 (B) converges to WS1,2 (B) n in the sense of Kuratowski-Mosco. To this end we have to prove that (B) un u weakly in W 1,2 (B), then u ∈ WS1,2 (B); (a) if WS1,2 n

238

Chapter 8. Domain convergence

(B) such that (b) for every u ∈ WS1,2 (B) there exists a sequence un ∈ WS1,2 n un → u in W 1,2 (B). To prove (a) note that by the Mazur theorem, the weak convergence implies that there is a sequence of ﬁnite convex combinations

m(N )

wN =

m(N )

αkN unk ,

αkN ∈ [0, 1],

k=1

αkN = 1,

k=1

such that wN → u converges strongly in W 1,2 (B) as N → ∞. It follows from Lemma 8.3.7 that for every k, there exists ϕnk ∈ C ∞ (B) with ϕnk − unk W 1,2 (B) ≤ 1/N,

ϕnk = 0 in a neighborhood of Snk .

Since S ⊂ Snk , each ϕnk vanishes in a neighborhood of S. Now set

m(N )

φN =

αkN ϕnk .

k=1 ∞

Obviously φN ∈ C (B) and vanishes in a neighborhood of S. Next,

m(N )

φN − uW 1,2 (B) ≤ wN − uW 1,2 (B) +

αkN ϕnk − unk W 1,2 (B)

k=1

≤ wN − uW 1,2 (B) + 1/N → 0 as N → ∞. Hence u ∈ WS1,2 (B). This proves claim (a). To prove (b), ﬁx u ∈ WS1,2 (B). Then there exists a sequence ϕn ∈ C ∞ (B) such that ϕn − uW 1,2 (B) ≤ 1/n,

ϕn = 0 in a neighborhood of S.

Since the distance between Sm and S tends to 0 as m → ∞, for every n there is mn such that ϕn = 0 in a neighborhood of Smn . Without loss of generality we can assume that mn → ∞ as n → ∞. It is clear that ϕn ∈ WS1,2 (B). Moreover, mn (B) for al j ≥ mn . Now set uj = ϕn since the sequence Sn decreases, ϕn ∈ WS1,2 j

for mn ≤ j ≤ mn+1 . We have uj ∈ WS1,2 (B) and j

uj − uW 1,2 (B) ≤ 1/n → 0 as j → ∞, which proves claim (b) and Proposition 8.3.6.

The last proposition in this section is an extension of the approximation Theorem 8.2.22 to functions depending on time. This result plays a crucial role in the proof of the domain stability of solutions to the initial-boundary value problem for compressible Navier-Stokes equations.

8.3. Applications to ﬂow around an obstacle

239

Proposition 8.3.9. Let B ⊂ Rd be a bounded domain with C 1 boundary and S B be a compact set. Furthermore assume that u ∈ L2 (0, T ; WS1,2 (B)). Then there is a sequence of functions ψn ∈ C ∞ (B) with the following properties: Each ψn vanishes in a neighborhood of S, satisﬁes 0 ≤ ψn ≤ 1, and

T

|u(x, t)| |∇ψn (x)| dxdt → 0

0

as n → ∞,

(8.3.1)

B\S

ψn (x) → 1

as n → ∞

for all x ∈ B \ S.

(8.3.2)

We split the proof into a sequence of lemmas. Lemma 8.3.10. Let un → u strongly in W 1,2 (B). Then |u| − |un | 2

W 1,2 (B)

≤ u − un 2W 1,2 (B) +

Fn2 (x) dx

(8.3.3)

B

for some Fn ∈ L2 (B), where Fn → 0 a.e. in B and |Fn | ≤ 2|∇u|, so in particular Fn → 0 in L2 (B). Proof. Since the mapping z → |z| is Lipschitz, |u| belongs to W 1,2 (B) and ∇|u| = |u|−1 u∇u for u = 0,

∇|u| = 0 for u = 0

(see [133, Thm. 2.1.11]). It is clear that |u| 1,2 ≤ uW 1,2 (B) , |un | W 1,2 (B) ≤ un W 1,2 (B) . W (B) Next we have ⎧ |un |−1 un (∇u − ∇un ) + (|u|−1 u − |un |−1 un )∇u ⎪ ⎪ ⎪ ⎨ for u = 0, un = 0, ∇|u| − ∇|un | = −1 ⎪ u∇u for u = 0, u |u| n = 0, ⎪ ⎪ ⎩ −1 −|un | un ∇un for u = 0, un = 0. It follows that where

∇|u| − ∇|un | ≤ |∇u − ∇un | + Fn ,

(8.3.4)

Fn = |u|−1 u − |un |−1 un |∇u| for u = 0 and un = 0,

and Fn = 0 otherwise. Since un converges to u a.e. in B we have −1 |u| u − |un |−1 un → 0 a.e. on {u = 0}. Since Fn vanishes on {u = 0}, we conclude that Fn → 0 a.e. in B. On the other hand obviously Fn ≤ 2|∇u|. Since Fn2 has the integrable majorant 4|∇u|2 and converges to 0 as n → ∞, the Lebesgue dominated convergence theorem

240

Chapter 8. Domain convergence

implies the strong convergence of Fn to 0 in L2 (B). It follows from (8.3.4) and the inequality |u| − |un | ≤ |u − un | that ∇|u| − ∇|un | 2 + |u| − |un | 2 dx |u| − |un | 2 1,2 = (B) W B Fn2 dx. ≤ u − un 2W 1,2 (B) + B

Hence Fn meets all requirements of the lemma. Lemma 8.3.11. If v ∈ WS1,2 (B) then |v| ∈ WS1,2 (B). Proof. There is a sequence of vector functions ϕn ∈ C ∞ (B) such that

ϕn vanishes in a neighborhood of S and ϕn → v strongly in W 1,2 (B). (8.3.5) By Lemma 8.3.10 we have

|v| − |ϕn | 2

W 1,2 (B)

≤ v −

ϕn 2W 1,2 (B)

Fn2 (x) dx,

+

(8.3.6)

B

where Fn → 0 in L2 (B). Letting n → ∞ shows that |ϕn | converges to |v| in W 1,2 (B). Since each |ϕn | vanishes in a neighborhood of S, we have |v| ∈ WS1,2 (B). Lemma 8.3.12. Let u ∈ L2 (0, T ; WS1,2 (B)). Then for any > 0 there exists a ﬁnite collection of disjoint measurable sets Ei ⊂ [0, T ], 1 ≤ i ≤ N , with i Ei = [0, T ], and functions ϕi ∈ C ∞ (B) vanishing in a neighborhood of S and such that T ϕ − |u| 2 1,2 dt ≤ 2 /T, (8.3.7) W (B) 0

where ϕ ∈ L2 (0, T ; W 1,2 (B)) is the simple function deﬁned by for t ∈ Ei ,

ϕ(x, t) = ϕi (x)

i = 1, . . . , N.

(8.3.8)

Proof. By Theorem 1.3.20 there is a sequence sn ∈ L2 (0, T ; WS1,2 (B)) of simple functions (see Deﬁnition 1.3.8) such that T sn − u2W 1,2 (B) dt → 0 as n → ∞. (8.3.9) S

0

Passing to a subsequence we can assume that sn (t) − u(t)W 1,2 (B) → 0 as n → ∞ for a.e. t ∈ [0, T ]. S

Applying Lemma 8.3.10 we obtain, for a.e. t ∈ [0, T ], |sn (t)| − |u(t)| 2 1,2 ≤ sn (t) − u(t)2W 1,2 (B) + Fn (t)2L2 (B) , (B) W

(8.3.10)

8.3. Applications to ﬂow around an obstacle

241

where Fn (t)L2 (B) → 0 as n → ∞ and Fn (t)L2 (B) ≤ 2u(t)W 1,2 (B) , so by the Lebesgue dominated convergence theorem,

T

Fn (t)2L2 (B) dt → 0 as n → ∞.

0

(8.3.11)

Next (8.3.10) yields

T

|sn (t)| − |u(t)| 2

W 1,2 (B)

0

≤ 0

T

sn (t) − u(t)2W 1,2 (B) dt +

0

T

Fn (t)2L2 (B) dt.

Letting n → ∞ and using (8.3.9), (8.3.11) we conclude that the left hand side of the last inequality tends to 0 as n → ∞. Hence for given > 0, there is n > 1 such that T |sn | − |u| 2 1,2 dt ≤ 2 /(4T ). (8.3.12) W

0

(B)

By Deﬁnition 1.3.8, the simple function sn : [0, T ] → WS1,2 (B) has a representation sn =

N

ci χEi ,

i=1

where χEi (t) are the characteristic functions of the pairwise disjoint measurable sets Ei ⊂ [0, T ], and ci ∈ WS1,2 (B). Thus we get |sn | =

N

|ci |χEi .

i=1

By Lemma 8.3.11 we have |ci | ∈ WS1,2 (B). Hence for every i there is a function ϕi ∈ CS∞ (B) such that ϕi − |ci | 1,2 ≤ /(2T ). W (B) Now set ϕ(x, t) =

ϕi (x)χEi (t).

i

It follows from (8.3.12), (8.3.13) and the inequality 2 2 ϕ − |u| 2 1,2 ≤ 2 |sn | − |u| W 1,2 (B) + 2ϕ − |sn | W 1,2 (B) W (B)

(8.3.13)

242

Chapter 8. Domain convergence

that 0

T

ϕ − |u| 2 1,2 dt (B) W

T

≤2

dt + 2 W 1,2 (B)

0

=2

|sn | − |u| 2

T

|sn | − |u| 2

W 1,2 (B)

0

dt + 2

2 + ≤ meas Ei ≤ 2 /T. 2T 2T 2 i

T

|sn | − ϕ2

W 1,2 (B)

0

dt

ϕi − |ci | 2 1,2 meas Ei (B) W i

2

Lemma 8.3.13. If u ∈ L2 (0, T ; WS1,2 (B)), then the function

T

|u(x, t)| dt

W (x) =

(8.3.14)

0

belongs to WS1,2 (B). Proof. It suﬃces to prove that for any > 0 there is Φ ∈ C ∞ (B) that vanishes in a neighborhood of S and Φ − W W 1,2 (B) ≤ . Let ϕ be given by Lemma 8.3.12. Deﬁne T N ϕ dt = ϕi meas Ei . Φ= 0

i=1

∞

Since each ϕi ∈ C (B) vanishes in a neighborhood of S, Φ belongs to C ∞ (B) and vanishes in a neighborhood of S. Next, by Theorem 1.3.15,

T

Φ − W W 1,2 (B) ≤ 0

√ ϕ − |u| 1,2 dt ≤ T W (B)

T

0

1/2 ϕ − |u| 2 1,2 dt , W (B)

and (8.3.7) yields Φ − W W 1,2 (B) ≤ .

Lemma 8.3.14. Let W be deﬁned in Lemma 8.3.13. Then there exists a sequence ψn ∈ C ∞ (B), n ≥ 1, such that each ψn vanishes in a neighborhood of S and satisﬁes 0 ≤ ψn ≤ 1 and W (x)|∇ψn (x)| dx → 0 as n → ∞, (8.3.15) B\S

ψn (x) → 1

as n → ∞

for all x ∈ B \ S.

Proof. Consider the sequence of open sets Vn = {x : dist(x, S) < 1/n}.

(8.3.16)

8.3. Applications to ﬂow around an obstacle We have Vn \ S ⊂ Vn−1 \ S and

n (Vn

243

\ S) = ∅. Hence

meas(Vn \ S) → 0 as n → ∞. Notice that W ∈ WS1,2 (B) can be extended to a function W ∈ W 1,2 (Rd ). Since W vanishes q.e. on S, i.e. W |S = 0, it follows from Theorem 8.2.22 that W ∈ W01,2 (Rd \ S). Denote by F the compact set cl B \ Vn . Applying to W statement (c) of Theorem 8.2.22 with K replaced by S and = 1/n we conclude that there is ψn ∈ C ∞ (Rd ) such that • 0 ≤ ψn ≤ 1 in Rd , • ψn = 1 on F , ψn = 0 in a neighborhood of S, • W (1 − ψn )W 1,2 (Rd ) ≤ 1/n. It immediately follows that ψn = 1 in B \ Vn ,

W (1 − ψn )W 1,2 (B) ≤ 1/n.

(8.3.17)

Next we have W |∇ψn | = W |∇(1 − ψn )| ≤ |∇W |(1 − ψn ) + |∇(W (1 − ψn ))|.

(8.3.18)

Since 1 − ψn = 0 on B \ Vn and ∇W = 0 a.e. on S we have 1/2 1/2 meas (Vn \ S) |∇W |(1 − ψn ) dx ≤ |∇W |dx ≤ |∇W |2 dx . Vn \S

B

B

On the other hand, (8.3.17) implies 1/2 |∇(W (1 − ψn ))| dx ≤ (meas B)1,2 |∇(W (1 − ψn ))|2 dx B √ ≤ cW (1 − ψn )W 1,2 (B) ≤ c/ n. Combining the results obtained with (8.3.18) we ﬁnally obtain 1/2 W |∇ψn | dx ≤ W W 1,2 (B) meas (Vn \ S) + cn−1/2 → 0 B

as n → ∞.

Proof of Proposition 8.3.9. It suﬃces to note that

T

|u(x, t)| |∇ψn (x)| dxdt =

0

B\S

and apply Lemma 8.3.14.

W (x)|∇ψn (x)| dx, B\S

244

Chapter 8. Domain convergence

8.4 S-compact classes of admissible obstacles In this section we present an example of an S-compact family of admissible obstacles. Such a class must satisfy two conditions. First, it should be suﬃciently large in order to meet practical requirements arising in real shape optimization problems. In particular, it should include the d-dimensional obstacles satisfying the double cone condition and the (d − 1)-dimensional Lipschitz manifolds. On the other hand a family of admissible obstacles should be deﬁned by clear geometrical constrains. The main diﬃculty is that there is no geometrical criterion for Kuratowski-Mosco convergence. Intuitively, it is clear that a compact set in the Kuratowski-Mosco topology should be in some sense “equi-thick”, but specifying such a requirement is a delicate task. A possible way to construct a family of admissible obstacles is the following. Let us assume that for every point x of an admissible obstacle S, there is a test set which is contained in S and thick at x. If we choose the test set as a Lipschitz image of a ﬁxed compact T ⊂ Rd which is thick at 0, then we arrive at the following deﬁnition. Deﬁnition 8.4.1. Let T ⊂ Rd be a compact set thick at 0, and B ⊂ Rd an arbitrary compact set. The family O(k, T , B ) consists of all compact sets S ⊂ B with the following property: for every x0 ∈ S, there exists a bi-Lipschitz bijection χ : Rd → Rd such that k −1 |y − y | ≤ |χ(y ) − χ(y )| ≤ k|y − y | for all y , y ∈ Rd , and x0 + χ T ⊂ S. χ(0) = 0,

(8.4.1)

Notice that in this deﬁnition the bijection χ depends on x0 , but the constant k does not. By Lemma 8.2.8 every obstacle in O(k, T , B ) is thick at each of its points. In accordance with Lemmas 8.2.10 and 8.2.12 we can choose T = (ω) in dimension three and T = I in dimension two. For this choice the class T includes all (d−1)-dimensional compact manifolds with Lipschitz boundaries and all compact sets S which are starlike with respect to three noncollinear points. The following theorem shows that O(k, T , B ) is compact with respect to S-convergence. Theorem 8.4.2. Every sequence Sn ∈ O(k, T , B ) contains a subsequence which S-converges to a set S ∈ O(k, T , B ). We split the proof into a sequence of lemmas. Lemma 8.4.3. If Sn ∈ O(k, T , B ) converges in the Hausdorﬀ metric to a compact set S, then S ∈ O(k, T , B ). Proof. Obviously S ⊂ B . Next choose x0 ∈ S. By the deﬁnition of the Hausdorﬀ convergence, there is a sequence xn ∈ Sn such that xn → x0 . Since Sn ∈ O(k, T , B ), there is a bi-Lipschitz bijection χn : Rd → Rd satisfying (8.4.1) such that xn + χn (T ) ⊂ Sn . Passing to a subsequence we can assume that χn converges

8.4. S-compact classes of admissible obstacles

245

uniformly on every compact set in Rd to a mapping χ : Rd → Rd . Obviously χ satisﬁes (8.4.1). Choose y ∈ T . We have Sn xn + χn (y) → x0 + χ(y). Since Sn → S in the Hausdorﬀ metric, we conclude that x0 + χ(y) ∈ S and hence x0 + χ(T ) ⊂ S. Therefore, S belongs to O(k, T , B ). Lemma 8.4.4. Let Z n : Rd → Rd be bi-Lipschitz bijections satisfying k −1 |x − x | ≤ |Z n (x ) − Z n (x )| ≤ k|x − x |

for all x , x ∈ Rd .

(8.4.2)

Assume that Z : Rd → Rd is a map such that Z n → Z uniformly on every bounded set in Rd . Then Z is a bijection satisfying (8.4.2) and the Jacobians Jn = det(Dx Z n ) converge weakly in Lr (D) to J = det(Dx Z) for all r ∈ [1, ∞) and all bounded measurable sets D ⊂ Rd . Proof. Letting n → ∞ in (8.4.2) we ﬁnd that Z satisﬁes (8.4.2) and hence it is a bi-Lipschitz bijection. In particular, the Jacobian matrices Dx Z n and Dx Z are uniformly bounded in L∞ (Rd ). Hence it suﬃces to prove that Jn (x)ϕ(x) dx = J(x)ϕ(x) dx for all ϕ ∈ C0∞ (Rd ). (8.4.3) lim n→∞

Rd

Rd

Notice that the mappings Z n and Z take homeomorphically Rd onto itself and have the bi-Lipschitz inverses (Z n )−1 and Z −1 . Obviously (Z n )−1 → Z −1 uniformly on every bounded set. Thus we get Jn (x)ϕ(x) dx = ϕ ◦ (Z n )−1 (z) dz Rd Rd ϕ ◦ Z −1 (z) dz = J(x)ϕ(x) dx. → Rd

Rd

Lemma 8.4.5. Let un u weakly in W 1,2 (Rd ). Let χn be a sequence of bi-Lipschitz bijections satisfying (8.4.1) and converging uniformly on every compact set to a bijection χ. Let Rd xn → x0 . Then the sequence vn (y) = un (xn + χn (y)) converges weakly in W 1,2 (Rd ) to v(y) = u(x0 + χ(y)). Proof. Since the sequence vn is obviously bounded in W 1,2 (Rd ), it contains a subsequence which converges weakly in W 1,2 (Rd ) to an element v ∗ ∈ W 1,2 (Rd ). Of course, this subsequence converges to v ∗ weakly in L2 (Rd ). Notice that vn v weakly in W 1,2 (R3 ) if and only if all such v ∗ , i.e. all limits of subsequences, coincide and are equal to v. Hence it suﬃces to show that vn v weakly in L2 (Rd ). Since vn is uniformly bounded in L2 (Rd ), it suﬃces to prove that vn · ϕ dy = v · ϕ dy for all ϕ ∈ C0∞ (Rd ). (8.4.4) lim n→∞

Rd

Rd

Denote by y = Z n (x) and y = Z(x) the inverse mappings of x = xn + χn (y) and x = x0 + χ(y), respectively. Since χn and χ satisfy (8.4.1) and χn → χ uniformly

246

Chapter 8. Domain convergence

on every compact set, the mappings Z n and Z meet all requirements of Lemma 8.4.4. On the other hand, we have un (x) = vn (Z n (x)) and u(x) = v(Z(x)), which leads to vn · ϕ dy = un (x) · ϕ(Z n (x))Jn (x) dy, d d R R (8.4.5) v · ϕ dy = u(x) · ϕ(Z(x))J(x) dy, Rd

Rd

where Jn and J are the Jacobians of Z n and Z. Since |Z n (x)| ≥ k −1 |x| − c,

|Z(x)| ≥ k −1 |x| − c,

c = sup |xn |,

and ϕ(y) is compactly supported in Rd , there is a bounded domain D ⊂ Rd such that the vector ﬁelds ϕ◦Z n and ϕ◦Z vanish outside of D. Moreover, ϕ◦Z n → ϕ◦Z uniformly on D. Recall that the embedding W 1,2 (Rd ) → L2 (D) is compact. Hence un → u strongly in L2 (D). Obviously un · ϕ ◦ Z n → u · ϕ ◦ Z strongly in L2 (D). On the other hand, by Lemma 8.4.4, Jn J weakly in L2 (D). Thus n lim un (x) · ϕ(Z (x))Jn (x) dy = u(x) · ϕ(Z(x))J(x) dy. n→∞

Rd

Rd

Now (8.4.5) yields the desired equality (8.4.4). Lemma 8.4.6. Let Sn ∈ O(k, T , B ) and let un ∈ W 1,2 (Rd ) satisfy un ∈ W01,2 (Rd \ Sn ),

un u

weakly in W 1,2 (Rd ).

Let χn : Rd → Rd be a sequence of bi-Lipschitz bijections satisfying (8.4.1) and converging to a bi-Lipschitz bijection χ uniformly on every bounded subset of Rd . Furthermore assume that there is a sequence Rd xn → x0 with xn + χn (T ) ⊂ Sn . Then the quasicontinuous representative of u vanishes q.e. on x0 + χ(T ), i.e., in accordance with Deﬁnition 8.2.21, u|x0 +χ(T ) = 0. Proof. First we show that vn (y) = un (xn + χn (y)) belongs to W)1,2 (Rd \ T ).

Observe that since un ∈ W01,2 (Rd \ Sn ), there is a sequence ϕk ∈ C0∞ (Rd ) with the following properties. Every ϕk vanishes in a neighborhood Uk of Sn and ϕk → un in W 1,2 (Rd ) as k → ∞. Notice that Uk xn + χn (T ). The inverse bi-Lipschitz homeomorphism Z n = (xn + χn )−1 takes Uk onto the open set Z n (Uk ) and xn + χn (T ) onto T . Obviously Z n (Uk ) is a neighborhood of T . Hence the Lipschitz function ϕk (xn + χn (y)) vanishes in a neighborhood of T . On the other hand, ϕk (xn + χn ) → un (xn + χn ) = vn as k → ∞ in W 1,2 (Rd )

since xn + χn is a bi-Lipschitz homeomorphism. It follows that vn ∈ W01,2 (Rd \ T ).

8.4. S-compact classes of admissible obstacles

247

By Lemma 8.4.5, the sequence vn converges weakly in W 1,2 (Rd ) to v = u ◦ (x0 + χ). Every closed subspace of the Hilbert space W 1,2 (Rd ) is weakly closed, since all Hilbert spaces are reﬂexive. In particular, W01,2 (Rd \ T ) is weakly closed. Therefore, v ∈ W01,2 (Rd \ T ). From Theorem 8.2.22 we now conclude that the quasicontinuous representative of v is equal to zero q.e. on T . Next, we show that if v is quasicontinuous, then so is u = v ◦ Z, where Z = (x0 + χ)−1 . First, as χ satisﬁes (8.4.1), Lemma 8.2.4 with χ replaced by x0 + χ shows that Cap(x0 + χ(G)) ≤ c(k) Cap G

for all open G ⊂ Rd .

Next, choose an open set D ⊂ Rd and > 0. Since v is quasicontinuous, there is an open set G with Cap G < /c(k) such that v is continuous on Z(D) \ G. It follows that u is continuous on D \ G∗ , where G∗ = x0 + χ(G). We also have Cap G∗ < . Hence from Deﬁnition 8.2.16 we deduce that u = v ◦ Z is quasicontinuous. Finally notice that by Lemma 8.2.4, the bi-Lipschitz homeomorphism Z takes zero capacity sets onto zero capacity sets. Since v = 0 q.e. on T and Z takes T onto x0 + χ(T ), it follows that u = v ◦ Z is zero q.e. on x0 + χ(T ). Lemma 8.4.7. Let B B and suppose Sn ∈ O(k, T , B ) converges in the Haus(B) converges dorﬀ metric to a compact set S. Furthermore assume that un ∈ WS1,2 n weakly in W 1,2 (B) to some u ∈ W 1,2 (B). Then u ∈ WS1,2 (B). Proof. By Corollary 8.3.4 it suﬃces to prove that the quasicontinuous representative of u is equal to zero q.e. on S. Let us consider the following construction. Since B ⊂ Rd is a bounded domain with C 1 boundary, there exists a bounded linear extension operator Ξ : W 1,2 (B) → W 1,2 (Rd ) which assigns to every u ∈ W 1,2 (B) its extension u ∈ W 1,2 (Rd ), such that uW 1,2 (Rd ) ≤ cuW 1,2 (B) , u = u on B. If u ∈ WS1,2 (B) then u ∈ W01,2 (Rd \ S), so we can set un = Ξun ∈ W01,2 (Rd \Sn ). Since the operator Ξ : W 1,2 (B) → W 1,2 (Rd ) is bounded, un u = Ξu weakly in W 1,2 (Rd ). The lemma will be proved if we show that the quasicontinuous representative of u is zero q.e. on S. To this end notice that in view of Theorem 8.2.18, the quasicontinuous representative of u is ﬁnely continuous q.e. in Rd . Hence it suﬃces to show that u(x0 ) = 0 at every point x0 ∈ S such that the quasicontinuous representative of u is ﬁnely continuous at x0 . Fix such an x0 ∈ S. Since Sn → S in the Hausdorﬀ metric, there is a sequence xn ∈ Sn which converges to x0 . Recall that Sn ∈ O(k, T , B ), so by Deﬁnition 8.4.1 there exists a bi-Lipschitz homeomorphism χn : Rd → Rd satisfying (8.4.1) such that xn + χn (T ) ⊂ Sn . Passing to a subsequence we can assume that χn converges to a bi-Lipschitz bijection χ uniformly on every bounded subset of Rd . Hence the sequences un , u, and χn meet all requirements of Lemma 8.4.6. Thus, the quasicontinuous representative of u is equal to zero q.e. on x0 + χ(T ).

248

Chapter 8. Domain convergence

Since u is ﬁnely continuous at x0 , by Proposition 8.2.15 there exists a set E such that u(x) = u(x0 ) and E is thin at x0 . (8.4.6) lim Rd \Ex→x0

Next, by Lemma 8.2.8, the set x0 + χ(T ) is thick at x0 . Lemma 8.2.7 shows that the set F = (x0 + χ(T )) \ E is thick at x0 . Since u = 0 q.e. on x0 + χ(T ), there is a set X of zero capacity such that u = 0 on F \ X. Since F is thick at x0 , Lemma 8.2.7 implies that F \ X is thick at x0 , and x0 is a limit point of F \ X. On the other hand, (F \ X) ∩ E = ∅, so (8.4.6) yields the desired equality u(x0 ) = limF \Xx→x0 u(x) = 0. Lemma 8.4.8. Let S ∈ O(k, T , B ) and suppose a sequence of compact sets Sn ⊂ B converges to S in the Hausdorﬀ metric. Let u ∈ WS1,2 (B). Then there is a sequence (B) which converges to u in W 1,2 (B). un ∈ WS1,2 n Proof. Since u ∈ WS1,2 (B), there is a sequence ϕk ∈ C ∞ (B) such that ϕk → u in W 1,2 (B) and ϕk vanishes in a neighborhood Gk of S. Without loss of generality we can assume that Gk+1 ⊂ Gk and k Gk = S. By the deﬁnition (8.1.1) of the Hausdorﬀ metric, for every k there is Nk such that Sn Gk for all n ≥ Nk . Hence we can take un = ϕk for all n ∈ [Nk , Nk+1 ). Proof of Theorem 8.4.2. Take any sequence Sn ∈ O(k, T , B ). Since the sets Sn are compact and contained in the compact set B , Lemma 8.1.1 yields the existence of a subsequence, still denoted by Sn , and a compact set S ⊂ B such that Sn → S in the Hausdorﬀ metric. By Lemma 8.4.7 for any sequence WS1,2 (B) n un u weakly in W 1,2 (B), the limit element u belongs to WS1,2 (B). Conversely, Lemma 8.4.8 implies that for any u ∈ WS1,2 (B) there exists a sequence un ∈ (B) which converges to u strongly in W 1,2 (B). From Deﬁnition 8.3.2 we thus WS1,2 n conclude that Sn → S in the Kuratowski-Mosco sense. The Kuratowski-Mosco convergence combined with the Hausdorﬀ convergence yields the S-convergence of Sn . It remains to note that in view of Lemma 8.4.3, the limit set S belongs to O(k, T , B ).

Chapter 9

Flow around an obstacle. Domain dependence 9.1 Preliminaries There are two applications of domain stability results for the governing equations of the in/out ﬂow problem: • The ﬁrst is to the existence of weak renormalized solutions with reasonable regularity assumptions on the data from the point of view of applications. This means that at the ﬁrst stage the existence is obtained under restrictive assumptions, in particular on the domain. Next, the assumptions are relaxed by applying the stability result. In this important direction we have a nice result: a closed obstacle is perfectly admissible for the existence of weak renormalized solutions for compressible Navier-Stokes equations. In other words, there is no need for artiﬁcial domain constraints—solutions for the governing equations exist even in the case of a closed obstacle. • The second is to shape optimization problems, in order to establish the existence of an optimal shape within a given family of admissible domains or obstacles, for a speciﬁc cost functional, with the state equation in the form of compressible Navier-Stokes equations. In this case the expected result would be a broad family of admissible obstacles such that the associated shape optimization problems enjoy solutions. But even for the simplest case of the Laplacian there is no satisfactory answer to the question, therefore we cannot expect to ﬁnd a better solution. We establish the existence of an optimal domain in the same class as in the case of the Laplacian. The result seems to be optimal, and unexpected. In this chapter it is shown that weak renormalized solutions to compressible Navier-Stokes equations are stable with respect to domain perturbations. In this

P. Plotnikov and J. Sokoáowski, Compressible Navier–Stokes Equations: Theory and Shape Optimization, Monografie Matematyczne 73, DOI 10.1007/978-3-0348-0367-0_9, © Springer Basel 2012

249

250

Chapter 9. Flow around an obstacle. Domain dependence

way the existence results are extended to more realistic data. The general framework is used for the analysis of shape optimization problems.

9.2 Existence theory In this chapter we consider in more detail the problem of a ﬂow around an obstacle, which is a particular case of problem (5.1.12) and is formulated as follows: Condition 9.2.1. Let B be an open, bounded hold-all domain in Rd with C ∞ boundary ∂B and Σ = ∂B × (0, T ). Furthermore a compact “obstacle” S is placed inside B, i.e., S is a compact subset of B. A given exterior vector ﬁeld U ∈ C ∞ (Rd × [0, T ]) splits the surface Σ into three disjoint parts (subsets) Σin , Σout and Γ deﬁned by the sign of the normal component of U, Σin = {(x, t) ∈ Σ : n(x) · U(x, t) < 0}, Γ = {(x, t) ∈ Σ : n(x) · U(x, t) = 0},

(9.2.1)

Σout = {(x, t) ∈ Σ : n(x) · U(x, t) > 0}, where n is the outward normal to ∂B. Hence in the topology of Σ, the compact set Γ separates the open sets Σin and Σout . With this notation we have Ω = B \ S,

Q = Ω × (0, T ),

ST = Σ ∪ (∂S × (0, T )).

(9.2.2)

We also assume that the given vector ﬁeld U vanishes near the obstacle, U(x, t) = 0 in a neighborhood of S × (0, T ).

(9.2.3)

Under these assumptions the problem of the ﬂow around an obstacle can be formulated in the same way as problem (5.1.12). Problem 9.2.2. Find a velocity ﬁeld u and a density function satisfying ∂t (u) + div( u ⊗ u) + ∇p(ρ) = div S(u) + f ∂t + div(u) = 0 in Q,

in Q,

(9.2.4a) (9.2.4b)

u=U

on ST , = ∞ on Σin × (0, T ), u|t=0 = U, |t=0 = ∞ in Ω.

Notice that u = 0 on ∂S × (0, T ).

(9.2.4c)

9.2. Existence theory

251

In Chapter 5 we studied weak renormalized solutions to this problem, but in this particular case we use a deﬁnition of renormalized solution which is stronger compared to Deﬁnition 5.1.4. In order to treat the case of nonsmooth obstacles we recall Deﬁnition 8.3.1 of the Sobolev space WS1,2 (B) of functions vanishing around an obstacle. Deﬁnition 9.2.3. A couple u ∈ L2 (0, T ; WS1,2 (B)),

∈ L∞ (0, T ; Lγ (B \ S))

is said to be a weak renormalized solution to problem (9.2.4) if it satisﬁes: • The kinetic energy is bounded, i.e., |u|2 ∈ L∞ (0, T ; L1 (B \ S)). • The velocity u satisﬁes the Dirichlet boundary condition on the lateral side of the cylinder, u = U on Σ. This condition makes sense since u ∈ L2 (0, T ; W 1,2 (Ω)) has a trace on the smooth surface Σ. • The extended velocity satisﬁes u ∈ L2 (0, T ; WS1,2 (B)). • The integral identity u · ∂t ξ + u ⊗ u : ∇ξ + p div ξ − S(u) : ∇ξ dxdt (B\S)×(0,T ) f · ξ dxdt + U · ξ dx = 0 (9.2.5) + (B\S)×(0,T )

(B\S)×{t=0}

holds for all vector ﬁelds ξ ∈ C ∞ ((B\S)×(0, T )) equal to 0 in a neighborhood of the lateral side (∂B ∪ ∂S) × (0, T ) and of the top (B \ S) × {t = T }. • The integral identity ϕ()∂t ψ + ϕ()u · ∇ψ + ψ(ϕ() − ϕ ()) div u dxdt (B\S)×(0,T ) ψϕ(∞ )U · n dΣ − ϕ(∞ )ψ dx (9.2.6) = Σin

(B\S)×{t=0}

holds for all ψ ∈ C ∞ ((B \ S) × (0, T )) vanishing in a neighborhood of the surface Σ \ Σin and in a neighborhood of the top (B \ S) × {t = T }, and for all C 1 functions ϕ : [0, ∞) → R such that lim sup(|ϕ()| + |ϕ ()|) < ∞.

(9.2.7)

→∞

Condition (9.2.7) means that ϕ has minimal admissible smoothness and it is bounded at inﬁnity.

252

Chapter 9. Flow around an obstacle. Domain dependence

The only diﬀerence between Deﬁnitions 5.1.4 and 9.2.3 is that in the latter we require the integral identity (9.2.6) to hold for all test functions ψ that vanish in a neighborhood of Σ \ Σin , while in Deﬁnition 5.1.4, only for those that vanish in a neighborhood of Σ \ Σin and of ∂S × (0, T ). Hence Deﬁnition 9.2.3 is stronger. Renormalized solutions in the sense of Deﬁnition 9.2.3 have the additional property that a solution to the mass balance equation (9.2.4b) can be extended to the obstacle S and the extended function also satisﬁes the transport equation. This property is used throughout this chapter, and it is speciﬁed by the following lemma. Lemma 9.2.4. Let u and meet all requirements of Deﬁnition 9.2.3, and deﬁne u∗ : B × (0, T ) → Rd and ∗ : B × (0, T ) → R+ by u∗ = u ∗ =

in (B \ S) × (0, T ),

in (B \ S) × (0, T ),

u∗ = 0

in S × (0, T ),

∗ (x, t) = ∞ (x, 0)

in S × (0, T ).

Then the integral identity B×(0,T )

ϕ(∗ )∂t ψ + ϕ(∗ )u∗ · ∇ψ + ψ(ϕ(∗ ) − ϕ (∗ )∗ ) div u∗ dxdt = ψϕ(∞ )U · n dΣ − (ϕ(∞ )ψ)(x, 0) dx

(9.2.8)

B

Σin

holds for all ψ ∈ C ∞ (Q) vanishing in a neighborhood of the surface Σ \ Σin and in a neighborhood of the top Ω × {T }, and for all smooth functions ϕ : [0, ∞) → R such that lim sup(|ϕ()| + |ϕ ()|) < ∞. (9.2.9) →∞

Proof. It is easy to see that ϕ(∗ )∂t ψ dxdt = − S×(0,T )

ϕ(∞ )ψ dx.

(9.2.10)

S×{t=0}

Notice that u∗ ∈ L2 (0, T ; WS1,2 (B)), in fact u∗ is simply u if u is considered as an element of WS1,2 (B). In particular, u∗ = 0 and ∇u∗ = 0 on S × (0, T ). Now from (9.2.6), (9.2.10), and the deﬁnition of ∗ we obtain (9.2.8). We are now in a position to prove the main result of this section, on existence of a weak renormalized solution to the boundary value problem (9.2.4). We assume that the ﬂow domain, the given data and the constitutive function p() satisfy the following conditions: Condition 9.2.5. and

• The function p : [0, ∞) → R+ has continuous ﬁrst derivative p () ≥ 0,

p(0) = p (0) = 0.

(9.2.11)

9.2. Existence theory

253

There are an exponent γ > d/2 and a constant cp > 0 such that for ∈ [0, ∞), γ ≤ cp (P () + 1),

γ ≤ cp (p() + 1),

p() ≤ cp (P () + 1),

(9.2.12)

and for ∈ [1, ∞), γ γ c−1 p ≤ p() ≤ cp ,

γ−1 c−1 ≤ p () ≤ cp γ−1 . p

(9.2.13)

In particular, (9.2.14) p() ≤ cp (γ + 1). - −1 Hence the energy density P () = 0 s p(s) ds is twice continuously diﬀerentiable on (0, ∞). • The pressure function has the representation p() = pc () + pb (), where pc is convex and pb is bounded. • B ⊂ Rd is a bounded domain with C ∞ boundary ∂B. The characteristic set Γ deﬁned in (9.2.1) and the given vector ﬁeld U satisfy the geometric condition lim sup σ −d meas Oσ < ∞, σ→0

Oσ = {(x, t) : dist((x, t), Γ) ≤ σ}.

(9.2.15)

• The initial and boundary data satisfy the conditions U ∈ C ∞ (Rd × (0, T )), f ∈ C(B × (0, T )),

∞ ∈ C 1 (Rd × (0, T )), U = 0 in a neighborhood of S × (0, T ).

(9.2.16)

Remark 9.2.6. Throughout, we denote by ce a generic constant depending only on γ,

diam Ω,

T,

∞ L∞ (ST ∪Ω×{0}) ,

UC 1 (Q) ,

f C(Q) ,

cp .

The following theorem, which is an analog of Theorem 7.4.1, gives the existence of renormalized solutions to problem (9.2.4). Theorem 9.2.7. Let γ > d/2 be given and suppose Condition 9.2.5 is satisﬁed. Furthermore, assume that ∂S is a C ∞ surface. Then problem (9.2.4) has a weak renormalized solution (u, ) which meets all requirements of Deﬁnition 9.2.3 and satisﬁes the estimate uL2 (0,T ;W 1,2 (Ω)) + |u|2 L∞ (0,T ;L1 (Ω)) + L∞ (0,T ;Lγ (Ω)) ≤ ce .

(9.2.17)

Moreover, for any cylinder Q Q, and for ϑ ∈ (0, min{2γd−1 − 1, 2−1 γ}), there is a constant c, depending only on ϑ, ce , and Q , such that γ+ϑ dxdt ≤ c. (9.2.18) Q

254

Chapter 9. Flow around an obstacle. Domain dependence

Proof. First observe that under the present assumptions, the ﬂow domain Ω and the given data U, ∞ satisfy all hypotheses of Theorem 7.4.1 and hence problem (9.2.4) has a weak solution which meets all requirements of Deﬁnition 5.1.4 and satisﬁes estimates (9.2.17), (9.2.18). Hence it remains to prove that satisﬁes the integral identity (9.2.6) for any ψ ∈ C ∞ (B × (0, T )) vanishing in a neighborhood of Σ \ Σin and of Ω × {T }. Fix such a ψ, and choose η ∈ C ∞ (R) such that η(t) = 1 for t > 2 and η(t) = 0 for t < 1. Set ηn (x) = η(n dist(x, S)) for x ∈ Ω = B \ S,

ηn (x) = 0 for x ∈ S.

Since ∂S is C 3 , for suﬃciently large n the function ηn belongs to C 3 (B), vanishes in the (1/n)-neighborhood of S and is 1 outside the (2/n)-neighborhood of S. In particular ψn = ψηn vanishes in a neighborhood of ∂Ω \ Σin and hence, by Deﬁnition 5.1.4, the integral identity

Q

ϕ()∂t ψn + ϕ()u · ∇ψn + ψn (ϕ() − ϕ ()) div u dxdt = ψn ϕ(∞ )U · n dΣ − (ϕ(∞ )ψn )(x, 0) dx Σin

(9.2.19)

Ω

holds for all suﬃciently large n and for all smooth ϕ : [0, ∞) → R such that lim sup(|ϕ()| + |ϕ ()|) < ∞. →∞

Since ηn is independent of t, the functions ψn and ∂t ψn are uniformly bounded and converge to ψ and ∂t ψ a.e. in Q = Ω × (0, T ). From the Lebesgue dominated convergence theorem we conclude that lim

n→∞

Q

ϕ()∂t ψn + ψn (ϕ() − ϕ ()) div u dxdt = ϕ()∂t ψ + ψ(ϕ() − ϕ ()) div u dxdt,

(9.2.20)

Q

and lim

n→∞

Σin

ψn ϕ(∞ )U · n dΣ − lim (ϕ(∞ )ψn )(x, 0) dx n→∞ Ω = ψϕ(∞ )U · n dΣ − (ϕ(∞ )ψ)(x, 0) dx. (9.2.21) Σin

Ω

Next we have ϕ()u · ∇ψn dxdt = ϕ()ηn u · ∇ψ dxdt + ϕ()ψu · ∇ηn dxdt. (9.2.22) Q

Q

Q

9.3. Main stability theorem

255

The Lebesgue dominated convergence theorem implies ϕ()ηn u · ∇ψ dxdt = ϕ()u · ∇ψ dxdt. lim n→∞

Q

(9.2.23)

Q

On the other hand, since u vanishes at ∂S we have dist(x, S)−1 |u|2 dxdt ≤ c(Ω)u2L2 (0,T ;W 1,2 (Ω)) , Q

hence

dist(x, S)−1 |u|2 dxdt = 0.

lim

n→∞

dist(x,∂S)≤2/n

This leads to |ϕ()ψu · ∇ηn | dxdt ≤ c(ϕ, ψ) |u| |∇ηn | dxdt ≤ cn Q

≤c

dxdt

1/2

dist(x,∂S)≤2/n

|u| dxdt dist(x,∂S)≤2/n

Q

≤c n

dist(x, S)−1 |u|2 dxdt

1/2

dist(x,∂S)≤2/n

dist(x, S)−1 |u|2 dxdt

1/2

→ 0 as n → ∞,

dist(x,∂S)≤2/n

which along with (9.2.23) and (9.2.22) implies ϕ()u · ∇ψn dxdt = ϕ()u · ∇ψ dxdt. lim n→∞

Q

(9.2.24)

Q

Letting n → ∞ in identity (9.2.19) and invoking (9.2.20), (9.2.21) and (9.2.24) yields (9.2.6).

9.3 Main stability theorem We are now in a position to prove the main result of this chapter on stability of solutions to the in/out ﬂow problem for compressible Navier-Stokes equations. Theorem 9.3.1. Let B ⊂ Rd be a bounded domain with C ∞ boundary, S B a compact set, and Sn a sequence of compact sets with Sn → S in the sense of Sconvergence deﬁned in Deﬁnition 8.3.5. Suppose the pressure function p() and the boundary and initial data (U, ∞ ) satisfy Condition 9.2.5. Let un : B×(0, T ) → Rd be vector ﬁelds and n : B × (0, T ) → R, n ≥ 1, be nonnegative functions satisfying un (x, t) = 0, un ∈ L

2

in Sn × (0, T ),

n (x, t) = ∞ (x, 0)

(0, T ; WS1,2 (B)), n

∞

n ∈ L (0, T ; L (B)). γ

(9.3.1) (9.3.2)

256

Chapter 9. Flow around an obstacle. Domain dependence

Assume that the couples (un , n ), n ≥ 1, are renormalized solutions to problem (9.2.4) (with ﬁxed initial and boundary data (U, ∞ )) in (B \ Sn ) × (0, T ), and meet all requirements of Deﬁnition 9.2.3. Furthermore, assume that un L2 (0,T ;W 1,2 (B)) + n |un |2 L∞ (0,T ;L1 (B)) + n L∞ (0,T ;Lγ (B)) ≤ ce , (9.3.3) Sn

for all suﬃciently large n, and for any compact Q (B \ S) × (0, T ) and θ ∈ (0, min{2γd−1 − 1, 2−1 γ}), there is a constant c, depending only on θ, ce , and Q , such that γ+θ dxdt ≤ c. (9.3.4) n Q

Then there are u ∈ and ∈ L∞ (0, T ; Lγ (Ω)) and subsequences, still denoted by un and n , with the properties L2 (0, T ; WS1,2 (B))

un u weakly in L2 (0, T ; W 1,2 (B)), n weakly in L∞ (0, T ; Lγ (B)), n → strongly in Lr ((B \ S) × (0, T ))

(9.3.5) for all 1 ≤ r < γ.

Moreover, there is b > 1 such that for all open S S, n un ⊗ un u ⊗ u n un u

weakly in L2 (0, T ; Lb (B \ S )),

weakly in L∞ (0, T ; L2γ/(γ+1) (B)).

(9.3.6)

For any compact set Ω B \ S, p(n ) → p()

in L1 (Ω × (0, T )).

(9.3.7)

The couple (u, ) is a renormalized solution to problem (9.2.4), meets all requirements of Deﬁnition 9.2.3, and satisﬁes estimates (9.3.3)–(9.3.4). The proof is based on the kinetic equation method. Our strategy is the following. First we prove that the distribution function f (x, t, s) associated with the sequence n satisﬁes the kinetic equations in the cylinder (B \ S) × (0, T ). This part of the proof imitates the proof of Theorem 7.4.1. Next we employ Proposition 8.3.9 to prove that the kinetic equations can be extended to the larger cylinder B × (0, T ). Then we apply Theorem 7.1.12 to ensure that f (x, t, ·) is a Heaviside step function, which yields the strong convergence of the densities n . Once the convergence of the densities is established, it is a routine task to prove that the limit (u, ) is a renormalized solution to compressible Navier-Stokes equations. Part 1. Kinetic equation. First of all we adopt the convention that the functions n are extended to B × (0, T ) by n (x, t) = ∞ (x, 0) in Sn × (0, T ).

(9.3.8)

9.3. Main stability theorem

257

Lemma 9.3.2. There are subsequences, still denoted by (un , n ), a vector ﬁeld u ∈ L2 (0, T ; WS1,2 (B)), a function ∈ L∞ (0, T ; Lγ (B)), and a Young measure μ ∈ L∞ w (Q; M(R)) such that for any function ϕ ∈ C0 (R), un u n ϕ(n ) ϕ

weakly in L2 (0, T ; W 1,2 (B)), weakly in L∞ (0, T ; Lγ (B)),

(9.3.9)

weakly in L∞ (B × (0, T )),

where ϕ(x, t) = μxt , ϕ. Proof. Since un is bounded in L2 (0, T ; W 1,2 (Ω)), passing to a subsequence we can assume that un ∈ L2 (0, T ; WS1,2 (Ω)) converges weakly in L2 (0, T ; W 1,2 (B)) n to some vector ﬁeld u. It remains to note that by Proposition 8.1.3 the limit u belongs to L2 (0, T ; WS1,2 (Ω)). The existence of follows from the boundedness of n in the space L∞ (0, T ; Lγ (B)) and weak compactness of bounded subsets of this space. The existence of the Young measure μ is a straightforward consequence of the fundamental Young measure theorem (Theorem 1.4.5). Denote as before by f (x, t, s) the distribution function of the Young measure μxt deﬁned by Lemma 9.3.2, f (x, t, s) = μxt (−∞, s]. Since the n are nonnegative, we have f (x, t, s) = 0 for s < 0 and

f (x, t, s) = μxt [0, s] for s ≥ 0.

(9.3.10)

The next lemma establishes general properties of the distribution function as well as the weak limit of the pressure function. Denote by p : B × (0, T ) → [0, ∞] the function deﬁned by p(x, t) = p(s) ds f (x, t, s), (x, t) ∈ B × (0, T ). [0,∞)

Lemma 9.3.3. Under the assumptions of Theorem 9.3.1, we have γ ψ(x, t) s ds f (x, t, s) dxdt ≤ lim ψγn dxdt, (9.3.11) n→∞ B×(0,T ) [0,∞) B×(0,T ) ψ(x, t) sθ+γ ds f (x, t, s) dxdt ≤ lim ψγ+θ dxdt, (9.3.12) n Q

n→∞

[0,∞)

Q

for all nonnegative ψ ∈ L∞ (B × (0, T )) and all Q Q, Q = (B \ S) × (0, T ). Moreover, for all ψ ∈ L∞ (Q ), ψp dxdt = lim ψp(n ) dxdt. (9.3.13) Q

n→∞

Q

258

Chapter 9. Flow around an obstacle. Domain dependence

Proof. Consider the nonnegative Carathéodory function F (x, t, s) = ψ(x, t)|s|γ . It follows from estimate (9.3.3) that F (x, t, n ) dxdt ≤ ce ψL∞ (B×(0,T )) . B×(0,T )

Hence the integrals of F (x, t, n ) over B × (0, T ) are uniformly bounded for n = 1, 2, . . . . Recall that f is the distribution function of the Young measure μxt associated with the sequence n , i.e. dμxt (s) = ds f (x, t, s). Hence we can apply the fundamental theorem on Young measures (Theorem 1.4.5) with Ω replaced by B × (0, T ) and vn replaced by n to obtain (9.3.11). Next consider the integrand F (x, t, s) = ψ(x, t)|s|θ+γ on a compact set Q Q. It follows from estimate (9.3.4) that F (x, t, n (x, t)) dxdt ≤ cψL∞ (Q) θ+γ dxdt ≤ ce ψL∞ (Q) . n Q

Q

Applying Theorem 1.4.5 with Ω replaced by Q and vn replaced by n we obtain (9.3.12). Finally choose Q Q and ψ ∈ L∞ (Q ). Consider the integrand F (x, t, s) = ψ(x, t)p(s). Since p ≤ c(1 + sγ ) for s ≥ 0, we have 1+θ/γ

|F (x, t, n )|1+θ/γ ≤ cψL∞ (Q ) (1 + γ+θ n ). Now estimate (9.3.4) yields 1+ϑ/γ |F (x, t, n )|1+θ/γ dxdt ≤ ce ψL∞ (Q ) 1 + Q

Q

γ+ϑ dxdt ≤ c, n

where c is independent of n. Since θ/γ > 0, the sequence F (x, t, n ) is equiintegrable in Q . Applying Theorem 1.4.5 we conclude that μxt , F (x, t, ·) dxdt = lim F (x, t, n (x, t)) dxdt, n→∞

Q

Q

which along with the identity μxt , F (x, t, ·) = ψ(x, t)

p(s) ds f (x, t, s) [0,∞)

yields (9.3.13).

In particular, the weak limits and p of the densities n and pressure functions p(n ) are determined as the expectation values (x, t) = s ds f (x, t, s), p(x, t) = p(s) ds f (x, t, s). (9.3.14) R

R

9.3. Main stability theorem

259

Lemma 9.3.4. Let ϕ ∈ C01 (R), let Sδ be the δ-neighborhood of S and let Qδ = (B \ Sδ ) × (0, T ) ⊂ Q. Then there exist subsequences, still denoted by n , un , such that as n → ∞, n un u ϕ(n )un ϕu n un ⊗ un u ⊗ u

weakly in L∞ (0, T ; L2γ/(γ+1) (B)), weakly in L2 (0, T ; L2 (B)),

(9.3.15)

weakly in L (0, T ; L (B \ Sδ )). 2

b

Here b is an arbitrary number from the interval (1, (1 − 2−1 a)−1 ), a = 2d−1 − γ −1 . Proof. By the extension lemma (Lemma 9.2.4), the functions n and un satisfy the mass and momentum balance equations ∂t n = div un in B × (0, T ), ∂t ϕn = div fn + hn ∂t (n un ) = div Vn + wn in (B \ Sn ) × (0, T ),

in B × (0, T ),

understood in the sense of distributions. Recall that un ∈ L2 (0, T ; WS1,2 (B)) vann ishes on Sn × (0, T ) and Sn ⊂ Sδ for all suﬃciently large n. Here, un = −n un , fn = −ϕ(n ) un , hn = (ϕ(n ) − n ϕ (n )) div un ,

wn = n f , Vn = S(un ) − p(n ) I − n un ⊗ un .

It follows from estimates (9.3.3) that the sequences un and n are bounded in L2 (0, T ; W 1,2 (B)) and L∞ (0, T ; Lγ (B)), respectively. On the other hand, (9.3.3) implies that (un , n ) are bounded energy functions in the sense of Section 4.2. Moreover their energies are bounded by the constant ce of (9.3.3) independent of n. Hence we can apply Corollary 4.2.2 to conclude that n un L∞ (0,T ;L2γ/(γ+1) (B)) ≤ ce . Consequently, the sequence un is bounded in L1 (B × (0, T )). Thus the functions n and the vector ﬁelds un satisfy Condition 4.4.1 of Theorem 4.4.2, and so n un u weakly in L2 (0, T ; Lm (Ω)) for m−1 > 2−1 + γ −1 − d−1 . Since the sequence n un is bounded in L∞ (0, T ; L2γ/(γ+1) (Ω)), it also converges weakly in L∞ (0, T ; L2γ/(γ+1) (Ω)). This leads to the ﬁrst relation in (9.3.15). Repeating these arguments we obtain the second relation. To prove the third relation, notice that B \ Sδ ⊂ B \ Sn for all suﬃciently large n. Next, by estimates (9.3.3) and the inequalities 0 ≤ p ≤ c(1 + γ ), the sequence Vn is bounded in L1 (Qδ ). On the other hand, since f is bounded, it follows from (9.3.3) that wn is bounded in L1 (Qδ ). Therefore, the sequences n , un , Vn , and wn satisfy all hypotheses of Theorem 4.4.2, which yields the last relation of (9.3.15). Now our task is to derive the kinetic equation for the distribution function f (x, t, s) in the cylinder (B \ S) × (0, T ). This is a consequence of two lemmas which can be considered as modiﬁcations of Lemmas 7.2.6–7.2.7.

260

Chapter 9. Flow around an obstacle. Domain dependence

Lemma 9.3.5. Let the hypotheses of Theorem 9.3.1 be satisﬁed and suppose Φ ∈ C(R+ ) vanishes for all suﬃciently large s. Then, possibly passing to a subsequence, we can assume that Φn div un Φ div u

weakly in L2 (Q),

(9.3.16)

Vn Φn −(λ + 1)Φ div u + Φp weakly in L (Q), 1 Φ div u = Φ div u + Φp − Φp . λ+1 2

(9.3.17) (9.3.18)

Here Φn = Φ(n ),

Vn = −(1 + λ) div un + p(n ).

Proof. Assume that Φ is extended to the semiaxis (−∞, 0), and the extended function belongs to Cc (R). It suﬃces to prove the lemma for Φ ∈ C0∞ (R) since C0∞ (R) is dense in C0 (R). First observe that the governing equations (9.2.4) for (un , n ) can be rewritten in the form ∂t (n un ) + div(n un ⊗ un ) = div Tn + n f in (B \ Sn ) × (0, T ), ∂t n + div(n un ) = 0 in (B \ Sn ) × (0, T ),

(9.3.19a) (9.3.19b)

where Tn = S(un ) − p(n ) I, .

(9.3.19c)

Moreover, since the couple (un , n ) is a renormalized solution to problem (9.2.4) we have the following equation for Φn = Φ(n ): ∂Φn + div(Φn un ) + n = 0 in (B \ Sn ) × (0, T ).

(9.3.19d)

Here n = (Φ (n )n − Φ(n )) div un .

(9.3.19e)

Now choose r satisfying 1 < r ≤ 1 + θγ −1 , q ∈ (1, 2) and take s = ∞. Next set Gn = 0, gn = gnϕ = 0. It follows from estimates (9.3.3) and (9.3.4) and the inequality p ≤ c(1 + γ ) that for any Q (B \ Sn ) × (0, T ), Tn L1 (B×(0,T )) ≤ c,

Tn Lr (Q ) ≤ c(Q ),

n L2 (B×(0,T )) ≤ ce .

(9.3.20)

The latter inequality is a consequence of estimate (9.3.3) for un and the boundedness of Φ − Φ. Now choose a suﬃciently small δ > 0 and denote by Sδ the δ-neighborhood of S. Since Sn → S in the Hausdorﬀ metric, we have (B \ Sδ ) × (0, T ) ⊂ (B \ Sn ) × (0, T ) for all suﬃciently large n. It follows that the triplet (n , un , Φn ) and the exponents s, r , q satisfy Condition 4.6.1 of Theorem 4.6.3 and, in view of representations

9.3. Main stability theorem

261

(9.3.19c) and (9.3.19e), they also meet all requirements of Theorem 4.7.1 in the cylinder (B \ Sδ ) × (0, T ). Next, by (9.3.4) the functions p(n ) are bounded in Lr (Q ) for any compact subset Q (B \ Sδ ) × (0, T ). On the other hand, in view of (9.3.13), lim ψ(x, t)p(n ) dxdt = ψ(x, t)p dxdt for all ψ ∈ L∞ (Q ). n→∞

Q

It follows that

Q

p(n ) p weakly in Lr (Q ).

(9.3.21)

Notice that p : B × (0, T ) → R is already deﬁned and is independent of Q . Thus Vn = −(1 + λ) div un + p(n ) V = −(1 + λ) div u + p

(9.3.22)

weakly in Lr (Q ) for every compact Q (B\Sδ )×(0, T ). Next, since Φ ∈ C0∞ (R), the convergences (9.3.9) imply Φn Φ weakly in L∞ (B × (0, T )). Applying Theorem 4.7.1 we ﬁnd that lim Vn Φn η dxdt = n→∞

(B\Sδ )×(0,T )

V Φη dxdt

(9.3.23)

(9.3.24)

(B\Sδ )×(0,T )

for all η ∈ C0∞ ((B \ Sδ ) × (0, T )). Notice that the integrand on the right hand side of (9.3.24) is deﬁned in B × (0, T ) and is independent of δ. Recall that p(s)Φ(s) belongs to Cc (R) and Φn p(n ) are bounded uniformly in n, which along with (9.3.9) implies Φn p(n ) Φp weakly in L∞ (B × (0, T )).

(9.3.25)

Together with (9.3.24), this implies that for any η ∈ C0∞ ((B \ Sδ ) × (0, T )), the limit lim ηΦn div un dxdt n→∞

Q

exists and is independent of η and δ. Since the sequence Φn div un is bounded in L2 (B × (0, T )), this implies the existence of some Φ div u ∈ L2 ((B \ Sδ ) × (0, T )) such that Φn div un Φ div u weakly in L2 ((B \ Sδ ) × (0, T )).

(9.3.26)

As this limit is independent of δ, this implies (9.3.16). Combining (9.3.25) with (9.3.26) we obtain −(1 + λ)Φ(n ) div un + Φ(n )p(n ) −(λ + 1)Φ div u + Φp weakly in L2 (Q),

262

Chapter 9. Flow around an obstacle. Domain dependence

which yields (9.3.17). It follows from (9.3.24) that the weak limit of Vn Φn exists and is independent of δ. Thus we get Φ div u = Φ div u +

1 Φp − Φp , λ+1

which is (9.3.18).

Lemma 9.3.6. Let the hypotheses of Theorem 9.3.1 be satisﬁed. Furthermore assume that ϕ ∈ C0∞ (R) and Φ(s) = ϕ (s)s − ϕ(s). Then the integral identity 1 ϕ(∂t ψ + ∇ψ · u) dxdt − ψ Φp − Φp dxdt λ+1 Q Q − ψΦ div u dxdt + (ψϕ(∞ ))(x, 0) dx −

Q

ψϕ(∞ )U · n dSdt = 0

ST

Ω

(9.3.27) holds for all ψ ∈ C ∞ (Q) vanishing in a neighborhood of Σ \ Σin , of S × (0, T ), and of Ω × {t = T }. Here ϕ= ϕ(s) ds f (x, t, s), p = p(s) ds f (x, t, s), [0,∞) [0,∞) (9.3.28) Φp = Φ(s)p(s) ds f (x, t, s), Φ = Φ(s) ds f (x, t, s). [0,∞)

[0,∞)

Proof. Choose ϕ ∈ C0∞ (R) and a test function ψ as in the statement. Since n is a renormalized solution to the boundary value problem (9.2.4) in the cylinder (B \ Sn ) × (0, T ) in the sense of Deﬁnition 9.2.3, we have (B\Sn )×(0,T )

ϕn (∂t ψ + ∇ψ · un ) − ψΦn div un dxdt (ψϕ(∞ ))(x, 0) dx − ψϕ(∞ )U · n dSdt = 0, + B\Sn

ST

where ϕn = ϕ(n ) and Φn = ϕ (n )n − ϕ(n ). It is important to notice that ψ vanishes in a neighborhood of Σ \ Σin , of Sn × (0, T ) and of (B \ Sn ) × {t = T } for all suﬃciently large n since Sn → S in the Hausdorﬀ metric. Letting n → ∞ and using (9.3.15) and (9.3.16) we obtain

ϕ(∂t ψ + ∇ψ · u) dxdt − ψΦ div u dxdt Q Q + (ψϕ(∞ ))(x, 0) dx − Ω

Substituting (9.3.18) we obtain (9.3.27).

ψϕ(∞ )U · n dSdt = 0. ST

9.3. Main stability theorem

263

Lemma 9.3.7. Under the assumptions of Theorem 9.3.1, let ψ ∈ C ∞ (B×(0, T )×R) satisfy in a neighborhood of ((Σ \ Σin ) × R) ∪ (B × {t = T } × R) ∪ (S × (0, T ) × R),

ψ=0

ψ(x, t, s) = 0

(9.3.29)

for all large s uniformly in (x, t).

Then the distribution function f deﬁned by (9.3.10) satisﬁes f ∂t ψ + f ∇x ψ · u − f ∂s ψ div u − ∂s ψ C[f ] dsdxdt Q R + f∞ (x, 0, s)ψ(x, 0, s) dsdx − f∞ (x, t, s)ψ(x, t, s)U · n dsdΣ = 0. R

Ω

Σin

R

(9.3.30) The nonlinear operator C[f ] is deﬁned by 1 C[f ] = (p(τ ) − p) dτ f (x, t, τ ), λ + 1 (−∞,s] p= p(s) ds f (x, t, s),

(9.3.31) (9.3.32)

R

and f∞ is given by f∞ (x, t, s) = 1

for s ≥ ∞ (x, t)

and

f∞ (x, t, s) = 0

otherwise.

(9.3.33)

Proof. As the proof uses the same arguments as in the proof of Lemma 7.2.8, we just give an outline. Choose η ∈ C0∞ (R) and set ∞ ∞ ϕ(s) = η(τ ) dτ, Φ(s) = sϕ (s) − ϕ(s) = −sη(s) − η(τ ) dτ. (9.3.34) s

s

First we derive a formula for ϕ. Repeating the arguments in the proof of Lemma 7.2.8 we obtain ϕ(x, t) = η(s)f (x, t, s) ds, ϕ∞ (x, t) = η(s)f∞ (x, t, s) ds, (9.3.35) R

R

and also ϕ = (sη) f (x, t, s) ds, Φ(x, t) = ϕ − ϕ(x, t) = sη (s)f (x, t, s) ds. (9.3.36) R

R

Next for a.e. (x, t) ∈ Q denote by F the distribution function of the Borel measure dνxt (s) = p(s)μxt (s) on R, p(τ ) dτ f (x, t, τ ). F (x, t, s) = (−∞,s]

264

Chapter 9. Flow around an obstacle. Domain dependence

Since p(s) is integrable with respect to the measure μxt for a.e. (x, t) ∈ Q, the function F (x, t, s) is monotone, right continuous in s and F (x, t, s) = 0 for s < 0,

F (x, t, s) → p(x, t) as s → ∞.

We have the following analogues of formulae (9.3.35)–(9.3.36): ϕ(s)p(s) ds f (x, t, s) = ϕ(s) ds F (x, t, s) = η(s)F (x, t, s) ds, R R R ϕ (s)sp(s) ds f (x, t, s) = ϕ (s)s ds F (x, t, s) = (sη(s)) F (x, t, s) ds, R

R

which gives

R

(ϕ (s)s − ϕ(s))p(s) ds f (x, t, s) = sη (s)F (x, t, s) ds R R η (s)s p(s) dτ f (x, t, τ ) ds. =

Φp =

R

(−∞,s]

Combining this formula with (9.3.36) we obtain Φp − Φp = η (s)s (p(s) − p) dτ f (x, t, τ ) ds R (−∞,s] = (λ + 1) sη (s)C[f ]) ds.

(9.3.37)

R

Let ς ∈ C ∞ (Q) vanish in a neighborhood of the set (Σ \ Σin ) ∪ (S × (0, T )) ∪ (Ω × {t = T }). Then ς and ϕ meet all requirements of Lemma 9.3.6 and hence 1 ϕ(∂t ς + ∇ς · u) dxdt − ς Φp − Φp dxdt λ+1 Q Q − ςΦ div u dxdt + (ςϕ(∞ ))(x, 0) dx − ςϕ(∞ )U · n dSdt = 0. (9.3.38) Q

ST

Ω

It follows from (9.3.35) that ϕ(∂t ς + ∇ς · u) dxdt = η(s)(∂t ς + ∇ς · u)f (x, t, s) dsdxdt, Q Q [0,∞) (ςϕ∞ )(x, 0) dx − ςϕ∞ U · n dSdt Ω ST ς(x, 0)η(s)f∞ (x, 0, s) dsdx − ςη(s)U · n f∞ (x, t, s) ds dSdt. = Ω

[0,∞)

ST

[0,∞)

(9.3.39)

9.3. Main stability theorem

265

Next, formulae (9.3.36) and (9.3.37) imply 1 ς Φp − Φp dxdt = ςsη (s)C[f ](x, t, s) dsdxdt, λ+1 Q Q [0,∞) ςΦ div u dxdt = ςsη (s)f (x, t, s) div u dsdxdt. Q

Q

(9.3.40)

[0,∞)

Substituting (9.3.39)–(9.3.40) into (9.3.38) we ﬁnally obtain f ∂t (ςη) + ∇x (ςη) · u dxdtds − s∂s (ςη)C[f ] dxdtds Q×[0,∞) Q×[0,∞) − s∂s (ςη)f div u dxdtds + (ςη)(x, 0, s)f∞ (x, 0, s) dxds Q×[0,∞) Ω×[0,∞) (ςη)f∞ U · n dSdtds = 0. (9.3.41) −

ST ×[0,∞)

Thus we have proved the desired identity (9.3.30) for all test functions ψ having the representation ψ(x, t, s) = ς(x, t)η(s). It remains to extend this result to all ψ(x, t, s) satisfying (9.3.29). To this end, apply the Fourier transform to ψ(x, t, s) with respect to variable s and argue as in the proof of Lemma 7.2.8 to obtain the desired identity (9.3.30). It follows from Lemma 9.3.7 that the distribution function f of the Young measure μ associated with the sequence n is a weak solution to the kinetic equation ∂t f + div(f u) − ∂s (sf div u) − ∂s C(f ) = 0 in Q × R. (9.3.42) The solution also satisﬁes in a weak sense the boundary and initial conditions f = f∞

at Σin × R,

f (x, 0, s) = f∞ (x, 0, s) on (B \ S) × R.

(9.3.43)

Our next task is to show that the solution obtained for the kinetic equation (9.3.42) is regular in the sense of Deﬁnition 7.1.8, i.e. it has the additional integrability properties pointed out in this deﬁnition. This results from the following three lemmas which are modiﬁcations of Lemmas 7.2.9–7.2.11. First we prove an identity which is a particular case of (9.3.18). For any continuous bounded function ϑ : Q → R set (9.3.44) φϑ (, x, t) = min{, ϑ(x, t)}. It follows from the deﬁnition of the distribution function f and from the fundamental theorem on Young measures (Theorem 1.4.5) that φϑ (·, n ) φϑ φϑ (x, t) = [0,∞)

weakly in L∞ (B × (0, T )), min{s, ϑ(x, t)} ds f (x, t, s).

where (9.3.45)

266

Chapter 9. Flow around an obstacle. Domain dependence

Lemma 9.3.8. Let ϑ ∈ C(Q), Q = (B \ S) × (0, T ). Then for any h ∈ Cc (Q), lim

n→∞

h φϑ (n ) div un − φϑ div u dxdt Q 1 lim = h φϑ (n )p(n ) − φϑ p dxdt. λ + 1 n→∞ Q

(9.3.46)

Proof. Let Q Q be a compact set such that supp h ⊂ Q . Notice that Q (B \ Sn ) × (0, T ) for all suﬃciently large n. Choose N > supQ ϑ(x, t) and δ > 0. Since φϑ is uniformly continuous in the cylinder Q × [0, N ], there are smooth functions ϕk : [0, N ] → R and ϑk : Q → R, 1 ≤ k ≤ n, such that the function φδ (x, t, ) =

n

ϑk (x, t)ϕk ()

(9.3.47)

k=1

approximates φϑ with error bounded by δ, sup |φϑ (x, t, ) − φδ (x, t, )| ≤ δ. Q×[0,N ]

Extend ϕk to [0, ∞) by setting ϕk (s) = ϕk (N ) for s > N . This gives an extension of φδ to the cylinder Q × [0, ∞) We keep the same notation for the extended functions. It follows from (9.3.44) that sup

|φϑ (x, t, ) − φδ (x, t, )| ≤ δ.

(9.3.48)

Q×[0,∞)

Let us prove that the desired equality (9.3.46) holds for the approximation φδ . Since the limits on both sides of (9.3.46) are linear in φϑ , it suﬃces to prove that for every k, lim

n→∞

hϑk ϕk (n ) div un − ϕk div u dxdt Q 1 lim = h ϕk (n )p(n ) − ϕk p dxdt. λ + 1 n→∞ Q

(9.3.49)

To this end notice that the Lipschitz function ϕk () − ϕk (N ) belongs to C0 (R+ ) and hence, by the convergences (9.3.9), we have

hϑk (ϕk (n ) − ϕk (N )) div un − (ϕk − ϕk (N )) div u dxdt n→∞ Q 1 = lim hϑk (ϕk (n ) − ϕk (N ))p(n ) − (ϕk − ϕk (N ))p dxdt. λ + 1 n→∞ Q lim

(9.3.50)

9.3. Main stability theorem

267

Recalling that div un converges to div u weakly in L2 (B×(0, T )) and using (9.3.18) we obtain lim hϑk ϕk (N ) div un − ϕk (N ) div u dxdt n→∞ Q 1 lim = hϑk ϕk (N )p(n ) − ϕk (N )p dxdt. λ + 1 n→∞ Q Combining this with (9.3.50) we arrive at (9.3.49). This completes the proof of the desired equality lim

n→∞

h φδ (n ) div un − φδ div u dxdt Q 1 lim = h φδ (n )p(n ) − φδ p dxdt λ + 1 n→∞ Q

(9.3.51)

for the approximation φδ . Next, by (9.3.3) we have div un L2 (Q ) + p(n )L1 (Q ) ≤ c, which gives h φϑ (n ) − φδ (n ) div un dxdt + h φϑ (n ) − φδ (n ) p(n ) dxdt ≤ cδ. Q

Q

(9.3.52) Next since p ∈ L1 (Q) and |φϑ − φδ | = |φϑ − φδ | ≤ δ, we have div u dxdt h φ − φ h φϑ − φδ p dxdt ≤ cδ. + ϑ δ Q

(9.3.53)

Q

Writing the desired conclusion (9.3.46) in the form lim H(x, t) dxdt = 0 n→∞

Q

with appropriate H, we observe that combining (9.3.51)–(9.3.53) gives −cδ ≤ lim inf H(x, t) dxdt ≤ lim sup H(x, t) dxdt ≤ cδ. n→∞

n→∞

Q

Q

Letting δ → 0 we obtain (9.3.46).

Now choose ϑ ∈ C(Q) and deﬁne Tϑ (x, t) = φϑ (x, t, ) − φϑ ,

where

(x, t) =

s ds f (x, t, s) [0,∞)

(9.3.54)

268

Chapter 9. Flow around an obstacle. Domain dependence

Recall (9.3.55)

φϑ (x, t) =

min{s, ϑ(x, t)} ds f (x, t, s), φϑ (x, t, ) = min s ds f (x, t, s), ϑ(x, t) . [0,∞)

(9.3.56)

[0,∞)

Lemma 9.3.9. Under the assumptions of Theorem 9.3.1 for every compact set Q Q = (B \ S) × (0, T ), there is a constant c independent of ϑ and Q such that Tϑ γ+1 Lγ+1 (Q ) ≤ lim sup n→∞

Q

| min{n , ϑ} − min{, ϑ}|γ+1 dxdt ≤ c.

(9.3.57)

Proof. Notice that the function s → φϑ (x, t, s) deﬁned by (9.3.54) is monotone and its derivative does not exceed 1. It follows that for any s ≤ s , |φϑ (x, t, s ) − φϑ (x, t, s )|1+γ ≤ (φϑ (x, t, s ) − φϑ (x, t, s ))(s − s ). γ

γ

(9.3.58)

Next, (9.2.13) implies s − s ≤ c(p(s ) − p(s )) for 1 ≤ s ≤ s . γ

γ

(9.3.59)

By (9.2.12), the pressure function p satisﬁes the estimate s ≤ c(p(s ) + 1) and is bounded on [0, 1], which leads to γ

s − s ≤ c(p(s ) − p(s )) + c for 0 ≤ s ≤ 1 ≤ s . γ

γ

(9.3.60)

Combining (9.3.59) and (9.3.60) we conclude that s − s ≤ c(p(s ) − p(s )) + c for 0 ≤ s ≤ s . γ

γ

(9.3.61)

From (9.3.58) we now obtain |φϑ (x, t, s ) − φϑ (x, t, s )|1+γ ≤ cp (φϑ (x, t, s ) − φϑ (x, t, s ))(p(s ) − p(s ) + c). (9.3.62) Now choose Q Q and h ∈ C0 (Q) such that 0 ≤ h ≤ 1 and h = 1 in Q . It follows from the deﬁnition of Tϑ that γ+1 |Tϑ | dxdt ≤ h|Tϑ |γ+1 dxdt Q Q h|φϑ (x, t, n ) − φϑ (x, t, )|1+γ dxdt. (9.3.63) = lim n→∞

Q

9.3. Main stability theorem

269

Now our task is to estimate the right hand side of this inequality. First we use (9.3.62) to obtain lim

n→∞

Q

h|φϑ (x, t, n ) − φϑ (x, t, )|1+γ dxdt h φϑ (x, t, n ) − φϑ (x, t, ) (p(n ) − p()) dxdt ≤ cp lim n→∞ Q + c lim h φϑ (x, t, n ) − φϑ (x, t, ) dxdt n→∞ Q = cp h I(x, t) dxdt + c h(φϑ − φϑ (x, t, )) dxdt, Q

(9.3.64)

Q

where I = φϑ p − φϑ p() − φϑ (·, ) p + φϑ (·, ) p(). Now rewrite the expression for I in the form I = φϑ p − φϑ p + φϑ − φϑ (·, ) p − p() .

(9.3.65)

Recall that, by Condition 9.2.5, the pressure function has a representation p() = pc () + pb () where pc is convex and pb is continuous and bounded, so there exists c such that |pb | ≤ c. Since pc is convex, we have pc ≥ pc (). It follows that p ≥ pc − c ≥ pc () − c ≥ p() − 2c,

so

p − p() ≥ −2c.

On the other hand, since φϑ (x, t, ·) is concave, we have φϑ ≤ φϑ (). Thus we get φϑ − φϑ () p − p() ≤ 2c φϑ () − φϑ , which along with (9.3.65) leads to I ≤ φϑ p − φϑ p + c φϑ () − φϑ . Combining this with (9.3.64) we arrive at lim

n→∞

h|φϑ (x, t, n ) − φϑ (x, t, )|1+γ dxdt Q ≤c h φϑ p − φϑ p dxdt + c h(φϑ () − φϑ ) dxdt, Q

(9.3.66)

Q

Next, the obvious inequality 0 ≤ φϑ (x, t, s) ≤ s implies 0 ≤ φϑ ≤ and 0 ≤ φϑ (x, t, ) ≤ . From this, (9.3.14) and Lemma 9.3.3, we obtain h(φϑ (x, t, ) − φϑ ) dxdt ≤ 2 h dxdt ≤ c. Q

Q

270

Chapter 9. Flow around an obstacle. Domain dependence

Combining this with (9.3.66) we arrive at h|φϑ (x, t, n ) − φϑ (x, t, )|1+γ dxdt ≤ c h φϑ p − φϑ p dxdt + c. lim →0

Q

Q

From this and Lemma 9.3.8 we obtain lim h|φϑ (x, t, n ) − φϑ (x, t, )|1+γ dxdt n→∞ Q ≤ c(λ + 1) h φϑ div u − φϑ div u dxdt + c, (9.3.67) Q

where

h φϑ div u − φϑ div u dxdt = lim

n→∞

Q

hφϑ (x, t, n )(div un − div u) dxdt. Q

Since φϑ (x, t, ) is bounded and ∇un ∇u weakly in L2 (Q), it follows that lim hφϑ (x, t, )(div un − div u) dxdt = 0. n→∞

Q

Combining the results obtained we arrive at h φϑ div u − φϑ div u dxdt Q = lim h φϑ (x, t, n ) − φϑ (x, t, ) div(un − u) dxdt. n→∞

(9.3.68)

Q

Next, the Young inequality implies c(λ + 1)h φϑ (x, t, n ) − φϑ (x, t, ) div(un − u) ≤ 2−1 h|φϑ (x, t, n ) − φϑ (x, t, )|γ+1 + 21/γ h{c(λ + 1)| div(un − u)|}(γ+1)/γ . Recall that the sequence div(un − u) is bounded in L2 (Q). From this and the inequality (γ + 1)/γ ≤ 2 we obtain lim sup | div(un − u)|(γ+1)/γ dxdt ≤ c, n→∞

Q

which gives c(λ + 1) lim

n→∞

h φϑ (x, t, n ) − φϑ (x, t, ) div(un − u) dxdt Q −1 ≤2 lim h|φϑ (x, t, n ) − φϑ (x, t, )|γ+1 dxdt + c. n→∞

Q

9.3. Main stability theorem

271

Recalling identity (9.3.68) we arrive at h φϑ div u − φϑ div u dxdt c(λ + 1) Q ≤ 2−1 lim h|φϑ (x, t, n ) − φϑ (x, t, )|γ+1 dxdt + c. n→∞

Q

Inserting this into (9.3.67) we get h|φϑ (x, t, n ) − φϑ (x, t, )|1+γ dxdt ≤ c, 2−1 lim n→∞

Q

which along with (9.3.63) leads to (9.3.57).

Lemma 9.3.10. Under the assumptions of Theorem 9.3.1, there is a constant c such that HL1+γ (Q) + sup Vυ L1 (Q) < c, (9.3.69) υ∈R+

where

min{s, υ}(p(s) − p) ds f (x, t, s),

Vυ (x, t) =

[0,∞)

(9.3.70) f (x, t, s)(1 − f (x, t, s)) ds.

H(x, t) = [0,∞)

Proof. It suﬃces to prove that for any compact subset Q Q, HL1+γ (Q ) + sup Vυ L1 (Q ) < c,

(9.3.71)

υ∈R+

with c independent of Q . Now recall the deﬁnitions of the functions Tϑ and : s ds f (x, t, s), ϑ(x, t) − min{s, ϑ(x, t)} ds f (x, t, s), Tϑ (x, t) = min [0,∞) [0,∞) (x, t) = s ds f (x, t, s). [0,∞)

Integration by parts in the Stieltjes integral yields min{s, ϑ(x, t)}ds f (x, t, s) = lim min{s, ϑ(x, t)}f (x, t, s) s→∞

[0,∞)

− lim min{s, ϑ(x, t)}f (x, t, s) − s→−0

ϑ

0

and

s ds f (x, t, s) = [0,∞)

f (x, t, s) ds

(9.3.72)

0

(x, t) =

ϑ

f (x, t, s) ds = ϑ −

0

∞

(1 − f (x, t, s)) ds.

(9.3.73)

272

Chapter 9. Flow around an obstacle. Domain dependence

It follows that

ϑ(x,t)

Tϑ (x, t) = Tϑ (x, t) =

f (x, t, s) ds 0 ∞

for (x, t) ≥ ϑ(x, t), (9.3.74)

(1 − f (x, t, s)) ds for (x, t) ≤ ϑ(x, t).

ϑ(x,t)

Next choose a sequence of nonnegative continuous functions ϑm such that ϑm → a.e. in Q. Substituting ϑm into (9.3.74) and letting m → ∞ we conclude that for a.e. (x, t) ∈ Q, (x,t) ∞ (9.3.75) Tϑm (x, t) → T = f (x, t, s) ds = (1 − f (x, t, s)) ds 0

(x,t)

as m → ∞. It now follows from Lemma 9.3.9 that there is a constant c such that T Lγ+1 (Q ) ≤ c. Noting that H ≤ T we arrive at the desired estimate for H in (9.3.69). To estimate Vυ , notice that by Lemma 9.3.8, 1 Vυ = w-lim φυ (n ) div un − φυ div u, n→∞ λ+1 where w-lim denotes the weak limit in L2 (Q ). Thus we get 1 Vυ = w-lim (φυ (n ) − φυ ()) div un − w-lim φυ (n ) − φυ () div u. n→∞ n→∞ λ+1 Now Lemma 9.3.9 yields Vυ L1 (Q ) ≤ c lim sup (div un L2 (Q ) + div uL2 (Q ) )φυ (n ) − φυ ()L2 (Q ) n→∞

≤ c lim sup φυ (n ) − φυ ()L2 (Q ) ≤ c. n→∞

Thus we have estimated Vυ as desired in (9.3.69).

By proving Lemma 9.3.10 we ﬁnish the ﬁrst part of the proof of the main stability theorem (Theorem 9.3.1) and arrive at the following: Conclusion. In view of Lemma 9.3.7 the distribution function f (x, t, s) deﬁned by Lemma 9.3.2 is a solution to the initial boundary value problem for the kinetic equation ∂t f + div(f u) − ∂s (sf div u) − ∂s C[f ] = 0 in (B \ S) × (0, T ) × R, f (x, t, s) = f∞ on Σin × R and (B \ S) × {t = 0} × R.

(9.3.76) (9.3.77)

Moreover, by Lemma 9.3.10 this solution is regular in the sense of Deﬁnition 7.1.8.

9.3. Main stability theorem

273

Part 2. Strong convergence of n . Now our task is to prove that under the assumptions of Theorem 9.3.1, the sequence of functions n converges a.e. in (B \ S) × (0, T ). We split the proof into three lemmas. The ﬁrst lemma shows that the integral identity (9.3.30) in Lemma 9.3.7, which deﬁnes a weak solution to the kinetic equation (9.3.76), is satisﬁed for a wider class of test functions than stated there. Lemma 9.3.11. Let ψ ∈ C ∞ (B × (0, T ) × R) satisfy the conditions ψ=0

in a neighborhood of ((Σ \ Σin ) × R) ∪ (B × {t = T } × R), for all large s uniformly in (x, t) ∈ B × (0, T ).

ψ(x, t, s) = 0

(9.3.78)

Then the distribution function f deﬁned by (9.3.10) satisﬁes (B\S)×(0,T )

R

f ∂t ψ + f ∇x ψ · u − f ∂s ψ div u − ∂s ψ C[f ] dsdxdt + f∞ (x, 0, s)ψ(x, 0, s) dsdx B\S

−

Σin

R

R

f∞ (x, t, s)ψ(x, t, s)U · n dsdΣ = 0. (9.3.79)

Here the operator f → C[f ] and the function f∞ are deﬁned by (9.3.31)–(9.3.33). Notice that the only diﬀerence between Lemmas 9.3.7 and 9.3.11 is that in Lemma 9.3.11 the test function ψ need not vanish in a neighborhood of S × (0, T ), but this diﬀerence is essential. Indeed if the test function is nonnull in a neighborhood of S × (0, T ), then a solution to the kinetic equation in B \ S × (0, T ) can be extended to S ×(0, T ) in such a way that the extension satisﬁes the integral identity (9.3.79). Proof of Lemma 9.3.11. By Lemma 9.3.7, (9.3.79) holds for all ψ which satisfy (9.3.78) and vanish in a neighborhood of S × (0, T ) × R. As u ∈ L2 (0, T, WS1,2 (B)), Proposition 8.3.9 yields a sequence of functions ψn ∈ C ∞ (B) that vanish in a neighborhood of S, satisfy 0 ≤ ψn ≤ 1, converge to 1 in B \ S and

T

|u| |∇ψn | dxdt = 0.

lim

n→∞

0

(9.3.80)

B\S

Let now ψ be an arbitrary function satisfying (9.3.78). It is clear that the functions ψψn , n ≥ 1, also satisfy (9.3.78) and vanish in a neighborhood of S × (0, T ). By Lemma 9.3.7 the identity (9.3.79) holds with ψ replaced by ψψn for all n ≥ 1.

274

Chapter 9. Flow around an obstacle. Domain dependence

Thus, with Ω = B \ S and Q = Ω × (0, T ), we get f ∂t ψ + f ∇x ψ · u − f ∂s ψ div u − ∂s ψ C[f ] ψn dsdxdt + Q R f∞ (x, 0, s)ψ(x, 0, s)ψn (x) dsdx − ψn f∞ (x, t, s)ψ(x, t, s)U · n dsdΣ Ω R Σin R =− f ψ∇x ψn · u dsdxdt. (9.3.81) Q

R

By (9.3.78), there is N < ∞ depending on ψ such that ψ(x, t, s) = 0 for |s| ≥ N . Thus we get f ∂t ψ + f ∇x ψ · u − f ∂s ψ div u ≤ c(1 + |u| + |∇u|)χN (s), (9.3.82) where χN is the characteristic function of the interval [−N, N ], and the constant c depends on ψ. It follows from (9.3.31) that |∂s ψ C[f ]| ≤ c(1 + p)χN (s).

(9.3.83)

From the boundedness of U we obtain |f∞ (x, 0, s)ψ(x, 0, s)| + |f∞ (x, t, s)ψ(x, t, s)U| ≤ cχN (s). Since |∇u| and p are integrable in B × (0, T ), it now follows from (9.3.82)–(9.3.83) that the integrands on the left hand side of (9.3.81) have an integrable majorant independent of n. The Lebesgue dominated convergence theorem yields f ∂t ψ + f ∇x ψ · u − f ∂s ψ div u − ∂s ψ C[f ] ψn dsdxdt Q R + f∞ (x, 0, s)ψ(x, 0, s)ψn (x) dsdx − ψn f∞ (x, t, s)ψ(x, t, s)U · n dsdΣ Ω R Σin R → f ∂t ψ + f ∇x ψ · u − f ∂s ψ div u − ∂s ψ C[f ] dsdxdt Q R + f∞ (x, 0, s)ψ(x, 0, s) dsdx − f∞ (x, t, s)ψ(x, t, s)U · n dsdΣ Ω

R

Σin

R

(9.3.84) as n → ∞. On the other hand, (9.3.80) implies f ψ∇x ψn · u dsdxdt ≤ |ψ| ds |u| |∇ψn (x)| dxdt Q R Ω×(0,T ) R |u| |∇ψn (x)| dxdt → 0 as n → ∞. (9.3.85) ≤c Ω×(0,T )

Letting n → ∞ in (9.3.81) and using (9.3.84)–(9.3.85) we obtain (9.3.79).

9.3. Main stability theorem

275

Now introduce the distribution function f ∗ : B × (0, T ) × R deﬁned by 0 f (x, t, s) in (B \ S) × (0, T ) × R, f ∗ (x, t, s) = (9.3.86) f∞ (x, 0, s) in S × (0, T ) × R. Recall that in view of (9.3.33), f∞ (x, 0, s) = 0 for s < ∞ (x, 0),

f∞ (x, 0, s) = 1 for s ≥ ∞ (x, 0). (9.3.87)

Lemma 9.3.12. Let ψ ∈ C ∞ (B × (0, T ) × R) satisfy condition (9.3.78). Then ∗ f ∂t ψ + f ∗ ∇x ψ · u − f ∗ ∂s ψ div u − ∂s ψ C[f ∗ ] dsdxdt B×(0,T ) R + f∞ (x, 0, s)ψ(x, 0, s) dsdx − f∞ (x, t, s)ψ(x, t, s)U · n dsdΣ = 0. B

R

Σin

R

(9.3.88) Proof. By (9.3.31) and (9.3.32), we have 1 ∗ ∗ ∗ p(τ ) dτ f (x, t, τ ) − p(τ ) dτ f (x, t, τ ) dτ f ∗ (x, t, τ ). C[f ] = λ + 1 (−∞,s] R (−∞,s] Notice that for x ∈ S we have f ∗ = f∞ , where f∞ is deﬁned by (9.3.87). It follows that for x ∈ S and s < ∞ (x, 0), ∗ p(τ ) dτ f (x, t, τ ) = dτ f ∗ (x, t, τ ) = 0, (−∞,s]

(−∞,s]

while for x ∈ S and s ≥ ∞ (x, 0), p(τ ) dτ f ∗ (x, t, τ ) = p(∞ (x, 0)), (−∞,s]

dτ f ∗ (x, t, τ ) = 1.

(−∞,s]

Moreover, for x ∈ S we have p= p(τ ) dτ f ∗ (x, t, τ ) = p(∞ (x, 0)). R

Combining the above we ﬁnd that C[f ∗ ] = 0 in S × (0, T ) × R. Noting that u = 0 a.e. in S × (0, T ) and f ∗ = f∞ is independent of t on this set, we arrive at ∗ f ∂t ψ + f ∗ ∇x ψ · u − f ∗ ∂s ψ div u − ∂s ψ C[f ∗ ] dsdxdt S×(0,T ) R = f ∗ ∂t ψ dxdtds = − f∞ (x, s)ψ(x, 0, t) dsdxdt. S×(0,T )

R

S

R

Combining this with (9.3.79) and noting that f = f ∗ in (B \ S) × (0, T ) × R we obtain the desired identity.

276

Chapter 9. Flow around an obstacle. Domain dependence

Lemma 9.3.13. The function f ∗ deﬁned by (9.3.86) satisﬁes the condition HL1+γ (B×(0,T )) + sup Vυ L1 (B×(0,T )) < ∞,

(9.3.89)

υ∈R+

where

Vυ (x, t) =

min{s, υ}(p(s) − p) ds f ∗ (x, t, s),

[0,∞)

∗

∗

f (x, t, s)(1 − f (x, t, s)) ds,

H(x, t) = [0,∞)

p= R

p(s) ds f ∗ (x, t, s).

Proof. It follows from (9.3.86)–(9.3.87) that f ∗ (x, t, ·) is a step function and takes only values 0 or 1 a.e. in S × (0, T ). It easily follows that H = Vυ = 0 in S × (0, T ). Since f = f ∗ in (B\S)×(0, T ), inequality (9.3.89) is a straightforward consequence of Lemma 9.3.10. Lemma 9.3.14. Let u and be deﬁned by Lemma 9.3.2. Then the sequence n converges to strongly in Lr ((B \ S) × (0, T )) for any r < γ. Moreover, for any compact set Q (B \ S) × (0, T ) and θ ∈ [0, min{2γ/d − 1, γ/2}), the sequence n converges to in Lγ+θ (Q ), and p(n ) converges to p() in L1+θ/γ (Q ). In particular, p = p() a.e. in (B \ S) × (0, T ). Proof. By Lemma 9.3.12, the distribution function f ∗ deﬁned by (9.3.86) is a weak solution to the initial boundary problem (7.1.45) in the cylinder B × (0, T ). Moreover, Lemma 9.3.13 implies that this solution is regular in the sense of Deﬁnition 7.1.8. It now follows from the hypotheses of Theorem 9.3.1 that f ∗ , u, the data ∞ , U and the domain B satisfy Condition 7.1.11 (with Ω replaced by B, and f replaced by f ∗ ) of Theorem 7.1.12. Hence there is a measurable function ∗ : B × (0, T ) → R such that f ∗ (x, t, s) = 0 for s < ∗ (x, t),

f ∗ (x, t, s) = 1 for s ≥ ∗ (x, t).

Notice that in view of (9.3.86), f ∗ is not equal to the distribution function f of the Young measure associated with the sequence n in the whole domain B × (0, T ), but only in (B \ O(S)) × (0, T ), where O(S) is an arbitrary neighborhood of the obstacle S. Thus we get ∗ ∗ s ds f (x, t, s) = s ds f (x, t, s) = (x, t) a.e. in (B \ O(S)) × (0, T ). (x, t) = R

R

In other words, the probability distribution generated by the Young measure associated with n is deterministic in (B \ O(S)) × (0, T ), and the associated Young measure is the Dirac measure concentrated at (x, t). It now follows from the general properties of Young measures that n → a.e. in (B \ O(S)) × (0, T ). Since O(S) is an arbitrary neighborhood of S, we obtain n → a.e. in (B \ S) × (0, T ). Since the sequence n is bounded in Lγ (B × (0, T )), Theorem 1.4.7 implies that n → in Lr ((B \ S) × (0, T )) for all 1 ≤ r < γ. It remains to note that the strong convergence of n in Lγ+θ (Q ) is a consequence of estimate (9.3.4).

9.3. Main stability theorem

277

Strong convergence of p(n ). The following lemma establishes the limit relation (9.3.7). Lemma 9.3.15. For any compact set Ω B \ S, in L1 (Ω × (0, T )).

p(n ) → p()

(9.3.90)

Proof. By Lemma 9.3.14 and the compactness criterion in Lp (Proposition 1.3.1), it suﬃces to show that the functions p(n ) are equi-integrable in Ω × (0, T ), i.e., for every ε > 0 there is δ > 0 independent of n such that p(n ) dxdt ≤ ε for all A ⊂ Ω × (0, T ) with meas A ≤ δ. (9.3.91) A

Choose 0 < h < T and set Q = Ω × (h, T − h). Then

p(n ) dxdt ≤ A

A∩Q

h

p(n ) dxdt +

T

+ 0

T −h

p(n ) dxdt. B\S

Since p ≤ c(1 + γ ) it follows from inequalities (9.3.3)–(9.3.4) that

h

T

p(n ) dxdt ≤ ch,

+ T −h

0

B\S

and A∩Q

p(n ) dxdt ≤ (meas A) θ

≤ (meas A) γ+θ

θ γ+θ

Q

Q

p(n )

γ+θ γ

(1 + n )γ+θ dxdt

dxdt γ γ+θ

γ γ+θ

≤ C(Ω , h)(meas A) γ+θ . θ

Here C = C(Ω , h) denotes a constant depending on Ω and h. Combining the above we arrive at θ p(n ) dxdt ≤ ch + C(Ω , h)(meas A) γ+θ . A

Setting h = ε/(2c) and δ = {ε/(2C(Ω , h))}1+γ/θ we obtain (9.3.91).

Completion of the proof of Theorem 9.3.1. Relations (9.3.5) in Theorem 9.3.1 directly follow from Lemmas 9.3.2 and 9.3.14. Lemmas 9.3.4 and 9.3.15 yield (9.3.6) and (9.3.7). It remains to prove that the limit functions (u, ) solve problem (9.2.4). First we show that they satisfy the momentum balance equations.

278

Chapter 9. Flow around an obstacle. Domain dependence

Lemma 9.3.16. Let (u, ) be deﬁned by Lemma 9.3.2. Then the integral identity u · ∂t ξ + u ⊗ u : ∇ξ + p() div ξ − S(u) : ∇ξ dxdt B×(0,T ) + f · ξ dxdt + (∞ U · ξ)(x, 0) dx = 0 (9.3.92) B×(0,T )

B

holds for all vector ﬁelds ξ ∈ C ∞ (B × (0, T )) equal to 0 in a neighborhood of the obstacle S × (0, T ), of the lateral surface ∂B × (0, T ), and of the top Ω × {T }. Proof. Choose ξ ∈ C ∞ (B × (0, T )) as in the statement. Notice that ξ vanishes in a neighborhood of Sn × (0, T ) for all suﬃciently large n since dH (Sn , S) → 0 as n → ∞. By hypothesis, n and un are weak solutions to the momentum balance equation (9.2.4a). The integral identity (9.2.5) yields n un · ∂t ξ dxdt + n ∇ξ : (un ⊗ un ) dxdt B×(0,T ) B×(0,T ) + ∇ξ : p(n ) I − S(un ) dxdt + n f · ξ dxdt B×(0,T ) Q ∞ (·, 0)U(·, 0) · ξ(·, 0) dx = 0. (9.3.93) + B

Since ξ vanishes in a neighborhood of S × (0, T ), the convergences (9.3.6) imply n un · ∂t ξ + n un ⊗ un − S(un ) : ∇ξ + n f · ξ dxdt lim n→∞ B×(0,T ) = u · ∂t ξ + u ⊗ u − S(u) : ∇ξ + f · ξ dxdt. (9.3.94) B×(0,T )

By the choice of the test vector ﬁeld ξ, there is a cylinder Ω ×(0, T ) with Ω B\S such that supp ξ ⊂ Ω × (0, T ). Now, letting n → 0 and applying Lemma 9.3.15 we obtain p(n ) div ξ dxdt = p() div ξ dxdt. (9.3.95) lim n→∞

B×(0,T )

Q

Finally, letting n → ∞ in (9.3.93) and invoking (9.3.94), (9.3.95) we obtain the desired integral identity (9.3.92). Lemma 9.3.17. The integral identity ϕ()∂t ψ + ϕ()u · ∇ψ + ψ ϕ() − ϕ () div u dxdt (B\S)×(0,T ) ψϕ(∞ )U · n dΣ − ϕ(∞ )ψ dx = Σin

(B\S)×{t=0}

(9.3.96)

9.3. Main stability theorem

279

holds for all test functions ψ ∈ C ∞ ((B \ S) × (0, T )) vanishing in a neighborhood of the surface Σ\Σin and of the top (B \S)×{t = T }, and for all smooth functions ϕ : [0, ∞) → R such that lim sup(|ϕ()| + |ϕ ()|) < ∞.

(9.3.97)

→∞

Proof. By the hypotheses of Theorem 9.3.1 the functions (un , n ) are renormalized solutions to the mass balance equations in (B \ Sn ) × (0, T ), and hence satisfy the identity (9.3.96) with S replaced by Sn . Since dH (Sn , S) → 0 as n → ∞, we have B×(0,T )

ϕ(n )∂t ψ + ϕ(n )un · ∇ψ + ψ ϕ(n ) − ϕ (n )n div un dxdt = ψϕ(∞ )U · n dΣ − ϕ(∞ )ψ dx (9.3.98) B×{t=0}

Σin

for all ψ ∈ C ∞ (B × (0, T )) vanishing in a neighborhood of Σ \ Σin , of B × {t = T }, and of S × (0, T ) and all suﬃciently large n. Fix such a ψ. It follows from the convergences (9.3.5) and the boundedness condition imposed on ϕ that ϕ(n ) and ϕ (n )n converge to ϕ() and ϕ (), respectively, strongly in Lr (B × (0, T )) for all r ∈ [1, ∞). Letting n → ∞ in (9.3.98) and recalling that un converges weakly in L2 (0, T ; W 1,2 (B)) we arrive at the integral identity (9.3.96). It remains to remove the restriction ψ = 0 in a neighborhood of S × (0, T ). For this purpose we use Proposition 8.3.9. Let ψn be a sequence of cut-oﬀ functions as in Proposition 8.3.9, and let ψ be an arbitrary test function satisfying the hypotheses of the lemma. Since ψn vanishes in a neighborhood of S × (0, T ) we may assume ψψn ∈ C ∞ (B × (0, T )). Moreover, each ψψn vanishes in a neighborhood of Σ \ Σin , of B × {t = T }, and of S × (0, T ). Substituting ψψn for ψ in (9.3.96) we obtain

ϕ()∂t ψ + ϕ()u · ∇ψ + ψ ϕ() − ϕ () div u ψn dxdt (B\S)×(0,T ) ψn ψϕ(∞ )U · n dΣ − ψn ϕ(∞ )ψ dx = Σin (B\S)×{t=0} − ψϕ()u · ∇ψn dxdt. (9.3.99) (B\S)×(0,T )

Since u ∈ L2 (0, T ; WS1,2 (B)), we can apply Proposition 8.3.9 to obtain ψϕ()u · ∇ψn dxdt ≤ c |u| |∇ψn | dxdt → 0 as n → ∞. (B\S)×(0,T )

(B\S)×(0,T )

Letting n → ∞ in (9.3.99) and recalling that the uniformly bounded functions ψn tend to 1 in B \ S, we arrive at the integral identity (9.3.96).

280

Chapter 9. Flow around an obstacle. Domain dependence

By Lemmas 9.3.16 and 9.3.17, the couple (u, ) meets all requirements of Deﬁnition 9.2.3 and is a renormalized solution to problem (9.2.4). The proof of Theorem 9.3.1 is complete.

Chapter 10

Existence theory in nonsmooth domains. Shape optimization, continuity of the work functional 10.1 Existence theory in nonsmooth domains In this section we employ the main stability theorem (Theorem 9.3.1) and Proposition 8.3.6 to prove that the ﬂow problem around an obstacle has a renormalized solution for every compact obstacle. The corresponding result is given by the following theorem, which is one of the main results of this monograph. Theorem 10.1.1. Let γ > d/2, assume that the pressure function p(), the ﬂow domain and the given data satisfy Condition 9.2.5, and S is an arbitrary compact set. Then problem (9.2.4) has a weak renormalized solution (u, ) which meets all requirements of Deﬁnition 9.2.3 and satisﬁes the estimate uL2 (0,T ;W 1,2 (Ω)) + |u|2 L∞ (0,T ;L1 (Ω)) + L∞ (0,T ;Lγ (Ω)) ≤ ce ,

(10.1.1)

where the constant ce depends only on d, UC 1 (Rd ×[0,T ]) , ∞ L∞ (Rd ×[0,T ]) , T , diam Ω, and the constant cp in Condition 9.2.5. Moreover, for every cylinder Q Q, and for θ ∈ [0, 2γ/d − 1), there is a constant c, depending only on θ, ce , and Q , such that Q

γ+θ dxdt ≤ c.

(10.1.2)

Proof. By Proposition 8.3.6, there exists a sequence of compact subsets Sn B S with C ∞ boundaries such that Sn −→ S (see Deﬁnition 8.3.5). It is easily seen that B, Sn , and the initial and boundary data U, ∞ meet all requirements of Theorem 9.2.7. Hence problem (9.2.4) in the cylinder (B \ Sn ) × (0, T ) has a renormalized

P. Plotnikov and J. Sokoáowski, Compressible Navier–Stokes Equations: Theory and Shape 281 Optimization, Monografie Matematyczne 73, DOI 10.1007/978-3-0348-0367-0_10, © Springer Basel 2012

282

Chapter 10. Existence theory in nonsmooth domains

solution (un , n ) in the sense of Deﬁnition 9.2.3. Moreover, by Theorem 9.2.7, this solution admits estimates (9.3.3) and (9.3.4). Hence the sequence (Sn , un , n ) meets all requirements of the stability theorem (Theorem 9.3.1), whose application completes the proof.

10.2 Continuity of the work functional In this section we show that the work functional W (see (3.3.1)) is well deﬁned on the set of renormalized solutions to problem (9.2.4) in ﬂow domains Ω = B \ S with compact obstacles S B. Moreover, we show that W is continuous with respect to S-convergence. It follows from (3.3.1) that

η (u · W)(x, T ) − (∞ U · W)(x, 0) dx

W (S, u, ) = −

T

+ 0

B\S

ηu · ∂t W + (u ⊗ u) − T : ∇(ηW) + ηf · W dxdt,

(10.2.1)

B\S

where T = ∇u + (∇u) + (λ − 1) div u I − p() I,

(10.2.2)

and W : Rd × (0, T ) → Rd is a smooth vector ﬁeld which characterizes the motion of the obstacle S. Recall that η ∈ C ∞ (B) is an arbitrary function such that η = 1 in a neighborhood of S and η = 0 in a neighborhood of ∂B. We cannot use (10.2.1) directly for renormalized solutions to problem (9.2.4) since the momentum u only belongs to L∞ (0, T ; L2γ/(γ+1) (B \ S)) and hence the vector ﬁeld (·, T )u(·, T ) is not correctly deﬁned as a product of two functions. The element (u)(·, T ) can be deﬁned by using the fact that for a.e. t the time derivative (u)t is a distribution on the space C0∞ (B \S). The following lemma allows us to cope with this diﬃculty. Lemma 10.2.1. Let U, ∞ , and a compact set S B satisfy all conditions of Theorem 10.1.1. Let (u, ) be the renormalized solution to problem (9.2.4) given by that theorem. Then the functions mh (x, T ) :=

1 h

T

(x, t)u(x, t) dt T −h

converge weakly in L2γ/(γ+1) (B \ S) to a function m0 (x, T ) as h → 0. Set (u)(x, T ) := m0 (x, T ).

(10.2.3)

Then (u)(·, T )L2γ/(γ+1) (B\S) ≤ cuL∞ (0,T ;L2γ/(γ+1) (B\S)) ≤ c.

(10.2.4)

10.2. Continuity of the work functional

283

Proof. Choose ϕ ∈ C0∞ (B \ S) and set υh (t) = h−1 (T − t) for T − h < t ≤ T,

υh (t) = 1 for t ≤ T − h.

(10.2.5)

Since ξ = υh (t)ϕ vanishes near the lateral surface ∂(B \ S) × [0, T ] and on the top {t = T }, we can insert ξ into the integral identity (9.2.5) to obtain

T

mh · ϕ dx = B\S

∞ U · ϕ(x, 0) dx,

υh (t)Pϕ dx dt + B\S

0

(10.2.6)

B\S

where Pϕ = (u ⊗ u − S(u)) : ∇ϕ + p() div ϕ + f · ϕ.

(10.2.7)

Since the vector ﬁeld f is bounded and p ≤ c(1 + γ ), we have |Pϕ | ≤ cϕC 1 (B\S) (|u|2 + |∇u| + |u| + γ + + 1). It now follows from (10.1.1) that

T

|∞ U · ϕ(x, 0)| dx + B\S

|Pϕ | dx dt ≤ cϕC 1 (B\S) . 0

B\S

On the other hand, we have |υh Pϕ | ≤ |Pϕ |, and υh Pϕ → Pϕ a.e. in (B \S)×(0, T ) as h → 0. The Lebesgue dominated convergence theorem yields

h→0

T

B\S

mh · ϕ dx =

lim

∞ U · ϕ(x, 0) dx.

Pϕ dx dt + + 0

B\S

(10.2.8)

B\S

Next, notice that in view of (10.1.1) the bounded energy functions and u satisfy all hypotheses of Corollary 4.2.2. Hence u ∈ L∞ (0, T ; L2γ/(γ+1) (B \ S)). As mh L∞ (0,T ;L2γ/(γ+1) (B\S))

1 ≤ h

T

T −h

(t)u(t)L2γ/(γ+1) (B\S) dt

≤ uL∞ (0,T ;L2γ/(γ+1) (B\S)) , the sequence mh is bounded in L2γ/(γ+1) (B \ S). Combining this with (10.2.8) we conclude that this sequence converges weakly in L2γ/(γ+1) (B \ S) as h → 0. Estimate (10.2.4) obviously follows from the deﬁnition of mh . Remark 10.2.2. The assertion of Lemma 10.2.1 stems from the fact that the time derivative ∂t (u) exists as a distribution deﬁned for a.e. t ∈ (0, T ) on the space C0∞ (B \ S). This easily follows from the momentum balance equation. Arguing as in the proof of Lemma 10.2.1 we conclude that 1 t (x, s)u(x, s) ds m0 (x, t) weakly in L2γ/(γ+1) (B \ S) as h → 0 h t−h

284

Chapter 10. Existence theory in nonsmooth domains

for any t ∈ (0, T ). Thus we can deﬁne (u)(x, t) := m0 (x, t)

(10.2.9)

for any t ∈ (0, T ). Hence the impulse moment u is uniquely deﬁned at each point of the interval (0, T ]. Moreover, one can prove that the mapping t → (u)(·, t) is continuous in the weak L2γ/(γ+1) (B \ S) topology. We point out that t → (u)(·, t) in (10.2.9) can be obtained from the product of the functions t → (·, t) and t → u(·, t) by a possible change of the value of this product on a set of measure zero with respect to the time variable. Therefore, for the ﬁxed value t = T the vector ﬁeld (u)(·, T ) deﬁned by (10.2.3) is just a notation and it cannot be recovered from just (·, T ) and u(·, T ). Remark 10.2.3. It follows from (10.2.3) and (10.2.8) that T Pϕ dx dt + ∞ U · ϕ(x, 0) dx. (u)(·, T ) = 0

B\S

(10.2.10)

B\S

It follows from Lemma 10.2.1 that the work functional W (S, u, ) is well deﬁned for all compact obstacles S and all renormalized solutions (u, ) deﬁned by Theorem 10.1.1. However, notice that the function (u)(T ) of the spatial variable in (10.2.1) is deﬁned by Lemma 10.2.1 as a weak trace of the moment (u)(x, t) on the top {t = T }. We are now in a position to prove the main result of this section on the continuity properties of the work functional. Assume that a sequence of compact obstacles Sn B and functions (un , n ) ∈ L2 (0, T ; WS1,2 (B)) × L∞ (0, T ; Lγ (B \ Sn )) satisfy the following condition: n S

Condition 10.2.4. (i) There is a compact set S B such that Sn −→ S as n → ∞ (see Deﬁnition 8.3.5). (ii) The couples (un , n ), deﬁned in the cylinder (B\Sn )×(0, T ), are renormalized solutions to problem (9.2.4) with boundary and initial data U, ∞ given by Theorems 9.3.1 and 10.1.1. (iii) There are (u, ) ∈ L2 (0, T ; WS1,2 (B)) × L∞ (0, T ; Lγ (B)) such that un u weakly in L2 (0, T ; W 1,2 (B)), n weakly in L∞ (0, T ; Lγ (B)). (iv) We have the convergence of volumes, lim meas Sn = meas S.

n→∞

(10.2.11)

Theorem 10.2.5. Let (un , n , Sn ) and (u, , S) satisfy Condition 10.2.4. Let W be a divergence free vector ﬁeld in C 1 ((B \ S) × (0, T )). Then (u, ) is a renormalized solution to problem (9.2.4), and W (Sn , un , n ) → W (S, u, )

as n → ∞.

(10.2.12)

10.2. Continuity of the work functional

285

As the sequence (un , n , Sn ) meets all requirements of the stability theorem (Theorem 9.3.1), the limit couple (u, ) is a renormalized solution to problem (9.2.4). Thus it suﬃces to prove (10.2.12). We split the proof into a sequence of lemmas. Lemma 10.2.6. Under the assumptions of Theorem 10.2.5, lim meas (S \ Sn ) ∪ (Sn \ S) = 0.

n→∞

Proof. For δ > 0, denote by S δ and Snδ the open δ-neighborhoods of the sets S and Sn , respectively. Since Sn converges to S in the Hausdorﬀ metric, the inclusion Sn ⊂ S δ holds for suﬃciently large n. For every such n, we have meas(S \ Sn ) ≤ meas(S δ \ Sn ) = meas S δ − meas Sn . Letting n → ∞ and using (10.2.11) we obtain lim sup meas (S \ Sn ) ≤ meas S δ − meas S.

(10.2.13)

n→∞

Since S is the intersection of the decreasing sequence {S δ }, we have limδ→0 meas S δ = meas S. Letting δ → 0 in (10.2.13) we ﬁnally get lim supn→∞ meas(S \ Sn ) = 0. It remains to show that limn→∞ meas(Sn \ S) = 0. To this end choose 0 < δ < ε. It follows from the deﬁnition (8.1.1) of the Hausdorﬀ metric that for all suﬃciently large n, we have Sn ⊂ Snδ ⊂ S ε , which gives Sn \ S ⊂ Snδ \ S ⊂ S ε \ S. Thus lim supn→∞ meas(Sn \ S) ≤ meas S ε − meas S. Letting ε → 0 we obtain limn→∞ meas(Sn \ S) = 0. Lemma 10.2.7. Under the assumptions of Theorem 10.2.5,

T

T

ηn f · W dxdt → B\Sn

0

ηf · W dxdt. 0

(10.2.14)

B\S

Proof. Choose ε > 0 and denote by S ε the ε-neighborhood of S. Since Sn → S in the Hausdorﬀ metric, we have B \ S ε ⊂ B \ Sn for suﬃciently large n. It follows from Condition 10.2.4 that (un , n , Sn ) and (u, , S) meet all requirements of the stability theorem (Theorem 9.3.1). Hence the functions n , extended as ∞ (·, 0) to Sn × (0, T ), are uniformly bounded in L∞ (0, T ; Lγ (B)) and converge a.e. in (B \ S) × (0, T ) to . In particular, n → in L1 ((B \ S) × (0, T )) and so

T

η(n − )f · W dxdt → 0 as n → ∞.

0

B\S

(10.2.15)

286

Chapter 10. Existence theory in nonsmooth domains

Notice that B \ Sn = ((B \ S) ∪ (S \ Sn )) \ (Sn \ S), which leads to

T

T

ηn f · W dxdt − 0

B\Sn

ηf · W dxdt

T

B\S

0

η(n − )f · W dxdt

= B\S

0

T

T

ηn f · W dxdt −

+ 0

S\Sn

0

Sn \S

ηn f · W dxdt.

By (10.2.15), the ﬁrst integral on the right hand side tends to zero as n → ∞, so lim sup n→∞

T

T

ηf · W dxdt

ηn f · W dxdt − B\Sn

0

= lim sup n→∞

T

0

ηn f · W dxdt − S\Sn

0

T

0

≤ c lim sup n→∞

B\S T

Sn \S

ηn f · W dxdt (10.2.16)

n dxdt, 0

SSn

where c = ηf · WC(B×(0,T )) < ∞ and denotes the symmetric diﬀerence of sets, S Sn = (S \ Sn ) ∪ (Sn \ S). Now the Hölder inequality implies T n dxdt ≤ n L∞ (0,T ;Lγ (SSn )) 1L1 (0,T ;Lγ/(γ−1) (SSn )) SSn

0

Since 1L1 (0,T ;Lγ/(γ−1) (SSn )) = T meas(S Sn )(γ−1)/γ , we obtain

T

0

SSn

n dxdt ≤ T n L∞ (0,T ;Lγ (SSn )) meas(S Sn )(γ−1)/γ .

By Condition 10.2.4(ii), the couple (un , n ) satisﬁes all conditions of Theorem 9.3.1 and hence admits the energy estimate (9.3.3). It follows that n L∞ (0,T ;Lγ (SSn )) ≤ ce < ∞, where ce is independent of n. Applying Lemma 10.2.6 we ﬁnally obtain T lim sup n dxdt ≤ T ce lim sup meas(S Sn )(γ−1)/γ = 0. n→∞

0

SSn

n→∞

Substituting this into (10.2.16) we arrive at (10.2.14).

Lemma 10.2.8. Under the assumptions of Theorem 10.2.5, for ϕ ∈ C 1 (B \ S) compactly supported in B \ S, lim (n un )(x, T ) · ϕ dx = (u)(x, T ) · ϕ dx. (10.2.17) n→∞

B\Sn

B\S

10.2. Continuity of the work functional

287

Proof. First we show that for any sequence hn → 0, T 1 n un · ϕ dxdt → (u)(x, T ) · ϕ dx as n → ∞. (10.2.18) hn T −hn B\Sn B\S Here the vector ﬁeld (u)(x, T ) is deﬁned by Lemma 10.2.1. Arguing as in the proof of Lemma 10.2.1 and noting that ϕ is compactly supported in B \ Sn for large n, we obtain, for all suﬃciently large n, T T 1 n un · ϕ dxdt = υhn (t)Pnϕ dx dt hn T −hn B\Sn 0 B\S (10.2.19) ∞ U · ϕ(x, 0) dx, + B\S

where υh is deﬁned by (10.2.5) and Pnϕ = (n un ⊗ un − S(un )) : ∇ϕ + p(n ) div ϕ + f · ϕ.

(10.2.20)

Notice that the functions υhn are uniformly bounded and converge everywhere to 1 as n → ∞. In particular, υhn → 1 strongly in any Lr (0, T ), r < ∞. On the other hand, Condition 10.2.4(iii) implies that −S(un ) : ∇ϕ + n f · ϕ −S(u) : ∇ϕ + f · ϕ 3/2

weakly in L T 0

((B \ S) × (0, T )). Thus we get

υhn (t) −S(un ) : ∇ϕ + n f · ϕ dxdt

B\Sn

T

→ 0

−S(u) : ∇ϕ + f · ϕ dxdt

as n → ∞. (10.2.21)

B\S

Next, as mentioned above, the sequence (un , n , Sn ) meets all requirements of Theorem 9.3.1. It follows that p(n ) → p() in L1 (Ω × (0, T )) for any measurable set Ω B \ S. Moreover, the matrix-valued functions n un ⊗ un converge weakly to u ⊗ u in L2 (0, T ; Lb (Ω )) for some b > 1. Setting Ω = supp ϕ leads to T υhn (t) n un ⊗ un : ∇ϕ + p(n ) div ϕ dxdt 0

B\Sn

T

→ 0

u ⊗ u∇ : ϕ + p() div ϕ dxdt

as n → ∞. (10.2.22)

B\S

Combining (10.2.21) and (10.2.22) we ﬁnally obtain T T 1 lim n un · ϕ dxdt = lim υhn (t)Pnϕ dx dt n→∞ hn T −h n→∞ 0 B\Sn B\S n T ∞ U · ϕ(x, 0) dx = Pϕ dx dt + ∞ U · ϕ(x, 0) dx, (10.2.23) + B\S

0

B\S

B\S

288

Chapter 10. Existence theory in nonsmooth domains

where Pϕ is deﬁned by (10.2.7). Combining (10.2.23) and identity (10.2.10) we obtain the desired relation (10.2.18). On the other hand, by Lemma 10.2.4 for every n there is hn > 0 such that hn < 1/n and

"

B\Sn

1 hn

T

T −hn

(n un )(x, T ) · ϕ(x) dx

# (x, t)u(x, t) dt · ϕ(x) dx −

B\Sn

≤ 1/n.

Substituting hn into (10.2.18) we arrive at (10.2.17).

Lemma 10.2.9. Under the assumptions of Theorem 10.2.5, η(n un )(x, T )·W(x, T ) dx = η(u)(x, T )·W(x, T ) dx. (10.2.24) lim n→∞

B\Sn

B\S

Proof. Choose 0 < δ < ε and denote the δ-neighborhood and ε-neighborhood of the limit obstacle S by S δ and S ε , respectively. Pick ς ∈ C0∞ (B) such that ς = 1 in B \ S ε ,

ς = 0 in S δ .

(10.2.25)

Since η vanishes near ∂B, the vector ﬁeld ϕ(·) = η ς W(·, T ) ∈ C 1 (B \ S) is compactly supported in B \ S, which along with Lemma 10.2.8 implies that for n → ∞, ςη(n un )(x, T ) · W(x, T ) dx − ςη(u)(x, T ) · W(x, T ) dx → 0. B\Sn

B\S

(10.2.26) Hence for t = T , η(n un ) · W dx − η(u) · W dx lim sup t=T n→∞ B\Sn B\S ≤ lim sup (1 − ς)η(n un ) · W dx − (1 − ς)η(u) · W dx n→∞

B\Sn

B\S

. t=T

(10.2.27) Next notice that Sn S ε for all suﬃciently large n and ς = 1 outside of S ε . For all such n we have (1 − ς)η(n un ) · W dx − (1 − ς)η(u) · W dx t=T B\Sn B\S = (1 − ς)η(n un ) · W dx − (1 − ς)η(u) · W dx t=T ε ε S \Sn S \S ≤c |(n un )(x, T )| dx + c |(u)(x, T )| dx, S ε \Sn

S ε \S

10.2. Continuity of the work functional

289

where c = (1 − ς)ηWC(B×(0,T )) . Inserting this in (10.2.27) we obtain lim sup η(n un ) · W dx − η(u) · W dx t=T n→∞ B\Sn B\S ≤ c lim sup |(n un )(x, T )| dx + c |(u)(x, T )| dx. (10.2.28) n→∞

S ε \Sn

S ε \S

Now our task is to show that the right hand side vanishes for n → ∞ and ε → 0. From Lemma 10.2.1 we conclude that (n un )(·, T )L2γ/(γ+1) (B\Sn ) + (u)(·, T )L2γ/(γ+1) (B\S) ≤ c, where c is independent of n. Next applying the Hölder inequality we obtain γ+1 γ−1 2γ |(u)(x, T )| dx ≤ c |(u)|2γ/(γ+1) dx meas(S ε \ S) 2γ S ε \S

S ε \S

≤ c meas(S ε \ S)(γ−1)/(2γ) . The same estimate holds for (n un ), |(n un )(x, T )| dx ≤ c meas(S ε \ Sn )(γ−1)/(2γ) . S ε \Sn

Inserting these into (10.2.28) we ﬁnally obtain lim sup η(n un )(x, T ) · W(x, T ) dx− n→∞ B\Sn η(u)(x, T ) · W(x, T ) dx B\S

≤ c lim sup meas(S ε \ Sn )(γ−1)/(2γ) + c meas(S ε \ S)(γ−1)/(2γ) . (10.2.29) n→∞

On the other hand, Lemma 10.2.6 and the inclusion S ε \ Sn ⊂ (S ε \ S) ∪ (S \ Sn )) give lim sup meas(S ε \ Sn )(γ−1)/(2γ) n→∞

(γ−1)/(2γ) ≤ lim sup meas(S ε \ S) + meas(S \ Sn ) ≤ meas(S ε \ S)(γ−1)/(2γ) . n→∞

Inserting this into (10.2.29) yields η(n un )(x, T ) · W(x, T ) dx lim sup n→∞ B\Sn − η(u)(x, T ) · W(x, T ) dx B\S

≤ 2c meas(S ε \ S)(γ−1)/(2γ) . It remains to note that the right hand side tends to 0 as ε → 0.

290

Chapter 10. Existence theory in nonsmooth domains

Remark 10.2.10. The results of Lemmas 10.2.6–10.2.9 require Condition 10.2.4. It is worth noting that the ﬁrst three conditions (i)–(iii) in 10.2.4 simply state that (Sn , un , n ) and (S, u, ) meet the conditions of the main stability theorem 9.3.1 and the existence theorem 10.1.1. The last condition (iv) is independent of the previous ones. This extra restriction means that the characteristic functions of the sets Sn converge in the L1 -norm, which is not in the spirit of the Hausdorﬀ and Kuratowski-Mosco convergences. It is important to notice that Lemmas 10.2.11– 10.2.14 below only require Condition 10.2.4(i)–(iii). S

(B)) converge Lemma 10.2.11. Let Sn −→ S B and let un ∈ L2 (0, T ; WS1,2 n 2 1,2 2 1,2 weakly in L (0, T ; W (B)) to u ∈ L (0, T ; W (B)). Then for any q ∈ [1, 2) and r ∈ [1, 2d/(d − 2)), un Lq (0,T ;Lr (S)) → 0

as n → ∞.

If in addition (un , n ) satisﬁes Condition 10.2.4(i)–(iii), then T n |un |2 dxdt → 0 as n → ∞.

(10.2.30)

(10.2.31)

S\Sn

0

Proof. Set

T

|un (x, t)| dt

un (x) =

for x ∈ B.

(10.2.32)

0

We have un L2 (B) ≤

√

T un L2 (0,T ;L2 (B)) ≤

√

T un L2 (0,T ;W 1,2 (B)) ≤ c.

(10.2.33)

Recall that a.e. in B, ∇|un | = |un |−1 un ∇un Hence ∇un 2L2 (B) ≤

B

for un = 0,

T

|∇un | dt 0

2

∇|un | = 0 otherwise.

dx ≤ T

T

|∇un |2 dt dx ≤ c. B

0

Altogether, un is bounded in W 1,2 (B). Since the embedding W 1,2 (B) → L2 (B) is compact, there is a subsequence unk which converges weakly in W 1,2 (B) and strongly in L2 (B) to some u ∈ W 1,2 (B). Moreover, any subsequence of un contains such a convergent subsequence (with limit depending on the subsequence, of course). Since unk ∈ L2 (0, T ; WS1,2 (B)), Lemma 8.3.13 implies that unk ∈ WS1,2 (B). nk nk Next notice that the obstacles Sn converge to S in the sense of Kuratowski-Mosco. It now follows from the weak convergence of unk that u ∈ WS1,2 (B) and, in particular, u = 0 on S. Recalling that unk converges to u strongly in L2 (B) and hence in L2 (S) we conclude that T |unk | dxdt = unk L1 (S) ≤ cunk L2 (S) → 0 as k → ∞. 0

S

10.2. Continuity of the work functional

291

Thus unk → 0 in L1 (S × (0, T )). This sequence is bounded in L2 (0, T ; W 1,2 (B)), so it is bounded in Lq (0, T ; Lr (S)) for all q ≤ 2 and r < 2d/(d−2). Hence unk → 0 in Lq (0, T ; Lr (S)) for all q ∈ [1, 2) and r ∈ [1, 2d/(d − 2)). Thus we have proved that any subsequence of un has a subsequence which tends to 0 in Lq (0, T ; Lr (S)). Hence the whole sequence tends to 0 in this space, which gives (10.2.30). It remains to prove (10.2.31). To this end notice that by Condition 10.2.4(ii), the functions (un , n ) satisfy all assumptions of Theorem 9.3.1 and, in particular, the energy estimate (9.3.3) with a constant ce independent of n. Next choose M > 0 and denote by En the set of all t ∈ (0, T ) such that un (·, t)W 1,2 (B) ≤ M. By (9.3.3) we have 2 un (·, t)W 1,2 (B) dt ≤ un (·, t)2W 1,2 (B) dt ≤ c2e , (0,T )\En

(0,T )

which along with un (·, t)W 1,2 (B) ≥ M on (0, T ) \ En leads to meas((0, T ) \ En ) ≤ c2e /M 2 . The energy estimate (9.3.3) yields n |un |2 L∞ (0,T ;L1 (B\Sn )) ≤ ce . Thus we get

n |un |2 dxdt B\Sn

(0,T )\En

≤ cn |un |2 L∞ (0,T ;L1 (B\Sn )) meas((0, T ) \ En ) ≤ ce · c2e /M 2 . (10.2.34) Next, the Hölder inequality and the energy estimate (see (9.3.3)) n L∞ (0,T ;Lγ (B\Sn )) ≤ ce imply En

n |un |2 dxdt S\Sn

1/γ

1−1/γ |un (·, t)|(2γ)/(γ−1) dx dt En B\Sn S = n (·, t)Lγ (B\Sn ) un (·, t)2Lr (S) dt ≤ ce un (·, t)2Lr (S) dt, ≤

En

n (·, t)γ dx

En

where r = 2γ/(γ − 1) < 2d/(d − 2) for γ > d/2.

292

Chapter 10. Existence theory in nonsmooth domains

Since the embedding W 1,2 (B) → Lr (B) is bounded, we have un (·, t)Lr (S) ≤ cM for every t ∈ En . This leads to n |un |2 dxdt ≤ ce M un (·, t)Lr (S) dt. (10.2.35) En

En

S\Sn

Combining (10.2.34) and (10.2.35) we deduce

T

n |un |2 dxdt ≤ ce M un L1 (0,T ;Lr (S)) + c3e /M. S\Sn

0

Letting n → ∞ and recalling (10.2.30) we obtain

T

n |un |2 dxdt ≤ c3e /M.

lim sup n→∞

0

S\Sn

Letting M → ∞ we ﬁnally arrive at (10.2.31).

Lemma 10.2.12. Assume that triplets (un , n , Sn ) and (u, , S) satisfy Condition 10.2.4(i)–(iii). Then

T

T

ηn un · ∂t W dxdt =

lim

n→∞

ηu · ∂t W dxdt.

B\Sn

0

(10.2.36)

B\S

0

(B)) and converge to Proof. Recall that the vector ﬁelds un belong to L2 (0, T ; WS1,2 n u weakly in L2 (0, T ; W 1,2 (B)). By Corollary 8.3.3 the limit vector ﬁeld u belongs to L2 (0, T ; WS1,2 (B)). It follows that un and u are zero a.e. on Sn × (0, T ) and S × (0, T ), respectively. Thus (10.2.36) can be equivalently rewritten as

T

T

ηn un · ∂t W dxdt → 0

B

ηu · ∂t W dxdt. 0

(10.2.37)

B

By Condition 10.2.4(ii), the triplets (un , n , Sn ) satisfy all hypotheses of Theorem 9.3.1. It follows from relation (9.3.6) in that theorem that the moments n un (extended by 0 onto Sn ) converge weakly in L∞ (0, T ; L2γ/(γ+1) (B)) to u. This obviously leads to (10.2.37). Lemma 10.2.13. Let (un , n , Sn ) and (u, , S) satisfy Condition 10.2.4(i)–(iii). Then T T Tn : ∇(ηW) dxdt → T : ∇(ηW) dxdt. (10.2.38) 0

B\Sn

0

B\S

Proof. Recall that the stress tensor T is deﬁned by (10.2.2) and consequently Tn = ∇un + (∇un ) + (λ − 1) div un I − p(n ) I.

(10.2.39)

10.2. Continuity of the work functional

293

(B)) and u ∈ L2 (0, T ; WS1,2 (B)), the vector ﬁelds un and Since un ∈ L2 (0, T ; WS1,2 n u vanish a.e. on Sn × (0, T ) and S × (0, T ), respectively. Hence ∇un = 0 a.e. on Sn ,

∇u = 0 a.e. on S.

Noting that ∇un converges to ∇u weakly in L2 (0, T ; L2 (B)) we obtain

T

(∇un + (∇un ) + (λ − 1) div un I) : ∇(ηW) dxdt

B\Sn

0

T

(∇u + (∇u) + (λ − 1) div u I) : ∇(ηW) dxdt.

→ 0

(10.2.40)

B\S

Recall that by the assumptions of Theorem 10.2.5, we have div W = 0, which leads to p I : ∇(ηW) = p∇η · W. Since ∇η vanishes near S and ∂B, ∇η ·W is compactly supported in some cylinder Ω × (0, T ) with Ω B \ S. Now Theorem 9.3.1 shows that p(n ) → p() in L1 (Ω × (0, T )). Thus we get

T

T

p(n ) I : ∇(ηW) dxdt = 0

B\Sn

T

→ 0

0

Ω

Ω

p(n )∇η · W dxdt

T

p() I : ∇(ηW) dxdt =

p()∇η · W dxdt. 0

B\S

Combining this relation with (10.2.40) we obtain (10.2.38).

Lemma 10.2.14. Let (un , n , Sn ) and (u, , S) satisfy Conditions 10.2.4(i)–(iii). Then T T n un ⊗un : ∇(ηW) dxdt → u⊗u : ∇(ηW) dxdt as n → ∞. 0

B\Sn

0

B\S

(10.2.41) Proof. By Condition 10.2.4, the functions (un , n ) satisfy all hypotheses of Theorem 9.3.1 and hence admit the energy estimate (9.3.3). It follows that the energy of (un , n ) is bounded from above by a constant ce independent of n. In particular, (un , n ) meet all requirements of Proposition 4.2.1. Corollary 4.2.3 of that proposition implies that there is an exponent b > 1, depending only on the adiabatic exponent γ and the space dimension d, such that n |un |2 L2 (0,T ;Lb (B\Sn )) ≤ c,

(10.2.42)

where c depends only on ce , b and the diameter of B. Moreover, it follows from the convergences (9.3.6) in Theorem 9.3.1 that for any Ω B \ S, n un ⊗ un u ⊗ u weakly in L2 (0, T ; Lb (Ω )).

(10.2.43)

294

Chapter 10. Existence theory in nonsmooth domains

Notice that Ω B \ Sn for all large n, since Sn → S in the Hausdorﬀ metric. Now choose ε > 0 and denote by S ε the ε-neighborhood of S. Then B \ S ε ⊂ B \ Sn for all suﬃciently large n. Hence for all such n, the sequence n un ⊗ un is deﬁned in (B \ S ε ) × (0, T ), bounded in L2 (0, T ; Lb (B \ S ε )), and converges weakly in L2 (0, T ; Lb (Ω )) to u ⊗ u for any compact Ω B \ S ε . Therefore n un ⊗ un converges weakly in L2 (0, T ; Lb (B \ S ε )) to u ⊗ u. Thus

T

T

n un ⊗un : ∇(ηW) dxdt →

u⊗u : ∇(ηW) dxdt. (10.2.44)

B\S ε

0

B\S ε

0

Let us consider the identity

T

T

n un ⊗ un : ∇(ηW) dxdt − 0

B\Sn

T

(S ε \S)\Sn T

T

n un ⊗ un : ∇(ηW) dxdt −

= 0

u ⊗ u : ∇(ηW) dxdt B\S

0

S ε \S

0 T

n un ⊗ un : ∇(ηW) dxdt −

+

B\S ε

0 T

u ⊗ u : ∇(ηW) dxdt B\S ε

0

u ⊗ u : ∇(ηW) dxdt

n un ⊗ un : ∇(ηW) dxdt.

+

(10.2.45)

S\Sn

0

It follows from Lemma 10.2.11 and boundedness of ∇(ηW) that

T

n un ⊗ un : ∇(ηW) dxdt → 0.

0

S\Sn

Inserting this along with (10.2.44) into (10.2.45) we obtain lim sup n→∞

T

T

u ⊗ u : ∇(ηW) dxdt

n un ⊗ un : ∇(ηW) dxdt − B\Sn

0

= lim sup n→∞

T

0

(S ε \S)\Sn

B\S

0

n un ⊗ un : ∇(ηW) dxdt

T

−

S ε \S

0

which leads to T lim sup n→∞

0

n un ⊗ un : ∇(ηW) dxdt −

B\Sn

≤ c lim sup n→∞

T

0

T

(S ε \S)\Sn

0

T

n |un |2 dxdt + c

u ⊗ u : ∇(ηW) dxdt,

u ⊗ u : ∇(ηW) dxdt

B\S

|u|2 dxdt, 0

S ε \S

(10.2.46)

10.2. Continuity of the work functional

295

where c = ∇(ηW)C(B×(0,T )) . Let us estimate the right hand side of this inequality. The Hölder inequality implies T n |un |2 dxdt 0

(S ε \S)\Sn

≤ n |un |2 L2 (0,T ;Lb (B\Sn )) 1L2 (0,T ;Lb/(b−1) ((S ε \S)\Sn )) . Next, (10.2.42) and

√ T meas((S ε \ S) \ Sn ))(b−1)/b √ ≤ T meas(S ε \ S)(b−1)/b

1L2 (0,T ;Lb/(b−1) ((S ε \S)\Sn )) = yield

T

0

n |un |2 dxdt ≤ c meas(S ε \ S)(b−1)/b .

(S ε \S)\Sn

(10.2.47)

Similar arguments give T S ε \S

0

|u|2 dxdt ≤ c meas(S ε \ S)(b−1)/b .

(10.2.48)

Inserting (10.2.47) and (10.2.48) into (10.2.46) we ﬁnally arrive at lim sup n→∞

T

n un ⊗ un : ∇(ηW) dxdt −

B\Sn

0

u ⊗ u : ∇(ηW) dxdt

B\S

≤ c meas(S ε \ S)(b−1)/b . It remains to note that the right hand side tends to 0 as ε → 0.

Proof of Theorem 10.2.5. We are now in a position to prove the desired relation 10.2.12, which will complete the proof of Theorem 10.2.5. To this end we have to substitute Sn , un and n into (10.2.1) and let n → ∞. By (10.2.1) we have W (Sn , un , n ) = W (0) (Sn , un , n ) + W (1) (Sn , un , n ), where

(S, u, ) = − (1) W (S, u, ) =

W

T

η(u)(x, T ) · W(x, T ) dx +

(0)

B\S

(10.2.49)

ηf · W dxdt, 0

B\S

η∞ (x, 0)U(x, 0) · W(x, 0) dx

B\S

T

+ 0

ηu · ∂t W + (u ⊗ u) − T : ∇(ηW) dxdt.

B\S

Letting n → ∞ and applying Lemmas 10.2.9 and 10.2.7 we obtain W (0) (S, u, ) = limn→∞ W (0) (Sn , un , n ). The proof is completed by applying Lemmas 10.2.11– 10.2.14 to conclude that W (1) (S, u, ) = limn→∞ W (1) (Sn , un , n ).

296

Chapter 10. Existence theory in nonsmooth domains

It is important to note that if W(·, T ) = f = 0 then W (0) (S, u, ) = 0. Hence to prove the continuity of W is suﬃces to show that W (1) (S, , u) = limn→∞ W (1) (Sn , un , n ). As mentioned above, this follows from Lemmas 10.2.11– 10.2.14 only. In these lemmas Condition 10.2.4(iv) is not required. Thus we get a stronger version of Theorem 10.2.5. Theorem 10.2.15. Let (un , n , Sn ) and (u, , S) satisfy Conditions 10.2.4(i)–(iii). Let W be a divergence free vector ﬁeld in C 1 ((B \ S) × (0, T )) with W(·, T ) = 0. Assume that f = 0. Then (u, ) is a renormalized solution to problem (9.2.4) and W (Sn , un , n ) → W (S, u, )

as n → ∞.

(10.2.50)

10.3 Applications of the stability theorem to optimization problems The main stability theorem 9.3.1 and the results on the continuity of the work functional form the theoretical base for studies of shape optimization in aerodynamics. In this section we present existence results for the problem of minimizing the work functional within a family of obstacles with a given volume. In the general case the problem can be formulated as follows: Assume that the state of the atmosphere and the plan of ﬂight are known, and deﬁned by given vector ﬁelds U, W, and the density distribution ∞ , which are ﬁxed data. We also assume that the volume of admissible obstacles is ﬁxed to be a constant V > 0. We want to ﬁnd an obstacle S from a speciﬁc admissible class such that the “shape” of the obstacle S minimizes the work of hydrodynamic forces acting on the moving body during the time interval (0, T ). In order to give an explicit formulation of this problem we have to deﬁne the set of admissible obstacles, some “technological” constraints, and a shape cost functional. The choice of admissible obstacles is restricted by physical and engineering constraints. Otherwise it should be as broad as possible. It is clear that a physically acceptable obstacle S should have a ﬁnite diameter. To bound it we ﬁx a compact set B ⊂ Rd and assume that all admissible obstacles S are contained in B . Next we ﬁx a hold-all domain B B . The set of admissible obstacles should be compact with respect to S-convergence in order to provide the continuity properties of ﬂow parameters as functions of the “shape” of S. One possible way, suﬃciently interesting from the mathematical point of view, is to assume that S ∈ O(k, T , B ) (see Deﬁnition 8.4.1). Moreover, the set of admissible obstacles should have the property that the mapping S → meas S is continuous with respect to S-convergence. This condition is fulﬁlled if we assume that the totality of characteristic functions χS of admissible obstacles is compact in the L1 (Rd ) topology. This obviously holds if the obstacles have uniformly bounded perimeters, or if their characteristic functions are bounded in some Sobolev space W s,r (Rd ) with s > 0.

10.3. Applications of the stability theorem to optimization problems

297

Deﬁnition 10.3.1. Let B B and V ∈ (0, meas B ] be given. We say that a set A ⊂ 2B is admissible if (i) Every element of A is a compact subset of B with meas S = V . (ii) A is a closed subset of O(k, T , B ) with respect to S-convergence. (iii) The family of characteristic functions χS of S ∈ A is compact in L1 (Rd ). The list of given data for the in/out ﬂow problem in the domain (B\S)×(0, T ) includes a vector ﬁeld U : B × (0, T ) → Rd , a nonnegative function ∞ : B × (0, T ) → R+ and a vector ﬁeld W : Rd × (0, T ) → Rd . For the ﬂow around a moving obstacle we have (see Section 2.5) ˙ ˙ W = U (t) U(t)x + a(t)) (10.3.1) where the mapping x → U(t)x + a(t) deﬁnes the evolution of a rigid body. In this case we also have U = −W on ∂B × (0, T ),

U = 0 in a neighborhood of B × (0, T ).

(10.3.2)

We can choose an arbitrary ﬁeld U satisfying conditions (10.3.2). Furthermore we assume that (10.3.3) U, W ∈ C ∞ (Ω × [0, T ]), ∞ ∈ C 1 (B × [0, T ]). By Theorem 10.1.1 for every S ∈ A and U, W, ∞ satisfying conditions (10.3.1)– (10.3.3), the in/out ﬂow problem (9.2.4) has a renormalized solution which meets all requirements of Deﬁnition 9.2.3 and satisﬁes estimates (10.1.1)–(10.1.2). Denote the totality of such solutions by R(S). The optimization problem can be formulated as follows: Problem 10.3.2. For ﬁxed U, W and ∞ satisfying conditions (10.3.1)–(10.3.3), ﬁnd an obstacle S ∈ A and (u, ) ∈ R(S) such that inf W (S, u, ) = inf W (S , u , ) . (10.3.4) S ∈A

(u , )∈R(S )

The following theorem gives an existence result for Problem 10.3.2:

Theorem 10.3.3. Let B B and A ⊂ 2B meet all requirements of Deﬁnition 10.3.1. Then Problem 10.3.2 has a solution S ∈ A. Proof. Since U, W and ∞ satisfy (10.3.1)–(10.3.3), for every S ⊂ B the given data satisfy all assumptions of the existence theorem 10.1.1 and Lemma 10.2.1. By Theorem 10.1.1, problem (9.2.4) has a renormalized solution (u, ) ∈ L∞ (0, T ; (B)) which satisﬁes estimates (10.1.1)–(10.1.2). MoreLγ (B \ S)) × L2 (0, T ; WS1,2 n over, by Lemma 10.2.1 the work functional W (S, u, ) is well deﬁned. Hence the set R(S) is not empty and we can deﬁne m = inf W (S, u, ) . inf S∈A

(u,)∈R(S)

298

Chapter 10. Existence theory in nonsmooth domains

Choose sequences Sn ∈ A and (un , n ) ∈ R(Sn ) such that W (Sn , un , n ) → m as n → ∞.

(10.3.5)

Since Sn ∈ A ⊂ O(k, T , B ), it follows from Theorem 8.4.2 that there is a subseS quence, still denoted by Sn , such that Sn → S. Moreover, by Deﬁnition 10.3.1, we have S ∈ A. By the existence theorem 10.1.1, (un , n ) satisﬁes the energy estimate (10.1.1) and hence we can assume, passing to a subsequence, that n weakly in L∞ (0, T ; Lγ (B)),

un u weakly in L2 (0, T ; W 1,2 (B)).

Recall that the functions n are extended to B × (0, T ) by setting n = ∞ (·, 0) on Sn × (0, T ), and un = 0 a.e. on Sn × (0, T ). Hence the triplets (Sn , un , n ) meet all requirements of the stability theorem 9.3.1. Consequently, (u, ) is a renormalized solution to problem (9.2.4) in (B \ S) × (0, T ) and so it belongs to R(S). It remains to note that by the continuity theorem 10.2.5, we have W (S, u, ) = limn→∞ W (Sn , un , n ) = m.

Chapter 11

Sensitivity analysis. Shape gradient of the drag functional 11.1 Introduction In this chapter we study domain dependence of stationary solutions to compressible Navier-Stokes equations. The main goal is to prove the existence and obtain a robust representation for the derivatives of solutions with respect to smooth deformations of the ﬂow domain. We apply these results to analysis of the drag minimization problem for the ﬂow around a moving body, and derive formulae for the variations of the drag with respect to the variations of the body. The stationary boundary value problem for compressible Navier-Stokes equations can be formulated as follows. Problem 11.1.1. Assume that a gas occupies the ﬂow domain Ω = B \ S, where B ⊂ R3 is a bounded hold-all domain, with an obstacle S in its interior. Find a couple (u, ) such that Δu + λ∇ div u = Re u · ∇u + div(u) = 0 in Ω, u = U on Σ, u = 0 = 0 on Σin .

Re ∇p() Ma2

in Ω,

on ∂S,

(11.1.1a) (11.1.1b) (11.1.1c) (11.1.1d)

where U is a given vector ﬁeld and Σ = ∂B. The inlet Σin and the outlet Σout are deﬁned by Σin = {x ∈ Σ : U · n < 0},

Σout = {x ∈ Σ : U · n > 0},

respectively. Here n stands for the outward normal to ∂(B \ S) = Σ ∪ ∂S. For

P. Plotnikov and J. Sokoáowski, Compressible Navier–Stokes Equations: Theory and Shape 299 Optimization, Monografie Matematyczne 73, DOI 10.1007/978-3-0348-0367-0_11, © Springer Basel 2012

300

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

simplicity of calculations we assume that the mass force equals zero, and 0 is a given positive constant. It is assumed that the body S moves through the gas with a constant speed U∞ , and that equations (11.1.1) are written in the moving frame attached to the body. In this framework the drag is the component parallel to U∞ of the power of the hydrodynamical force acting on the body. It equals Re−1 J(Ω), where we denote by J(Ω) the drag functional given by the boundary and volume integrals of the form

Re J(Ω) = −U∞ · ∇u + (∇u) + (λ − 1) div u I − pI · n dS Ma2 ∂S Re U∞ · ∇u + (∇u) + (λ − 1) div u I − pI ∇η + Re ηu∇u dx, =− Ma2 Ω

where η ≡ 1 in the vicinity of S, and η ≡ 0 in the vicinity of the boundary Σ = ∂B of the hold-all domain B. Notice that the latter integral is independent of the choice of η. For ﬁxed B, U, U∞ , and 0 the drag depends only on the shape of the body S. The minimization of the drag is an important applied problem. If Sε , ε ∈ R, is a one-parameter family of deformations of S with S0 = S, then the functional J(Ωε ), Ωε = B \ Sε becomes a function of ε. The study of this function for all possible deformations of S and the calculation of the ﬁrst order derivative dJ(Ωε )/dε is the main task of shape sensitivity analysis performed in this chapter.

11.1.1 Change of variables in Navier-Stokes equations In order to perform the shape sensitivity analysis in three space dimensions we choose a vector ﬁeld T ∈ C 2 (R3 )3 vanishing in a neighborhood of Σ, and introduce the mapping y = x + εT(x), which deﬁnes the perturbation Sε = y(S) of the obstacle S. For small ε, the mapping x → y takes diﬀeomorphically the ﬂow region Ω = B \ S onto Ωε = B \ Sε . In the perturbed domain Problem 11.1.1 reads Problem 11.1.2. Find a solution (¯ uε , ¯ε ), to the following boundary value problem posed in the variable domain Ωε = B \ Sε , for the shape parameter ε ∈ (−δ, δ) with δ > 0: ¯ ε = Re ¯ε u ¯ ε · ∇¯ uε + Δ¯ uε + λ∇ div u ¯ ε ) = 0 in B \ Sε , div(¯ ε u ¯ ε = 0 on ∂Sε , ¯ ε = U on Σ, u u ¯ε = 0 on Σin .

Re ∇p(¯ ε ) Ma2

in B \ Sε ,

(11.1.2a) (11.1.2b) (11.1.2c) (11.1.2d)

11.1. Introduction

301

The shape functional in the variable domain B \ Sε is written in the form ¯ ε ∇¯ η ¯ε u uε dx

J(Ωε ) = −Re U∞ ·

B\Sε

− U∞ ·

¯ε I − uε ) + (λ − 1) div u ∇¯ uε + (∇¯

B\Sε

Re p(¯ )I · ∇η dx. ε Ma2 (11.1.3)

Now, we perform a change of variables in equations (11.1.2) as well as in the drag functional (11.1.3) in order to reduce Problem 11.1.2 in the variable domain Ωε , depending on a small parameter ε, to a problem in the ﬁxed domain Ω, named the reference domain. Denote by M the Jacobi matrix of the mapping x → x + εT(x) and by g the determinant of M, i.e., M(x) = I + εD T(x),

g(x) = det(I + εD T(x)).

(11.1.4)

Further the notation N stands for the adjugate matrix N = (det M) M−1 with det N = g2 .

(11.1.5)

Deﬁnition 11.1.3. • In shape sensitivity analysis the material derivatives of solutions to stationary Navier-Stokes equations are obtained by the diﬀerentiation of the mapping ε → (¯ uε (x + εT(x)), ¯ε (x + εT(x))),

x ∈ Ω.

• The shape derivatives of solutions to stationary Navier-Stokes equations are deﬁned by derivatives of the mapping ε → (¯ uε (y), ¯ε (y)),

y ∈ Ωε .

• The shape gradient of the drag functional for stationary Navier-Stokes equations is given by the derivative of the function ε → J(Ωε ). Introduce the functions ¯ ε (x + εT(x)), uε (x) = N(x) u

ε (x) = ¯ε (x + εT(x)),

¯ ε. The function uε is called the Piola transformation of u

x ∈ Ω.

(11.1.6)

302

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

Lemma 11.1.4. Let (¯ uε (y), ¯ε (y)) be a solution to Problem 11.1.2. Then the couple (uε (x), ε (x)) deﬁned by (11.1.6) satisﬁes Re Δuε + ∇ λg−1 div uε − 2 p(ε ) Ma = A (uε ) + Re B(ε , uε , uε ) div(ε uε ) = 0 in Ω, uε = U on Σ, uε = 0 on ∂S, ε = 0

in Ω,

(11.1.7a) (11.1.7b) (11.1.7c) (11.1.7d)

on Σin .

Here, the linear operator A and the trilinear form B are deﬁned by A (u) = Δu − N− div g−1 NN ∇(N−1 u) , B(, u, w) = N− (u ∇(N−1 w)).

(11.1.8)

˜(x) = a(y(x)). For any φ ∈ C 1 (Ωε ) Proof. Let a ∈ C 1 (Ωε ) be a vector ﬁeld and a − 1 1 = φ(y(x)). It follows that the we have (∇y φ)(y(x)) = (M ∇x φ)(x), where φ(x) identities

˜ det M dx = (divy a)(y(x)) φ(x) divy aφ(y) dy = − a · ∇y φ dy Ω Ωε Ωε ˜ det M dx = ˜ dx ˜ · M− ∇x φ(x) ˜) φ(x) =− a divx ((det M) M−1 a Ω Ω ˜ dx divx (N˜ a) φ(x) = Ω

hold true for all φ ∈ C0∞ (Ωε ). This leads to (divy a)(y(x)) = g−1 divx N(x) a(y(x)) .

(11.1.9)

Setting a(y) = ¯ε (y)¯ uε (y) and noting that, by (11.1.6), N(x)a(y(x)) = ε (x)uε (x) we obtain 0 = divy (¯ ε (y)¯ uε (y)) ≡ g−1 div(ε (x)uε (x)), which implies the modiﬁed mass balance equation (11.1.7b). Repeating these arguments gives (divy (¯ uε ))(y(x)) = g−1 (x) divx uε (x). From this and the identity M− = g−1 N we obtain Re Re −1 −1 ¯ε − ∇y λ divy u p(¯ ε ) = g N ∇x λg divx uε − p(ε ) . Ma2 Ma2 (11.1.10)

11.1. Introduction

303

¯ ε . We have Next set a = ∇y u ˜ = (∇y u ¯ ε )(y(x)) = M− ∇x u ¯ ε (y(x)) = M− ∇x (N−1 uε (x)) a = g−1 N ∇x (N−1 uε (x)), which along with (11.1.9) yields ¯ ε )(y(x)) = g−1 divx g−1 NN ∇x (N−1 uε ) (x). divy (∇y u In view of the identity Δ = div ∇ we have Δ¯ uε (y(x)) = g−1 (x) div g−1 NN ∇(N−1 uε ) (x) = g−1 (x)N (Δuε − A (uε ))(x).

(11.1.11)

¯ ε ∇¯ uε )i of the vector u uε satisfy Next note that the components (¯ uε ∇¯ ¯ ε (y(x)) · ∇y u ¯ ε (y(x)) · M− (x)∇x u ¯ε,i (y(x)) uε )i (y(x)) = u ¯ε,i (y(x)) = u (¯ uε ∇¯ ¯ ε (y(x)) · g−1 (x)N (x)∇x u =u ¯ε,i (y(x)) = g−1 (x)N(x)¯ ¯ε,i (y(x)) = g−1 uε (x) · ∇x (N−1 (x)uε (x))i . uε (y(x)) · ∇x u This gives ¯ ε ∇¯ uε ) (y(x)) = g−1 (x)N (x)B(ε (x), uε (x), uε (x)). (¯ ε u

(11.1.12)

Substituting (11.1.10)–(11.1.12) into (11.1.2a) and multiplying both sides of the resulting equality by gN− , we obtain (11.1.7a). Corollary 11.1.5. The expression for the drag functional J(Ωε ) in the reference domain reads ε uε ∇(N−1 uε ) η dx J(Ωε ) = −Re U∞ · Ω $ −1 . g N ∇(N−1 uε ) + ∇(N−1 uε ) N − div uε I − qε I N ∇η dx − U∞ · Ω

(11.1.13)

where η is an arbitrary smooth function such that η ≡ 1 in a neighborhood of S and η ≡ 0 in a neighborhood of Σ, and the eﬀective viscous pressure qε is given by qε =

Re p(ε ) − λg−1 div uε , Ma2

(11.1.14)

By abuse of notation we will omit the subscript ε and write (u, ) instead of (uε , ε ). Thus we come to the following problem:

304

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

Problem 11.1.6. Find (u, ) such that Re −1 Δu + ∇ λg div u − p() = A (u) + Re B(, u, u) Ma2 div(u) = 0 in Ω, u = U on Σ, u = 0 on ∂S, = 0 on Σin . Here g =

√

in Ω,

(11.1.15a) (11.1.15b) (11.1.15c) (11.1.15d)

det N, and the operators A and B are deﬁned by (11.1.8).

Observe that equations (11.1.15) and the functional (11.1.13) depend only on the matrix N and do not depend on ε directly. In this framework, the fact that N is the adjugate matrix to the Jacobi matrix I + εD T is not so important. Hence our task is to show that Problem 11.1.6 is well-posed and its solution is diﬀerentiable with respect to N. We begin by analysing the well-posedness.

11.1.2 Extended problem. Perturbation theory There are numerous papers dealing with the homogeneous boundary value problem for stationary compressible Navier-Stokes equations with small data. We just recall that there are three diﬀerent approaches to this problem proposed in [11], [103], and [96], respectively. The basic results on the local existence and uniqueness of strong solutions are assembled in [101]. For an interesting overview see [104]. The in/out ﬂow boundary value problems were studied in [69]–[70], where the local existence and uniqueness results were obtained in the two-dimensional case under the assumption that the velocity u is close to a given constant vector. The case of small Reynolds and Mach numbers was treated in [108]. The diﬃculties that appear include: • The problem of total mass control. It is important to notice that, in contrast to the case of zero velocity boundary conditions when the total mass of the gas is prescribed, for in/out ﬂow the problem of controlling the total mass of the gas remains essentially unsolved. • The problem of singularities developed by solutions at the interface between Σin and Σ \ Σin . • The formation of a boundary layer near the inlet for small Mach numbers. The main idea of the approach developed in [11], [96] and [108] is to express div u in terms of , next to substitute this expression in the mass balance equations, thus reducing the original problem to a boundary value problem for a transport equation. Unfortunately, the resulting transport equation contains nonlocal operators. However this scheme gives the existence and uniqueness of strong solutions for Re, Ma2 1.

11.1. Introduction

305

In this book we use the simple algebraic scheme proposed in [103]. The basic idea is to introduce the eﬀective viscous pressure q and a large parameter σ0 , Re p() − λg−1 div u, Ma2 and write equations (11.1.15) in the form q :=

σ0 =

Re , λMa2

Δu − ∇q = A (u) + Re B(, u, u) in Ω, gq in Ω, div u = gσ0 p() − λ gq u · ∇ + gσ0 p() = in Ω, λ u = U on Σ, u = 0 on ∂S, = 0 on Σin ,

(11.1.16)

(11.1.17a) (11.1.17b) (11.1.17c) (11.1.17d) (11.1.17e)

System (11.1.17) consists of perturbed Stokes equations (11.1.17a)–(11.1.17b) for (u, q) and a perturbed transport equation (11.1.17c) for . It is easy to see that equations (11.1.17b) and (11.1.17c) are equivalent to (11.1.16) and the mass balance equation div(u) = 0. Therefore, equations (11.1.17) are equivalent (11.1.15). It was shown in [103] and [107] that problem (11.1.17) has a strong solution if the parameters satisfy the conditions λ 1,

Re 1,

σ0 1,

which are more restrictive than the standard conditions Re 1, Ma2 1 adopted in [11], [96], and [108]. Perturbations. We will look for a local solution to problem (11.1.15) which is close to an approximate solution (u , q , ). In order to deﬁne such a solution notice that for small Mach and Reynolds numbers equations (11.1.17a)–(11.1.17b) are close to Stokes equations and the density is close to 0 . Thus we take = 0

(11.1.18)

and choose (u , q ) as the solution to the following boundary problem for the Stokes equation: Δu − ∇q = 0, div u = 0 in Ω, u = U on Σ, u = 0 on ∂S, Πq = q ,

(11.1.19)

1 q dx. (11.1.20) meas Ω Ω The solution to problem (11.1.15) is decomposed into the approximate solution (u , q , ) and small perturbations (v, π, ϕ, m), where

Πq := q −

u = u + v,

= + ϕ,

q = q + λσ0 p( ) + π + λm,

(11.1.21)

with some unknown functions ϑ = (v, π, ϕ) and an unknown constant m.

306

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

Remark 11.1.7. We point out that our decomposition (11.1.21) holds exclusively in the reference domain Ω = B \ S. The unknown functions ϑ and the unknown constant m contain the necessary and suﬃcient information on the shape dependence of solutions to the equations (11.1.15) upon the shape parameter ε → 0. This information is used in order to determine the material derivatives of the solutions to the model (11.1.2) with respect to the shape perturbation. However, the Stokes boundary value problem (11.1.19) deﬁned in the reference domain Ω is meaningless for the model (11.1.2) deﬁned in the variable domain Ωε . Substituting decomposition (11.1.21) into (11.1.17) we obtain the following boundary value problem for ϑ = (v, π, ϕ): Δv − ∇π = A (u) + Re B(, u, u) in Ω, σ ϕ − Ψ[ϑ] − m in Ω, div v = g u · ∇ϕ + σϕ = Ψ1 [ϑ] + mg in Ω, v = 0 on ∂Ω, ϕ = 0 on Σin , Ππ = π, where σ q + π − Ψ[ϑ] = H(ϕ), λ p ( ) σ = σ0 p ( ) ,

(11.1.22a) (11.1.22b) (11.1.22c) (11.1.22d)

σ 2 Ψ1 [ϑ] = g Ψ[ϑ] − ϕ + σϕ(1 − g), (11.1.22e)

H(ϕ) = p( + ϕ) − p( ) − p ( )ϕ,

(11.1.22f)

the vector ﬁeld u and the function are given by (11.1.21). Determination of m. Extended system. Finally, we specify the constant m. In our framework, in contrast to the case of homogeneous boundary value problem, the answer to the question how m can be determined is not trivial. Note that since div u has null mean value, the right hand side of equation (11.1.22c) must satisfy the compatibility condition σ g dx = g ϕ − Ψ[ϑ] dx, m Ω Ω which formally determines m. However, this choice of m leads to essential mathematical diﬃculties. To make this issue clear note that in the simplest case g = 1 2 −1 we have m = −1 ), and the principal part of equations σ(I − Π)ϕ + O(|ϑ| ) + O(λ (11.1.22) becomes ⎞⎛ ⎛ ⎞ ⎛ ⎞ ⎛ Δv − ∇π ⎞ Δ −∇ 0 v 0 σ σ ⎟ ⎟⎝ ⎜ div ⎜ 0 − π ⎠ + ⎝ m ⎠ ∼ ⎝ div v − Πϕ ⎠ . ⎠ ⎝ ϕ −m u∇ϕ + σΠϕ 0 0 u∇ + σ

11.1. Introduction

307

Hence, the question of invertibility of the principal part of equations (11.1.22) can be reduced to the question of solvability of the boundary value problem for the nonlocal transport equation u · ∇ϕ + σΠϕ = f

in Ω,

ϕ = 0 on Σin .

(11.1.23)

The following simple example shows that there is a drastic diﬀerence between properties of solutions to the nonlocal problem (11.1.23) and solutions to the local boundary value problem u · ∇ϕ + σϕ = f

in Ω,

ϕ = 0 on Σin .

Example 11.1.8. Let us consider the one-dimensional boundary value problems ∂x1 ϕ¯ + σ ϕ¯ = 1, and

1

∂x1 ϕ + σ ϕ −

0 ≤ x1 ≤ 1,

ϕ(x1 ) dx1 = 1,

ϕ(0) ¯ = 0,

0 ≤ x1 ≤ 1,

ϕ(0) = 0.

0

Simple calculations give ϕ(x ¯ 1) =

1 (1 − e−σx1 ), σ

ϕ(x1 ) =

1 − e−σx1 . 1 − e−σ

Hence ϕ¯ = O(σ −1 ) but ϕ = O(1) as σ → ∞. Thus for the nonlocal transport equation, the advantage of having a large parameter σ is lost. We point out that this diﬃculty disappears for the homogeneous boundary value problem, i.e., for homogeneous Dirichlet conditions for the velocity, since in this case the mean value of over Ω is prescribed and ϕ = Πϕ. The diﬃculty of determining m is only present in the in/out ﬂow problem. In order to cope with it we introduce an extra unknown function ζ satisfying the adjoint transport equation and add an extra equation for m. By doing so we split the problem into the problem of determining ϑ = (v, π, ϕ) and the problem of ﬁnding m. The augmented problem has the following form: Problem 11.1.9. Given (u , q , ), ﬁnd (v, π, ϕ) and a constant m, depending on ε, such that Δv − ∇π = A (u) + ReB(, u, u) in Ω, σ ϕ − Ψ[ϑ] − m in Ω, div v = g u · ∇ϕ + σϕ = Ψ1 [ϑ] + mg in Ω, v = 0 on ∂Ω, ϕ = 0 on Σin , Ππ = π.

(11.1.24a)

The constant m is determined from the following relations: −1 −1 m = κ (−1 Ψ [ϑ] ζ −gΨ[ϑ]) dx, κ = g(1−ζ − ζϕ) dx , (11.1.24b) 1 Ω

Ω

308

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

where the auxiliary function ζ is a solution to the adjoint boundary value problem − div(uζ) + σζ = σg

in Ω,

ζ=0

on Σout .

(11.1.24c)

Here we denote q + π σ H(ϕ), − λ p ( ) H(ϕ) = p( + ϕ) − p( ) − p ( )ϕ, σ Ψ1 [ϑ] = g Ψ[ϑ] − ϕ2 + σϕ(1 − g), − A (u) = Δu − N div g−1 NN ∇(N−1 u) , Ψ[ϑ] =

B(, u, u) = N− (u ∇(N−1 u)),

(11.1.24d)

(11.1.24e)

with the velocity ﬁeld u := u + v and the density := + ϕ. Equations (11.1.24) deﬁne a complete system of diﬀerential equations and boundary conditions for the functions ϑ, ζ and the constant m. It is easy to check that every solution to (11.1.24) also satisﬁes equations (11.1.22). The important feature of equations (11.1.24) is that the right hand side in the expression (11.1.24b) for m is small for large σ, λ and suﬃciently small ϕ, provided that κ is bounded from above. We employ this fact in the next section devoted to the existence theory.

11.2 Existence of solutions 11.2.1 Function spaces. Results Function spaces. To establish existence results for problem (11.1.22), we begin with the deﬁnition of an admissible class of solutions to this problem. Notice that we are looking for strong solutions, which means that in equations (11.1.22) the unknown functions and their derivatives are integrable, i.e., these equations are satisﬁed a.e. in Ω. To this end it suﬃces to assume that v ∈ W 2,2 (Ω) and (π, ϕ, ζ) ∈ W 1,2 (Ω). Next, we are looking for solutions such that (π, ϕ, ζ) are continuous and v is continuously diﬀerentiable in Ω. In order to satisfy this demand it suﬃces to take v ∈ W s+1,r (Ω) and (π, ϕ, ζ) ∈ W s,r (Ω) with sr > 3. To address both issues we introduce the Banach spaces X s,r = W s,r (Ω) ∩ W 1,2 (Ω), Y s,r = W s+1,r (Ω) ∩ W 2,2 (Ω),

0 < s < 1 < r < ∞,

(11.2.1)

equipped with the norms uX s,r = uW s,r (Ω) + uW 1,2 (Ω) ,

uY s,r = uW s+1,r (Ω) + uW 2,2 (Ω) .

11.2. Existence of solutions

309

Throughout of the rest of this chapter we intensively exploit the fact that the spaces X s,r and Y s,r are reﬂexive. The proof of this fact is nontrivial and is given below. It is well known (see [2, Thm. 3.5]) that for any bounded domain Ω with C 1 boundary, any integer m ≥ 0 and 1 < r < ∞, the Sobolev space W m,r (Ω) is reﬂexive. The space W01,r (Ω) is obviously reﬂexive as a closed subspace of a reﬂexive Banach space (see [2, Thm. 1.21]). Lemma 11.2.1. Let Ω be a bounded domain with C 1 boundary and 0 < s < 1 < r < ∞. Then the spaces W s,r (Ω) and W s+1,r (Ω) are reﬂexive. Proof. First observe that W 1,r (Ω) and W01,r (Ω) are dense in Lr (Ω) since C0∞ (Ω) is dense in Lr (Ω). By Lemma 1.1.12, recalling that W s,r (Ω) = [Lr (Ω), W 1,r (Ω)]s,r we obtain (W s,r (Ω)) = [Lr (Ω) , (W 1,r (Ω)) ]s,r . (11.2.2) Next notice that W01,r (Ω) ⊂ W 1,r (Ω) ⊂ Lr (Ω). Hence Lr (Ω) ⊂ (W 1,r (Ω)) ⊂ (W01,r (Ω)) . In other words Lr (Ω) and (W 1,r (Ω)) are linear subspaces of (W01,r (Ω)) . Recall (see Section 1.5) that (W01,r (Ω)) = W −1,r (Ω) is the completion of Lr (Ω) = 1,r r r 1,r L (Ω) in the (W0 (Ω)) -norm. Hence the intersection L (Ω) ∩ (W (Ω)) = Lr (Ω) is dense in the ambient space (W01,r (Ω)) . Therefore we may apply Lemma 1.1.12 to obtain ((W s,r (Ω)) ) ≡ ([Lr (Ω) , (W 1,r (Ω)) ]s,r ) = [(Lr (Ω) ) , ((W 1,r (Ω)) ) ]s,r = [Lr (Ω), W 1,r (Ω)]s,r = W s,r (Ω), since (Lr (Ω) ) = Lr (Ω) and ((W 1,r (Ω)) ) = W 1,r (Ω). Thus W s,r (Ω) is reﬂexive. Since W s+1,r (Ω) is the interpolation space [W 1,r (Ω), W 2,r (Ω)]s,r and the spaces W 1,r (Ω), W 2,r (Ω) are reﬂexive, we can apply the same arguments to conclude that W s+1,r (Ω) is reﬂexive. Lemma 11.2.2. For any functional f ∈ (X s,r ) there are continuous functionals g ∈ (W s,r (Ω)) and h ∈ (W 1,2 (Ω)) such that f, ϕ = g, ϕ + h, ϕ

for all ϕ ∈ X s,r .

Proof. Applying Lemma 1.1.11 with A0 = Lr (Ω) and A1 = W 1,r (Ω) we ﬁnd that W 1,r (Ω) is dense in the interpolation space W s,r (Ω). In particular, C ∞ (Ω) is dense in W s,r (Ω), since it is dense in W 1,r (Ω). Next, C ∞ (Ω) is dense in W 1,2 (Ω). As C ∞ (Ω) ⊂ W s,r (Ω) ∩ W 1,2 (Ω), we conclude that the latter intersection is dense in both W s,r (Ω) and W 1,2 (Ω). Moreover both these spaces contain W 1,m (Ω) for m > 2 + r, and hence their duals are embedded in (W 1,m (Ω)) . By Theorem 1.1.1,

310

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

for any functional f ∈ (W s,r (Ω) ∩ W 1,2 (Ω)) there are continuous functionals g ∈ (W s,r (Ω)) and h ∈ (W 1,2 (Ω)) such that f, ϕ = g, ϕ + h, ϕ for all ϕ ∈ W s,r (Ω) ∩ W 1,2 (Ω). It remains to recall that X s,r = W s,r (Ω) ∩ W 1,2 (Ω).

Lemma 11.2.3. Let Ω be a bounded domain with C 1 boundary and 0 < s < 1 < r < ∞. Then the spaces X s,r and Y s,r are reﬂexive. Proof. We give the proof only for X s,r . The same proof works for Y s,r . In view of Theorem 1.1.3, it suﬃces to show that X s,r (Ω) is weakly complete. Choose a sequence ϕn ∈ X s,r , n ≥ 1, such that f, ϕn converges in R for all f ∈ (X s,r ) . We need to show that ϕn converges weakly to some ϕ ∈ X s,r . Since X s,r is compactly embedded in C(Ω), ϕn converges in C(Ω), there is ϕ ∈ C(Ω) such that ϕn converges uniformly in Ω to ϕ. Next, since {ϕn } is bounded in W s,r (Ω) and W 1,2 (Ω), there is a subsequence {ϕm } ⊂ {ϕn } such that ϕm ϕ∗ weakly in W s,r (Ω) and ϕm ϕ∗∗ weakly in W 1,2 (Ω) as m → ∞. Obviously ϕ∗ = ϕ∗∗ = ϕ and ϕ ∈ X s,r . Hence ϕm ϕ weakly in W s,r (Ω) and W 1,2 (Ω). Repeating this argument we conclude that every subsequence of {ϕn } contains a subsequence which converges weakly to ϕ in W s,r (Ω) and W 1,2 (Ω). Hence the whole sequence {ϕn } converges weakly to ϕ in both W s,r (Ω) and W 1,2 (Ω). It remains to show that ϕn ϕ weakly in X s,r . To this end, choose f ∈ (X s,r ) . In view of Lemma 11.2.2, there are g ∈ (W s,r (Ω)) and h ∈ (W 1,2 (Ω)) such that f = g + h. Hence f, ϕn = g, ϕn + h, ϕn → g, ϕ + h, ϕ = f, ϕ. Therefore, ϕn converges weakly to ϕ in X s,r .

While W s,r (Ω) and W 1,2 (Ω) are separable, we cannot conclude from this that X s,r is separable. However, the dual spaces (X s,r ) and (Y s,r ) are separable: Lemma 11.2.4. Let Ω ⊂ Rd be a bounded domain with C 1 boundary and 0 < s < 1 < r < ∞. Then (X s,r ) and (Y s,r ) are separable. Proof. We prove the separability of (X s,r ) . The same proof works for (Y s,r ) . Notice that (W s,r (Ω)) is the interpolation space [Lr (Ω) , (W 1,r (Ω)) ]s,r . The space Lr (Ω) = Lr (Ω) is separable, so contains a countable dense set. By Lemma 1.1.11, Lr (Ω) is dense in the interpolation space (W s,r (Ω)) . Hence (W s,r (Ω)) contains a countable dense subset and consequently is separable. Also, by [2, Thm. 3.8], the space (W 1,2 (Ω)) is separable. Next, by Lemma 11.2.2 we have (X s,r ) = (W s,r (Ω)) + (W 1,2 (Ω)) . Obviously (X s,r ) is separable as a sum of separable spaces.

11.2. Existence of solutions

311

Existence theory. Deﬁne the Banach space E = (Y s,r )3 × (X s,r )2 ,

(11.2.3)

and denote by Bτ ⊂ E the closed ball of radius τ centered at 0. Next, note that for sr > 3, elements of the ball Bτ satisfy the inequality vC 1 (Ω) + πC(Ω) + ϕC(Ω) ≤ ce (r, s, Ω)ϑE ≤ ce τ,

(11.2.4)

where the norm in E is deﬁned by ϑE = vY s,r + πX s,r + ϕX s,r . Further, c denotes generic constants, which are diﬀerent in diﬀerent places and depend only on Ω, U, σ and r, s. We assume that the ﬂow domain and the given data satisfy the following conditions. Condition 11.2.5. • ∂Ω is a closed surface of class C 3 and the set Γ = cl Σin ∩ Σ \ Σin is a closed C 3 one-dimensional manifold such that Σ = Σin ∪ Γ ∪ Σout . • The vector ﬁeld U ∈ C 3 (∂Ω) satisﬁes U · n dΣ = 0. ∂Ω

Moreover, we can assume that it is extended to R3 in such a way that the extension U ∈ C 3 (R3 ) vanishes in a neighborhood of S. • There is a positive constant c such that U · ∇(U · n) > c > 0 on Γ. Since the vector ﬁeld U is tangent to ∂Ω on Γ, the left hand side is well deﬁned. The following existence theorem is the main result of this section. Theorem 11.2.6. Assume that the surface Σ and the given vector ﬁeld U satisfy Condition 11.2.5. Furthermore, let numbers r and s satisfy 0 < s < 1,

1 < r < ∞,

2s − 3r−1 < 1,

sr > 3.

(11.2.5)

Then there exists σ ∗ > 1, depending only on U, Ω and s, r, with the following property. For every σ > σ ∗ there are positive τ ∗ and c, depending only on U, Ω, r, s, and σ, such that whenever τ ∈ (0, τ ∗ ],

λ−1 , Re ∈ (0, τ 2 ],

N − IC 2 (Ω) ≤ τ 2 ,

(11.2.6)

Problem 11.1.9 has a solution ϑ ∈ Bτ , ζ ∈ X s,r , m ∈ R. The auxiliary function ζ and the constants κ, m admit the estimates ζX s,r + |κ| ≤ c,

|m| ≤ cτ < 1.

(11.2.7)

312

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

Let ΘN ⊂ Bτ × W s,r (Ω) × R be the set of all solutions (ϑ, ζ, m) = (v, π, ϕ, ζ, m) corresponding to a matrix-valued function N. Then for every N in the ball B(τ 2 ) = {N : I − NC 2 (Ω) ≤ τ 2 } there is a nonempty subset ΘN ⊂ ΘN such that

s+1,r (Ω) × W s,r (Ω)3 × R. N∈B(τ 2 ) ΘN is relatively compact in W The remaining part of this section is devoted to the proof of Theorem 11.2.6. The proof falls into ﬁve steps.

11.2.2 Step 1. Estimates of composite functions We begin by proving the following lemmas which furnish regularity properties of composite functions. Lemma 11.2.7. Let Ω ⊂ R3 be a bounded domain with C 1 boundary and 0 < s < 1 < r < ∞. Then there is c, depending on Ω, s, r, such that uvW s,r (Ω) ≤ cuL∞ (Ω) vW s,r (Ω) + cvL∞ (Ω) uW s,r (Ω) , uvX s,r ≤ cuL∞ (Ω) vX s,r + cvL∞ (Ω) uX s,r .

(11.2.8) (11.2.9)

If, in addition, sr > 3, then uvX s,r ≤ cuX s,r vX s,r .

(11.2.10)

Proof. Obviously uvLr (Ω) ≤ uL∞ (Ω) vLr (Ω) + vL∞ (Ω) uLr (Ω) .

(11.2.11)

Next we have r |uv|s,r,Ω ≡ |u(x)v(x) − u(y)v(y)|r |x − y|−3−sr dxdy Ω Ω |u(x)|r |v(x) − v(y)|r + |v(y)|r |u(x) − u(y)|r |x − y|−3−sr dxdy ≤c Ω Ω |v(x) − v(y)|r |x − y|−3−sr dxdy ≤ curL∞ (Ω) Ω Ω r + cvL∞ (Ω) |u(x) − u(y)|r |x − y|−3−sr dxdy Ω

Ω

= curL∞ (Ω) |v|rs,r,Ω + cvrL∞ (Ω) |u|rs,r,Ω .

(11.2.12)

Combining (11.2.2) and (11.2.11), and recalling the equality uW s,r (Ω) = uLr (Ω) + |u|s,r,Ω , we obtain uvW s,r (Ω) ≤ cuL∞ (Ω) vW s,r (Ω) + cvL∞ (Ω) uW s,r (Ω) , which is (11.2.8). The identity ∇(uv) = u∇v + v∇u implies ∇(uv)L2 (Ω) ≤ uL∞ (Ω) ∇vL2 (Ω) + vL∞ (Ω) ∇uL2 (Ω) ,

(11.2.13)

11.2. Existence of solutions

313

and uvW 1,2 (Ω) ≤ cuL∞ (Ω) vW 1,2 (Ω) + cvL∞ (Ω) uW 1,2 (Ω) , which along with (11.2.13) leads to (11.2.9). Since for sr > 3 the space W s,r (Ω) is continuously embedded in L∞ (Ω), we have uL∞ (Ω) ≤ cuW s,r (Ω) ≤ cuX s,r for all u ∈ X s,r . Combining this with (11.2.9) we obtain (11.2.10). Let us consider functions u, v : Ω → BK , where BK = {x : |x| ≤ K} ⊂ R3 is the ball of radius K centered at 0. Lemma 11.2.8. Assume that u, v ∈ X s,r , s ∈ (0, 1), sr > 3, and f ∈ C 2 (Ω × BK ). Then f(·, u)X s,r ≤ c(r, s)fC 1 (Ω×BK ) (1 + uX s,r ), f(·, v) − f(·, u)X s,r ≤ c(r, s)fC 2 (Ω×BK ) (1 + uX s,r + vX s,r )u − vX s,r .

(11.2.14) (11.2.15)

Proof. We begin by proving (11.2.14). Notice that for every function f we have f X s,r ≤ f L∞ (Ω) + |f |s,r,Ω + ∇f L2 (Ω) . Since the function |f(x, u(x))| is bounded by fC(Ω×BK ) , it suﬃces to estimate |f|s,r,Ω and ∇fL2 (Ω) . We have |f(x, u(x)) − f(y, u(y))|r ≤ c(r)frC 1 (Ω×BK ) (|x − y|r + |u(x) − u(y)|r ), which along with −3 − rs + r > −3 gives |f(·, u)|rs,r,Ω = |f(x, u(x)) − f(y, u(y))|r |x − y|−3−rs dxdy Ω

Ω

≤ cfrC 1 (Ω×BK ) × |x − y|−3−rs+r dxdy + |u(x) − u(y)|r |x − y|−3−rs dxdy ≤

Ω Ω r cfC 1 (Ω×BK ) (1

+

|u|rs,r,Ω )

≤

Ω Ω r c|fC 1 (Ω×BK ) (1

+ urX s,r ).

(11.2.16)

Next we have |∇f(x, u)| ≤ cfC 1 (Ω×BK ) (1 + |∇u|), which yields ∇f(·, u)L2 (Ω) ≤ cfC 1 (Ω×BK ) (1 + uW 1,2 (Ω) ) ≤ cfC 1 (Ω×BK ) (1 + uX s,r ). Combining this with (11.2.16) we obtain (11.2.14). Let us turn to the proof of (11.2.15). We have 1 ∇u f(x, tu + (1 − t)v) dt. f(x, u) − f(x, v) = (u − v) · 0

It follows from (11.2.14) that ∇u f(·, tu+(1−t)v)X s,r ≤ cfC 2 (Ω×BK ) (1+tuX s,r +(1−t)vX s,r ). (11.2.17)

314

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

Thus we get

1

∇u f(·, tu + (1 − t)v) dt

0

≤ cf

X s,r

C 2 (Ω×B

K)

≤

1

∇u f(·, tu + (1 − t)v)X s,r dt

0

1

(1 + tuX s,r + (1 − t)vX s,r ) dt

0

≤ cfC 2 (Ω×BK ) (1 + uX s,r + vX s,r ),

(11.2.18)

From this and estimate (11.2.10) we ﬁnally obtain f(·, v) − f(·, u)X s,r ≤ cv − uX s,r

1

∇u f(·, tu + (1 − t)v) dt

X s,r

0

≤ c(r, s)fC 2 (Ω×BK ) v − uX s,r (1 + uX s,r + vX s,r ).

11.2.3 Step 2. Stokes equations In this section we establish the existence and uniqueness results in the Banach space Y s,r × X s,r for the boundary value problem for Stokes equations Δv − ∇π = F, v = 0 on ∂Ω,

div v = ΠG in Ω, Ππ = π.

(11.2.19)

Recall notation (11.2.1) for the Banach spaces X s,r and Y s,r and introduce the Banach space Z s,r = W s−1,r (Ω) ∩ L2 (Ω),

0 < s < 1 < r < ∞.

(11.2.20)

While the Banach spaces X s,r and Y s,r consist of functions of real variables and their structure is clear, the structure of the space Z s,r is more complicated and extra explanations are in order. By Lemma 1.5.1 the space W s−1,r (Ω) is the dual space to W01−s,r (Ω), where r = r/(r − 1). Moreover, W s−1,r (Ω) is the completion of Lr (Ω) in the norm 1 f W s−1,r (Ω) = sup f u dx. u 1−s,r 1−s,r Ω W (Ω) 0 =u∈W (Ω) 0

0

We can identify f ∈ Lr (Ω) with the continuous functional f u dx ∈ R Lr (Ω) u → Ω

and, by doing so, identify Lr (Ω) with the dual space Lr (Ω) . Thus W s−1,r (Ω) can be regarded as the completion of Lr (Ω) in the corresponding norm. Hence Lr (Ω) is dense in W s−1,r (Ω). We identify L2 (Ω) and L2 (Ω) since L2 (Ω) is a Hilbert

11.2. Existence of solutions

315

space. Thus we get Z s,r = (W01−s,r (Ω)) ∩ L2 (Ω) . Next, for t > max{r , 2} we have W01,t (Ω) ⊂ W01−s,r (Ω) and W01,t (Ω) ⊂ L2 (Ω). It follows that

W s−1,r (Ω) ⊂ W −1,t (Ω) = (W01,t (Ω)) ,

t = t/(t − 1),

and L2 (Ω) ⊂ W −1,t (Ω). Hence the intersection Z s,r = (W01−s,r (Ω)) ∩ L2 (Ω) makes sense as a linear subspace of the Banach space W −1,t (Ω). Equipped with the standard norm of the intersection (see [49, Ch. 1]), f Z s,r = f W s−1,r (Ω) + f L2 (Ω) , Z s,r becomes a Banach space. Notice that it is not empty since both W s−1,r (Ω) and L2 (Ω) contain C0∞ (Ω). Now we can formulate the main result of this section. Lemma 11.2.9. Let Ω ⊂ R3 be a bounded domain with ∂Ω ∈ C 2 and (F, G) ∈ Z s,r × X s,r (0 < s < 1 < r < ∞). Then the boundary value problem (11.2.19) has a unique solution (v, π) ∈ Y s+1,r (Ω) × X s,r (Ω) such that vY s,r + πX s,r ≤ c(Ω, r, s)(F Z s,r + GX s,r ). Proof. This is an obvious consequence of Lemma 1.7.9.

(11.2.21)

11.2.4 Step 3. Transport equations In this section we collect the auxiliary results concerning existence and uniqueness of solutions to the following boundary value problems for transport equations: L ϕ := u∇ϕ + σϕ = f

in Ω,

∗

∗

∗

∗

L ϕ := − div(ϕ u) + σϕ = f

ϕ = 0 on Σin , in Ω,

∗

ϕ = 0 on Σout .

(11.2.22) (11.2.23)

∗

Bounded functions ϕ, ϕ are called weak solutions to problems (11.2.22), (11.2.23), respectively, if the integral identities ∗ ∗ ∗ (ϕL φ − f φ ) dx = 0, (ϕ∗ L φ − f φ) dx = 0, (11.2.24) Ω

Ω ∗

hold true for all test functions φ , φ ∈ C(Ω) ∩ W 1,1 (Ω), respectively, such that φ∗ = 0 on Σout and φ = 0 on Σin . Our considerations are based on the following two lemmas, which follow from the general theory of transport equations developed in Chapter 12. Lemma 11.2.10. Suppose that a bounded domain Ω = B \ S ⊂ Rd and a vector ﬁeld U : ∂Ω → Rd satisfy Condition 11.2.5. Furthermore assume that a vector ﬁeld u ∈ C 1 (Ω) satisﬁes the boundary condition u=U ∞

on ∂Ω.

Then for every σ > 0 and f ∈ L (Ω) problem (11.2.22) has a unique weak solution ϕ ∈ L∞ (Ω) satisfying ϕL∞ (Ω) ≤ σ −1 f L∞ (Ω) .

316

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

Proof. It follows from Condition 11.2.5 that Γ = cl Σin ∩ (Σ \ Σin ) is a onedimensional manifold of class C 3 . Hence u and Ω meets all requirements of Theorem 12.1.2 with c = σ and r = ∞, and the lemma follows directly from that theorem. Lemma 11.2.11. Assume that ∂Ω and U satisfy Condition 11.2.5, and the vector ﬁeld u belongs to C 1 (Ω) and satisﬁes u=U

on ∂Ω.

(11.2.25)

Furthermore, let s and r be constants satisfying 0 < s ≤ 1,

1 < r < ∞,

2s − 3r−1 < 1.

(11.2.26)

Then there are positive constants σ ∗ > 1 and C, which depend on ∂Ω, U, s, r, uC 1 (Ω) and are independent of σ, such that for any σ > σ ∗ and f ∈ W s,r (Ω) ∩ L∞ (Ω), problem (11.2.22) has a unique solution ϕ ∈ W s,r (Ω) ∩ L∞ (Ω) satisfying ϕW s,r (Ω) ≤ Cσ −1 f W s,r (Ω) + Cσ −1+α f L∞ (Ω) ϕW s,r (Ω) ≤ Cσ

−1

f W s,r (Ω) + Cσ

−1+α

for sr = 1, 2, 1/r

(1 + log σ)

f L∞ (Ω)

for sr = 1, 2. (11.2.27)

Here, the accretivity defect α is deﬁned by α(s, r) = max{0, s − r−1 , 2s − 3r−1 }.

(11.2.28)

If, in addition, sr > 3 then for any σ > σ ∗ and f ∈ W s,r (Ω), problem (11.2.22) has a unique solution ϕ ∈ W s,r (Ω) ∩ L∞ (Ω) satisfying ϕW s,r (Ω) ≤ Cσ −1+α f W s,r (Ω) ϕW s,r (Ω) ≤ Cσ

−1

Cσ

−1+α

for sr = 1, 2,

(1 + log σ)1/r f W s,r (Ω)

for sr = 1, 2.

(11.2.29)

Proof. It follows from Condition 11.2.5 that ∂Ω and U also satisfy Condition 12.2.2 and hence meet all requirements of Theorem 12.2.3. Therefore, the lemma is a straightforward consequence of that theorem. Notice only that for sr > 3 the space W s,r (Ω) is continuously embedded is C(Ω) and hence estimate (11.2.29) follows from (11.2.27). The following proposition on solvability of problem (11.2.22) is the main result of this section. Proposition 11.2.12. Assume that Σ and U satisfy Condition 11.2.5, the exponents s, r satisfy the inequalities 0 < s ≤ 1,

1 < r < ∞,

2s − 3r−1 < 1,

sr > 3,

(11.2.30)

11.2. Existence of solutions

317

and the vector ﬁeld u ∈ Y s,r (Ω) satisﬁes the boundary condition u=U

on Σ,

u=0

on ∂S.

(11.2.31)

Then there are constants σ0 > 1 and C > 0, depending only on Ω, U, s, r, and uY s,r , such that: For any σ > σ0 and f ∈ X s,r , problem (11.2.22) has a unique solution ϕ ∈ X s,r satisfying ϕX s,r ≤ Cf X s,r ,

ϕL∞ (Ω) ≤ σ −1 f L∞ (Ω) ,

(11.2.32)

and problem (11.2.23) has a unique solution ϕ∗ ∈ X s,r satisfying ϕ∗ X s,r ≤ Cf X s,r ,

ϕ∗ L∞ (Ω) ≤ (σ − div uC(Ω) )−1 f L∞ (Ω) . (11.2.33)

Proof. We begin by proving (11.2.32). Since sr > 3, the space Y s,r = W s+1,r (Ω) ∩ W 2,2 (Ω) is continuously embedded in C 1 (Ω), which yields uC 1 (Ω) ≤ c(Ω, s, r)uY s,r . Hence Ω, s, r, and u meet all requirements of Lemma 11.2.11. It follows that there are σ ∗ > 1 and c0 > 0, depending only on Ω, U, s, r, and uY s,r , such that for all σ > σ ∗ and f ∈ W s,r (Ω), problem (11.2.22) has a unique solution ϕ ∈ W s,r (Ω) satisfying ϕW s,r (Ω) ≤ c0 σ −1 f W s,r (Ω) + c0 σ −1+α f L∞ (Ω) ,

(11.2.34)

where α ∈ (0, 1) is deﬁned by (11.2.28). On the other hand, Ω, U, u, and the exponents s = 1, r = 2 also satisfy all conditions of Lemma 11.2.11. Therefore, there are σ ∗ > 1 and c∗ > 0, depending only on Ω, U, and uY s,r , such that for all σ > σ ∗ and f ∈ W 1,2 (Ω), problem (11.2.22) has a unique solution ϕ ∈ W 1,2 (Ω) satisfying ϕW 1,2 (Ω) ≤ c∗ σ −1 f W 1,2 (Ω) + c∗ σ −1/2 (1 + log σ)1/2 f L∞ (Ω) .

(11.2.35)

Next ﬁx σ0 > max{σ ∗ , σ ∗ } and set K(σ) = min{σ −1+α , σ −1/2 (1 + log σ)1/2 } > σ −1 . Recall that X s,r = W s,r (Ω) ∩ W 1,2 (Ω), which along with rs > 3 implies f L∞ (Ω) ≤ c(Ω, s, r)f X s,r . Combining (11.2.34) and (11.2.35), we conclude that for all σ > σ0 and f ∈ X s,r , problem (11.2.22) has a unique solution ϕ ∈ X s,r such that ϕX s,r ≤ 2(c∗ + c0 )K(σ) f X s,r .

(11.2.36)

Since K decreases in σ, we may take C = 2(c0 + c∗ )K(σ0 ) to obtain the ﬁrst estimate in (11.2.32). The second follows from Lemma 11.2.10 since ϕ ∈ X s,r ⊂ L∞ (Ω) is a strong and weak solution to problem (11.2.22).

318

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

Let us turn to the proof of (11.2.33). Consider the auxiliary boundary value problem (11.2.37) −u∇ϕ∗ + σϕ∗ = f in Ω, ϕ∗ = 0 on Σout . It follows from Condition 11.2.5 that Γ = cl Σout ∩ (Σ \ Σout ). Hence Ω and the vector ﬁeld −U satisfy Condition 11.2.5 with Σin replaced by Σout . It follows that Ω, −U, −u, and the exponents s, r meet all requirements of Lemma 11.2.11. Arguing as in the proof of (11.2.32) we conclude that there are constants c and σ0 > 1, depending only on Ω, U, and uY s,r , such that for all σ > σ0 and f ∈ X s,r , problem (11.2.37) has a unique solution ϕ∗ ∈ X s,r such that ϕ∗ X s,r ≤ cK(σ)f X s,r ,

ϕ∗ L∞ (Ω) ≤ σ −1 f L∞ (Ω) .

(11.2.38)

Introduce the linear operator A : X s,r → X s,r which assigns to every f ∈ X s,r the solution ϕ∗ to problem (11.2.37). It follows from (11.2.38) that Af X s,r ≤ cK(σ)f X s,r .

(11.2.39)

Now we can rewrite equations (11.2.23) in the form ϕ∗ = A(ϕ∗ div u) + Af.

(11.2.40)

Since div uX s,r ≤ cuY s,r , it follows from (11.2.39) and Lemma 11.2.8 that A(ϕ∗ div u)X s,r ≤ cK(σ) ϕ∗ div uX s,r ≤ cK(σ)uY s,r ϕ∗ X s,r Notice that K(σ) → 0 as σ → ∞. Hence we can choose σ0 > 0, depending only on Ω, U, s, r, and uY s,r , so large that cK(σ)uY s,r < 1/2 for all σ > σ0 . For such σ, the mapping ϕ∗ → A(ϕ∗ div u) is a contraction in X s,r and its norm is less than 1/2. By the Banach ﬁxed point theorem (the contraction mapping principle), equation (11.2.40) has a unique solution ϕ∗ ∈ X s,r such that ϕ∗ X s,r ≤ 2Af X s,r ≤ 2cK(σ)f X s,r . Since K decreases in σ, these inequalities lead to the ﬁrst estimate in (11.2.33) with the constant C = 2K(σ0 ). It remains to note that equation (11.2.23) can be written in the form −u · ∇ϕ∗ + σϕ∗ = ϕ∗ div u + f and estimates (11.2.38) for solutions to problem (11.2.37) lead to the inequalities ϕ∗ L∞ (Ω) ≤ σ −1 div uC(Ω) ϕ∗ L∞ (Ω) + σ −1 f L∞ (Ω) , which imply the second estimate in (11.2.33).

11.2. Existence of solutions

319

11.2.5 Step 4. Fixed point scheme. Estimates Fixed point scheme We solve Problem 11.1.9 by an application of the Tikhonov ﬁxed point theorem in the following framework: Fix τ ∈ (0, 1) and choose a matrix-valued function N ∈ C 1 (Ω) such that I − NC 1 (Ω) ≤ τ . Next choose ϑ = (v, π, ϕ) ∈ Bτ , and denote u := u + v,

q := q + π,

(11.2.41a)

:= + ϕ.

Consider the following problem for new unknowns ϑ1 = (v1 , π1 , ϕ1 ): u · ∇ϕ1 + σϕ1 = Ψ1 [ϑ] + mg in Ω,

ϕ1 = 0 on Σin ,

(11.2.41b)

Δv1 − ∇π1 = A (u) + Re B(, u, u) ≡ F [ϑ] in Ω,

(11.2.41c)

−1 Π(gσϕ1

div v1 = v1 = 0 on ∂Ω,

− g Ψ[ϑ] − gm ) in Ω, π1 − Ππ1 = 0.

Here m is given by m = κ (−1 Ψ1 [ϑ] ζ −gΨ[ϑ]) dx,

κ=

Ω

(11.2.41d) (11.2.41e) −1

g(1−ζ −−1 ζϕ) dx

, (11.2.41f)

Ω

where the auxiliary function ζ is a solution to the adjoint boundary value problem − div(uζ) + σζ = σg in Ω,

ζ = 0 on Σout .

(11.2.41g)

Notice that m and ζ are completely deﬁned by ϑ. Relations (11.2.41) form a complete system of equations and boundary conditions for ϑ1 and deﬁne a mapping Ξ : ϑ → ϑ1 . We claim that for a suitable choice of the constant τ , Ξ is an automorphism of the ball Bτ . Our strategy is the following. First we estimate the right hand sides of equations (11.2.41) in Z s,r and X s,r norms. Then we apply Lemma 11.2.9 and Proposition 11.2.12 to estimate ϑ1 = Ξ(ϑ). Auxiliary lemmas Lemma 11.2.13. Let Ω be a bounded domain with C 1 boundary and 0 < s < 1 < r < ∞. Then the operators div : W s,r (Ω) → W s−1,r (Ω) and ∇ : W s,r (Ω) → W s−1,r (Ω) are continuous. In particular, there is c, depending only on Ω, s, r, such that div uW s−1,r (Ω) ≤ cuW s,r (Ω) ,

∇uW s−1,r (Ω) ≤ cuW s,r (Ω) .

Proof. It follows from the deﬁnitions (see Section 1.5) that the mappings div : Lr (Ω) → W −1,r (Ω) and div : W 1,r (Ω) → Lr (Ω) are continuous. Since W s,r (Ω) = [Lr (Ω), W 1,r (Ω)]s,r and W s−1,r (Ω) = [Lr (Ω), W −1,r (Ω)]1−s,r ≡ [W −1,r (Ω), Lr (Ω)]s,r

320

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

are interpolation spaces, the continuity of div : W s,r (Ω) → W s−1,r (Ω) follows from the interpolation lemma 1.1.13. The same arguments imply the continuity of ∇ : W s,r (Ω) → W s−1,r (Ω). Lemma 11.2.14. Let Ω be a bounded domain with C 1 boundary and 0 < s < 1 < r < ∞. Then for every matrix-valued function M ∈ C 1 (Ω) and w ∈ W s−1,r (Ω), MwW s−1,r (Ω) ≤ c(Ω, s, r)MC 1 (Ω) wW s−1,r (Ω) .

Proof. Lemma 1.5.1 implies that W s−1,r (Ω) is the dual space (W01−s,r (Ω)) . By deﬁnition, we have

Mw, ϕ := w, M ϕ for all ϕ ∈ W01−s,r (Ω),

where ·, · is the duality pairing between (W01−s,r (Ω)) and W01−s,r (Ω). Obviously the mapping ϕ → M ϕ is continuous from Lr (Ω) to Lr (Ω) and from W01,r (Ω) to W01,r (Ω). Moreover, we have M ϕLr (Ω) ≤ cM C 1 (Ω) ϕLr (Ω) , M ϕW 1,r (Ω) ≤ cM C 1 (Ω) ϕW 1,r (Ω) . 0

0

Since W01−s,r (Ω) = [Lr (Ω), W01,r (Ω)]1−s,r is an interpolation space, it follows from Lemma 1.1.13 that M ϕW 1−s,r (Ω) ≤ cM C 1 (Ω) ϕW 1−s,r (Ω) . 0

0

Thus we get |Mw, ϕ| = |w, M ϕ| ≤ wW s−1,r (Ω) M ϕW 1−s,r (Ω) 0

≤ cM C 1 (Ω) wW s−1,r (Ω) ϕW 1−s,r (Ω) . 0

It remains to note that MwW s−1,r (Ω) =

sup ϕ

|Mw, ϕ|.

=1 1−s,r W0 (Ω)

Lemma 11.2.15. Let N ∈ C 2 (Ω) satisfy I−NC 2 (Ω) ≤ τ 2 . Then there are absolute constants τ0 > 0 and c > 0 such that for all τ ∈ (0, τ0 ), I − N−1 C 2 (Ω) ≤ cτ 2 ,

1 − g±1 C 2 (Ω) ≤ cτ 2 .

(11.2.42)

Proof. Observe that ABC 2 (Ω) ≤ cAC 2 (Ω) BC 2 (Ω) for all matrix-valued functions A and B. Here c depends only on the dimension of R3 , i.e., it is an absolute constant. From this and the identity (I − N)k N−1 − I = k≥1

11.2. Existence of solutions

321

we obtain N−1 − IC 2 (Ω) ≤

ck−1 I − NkC 2 (Ω) ≤

k≥1

ck−1 τ 2k = τ 2 /(1 − cτ 2 ).

k≥1

Choosing τ0 suﬃciently small we obtain the ﬁrst √ estimate in (11.2.42). In order to prove the second it suﬃces to note that g = det N is an analytic function of the entries of N, and g = 1 for N = I. Let us turn to equations (11.2.41b)–(11.2.41e). Our task is to estimate their right hand sides and, by doing so, estimate their solutions. Estimates of right hand sides in (11.2.41b)–(11.2.41e) Lemma 11.2.16. Let σ > 1,

ϑ ∈ Bτ ,

0 < τ ≤ τ0 < 1,

(11.2.43)

where τ0 is given by Lemma 11.2.15. Then there is c > 0, depending only on Ω, U and s, r, such that the functions Ψ, Ψ1 , A , and B, given by (11.1.24d) and (11.1.24e) satisfy A (u) + Re B(, u, u)Z s,r ≤ cτ 2 ,

(11.2.44)

Ψ[ϑ]X s,r ≤ cστ ,

(11.2.45)

Ψ1 [ϑ]X s,r ≤ cστ .

(11.2.46)

2

2

Proof. We begin by proving (11.2.44). It follows from (11.1.8) that A (u) = (I − N− ) div(∇u) + N− div (I − A)∇u + A∇(Bu) , where

A = g−1 NN ,

B = I − N−1 .

Applying Lemmas 11.2.13 and 11.2.14 we obtain A (u)W s−1,r (Ω) ≤ cτ 2 ∇uW s,r (Ω) + c(I − A)∇uW s,r (Ω) + A∇ Bu)W s,r (Ω) . Since sr > 3, estimate (11.2.8) implies (I − A)∇uW s,r (Ω) + A∇ Bu)W s,r (Ω)

≤ cI − AW s,r (Ω) ∇uW s,r (Ω) + cAW s,r (Ω) ∇ Bu)W s,r (Ω) ≤ cI − AC 2 (Ω) ∇uW s,r (Ω) + cAC 2 (Ω) ∇ Bu)W s,r (Ω) . (11.2.47)

Notice that I − A = (I − N) − (N − I)N − (g−1 − 1)NN .

322

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

Lemma 11.2.15 implies I − AC 2 (Ω) + BC 2 (Ω) ≤ cτ 2 ,

AC 2 (Ω) ≤ c.

Combining the estimates obtained we arrive at A (u)W s−1,r (Ω) ≤ cτ 2 ∇uW s,r (Ω) + c∇ Bu)W s,r (Ω) . On the other hand, the identity ∇(Bu) = ∇Bu + ∇uB , where ∇(Bu)ij = ∂xi (Bjk uk ), (∇Bu)ij = ∂xi (Bjk )uk , and estimate (11.2.42) imply ∇(Bu)W s,r (Ω) ≤ ∇BW s,r (Ω) uW s,r (Ω) + cBW s,r (Ω) ∇uW s,r (Ω) ≤ cBC 2 (Ω) (∇uW s,r (Ω) + uW s,r (Ω) ) ≤ cτ 2 (∇uW s,r (Ω) + uW s,r (Ω) ).

(11.2.48)

Thus we get A (u)W s−1,r (Ω) ≤ cτ 2 (∇uW s,r (Ω) + uW s,r (Ω) ) ≤ cτ 2 uW s+1,r (Ω) . We also have the obvious estimate A (u)L2 (Ω) ≤ I − NC 2 (Ω) uW 2,2 (Ω) ≤ cτ 2 uW 2,2 (Ω) . On the other hand, since = const and u ∈ C 2 (Ω), equalities (11.2.41a) imply X s,r + uX s,r + ∇uX s,r ≤ c + ϕX s,r + vX s,r + vY s,r ≤ c + cτ ≤ 2c.

(11.2.49)

Recalling the identity uY s+1,r = uW s+1,r (Ω) + uW 2,2 (Ω) we obtain A (u)Z s,r = A (u)W s−1,r (Ω) + A (u)L2 (Ω) ≤ cτ 2 uY s+1,r ≤ cτ 2 .

(11.2.50)

Next, since X s,r → Z s,r is continuous, the product estimate (11.2.8) implies Re B(, u, u)Z s,r ≤ Re B(, u, u)X s,r ≤ c Re X s,r uX s,r ∇uX s,r . From this, (11.2.49), and the inequality Re ≤ τ 2 we obtain Re B(, u, u)Z s,r ≤ cτ 2 . Combining this estimate with (11.2.50) we arrive at (11.2.44).

11.2. Existence of solutions

323

Our next task is to estimate the function Ψ[ϑ] =

q + π σ H(ϕ), − λ p ( )

H(ϕ) = p( + ϕ) − p( ) − p ( )ϕ.

Since p ∈ C 3 (Ω), it follows by Taylor expansion that H admits a representation H = f(ϕ)ϕ2 , where f is a C 1 function on the real line. It follows from the product estimates in Lemma 11.2.7 and from the estimates for composite functions in Lemma 11.2.8 that HX s,r ≤ cf(ϕ)X s,r ϕ2X s,r ≤ c(1 + ϕX s,r )ϕ2X s,r . Since ϑ = (v, π, ϕ) ∈ Bτ , we have ϕX s,r ≤ τ and πX s,r ≤ cτ . Recalling the inequality λ−1 ≤ τ 2 < 1 we obtain the estimate Ψ[ϑ]X s,r ≤

1 c (c + πX s,r ) + cσHX s,r ≤ + cστ 2 ≤ cτ 2 + cστ 2 , λ λ

which implies (11.2.45) since σ > 1. Finally, let us estimate the function σ Ψ1 [ϑ] = g Ψ[ϑ] − ϕ2 + σϕ(1 − g). By Lemma 11.2.15, for τ ∈ (0, τ0 ] we have gC 2 (Ω) ≤ cN3C 2 (Ω) ≤ c, 1 − gC 2 (Ω) ≤ cN2C 2 (Ω) I − NC 2 (Ω) ≤ cτ 2 ,

(11.2.51)

so that Ψ1 [ϑ]X s,r ≤ cX s,r Ψ[ϑ]X s,r + cσϕ2X s,r + cστ 2 ϕX s,r ≤ cτ 2 + τ 2 + cστ 2 . Since σ > 1 this obviously yields (11.2.46).

Estimates of solutions to problem (11.2.41) Lemma 11.2.17. Let τ ∈ (0, τ0 ], where τ0 is given by Lemma 11.2.15, and m ∈ R be given by (11.2.41f). Furthermore, assume that s, r satisfy (11.2.5). Then there are σ1 > 1 and c > 0, depending only on Ω, U and s, r, with the following property. For every σ > σ1 and ϑ ∈ Bτ , problem (11.2.41) has a unique solution ϑ1 = (v1 , π1 , ϕ1 ) ∈ E, where E is given by (11.2.3), such that ϕ1 X s,r + v1 Y s,r + π1 X s,r ≤ cστ 2 + c|m|.

(11.2.52)

324

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

Proof. First we estimate the solution ϕ1 to problem (11.2.41b)–(11.2.41e). By (11.2.41a), uY s,r ≤ u C 2 (Ω) + vY s,r ≤ c + cvY s,r ≤ c + cτ ≤ 2c. From this and the assumptions of Theorem 11.2.6 we conclude that u and s, r meet all requirements of Proposition 11.2.12. Hence there exist C > 0 and σ1 > 1, depending only on Ω, U and s, r, such that for all σ > σ1 problem (11.2.41b)– (11.2.41e) has a unique solution ϕ1 ∈ X s,r satisfying ϕ1 X s,r ≤ CΨ1 [ϑ] + mgX s,r . It now follows from (11.2.46) that ϕ1 X s,r ≤ cτ 2 + cστ 2 + c|m| ≤ cστ 2 + c|m|

(11.2.53)

since σ > 1. Next we estimate the solution v1 , π1 to problem (11.2.41c)–(11.2.41e). Notice that the right hand side of the equation for div v1 in (11.2.41d) admits the estimate −1 Π(gσϕ1 − g Ψ[ϑ] − gm )X s,r ≤ cσϕ1 X s,r + cΨ[ϑ]X s,r + c|m|. From this and (11.2.45), (11.2.53) we obtain 2 2 −1 Π(gσϕ1 − g Ψ[ϑ] − gm )X s,r ≤ cτ + cστ + c|m|.

Combining this with (11.2.44) and applying Lemma 11.2.9 we arrive at v1 Y s,r + π1 X s,r ≤ cστ 2 + c|m|.

(11.2.54)

Estimates (11.2.54) and (11.2.53) imply that for all σ > σ1 > 1 and τ ∈ (0, τ0 ], problem (11.2.41) has a unique solution satisfying (11.2.52). Estimate of m Proposition 11.2.18. Let all assumptions of Theorem 11.2.6 be satisﬁed and σ1 be given by Lemma 11.2.17. Then there is σ2 ≥ σ1 , depending only on Ω, U, and s, r, with the following property. For every σ > σ2 , there is τ1 ∈ (0, τ0 ], depending only on Ω, U, s, r, and σ, such that for all ϑ ∈ Bτ with τ ∈ (0, τ1 ) and α ∈ (0, s − 3/r), ζX s,r = ζW s,r (Ω) + ζW 1,2 (Ω) ≤ cσ, κ ≤ cσ |m| ≤ cσ

1/α

(11.2.56)

,

1/α+1

(11.2.55)

2

τ ,

where c depends on σ, Ω, U, s, r, and α.

(11.2.57)

11.2. Existence of solutions

325

Proof. Observe that Ω, U, u, and the exponents s, r meet all requirements of Proposition 11.2.12. On the other hand, equations (11.2.41g) are equivalent to (11.2.23) with ϕ∗ and f replaced by ζ and σ. In view of Proposition 11.2.12, there exist c > 0 and σ2 > 1, depending only on Ω, U, and s, r, such that for all σ > σ2 problem (11.2.41g) has a unique solution ζ ∈ X s,r satisfying (11.2.55). We can assume that σ2 ≥ σ1 . Since sr > 3, the embedding W s,r (Ω) → C α (Ω) is bounded for 0 < α < s − 3/r, which yields ζC α (Ω) ≤ cσ

for all σ > σ2 .

(11.2.58)

Next, since div u = div v satisﬁes div vC(Ω) ≤ vC 1 (Ω) ≤ cvY s,r ≤ cτ, Proposition 11.2.12 implies ζL∞ (Ω) ≤ σ/(σ − div vC(Ω) ) ≤ cσ(σ − cτ )−1 ≤ (1 − cτ )−1 . Choose τ1 > 0, depending only on Ω, U and s, r, so small that cτ1 < 1/2. We have (11.2.59) ζL∞ (Ω) ≤ 2 for all τ ∈ (0, τ1 ]. Next, formula (11.2.41f) for m and estimates (11.2.45), (11.2.46), (11.2.59) imply |m| ≤ |κ|c(Ψ[ϑ]X s,r + Ψ1 [ϑ]X s,r ) ≤ |κ|(cτ 2 + cστ 2 ).

(11.2.60)

It remains to estimate the constant κ. By (11.2.41f), −1 κ = (1 − ζ) dx + (g − 1)ζ dx − g−1 ζϕ dx. Ω

Since the embedding X

Ω

s,r

Ω

→ C(Ω) is bounded we have ϕC(Ω) ≤ cϕX s,r ≤ cτ.

On the other hand, inequalities (11.2.51) and (11.2.59) imply |ζ| ≤ 2, Thus we get κ

−1

|1 − g| ≤ cτ 2 .

≥

(1 − ζ) dx − cτ 2 − cτ. Ω

Now represent 1 − ζ in the form 1 − ζ = (1 − ζ)+ + (1 − ζ)− , where (1 − ζ)+ = max{1 − ζ, 0},

(1 − ζ)− = min{1 − ζ, 0}.

Our further considerations are based on the following two lemmas.

(11.2.61)

326

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

Lemma 11.2.19. There is a constant c, depending only on Ω, U and s, r, such that for all τ ∈ (0, τ0 ) and σ > σ1 , where τ0 is given by Lemma 11.2.15 and σ1 is given by Lemma 11.2.17, we have (1 − ζ)− C(Ω) ≤ cσ −1 τ + cτ 2 .

(11.2.62)

Proof. Since the mapping ζ → (1 − ζ)− is Lipschitz and ζ ∈ W 1,2 (Ω) is a continuous function, (1 − ζ)− is a continuous function in W 1,2 (Ω). Moreover (see [133, Corollary 2.1.8]), we have 0 −∇ζ a.e. on {x : 1 − ζ(x) < 0}, − ∇(1 − ζ) = 0 a.e. on {x : 1 − ζ(x) ≥ 0}. On the other hand, ζ is a strong solution to equation (11.2.41g), which along with the equality div u = div v yields −u · ∇ζ = ζ div v + ζ(g − 1) + (1 − ζ)σ Since (1 − ζ)− = 1 − ζ on {x : 1 − ζ < 0} and (1 − ζ)− = 0 on {x : 1 − ζ ≥ 0}, we have −u · ∇(1 − ζ)− + σ(1 − ζ)− = f a.e. in Ω, f = −ζ div v + σ(g − 1) if 1 − ζ < 0 and

(11.2.63) f = 0 if 1 − ζ < 0.

Moreover, since ζ = 0 on Σout , the function (1 − ζ)− vanishes on Σout . Now choose φ ∈ C 1 (Ω) vanishing in a neighborhood of Σ \ Σout . Multiplying (11.2.63) by φ and integrating the result over Ω we obtain (1 − ζ)− (div(φu) + σφ − φf ) dx = 0. Ω

Hence (1 − ζ)− ∈ C(Ω) is a bounded weak solution to problem (11.2.22) with u, U, and Σin replaced by −u, −U, and Σout , respectively. Since −u, −U, and Ω meet all requirements of Lemma 11.2.10, we obtain (1 − ζ)− L∞ (Ω) ≤ σ −1 f L∞ (Ω) ≤ 2σ −1 div vL∞ (Ω) + g − 1C 2 (Ω) ≤ 2σ −1 vC 1 (Ω) + τ 2 ≤ cσ −1 vY s,r + τ 2 ≤ cσ −1 τ + τ 2 .

Lemma 11.2.20. Let σ2 > 1, α, and τ1 be given by Proposition 11.2.18. Then there is c > 0, depending only on Ω, U, s, r, and α, such that for all σ > σ2 , (1 − ζ)+ dx ≥ c−1 σ −1/α . (11.2.64) Ω

Proof. Since ζ vanishes at Σout , inequality (11.2.58) implies |ζ(x)| ≤ cσρα

for all x with dist(x, Σout ) ≤ ρ.

11.2. Existence of solutions

327

Choosing ρ = (cσ)−1/α , where c is the constant from (11.2.58), we obtain |ζ(x)| ≤ 1/2 if

dist(x, Σout ) ≤ ρ.

Denote by O(ρ) the ρ-neighborhood of Σout in R3 . Since Σout is a C 1 nonempty manifold, there is a constant c1 , depending only on Ω, such that meas(O(ρ) ∩ Ω) ≥ c−1 . Thus we get O(ρ)∩Ω

−1/α (1 − ζ) dx ≥ 2c−1 . 1 (cσ)

Next, as the nonnegative function (1 − ζ)+ coincides with 1 − ζ in O(ρ) ∩ Ω, we get −1/α (1 − ζ)+ dx ≥ (1 − ζ) dx ≥ 2c−1 . 1 (cσ) O(ρ)∩Ω

Ω

We are now in a position to estimate the constant κ. Substituting (11.2.62) and (11.2.64) into (11.2.61) we obtain |(1 − ζ)− | dx − cτ 2 − cτ κ −1 ≥ (1 − ζ)+ dx − cτ 2 − cτ ≥ (1 − ζ)+ dx − ≥c

Ω −1 −1/α

σ

Ω

− cσ

−1

Ω

τ − cτ − cτ. 2

Since σ > 1 and τ < 1 we can rewrite this inequality in the form κ −1 ≥ c−1 σ −1/α (1 − cτ σ 1/α ).

(11.2.65)

Now ﬁx σ > σ1 and choose τ ∗ > 0 so small that cτ σ 1/α ≤ 1/2 for all τ ∈ (0, τ ∗ ). For such τ we have 0 ≤ κ ≤ 2c1 σ 1/α , which gives (11.2.56). From this and (11.2.60) we obtain |m| ≤ cσ 1/α (τ 2 + στ 2 ),

which leads to (11.2.57) since σ > 1.

We are now in a position to prove that for small τ the mapping Ξ takes the ball Bτ into itself. The corresponding result is given by Lemma 11.2.21. Under the assumptions of Theorem 11.2.6, there exists σ ∗ > 1, depending only on U, Ω and s, r, with the following property. For every σ > σ ∗ there are positive τ ∗ and c, depending only on U, Ω, r, s, and σ, such that whenever τ ∈ (0, τ ∗ ],

λ−1 , Re ∈ (0, τ 2 ],

the mapping Ξ takes the ball Bτ into itself.

N − IC 2 (Ω) ≤ τ 2 ,

(11.2.66)

328

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

Proof. Set σ ∗ = max{σ1 , σ2 }, where σi are given by Lemma 11.2.17 and Proposition 11.2.18. Next, take τ2 = min{τ0 , τ1 }, where τi are deﬁned by Lemma 11.2.15 and Proposition 11.2.18. It follows from (11.2.52) and (11.2.57) that for σ > σ ∗ and τ ∈ (0, τ2 ), the solution ϑ1 := Ξ(ϑ) to problem (11.2.41) satisﬁes ϕ1 2X s,r + v1 Y s,r + π1 X s,r ≤ cστ 2 + cσ 1/α+1 τ 2 . Since σ is ﬁxed, we can choose τ ∗ ∈ (0, τ2 ] so small that cστ + cσ 1/α+1 τ < 1 for all τ ∈ (0, τ ∗ ]. It remains to note that for such τ , ϑ1 E = ϕ1 X s,r + v1 Y s,r + π1 X s,r < τ.

11.2.6 Step 5. Proof of Theorem 11.2.6 Sequential weak continuity of Ξ. It follows from Lemma 11.2.21 that for a ﬁxed σ > σ ∗ and all τ ∈ (0, τ ∗ ) the mapping Ξ takes the ball Bτ ⊂ E into itself. In order to apply the ﬁxed point theory we show that Ξ is sequentially weakly continuous. To do so, assume that all assumptions of Lemma 11.2.21 are satisﬁed and N, λ, Re, τ satisfy condition (11.2.66). Choose a sequence ϑn ∈ Bτ such that ϑn = (un , πn , ϕn ) converges weakly in E = (Y s,r )3 × X s,r × X s,r . It follows from Lemma 11.2.3 that E is reﬂexive as a product of reﬂexive spaces. Hence there is ϑ ∈ E such that ϑn ϑ weakly in E. Since the ball Bτ is closed and convex, ϑ belongs to Bτ . By Lemma 11.2.21, the corresponding elements ϑ1,n = Ξ(ϑn ) belong to the ball Bτ , and the sequence of functions ζn is bounded in X s,r . There are subsequences {ϑ1,j } ⊂ {ϑ1,n } and {ζj } ⊂ {ζn } such that ϑ1,j converges weakly in E to some ϑ1 ∈ Bτ and ζj converges weakly in X s,r to some ζ ∈ X s,r . Since the embedding E → C(Ω) is compact, we have ϑn → ϑ, ϑ1,j → ϑ1 in C(Ω), and ∇ζj ∇ζ

weakly in L2 (Ω),

ζj → ζ

in C(Ω).

Substituting ϑj and ϑ1,j into equations (11.2.41) and letting j → ∞ we ﬁnd that the limits ϑ and ϑ1 also satisfy these equations. Thus, ϑ1 = Ξ(ϑ). In view of Lemma 11.2.21, for given ϑ ∈ Bτ , problem (11.2.41) has a unique solution ϑ1 ∈ E. Hence all weakly convergent subsequences of ϑ1,n have the common limit ϑ1 . Therefore, the whole sequence ϑ1,n = Ξ(ϑn ) converges weakly to Ξ(ϑ). Hence the mapping Ξ : Bτ → Bτ is sequentially weakly continuous. Notice that, by Lemma 11.2.4, E = (Y s,r )3 × (X s,r )2 is separable as a Cartesian product of separable spaces. Therefore we can apply the Tikhonov ﬁxed point theorem 1.1.18 to conclude that there is ϑ ∈ B(τ ) such that ϑ = Ξ(ϑ). Proof of Theorem 11.2.6. It remains to prove that the ﬁxed point ϑ = Ξ(ϑ) ∈ E satisﬁes the original equations (11.1.24). It follows from the deﬁnition of the

11.2. Existence of solutions

329

mapping Ξ and equations (11.2.41) that ϑ and the corresponding function ζ serve as a strong solution to the boundary value problem u := u + v,

q := q + π, := + ϕ, u · ∇ϕ + σϕ = Ψ1 [ϑ] + mg in Ω, ϕ = 0 on Σin ,

(11.2.67) (11.2.68)

Δv − ∇π = A (u) + Re B(, u, u) ≡ F [ϑ] in Ω, (11.2.69) div v = −1 Π(gσϕ − g Ψ[ϑ] − gm ) in Ω, v = 0 on ∂Ω, π − Ππ = 0, −1 m = κ (−1 Ψ [ϑ] ζ − gΨ[ϑ]) dx, κ = g(1 − ζ − −1 , 1 ζϕ) dx Ω

Ω

− div(uζ) + σζ = σg in Ω,

(11.2.70) (11.2.71)

ζ = 0 on Σout .

The only diﬀerence between problems (11.2.67)–(11.2.71) and (11.1.24) is the presence of the projection Π on the right hand side of (11.2.69). Hence, it suﬃces to show that −1 Π(−1 gσϕ − gΨ[ϑ] − gm) = gσϕ − gΨ[ϑ] − gm. By expression (1.7.6) for Π, this means that the function Υ = −1 gσϕ − gΨ[ϑ] − gm has zero mean over Ω, i.e.,

(11.2.72)

Υ(x) dx = 0. Ω

In order to prove (11.2.72), we note that ϕ is a strong solution to problem (11.2.68). Next, both ζ and ϕ are continuous and belong to W 1,2 (Ω). Moreover, ζ vanishes on Σout . Multiplying (11.2.68) by ζ, integrating the result by parts, and recalling (11.2.71) we obtain ϕg dx = ζ(Ψ1 [ϑ] + mg) dx. (11.2.73) σ Ω

Ω

Next, multiplying the formula for m in (11.2.70) by κ and noting that 1 + −1 ϕ= / we obtain −1 m g(1 − −1 ζ) dx = Ψ [ϑ]ζ dx − gΨ[ϑ] dx. (11.2.74) 1 Ω

Ω

Ω

It follows from the expression for Υ that −gΨ[ϑ] = Υ + gm − −1 σgϕ, which leads to

−

gΨ[ϑ] dx =

Ω

Ω

g dx −

Υ dx + m Ω

Ω

−1 σgϕ dx.

330

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

Substituting this into (11.2.74) we obtain −1 0= −1 (Ψ [ϑ] + mg)ζ dx − σgϕ dx + Υ dx. 1 Ω

Ω

Ω

Since = const, it now follows from (11.2.73) that −1 −1 σ ϕg dx − σgϕ dx + Υ dx, 0= Ω

Ω

Ω

which implies (11.2.72). Hence the ﬁxed point ϑ ∈ Bτ is a solution to problem (11.1.24). Next, estimates (11.2.7) for ζ, m and κ follow from Proposition 11.2.18. Let us prove that the set of solutions (ϑ, ζ, m) ∈ Bτ × X s,r × R to Problem 11.1.9 contains a subset which is relatively compact in W s+1,r (Ω) × W s,r (Ω)3 × R. To this end choose s0 > s such that s0 and r satisfy inequalities (11.2.5) with s replaced by s0 . Now we can choose σ ∗ > 1 and τ ∗ (σ) such that whenever σ > σ ∗ , τ ∈ (0, τ ∗ (σ)] and λ−1 , Re ∈ (0, τ 2 ],

N − IC 2 (Ω) ≤ τ 2 ,

Problem 11.1.9 has a solution (ϑ, ζ, m) ∈ Bτ0 × W s0 ,r (Ω) × R, where Bτ0 = {ϑ = (v, π, ϕ) : vY s0 ,r + πX s0 ,r + ϕX s0 ,r ≤ τ ≤ 1}. Notice that Bτ0 ⊂ Bτ . Since the embeddings Y s0 ,r → W s0 +1,r (Ω),

X s0 ,r → W s0 ,r (Ω)

are continuous, Bτ0 is a bounded subset of W s0 +1,r (Ω) × W s0 ,r (Ω)2 . On the other hand, estimate (11.2.7) with s replaced by s0 implies that the functions ζ and constants m are bounded in X s0 ,r × R. Hence the constructed set of solutions (ϑ, ζ, m) to Problem 11.1.9 is bounded in W s0 +1,r (Ω) × W s0 ,r (Ω)3 × R. It remains to note that as s0 > s, this set is relatively compact in W s+1,r (Ω) × W s,r (Ω)3 × R.

11.3 Dependence of solutions on N 11.3.1 Problem formulation. Basic equations. Transposed problem Theorem 11.2.6 yields the existence of solutions (may be not unique) ϑ ∈ Bτ to Problem 11.1.9 for all suﬃciently small Re, λ, σ −1 and all matrices N close to I. Recall that N is completely deﬁned by a deformation x + εT(x) and is given by (11.1.5). The main goal of the sensitivity analysis is to prove diﬀerentiability of solutions to Problem 11.1.9 with respect to ε. The ﬁrst step in this direction is to investigate the dependence of these solutions on the matrix N. At this stage the special structure of N is not important and the results will be valid for an arbitrary smooth matrix-valued function N(x), x ∈ Ω. In this section we prove that, under

11.3. Dependence of solutions on N

331

the assumptions of Theorem 11.2.6, the solution to Problem 11.1.9 is unique and the mapping N → ϑ is Lipschitz in a weak norm. The simplest way to do this is to take two diﬀerent matrices Ni , i = 0, 1, and to compare the corresponding solutions (ϑi , ζi , mi ) ∈ E × X s,r × R (with ϑi = (vi , πi , ϕi )) to Problem 11.1.9. Denote by w = v0 − v1 , ω = π0 − π1 , ψ = ϕ0 − ϕ1 , ξ = ζ0 − ζ1 , n = m0 − m1 ,

(11.3.1)

the diﬀerence between these solutions. It follows from (11.1.24) that the functions (w, ω, ψ, ξ) and the constant n satisfy u0 ∇ψ + σψ = −w · ∇ϕ1 + b11 ψ + b12 ω + b13 n + b10 d in Ω,

(11.3.2a)

Δw − ∇ω = A0 (w) + Re C (ψ, w) + D in Ω, div w = b21 ψ + b22 ω + b23 n + b20 d in Ω,

(11.3.2b)

− div(u0 ξ) + σξ = div(ζ1 w) + σd in Ω,

(11.3.2c)

w = 0 on ∂Ω, ω − Πω = 0,

ψ = 0 on Σin , ξ = 0 on Σout , n = κ0 (b31 ψ + b32 ω + b34 ξ + b30 d) dx.

(11.3.2d)

Ω

Here the coeﬃcients are given by σg0 (ϕ0 + ϕ1 ) + g0 m1 , σ = 1 Ψ[ϑ1 ] − ϕ21 − σϕ1 + m1 1 ,

b11 = σ(1 − g0 ) + g0 Ψ[ϑ0 ] − g0 1 Φ(ϕ0 , ϕ1 ) − b12 = λ−1 1 g0 , b13 = g0 0 , b10 σ b21 = g0 + Φ(ϕ0 , ϕ1 ) , b22 = −g0 /λ,

b20 = σϕ1 −1 − Ψ[ϑ1 ] − m1 , g0 σ −1 = ζ1 σ(1 − g0 ) + g0 Ψ[ϑ0 ] − g0 1 Φ(ϕ0 , ϕ1 ) − (ϕ0 + ϕ1 )

b23 = −g0 , b31

+ g0 Φ(ϕ0 , ϕ1 ) +

m1 g0 −1 ζ1 ,

b32 = g0 λ−1 (−1 ζ1 1 − 1),

−1 b34 = −1 Ψ1 [ϑ0 ] + m1 g0 (1 + ϕ0 ),

−1 b30 = −1 ζ1 (b10 − m1 1 ) − Ψ[ϑ1 ] − m1 (1 − ζ1 − ζ1 ϕ1 ), 1 H (ϕ0 s + ϕ1 (1 − s)) ds, Φ(ϕ0 , ϕ1 ) = (p ( ) )−1 σ 0 −1 g0 (1 − ζ0 − −1 , κ0 = ζ0 ϕ0 ) dx Ω

the linear operator C is given by C (ψ, w) = B0 (ψ, u0 , u0 ) + B0 (1 , w, u0 ) + B0 (1 , u1 , w),

(11.3.3)

332

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

the linear operators Ai and trilinear forms Bi are given by −1 Ai (u) = Δu − N− div g−1 i i Ni Ni ∇(Ni u) , −1 Bi (, u, w) = N− i (u ∇(Ni w)),

(11.3.4)

the vector ﬁelds ui and functions i are given by ui = u + vi ,

i = + ϕi ,

i = 0, 1,

the functions D and d are given by D = A0 (u1 ) − A1 (u1 ) + Re(B0 (1 , u1 , u1 ) − B1 (1 , u1 , u1 )), & & d = g0 − g1 ≡ det N0 − det N1 .

(11.3.5)

We consider D and d as given functions, and equalities (11.3.2) as a linear system of equations and boundary conditions for unknowns w, ω, ψ, ξ, and n. Our goal is to establish the well-posedness of problem (11.3.2) and to estimate its solutions in terms of D and d. The main diﬃculty here is that we cannot apply Lemmas 11.2.10 and 11.2.11 to the transport equation (11.3.2a) since the term w · ∇ϕ1 in (11.3.2a) is neither smooth nor bounded. However, as shown in Lemma 11.3.7 below, this term belongs to some dual space. Thus we come to the idea to consider (w, ω, ψ, ξ) as a very weak solution to problem (11.3.2) (see [50, Ch. IV.9] for the theory of very weak solutions of Stokes equations). The construction of very weak solutions to a boundary value problem is built upon the analysis of the transposed problem, which in our case is formulated as follows. For a given vector ﬁeld H, functions G, F , M and a constant e ﬁnd a vector ﬁeld h, functions g, ς, υ, and a constant l satisfying Δh − ∇g = F (h) + Re H (h) − ς∇ϕ1 − ζ1 ∇υ + H in Ω, div h = Π(b22 g + b12 ς + b32 l) + ΠG in Ω,

(11.3.6a)

− div(u0 ς) + σς = Re M (h) + b11 ς + b21 g + b31 l + F u0 ∇υ + συ = b34 l + M in Ω,

(11.3.6b)

in Ω,

h = 0 on ∂Ω, g − Πg = 0,

ς = 0 on Σout , υ = 0 on Σin , l = κ0 (b13 ς + b23 g) dx + e,

(11.3.6c)

Ω

where the coeﬃcients bij and the constant κ0 are deﬁned by (11.3.3), and the linear operators F , H , M are given by −1 div g0 N0 N F (h) = Δh − N− 0 ∇(N0 h) , 0 −1 − H (h) = 1 ∇(N−1 div(1 u1 ⊗ (N−1 0 u0 )N0 h − N0 0 h)),

M (h) =

(u0 ∇(N−1 0 u0 ))

·

N−1 0 h.

(11.3.7)

11.3. Dependence of solutions on N

333

Remark 11.3.1. Here we use the following notation. If a is a vector with components aj and A is a matrix with entries Aij , then ∇a denotes the matrix with entries (∇a)ij = ∂xi aj , and div A denotes the vector with components (div A)j = ∂xi Aij . Our strategy is the following. First we establish the existence and uniqueness of weak and strong solutions to problem (11.3.6). Using this we show that the differences (w, ω, ψ, ξ) given by (11.3.1) are a very weak solution to problem (11.3.2). Next we prove that the existence result for the transposed problem implies the desired estimates for a very weak solution to problem (11.3.2). We start by proving the well-posedness of problem (11.3.6).

11.3.2 Transposed problem In this section we prove the existence and uniqueness of solutions to problem (11.3.6). We assume that the following conditions are satisﬁed. Condition 11.3.2. • The C 3 surface ∂Ω and the vector ﬁeld U ∈ C 3 (Ω) satisfy Condition 11.2.5. • The exponents s and r satisfy 1/2 < s < 1,

r < ∞,

2s − 3r−1 < 1,

sr > 3,

(1 − s)r > 3. (11.3.8)

• σ ∗ > 1 and τ ∗ (σ) ≤ 1 are given by the requirements of Theorem 11.2.6. • σ > σ ∗ and τ ∈ (0, τ ∗ (σ)]. • We have

λ−1 , Re ∈ (0, τ 2 ],

Ni − IC 2 (Ω) ≤ τ 2 .

(11.3.9)

• τ ∈ (0, τ0 ], where τ0 is the absolute constant given by Lemma 11.2.15, so ±1 2 N±1 i − IC 2 (Ω) + gi − 1C 2 (Ω) ≤ cτ .

(11.3.10)

• ϑi = (vi , πi , ϕi ) ∈ Bτ , ζi ∈ W s,r (Ω), mi ∈ R, i = 0, 1, are solutions to Problem 11.1.9 and satisfy inequalities (11.2.7). • ui = u + vi and i = + ϕi . Throughout this section we will use the following notation for products of Sobolev spaces. Deﬁnition 11.3.3. For every s ∈ (0, 1) and r ∈ (1, ∞), we set U s,r = (W s−1,r (Ω))3 × (W s,r (Ω))3 × R, V s,r = (W s+1,r (Ω))3 × (W s,r (Ω))3 × R,

334

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

and E s,r = (Z s,r )3 × (X s,r )3 × R, F s,r = (Y s,r )3 × (X s,r )3 × R. We are now in a position to formulate the main result of this section on existence and uniqueness of solutions to the transposed problem. Theorem 11.3.4. Let Condition 11.3.2 be satisﬁed. Then there are c and σc ≥ σ ∗ , depending only on Ω, U, s, r, and τc (σ) ∈ (0, min{τ0 , τ ∗ (σ)}], depending on Ω, U, s, r, and σ, with the following property. If σ > σc

and

0 < τ ≤ τc (σ),

then for every f = (H, G, F, M, e) ∈ U s,r , problem (11.3.6) has a unique solution h = (h, g, ς, υ, l) ∈ V s,r which satisﬁes hV s,r ≤ cfU s,r .

(11.3.11)

hF s,r ≤ cfE s,r .

(11.3.12)

If, in addition, f ∈ E s,r , then

Remark 11.3.5. In the case f ∈ E s,r the solution h to the transposed problem (11.3.6) is understood in the strong sense, which creates no diﬃculties. The case f ∈ Us,r needs clariﬁcation. In this case we assume that the right hand side of the Stokes equations (11.3.6a) belongs to W s−1,r (Ω) × W s,r (Ω) and (h, g) ∈ W s+1,r (Ω) × W s,r (Ω) satisﬁes (11.3.6a) in the sense of Lemma 1.7.9. As for the transport equations (11.3.6b), we assume that they are satisﬁed in the weak sense (see Section 11.2.4). The rest of this section is devoted to the proof of Theorem 11.3.4. We split it into four steps. Step 1. Multiplicators. We ﬁrst show that for ϕ, ς ∈ W s,r (Ω), the quantity ς∇ϕ belongs to the dual space W s−1,r (Ω). This is based on the following lemma on multiplicators in fractional Sobolev spaces. Lemma 11.3.6. Let the exponents s and r satisfy 1/2 < s < 1,

r < ∞,

rs > 3,

(1 − s)r > 3,

(11.3.13)

and suppose functions ς ∈ W s,r (R3 ), w ∈ W 1−s,r (R3 ), r = r/(r − 1), are compactly supported in a ball B ⊂ R3 . Then ς wW 1−s,r (R3 ) ≤ cwW 1−s,r (R3 ) ςW s,r (R3 ) , where c depends on s, r, and the diameter of B.

(11.3.14)

11.3. Dependence of solutions on N

335

Proof. Since ς and w are supported in B, we have ς wW 1−s,r (R3 ) = ς wLr (B) 1/r + |x − y|−3−(1−s)r |ς w(x) − ς w(y)|r dxdy . B×B

The identity ς w(x) − ς w(y) = ς(x)(w(x) − w(y)) + w(y)(ς(x) − ς(y)) implies ς wW 1−s,r (R3 ) ≤ ς wLr (B) 1/r + |x − y|−3−(1−s)r |ς(x)|r |w(x) − w(y)|r dxdy B×B

|x − y|−3−(1−s)r |w(y)|r |ς(x) − ς(y)|r dxdy

+

1/r

B×B

By an embedding theorem (see Section 1.5), we have |ς(x)| ≤ cςW s,r (R3 )

in R3 for sr > 3,

which gives ς wLr (B) +

|x − y|−3−(1−s)r |ς(x)|r |w(x) − w(y)|r dxdy

1/r

B×B

≤ cςW s,r (R3 ) wLr (B) +

|x − y|−3−(1−s)r |w(x) − w(y)|r dxdy

1/r

B×B

≤ cwW 1−s,r (R3 ) ςW s,r (R3 ) . Thus we get ς wW 1−s,r (R3 ) ≤ cwW 1−s,r (R3 ) ςW s,r (R3 ) 1/r + |x − y|−3−(1−s)r |w(y)|r |ς(x) − ς(y)|r dxdy . (11.3.15) B×B

In order to estimate the integral on the right hand side, we use the identity

|x − y|−3−(1−s)r |w(y)|r |ς(x) − ς(y)|r = |x − y|−α |w(y)|r (|x − y|−3−sr )r /r |ς(x) − ς(y)|r , where α = 3 + (1 − s)r − (3 + sr)r /r = 3

r r−2 + (1 − 2s) . r−1 r−1

336

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

Applying the Hölder inequality we obtain 1/r |x − y|−3−(1−s)r |w(y)|r |ς(x) − ς(y)|r dxdy B×B

≤

|x − y|−αp |w(y)|pr | dxdy

1/(pr )

B×B

×

≤c

|x − y|−3−sr |ς(x) − ς(y)|r dxdy

B×B

|x − y|−αp |w(y)|pr dxdy

1/(pr )

1/r

ςW s,r (R3 ) , (11.3.16)

B×B

where

1/p = 1 − r /r,

p = (r − 1)/(r − 2),

pr = r/(r − 2).

Notice that r > 2 since sr > 3. We have pα = 3 + (1 − 2s)r/(r − 2) < 3 since which gives

s > 1/2,

|x − y|−αp dx ≤ c.

B

Inserting this in (11.3.16) we obtain

1/r |x − y|−3−(1−s)r |w(y)|r |ς(x) − ς(y)|r dxdy B×B (r−2)/r ≤c |w(y)|r/(r−2) dxdy ςW s,r (R3 ) . (11.3.17) B×B

Next an embedding theorem (see Section 1.5) implies wLt (R3 ) ≤ cwW 1−s,r (R3 ) for

1/t = 1/r − (1 − s)/3,

t=

r r−1−

(1−s)r 3

.

Since (1 − s)r > 3, we have t > r/(r − 2), which leads to wLr/(r−2) (B) ≤ cwLt (R3 ) ≤ cwW 1−s,r (R3 ) . Inserting this into (11.3.17) we obtain

|x − y|−3−(1−s)r |w(y)|r |ς(x) − ς(y)|r dxdy

1/r

B×B

≤ cwW 1−s,r (R3 ) ςW s,r (R3 ) . Combining this estimate with (11.3.15) we arrive at (11.3.14).

11.3. Dependence of solutions on N

337

Lemma 11.3.7. Let Ω ⊂ R3 be a bounded domain with C 1 boundary. Let the exponents s ∈ (0, 1) and r ∈ (1, ∞) satisfy the inequalities 1/2 < s < 1,

r < ∞,

(1 − s)r > 3,

sr > 3,

(11.3.18)

and let ϕ, ς ∈ W s,r (Ω) ∩ W 1,2 (Ω) and w ∈ W01−s,r (Ω). Then there is a constant c, depending only on s, r, and Ω, such that the trilinear form ςw · ∇ϕ dx B(w, ϕ, ς) = Ω

satisﬁes |B(w, ϕ, ς)| ≤ cwW 1−s,r (Ω) ϕW s,r (Ω) ςW s,r (Ω) , 0

(11.3.19)

and can be continuously extended to B : W01−s,r (Ω)3 × W s,r (Ω)2 → R. In particular, ς∇ϕ ∈ W s−1,r (Ω)

ς∇ϕW s−1,r (Ω) ≤ cϕW s,r (Ω) ςW s,r (Ω) .

and

(11.3.20)

Proof. Let B be a ball in R such that Ω B. Since ∂Ω is C , the functions ϕ and ς have extensions ϕ, ς ∈ W s,r (R3 ) ∩ W 1,2 (R3 ) such that ϕ and ς are compactly supported in B and 3

1

ϕW s,r (R3 ) ≤ cϕW s,r (Ω) ,

ςW s,r (R3 ) ≤ cςW s,r (Ω) ,

where c depends on s, r and B. By the deﬁnition of W01,r (Ω) (see Section 1.5), the extension operator T : W01,r (Ω) → W 1,r (R3 ) deﬁned by Tw = w

T w = 0 in R3 \ Ω

in Ω and

is continuous. Obviously it maps continuously Lr (Ω) to Lr (R3 ). Since W01−s,r (Ω) = [Lr (Ω), W01,r (Ω)]1−s,r and W 1−s,r (R3 ) = [Lr (R3 ), W 1,r (R3 )]1−s,r are inter polation spaces, Lemma 1.1.13 shows that T : W01−s,r (Ω) → W 1−s,r (R3 ) is also continuous. Therefore, for every w ∈ W01−s,r (Ω), the extension w := T w satisﬁes wW 1−s,r (R3 ) ≤ c(s, r )wW 1−s,r (Ω) . 0

Obviously we have

(11.3.21)

B(w, ϕ, ς) = R3

w · ∇ϕ ς dx.

It follows that |B(w, ϕ, ς)| ≤ w ςW 1−s,r (R3 ) ∇ϕW s−1,r (R3 ) ≤ cw ςW 1−s,r (R3 ) ϕW s,r (R3 ) ≤ cw ςW 1−s,r (R3 ) ϕW s,r (Ω) . Application of Lemma 11.3.6 completes the proof.

338

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

Step 2. Estimates of the operators F , H , M and the coeﬃcients bij . Our next task is to estimate the norms of the diﬀerential operators F , H , M appearing in equations (11.3.6) and next to estimate the coeﬃcients of these equations. Recall that F , H , M are deﬁned by (11.3.7). We begin by estimating the operator F . Lemma 11.3.8. Let Condition 11.3.2 be satisﬁed. Then there is a constant c > 0, depending only on s, r, and Ω, such that F (h)W s−1,r (Ω) ≤ cτ 2 hW s+1,r (Ω) F (h)Z s,r ≤ cτ hY s,r 2

for all h ∈ W s+1,r (Ω),

for all h ∈ Y

s,r

.

(11.3.22) (11.3.23)

Proof. We have − −1 F (h) = div (I − S)∇h + S∇((I − N−1 0 )h) + ∇(N0 ) · N0 S∇(N0 h) , where

S = g0 N− 0 N0 N0 ,

and the notation ∇(A) · B stands for the column vector with components (∇(A) · B)i = ∂xj (A)im (B)mj . Estimates (11.3.10) yield 2 I − N−1 0 C 2 (Ω) ≤ cτ .

I − SC 2 (Ω) ≤ cτ 2 ,

(11.3.24)

On the other hand, Lemma 11.2.13 implies div (I − S)∇h + S∇((I − N−1 )h) 0

W s−1,r (Ω)

≤ c(I − S)∇h + S∇((I − N−1 0 )h)W s,r (Ω) . By (11.3.24), we have (I − S)∇h + S∇((I − N−1 0 )h)W s,r (Ω) ≤ cI − SC 2 (Ω) ∇hW s,r (Ω) + cSC 2 (Ω) ∇((I − N−1 0 )h)W s,r (Ω) ≤ cτ 2 ∇hW s,r (Ω) + c∇((I − N−1 0 )h)W s,r (Ω) ≤ cτ 2 hW s+1,r (Ω) + c(I − N−1 0 )hW s+1,r (Ω) 2 ≤ c(τ 2 + I − N−1 0 C 2 (Ω) )hW s+1,r (Ω) ≤ cτ hW s+1,r (Ω) .

Thus we get div (I − S)∇h + S∇((I − N−1 )h) s−1,r ≤ cτ 2 hW s+1,r (Ω) . 0 W (Ω)

(11.3.25)

Next, it follows from (11.3.24) that − − I)C 1 (Ω) ≤ N− − IC 2 (Ω) ≤ cτ 2 . ∇(N− 0 )C 1 (Ω) = ∇(N0 0

11.3. Dependence of solutions on N

339

Applying Lemmas 11.2.13 and 11.2.14 we obtain −1 ∇(N− 0 ) · (N0 S∇(N0 h))W s−1,r (Ω) −1 ≤ ∇(N− 0 )C 1 (Ω) N0 S∇(N0 h)W s−1,r (Ω) −1 ≤ cτ 2 N 0 SC 1 (Ω) ∇(N0 h)W s−1,r (Ω) 2 2 ≤ cτ 2 N−1 0 hW s,r (Ω) ≤ cτ hW s,r (Ω) ≤ cτ hW s+1,r (Ω) . (11.3.26)

Combining (11.3.25) and (11.3.26) we arrive at (11.3.22). Next, estimates (11.3.24) imply F (h)L2 (Ω) ≤ cτ 2 hW 2,2 (Ω) for all h ∈ W 2,2 (Ω). Combining this with (11.3.22) and recalling F (h)Z s,r = F (h)W s−1,r (Ω) + F (h)L2 (Ω) , hY s,r = hW s+1,r (Ω) + hW 2,2 (Ω)

we arrive at (11.3.23).

Lemma 11.3.9. Let Condition 11.3.2 be satisﬁed. Then there is a constant c > 0, depending only on s, r, Ω, and U, such that H (h)W s−1,r (Ω) ≤ chW s+1,r (Ω) H (h)

Z s,r

≤ chY

s,r

for all h ∈ W s+1,r (Ω),

for all h ∈ Y

s,r

.

(11.3.27) (11.3.28)

Proof. Since W s,r (Ω) is a Banach algebra, we have −1 −1 −1 1 ∇(N−1 0 u0 )N0 hW s−1,r (Ω) ≤ 1 ∇(N0 u0 )N0 hW s,r (Ω) −1 ≤ c1 W s,r (Ω) ∇(N−1 0 u0 )W s,r (Ω) N0 hW s,r (Ω)

≤ cN−1 0 u0 W s+1,r (Ω) hW s,r (Ω) ≤ cu0 W s+1,r (Ω) hW s,r (Ω) ≤ chW s,r (Ω) .

(11.3.29)

Next applying Lemmas 11.2.13 and 11.2.14 we obtain −1 N− div(1 u1 ⊗ (N−1 0 0 h))W s−1,r (Ω) ≤ c div(1 u1 ⊗ (N0 h))W s−1,r (Ω) −1 ≤ c1 u1 ⊗ (N−1 0 h)W s,r (Ω) ≤ c1 u1 W s,r (Ω) N0 hW s,r (Ω)

≤ cN−1 0 hW s,r (Ω) ≤ chW s,r (Ω) .

(11.3.30)

Combining (11.3.29) and (11.3.30) we arrive at (11.3.27). It remains to prove inequality (11.3.28). Since H (h)Z s,r = H (h)W s−1,r (Ω) + H (h)L2 (Ω) , hY s,r = hW s+1,r (Ω) + hW 2,2 (Ω) ,

340

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

it suﬃces to show that H (h)L2 (Ω) ≤ chY s,r .

(11.3.31)

Since X s,r is continuously embedded in C(Ω) and Y s,r is continuously embedded in C 1 (Ω), we have 1 C(Ω) ≤ c1 X s,r ≤ c,

ui C 1 (Ω) ≤ cui Y s,r ≤ c.

(11.3.32)

Thus we get −1 −1 −1 1 ∇(N−1 0 u0 )N0 hL2 (Ω) ≤ 1 C(Ω) ∇(N0 u0 )C(Ω) N0 hL2 (Ω)) −1 ≤ c1 W s,r (Ω) ∇(N−1 0 u0 )W s,r (Ω) N0 hW s,r (Ω)

≤ cN−1 0 u0 W s+1,r (Ω) hW s,r (Ω) ≤ cu0 W s+1,r (Ω) hW s,r (Ω) ≤ chW s,r (Ω) ≤ chY s,r .

(11.3.33)

Next we have −1 div(1 u1 ⊗ (N−1 N− 0 0 h))L2 (Ω) ≤ c1 u1 ⊗ (N0 h)W 1,2 (Ω)

≤ c1 u1 W 1,2 (Ω) hL∞ (Ω) + chW 1,2 (Ω) . (11.3.34) Since the embedding W 1,2 (Ω) → X s,r is continuous, inequalities (11.3.32) imply 1 u1 W 1,2 (Ω) ≤ c1 W 1,2 (Ω) ≤ c. From this and (11.3.34) we obtain N− div(1 u1 ⊗ (N−1 0 0 h))L2 (Ω) ≤ chL∞ (Ω) + chW 1,2 (Ω) ≤ chW 2,2 (Ω) . Combining this with (11.3.33) we arrive at H (h)L2 (Ω) ≤ chY s,r , which along with (11.3.27) yields (11.3.31).

Lemma 11.3.10. Let Condition 11.3.2 be satisﬁed. Then there is c > 0, depending only on s, r, and Ω, such that M (h)W s,r (Ω) ≤ chW s,r (Ω) for all h ∈ W s+1,r (Ω), M (h)X s,r ≤ chY s,r for all h ∈ Y s,r .

(11.3.35) (11.3.36)

Proof. The proof imitates the proofs of Lemmas 11.3.8 and 11.3.9. Recall from (11.3.7) that −1 M (h) = u0 ∇(N−1 0 u0 ) · N0 h.

11.3. Dependence of solutions on N

341

We have −1 −1 −1 u0 ∇(N−1 0 u0 ) · N0 hW s,r (Ω) ≤ u0 ∇(N0 u0 )W s,r (Ω) N0 hW s,r (Ω) −1 ≤ cu0 W s,r (Ω) ∇(N−1 0 u0 )W s,r (Ω) N0 hW s,r (Ω)

≤ cN−1 0 u0 W s+1,r (Ω) hW s,r (Ω) ≤ cu0 W s+1,r (Ω) hW s,r (Ω) ≤ chW s,r (Ω) ,

(11.3.37)

which leads to (11.3.35). Next we have −1 u0 ∇(N−1 0 u0 )N0 hW 1,2 (Ω) −1 ≤ cu0 ∇(N−1 0 u0 )L∞ (Ω) hW 1,2 (Ω) + u0 ∇(N0 u0 )W 1,2 (Ω) hL∞ (Ω)

≤ chW 1,2 (Ω) + u0 ∇(N−1 0 u0 )W 1,2 (Ω) hL∞ (Ω) . On the other hand, inequalities (11.3.32) imply −1 −1 u0 ∇(N−1 0 u0 )W 1,2 (Ω) ≤ c∇(N0 u0 )W 1,2 (Ω) ≤ cN0 u0 W 2,2 (Ω) ≤ cu0 Y s,r ≤ c.

Thus we get −1 u0 ∇(N−1 0 u0 )N0 hW 1,2 (Ω) ≤ chW 1,2 (Ω)) + hL∞ (Ω)) ≤ chY s,r .

Combining this with (11.3.35) we obtain (11.3.36). The next lemma gives estimates for the coeﬃcients of equations (11.3.6).

Lemma 11.3.11. Let Condition 11.3.2 be satisﬁed and bij be given by (11.3.3). Then b11 X s,r + b10 X s,r + b20 X s,r + b31 X s,r + b34 X s,r + b30 X s,r + b22 X s,r + b32 X s,r ≤ cτ, b13 X s,r + b23 X s,r + b21 X s,r ≤ c,

(11.3.38)

where c depends on Ω, U, s, r, and σ. Proof. It follows from (11.3.2) that g0 − 1C 2 (Ω) ≤ cτ 2 .

(11.3.39)

Next, in view of Condition 11.3.2, the functions ϑi ∈ Bτ , and σ > σ ∗ , τ ∈ (0, τ ∗ (σ)] meet all requirements of Lemma 11.2.16. Hence we obtain the following estimates for the functions Ψ and Ψ1 given by (11.1.24d): Ψ[ϑ]X s,r ≤ cστ 2 ,

Ψ1 [ϑ]X s,r ≤ cστ 2 .

(11.3.40)

342

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

Next notice that the function H deﬁned by (11.1.24d) has the derivative H (ϕ) = p ( + ϕ) − p ( ) which belongs to C 2 (R) and vanishes at 0. Hence Φ(ϕ0 , ϕ1 ), deﬁned in (11.3.3), also belongs to C 2 (R2 ) and vanishes at 0. Thus we get Φ(ϕ0 , ϕ1 ) = ϕ0 Φ0 (ϕ0 , ϕ1 ) + ϕ1 Φ1 (ϕ0 , ϕ1 ), where Φi ∈ C 1 (R2 ), i = 0, 1. Since ϑi = (vi , πi , ϕi ), i = 0, 1, belong to the ball Bτ , we have (11.3.41) ϕi X s,r ≤ τ. It follows from this and Lemmas 11.2.7 and 11.2.8 that Φ(ϕ0 , ϕ1 )X s,r ≤ cϕ0 X s,r Φ0 (ϕ0 , ϕ1 )X s,r + cϕ1 X s,r Φ1 (ϕ0 , ϕ1 )X s,r ≤ cτ. (11.3.42) Finally note that by Condition 11.3.2, mi and ζi , i = 0, 1, satisfy all requirements of Theorem 11.2.6 and hence satisfy ζi X s,r + |κ0 | ≤ c,

|mi | ≤ cτ.

(11.3.43)

Applying Lemma 11.2.7 to formulae (11.3.3) for the coeﬃcients bkl and using the estimates (11.3.39)–(11.3.43) we obtain b11 X s,r +b10 X s,r +b20 X s,r +b31 X s,r +b34 X s,r +b30 X s,r ≤ cτ. (11.3.44) Since λ−1 ∈ (0, τ 2 ], we have b22 X s,r + b32 X s,r ≤ cλ−1 ≤ cτ.

(11.3.45)

On the other hand, it follows from (11.3.3) that the coeﬃcients b13 , b21 , b23 are not small and admit the estimate b13 X s,r + b21 X s,r + b23 X s,r ≤ c. Combining this with (11.3.44) and (11.3.45) we arrive at (11.3.38).

Step 3. Truncated problem. It follows from Lemmas 11.3.7–11.3.11 that some terms in equations (11.3.6) are small while τ is small. If we neglect these terms, we come to the following truncated problem: Δh − ∇g = H − ζ1 ∇υ div h = ΠG

in Ω,

in Ω,

−u0 ∇ς + σς = b21 g + F u0 ∇υ + συ = M in Ω,

in Ω,

(11.3.46a) (11.3.46b)

h = 0 on ∂Ω, g − Πg = 0,

ς = 0 on Σout , υ = 0 on Σin , l = κ0 (b13 ς + b23 g) dx + e, Ω

where the coeﬃcients bij and the constant κ0 are deﬁned by (11.3.3).

(11.3.46c)

11.3. Dependence of solutions on N

343

Lemma 11.3.12. Let Condition 11.3.2 be satisﬁed. Then there are constants σ3 ≥ σ ∗ and c, depending only on Ω, U, s, r, with the following property. If σ > σ3 , then for every f = (H, G, F, M, e) ∈ U s,r , problem (11.3.6) has a solution h = (h, g, ς, υ, l) ∈ V s,r which satisﬁes hV s,r ≤ cfU s,r .

(11.3.47)

hF s,r ≤ cfE s,r .

(11.3.48)

If, in addition, f ∈ E s,r , then

Proof. Observe that the truncated system (11.3.12) is in triangular form. Hence we can solve it step by step starting with the boundary value problem for υ. By (11.3.32), we have u0 C 1 (Ω) ≤ cu0 Y s,r ≤ c. It follows from this and Condition 11.3.2 that Ω, u0 and s, r meet all requirements of Lemma 11.2.11 and Proposition 11.2.12. Applying Lemma 11.2.11 we ﬁnd that there is σ3 > 1, depending only on Ω, U, and s, r, with the following property. For every σ > σ3 and M ∈ W s,r (Ω) the boundary value problem u0 ∇υ + συ = M

in Ω,

υ = 0 on Σin

has a solution satisfying υW s,r (Ω) ≤ cM W s,r (Ω) .

(11.3.49)

If, in addition, M ∈ X s,r , then by Proposition 11.2.12, υX s,r ≤ cM X s,r .

(11.3.50)

Here c depends only on Ω, U, and s, r. Obviously we can take σ3 ≥ σ ∗ . Let us turn to the Stokes equations (11.3.46a). Notice that in view of Condition (11.3.2), the functions (ϑi , ζi ) solve Problem 11.1.9 and meet all requirements of Theorem 11.2.6. In particular, we have ζ1 X s,r ≤ c. Since X s,r = W s,r (Ω) ∩ W 1,2 (Ω), we can apply Lemma 11.3.7 to obtain ζ1 ∇υW s−1,r (Ω) ≤ cζ1 W s,r (Ω) υW s,r (Ω) ≤ cζ1 X s,r υW s,r (Ω) ≤ cυW s,r (Ω) .

(11.3.51)

Next, since ζC(Ω) ≤ cζ1 X s,r ≤ c, we have ζ1 ∇υL2 (Ω) ≤ cζ1 C(Ω) ∇υL2 (Ω) ≤ cυW 1,2 (Ω) ≤ cυX s,r . Combining this with (11.3.51) and noting that Z s,r = W 1−s,r (Ω)∩L2 (Ω) we arrive at ζ1 ∇υZ s,r ≤ cυX s,r . (11.3.52)

344

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

Inequalities (11.3.49)–(11.3.52) imply ζ1 ∇υW s−1,r (Ω) ≤ cM W s,r (Ω) ,

ζ1 ∇υZ s,r ≤ cM X s,r .

(11.3.53)

In view of Lemma 1.7.9, for H − ζ1 ∇υ ∈ W s−1,r (Ω) and G ∈ W s,r (Ω), equations (11.3.46a) have a solution (h, g) ∈ W s+1,r (Ω) × W s,r (Ω) satisfying conditions (11.3.46c). This solution satisﬁes hW s+1,r (Ω) + gW s,r (Ω) ≤ c(H − ζ1 ∇υW s−1,r (Ω) + GW s,r (Ω) ). If, in addition, H ∈ Z s,r and G ∈ X s,r , then by Lemma 11.2.9, hY s,r + gX s,r ≤ c(Ω, r, s)(H − ζ1 ∇υZ s,r + GX s,r (Ω) ). From this and (11.3.53) we obtain hW s+1,r (Ω) + gW s,r (Ω) ≤ c(HW s−1,r (Ω) + GW s,r (Ω) + M W s,r (Ω) ), hY s,r + gX s,r ≤ c(HZ s,r + GX s,r + M X s,r ).

(11.3.54) (11.3.55)

Now the functions h, g, υ are deﬁned and we can solve the transport equations (11.3.46b) for ς. Notice that, by Condition 11.3.2, U and Ω satisfy Condition 11.2.5. In particular, −U and Ω also satisfy Condition 11.2.5. Hence −u0 and Ω meet all requirements of Lemma 11.2.11 and Proposition 11.2.12 with Σin replaced by Σout . Hence we can choose σ3 , depending only on Ω, U, and s, r, such that for σ > σ3 and F, g ∈ W s,r (Ω) (respectively F, g ∈ X s,r ), the boundary value problem −u0 ∇ς + σς = b21 g + F in Ω, ς = 0 on Σout has a solution satisfying ςW s,r (Ω) ≤ cF W s,r (Ω) + cb21 gW s,r (Ω) ≤ cF W s,r (Ω) + cgW s,r (Ω) ≤ c(F W s,r (Ω) + HW s−1,r (Ω) + GW s,r (Ω) + M W s,r (Ω) ), (11.3.56) ςX s,r ≤ cF X s,r + cb21 gX s,r ≤ cF X s,r + cgX s,r ≤ c(HY s,r + F X s,r + GX s,r + M X s,r ).

(11.3.57)

Thus we have deﬁned the functions h, g, ς and υ. It remains to note that the constant l is completely determined by ς and g. Moreover, relation (11.3.46c) along with estimates (11.3.38) and (11.3.43) yields |l| ≤ c(gL1 (Ω) + ςL1 (Ω) ). Combining this with estimates (11.3.49), (11.3.50) and (11.3.54)–(11.3.57) we obtain (11.3.47) and (11.3.48).

11.3. Dependence of solutions on N

345

Step 4. Proof of Theorem 11.3.4. We prove Theorem 11.3.4 by applying the contraction mapping principle. To this end we write the equations and boundary conditions (11.3.6) in the form of an operator equation. Introduce the linear diﬀerential operator ⎛ ⎞ ⎛ ⎞ h F (h) + Re H (h) − ς∇ϕ ⎜ g ⎟ ⎜ Π(b22 g + b12 ς + b32 l) ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ V : ⎜ ς ⎟ → ⎜ ⎜ Re M (h) + b11 ς + b31 l ⎟ . ⎝ υ ⎠ ⎝ ⎠ b34 l l 0 Lemma 11.3.13. Let Condition 11.3.2 be satisﬁed. Then there is a constant c, depending only on Ω, U, and s, r, such that for h = (h, g, ς, υ, l), V hU s,r ≤ cτ hV s,r (Ω) , V hEs,r ≤ cτ hF s,r .

(11.3.58) (11.3.59)

Proof. Let us prove (11.3.58). Set h = (h, g, ς, υ, l) = Vh. Our ﬁrst task is to estimate h. By Condition 11.3.2, we have ϑi ∈ Bτ and hence ϕ1 X s,r ≤ τ . Since the embeddings X s,r → W s,r (Ω) → C(Ω) are continuous, ϕ1 W s,r (Ω) ≤ ϕ1 X s,r ≤ cτ

and

ςC(Ω) ≤ ςX s,r .

(11.3.60)

Application of Lemma 11.3.7 gives ς∇ϕ1 W s−1,r (Ω) ≤ cςW s,r (Ω) ϕ1 W s,r (Ω) ≤ cτ ςW s,r (Ω) .

(11.3.61)

Since X s,r = W s,r (Ω) ∩ W 1,2 (Ω), inequality (11.3.60) yields ς∇ϕ1 L2 (Ω) ≤ cςC(Ω) ∇ϕ1 L2 (Ω) ≤ cϕ1 X s,r ςX s,r ≤ cτ ςX s,r . Combining this with (11.3.61) and noting that Z s,r = W 1−s,r (Ω)∩L2 (Ω) we arrive at (11.3.62) ς∇ϕ1 Z s,r ≤ cςX s,r . Since Re ∈ (0, τ 2 ] and τ ≤ 1 estimates (11.3.22), (11.3.27) and (11.3.61) imply hW s−1,r (Ω) ≤ F (h)W s−1,r (Ω) + Re H (h)W s−1,r (Ω) + ς∇ϕ1 W s−1,r (Ω) ≤ cτ hW s+1,r (Ω) + cτ ςW s,r (Ω) .

(11.3.63)

Finally, estimates (11.3.23), (11.3.28) and (11.3.62) yield hZ s,r ≤ F (h)Z s,r + Re H (h)Z s,r + ς∇ϕ1 Z s,r ≤ cτ hY s,r + cτ ςX s,r .

(11.3.64)

346

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

Let us estimate g. Recalling that W s,r (Ω) and X s,r are Banach algebras and applying Lemma 11.3.11 we obtain gW s,r (Ω) ≤ c(b22 W s,r (Ω) gW s,r (Ω) + b12 W s,r (Ω) ςW s,r (Ω) + |l|b22 W s,r (Ω) ) (11.3.65) ≤ cτ (gW s,r (Ω) + ςW s,r (Ω) + |l|) and gX s,r ≤ c(b22 X s,r gX s,r + b12 X s,r ςX s,r + |l| b22 X s,r ) ≤ cτ (gX s,r + ςX s,r + |l|). (11.3.66) Next applying Lemmas 11.3.10 and 11.3.11 we get ςW s,r (Ω) ≤ c(Re M (h)W s,r (Ω) + b11 W s,r (Ω) ςW s,r (Ω) + |l| b31 W s,r (Ω) ) ≤ cτ (hW s+1,r (Ω) + ςW s,r (Ω) + |l|) (11.3.67) and ςX s,r ≤ c(Re M (h)X s,r + b11 X s,r ςX s,r + |l| b31 X s,r ) (11.3.68) ≤ cτ (hY s,r + ςX s,r + |l|). Finally, from Lemma 11.3.11 we obtain υW s,r (Ω) ≤ c|l| b34 W s,r (Ω) ≤ cτ |l|, υX s,r ≤ c|l| b34 X s,r ≤ cτ |l|.

(11.3.69)

Combining estimates (11.3.63)–(11.3.69) we obtain (11.3.58) and (11.3.59).

We are now in a position to complete the proof of Theorem 11.3.4. Set σc = min{σ ∗ , σ3 }, where σ ∗ is a constant from Condition 11.3.2 and σ3 is given by Lemma 11.3.12. Let σ satisfy Condition 11.3.2 and σ > σc . Denote by U the mapping which assigns to every vector (H, G, F, M, l) the solution (h, g, ς, υ, l) of the truncated problem (11.3.46). Thus the transposed problem (11.3.6) can be written in the form of the operator equation (Id − U V) h = U f.

(11.3.70)

In view of Lemmas 11.3.12 and 11.3.13, the operators U : U s,r → V s,r ,

U V : V s,r → V s,r

are continuous. Moreover, inequalities (11.3.47) and (11.3.58) imply UL (U s,r →V s,r ) ≤ cU ,

U VL (V s,r →V s,r ) ≤ cV τ.

(11.3.71)

Next, it also follows from Lemmas 11.3.12 and 11.3.13 that the operators U : E s,r → F s,r ,

U V : F s,r → F s,r

11.3. Dependence of solutions on N

347

are continuous. Inequalities (11.3.48) and (11.3.59) imply UL (E s,r →F s,r ) ≤ cE ,

U VL (F s,r →F s,r ) ≤ cF τ.

(11.3.72)

The constants cU , cV , cE , and cF are completely determined by the constant c in inequalities (11.3.47), (11.3.58), (11.3.59) and depend only on Ω, U , s, r, and σ. Notice that, by Condition 11.3.2, the parameter τ satisﬁes 0 < τ ≤ τ ∗ (σ) and τ ≤ τ0 , where τ ∗ (σ) depends on Ω, U , s, r, and σ, while τ0 is an absolute constant. Now choose a constant τc such that 0 < τc ≤ min{τ0 , τ ∗ (σ)},

τc cV + τc cF < 1/2.

It follows from (11.3.71), (11.3.72) that for τ ∈ (0, τc ] the norms U VL (V s,r →V s,r ) andU VL (F s,r →F s,r ) are less than 1/2. By the contraction mapping principle, for such τ and any f ∈ U s,r (resp. f ∈ E s,r ), the operator equation (11.3.70) has a unique solution h ∈ V s,r (resp. h ∈ F s,r ). This solution satisﬁes the estimates (11.3.58), (11.3.59). It remains to note that the operator equation (11.3.70) is equivalent to the transposed problem (11.3.6).

11.3.3 Special dual space Recall that our goal is to establish well-posedness for the linear boundary value problem (11.3.2) in dual spaces. In this section we consider in detail the properties of the Banach space dual to W s,r (Ω). For every 1 < r < ∞ and v ∈ Lr (Ω) deﬁne the functional v(x)u(x) dx, u ∈ Lr (Ω), r = r/(r − 1). Lv (u) = v, u = Ω

By the Hölder inequality, Lv ∈ Lr (Ω) . Moreover, in view of Theorem 1.2.27, Lv Lr (Ω) = vLr (Ω) , and the mapping v → Lv deﬁnes an isometry of Lr (Ω) onto Lr (Ω) . We will identify Lr (Ω) and Lr (Ω) . Since W s,r (Ω), s > 0, is con r tinuously embedded into L (Ω), Lv is a continuous functional on W s,r (Ω) with the ﬁnite norm Lv (W s,r (Ω)) :=

|Lv (u)| . u W s,r (Ω) (Ω)\{0}

sup

u∈W

s,r

Deﬁnition 11.3.14. For every s ∈ (0, 1] and r ∈ (1, ∞), the Banach space W−s,r (Ω) is the completion of Lr (Ω) in the norm vW−s,r (Ω) = Lv (W s,r (Ω)) . Lemma 11.3.15. Let Ω be a bounded domain in Rd with C 1 boundary and s ∈ (0, 1], r ∈ (1, ∞). Then W−s,r (Ω) is algebraically and topologically isomorphic to the

348

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

dual space (W s,r (Ω)) and can be identiﬁed with it. Moreover, W−s,r (Ω) is the interpolation space

W−s,r (Ω) = [Lr (Ω), W−1,r (Ω)]s,r = ([Lr (Ω), W 1,r (Ω)]s,r ) .

(11.3.73)

Proof. By deﬁnition, W−s,r (Ω) is the completion of the linear space L = Lr (Ω) of functionals Lv , v ∈ Lr (Ω), in the (W s,r (Ω)) -norm. Hence W−s,r (Ω) is a closed subspace of (W s,r (Ω)) . To prove that W−s,r (Ω) coincides with (W s,r (Ω)) , it suﬃces to show that L is dense in (W s,r (Ω)) . We start with the case s = 1. If L is not dense in (W 1,r (Ω)) , then there exists T ∈ (W 1,r (Ω)) such that T = 0 and T (Lv ) = 0 for all v ∈ Lr (Ω). Since 1,r the space W (Ω) is reﬂexive (see [2, Thm. 3.5]), there is ϕ ∈ W 1,r (Ω) such that

T (f ) = f (ϕ) for all f ∈ (W 1,r (Ω)) . (11.3.74)

ϕW 1,r (Ω) = T (W 1,r (Ω)) = 0 and Thus we get T (Lv ) = Lv (ϕ) =

vϕ dx = 0 for all v ∈ Lr (Ω). Ω

Hence ϕ = 0, which contradicts (11.3.74). It follows that L is dense in (W 1,r (Ω)) . Hence W−1,r (Ω) = (W 1,r (Ω)) . Let us turn to the case s ∈ (0, 1). Notice that W 1,r (Ω) is dense in Lr (Ω). Therefore, we can apply Lemma 1.1.12 to obtain

([Lr (Ω), W 1,r (Ω)]s,r ) = [Lr (Ω) , (W 1,r (Ω)) ]s,r .

(11.3.75)

Since, [Lr (Ω), W 1,r (Ω)]s,r = W s,r (Ω), we have

(W s,r (Ω)) = [Lr (Ω) , (W 1,r (Ω)) ]s,r .

(11.3.76)

Next, the inclusion W 1,r (Ω) ⊂ Lr (Ω) implies Lr (Ω) ⊂ (W 1,r (Ω)) . From this and Lemma 1.1.11 we ﬁnd that L = Lr (Ω) is dense in [Lr (Ω) , (W 1,r (Ω)) ]s,r and hence in (W s,r (Ω)) . Thus (W s,r (Ω)) = W−s,r (Ω). It remains to note that relations (11.3.73) follow from (11.3.75) and (11.3.76).

11.3.4 Very weak solutions This step is crucial for further analysis. We replace equations (11.3.2) by an integral identity, which leads to the notion of a very weak solution to (11.3.2). For s ∈ (0, 1), r ∈ (1, ∞), denote by ·, ·1 the duality pairing between W s−1,r (Ω) and W01−s,r (Ω), and by ·, ·0 the duality pairing between W−s,r (Ω) and W s,r (Ω). Deﬁnition 11.3.16. Let Condition 11.3.2 be satisﬁed and σ > σc , τ ∈ (0, τc (σ)], where σc and τc (σ) are given by Theorem 11.3.4. The vector ﬁeld w ∈ W01−s,r (Ω),

11.3. Dependence of solutions on N

349

functionals (ω, ψ, ξ) ∈ W−s,r (Ω)3 and constant n are said to be a very weak solution to problem (11.3.2) if whenever (H, G, F, M, e) ∈ W s−1,r (Ω)×(W s,r (Ω))3 ×R, H, w1 + ω, G0 + ψ, F 0 + ξ, M0 + κ0−1 ne h · D + g b20 d + ς b10 d + υ σd + l b30 d dx, (11.3.77) = Ω

where (h, g, ς, υ, l) is the solution to the transposed problem (11.3.6). Condition 11.3.17. • The C 3 surface ∂Ω and the vector ﬁeld U ∈ C 3 (Ω) satisfy Condition 11.2.5. • The exponents s and r satisfy 1/2 < s < 1,

r < ∞,

2s − 3r−1 < 1,

sr > 3,

(1 − s)r > 3.

• σc > 1 and τc (σ) ∈ (0, 1] satisfy the hypotheses of of Theorem 11.3.4. In particular τc (σ) ≤ τ0 , where τ0 is given by Lemma 11.2.15. • σ > σc and τ ∈ (0, τc (σ)]. • We have

λ−1 , Re ∈ (0, τ 2 ],

Ni − IC 2 (Ω) ≤ τ 2 .

• ϑi = (vi , πi , ϕi ) ∈ Bτ , ζi ∈ W s,r (Ω), mi ∈ R, i = 0, 1, are solutions to Problem 11.1.9 and satisfy inequalities (11.2.7). • ui = u + vi and i = + ϕi . We set w = v 0 − v1 ,

ω = π0 − π1 ,

ψ = ϕ0 − ϕ1 ,

ξ = ζ0 − ζ1 ,

n = m 0 − m1 .

The following theorem is the main result of this section. Theorem 11.3.18. Assume that Condition 11.3.17 is satisﬁed. Then (w, ω, ψ, ξ, n) ∈ W01−s,r (Ω) × W−s,r (Ω)3 × R is a very weak solution to problem (11.3.2) and wW 1−s,r (Ω) + ωW−s,r (Ω) + ψW−s,r (Ω) + ξW−s,r (Ω) + |n| 0

≤ c(DL1 (Ω) + dL1 (Ω) ), (11.3.78) where c depends on Ω, U, s, r, and σ. Proof. The proof falls into two steps.

350

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

Step 1. First we deduce the Green formula for problem (11.3.2). Observe that under the assumptions of Theorem 11.3.18 the embeddings Y s,r → W s+1,r (Ω) → C 1 (Ω),

X s,r → W s,r (Ω) → C(Ω)

(11.3.79)

are continuous. Hence ui , vi , w ∈ Y s,r = W s+1,r (Ω) ∩ W 2,2 (Ω) are continuously diﬀerentiable and belong to W 2,2 (Ω). In its turn, the functions (i , πi , ϕi , ζi ) and (ω, ψ, ξ) belong to X s,r . Hence they are continuous and belong to W 1,2 (Ω). It follows that equations (11.3.2) are satisﬁed in the strong sense. Next choose h ∈ Y s,r , (g, ς, υ) ∈ X s,r , and l ∈ R such that h = 0 on ∂Ω,

ς = 0 on Σout ,

υ = 0 on Σin .

Notice that h is continuously diﬀerentiable and belongs to W 2,2 (Ω), while g, ς, and υ are continuous and belong to W 1,2 (Ω). Multiplying the Stokes equations (11.3.2b) by h and g respectively, integrating the result by parts over Ω, and combining the integral identities obtained we arrive at w · (Δh − ∇g) dx + ω div h = h · A0 (w) + Re C (ψ, w) + D dx Ω Ω Ω g(b21 ψ + b22 ω + b23 n + b20 d) dx. + Ω

(11.3.80)

It follows from (11.3.4) and (11.3.7) that h · (A0 (w) + Re C (ψ, w)) dx = w · (F (h) + Re H (h)) dx + ψ Re M (h) dx. Ω

Ω

Ω

Inserting this into (11.3.80) we obtain w · Δh − ∇g − F (h) − Re H (h) dx + ω(div h − b22 g) dx Ω Ω ψ(Re M (h) + b21 g) dx − (nb23 g + b20 d g) dx = h · D. (11.3.81) − Ω

Ω

Ω

Next, multiplying the transport equation (11.3.2a) by ς, integrating the result by parts over Ω, and noting that ψς vanishes at ∂Ω we arrive at ψ − div(u0 ς) + σς − b11 ς dx + w · (ς∇ϕ1 ) dx Ω Ω − (ωb12 ς + nb13 ς + b10 dς) dx = 0. (11.3.82) Ω

Multiplying the transport equation (11.3.2c) by υ, integrating the result by parts over Ω, and noting that ξυ vanishes at ∂Ω we obtain ξ u0 ∇υ + συ dx + w · (ζ1 ∇υ) dx − σdυ dx = 0. (11.3.83) Ω

Ω

Ω

11.3. Dependence of solutions on N

351

Finally, multiplying the formula for n in (11.3.2d) by κ0−1 (meas Ω)−1 l and integrating over Ω, we arrive at κ0−1 nl − (ψb31 l + ωb32 l + ξb34 l + b30 dl) dx = 0. (11.3.84) Ω

Adding (11.3.81)–(11.3.84) we arrive at the Green formula for problem (11.3.2):

w · Δh − ∇g − F (h) − Re H (h) + ς∇ϕ1 + ζ1 ∇υ dx Ω + ω(div h − b22 g − b12 ς − b32 l) dx Ω ψ − div(u0 ς) + σς − Re M (h) − b21 g − b11 ς − b31 l dx + Ω ξ u0 ∇υ + συ − b34 l dx + n κ0−1 l − (b23 g + b13 ς) dx + Ω Ω = (h · D + gb20 d + ςb10 d + υσd + lb30 d) dx. (11.3.85) Ω

Notice that h is continuously diﬀerentiable and vanishes on ∂Ω, hence Π div h = div h. On the other hand, in view of (11.3.2d), we have Πω = ω. From this and the symmetry of the operator Π given by (1.7.6) we conclude that Ω

ω(div h − b22 g − b12 ς − b32 l) dx ω div h − Π(b22 g + b12 ς + b32 l) dx. (11.3.86) = Ω

Step 2.

Recall the notation for the Banach spaces

U s,r = W s−1,r (Ω) × (W s,r (Ω))3 × R, E s,r = Z s,r × (X s,r )3 × R,

V s,r = W s+1,r (Ω) × (W s,r (Ω))3 × R,

F s,r = Y s,r × (X s,r )3 × R.

Introduce also the temporary notation f = (H, G, F, M, e),

h = (h, g, ς, υ, l).

Now our task is to prove that equality (11.3.77) holds for every f ∈ U s,r . The proof is based on the following Lemma 11.3.19. Under the assumptions of Theorem 11.3.18, Lr (Ω) × (W 1,r (Ω))3 × R

is contained in E s,r and is dense in U s,r .

352

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

Proof. Since r > 2, we have the inclusions Lr (Ω) ⊂ W s−1,r (Ω)∩L2 (Ω) = Z s,r

and

W 1,r (Ω) ⊂ W s,r (Ω)∩W 1,2 (Ω) = X s,r .

Therefore, Lr (Ω) × (W 1,r (Ω))3 × R is contained in E s,r . Let us prove that it is dense in U s,r . Notice that W s−1,r (Ω) is obtained by completing Lr (Ω) in the dual norm deﬁned by (1.5.5). Hence Lr (Ω) is dense in W s−1,r (Ω). It remains to prove that W 1,r (Ω) is dense in W s,r (Ω). By deﬁnition, W s,r (Ω) is the interpolation space [Lr (Ω), W 1,r (Ω)]s,r . Moreover, we have W 1,r (Ω) ⊂ Lr (Ω). Hence Lemma 1.1.11 shows that W 1,r (Ω) is dense in W s,r (Ω). Therefore, Lr (Ω) × (W 1,r (Ω))3 × R is dense in U s,r . Now choose f = (H, G, F, M, e) ∈ U s,r . In view of Lemma 11.3.19, there is a sequence fn = (Hn , Gn , Fn , Mn , en ) ∈ Lr (Ω) × (W 1,r (Ω))3 × R,

n ≥ 1,

such that fn → f

in U s,r

as n → ∞.

(11.3.87)

It follows from Condition 11.3.17 that the matrices Ni , parameters Re, λ and σ, τ , and functions ϑi satisfy all hypotheses of Theorem 11.3.4. Hence problem (11.3.6) with the given f and fn has solutions h = (h, g, ς, υ, l) ∈ V s,r ,

hn = (hn , gn , ςn , υn , ln ) ∈ F s,r ⊂ V s,r ,

satisfying hV s,r ≤ cfU s,r ,

h − hn V s,r ≤ cf − fn U s,r → 0 as n → ∞,

(11.3.88)

where c depends only on Ω, U, s, r, and σ. It follows from equations (11.3.6) that the Green formula (11.3.85) holds for hn . Substituting hn into (11.3.85), using identity (11.3.86), and recalling equations (11.3.6) we obtain Ω

(w · Hn + ωΠGn + ψFn + ξMn ) dx + κ0−1 nen = (hn · D + gn b20 d + ςn b10 d + υn σd + ln b30 d) dx. Ω

Since the operator Π is symmetric and Πω = ω we can rewrite this identity in the equivalent form Ω

(w · Hn + ωGn + ψFn + ξMn ) dx + κ0−1 nen = (hn · D + gn b20 d + ςn b10 d + υn σd + ln b30 d) dx. (11.3.89) Ω

11.3. Dependence of solutions on N

353

Next we show that the integral on the left hand side of this equality may be replaced by a duality pairing. To do this we have to prove that (Hn , ω, ψ, ξ) and (w, Gn , Fn , Mn ) belong to dual spaces. Observe that Condition 11.3.17 implies r > 3 and hence 1 < r = r/(r − 1) < r. It follows that

ω, ψ, ξ ∈ W s,r (Ω) ⊂ Lr (Ω) ⊂ W−s,r (Ω) and which yields

Hn ∈ Lr (Ω) ⊂ W s−1,r (Ω),

(Hn , ω, ψ, ξ) ∈ W s−1,r (Ω) × (W−s,r (Ω))3 .

(11.3.90)

Next, embedding (11.3.79) implies that w ∈ Y belongs to C (Ω). Moreover, equations (11.3.2) imply that w vanishes on ∂Ω and hence w ∈ W01,r (Ω). Re call that W01−s,r (Ω) is the interpolation space [Lr (Ω), W01,r (Ω)]1−s,r , and that W01,r (Ω) ⊂ Lr (Ω). Lemma 1.1.11 shows that W01,r (Ω) ⊂ W01−s,r (Ω). It follows that w ∈ W01−s,r (Ω). On the other hand, Gn , Fn , Mn ∈ W s,r (Ω). Thus s,r

(w, Gn , Fn , Mn ) ∈ W01−s,r (Ω) × (W s,r (Ω))3 .

1

(11.3.91)

Obviously, W s−1,r (Ω) × (W−s,r (Ω))3 = (W01−s,r (Ω) × (W s,r (Ω))3 ) . From this and (11.3.90), (11.3.91) we conclude that the integral on the left hand side of (11.3.89) can be replaced by a duality pairing. Thus we get Hn , w1 + ω, Gn 0 + ψ, Fn 0 + ξ, Mn 0 + κ0−1 nen hn · D + gn b20 d + ςn b10 d + υn σd + ln b30 d dx. (11.3.92) = Ω

Now our task is to pass to the limit as n → ∞ in (11.3.92) and, by doing so, to derive identity (11.3.77). Relation (11.3.87) implies Hn → H in W s−1,r (Ω) and

(Gn , Fn , Mn ) → (G, F, M ) in W s,r (Ω),

which leads to Hn , w1 + ω, Gn 0 + ψ, Fn 0 + ξ, Mn 0 + κ0−1 nen → H, w1 + ω, G0 + ψ, F 0 + ξ, M 0 + κ0−1 ne. In view of (11.3.79), the embedding V s,r → C 1 (Ω) × (C(Ω))3 × R is continuous. It follows from this and (11.3.88) that hn , gn , ςn υn converge to h, g, ς, υ uniformly in Ω. Since D, d ∈ L2 (Ω), the Lebesgue dominated convergence theorem yields Ω

hn · D + gn b20 d + ςn b10 d + υn σd + ln b30 d dx h · D + g b20 d + ς b10 d + υ σd + l b30 d dx. → Ω

354

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

Letting n → ∞ in (11.3.92) we arrive at (11.3.77). Hence (w, ω, ψ, ξ, n) is a very weak solution to problem (11.3.2). It remains to prove estimate (11.3.78). It suﬃces to show that for all f = (H, G, F, M, e) ∈ U s,r , H, w1 + ω, G0 + ψ, F 0 + ξ, M 0 + κ −1 nen 0

≤ c(DL1 (Ω) + dL1 (Ω) ) fU s,r . (11.3.93) Notice that in view of Lemma 11.3.11 and embedding (11.3.79), the coeﬃcients b23 and b13 are bounded by a constant depending only on Ω, U, s, r, and σ. From this and (11.3.77) we obtain H, w1 + ω, G0 + ψ, F 0 + ξ, M 0 + κ −1 ne 0

≤ c(DL1 (Ω) + dL1 (Ω) ) hC(Ω)6 ×R , (11.3.94) where h = (h, g, ς, υ, l) is the solution to problem (11.3.6) corresponding to f = (H, G, F, M, e). Since the embedding V s,r → C 1 (Ω) × (C(Ω))3 × R is continuous, inequality (11.3.88) implies hC(Ω)6 ×R ≤ chV s,r ≤ cfU s,r .

Inserting this into (11.3.94) we arrive at (11.3.93).

11.3.5 Uniqueness and existence. Main theorem In this section we assemble the obtained results on existence, uniqueness, and compactness properties of solutions to Problem 11.1.9 in the form of the following general theorem. Theorem 11.3.20. Assume that the surface Σ and the given vector ﬁeld U satisfy Condition 11.2.5. Furthermore, let numbers r and s satisfy 1/2 < s < 1,

r < ∞,

2s − 3r−1 < 1,

sr > 3,

(1 − s)r > 3.

(11.3.95)

Then there exists σ > 1, depending only on U, Ω and s, r, with the following property. For every σ > σ there are positive τ (σ) and c, depending only on U, Ω, r, s, and σ, such that whenever τ ∈ (0, τ ],

λ−1 , Re ∈ (0, τ 2 ],

N − IC 2 (Ω) ≤ τ 2 ,

(11.3.96)

Problem 11.1.9 has a unique solution ϑ ∈ Bτ , ζ ∈ X s,r , m ∈ R. The auxiliary function ζ and the constants κ, m satisfy ζX s,r + |κ| ≤ c,

|m| ≤ cτ < 1.

(11.3.97)

The totality of solutions {ϑ, ζ, m} to Problem 11.1.9 corresponding to all matrixvalued functions N in the ball B(τ 2 ) = {N : I − NC 2 (Ω) ≤ τ 2 } is a relatively compact subset of W s+1,r (Ω) × W s,r (Ω)3 × R.

11.4. Shape derivative

355

Moreover, if ϑi ∈ Bτ , ζi , mi , i = 0, 1, are solutions to Problem 11.1.9 with Ni ∈ B(τ 2 ), then the transposed problem (11.3.6) has a unique solution satisfying inequality (11.3.11). Proof. Set σ = max{σ ∗ , σc } and τ (σ) = min{τ ∗ (σ), τc (σ), τ0 }, where the constants σ ∗ and τ ∗ (σ) are given by Theorem 11.2.6, σc and τc (σ) are given by Theorem 11.3.4, and τ0 is given by Lemma 11.2.15. For such a choice, the hypotheses of Theorem 11.3.20 are stronger than those of Theorems 11.2.6, 11.3.4 and 11.3.18. Applying Theorem 11.2.6 we conclude that if σ > σ and if τ , λ, Re, N satisfy condition (11.3.96) and τ ∈ [0, τ ), then Problem 11.1.9 has a solution ϑ ∈ Bτ , ζ ∈ X s,r , m ∈ R, which satisﬁes estimate (11.3.97). Let us prove that this solution is unique. If for some N, the problem has two solutions (ϑi , ζi , mi ), i = 0, 1, with ϑi ∈ Bτ , then the corresponding diﬀerences w, ψ, ω, ξ satisfy all hypotheses of Theorem 11.3.78 with d = 0 and D = 0. Therefore, the elements w, ψ, ω, ξ are all 0. Hence the solutions (ϑi , ζi , mi ) coincide. It follows that the set ΘN ⊂ Bτ ×X s,r ×R of solutions (ϑ, ζ, m) corresponding to a matrix N has only one element. Applying Theorem 11.2.6 we conclude that the totality N∈B(τ 2 ) ΘN of all such solutions is relatively compact in the product W s+1,r (Ω) × (W s,r (Ω))3 × R.

11.4 Shape derivative 11.4.1 Preliminaries. Results In the previous sections we developed the theory of the perturbation problem 11.1.9. In our analysis the structure of the matrix N did not play any role, and the results obtained hold true for an arbitrary smooth N close to the unit matrix. But in the original formulation of the domain dependence problem given in Section 11.1, the matrix N is determined by the family of deformations x → x + εT(x) of the unperturbed ﬂow domain Ω. By (11.1.4), N is the adjugate matrix of the Jacobi matrix of the mapping Id + εT, N(ε) = g(ε)(I + εD T)−1 ,

g(ε) = det(I + εD T).

(11.4.1)

In this framework, the solution to Problem 11.1.9 becomes a function of the parameter ε. Investigating the diﬀerentiability of this function is the main goal of sensitivity analysis. In this section we consider this question in detail and show that the relevant derivative exists in a weak sense. In order to give a rigorous formulation of this fact we need some preliminary results which are given in the following auxiliary lemmas. Lemma 11.4.1. There are ε1 > 0 and c, c1 , depending only on TC 2 (Ω) , such that for every ε ∈ [0, ε1 ], N(ε)±1 = I ± εD + ε2 D± ,

g(ε)±1 = 1 ± ε div T + ε2 g± ,

(11.4.2)

356

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

where D = div T I − DT and D± C 2 (Ω) + g± C 2 (Ω) ≤ c.

(11.4.3)

The deviations N(ε)±1 − I and g(ε)±1 − 1 satisfy N(ε)±1 − IC 2 (Ω) + g(ε)±1 − 1C 2 (Ω) ≤ c1 ε.

(11.4.4)

Proof. Direct calculations show that N(ε) = I + εD + ε2 D+ where D+ = [∂x2 T × ∂x3 T, ∂x3 T × ∂x1 T, ∂x1 T × ∂x2 T] . It follows that D+ satisﬁes (11.4.3). Next we have N(ε)−1 = 1 − εD + ε2 D−

where

D− = −D+ +

(−1)k εk (D + εD+ )k+2 .

k≥0

Obviously, there is ε0 , depending on TC 2 (Ω) , such that the series above converges in C 2 (Ω) for all ε ∈ [0, ε0 ]. This leads to estimate (11.4.3) for D− . Next, the determinant g(ε) has a representation g(ε) = 1 + ε div T + ε2 g+ , where g+ is a cubic polynomial of the entries of the Jacobi matrix DT. Arguing as before we obtain the estimates (11.4.3) for the functions g± . It remains to note that (11.4.4) is a straightforward consequence of (11.4.3). Lemma 11.4.1 implies that for suﬃciently small ε the matrices N(ε) satisfy the assumption of Theorem 11.3.20. Hence we can apply that theorem to analyze solutions to Problem 11.1.9. To be more precise we formulate explicit conditions on the ﬂow domain and given data, which provide the existence and uniqueness of solutions to Problem 11.1.9 with the matrix N = N(ε). We assume that the following condition will be satisﬁed throughout the section. Condition 11.4.2. • The C 3 surface ∂Ω and the vector ﬁeld U ∈ C 3 (Ω) satisfy Condition 11.2.5. • The exponents s and r satisfy 1/2 < s < 1,

r < ∞,

2s − 3r−1 < 1,

sr > 3,

(1 − s)r > 3.

• σ > 1 and τ (σ) ∈ (0, 1] satisfy the hypotheses of of Theorem 11.3.20. • σ > σ and τ ∈ (0, τc (σ)], τc (σ) is given by Theorem 11.3.4, and λ−1 , Re ∈ (0, τ 2 ], 2 • ε ∈ [0, ε ], where ε = min{c−1 1 τ , ε1 } with the constants c1 and ε1 given by Lemma 11.4.1.

11.4. Shape derivative

357

Lemma 11.4.3. Let Condition 11.4.2 be satisﬁed. Then for every ε ∈ [0, ε ] and N = N(ε) Problem 11.1.9 has a unique solution (ϑ(ε), ζ(ε), m(ε)) ∈ Bτ ×X s,r ×R. The totality of all such solutions is a relatively compact subset of Bτ × X s,r × R. Proof. By Lemma 11.4.1 and the choice of ε , the inequality N(ε) − IC 2 (Ω) ≤ τ 2 holds for all ε ∈ [0, ε ]. It follows from Condition 11.4.2 that Ω, U, s, r, λ, Re and N = N(ε) satisfy all hypotheses of Theorem 11.3.20. Application of this theorem completes the proof. By abuse of notation we will write v = v(0),

π = π(0),

ϕ = ϕ(0),

ζ = ζ(0),

(11.4.5)

m = m(0).

By Lemma 11.4.3, the quantities w(ε) = v − v(ε), ω(ε) = π − π(ε), ξ(ε) = ζ − ζ(ε), n(ε) = m − m(ε)

ψ(ε) = ϕ − ϕ(ε),

are well deﬁned and (w(ε), ω(ε), ψ(ε), ξ(ε), n(ε)) belongs to the Banach space F s,r given by Deﬁnition 11.3.3. Now set wε = −ε−1 w(ε), ξε = −ε−1 ξ(ε),

ωε = −ε−1 ω(ε),

ψε = −ε−1 ψ(ε),

(11.4.6)

nε = −ε−1 n(ε).

With this notation the material derivatives of the solution to Problem 11.1.9 at ε = 0 are given by w = lim wε , ε→0

ω = lim ωε , ε→0

ψ = lim ψε , ε→0

ξ = lim ξε , ε→0

n = lim nε , ε→0

provided that the limits exist in some sense. Remark 11.4.4. Let us point out that according to Deﬁnition 11.1.3 the material derivative for the velocity ﬁeld in the reference domain, deﬁned by u := N(ε)−1 (u + v(ε)), obviously takes the form u˙ := w − D(u + v(0)). Thus, the element w is only a part of the material derivative for the velocity ﬁeld we are looking for. However, in the following the element w is called the material derivative. It is easy to derive formally the equations for the material derivatives by substituting ϑ = ϑ(ε) and N = N(ε) into the basic equations (11.1.24) and differentiating the result with respect to ε at zero. This formal procedure gives the following system of equations and boundary conditions for the material derivatives

358

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

w, (ω, ψ, ζ), and n: u∇ψ + σψ = −w · ∇ϕ1 + a11 ψ + a12 ω + a13 n + a10 d0 Δw − ∇ω = Reu∇w + Re w∇u + Re ψu∇u + D0 div w = a21 ψ + a22 ω + a23 n + a20 d0 in Ω,

in Ω, in Ω,

− div(uξ) + σξ = div(ζw) + σd0 in Ω, w = 0 on ∂Ω, ψ = 0 on Σin , ξ = 0 on Σout , ω − Πω = 0, n = κ0 (a31 ψ + b32 ω + b34 ξ + b30 d) dx.

(11.4.7a) (11.4.7b) (11.4.7c) (11.4.7d) (11.4.7e)

Ω

Here the coeﬃcients aij , κ and the functions D0 , d0 are given by 2σ ϕ + m, a12 = λ−1 , a13 = , σ = Ψ[ϑ] − ϕ2 − σϕ + m, a21 = σ −1 , a22 = −λ−1 , = −1, a20 = σϕ−1 (11.4.8) − Ψ[ϑ] − m, 2σ = −1 ϕ0 + m−1 ζ Ψ[ϑ] − ζ, −1 = λ−1 (−1 a34 = −1 ζ − 1), Ψ1 [ϑ] + m(1 + ϕ),

a11 = Ψ[ϑ] − a10 a23 a31 a32

−1 a30 = −1 ζ(a10 − m) − Ψ[ϑ] − m(1 − ζ − ζϕ), −1 (1 − ζ − −1 , κ= ζϕ) dx Ω

D0 = D Δu + Δ(Du) + div(T∇u) − Re (D u∇u + u∇(Du)), d0 = div T I,

T = div T I − DT − (DT) ,

D = div T I − DT.

(11.4.9)

In these formulae ϑ = (v, π, ϕ), ζ and the constant m are given by (11.4.5), and u = u + v, = + ϕ. It is important to note that the corresponding transposed problem is formulated as follows: For given f = (H, G, F, M, e), ﬁnd h = (h, g, ς, υ, l) satisfying Δh − ∇g = Re H0 (h) − ς∇ϕ − ζ∇υ + H in Ω, div h = Π(a22 g + a12 ς + a32 l) + ΠG in Ω, − div(uς) + σς = Re M0 (h) + a11 ς + a21 g + a31 l + F u∇υ + συ = a34 l + M in Ω,

(11.4.10a) in Ω,

h = 0 on ∂Ω, g − Πg = 0,

ς = 0 on Σout , υ = 0 on Σin , l = κ (a13 ς + a23 g) dx + e.

(11.4.10b) (11.4.10c) (11.4.10d) (11.4.10e)

Ω

Here the operators H0 , M0 are given by H0 (h) = ∇(u)h − div(u ⊗ h),

M0 (h) = (u∇u) · h.

(11.4.11)

11.4. Shape derivative

359

Notice that the linearized equations (11.4.7) are obtained by formal diﬀerentiation. It is not clear whether the limits w, (ω, ψ, ζ), and n really exist and satisfy (11.4.7). We claim that they exist in a weak sense and the material derivatives are deﬁned as elements of the space W 1−s,r (Ω) × (W−s,r (Ω))3 × R. Moreover, we claim that the material derivatives are given by a very weak solution to the linear problem (11.4.7). Deﬁnition 11.3.16 of a very weak solution requires the well-posedness of the transposed problem (11.4.10) associated with the linearized equations (11.4.7). This fact results from the following Lemma 11.4.5. Let Condition 11.4.2 be satisﬁed. Then for every (H, G, F, M, e) ∈ U s,r , problem (11.4.10) has a unique solution h = (h, g, ς, υ, l) ∈ V s,r . Proof. Notice that problem (11.4.10) is a particular case of problem (11.3.6) with N0 = I, g0 = 1 and ϑi = ϑ(0),

ζi = ζ(0),

mi = m(0),

ui = u + v(0),

i = + ϕ(0)

for i = 0, 1. Hence the statement is a straightforward consequence of Theorem 11.3.20. Now we are in a position to formulate the main result of this section. Theorem 11.4.6. Let Condition 11.4.2 be satisﬁed. Then there are (w, ω, ψ, ξ, n) ∈ W01−s,r (Ω) × (W−s,r (Ω))3 × R such that

wε w weakly in W01−s,r (Ω),

(ωε , ψε , ξε ) (ω, ψ, ξ) weakly in W−s,r (Ω),

and nε → n in R as ε → 0. Moreover, for every (H, G, F, M, e) ∈ U s,r , H, w1 + ω, G0 + ψ, F 0 + ξ, M0 + κ −1 ne h · D0 + g a20 d0 + ς a10 d0 + υ σd0 + l a30 d0 dx. (11.4.12) = Ω

Here h = (h, g, ς, υ, l) ∈ V s,r is the solution to problem (11.4.10) given by Lemma 11.4.6, κ is given by (11.4.8), and D0 and d0 are given by (11.4.9). The next section is devoted to the proof of this theorem.

11.4.2 Proof of Theorem 11.4.6 We split the proof into a sequence of lemmas. The ﬁrst is just a particular case of Theorem 11.3.18. In order to formulate it we introduce the following notation. Assume that N(ε), g(ε) are deﬁned by (11.4.1), and ϑ(ε), ζ(ε), m(ε) are given by Lemma 11.4.3. Next set u = u + v,

u(ε) = u + v(ε),

= + ϕ,

(ε) = + ϕ(ε).

(11.4.13)

360

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

Introduce the functions bε,ij , D(ε), d(ε) and operators Hε , M0 by bε,11 = Ψ[ϑ] − (ε)Φ(ϕ, ϕ(ε)) − bε,12 = λ−1 (ε), bε,13 = , bε,10

σ (ϕ + ϕ(ε)) + m(ε), σ = (ε)Ψ[ϑ(ε)] − ϕ(ε)2 − σϕ(ε) + m(ε)(ε),

σ + Φ(ϕ, ϕ(ε)), bε,22 = −λ−1 , = −1, bε,20 = σϕ(ε)−1 − Ψ[ϑ(ε)] − m(ε), σ −1 = ζ(ε) Ψ[ϑ] − (ε)Φ(ϕ, ϕ(ε)) − (ϕ + ϕ(ε))

bε,21 = bε,23 bε,31

(11.4.14)

− Φ(ϕ, ϕ(ε)) + m(ε)−1 ζ(ε), bε,32 = λ−1 (−1 ζ(ε)(ε) − 1),

−1 bε,34 = −1 Ψ1 [ϑ] + m(ε)(1 + ϕ),

−1 bε,30 = −1 ζ(ε)(bε,10 − m(ε)(ε)) − Ψ[ϑ(ε)] − m(ε)(1 − ζ(ε) − ζ(ε)ϕ(ε)), 1 Φ(ϕ0 , ϕ1 ) = (p ( ) )−1 σ H (ϕ0 s + ϕ1 (1 − s)) ds, 0 −1 (1 − ζ − −1 ζϕ) dx , H (ϕ) = p ( + ϕ) − p ( ), κ = Ω

Hε (h) = (ε)∇u h − div((ε)u(ε) ⊗ h), M0 (h) = (u∇u) · h, D(ε) = N(ε)− div g(ε)−1 N(ε)N(ε) ∇(N(ε)−1 u(ε)) − Δu(ε) + Re (ε) u(ε)∇u(ε) − N(ε)− u(ε)∇(N(ε)−1 u(ε)) ,

(11.4.16)

d(ε) = 1 − g(ε).

(11.4.17)

(11.4.15)

Let us consider the transposed problem Δhε − ∇gε = Re Hε (hε ) − ςε ∇ϕ(ε) − ζ(ε)∇υε + H in Ω, div hε = Π(bε,22 gε + bε,12 ςε + bε,32 lε ) + ΠG in Ω,

(11.4.18a)

− div(uςε ) + σςε = Re M (hε ) + bε,11 ςε + bε,21 gε + bε,31 lε + F u∇υ + συ = bε,34 l + M in Ω, hε = 0 on ∂Ω, gε − Πgε = 0,

in Ω,

ςε = 0 on Σout , υε = 0 on Σin , l = κ (bε,13 ςε + bε,23 gε ) dx + e.

(11.4.18b)

(11.4.18c)

Ω

Lemma 11.4.7. Let Condition 11.4.2 be satisﬁed. Then for each f = (H, G, F, M, e) ∈ U s,r , problem (11.4.18) has a unique solution hε = (hε , gε , ςε , υε , lε ) ∈ V s,r . This solution satisﬁes the estimate hε V s,r ≤ cfU s,r ,

(11.4.19)

11.4. Shape derivative

361

where c is independent of ε. Moreover, the functions (wε , ωε , ψε , ξε , nε ) given by (11.4.6) belong to the class W01−s,r × (W−s,r (Ω))3 × R and satisfy the identity H, wε 1 + ωε , G0 + ψε , F 0 + ξε , M0 + κ −1 nε e 1 hε · D(ε) + gε bε,20 d(ε) + ςε bε,10 d(ε) + υε σd(ε) + lε bε,30 d(ε) dx, =− ε Ω (11.4.20) where (hε , gε , ςε , υε , lε ) is the solution to problem (11.4.18). Proof. Observe that in view of Lemma 11.4.1 and Condition 11.4.2, the inequality N(ε)−IC 2 (Ω) ≤ τ 2 holds for all ε ∈ [0, ε ]. Hence Ω, U, s, r, λ, Re and N = N(ε) satisfy all hypotheses of Theorem 11.3.20. Moreover, the functions and matrices N0 = I, N1 = N(ε), g0 = 1, (ϑ0 , ζ0 , m0 ) = (ϑ, ζ, m), (ϑ1 , ζ1 , m1 ) = (ϑ(ε), ζ(ε), m(ε)), u0 = u + v, u1 = u(ε), 0 = + ϕ = , 1 = (ε)

(11.4.21)

satisfy all hypotheses of Theorems 11.3.4 and 11.3.18. In particular, formulae (11.4.14)–(11.4.15) imply that problem (11.4.18) is a special case of (11.3.6). Therefore the solvability of problem (11.4.18) and estimates (11.4.19) follow directly from Theorem 11.3.4. Next, applying Theorem 11.3.18 we ﬁnd that the diﬀerences (w(ε), ω(ε), ψ(ε), ξ(ε), n(ε)) satisfy identity (11.3.77) with D and d replaced by D(ε) and d(ε), bij replaced by bε,ij , and (h, g, ς, υ, l) replaced by (hε , gε , ςε , υε , lε ). Thus we get H, w(ε)1 + ω(ε), G0 + ψ(ε), F 0 + ξ(ε), M0 + κ −1 n(ε)e = hε · D(ε) + gε bε,20 d(ε) + ςε bε,10 d(ε) + υε σd(ε) + lε bε,30 d(ε) dx, Ω

Substituting expressions (11.4.6) into this identity we arrive at (11.4.20).

In order to prove Theorem 11.4.6 we have to pass to the limit in (11.4.20) as ε → 0. To this end, we investigate all terms in this identity and begin with the evaluation of D(ε) and d(ε) given by (11.4.16). Lemma 11.4.8. Let Condition 11.4.2 be satisﬁed. Then D(ε) = −εD1 (ε) + ε2 D2 (ε),

d(ε) = −ε div T + ε2 d2 (ε),

(11.4.22)

where D1 (ε) = div(T∇u(ε)) + D Δu(ε) + Δ(Du(ε)) − Re (ε) D u(ε)∇u(ε) + u(ε)∇(Du(ε)) ,

T = div T I − DT − (DT) ,

D = div T I − DT.

(11.4.23)

362

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

The remainders D2 (ε) and d2 (ε) satisfy the estimates D2 (ε)L2 (Ω) + d2 (ε)C(Ω) ≤ c,

(11.4.24)

where c is independent of ε. Proof. By Lemma 11.4.1, we have S(ε) := g(ε)−1 N(ε)N(ε) = (1 − ε div T + ε2 g− )(I + εD + εD+ )(I + εD + εD+ ) = I − εT + ε2 S2 (ε), where the matrix S2 satisﬁes the estimate S2 (ε)C 2 (Ω) ≤ c.

(11.4.25)

Substituting S(ε) and decompositions (11.4.2) into (11.4.16) we arrive at expansion (11.4.22) with D2 (ε) = (D− ) div S(ε)∇(N(ε)−1 u(ε)) + N(ε)− div S2 (ε)∇(N(ε)−1 u(ε)) + N(ε)− div S(ε)∇(D− (ε)u(ε)) + D Δ(Du(ε)) − D div(T∇u(ε)) − div T∇(Du(ε)) + Re (ε) D u(ε)∇(Du(ε)) − (D− ) u(ε)∇(N(ε)−1 u(ε)) − (D− ) u(ε)∇(N(ε)−1 u(ε)) − εD div T∇(Du(ε)) . (11.4.26) It follows from Lemma 11.4.3 that ϑ(ε) = (v(ε), ω(ε), ϕ(ε)) belongs to the ball Bτ and hence v(ε)Y s,r + ϕ(ε)X s,r ≤ τ. Since Y s,r = W 1+s,r (Ω) ∩ W 2,2 (Ω) and X s,r = W s,r (Ω) ∩ W 1,2 (Ω), sr > 3, the functions u(ε) = u + v(ε) and (ε) = + ϕ(ε) satisfy u(ε)W 2,2 (Ω) + (ε)C(Ω) ≤ c,

(11.4.27)

where c is independent of ε. It follows from this, expression (11.4.26), and estimates (11.4.3), (11.4.25) that D2 (ε) satisﬁes (11.4.24). It remains to note that d2 (ε) = −g+ (ε) and inequality (11.4.24) for d2 is a direct consequence of (11.4.3). Lemma 11.4.9. Let Condition 11.4.2 be satisﬁed and (wε , ωε , ψε , nε ) be given by (11.4.6). Then there exists c independent of ε such that wε W 1−s,r (Ω) + ωε W−s,r (Ω) + ψε W−s,r (Ω) + ξε W−s,r (Ω) + |nε | ≤ c. 0

(11.4.28)

11.4. Shape derivative

363

Proof. Recall the notation U s,r = W s−1,r (Ω) × (W s,r (Ω))3 × R,

V s,r = W s+1,r (Ω) × (W s,r (Ω))3 × R,

It suﬃces to show that for all f = (H, G, F, M, e) ∈ U s,r , H, wε 1 + ωε , G0 + ψε , F 0 + ξε , Mn 0 + κ −1 nε e ≤ cfU s,r . 0

(11.4.29)

Notice that in view of Lemma 11.3.11 and the embedding X s,r → C(Ω), the coeﬃcients bε,23 and bε,13 are bounded by a constant depending only on Ω, U, s, r, and σ. From this and (11.4.20) we obtain H, wε 1 + ωε , G0 + ψε , F 0 + ξε , M 0 + κ −1 nε e 0

≤ ε−1 (D(ε)L1 (Ω) + d(ε)L1 (Ω) )hε C(Ω)6 ×R , (11.4.30) where hε = (hε , gε , ςε , υε , lε ) ∈ V s,r is the solution to problem (11.4.18). Since the embedding V s,r → C 1 (Ω)×(C(Ω))3 ×R is continuous, inequality (11.4.19) implies hε C(Ω)6 ×R ≤ chε V s,r ≤ cfε U s,r . On the other hand, Lemma 11.4.8 yields ε−1 D(ε)L2 (Ω) + ε−1 d(ε)C(Ω) ≤ c.

Inserting these results into (11.4.30) we arrive at (11.4.29).

Although Lemma 11.4.3 states the existence and uniqueness of solutions (ϑ(ε), ζ(ε), m(ε)) to the basic Problem 11.1.9, it says nothing about the dependence of these solutions on ε. Now we are in a position to prove they are continuous at ε = 0. Lemma 11.4.10. Let Condition 11.4.2 be satisﬁed. Then (ϑ(ε), ζ(ε), m(ε)) → (ϑ, ζ, m)

in Y s,r × (X s,r )3 × R

as ε → 0.

Proof. Recall that F s,r = Y s,r ×(X s,r )3 ×R and introduce the temporary notation q(ε) = (ϑ(ε), ζ(ε), m(ε)) ≡ (v(ε), ω(ε), ϕ(ε), ζ(ε), m(ε)), q = (ϑ, ζ, m) ≡ (v, ω, ϕ, ζ, m), T

s,r

= W01−s,r (Ω) × (W−s,r (Ω))3 × R.

We must prove that q(ε) → q in F s,r . In view of Lemma 11.4.3, the sequence q(ε), ε ∈ (0, ε ], is relatively compact in F s,r . Hence it contains a subsequence, still denoted by q(ε), which converges in F s,r to some q∗ . Notice that v(ε) → v∗ in Y s,r and hence v(ε) → v∗ in C 1 (Ω). Since v(ε) vanishes at ∂Ω, it follows that v(ε) → v∗ in W01,r (Ω). On the other hand, W01,r (Ω) is continuously embedded and dense in

364

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

the interpolation space W01−s,r (Ω) = [Lr (Ω), W01,r (Ω)]1−s,r . Therefore, we have v(ε) → v∗ in W 1−s,r (Ω). Next, (ω(ε), ϕ(ε), ζ(ε)) → (ω ∗ , ϕ∗ , ζ ∗ ) in W s,r (Ω) and hence in Lr (Ω). Since r > r , the space Lr (Ω) is continuously embedded in Lr (Ω). In its turn, Lr (Ω) is continuously embedded and dense in W−s,r (Ω) = [Lr (Ω), W −1,r (Ω)]s,r . Hence (ω(ε), ϕ(ε), ζ(ε)) → (ω ∗ , ϕ∗ , ζ ∗ ) in W−s,r (Ω). Thus q(ε) → q∗ in T s,r . On the other hand, formula (11.4.6) and Lemma 11.4.9 imply q − q(ε) = ε(wε , ωε , ψε , nε ) → 0 in T s,r . It follows that the limit q∗ = q is independent of the choice of a converging subsequence of q(ε). Hence the whole sequence q(ε) converges to q in T s,r . Our last task is to pass to the limit as ε → 0 in the transposed equations (11.4.18). We begin with the study of the coeﬃcients of these equations. Lemma 11.4.11. Let Condition 11.4.2 be satisﬁed. Let bε,ij , Hε , and D(ε), d(ε) be given by (11.4.14). Then bε,ij → aij

in X s,r ,

H ε → H0

in L(W s+1,r (Ω) → W s−1,r (Ω)),

−ε

−1

D(ε) → D0

2

in L (Ω),

−ε

−1

d(ε) → d0

(11.4.31) in C(Ω),

where aij are given by (11.4.8), D0 and d0 are given by (11.4.9), and H0 is given by (11.4.11). Proof. It follows from (11.4.14) that bε,ij are C 3 functions of ϑ, ζ and ϑ(ε), ζ(ε) deﬁned over the whole space and aij = b0,ij . On the other hand, in view of Lemma 11.4.10, ϑ(ε) and ζ(ε) converge to ϑ and ζ in X s,r as ε → 0. It follows from Lemma 11.2.8 that the composition operator, which assigns to each element of X s,r the composition of C 2 function with this element, is continuous in X s,r . Hence the ﬁrst relation in (11.4.31) is a straightforward consequence of Lemma 11.2.8. Let us estimate the norm of Hε − H0 . Choose h ∈ W s+1,r (Ω) such that hW s+1,r (Ω) ≤ 1. We have Hε (h) − H0 (h) = (ϕ(ε) − ϕ)∇u h + div ((ε)u(ε) − u) ⊗ h . Here we use identities (11.4.13). Applying Lemma 11.2.13 to the last term on the right hand side we arrive at Hε (h) − H0 (h)W s−1,r (Ω) ≤ cϕ(ε) − ϕ)∇u hW s,r (Ω) + c((ε)u(ε) − u) ⊗ hW s,r (Ω) . (11.4.32) Since W s,r (Ω) is a Banach algebra and ∇uW s,r (Ω) ≤ cuW s+1,r (Ω) ≤ c, we have (ϕ(ε) − ϕ)∇u hW s,r (Ω) ≤ cϕ(ε) − ϕW s,r (Ω) ∇uW s,r (Ω) hW s,r (Ω) ≤ cϕ(ε) − ϕW s,r (Ω) .

(11.4.33)

11.4. Shape derivative

365

Next we have ((ε)u(ε) − u) ⊗ hW s,r (Ω) ≤ c(ε)u(ε) − uW s,r (Ω) hW s,r (Ω) ≤ c(ε)u(ε) − uW s,r (Ω) . Using the identity (ε)u(ε) − u = (ϕ(ε) − ϕ)u(ε) + (v(ε) − v) and noting that u(ε), (ε) are uniformly bounded in W s,r (Ω) we obtain (ε)u(ε) − uW s,r (Ω) ≤ cϕ(ε) − ϕW s,r (Ω) + cv(ε) − vW s,r (Ω) . Thus we get ((ε)u(ε)−u)⊗hW s,r (Ω) ≤ cϕ(ε)−ϕW s,r (Ω) +cv(ε)−vW s,r (Ω) . (11.4.34) Inserting (11.4.33) and (11.4.34) into (11.4.32) leads to Hε (h) − H0 (h)W s−1,r (Ω) ≤ cϕ(ε) − ϕW s,r (Ω) + cv(ε) − vW s,r (Ω) → 0 as ε → 0. It follows that Hε (h) → H0 (h) in W s−1,r (Ω) uniformly on the unit ball in W s+1,r (Ω), which leads to the second relation in (11.4.31). It remains to prove the third. Lemma 11.4.8 and expression (11.4.9) for D0 imply (11.4.35) D0 + ε−1 D(ε) = D3 (ε) + Re D4 (ε) + εD2 (ε), where D3 (ε) = div T∇(u − u(ε)) + D Δ(u − u(ε)) + Δ D(u − u(ε)) , D4 (ε) = (ε) D u(ε)∇u(ε) + u(ε)∇(Du(ε)) − D u∇u + u∇(Du) , T = div T I − DT − (DT) ,

D = div T I − DT.

Lemma 11.4.10 and identities (11.4.13) imply that u(ε) → u in Y s,r = W s+1,r (Ω)∩ W 2,2 (Ω). Since sr > 3, the embedding W s+1,r (Ω) → C 1 (Ω) is continuous. Thus u(ε) → u in W 2,2 (Ω) and in C 1 (Ω) as ε → 0. Next, (ε) → in X s,r = W s,r (Ω) ∩ W 1,2 (Ω). Since the embedding W s,r (Ω) → C(Ω) is continuous, we have (ε) → in W 1,2 (Ω) and in C(Ω) as ε → 0. It follows that D3 (ε) → 0 in L2 (Ω) and

D4 (ε) → 0 in C(Ω) as ε → 0.

366

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

Estimate (11.4.24) implies that εD2 (ε)L2 (Ω) ≤ cε → 0 as ε → 0. Letting ε → 0 in (11.4.35) we obtain D0 + ε−1 D(ε) → 0 in L2 (Ω) as ε → 0. It remains to note that in view of (11.4.24), d0 + ε−1 d(ε) = εd2 (ε) → 0 in C(Ω) as ε → 0.

The next lemma shows that solutions to the transposed problem converge to solutions of problem (11.4.10) as ε → 0. Lemma 11.4.12. Let Condition 11.4.2 be satisﬁed. Let hε = (hε , gε , ςε , υε , lε ), ε ∈ (0, ε ], be the solutions to problem (11.4.18). Then hε h

weakly in W s+1,r (Ω) × (W s,r (Ω))3 × R,

where h = (h, g, ς, υ, l) is the solution to problem (11.4.10). Proof. We have to pass to the limit as ε → 0 in equations (11.4.18). Recall the notation V s,r = W s+1,r (Ω) × (W s,r (Ω))3 × R. By Lemma 11.2.13 the space V s,r is reﬂexive. In view of Lemma 11.4.7 the sequence hε is bounded in V s,r . Hence there are h ∈ V s,r and a subsequence, still denoted by hε , such that hε converges weakly to h as ε → 0. Let us prove that h satisﬁes equations (11.4.10). We begin with the Stokes equations (11.4.18a) which are the integral part of problem (11.4.18). These equations can be written in the form Δhε − ∇gε = Qε , div hε = ΠPε hε = 0 on ∂Ω, gε = Πgε ,

in Ω,

(11.4.36)

where Qε = Re Hε (hε ) − ςε ∇ϕ(ε) − ζ(ε)∇υε + H, Pε = bε,22 gε + bε,12 ςε + bε,32 lε + G. Set Q = Re H0 (h) − ς∇ϕ − ζ∇υ + H,

P = a22 g + a12 ς + a32 l + G.

We have Q − Qε = Re(H0 − Hε )(hε ) − ςε ∇(ϕ − ϕ(ε)) − (ζ − ζ(ε))∇υε + Re H0 (h − hε ) − (ς − ςε )∇ϕ − ζ∇(υ − υε ). Lemma 11.4.11 implies (H0 − Hε )(hε )W s−1,r (Ω) ≤ H0 − Hε L(W s+1,r (Ω)→W s−1,r (Ω)) hε W s+1,r (Ω) ≤ cH0 − Hε L(W s+1,r (Ω)→W s−1,r (Ω)) → 0 as ε → 0. (11.4.37)

11.4. Shape derivative

367

In view of Lemma 11.3.7 the bilinear mapping (W s,r (Ω))2 (ϕ, ψ) → ψ∇ϕ ∈ W s−1,r (Ω) is continuous. It follows from this and Lemma 11.4.10 that ςε ∇(ϕ − ϕ(ε))W s−1,r (Ω) + (ζ − ζ(ε))∇υε W s−1,r (Ω) ≤ cϕ − ϕ(ε)W s,r (Ω) ςε W s,r (Ω) + cζ − ζ(ε)W s,r (Ω) υε W s,r (Ω) ≤ cϕ − ϕ(ε)W s,r (Ω) + cζ − ζ(ε)W s,r (Ω) → 0 as ε → 0. (11.4.38) The operator H0 is a special case of the operator H given by (11.3.7) with N0 = I and u0 = u. Lemma 11.3.10 shows that H0 : W s+1,r (Ω) → W s,r (Ω) is continuous, and hence is continuous in the weak topology, which leads to H0 (h − hε ) 0 weakly in W s−1,r (Ω) as ε → 0.

(11.4.39)

Next, Lemma 11.3.7 implies that the trilinear form

(W s,r (Ω))2 × W01−s,r (Ω) (ϕ, ψ, w) → ψ∇ϕ, w1 ∈ R,

where ·, ·1 denotes the duality pairing between W s−1,r (Ω) and W01−s,r (Ω), is continuous. Hence for ﬁxed ϕ ∈ W s,r (Ω) and w ∈ W0s,r (Ω) the mapping ψ → ψ∇ϕ, w1 deﬁnes a continuous linear functional on W s,r (Ω). Since ς − ςε 0 weakly in W s,r (Ω), it follows that

(ς − ςε )∇ϕ, w1 → 0 for all w ∈ W01−s,r (Ω) as ε → 0. This means that (ς − ςε )∇ϕ → 0 weakly in W s−1,r (Ω). Notice that W s−1,r (Ω) is reﬂexive as the dual space to the reﬂexive space W01−s,r (Ω). Thus we get (ς − ςε )∇ϕ 0 weakly in W s−1,r (Ω) as ε → 0.

(11.4.40)

The same arguments give ζ∇(υ − υε ) 0 weakly in W s−1,r (Ω) as ε → 0. Combining this with (11.4.38)–(11.4.40) we obtain Qε Q weakly in W s−1,r (Ω) as ε → 0.

(11.4.41)

Let us consider the sequence Pε . In view of Lemma 11.4.11, the coeﬃcients bε,22 , bε,12 , bε,32 converge to a22 , a12 , a32 in W s,r . On the other hand gε , ςε converge to g, ς weakly in W s,r . Hence bε,22 gε + bε,12 ςε + bε,32 lε → a22 g + a12 ς + a32 l

weakly in W s−1,r (Ω) as ε → 0,

368

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

which leads to Pε P

weakly in W s−1,r (Ω) as ε → 0.

(11.4.42)

Let us turn to the Stokes problem (11.4.36). Lemma 1.7.9 implies that the operator T : W s−1,r (Ω) × W s,r (Ω) → W s+1,r (Ω) × W s,r (Ω), which assigns to a couple (Qε , Pε ) the solution (hε , gε ) of problem (11.4.36), is continuous and hence it is continuous in the weak topology. From this and (11.4.41), (11.4.42) we conclude that the weak limit (h, g) of the solutions to problem (11.4.36) satisﬁes Δh − ∇g = Q, h = 0 on ∂Ω,

div h = ΠP g = Πg,

in Ω,

which coincides with (11.4.10a). It remains to prove that ς and υ satisfy the transport equations (11.4.10b) and (11.4.10c). Notice that these equations are understood in the weak sense. Let us consider the boundary value problem (11.4.18b), (11.4.18c) for the function ςε . Recall that ςε is a weak solution to this problem if and only if the integral identity ςε u∇φ+φ σςε −Re M (hε )−bε,11 ςε −bε,21 gε −bε,31 lε −F dx = 0 (11.4.43) Ω

holds for every φ ∈ C 1 (Ω) vanishing in a neighborhood of ∂Ω \ Σout . Since the embedding W s+1,r (Ω) → W s,r (Ω) is compact, the sequence hε converges to h strongly in W s,r (Ω). On the other hand, Lemma 11.3.10 implies that the operator M0 : W s,r (Ω) → W s,r (Ω) is continuous. Thus we get M0 (hε ) → M0 (h) in W s,r (Ω) as ε → 0. It follows that M0 (hε ) → M0 (h) in C(Ω). Since the embedding W s,r (Ω) → C(Ω) is compact the functions (gε , ςε , υε ) converge to (g, ς, υ). In view of Lemma 11.4.11, the functions bε,ij converge to aij in C(Ω). Letting ε → 0 in (11.4.43) we arrive at ςu∇φ + φ σς − Re M (h) − a11 ς − a21 g − a31 l − F dx = 0, Ω

which means that ς satisﬁes the transport equation (11.4.10b) and the boundary condition (11.4.10d). The same arguments show that υ also satisﬁes (11.4.10c), (11.4.10d). Obviously we have (bε,13 ςε + bε,23 gε ) dx → (a13 ς + a23 g) dx as ε → 0 and g = Πg. Ω

Ω

Assembling the results obtained we conclude that h = (w, g, ς, υ, l) satisﬁes equations (11.4.10). Thus we have proved that every weak limit h ∈ V s,r of the sequence hε = (wε , gε , ςε , υε , lε ) is a solution to the boundary value problem (11.4.10). Since such a solution is unique, it follows that the whole sequence hε converges weakly in V s,r to h.

11.4. Shape derivative

369

Proof of Theorem 11.4.6. We are now in a position to complete the proof of Theorem 11.4.6. Observe that in view of Lemma 11.4.9, the sequence (wε , ωε , ψε , ξε , nε ) given by (11.4.6) is bounded in W01−s,r (Ω) × (W−s,r (Ω))3 × R. Since this space is reﬂexive we may assume, passing to a subsequence if necessary, that there are (w, ω, ψ, ξ, n) ∈ W01−s,r (Ω) × (W−s,r (Ω))3 × R such that

wε w weakly in W01−s,r (Ω),

(ωε , ψε , ξε ) (ω, ψ, ξ) weakly in W−s,r (Ω).

On the other hand, Lemma 11.4.7 implies that for (H, G, F, M, e) ∈ W s−1,r (Ω) × (W s,r (Ω))3 × R, we have H, wε 1 + ωε , G0 + ψε , F 0 + ξε , M0 + κ −1 nε e 1 hε · D(ε) + gε bε,20 d(ε) + ςε bε,10 d(ε) + υε σd(ε) + lε bε,30 d(ε) dx, =− ε Ω (11.4.44)

where (hε , gε , ςε , υ, lε ) is the solution to problem (11.4.18). Since W01−s,r (Ω) × (W−s,r (Ω))3 × R is dual to W s−1,r (Ω) × (W s,r (Ω))3 × R we can pass to the limit on the left hand side of this identity to obtain H, wε 1 + ωε , G0 + ψε , F 0 + ξε , M0 + κ −1 nε e → H, w1 + ω, G0 + ψ, F 0 + ξ, M0 + κ0−1 ne as ε → 0. Since the embedding of V s,r = W s+1,r (Ω)×(W s,r (Ω))3 ×R into C(Ω)6 ×R is continuous, it follows from Lemma 11.4.12 that (hε , gε , ςε , υ, lε ) → (h, g, ς, υ, l) in C(Ω)6 × R, Here (h, g, ς, υ, l) ∈ V s,r is the solution to problem (11.4.10) given by Lemma 11.4.6. Next, Lemma 11.4.11 implies bε,ij → aij

in C(Ω),

−ε−1 D(ε) → D0

in L2 (Ω),

−ε−1 d(ε) → d0

in C(Ω).

Hence we can pass to the limit on the right hand side of (11.4.44) to obtain 1 hε · D(ε) + gε bε,20 d(ε) + ςε bε,10 d(ε) + υε σd(ε) + lε bε,30 d(ε) dx − ε Ω h · D0 + g a20 d0 + ς a10 d0 + υ σd0 + l a30 d dx. → Ω

Letting ε → 0 in (11.4.44) we ﬁnally obtain H, w1 + ω, G0 + ψ, F 0 + ξ, M0 + κ −1 ne h · D0 + g a20 d0 + ς a10 d0 + υ σd0 + l a30 d0 dx, (11.4.45) = Ω

370

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

where (h, g, ς, υ, l) is a solution to problem (11.4.10). In view of Lemma 11.4.6 such a solution exists and is unique. Thus we have proved that every weak limit (w, ω, ψ, ξ, n) of the sequence (wε , ωε , ψε , ξε , nε ) satisﬁes identity (11.4.45). It remains to show that this sequence has a unique weak limit point, and hence converges weakly to this point. Assume that there are (w, ω, ψ, ξ, n) and (w , ω , ψ , ξ , n ) in W01−s,r (Ω) × (W−s,r (Ω))3 × R which satisfy (11.4.45). Then s−1,r for all (H, G, F, M, e) ∈ W (Ω) × (W s,r (Ω))3 × R, H, w − w 1 + ω − ω , G0 + ψ − ψ , F 0 + ξ − ξ , M0 + κ −1 (n − n )e = 0.

Since W s−1,r (Ω) × (W s,r (Ω))3 × R is dual to W01−s,r (Ω) × (W−s,r (Ω))3 × R, it follows that (w , ω , ψ , ξ , n ) = (w, ω, ψ, ξ, n).

11.4.3 Conclusion. Material and shape derivatives Recall that our main goal was to investigate the domain perturbations inﬂuence on solutions to Navier-Stokes equations. In order to specify the domain perturbations we considered the one-parameter family of domains Ωε = (Id + εT)(Ω), where T : R3 → R3 is an arbitrary compactly supported smooth mapping. Thus we have got two boundary value problems. The ﬁrst is the boundary value problem (11.1.1) for (u, ) in the unperturbed domain Ω, and the second is the boundary value problem (11.1.2) for (¯ uε , ¯ε ) in the domain Ωε . We stress that u(x) and (x) ¯ ε (y) are the velocity and density calculated at the physical point x ∈ Ω, and u and ¯ε (y) are the perturbed velocity and density calculated at the physical point y ∈ Ωε . A comparison between perturbed and unperturbed quantities is diﬃcult since they are deﬁned in diﬀerent domains. In order to cope with this diﬃculty we introduced the Piola transform uε and the transformed density ε deﬁned in Ω and given by ¯ ε (x + εT(x)), uε (x) = N(ε)(x) u

ε (x) = ¯ε (x + εT(x)),

x ∈ Ω,

where N(ε) = det(I + εDT) (I + εDT)−1 . In this setting we refer to Ω as the reference domain in order to emphasize that for ε = 0, points of this domain cannot be regarded as points of the “physical” Euclidean space. The change of independent variables in (11.1.2) leads to Problem 11.1.6 for the transformed functions and next to Problem 11.1.9 for their perturbations in the reference domain. Theorem 11.3.20 and Lemma 11.4.3 state the existence of a one-parameter family of solutions ϑ(ε) = (v(ε), π(ε), ϕ(ε)) to Problem 11.1.9. In its turn, the functions u(ε) = u + v(ε),

(ε) = + ϕ(ε),

11.5. Shape derivative of the drag functional. Adjoint state

371

where (u , ) is the approximate solution given by (11.1.18), (11.1.19), are a solution to Problem 11.1.6. Theorem 11.4.6 shows that there are w ∈ W01−s,r (Ω) and ψ ∈ W−s,r (Ω) such that

ε−1 (u(ε)−u) w weakly in W01−s,r (Ω), ε−1 ((ε)−) ψ weakly in W−s,r (Ω); w and ψ are named material derivatives of solutions to Problem 11.1.6 and are denoted by ˙ u(0) := w, (0) ˙ := ψ. These artiﬁcial quantities have no physical meaning because they express the derivative of the Piola transform of the velocity and the derivative of the density in the nonphysical reference domain. The real shape derivatives of the velocity and density at a ﬁxed point of the real physical space are deﬁned as follows: u (0) := lim ε−1 (¯ uε − u), ε→0

(0) := lim ε−1 (¯ ε − u). ε→0

It is easy to establish formally the connection between the material and shape derivatives using the formulae ¯ ε = (N(ε))−1 ◦ (Id + εT)−1 u(ε) ◦ (Id + εT)−1 , u

¯ε = (ε) ◦ (Id + εT)−1 .

Direct calculations show that the shape derivatives can be recovered from the material derivatives: ˙ − ∇uT + DT u − div T u, u (0) = u(0)

(0) = (0) ˙ − ∇ · T.

Since u ∈ W 2,2 (Ω) and ∈ W 1,2 (Ω), all terms in the right hand sides are well deﬁned. On the other hand, the justiﬁcation of the boundary value problem obtained for the shape derivatives is a delicate task for Navier-Stokes equations [115].

11.5 Shape derivative of the drag functional. Adjoint state Theorem 11.3.20 and Lemma 11.4.3 imply the existence and uniqueness of solutions to Problem 11.1.9 associated with the matrices N(ε). These solutions generate the solutions u(ε) = u + v(ε), (ε) = + ϕ(ε), q(ε) = q + λσ0 p( ) + π(ε) + λm(ε)

(11.5.1)

to Problem 11.1.6. Substituting (11.5.1) into expression (11.1.13) for the drag functional J(Ωε ) we obtain the representation for the drag functional as a function of the parameter ε: J(Ωε ) = U∞ · J1 (Ωε ) + U∞ · J2 (Ωε ),

(11.5.2)

372

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

where

(ε)u(ε)∇ N−1 u(ε) η dx, J1 (Ωε ) = −Re 4Ω g(ε)−1 N(ε) ∇ N(ε)−1 u(ε) J2 (Ωε ) = − Ω 5 + ∇ N(ε)−1 u(ε) N − div u(ε)I − q(ε)I N(ε) ∇η dx.

(11.5.3)

Recall that Ωε = B \ Sε , where B is a ﬁxed hold-all domain and Sε = (Id + εT)(S) is the compact body placed in B. While the solutions v(ε), ϕ(ε), π(ε), m(ε) to Problem 11.1.6 have no clear physical meaning, the drag Re−1 J(Ωε ) is a real physical quantity. It equals the power developed by the hydrodynamical force acting on the body Sε . The calculation of the drag and its variations with respect to perturbations of S is a problem of practical signiﬁcance. In this section we derive an explicit formula for the derivative 1 (11.5.4) dJ(Ω)[T] := lim J(Ωε ) − J(Ω) . ε→0 ε Notice that the perturbed domains Ωε = (Id + εT)(Ω) are generated by the deformations Id + εT of the ambient space R3 . Moreover, Ωε can be represented as the shift of Ω along the integral lines of the vector ﬁeld T(x, ε) := (T ◦ (Id + εT)−1 )(x) (see [126] for details). From this viewpoint the quantity (11.5.4) is nothing but the Gâteaux derivative of the shape functional J(Ω) in the direction T. It is not too diﬃcult to calculate dJ(Ω)[T] in terms of the material derivatives of solutions to Problem 11.1.9. The corresponding result is given in the following theorem, which is a straightforward consequence of Theorem 11.4.6. Theorem 11.5.1. Let all hypotheses of Theorem 11.4.6 be satisﬁed. Then dJ(Ω)[T] is well deﬁned and has the representation dJ(Ω)[T] = Lu (T) + Le (T).

(11.5.5)

Here the form Lu is given by Lu (T) = B, w1 + ω, A0 + ψ, C0 ,

(11.5.6)

where the material derivatives w, ω, and ψ are given by Theorem 11.4.6 and B = ΔηU∞ + Re (u · ∇η)U∞ − Re η ∇uU∞ , A = ∇η · U∞ , C = −Re η(u∇u) · U∞ ,

(11.5.7)

and the form Le is given by ∇u + (∇u) − div uI DT − D ∇u − ∇(Du) Le (T) = U∞ · Ω (u · ∇η) Du dx, (11.5.8) − D ∇u − ∇(Du) + qD ∇η dx − Re U∞ · Ω

where D = div T I − DT.

11.5. Shape derivative of the drag functional. Adjoint state

373

Proof. It follows from (11.4.6) and (11.5.1) that u(ε) − u = εwε ,

q(ε) − q = εωε + ελnε ,

(ε) − = εψε .

Substituting these relations into (11.5.2) we obtain 1 (J(Ωε ) − J(Ω)) = Lε,u + Lε,e , ε

(11.5.9)

where Lε,u = −ReU∞ · (u∇wε + ψε u∇uε + ε wε ∇uε )η dx Ω ∇wε + (∇wε ) − div wε I ∇η dx + U∞ · (ωε + λnε )∇η dx, − U∞ · Ω

Ω

(11.5.10)

and εLε,e = Re U∞ · + U∞ ·

Ω

(ε)u(ε)∇ (I − N(ε)−1 )u(ε) η dx $ . ∇(u(ε)) + ∇(u(ε)) − div u(ε)I − q(ε)I ∇η dx

Ω 4

g(ε)−1 N(ε) ∇ N(ε)−1 u(ε) Ω 5 + ∇ N(ε)−1 u(ε) N − div u(ε)I − q(ε)I N(ε) ∇η dx. − U∞ ·

(11.5.11)

Notice that u, u(ε), wε are continuously diﬀerentiable and belong to W 2,2 (Ω). In its turn, q, q(ε), , (ε), ψε are continuous and belong to W 1,2 (Ω). Moreover, wε vanishes at ∂Ω and div(u) = 0 in Ω. Hence we can integrate by parts to obtain ηu∇wε dx = − (∇η · u)wε dx, Ω Ω Δηwε dx. ∇wε + (∇wε ) − div wε I ∇η dx = − Ω

Ω

Since η vanishes in a neighborhood of ∂B and equals 1 in a neighborhood of ∂S, we have ∇η dx = n dS = 0. ∂S

Ω

Inserting the results obtained into (11.5.10) we arrive at Lε,u = (Bε · wε + ωε A + ψε Cε ) dx, Ω

374

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

where Bε = ΔηU∞ + Re (u · ∇η)U∞ − Re η (ε) ∇u(ε)U∞ , A = ∇η · U∞ , Cε = −Re η(u∇u(ε)) · U∞ . It follows from (11.5.1), Lemma 11.4.10, and the continuity of the embeddings Y s,r → C 1 (Ω) and X s,r → C(Ω) that Bε → B in C(Ω) as ε → 0. In particular, Bε ∈ C(Ω) → Lr (Ω) → W s−1,r (Ω) converges to B in W s−1,r (Ω). On the other hand, in view of Theorem 11.4.6 the functions wε converge to w weakly in W01−s,r (Ω). Thus we get Bε · wε dx = Bε , wε 1 → B, w1 as ε → 0. (11.5.12) Ω

Next, it follows from Theorem 11.4.6 and (11.5.1) that uε → u in Y s,r and hence ∇uε → ∇u in X s,r → W s,r (Ω). Since , u ∈ W s,r (Ω) and W s,r (Ω) is a Banach algebra, we get Cε = −Re η(u∇u(ε)) · U∞ → −Re η(u∇u) · U∞ = C

in W s,r (Ω).

Notice that by Theorem 11.4.6, ψε → ψ in W−s,r (Ω), which leads to ψε Cε dx = ψε , Cε 0 → ψ, C0 as ε → 0.

(11.5.13)

Ω

Finally, notice that A ∈ C 1 (Ω) ⊂ W s,r (Ω) and ωε → ω in W−s,r (Ω), which gives ωε A dx = ωε , A0 → ω, A0 as ε → 0. (11.5.14) Ω

Combining (11.5.12)–(11.5.14) we obtain Lε,u (T) → Lu (T) as ε → 0.

(11.5.15)

In order to pass to the limit in the expression for Lε,e notice that uε are bounded in C 1 (Ω) and q(ε), (ε) are bounded in C(Ω). Substituting representations (11.4.2) for N(ε) and g(ε) into (11.5.11) we obtain ∇u(ε) + (∇u(ε)) − div u(ε)I DT Lε,e (T) = U∞ · Ω − D ∇u(ε) − ∇(Du(ε)) − D ∇u(ε) − ∇(Du(ε)) + q(ε)D ∇η dx (ε)(u(ε) · ∇η) Du(ε) dx + O(ε). − Re U∞ · Ω

11.5. Shape derivative of the drag functional. Adjoint state

375

As mentioned above, u(ε) → u in C 1 (Ω) and (q(ε), (ε)) → (q, ) in C(Ω). Thus we get Lε,e (T) → Le (T) as ε → 0. (11.5.16) Letting ε → 0 in (11.5.9) and using (11.5.15), (11.5.16) we arrive at (11.5.5).

Remark 11.5.2. Recall that η ∈ C ∞ (Ω) is an arbitrary function such that η = 1 in a neighborhood of ∂S and η = 0 in a neighborhood of Σ = ∂B. In particular, ∇η = 0 in a neighborhood of ∂Ω. Notice that the functionals J(Ω) and dJ(Ω)[T] are independent of the choice of η. On the other hand, T ∈ C0∞ (R3 ) is an arbitrary vector ﬁeld vanishing in a neighborhood of Σ. Hence for a given T we can always choose η such that Le (T) = 0. Formulae similar to (11.5.5) are widely used for solving shape optimization problems. The standard simple gradient scheme in shape optimization is the following. We choose an arbitrary compact set S ⊂ B as a starting point of the iteration process. Next we look for a vector ﬁeld T such that dJ(Ω)[T] < 0. The set Sε = (Id + εT)(S) may be considered as the successive iteration step. Then we take Ω = Ω(ε) and repeat the calculations. We may hope that after a number of steps we obtain a conﬁguration close to an optimum. Realization of such a scheme requires eﬃcient methods for calculation of the ﬁrst order shape derivative dJ. We refer the reader to [117, 64] for preliminary numerical results of drag minimization within this framework, with the use of numerical iterations [59] for the ﬁxed point, as well as ﬁnite elements [117] or ﬁnite volumes [64]. From the viewpoint of numerical methods, however, in general representation (11.5.5) falls far short of being a good basis for calculations, in particular when using the level set method [47]. Further we refer to basic Problem 11.1.9 with N = I (unperturbed case) as the state equations. Solutions ϑ = (v, π, ϕ), ζ, m to this problem are called the state variables. The state variables completely describe the ﬂow in the unperturbed domain Ω and are independent of the vector ﬁeld T. Theorem 11.5.1 shows that the derivative of the functional J(Ωε ) consists of two parts, Le and Lu . The geometric part Le is simply the integral of a linear form of T over Ω. The coeﬃcients of this form are deﬁned by the state variables. Hence it is not diﬃcult to calculate Le for all smooth T. In contrast, the dynamical part Lu depends on T in a complicated manner. In order to calculate Lu we have to ﬁnd the very weak solution to the linearized problem and then to substitute this solution into (11.5.6). Notice that the very weak solution is given by Theorem 11.4.6 and depends on T in some implicit way. If we would like to ﬁnd Lu for another vector ﬁeld T, then we have to repeat all calculations. Fortunately, the situation can be drastically improved if we represent Lu via the so-called adjoint state. Adjoint state. It is a remarkable fact of optimization theory that we do not need the material or shape derivatives in order to calculate the ﬁrst order shape derivative dJ. Moreover we can express dJ as the integral of a linear form of T

376

Chapter 11. Sensitivity analysis. Shape gradient of the drag functional

with coeﬃcients depending only on the state variables and the adjoint state which is deﬁned as follows. Deﬁnition 11.5.3. The adjoint state is a solution (h, g, ς, υ, l) ∈ W s+1,r (Ω) × (W s,r (Ω))3 × R to the transposed problem Δh − ∇g = Re H0 (h) − ς∇ϕ − ζ∇υ + B in Ω, div h = Π(a22 g + a12 ς + a32 l) + ΠA in Ω, − div(uς) + σς = Re M0 (h) + a11 ς + a21 g + a31 l + C

in Ω,

u∇υ + συ = a34 l in Ω, h = 0 on ∂Ω, ς = 0 on Σout , υ = 0 on Σin , g − Πg = 0, l = κ (a13 ς + a23 g) dx, Ω

where the coeﬃcients aij are given by (11.4.8), the functions B, A, and C are given by (11.5.7), and H0 (h) = ∇(u)h − div(u ⊗ h),

M0 (h) = (u∇u) · h.

The coeﬃcients and given functions are deﬁned by the state variables. Hence the adjoint state is completely deﬁned by the state variables and is independent of T. Notice that B ∈ W s−1,r (Ω) and A, C ∈ W s,r (Ω). Hence we may apply Lemma 11.4.5 to conclude that the boundary value problem for the adjoint state has a unique solution in the class W s+1,r (Ω) × (Ws,r (Ω))3 × R. Therefore, the adjoint state is well deﬁned and it is unique. The following theorem, which is the main result of this section, shows that Lu can be represented as an integral of a linear form of T. Theorem 11.5.4. Under the assumptions of Theorem 11.5.1, h · D0 + g a20 d0 + ς a10 d0 + υ σd0 + l a30 d0 dx, Lu (T) = Ω

where (h, g, ς, υ, l) is the adjoint state and D0 = D Δu + Δ(Du) + div(T∇u) − Re D u∇u + u∇(Du) , T = div T I − DT − (DT) ,

d0 = div T I,

D = div T I − DT.

Proof. Choosing (H, G, F, M, e) = (B, A, C, 0, 0) and applying Theorem 11.4.6 we arrive at Lu (T) = B, w1 + ω, A0 + ψ, C0 h · D0 + g a20 d0 + ς a10 d0 + υ σd0 + l a30 d0 dx. = Ω

Chapter 12

Boundary value problems for transport equations 12.1 Introduction The ﬁrst order scalar diﬀerential equation u · ∇ϕ + bϕ = f,

(12.1.1)

which is called the transport equation, is one of the basic equations of mathematical physics. It is widely used for mathematical modeling of mass and heat transfer and plays an important role in kinetic theory of such phenomena. In the framework of compressible Navier-Stokes equations the most important examples of transport equations are given by the stationary mass balance equation, div(ϕu) = 0,

(12.1.2)

and the relaxed mass balance equation div(ϕu) + αϕ = h,

(12.1.3)

where the given vector ﬁeld u belongs to the Sobolev space W 1,2 (Ω) and satisﬁes u = U on ∂Ω.

(12.1.4)

A typical boundary value problem for the transport equation can be formulated as follows: Let Ω ⊂ R3 be a bounded domain. Split the boundary of Ω into three disjoint parts, the inlet Σin , the outgoing set Σout , and the characteristic set Σ0 , deﬁned by Σin = {x ∈ ∂Ω : U·n < 0},

Σout = {x ∈ ∂Ω : U·n > 0},

Σ0 = ∂Ω\(Σin ∪Σout ). (12.1.5)

P. Plotnikov and J. Sokoáowski, Compressible Navier–Stokes Equations: Theory and Shape 377 Optimization, Monografie Matematyczne 73, DOI 10.1007/978-3-0348-0367-0_12, © Springer Basel 2012

378

Chapter 12. Transport equations

Let Γ = ∂Ω \ Σin ∩ cl Σin ⊂ Σ0 be the interface between the inlet and the rest of ∂Ω. The problem is to ﬁnd a solution ϕ to the diﬀerential equation (12.1.1) such that ϕ takes a prescribed value ϕ at the inlet Σin , ϕ = ϕ

on Σin .

(12.1.6)

The simplest way to ﬁnd a solution to problem (12.1.1), (12.1.6) is to apply the method of characteristics which can be described as follows. Suppose for a moment that the vector ﬁeld u has continuous derivatives and does not vanish in Ω. Suppose in addition that for each x∗ ∈ Ω, there is a unique C 1 integral curve x = x(s), 0 ≤ s ≤ s∗ , of the vector ﬁeld u, i.e., a solution of the ODE dx = u(x), ds such that x(s∗ ) = x∗ and x(0) ∈ Σin . Then the solution to problem (12.1.1), (12.1.6) is given by ϕ(x∗ ) = φ(s∗ ) where φ is a solution to the Cauchy problem dφ(s) + b(x(s))φ(s) = f (x(s)), s ∈ [0, s∗ ], φ(0) = ϕb (x(0)). ds The method of characteristics does not work if the totality of integral curves has a complicated structure, for example if u has rest points within Ω, and when u is not smooth and therefore the integral curves are not well deﬁned. To address the ﬁrst issue note that the theory of linear transport equations is part of a general theory of elliptic-parabolic equations also known as second order equations with nonnegative quadratic form. The theory deals with general second order partial diﬀerential equations −

n

aij ∂xi ∂xj + u · ∇ + b = h

(12.1.7)

i,j=1

under the assumption n

aij (x)ξi ξj ≥ 0 for all x ∈ Ω and ξ ∈ R3 .

i,j=1

Boundary value problems for second order PDE’s with nonnegative quadratic form were studied by many authors starting from the pioneering papers of Fichera [41], Kohn and Nirenberg [66], and Ole˘ınik and Radkevich [102]. In this section we give a short review of available results. Weak solutions. We start with a discussion of the theory of weak solutions to the boundary value problem for the transport equation. Without loss of generality we may assume that ϕb = 0 and so the boundary condition takes the form ϕ = 0 on Σin .

(12.1.8)

12.1. Introduction

379

Deﬁnition 12.1.1. A function ϕ ∈ L1 (Ω) is a weak solution to problem (12.1.1), (12.1.8) if the integral identity (ϕL ∗ ζ − f ζ) dx = 0 (12.1.9) Ω

holds for all ζ ∈ C 1 (Ω) vanishing on Σout . Here the adjoint operator L ∗ is deﬁned by (12.1.10) L ∗ ζ := − div(uζ) + bζ. The ﬁrst result on the existence of weak solutions to the general equation (12.1.7) is due to Fichera. A complete theory of integrable weak solutions to elliptic-parabolic equations was developed by Ole˘ınik and Radkevich. The following theorem on existence and uniqueness of weak solutions to (12.1.1), (12.1.8) is a particular case of Ole˘ınik’s results (see Theorems 1.5.1 and 1.6.2 in [102]). Theorem 12.1.2. Let Ω ⊂ R3 be a bounded domain with C 2 boundary and let 1 < r ≤ ∞. Assume that the vector ﬁeld u ∈ C 1 (Ω) and the function b ∈ C(Ω) satisfy δ = inf (b(x) − r−1 div u(x)) > 0. x∈Ω

Then problem (12.1.1), (12.1.8) has a weak solution ϕ ∈ Lr (Ω) satisfying ϕLr (Ω) ≤ δ −1 f Lr (Ω) .

(12.1.11)

If, in addition, r > 3 and Γ = cl(Σout ∪ Σ0 ) ∩ cl Σin is a one-dimensional C 1 manifold, then there is a unique weak solution ϕ ∈ Lr (Ω). Moreover, in [102] it is shown that weak solutions are continuous at interior points of Σin and take the boundary value in the classical sense. Remark 12.1.3. The existence of solutions satisfying (12.1.11) was proved in [102] by using the vanishing viscosity method. In view of (12.1.11) the solution obtained by this method is unique. However problem (12.1.1), (12.1.8) may have other solutions which do not satisfy (12.1.11). Hence the problem of uniqueness is not trivial. Strong solutions. The question of regularity of solutions to boundary value problems for transport equations is diﬃcult. All known results [66], [102] relate to the case of multi-connected domains with isolated inlet. We illustrate the theory by two theorems. The ﬁrst is a consequence of a general result of Kohn and Nirenberg (see [66]) on solvability of boundary value problems for elliptic-parabolic equations. Theorem 12.1.4. Let Ω ⊂ Rd be a bounded domain with C ∞ boundary, u, b ∈ C ∞ (Ω), and k ≥ 1 an integer. Furthermore, assume that Γ = cl Σin ∩ cl(Σout ∪ Σ0 ) = ∅

(12.1.12)

380

Chapter 12. Transport equations

and b > c0 > 0, where c0 is a large constant depending only on Ω, uC 3 (Ω) , bC 3 (Ω) , and k. Then for any f ∈ W 2,k (Ω) problem (12.1.1), (12.1.8) has a unique solution satisfying ϕW 2,k (Ω) ≤ C(k, Ω, u, b)f W 2,k (Ω) . The second result is a consequence of [102, Thm. 1.8.1]. Theorem 12.1.5. Assume that Ω is a bounded domain with C 2 boundary and u, b, f ∈ C 1 (R3 ). Furthermore, suppose that: • The vector ﬁeld U = u|∂Ω and the manifold Γ satisfy condition (12.1.12). • There is Ω Ω such that " ∂ui 1 ∂ui # 1 b(x) − sup | div u| + sup ∂xj + 2 sup ∂xj > 0. 2 i j Ω j =i

j =i

Then every weak solution ϕ ∈ L∞ (Ω) to problem (12.1.1), (12.1.8) satisﬁes the Lipschitz condition in cl Ω. If the vector ﬁeld u satisﬁes the nonpermeability condition u · n = 0 on ∂Ω, then Σin = Σout = ∅ and we do not need boundary conditions. This particular case was investigated in detail by Beirão da Veiga [13] and Novotný [96], [97]. The case Γ = ∅ is still poorly investigated. In the next sections we develop an existence theorem for problem (12.1.1), (12.1.8) in fractional Sobolev space under the assumption that Γ satisﬁes some natural geometric conditions.

12.2

Existence theory in Sobolev spaces

The main diﬃculty of the theory of the boundary value problem for the transport equation in the case of nonempty interface Γ is that a solution may develop singularities at Γ. To make things clear consider the following simple example. Example 12.2.1. Consider the boundary value problem ∂x1 ϕ(x) = f (x) in Ω,

ϕ = 0 on Σin ,

(12.2.1)

in the plane domain Ω = {x = (x1 , x2 ) : x2 > (x21 − 1)2 , x1 ∈ R} ⊂ R2 . Assume that f ∈ C 1 (Ω), i.e., f is C 1 on cl Ω in the notation of Section 1.2.1. In this case the inlet is the union of two arcs, Σin = {x : x2 = (x21 − 1)2 , x1 ∈ (−∞, −1)} ∪ {x : x2 = (x21 − 1)2 , x1 ∈ (0, 1)},

12.2. Existence theory in Sobolev spaces

381

and the characteristic set consists of three points Σ0 = Γ = {(−1, 0)} ∪ {(0, 1)} ∪ {(1, 0)}. For α > 1 the intersection of the line {x2 = α} and Ω is the interval 1/2

1/2

I = {−(1 + x2 )1/2 < x1 < (1 + x2 )1/2 } Hence for x2 > 1 the solution to problem (12.2.1) is given by x1 ϕ(x) = √ √ f (s, x2 ) ds for x1 ∈ I. −

1+ x2

For 0 < α ≤ 1 the intersection of the line {x2 = α} and Ω consists of the disjoint intervals I − : −(1 + x2 )1/2 < x1 < −(1 − x2 )1/2 , I + : (1 − x2 )1/2 < x1 < (1 + x2 )1/2 . 1/2

1/2

1/2

1/2

Hence for 0 < x2 ≤ 1 the solution to problem (12.2.1) is given by x1 ϕ(x) = √ √ f (s, x2 ) ds for x1 ∈ I − −

and

1+ x2

x1 ϕ(x) = √

√ 1− x2

f (s, x2 ) ds

for x1 ∈ I + .

√ It follows from these formulae that ϕ has a jump across the segment (0, 2) × {1} emanating from the point (0, 1) ∈ Γ. On the other hand, ϕ is continuous at the −1/2 points (±1, 0) ∈ Γ, but the gradient of ϕ develops a singularity, ∇ϕ ∼ x2 , at these points. Hence there is essential diﬀerence between the behavior of the solution at (0, 1) and (±1, 0). Calculations show that in our case, the vector ﬁeld U = (1, 0) satisﬁes U · ∇(U · n) < 0 at (0, 1) and

U · ∇(U · n) > 0 at (±1, 0).

Hence imposing the condition U · ∇(U · n) > 0 on Γ prevents the formation of discontinuities of solutions to the boundary value problem (12.2.1), but the derivatives of solutions have singularities at Γ in any case. In view of the above discussion we assume that a characteristic set Γ ⊂ ∂Ω and a given vector ﬁeld U satisfy the following condition, referred to as the emergent vector ﬁeld condition. Condition 12.2.2. Assume that ∂Ω is a closed surface of class C 3 and the set Γ is a closed C 3 one-dimensional manifold. Assume also that U ∈ C 3 (∂Ω). Moreover, there is a positive constant c such that U · ∇(U · n) > c > 0 on Γ.

(12.2.2)

Since the vector ﬁeld U is tangent to ∂Ω on Γ, the left hand side of (12.2.2) is well deﬁned.

382

Chapter 12. Transport equations

This condition is obviously fulﬁlled for all strictly convex domains and constant vector ﬁelds. It has a simple geometric interpretation: U · n only vanishes up to the ﬁrst order at Γ, and for each point P ∈ Γ, the vector U(P ) points to the part of ∂Ω where U is an exterior vector ﬁeld. Note that the emergent vector ﬁeld condition plays an important role in theory of the oblique derivative problem for elliptic equations (see [58]). The following theorem is the ﬁrst main result of this chapter. Let us consider the following boundary value problems for linear transport equations: L ϕ := u∇ϕ + σϕ = f L ∗ ϕ∗ := − div(ϕ∗ u) + σϕ∗ = f

in Ω,

ϕ = 0 on Σin ,

(12.2.3)

in Ω,

ϕ∗ = 0 on Σout .

(12.2.4)

Bounded functions ϕ, ϕ∗ are called weak solutions to problems (12.2.3), (12.2.4), respectively, if the integral identities (ϕL∗ ζ ∗ − f ζ ∗ ) dx = 0, (ϕ∗ Lζ − f ζ) dx = 0 (12.2.5) Ω

Ω

hold for all ζ ∗ , ζ ∈ C(Ω) ∩ W 1,1 (Ω), respectively, such that ζ ∗ = 0 on Σout and ζ = 0 on Σin . Theorem 12.2.3. Let Ω ⊂ R3 be a bounded domain. Assume that ∂Ω and U satisfy Condition 12.2.2, and the vector ﬁeld u belongs to C 1 (Ω) and satisﬁes u=U

on ∂Ω.

(12.2.6)

Furthermore, let s and r be constants satisfying 0 < s ≤ 1,

κ := 2s − 3r−1 < 1.

1 < r < ∞,

(12.2.7)

Then there are positive constants σ ∗ > 1 and C, which depend on ∂Ω, U, s, r, uC 1 (Ω) and are independent of σ, such that for any σ > σ ∗ and f ∈ W s,r (Ω) ∩ L∞ (Ω), problem (12.2.3) has a unique solution ϕ ∈ W s,r (Ω) ∩ L∞ (Ω) satisfying ϕW s,r (Ω) ≤ Cσ −1 f W s,r (Ω) + Cσ −1+α f L∞ (Ω) ϕW s,r (Ω) ≤ Cσ

−1

f W s,r (Ω) + Cσ

−1+α

for sr = 1, 2, 1/r

(1 + log σ)

f L∞ (Ω)

for sr = 1, 2. (12.2.8)

Here, the accretivity defect α is deﬁned by α(s, r) = max{0, s − r−1 , 2s − 3r−1 }.

(12.2.9)

12.3 Proof of Theorem 12.2.3 Our strategy is as follows. First, we show that in the vicinity of each P ∈ Σin there exist normal coordinates (y1 , y2 , y3 ) such that u · ∇x = e1 ∇y . Hence, existence

12.3. Proof of Theorem 12.2.3

383

of solutions to the transport equation in the neighborhood of Σin reduces to a boundary value problem for the model equation ∂y1 ϕ + σϕ = f . Next, we prove that the boundary value problem for the model equation has a unique solution in a fractional Sobolev space, which leads to the existence and uniqueness of solutions in the neighborhood of the inlet set. Using the existence of local solutions we reduce problem (12.2.3) to the problem for a modiﬁed equation, which does not require boundary data. Application of well known results on solvability of elliptichyperbolic equations in the case Γ = ∅ ﬁnally gives the existence and uniqueness of solutions to problem (12.2.3).

12.3.1 Normal coordinates First, we introduce some notation which is used throughout this section. For any a > 0 we denote by Qa the cube

and by Q+ a the slab

Qa := [−a, a]3

(12.3.1)

2 Q+ a := [−a, a] × [0, a]

(12.3.2)

in the space of points y = (y1 , y2 , y3 ) ∈ R . We write Y for (y2 , y3 ) so that y = (y1 , Y ). 3

Deﬁnition 12.3.1. A standard parabolic neighborhood associated with the constant c0 is a compact subset of a slab Q+ a , deﬁned by − + Pa = {y = (y1 , Y ) ∈ Q+ a : a (Y ) ≤ y1 ≤ a (Y )},

(12.3.3)

where a± : [−a, a] × [0, a] → R are continuous, piecewise C 1 functions satisfying −a ≤ a− (Y ) ≤ 0 ≤ a+ (Y ) ≤ a, √ √ −c0 y3 ≤ a− (Y ) ≤ a+ (Y ) ≤ c0 y3 , √ |∂y2 a± (Y )| ≤ c0 , |∂y3 a± (Y )| ≤ c0 / y3 .

(12.3.4)

Denote by Σyin and Σyout the surfaces Σyin = {y : Y ∈ Qin , y1 = a− (Y )}, Σyout = {y : Y ∈ Qout , y1 = a+ (Y )}, where Qin = {Y : a− (Y ) > −a} and Qout = {Y : a+ (Y ) < a}. It is clear that ∂Pa = (∂Qa ∩ ∂Pa ) ∪ Σyin ∪ Σyout . Lemma 12.3.2. Suppose that Γ = cl Σin ∩ cl (Σout ∪ Σ0 ) is a C 3 manifold, and Γ and the vector ﬁeld U ∈ C 3 (∂Ω) satisfy Condition 12.2.2. Let u ∈ C 1 (R3 ) be a compactly supported vector ﬁeld such that u = U on ∂Ω. Denote M = uC 1 (R3 ) . Then there are positive constants a, c, C, ρ, and R, depending only on M , ∂Ω, and U, with the properties:

384

Chapter 12. Transport equations

(P1) For any point P ∈ Γ, there exists a diﬀeomorphism y → x(y) of the cube Qa onto a neighborhood OP of P which satisﬁes ∂y1 x(y) = u(x(y)) and

xC 1 (Qa ) + x−1 C 1 (OP ) ≤ C,

in Qa ,

(12.3.5)

|x(y)| ≤ C|y|.

(12.3.6)

(P2) There is a standard parabolic neighborhood Pa associated with the constant c such that x(Pa ) = OP ∩ Ω,

x(Σyin ) = Σin ∩ OP ,

x(Σyout ) = Σout ∩ OP .

(12.3.7)

(P3) Denote by Ga ⊂ Pa the domain Ga = {y = (y1 , Y ) ∈ Pa : Y ∈ Qin },

(12.3.8)

and by BP (ρ) the ball |x − P | ≤ ρ. Then BP (ρ) ∩ Ω ⊂ x(Ga ) ⊂ OP ∩ Ω ⊂ BP (R) ∩ Ω.

(12.3.9)

The following lemma shows the existence of normal coordinates in the vicinity of points of the inlet Σin . Lemma 12.3.3. Let u and U meet all requirements of Lemma 12.3.2 and suppose −U(P ) · n > N > 0. Then there are b > 0 and C > 0, depending only on N , Ω and M = uC 1 (Ω) , with the following properties. There exists a diﬀeomorphism y → x(y) of the cube Qb = [−b, b]3 onto a neighborhood OP of P which satisﬁes ∂y3 x(y) = u(x(y)) and

in Qb ,

x(y1 , y2 , 0) ∈ ∂Ω ∩ OP

xC 1 (Qb ) + x−1 C 1 (OP ) ≤ C,

for |y2 | ≤ a,

x(0) = P.

(12.3.10) (12.3.11)

Moreover, BP (ρi ) ∩ Ω ⊂ x(Qb ∩ {y3 > 0}) ⊂ BP (Ri ) ∩ Ω for ρi = C

−1

(12.3.12)

b and Ri = Cb.

Lemmas 12.3.2 and 12.3.3 are proved in Section 12.3.6.

12.3.2 Model equation Let Pa be a standard parabolic neighborhood associated with the constant c0 (see Deﬁnition 12.3.1). Consider the boundary value problem ∂y1 ϕ(y) + σϕ(y) = f (y) in Pa ,

ϕ(y) = 0 for y1 = a− (Y ).

(12.3.13)

12.3. Proof of Theorem 12.2.3

385

Lemma 12.3.4. Suppose that the exponents r, s and the accretivity defect α meet all requirements of Theorem 12.2.3, σ > 1, and f ∈ W s,r (Pa ) ∩ L∞ (Pa ). Then there is a constant c, depending only on a, c0 , r, s, such that problem (12.3.13) has a solution that satisﬁes ϕW s,r (Pa ) ≤ c σ −1 f W s,r (Pa ) + ℘(s, r, σ)f L∞ (Pa ) , (12.3.14) ϕL∞ (Pa ) ≤ σ −1 f L∞ (Pa ) , where

0 σ −1+α ℘(s, r, σ) = σ −1+α (1 + log σ)1/r

for rs = 1, 2, for rs = 1, 2.

The proof is given in Appendix 12.3.7. Next, consider the boundary value problem ∂y3 ϕ + σϕ = f

2 in Q+ a = [−a, a] × [0, a],

ϕ(y) = 0 for y3 = 0.

(12.3.15)

Lemma 12.3.5. Suppose that the exponents r, s and the accretivity defect α meet all requirements of Theorem 12.2.3 and σ > 1. Then for any f ∈ W s,r (Pa ) ∩ L∞ (Pa ), problem (12.3.15) has a unique solution satisfying ϕW s,r (Q+ . (12.3.16) ≤ c(r, s, a) σ −1 f W s,r (Q+ + ℘(s, r, σ)f L∞ (Q+ a) a) a)

Proof. The proof of Lemma 12.3.4 can be used also in this simple case.

12.3.3 Local estimates It follows from the assumptions of Theorem 12.2.3 that u and Ω meet all requirements of Theorem 12.1.2. In view of this theorem for every σ > div uC(Ω) and f ∈ L∞ (Ω) problem (12.2.3) has a unique weak solution ϕ ∈ L∞ (Ω). In this section we estimate the W s,r -norm of ϕ in a neighborhood of an arbitrary point of Σin . It follows from the assumptions of Theorem 12.2.3 that the vector ﬁeld u and the manifold Γ satisfy all assumptions of Lemma 12.3.2. Therefore, there exist positive numbers a, ρ and R, depending only on Ω and uC 1 (Ω) , such that for all P ∈ Γ, the canonical diﬀeomorphism x : Qa → OP is well deﬁned and meets all requirements of Lemma 12.3.2. The following lemma is the main result of this section. Lemma 12.3.6. Assume that r, s, α and U are as in Theorem 12.2.3 and uC 1 (Ω) ≤ M . Then there exists σ ∗ > 1, depending on Ω, s, r, and M , such that for any f ∈ W s,r (Ω) ∩ L∞ (Ω) and σ > σ ∗ , the weak solution ϕ ∈ L∞ (Ω) to problem (12.2.3) deﬁned by Theorem 12.1.2 satisﬁes ϕW s,r (BP (ρ)∩Ω) ≤ c(σ −1 f W s,r (BP (R)∩Ω) + ℘(s, r, σ)f L∞ (BP (R)∩Ω) ), ϕL∞ (BP (ρ)∩Ω) ≤ σ −1 f L∞ (BP (R)∩Ω) , (12.3.17)

386

Chapter 12. Transport equations

where the constant c depends only on ∂Ω, U, M , s, and where r, ρ, R are deﬁned by Lemma 12.3.2. Proof. The proof falls into three steps. Step 1. Let Ga be the domain deﬁned in Lemma 12.3.2. Denote by Da ⊂ Ω the domain x(Ga ) and set M(y) = x (y),

|M|(y) = det x (y),

y ∈ Ga .

In view of Lemma 12.3.2 we have M, ∂y1 M ∈ C 1 (Ga ). Next choose η ∈ C 1 (Da ) and set ζ(y) = η(x(y)). Our ﬁrst task is to show that for all h ∈ L1 (Da ), (12.3.18) h(x) div(η(x)u(x)) dx = h(x(y))∂y1 ζ(y)|M|(y) dy. Da

Ga

Since the functions div(ηu) and ∂y1 (|M| ζ) are continuous in cl Da and cl Ga respectively, it suﬃces to prove (12.3.18) for h ∈ C0∞ (Da ). It is easy to check that h ◦ x ∈ C 1 (Ga ) is compactly supported in Ga and (∇x h)(x(y)) = (M−1 ) ∇y h(x(y)). Thus we get h(x) div(η(x)u(x)) dx = − η(x)∇x h(x) · u(x) dx Da Da =− η(x(y))∇y h(x(y)) · M−1 u(x(y)) |M| dy Ga =− ζ(y)∇y h(x(y)) · |M|M−1 u(x(y)) dy. Ga

(12.3.19) Notice that the matrix M has columns ∂yi x(y), i = 1, 2, 3, and hence $ . |M|M−1 = ∂y2 x × ∂y3 x, ∂y3 x × ∂y1 x, ∂y1 x × ∂y2 x , |M| = (∂y2 x × ∂y3 x) · ∂y1 x. On the other hand, Lemma 12.3.2 yields u(x(y)) = ∂y1 x(y). It follows that |M|M−1 u(x(y)) = (|M|, 0, 0) = |M|e1 . Lemma 12.3.2 also implies that ∂y1 |M| belongs to C 1 (Ga ). Thus we get −

ζ(y)∇y h(x(y)) · |M|M−1 u(x(y)) dy Ga =− ζ(y)∂y1 h(x(y)) |M| dy = Ga

Ga

Combining this with (12.3.19) we arrive at (12.3.18).

h(x(y))∂y1 (|M|ζ(y)) dy.

12.3. Proof of Theorem 12.2.3

387

Step 2. Let ϕ be a bounded weak solution to problem (12.2.3). Let us consider the restriction of ϕ to Da and set ϕ(y) = ϕ(x(y)),

f (y) = f (x(y)),

y ∈ Ga .

Now our task is to show that ∂y1 ϕ belongs to L∞ (Ga ) and satisﬁes the equations ∂y1 ϕ + σϕ = f

in Ga ,

ϕ = 0 on Σyin .

(12.3.20)

To this end choose ζ ∈ C 1 (Ga ) vanishing in a neighborhood of ∂Ga \ Σyin . In view of Lemma 12.3.2 (equalities (12.3.7), (12.3.8)) we have Σin ∩ ∂Da = Σyin ∩ Ga . Hence the function η = ζ ◦ x−1 belongs to C 1 (Da ) and vanishes in a neighborhood of ∂Da \ Σin . Extend η by zero to Ω. The extension belongs to C 1 (Ω) and vanishes in a neighborhood of ∂Ω \ Σin . Since ϕ is a bounded weak solution to problem (12.2.3), we have (ϕ div(ηu) − σϕη + f η) dx = (ϕ div(ηu) − σϕη + f η) dx = 0. Da

Ω

From this and identity (12.3.18) with h replaced by ϕ we obtain (ϕ∂y1 (ζ|M|) − σϕζ|M| + f ζ|M|) dy = 0.

(12.3.21)

Ga

Next, the open set Ga can be covered by a countable collection of cubes Cn = In3 ,

In = [αn , βn ],

n ≥ 1.

Denote by C01 (In ) ⊂ C 1 (In ) the Banach space which consists of functions υ ∈ C 1 (In ) vanishing at the boundary of In . Since C01 (In ) is a separable Banach space, it contains a countable dense subset {υm,n }m≥1 . Now choose ς ∈ C0∞ (In2 ) and set ζ(y1 , Y ) = ς(Y )υm,n (y1 ). Assume that ζ is extended by zero to Ga . It is clear that the extension belongs to C 1 (Ga ) and is compactly supported in Ga . Substituting ζ in (12.3.21) we obtain ς(Y )Ξm,n (Y ) dY = 0, 2 In

where Ξm,n (Y ) =

In

ϕ(y1 , Y )∂y1 υm,n (y1 )|M(y1 , Y )| + f − σϕ (y1 , Y ) υm,n (y1 )|M(y1 , Y )| dy1 .

Since ς(Y ) is an arbitrary function of class C0∞ (In2 ), we conclude that there is a set Em,n ⊂ R2 of zero measure such that Ξm,n (Y ) = 0 for all Y ∈ In2 \ Em,n . It

388

Chapter 12. Transport equations

is clear that the set E = m,n Em,n has zero measure and Ξm,n (Y ) = 0 for every Y ∈ In2 \ E and all m ≥ 1. Since the υm,n are dense in C01 (In ), the identity In

(ϕ∂y1 (υ|M|) − σϕυ|M| + f υ|M|)(y1 , Y ) dy1 = 0

(12.3.22)

holds for all Y ∈ In2 \ E and all υ ∈ C01 (In ). Since |M|−1 (·, Y ) belongs to C 1 (In ) the function υ/|M|(·, Y ) belongs to C01 (In ). Hence we can replace υ by υ/|M| in (12.3.22). Therefore, In

(ϕ∂y1 υ − σϕυ + f υ)(Y, y1 ) dy1 = 0

for all Y ∈ In2 \ E and all υ ∈ C01 (In ). This means that for every Y ∈ In2 \ E the function ϕ(·, Y ) has a bounded derivative satisfying ∂y1 ϕ = −σϕ + f

in Cn .

Recalling that the sets Cn cover Ga we obtain ∂y1 ϕ = −σϕ + f

a.e. in Ga .

(12.3.23)

Since ϕ and f are bounded in Ga , it follows that ∂y1 ϕ is bounded in Ga . In view of Lemma 12.3.2 the open connected set Ga is a union of segments parallel to the y1 axis, Ga = (a− (Y ), a+ (Y )) × {Y }. (12.3.24) Y ∈Qin

Since ∂y1 ϕ is bounded there exists a bounded limit ϕ− (Y ) =

lim

y1 →a− (Y )

ϕ(y1 , Y ),

Y ∈ Qin .

Now we turn to the integral identity (12.3.21). Integrating by parts and using (12.3.23) we obtain ζ(a− (Y ), Y )ϕ− (Y ) dY = 0 Qin

for all ζ ∈ C 1 (Ga ) vanishing in a neighborhood of ∂Ga \ Σyin = Ga \ {(a− (Y ), Y ) : Y ∈ Qin }. Hence ϕ− = 0, which along with (12.3.23) yields (12.3.20). Finally notice that by (12.3.20), ϕ(y1 , Y ) =

y1

a− (Y

eσ(t−y1 ) f (t, Y ) dt, )

Y ∈ Qin , a− (Y ) ≤ y1 ≤ a+ (Y ). (12.3.25)

12.3. Proof of Theorem 12.2.3 Step 3.

389

Consider the boundary value problem ∂y1 ϕ˜ + σ ϕ˜ = f

in Pa ,

ϕ(y) ˜ = 0 for y1 = a− (Y ),

(12.3.26)

in the parabolic neighborhood Pa ⊃ Ga given by Lemma 12.3.2. Obviously, y1 eσ(t−y1 ) f (t, Y ) dt, Y ∈ [−a, a]2 , a− (Y ) ≤ y1 ≤ a+ (Y ). ϕ(y ˜ 1, Y ) = a− (Y )

Since Qin ⊂ [−a, a]2 , it follows from (12.3.24) and (12.3.25) that ϕ = ϕ˜ in Ga . Next Lemma 12.3.2 implies BP (ρ)∩Ω ⊂ Da = x(Ga ). Since x is a diﬀeomorphism, we conclude that ϕW s,r (BP (ρ)∩Ω) ≤ ϕW s,r (Da ) ≤ cϕW s,r (Ga ) ≤ ϕ ˜ W s,r (Pa ) .

(12.3.27)

On the other hand, it follows from Lemma 12.3.4 and the inclusion X(Pa ) ⊂ Ω ∩ BP (ρ) that ϕ ˜ W s,r (Pa ) ≤ c(σ −1 f W s,r (Pa ) + ℘(s, r, σ)f L∞ (Pa ) ) ≤ c(σ −1 f W s,r (Ω∩BP (ρ)) + ℘(s, r, σ)f L∞ (Ω∩BP (ρ)) ). Here, the constant c depends only on M , r, s, U and ∂Ω. Combining these estimates with (12.3.27) we obtain the ﬁrst inequality in (12.3.17). The second obviously follows from (12.3.25) and the estimates f L∞ (Ga ) = f L∞ (Da ) ≤ f L∞ (Ω∩BP (ρ)) .

To formulate a similar result for the interior points of the inlet we deﬁne Σin = {x ∈ Σin : dist(x, Γ) ≥ ρ/3},

(12.3.28)

where the constant ρ is given by Lemma 12.3.2. It is clear that inf −U(P ) · n(P ) ≥ N > 0,

P ∈Σin

where the constant N depends only on M , U, and ∂Ω. It follows from Lemma 12.3.3 that there are positive numbers b, ρi and Ri such that for each P ∈ Σin , the canonical diﬀeomorphism x : Qb → OP ⊂ BP (Ri ) is well deﬁned and satisﬁes the hypotheses of Lemma 12.3.3. Lemma 12.3.7. Suppose that the exponents s, r satisfy condition (12.2.7) and that uC 1 (Ω) ≤ M . Then there exists σ ∗ > 1, depending on Ω, s, r, and M , such that for any f ∈ W s,r (Ω) ∩ L∞ (Ω) and σ > σ ∗ , the weak solution ϕ ∈ L∞ (Ω) to problem (12.2.3) deﬁned by Theorem 12.1.2 satisﬁes ϕW s,r (BP (ρi )∩Ω) ≤ c σ −1 f W s,r (BP (Ri )∩Ω) + ℘(s, r, σ)f L∞ (BP (Ri )∩Ω) , ϕL∞ (BP (ρi )∩Ω) ≤ σ −1 f L∞ (BP (Ri )∩Ω) . (12.3.29) where c depends on Σ, M , U and the exponents s, r.

390

Chapter 12. Transport equations

Proof. Using the normal coordinates given by Lemma 12.3.3 and repeating the proof of Lemma 12.3.6 we obtain ∂y3 ϕ + σϕ = f

in Qb ,

ϕ = 0 for y3 = 0,

where ϕ = ϕ ◦ x and f = f ◦ x. Applying Lemma 12.3.4 and arguing as in the proof of Lemma 12.3.6 we obtain (12.3.29).

12.3.4 Estimates near the inlet We are now in a position to prove local estimates of the solution for the boundary value problem (12.2.3) near the inlet. Let Ωt be the tubular neighborhood of the set Σin , Ωt = {x ∈ Ω : dist(x, Σin ) < t}. Lemma 12.3.8. Let t = min{ρ/2, ρi /2} and T = max{R, Ri }, where the constants ρ, ρi and R, Ri are deﬁned by Lemmas 12.3.2 and 12.3.3, respectively. Then there exists σ ∗ > 1, depending on Ω, s, r, and uC 1 (Ω) , such that for any f ∈ W s,r (Ω) ∩ L∞ (Ω) and σ > σ ∗ , the weak solution ϕ ∈ L∞ (Ω) to problem (12.2.3) satisﬁes ϕW s,r (Ωt ) ≤ C σ −1 f W s,r (ΩT ) + ℘(s, r, σ)f L∞ (ΩT ) , (12.3.30) ϕL∞ (Ωt ) ≤ σ −1 f L∞ (ΩT ) . where the constant C depends only on M , ∂Ω, U and the exponents s, r. Proof. There exists a covering of the characteristic manifold Γ by a ﬁnite collection of balls BPi (ρ/4), 1 ≤ i ≤ m, Pi ∈ Γ. Obviously, the balls BPi (ρ) cover the set VΓ = {x ∈ Ω : dist(x, Γ) < ρ/2}. By Lemma 12.3.6 for any P ∈ Γ, the solution to problem (12.2.3) is uniquely determined in some neighborhood of P containing the ball BP (ρ). Hence, it suﬃces to prove estimates (12.3.30). To this end recall that by deﬁnition (see (1.5.1)) uW s,r (Ω) = uLr (Ω) + |u|s,r,Ω ,

where |u|rs,r,Ω

|x − y|−3−rs |u(x) − u(y)|r dxdy.

= Ω×Ω

We have

|ϕ|rs,r,VΓ =

2 VΓ

≤

|x − y|−3−rs |ϕ(x) − ϕ(y)|r dxdy |x − y|−3−rs |ϕ(x) − ϕ(y)|r dxdy

(VΓ )2 ∩{|x−y|<ρ/2} + cρ−3−rs ϕrL∞ (VΓ )

meas (VΓ )2 .

(12.3.31)

12.3. Proof of Theorem 12.2.3

391

Since any pair of points x, y ∈ VΓ with |x − y| < ρ/2 belongs to some ball BPi (ρ), the ﬁrst term on the right hand side above does not exceed |x − y|−3−rs |ϕ(x) − ϕ(y)|r dxdy = |ϕ|rs,r,BP (ρ)∩Ω , (BPi (ρ)∩Ω)2

i

i

i

which leads to ϕW s,r (VΓ ) = |ϕ|s,r,VΓ + ϕLr (VΓ ) ≤c ϕW s,r (BPi (ρ)∩Ω) + cϕL∞ (VΓ ) ,

(12.3.32)

i

where c depends on s, r and ρ, i.e., on s, r, U, ∂Ω and M . By Lemma 12.3.6 in each of these balls, the solution to problem (12.2.3) satisﬁes (12.3.17), so ϕW s,r (VΓ ) ≤ c℘(s, r, σ) f L∞ (BPi (R)∩Ω) + cσ

−1

i

f W s,r (BPi (R)∩Ω) + cϕL∞ (VΓ ) .

(12.3.33)

i

On the other hand, ϕL∞ (BPi (ρ)∩Ω) ≤ σ −1 f L∞ (BPi (R)∩Ω) . Moreover, since BPi (R) ∩ Ω ⊂ ΩT we have ℘(s, r, σ)

f rL∞ (BP

i

i

(R)∩Ω)

+ σ −1

f W s,r (BPi (R)∩Ω)

i

≤ m℘(s, r, σ)f L∞ (ΩT ) + mσ −1 f W s,r (ΩT ) . From this and (12.3.33), we ﬁnally obtain the estimates for the solution to problem (12.2.3) in the neighborhood of the characteristic manifold Γ, ϕW s,r (VΓ ) ≤ c℘(s, r, σ)f L∞ (ΩT ) + cσ −1 f W s,r (ΩT ) ,

(12.3.34)

where c depends only on M , ∂Ω, U and s, r. Our next task is to obtain a similar estimate in the neighborhood of the compact set Σin ⊂ Σin given by (12.3.28). To this end, we deﬁne Vin = {x ∈ Ω : dist(x, Σin ) < ρi /2}. Let BPk (ρi /4), 1 ≤ k ≤ m, be a minimal collection of balls of radius ρi /4 covering Σin . It is clear that the sets BPk (ρi ) ∩ Ω cover Vin . Arguing as in the proof of (12.3.32) we obtain ϕW s,r (Vin ) ≤ ϕW s,r (BPk (ρi )∩Ω) + cϕL∞ (Vin ) . k

From this and Lemma 12.3.7 we obtain f W s,r (BPk (Ri )∩Ω) + c℘(s, r, σ) f L∞ (BPk (Ri )∩Ω) . ϕW s,r (Vin ) ≤ cσ −1 k

k

392

Chapter 12. Transport equations

Thus, we get ϕW s,r (Vin ) ≤ cσ −1 f W s,r (ΩT ) + c℘(r, s, σ)f L∞ (ΩT ) . Since VΓ and Vin cover Ωt , this inequality along with (12.3.34) yields (12.3.30).

12.3.5 Partition of unity Let us turn to the analysis of the general problem L ϕ := u · ∇ϕ + σϕ = f

in Ω,

ϕ = 0 on Σin .

(12.3.35)

We split the weak solution ϕ ∈ L∞ (Ω) to problem (12.3.35) into two parts, namely the local solution, deﬁned by Lemma 12.3.8, and the remainder vanishing near the inlet. To this end ﬁx η ∈ C ∞ (R) such that 0 ≤ η ≤ 3,

η(u) = 0 for u ≤ 1 and

η(u) = 1 for u ≥ 3/2,

and introduce the one-parameter family of smooth functions dist(y, Σin ) 1 2(x − y) χt (x) = η dy, t3 R3 t t

(12.3.36)

(12.3.37)

where ∈ C ∞ (R3 ) is a standard mollifying kernel with supp ⊂ {|y| ≤ 1}. It follows that χt is a C ∞ function with |∇χt | ≤ c/t and χt (x) = 0

for dist(x, Σin ) ≤ t/2,

χt (x) = 1

for dist(x, Σin ) ≥ 2t. (12.3.38)

The functions χt/2 and 1 − χt/2 form a partition of unity in Ω. Now, let t be as in Lemma 12.3.8 and write ϕ(x) = (1 − χt/2 (x))ϕ(x) + φ(x).

(12.3.39)

By (12.3.38) and Lemma 12.3.8, φ ∈ L∞ (Ω) vanishes in Ωt/4 and satisﬁes in a weak sense the equations u · ∇φ + σφ = χt/2 f + ϕu · ∇χt/2 =: F

in Ω,

φ = 0 on Σin .

˜(x) = χt/8 (x)u(x). It is easy to see that Next, introduce a new vector ﬁeld u χt/8 = 1 on the support of φ, and hence φ is also a weak solution to the modiﬁed transport equation ˜ · ∇φ + σφ = F in Ω. L˜φ := u (12.3.40) The advantage gained here is that the topology of integral curves of the modiﬁed ˜ drastically diﬀers from that of u. The corresponding outgoing and vector ﬁeld u ˜ in = ∅. In particular, equacharacteristic sets have a diﬀerent structure and Σ tion (12.3.40) does not require boundary conditions. Finally, the C 1 -norm of the modiﬁed vector ﬁeld can be bounded by ˜ uC 1 (Ω) ≤ M (1 + c1 t−1 ),

(12.3.41)

In the following lemma the existence and uniqueness of solutions to the modiﬁed equation is established.

12.3. Proof of Theorem 12.2.3

393

Lemma 12.3.9. Suppose that σ > σ ∗ + 1,

σ ∗ = 16M (1 + c1 t−1 ) + 16,

M = uC 1 (Ω) ,

(12.3.42)

and furthermore 0 ≤ s ≤ 1, r > 1. Then for any F ∈ W s,r (Ω) ∩ L∞ (Ω), equation (12.3.40) has a unique weak solution φ ∈ W s,r (Ω) ∩ L∞ (Ω) such that φW s,r (Ω) ≤ cσ −1 F W s,r (Ω) ,

φL∞ (Ω) ≤ σ −1 F L∞ (Ω) ,

(12.3.43)

where c depends only on r. Proof. We can assume that F ∈ C 1 (Ω). By (12.3.41) and (12.3.42), the vector ˜ and σ meet all requirements of Theorem 12.1.5. Hence, equation (12.3.40) ﬁelds u has a unique solution φ ∈ W 1,∞ (Ω). For i = 1, 2, 3 and τ > 0, deﬁne the ﬁnite diﬀerence operator 1 δiτ φ = (φ(x + τ ei ) − φ(x)). τ It can be easily seen that ˜ · ∇δiτ φ + σδiτ φ = δiτ F − δiτ u ˜ · ∇φ(x + τ ei ) in Ω ∩ (Ω − τ ei ). u

(12.3.44)

˜ in = ∅, Next, set ηh (x) = η(dist(x, ∂Ω)/h), where η is deﬁned by (12.3.36). Since Σ the inequality g˜ u · ∇ηh (x) dx ≤ 0

lim sup h→0

(12.3.45)

Ω

holds for all nonnegative functions g ∈ L∞ (Ω). Choosing h > τ , multiplying both sides of equation (12.3.44) by ηh |δiτ φ|r−2 δiτ φ and integrating the result over Ω ∩ (Ω − τ ei ) we obtain 1 ˜ · ∇ηh dx ˜ ηh |δiτ φ| σ − div u dx − |δiτ φ|r u r Ω∩(Ω−τ ei ) Ω∩(Ω−τ ei ) ˜ · ∇φ(x + τ ei ) ηh |δiτ φ|r−2 δiτ φ dx. = δiτ F − δiτ u

r

Ω∩(Ω−τ ei )

Letting τ → 0 and then h → 0 and using inequality (12.3.45) we obtain 1 ˜ dx ≤ (∂xi F − ∂xi u ˜ · ∇φ)|∂xi φ|r−2 ∂xi φ dx. (12.3.46) |∂xi φ|r σ − div u r Ω Ω Next, note that i

˜ · ∇φ∂xi φ|r−2 ∂xi φ ≤ 3˜ ∂xi u uC 1 (Ω) |∂xi φ|r . i

On the other hand, since 1/r + 3 ≤ 4, inequalities (12.3.41) and (12.3.42) imply σ − (1/r + 3)˜ uC 1 (Ω) ≥ σ − σ ∗ ≥ 1.

394

Chapter 12. Transport equations

From this we conclude that (σ − σ ∗ ) |∂xi φ|r dx ≤ |∂xi φ|r−1 |∂xi F | dx ≤ c∇φr−1 Lr (Ω) ∇F Lr (Ω) , Ω

i

i

Ω

which leads to ∇φLr (Ω) ≤ c(r)σ −1 ∇F Lr (Ω)

for σ > σ ∗ (M, r).

(12.3.47)

Next, multiplying both sides of (12.3.40) by |φ|r−2 ηh and integrating the result over Ω we get ˜)ηh |φ|r dx − ˜ · ∇ηh dx = (σ − r−1 div u |φ|r u F ηh |φ|r−2 φ dx. Ω

Ω

Ω

The passage h → 0 gives the inequality ˜)|φ|r dx ≤ (σ − r−1 div u |F | |φ|r−1 dx. Ω

Recalling that σ − r

−1

Ω ∗

˜ ≥ σ − σ , we ﬁnally obtain div u φLr (Ω) ≤ c(r)σ −1 F Lr (Ω) .

(12.3.48)

Inequalities (12.3.47) and (12.3.48) imply (12.3.43) for s = 0, 1. Consequently, for σ > σ ∗ , the linear operator L˜−1 : F → φ is continuous between the Banach spaces Lr (Ω) and W 1,r (Ω) and its norm does not exceed c(r)σ −1 . Recall that W s,r (Ω) is the interpolation space [Lr (Ω), W 1,r (Ω)]s,r . From interpolation theory we thus conclude that (12.3.43) holds for all s ∈ [0, 1]. It remains to note that the L∞ estimate follows directly from Theorem 12.1.2. We are now in a position to complete the proof of Theorem 12.2.3. Fix σ > σ ∗ +1, where the constant σ ∗ depends only on Σ, U and uC 1 (Ω) , and it is deﬁned by (12.3.42). We can assume that f ∈ C 1 (Ω). The existence and uniqueness of a weak bounded solution for σ > σ ∗ follows from Theorem 12.1.2. Therefore, it suﬃces to prove estimate (12.2.8) for ϕW s,r (Ω) . Since W s,r (Ω) ∩ L∞ (Ω) is a Banach algebra, representation (12.3.39) together with inequality (12.3.38) implies ϕW s,r (Ω) ≤ c(ϕW s,r (Ωt ) + ϕL∞ (Ωt ) ) + cφW s,r (Ω) .

(12.3.49)

On the other hand, Lemma 12.3.9 along with (12.3.40) yields φW s,r (Ω) ≤ cσ −1 F W s,r (Ω) ≤ cσ −1 χt/2 f W s,r (Ω) + σ −1 ϕu · ∇χt/2 W s,r (Ω) ≤ cσ −1 (f W s,r (Ω) + ϕW s,r (Ωt ) ) since ∇χt/2 ∈ C0∞ (Ωt ). Substituting these estimates into (12.3.49) we arrive at ϕW s,r (Ω) ≤ c σ −1 ϕW s,r (Ωt ) + ϕL∞ (Ωt ) + σ −1 f W s,r (Ωt ) + σ −1 f L∞ (Ω) , which along with (12.3.30) leads to estimate (12.2.8), and the theorem follows.

12.3. Proof of Theorem 12.2.3

395

12.3.6 Proofs of Lemmas 12.3.2 and 12.3.3 Proof of Lemma 12.3.2. We start with the proof of (P1). Choose an arbitrary P ∈ Γ and recall the emergent ﬁeld Condition 12.2.2. Without loss of generality we can assume that Cartesian coordinates (x1 , x2 , x3 ) have origin at P and U(P ) = (U, 0, 0) with U = |U(P )| and n(P ) = (0, 0, −1). It follows from this and Condition 12.2.2 that there is a neighborhood O = [−k, k]2 × [−t, t] of P such that the intersections ∂Ω ∩ O and Γ ∩ O are deﬁned by F0 (x) ≡ x3 − F (x1 , x2 ) = 0,

∇F0 (x) · U(x) = 0,

and Ω ∩ O is the epigraph {F0 > 0} ∩ O. The function F satisﬁes F C 2 ([−k,k]2 ) ≤ K,

F (0, 0) = 0,

∇F (0, 0) = 0,

(12.3.50)

where the constants k, t < 1 and K > 1 depend only on the curvature of ∂Ω and are independent of the point P . Since the vector ﬁeld U is transversal to Γ, the manifold Γ ∩ O admits a parameterization x = g(y2 ), y2 , F (g(y2 ), y2 ) (12.3.51) such that g(0) = 0 and gC 2 ([−k,k]) ≤ C, where the constant C > 1 depends only on Ω and U. With this notation, the inequality (12.2.2) implies the existence of positive constants N ± independent of P such that for x ∈ ∂Ω given by the condition F0 (x1 , x2 , x3 ) = x3 − F (x1 , x2 ) = 0, we have N − x1 − g(x2 ) ≤ −∇F0 (x) · U(x) ≤ N + x1 − g(x2 ) for x1 > g(x2 ), −N − x1 − g(x2 ) ≤ ∇F0 (x) · U(x) ≤ −N + x1 − g(x2 ) for x1 < g(x2 ). (12.3.52) Consider the Cauchy problem ∂y1 x = u(x(y)) in Qa , x1 (y) = g(y2 ), x2 (y) = y2 for y1 = 0, x3 = F (g(y2 ), y2 ) + y3 for y1 = 0.

(12.3.53)

We can assume that 0 < a < k < 1. For any such a, problem (12.3.53) has a unique solution in C 1 (Qa ). Denote F(y) = Dy x(y). Calculations show that ⎞ ⎛ ⎞ ⎛ U g (0) 0 g (y2 ) 0 u1 1 0 ⎠, 1 0 ⎠ , F(0) = ⎝ 0 F0 := F(y)|y1 =0 = ⎝ u2 0 0 1 u3 ∂y2 F (g(y2 ), y2 ) 1 which implies

F(0)±1 ≤ C/3,

F0 (y) − F(0) ≤ ca.

(12.3.54)

396

Chapter 12. Transport equations

Diﬀerentiation of (12.3.53) leads to an ordinary diﬀerential equation for F, ∂y1 F = Dy u(x)F,

F|y1 =0 = F0 .

From this we get ∂y1 F − F0 ≤ M (F − F0 + F0 ), and hence F − F0 ≤ c(M )F0 a. Combining this with (12.3.54), we arrive at F(y) − F(0) ≤ ca.

(12.3.55)

This inequality along with the implicit function theorem implies the existence of a > 0, depending only on M and Ω, such that the mapping x = x(y) takes diﬀeomorphically the cube Qa onto some neighborhood of the point P and satisﬁes inequalities (12.3.6). Let us turn to the proof of (P2). Observe that the manifold x−1 (∂Ω ∩ O) is deﬁned by Φ0 (y) := x3 (y) − F (x1 (y), x2 (y)) = 0, y ∈ Qa . Let us show that Φ0 is strictly monotone in y3 and has opposite signs on the faces y3 = ±a. To this end note that the formula for F(0) along with (12.3.55) implies |∂y3 x3 (y) − 1| + |∂y3 x1 (y)| + |∂y3 x2 (y)| ≤ ca in Qa . Thus, we get 1 − ca ≤ ∂y3 Φ0 (y) = ∂y3 x3 (y) − ∂xi F (x1 , x2 )∂y3 xi (y) ≤ 1 + ca. It follows from (12.3.55) that for y3 = 0, we have |x3 (y)| ≤ ca|y|, which along with (12.3.6) yields the estimate |Φ0 (y)| ≤ |x3 (y)| + |F (x(y))| ≤ ca|y| + KC|y|2 ≤ ca2

for y3 = 0.

Hence, there is a positive a, depending only on M and Ω, such that 1/2 ≤ ∂y3 Φ0 (y) ≤ 2,

±Φ0 (y1 , y2 , ±a) > 0,

for all y ∈ Qa . Therefore, the equation Φ0 (y) = 0 has a unique solution y3 = Φ(y1 , y2 ) in the cube Qa . Moreover, Φ ∈ C 1 ([−a, a]2 ) vanishes for y1 = y3 = 0. Thus, we get Pa := x−1 (O ∩ Ω) = {Φ(y1 , y2 ) < y3 < a, |y1 |, |y2 | ≤ a}. Note that |u(x(y)) − U e1 | ≤ M |x(y)| ≤ Ca. Therefore, we can choose a = a(M, Ω) such that 2U/3 ≤ u1 ≤ 4U/3 and C|u2 | ≤ U/3 in Qa . Recall that x1 (y) − g(x2 (y)) vanishes at the plane y1 = 0 and ∂y1 [x1 (y) − g(x2 (y))] = u1 (y) − g (x2 (y))u2 (y).

12.3. Proof of Theorem 12.2.3

397

We deduce that for a suitable choice of a, |y1 |U/3 ≤ |x1 (y) − g(x2 (y))| ≤ |y1 |5U/3 for y ∈ Qa .

(12.3.56)

Equations (12.3.53) imply the identity ∂y1 Φ0 (y) ≡ ∇F0 (x(y)) · u(x(y)) = ∇F0 (x(y)) · U(x(y)) for Φ0 (y) = 0. Combining this with (12.3.52) and (12.3.56), we obtain the estimates |y1 |N − U/3 ≤ |∂y1 Φ0 (y)| ≤ |y1 |N + U 5/3, which, along with the identity, ∂y1 Φ = −∂y1 Φ0 (∂y3 Φ0 )−1 yields the inequalities −c < ∂y1 Φ(y1 , y2 ) ≤ cy1 for −a < y1 < 0, cy1 < ∂y1 Φ(y1 , y2 ) ≤ c for 0 < y1 < a, |∂y2 Φ(y1 , y2 )| ≤ c,

0 ≤ Φ(y1 , y2 ) ≤

(12.3.57)

cy12 .

It is clear that for a suﬃciently small, depending only on U and Ω, the functions Φ± (y2 ) = Φ(±a, y2 ) satisfy ca2 ≤ Φ± (y2 ) < a. Set Qin = {Y = (y2 , y3 ) ∈ [−a, a] × [0, a] : 0 < y3 < Φ− (y2 )}, Qout = {Y = (y2 , y3 ) ∈ [−a, a] × [0, a] : 0 < y3 < Φ+ (y2 )}. It follows from (12.3.57) that for every Y ∈ Qin (resp. Y ∈ Qout ) the equation y3 = Φ(y1 , y2 ) has a unique solution y1 = a− (Y ) < 0 (resp. y1 = a+ (Y ) > 0). We adopt the convention that a± (Y ) = ±a for y3 > Φ± (y2 ). It remains to note that, by (12.3.57), the functions a± meet all requirements of Lemma 12.3.2. Proof of Lemma 12.3.3. The proof is similar to that of Lemma 12.3.2. We may assume that Cartesian coordinates (x1 , x2 , x3 ) are centered at P and n = e3 . By the smoothness of ∂Ω, there is a neighborhood O = [−k, k]2 × [−t, t] such that the manifold ∂Ω ∩ O is deﬁned by x3 = F (x1 , x2 ),

F (0, 0) = 0,

|∇F (x1 , x2 )| ≤ K(|x1 | + |x2 |).

The constants k, t and K depend only on Ω. Consider the initial value problem ∂y3 x = u(x(y)) in Qa ,

x|y3 =0 = (y1 , y2 , F (y1 , y2 )).

(12.3.58)

We can assume that 0 < b < k < 1. For any such b, problem (12.3.58) has a unique solution in C 1 (Qb ). Next, note that for y3 = 0 we have |x(y)| ≤ (K + 1)|y|,

|u(x(y)) − u(0)| ≤ M (K + 1)|y|.

(12.3.59)

398

Chapter 12. Transport equations

Denote F(y) = Dy x(y). Calculations show that ⎛ ⎞ ⎛ ⎞ 1 0 u1 1 0 u1 (P ) F0 := F(y)|y3 =0 = ⎝ 0 1 u2 ⎠ , F(0) = ⎝ 0 1 u2 (P ) ⎠ , 0 0 Un 0 0 u3 where Un = −U(P ) · n, which along with (12.3.59) implies F(0)±1 ≤ C/3,

F0 (y) − F(0) ≤ cb.

(12.3.60)

Next, diﬀerentiation of (12.3.58) with respect to y leads to ∂y1 F = Dy u(x)F,

F|y3 =0 = F0 .

Arguing as in the proof of Lemma 12.3.2 we obtain F − F0 ≤ c(M )F0 b. Combining this with (12.3.60), we arrive at F(y) − F(0) ≤ cb. From this and the implicit function theorem we conclude that there is b > 0, depending only on M and Ω, such that the mapping x = x(y) takes diﬀeomorphically the cube Qb onto some neighborhood of P , and satisﬁes inequalities (12.3.11). Inclusions (12.3.12) easily follow from (12.3.11).

12.3.7 Proof of Lemma 12.3.4 Throughout the section, c, C stand for various constants depending only on the domain Pa and the exponents s, r. Furthermore, for any y = (y1 , y2 , y3 ) and z = (z1 , z2 , z3 ), Y and Z stand for (y2 , y3 ) and (z2 , z3 ), respectively. The existence and uniqueness of solutions to problem (12.3.13) is obvious. Multiplying (12.3.13) by |ϕ|r−1 ϕ and integrating the result over Pa we obtain ϕLr (Pa ) ≤ σ −1 f Lr (Pa )

for r < ∞.

(12.3.61)

Letting r → ∞ we conclude that (12.3.61) holds for r = ∞. Let us turn to the proof of inequality (12.3.14), beginning with the case s = 1. First, we derive the estimates for ∂yk ϕ. We have the representation ∂yk ϕ = ϕ + ϕ , where 0 − −eσ(a (Y )−y1 ) ∂yk a− (Y )f (a− (Y ), Y ) for k = 2, 3, ϕ (y) = − eσ(a (Y )−y1 ) (Y )f (a− (Y ), Y ) for k = 1, and ϕ is a solution to the boundary value problem ∂y1 ϕ + σϕ = ∂yk f

in Pa ,

ϕ (y) = 0 for y1 = a− (Y ).

It follows from (12.3.61) that ϕ Lr (Pa ) ≤ σ −1 f W 1,r (Pa ) . On the other hand, inequalities (12.3.4) yield the estimate

a+ (Y ) a− (Y

)

−r/2

|ϕ (y1 , Y )|r dy1 ≤ c(r)σ −1 f rL∞ (Pa ) y3

(1 − erσ(a

−

(Y )−a+ (Y ))

).

12.3. Proof of Theorem 12.2.3

399

Since 0 ≤ a+ − a− ≤ cy3

1/2

ϕ rLr (Pa )

we conclude that a √ −r/2 ≤ cσ −1 f rL∞ (Pa ) y3 (1 − e−cσ y3 ) dy3 .

(12.3.62)

0

We have a √ −r/2 σ −1 y3 (1 − e−cσ y3 ) dy3 0

aσ 2

= σ r−3

√ −c t

t−r/2 (1 − e

0

⎧ r−3 ⎪ for 2 < r < 3, ⎨cσ −1 ) dt ≤ cσ (1 + log σ) for r = 2, ⎪ ⎩ −1 for 1 < r < 2. cσ

Thus, we get

ϕ Lr (Pa )

0 cσ −1+α ≤ cf L∞ (Pa ) cσ −1+α (1 + log σ)1/r

for r ∈ (1, 2) ∪ (2, 3), for r = 2,

where α = max{0, 1 − 1/r, 2 − 3/r}. Combining the estimates for ϕ and ϕ we obtain (12.3.14). The proof of (12.3.14) for 0 < s < 1 is more complicated. By (12.3.61), it suﬃces to estimate the seminorm 1/r |ϕ|s,r,Pa = |ϕ(z) − ϕ(y)|r |z − y|−3−rs dydz . Pa2

Since this expression is invariant with respect to the change (y, z) → (z, y), we have |ϕ(z) − ϕ(y)|r |z − y|−3−rs dydz, (12.3.63) |ϕ|s,r,Pa ≤ (2I)1/r , I = Da

where Da = {(y, z) ∈ (Pa )2 : a− (Z) ≤ a− (Y )}. It is easy to see that ϕ(z) − ϕ(y) = ϕ(z1 , Z) − ϕ(y1 , Z) +

y1

a− (Z)

eσ(x1 −y1 ) f (x1 , Z) dx1

eσ(x1 −y1 ) (f (x1 , Z) − f (x1 , Y )) dx1 = I1 + I2 + I3 . (12.3.64)

+ a− (Y

a− (Y )

)

Hence, our task is to estimate the integrals Jk = |Ik |r |z − y|−3−rs dydz,

k = 1, 2, 3.

(12.3.65)

Da

The evaluation falls naturally into three steps and it is based on the following proposition.

400

Chapter 12. Transport equations

Proposition 12.3.10. If r, s > 0 and i = j = k, i = k, then |z − y|−3−rs dyi dyj ≤ c(r, s)|zk − yk |−1−rs . [−a,a]2

Proof. Note that the left hand side is equal to (−3−rs)/2 |zi − yi |2 + |zj − yj |2 dyi dyj 1+ 2 2 |z − y | |z k k k − yk | [−a,a]2 ≤ c(r, s)|zk − yk |−1−rs (1 + |yi |2 + |yj |2 )−(3+rs)/2 dyi dyj .

−1−rs

|zk − yk |

R2

Step 1. By the extension principle, the right hand side f has an extension over R3 which vanishes outside the cube Q3a and satisﬁes f W s,r (R3 ) ≤ cf W s,r (Qa ) ,

f L∞ (R3 ) ≤ cf L∞ (Qa ) .

(12.3.66)

Next, recall that a− (Z) ≤ y1 , z1 ≤ a for all (y, z) ∈ Da . From this and Proposition 12.3.10 we obtain |ϕ(z1 , Z) − ϕ(y1 , Z)|r M1 (z, y1 ) dy1 dz1 dZ J1 ≤ [−a,a]2 [a− (Z),a]2 ≤c |ϕ(z1 , Z) − ϕ(y1 , Z)|r |z1 − y1 |−1−rs dy1 dz1 dZ [−a,a]2

[a− (Z),a]2

where we denote

M1 (z, y1 ) =

[−a,a]2

|z − y|−3−rs dy2 dy3 ≤ c|z1 − y1 |−1−rs .

As the right hand side of the last inequality is invariant with respect to (y1 , z1 ) → (z1 , y1 ), we have J1 ≤ c(r, s) |ϕ(z1 , Z) − ϕ(y1 , Z)|r |y1 − z1 |−1−rs dy1 dz1 dZ, [−a,a]2

D(Z)

(12.3.67) where D(Z) = {(y1 , z1 ) : a− (Z) ≤ z1 ≤ y1 ≤ a}. Next, for all (y1 , z1 ) ∈ D(Z), y1 z1 σ(t−y1 ) ϕ(y1 , Z) = e f (t, Z) dt = eσ(t−z1 ) f (t + ξ, Z) dt, a− (Z)

a− (Z)−ξ

where ξ = y1 − z1 > 0. Thus, we get z1 eσ(t−z1 ) (f (t + ξ, Z) − f (t, Z)) dt ϕ(y1 , Z) − ϕ(z1 , Z) = a− (Z)

a− (Z)

+ a− (Z)−ξ

eσ(t−z1 ) f (t + ξ, Z) dt = I11 + I12 .

(12.3.68)

12.3. Proof of Theorem 12.2.3

401

Since f is extended to R3 and vanishes outside [−a, a]3 , we have the estimate

|I11 |r |y1 − z1 |−1−rs dy1 dz1 ≤

D(Z)

a

a− (Z)

2a

|M (z1 , ξ, Z)|r dz1 dξ,

(12.3.69)

0

where M (z1 , ξ, Z) = ξ

−s−1/r

z1

a− (Z)

eσ(t−z1 ) (f (t + ξ, Z) − f (t, Z)) dt.

It is easy to see that M satisﬁes for z1 ∈ (a− (Z), a),

∂z1 M + σM = K

M = 0 for z1 = a− (Z),

where K(z1 , ξ, Z) = ξ −s−1/r (f (z1 + ξ, Z) − f (z1 , Z)). Multiplying this equation by |M |r−2 M and integrating the result over (a− (Z), a) we arrive at

a

|M | dz1 ≤

a

r

σ a− (Z)

≤

a− (Z) a

|M |r−1 |K| dz1

a− (Z)

|M |r dz1

1−1/r

a a− (Z)

|K|r dz1

1/r

,

which gives

a

|M (z1 , ξ, Z)| dz1 ≤ σ r

a− (Z)

−r −1−rs

a

ξ

a− (Z)

|f (z1 + ξ, Z) − f (z1 , Z)|r dz1 .

Combining this inequality with (12.3.69) we obtain the following estimate for the term I11 on the right hand side of (12.3.68):

[−a,a]2

|I11 |r |y1 − z1 |−1−rs dy1 dz1 dZ

D(Z)

≤ σ −r ≤σ

−r

[−a,a]2

R4

a a− (Z)

2a

ξ −1−rs |f (z1 + ξ, Z) − f (z1 , Z)|r dz1 dξdZ

0

|f (y1 , Z) − f (z1 , Z)|r |y1 − z1 |−1−rs dy1 dz1 dZ

≤ cσ −r f rLr (R2 ;W s,r (R)) ≤ cσ −r f rW s,r (R3 ) .

(12.3.70)

To estimate I12 note that |I12 | =

a− (Z)

a− (Z)−ξ

− eσ(t−z1 ) f (t + ξ, Z) dt ≤ f L∞ (Q3a ) eσ(a (Z)−z1 ) σ −1 (1 − e−σξ ),

402

Chapter 12. Transport equations

which gives

|I12 |r |y1 − z1 |−1−rs dy1 dz1

D(Z)

≤ cσ ≤ cσ

−r

f rL∞ (Q3a )

−1−r+rs

We conclude that [−a,a]2

D(Z)

a

rσ(a− (Z)−z1 )

e a− (Z)

f rL∞ (Q3a )

0

∞

3a

dz1

ξ −1−rs (1 − e−σξ )r dξ

0

ξ −1−rs (1 − e−ξ )r dξ ≤ cσ −1−r+rs f rL∞ (Q3a ) .

|I12 |r |y1 − z1 |−1−rs dy1 dz1 dZ ≤ cf rL∞ (Q3a ) σ −1−r+rs .

Inserting this together with (12.3.70) in (12.3.67), and recalling (12.3.66), we ﬁnally obtain 1/r J1 ≤ c σ −1 f W r,s (Qa ) + σ −1+(s−1/r) f L∞ (Qa ) . (12.3.71) Step 2.

Our next task is to estimate J2 . It follows from (12.3.64) that |I2 | ≤ cf L∞ (Pa ) σ −1 eσ(a

−

(Y )−y1 )

−

1 − eσ(a

(Z)−a− (Y ))

.

Next, inequalities (12.3.4) imply, for all (Y, Z) ⊂ Da , y2 y3 ∂t a− (t, z3 ) dt + ∂t a− (y2 , t) dt 0 ≤ a− (Y ) − a− (Z) = z2 z3 y3 √ √ ≤ c|y2 − z2 | + c t−1/2 dt ≤ c|z2 − y2 | + c| z3 − y3 |. z3

Thus we get −

1 − eσ(a

(Z)−a− (Y ))

≤ 1 − e−c|z2 −y2 |−c|

√

√ z3 − y3 |

≤ 1 − e−c|z2 −y2 | + 1 − e−c|

√

√ z3 − y3 |

It follows that |I2 |r |z − y|−3−rs ≤ cσ −r f rL∞ (Qa ) (J23 + J22 ), where J2 = Da √ √ − J23 = erσ(a (Y )−y1 ) (1 − e−cσ| y3 − z3 | )r |z − y|−3−rs dydz, Da − J22 = erσ(a (Y )−y1 ) (1 − e−cσ|y2 −z2 | )r |z − y|−3−rs dydz. Da

.

(12.3.72)

12.3. Proof of Theorem 12.2.3

403

It follows from the obvious inclusion Da ⊂ {(y, z) : y2 , z1 , z2 ∈ (−a, a), y3 , z3 ∈ (0, a), a− (Y ) < y1 < a+ (Y )} that J23 ≤ [0,a]2

a

a

4

a+ (Y )

a− (Y )

−

M2 (y, z3 )erσ(a

where M2 (y, z3 ) =

5 √ √ dy1 dy2 (1 − e−cσ| y3 − z3 | )r dy3 dz3 ,

(Y )−y1 )

|z − y|−3−rs dz1 dz2 .

[0,a]2

Next, Proposition 12.3.10 implies M2 (y, z3 ) ≤ c|z3 − y3 |−1−rs . From this and the √ inequality |a− (Y ) − a+ (Y )| ≤ c y3 we obtain

a+ (Y )

a− (Y )

M2 (y, z3 )e

rσ(a− (Y )−y1 )

dy1 ≤ c|y3 − z3 |

≤ cσ −1 |y3 − z3 |−1−rs (1 − erσ(a

−

(Y )−a+ (Y ))

which leads to J23 ≤ cσ

−1

−1−rs

a+ (Y )

erσ(a

−

(Y )−y1 )

a− (Y )

dy1

) ≤ cσ −1 |y3 − z3 |−1−rs (1 − e−cσ

√

y3

),

a

(12.3.73)

S(z3 ) dz3 , 0

where

a

S(z3 ) =

(1 − e−cσ

√

y3

)(1 − e−cσ|

√

√ y3 − z3 | r

) |y3 − z3 |−1−rs dy3 .

0

& The change of variable t = y3 /z3 − 1 gives ∞ −rs (1 − e−cμ(t+1) )(1 − e−cμ|t| )r |t(t + 2)|−1−rs (t + 1) dt S(z3 ) ≤ cz3 ≤ cz3−rs

−1 1

(1 − e−cμ|t|−cμ )(1 − e−cμ|t| )r |t|−1−rs dt ∞ (1 − e−cμt )r+1 t−1−2rs dt + cz3−rs

≤ cz3−rs

−1

1

1

(1 − e−cμt−cμ )(1 − e−cμt )r t−1−rs dt 0 ∞ −rs + cz3 (1 − e−cμt )r+1 t−1−2rs dt,

√

1

where μ = σ z3 . Another change of variable τ = μt along with the identity z3−rs = σ 2rs μ−2rs yields S(z3 ) ≤ cσ 2rs (S1 (μ) + S∞ (μ)),

404

Chapter 12. Transport equations

where −rs

S1 (μ) = μ S∞ (μ) =

∞

μ

(1 − e−cτ −cμ )(1 − e−cτ )r τ −1−rs dτ,

0

(1 − e−cτ )r+1 τ −1−2rs dτ.

μ

Note that the inequality 0 < s < 1 guarantees the convergence of these integrals. Inserting this estimate into (12.3.73), we obtain J23 ≤ cσ

2rs−1

a

√ σ a

2rs−3

(S1 (μ) + S∞ (μ)) dz3 = cσ

(S1 (μ) + S∞ (μ))μ dμ.

0

0

(12.3.74) Since (1 − e−cτ −cμ )(1 − e−cτ )r ≤ cτ r+1 + cμτ r and (1 − e−cτ )r+1 ≤ cτ r+1 , we have μ (τ r−rs + μτ r−rs−1 ) dτ ≤ cμr−2rs+2 , μS1 (μ) ≤ cμ1−rs 0

1

μS∞ (μ) ≤ cμ

τ r−2rs dτ + cμ

μ

∞

τ −1−2rs dτ ≤ cμ + cμr−rs+2

1

√ for all μ ∈ (0, a). Next we have μS1 (μ) ≤ cμ

μS∞ (μ) ≤ cμ

∞

√ a

1−rs

τ

r−1−rs

1−rs

dτ + cμ

0

μ √ a

τ −1−rs dτ ≤ cμ1−rs ,

τ −1−2rs dτ ≤ cμ1−2rs

μ

√ for all μ ∈ ( a, ∞). From the inequality r − 2rs + 2 = −1 + r(1 − 2s + 3r−1 ) > −1 we thus conclude that the integrals in (12.3.74) converge at 0, and are ﬁnite for each ﬁnite σ. Hence, for all σ > 1, J23 ≤ cσ

√ a r−rs+2

2rs−3

(μ + μ 0

) dτ + cσ

2rs−3

√ σ a

√

(μ1−rs + μ1−2rs ) dμ a

0 for sr = 1, 2, σ 2rs−3 + σ rs−1 ≤c 2rs−3 rs−1 (σ +σ )(1 + log σ) for sr = 1, 2. Since 2rs − 3, rs − 1 ≤ rα, we conclude that for all σ > 1, 1/r

J23 ≤ cσ α for sr = 1, 2,

1/r

J23 ≤ cσ α (1 + log σ)1/r for sr = 1, 2,

Let us estimate the quantity J22 . We have M3 (Y, z2 )(1 − e−cσ|y2 −z2 | )r dy3 dy2 dz2 , J22 ≤ [−a,a]2 ×[0,a]

(12.3.75)

12.3. Proof of Theorem 12.2.3

405

where M3 (Y, z2 ) =

−

eσr(a

Since

(Y )−y1 )

[−a,a]2 ×[0,a]

|z − y|−3−rs dz1 dy1 dz3 .

|z − y|−3−rs dz1 dz3 ≤ c|y2 − z2 |−1−rs ,

[−a,a]×[0,a]

we have M3 (Y, z2 ) ≤ cσ −1 |y2 − z2 |−1−rs , which yields J22 ≤ cσ −1 = cσ

rs−1

∞

(1 − e−cσ|y2 −z2 | )r |y2 − z2 |−rs−1 dy2

−∞ ∞

−∞

(1 − e−c|t| )r |t|−1−rs dt ≤ cσ rs−1 .

Since σ rs−1 ≤ cσ rα for all σ ≥ 1, from (12.3.75) and (12.3.72) we conclude that 0 for sr = 1, 2, cf L∞ (Pa ) σ −1+α 1/r J2 ≤ (12.3.76) −1+α 1/r cf L∞ (Pa ) σ (1 + log σ) for sr = 1, 2. Step 3.

Observe that I3 (y1 , Y, Z) deﬁned by (12.3.64) satisﬁes ∂y1 I3 + σI3 = K3

for a− (Y ) < y1 < a,

I3 (a− (Y ), Y, Z) = 0,

where K3 (y1 , Y, Z) = f (y1 , Z) − f (y1 , Y ). Multiplying this equation by |I3 |r−2 I3 and integrating the result over (a− (Y ), a) we arrive at a a |I3 |r dy1 ≤ |I3 |r−1 |K3 | dy1 σ a− (Y )

a− (Y ) a

≤ which leads to

a

a− (Y )

a− (Y )

|I3 |r dy1 ≤ σ −r

|I3 |r dy1

1−1/r

a

a− (Y )

|K3 |r dy1

1/r

|f (y1 , Z) − f (y1 , Y )|r dy1 . [−a,a]

Since a− (Y ) ≤ y1 for all (y, z) ∈ Da , taking into account the inequality |z − y|−3−rs dz1 ≤ c|Y − Z|−2−rs [−a,a]

,

406

Chapter 12. Transport equations

we conclude that |I3 |r |z − y|−3−rs dydz J3 = Da −r ≤ cσ |f (y1 , Z) − f (y1 , Y )|r |Y − Z|−2−rs dY dZ dy1 ≤ cσ

−r

[−a,a] [−a,a]4 r f Lr (−a,a;W s,r ([−a,a]2 ))

≤ cσ −r f rW s,r (Qa ) .

(12.3.77)

Inserting estimates (12.3.71), (12.3.76), and (12.3.77) into the inequality 1/r

|ϕ|s,r,Pa ≤ J1

1/r

+ J2

1/r

+ J3

we conclude that under the assumptions of the lemma, 0 c σ −1 f W s,r (Pa ) + σ −1+α f L∞ (Pa ) −1 |ϕ|s,r,Pa ≤ c σ f W s,r (Pa ) + σ −1+α (1 + log σ)1/r f L∞ (Pa ) which completes the proof.

for sr = 1, 2, for sr = 1, 2,

Chapter 13

Appendix 13.1 Proof of Lemma 2.4.1 Choose an arbitrary vector ﬁeld ϕ such that ϕ ∈ C0∞ (Rd × R),

supp ϕ ⊂ D =

Ωt × {t}.

(13.1.1)

t

Multiplying (2.1.6a) by ϕ and integrating by parts we arrive at " ui D

∂ϕi ∂ϕi 1 ∂ϕi + ui uj − Sij (u) ∂t ∂xj Re ∂xj # 1 ∂ϕi 1 + p + 2 fi ϕi dxdt = 0. (13.1.2) Ma2 ∂xi Frm

Noting that ∂xi = Uip ∂yp we have ∂ui ∂uj ∂up ∂ϕi Sij (u) = Ujk + Uik + (λ − 1)Upk δij Ujn , ∂yk ∂yk ∂yk ∂yn ∂ϕi ∂ϕi ∂ϕi ∂ϕi ui u j +p = ui uj Ujk + p Uik . ∂xj ∂xi ∂yk ∂yk

(13.1.3) (13.1.4)

Next, set u = U(t) w,

ϕ = U(t) ψ,

1 f = U F. Fr2m

Since Uαβ Uγβ = Uβα Uβγ = δαγ , we have Ujk Ujn

∂wq ∂ψp ∂wq ∂ψq ∂ui ∂ϕi = Ujk Ujn Uip Uiq = . ∂yk ∂yn ∂yk ∂yn ∂yk ∂yk

P. Plotnikov and J. Sokoáowski, Compressible Navier–Stokes Equations: Theory and Shape 407 Optimization, Monografie Matematyczne 73, DOI 10.1007/978-3-0348-0367-0_13, © Springer Basel 2012

408

Chapter 13. Appendix

The same arguments give Uik Ujn and Upk Uin

∂uj ∂ϕi ∂wj ∂ψi = ∂yk ∂yn ∂yi ∂yj

∂up ∂ϕi ∂wl ∂ψm ∂wj ∂ψi = Upk Uin Upl Uim = . ∂yk ∂yn ∂yk ∂yn ∂yj ∂yi

Substituting these into (13.1.3) gives Sij (u)

∂ϕi ∂ψi = Sij (w) . ∂xj ∂yj

(13.1.5)

Arguing as before we obtain ui uj Ujk

∂ϕi ∂ψm ∂ψp ∂ψi = Uip Ujq Ujk Uim wp wq = wp wq = wi wj , ∂yk ∂yk ∂yq ∂yj ∂ϕi ∂ψn ∂ψi Uik = Uik Uin = . ∂yk ∂yk ∂yi

Substituting these along with (13.1.5) into (13.1.3)–(13.1.4) and recalling the notation for ρ we obtain "

# ∂ϕi 1 ∂ϕi 1 ∂ϕi 1 Sij (u) − + p + f ϕ i i dx dt ∂xj Re ∂xj Ma2 ∂xi Fr2m D " # ∂ψi 1 ∂ψi 1 ∂ψi − + p + ρ Fi ψi dy dt. = Sij (w) ρ wi wj ∂yj Re ∂yj Ma2 ∂yi Ω0 ×R ui u j

(13.1.6)

Next, note that ∂ϕi ∂ϕi ∂yk ∂ϕi = + . ∂t x=const ∂t y=const ∂t x=const ∂yk We have y = U (x − a). From this and the identity U˙ U + U˙ U = 0 we obtain ∂yk ˙ + a) ˙ = −W. = U˙ (x − a) − U a˙ = U˙ Uy − U a˙ = −U (Uy ∂t x=const Thus we get

∂ϕi ∂ϕi ∂ϕi = − Wk , ∂t x=const ∂t y=const ∂yk

13.1. Proof of Lemma 2.4.1

and

ui D

∂ϕi dxdt = ∂t

409

Ω0 ×R

ρui

∂ϕi ∂ϕi − Wk ∂t ∂yk

(13.1.7)

dydt.

Next, we have the equality ∂ϕi ∂ ∂ϕi ∂ ui − Wk Uip ψp − Wk = Uin wn Uip ψp ∂t ∂yk ∂t ∂yk ∂ψi ∂ψi ∂Uip − wi W k , = wi + wn ψp Uin ∂t ∂yk ∂t which along with Uin gives

ρui

∂ϕi ∂ϕi − Wj ∂t ∂yj

∂Uip ∂Wn = ∂t ∂yp

= ρwi

∂ψi ∂ψi − Wj ∂t ∂yj

+ ρwj ψi

∂Wj . ∂yi

Substituting this identity into (13.1.7) and combining the result with (13.1.6) and (13.1.2) we ﬁnally obtain " D

# ∂ϕi ∂ϕi 1 ∂ϕi 1 ∂ϕi 1 + ui uj Sij (u) ui − + p + 2 fi ϕi dx dt ∂t ∂xj Re ∂xj Ma2 ∂xi Frm " ∂ψi ∂ψi 1 ∂ψi ∂ψi − = − Wj Sij (w) ρwi + ρ wi wj ∂t ∂yj ∂yj Re ∂yj Ω0 ×R # ∂Wj 1 ∂ψi dy dt = 0, (13.1.8) + p + ρ Fi ψi + ρwj ψi ∂yi Ma2 ∂yi

where D is given by (13.1.1). Our next task is to obtain an equation for the deviation v = w − W. Substituting this equality into (13.1.8) and noting that S(W) = 0 we obtain "

Ω0 ×R

ρvi

∂ψi ∂ψi 1 ∂ψi 1 ∂ψi − + p + ρ vi vj Sij (v) 2 ∂t ∂yj Re ∂yj Ma ∂yi

# ∂Wj dy dt + ρ Fi ψi + ρvj ψi ∂yi " # ∂ψi ∂ψi ∂Wj + ρ ψ i Wj + + ρ Wi vj ρWi dy dt = 0. (13.1.9) ∂t ∂yj ∂yi Ω0 ×R

Let us turn to the mass balance equation. Multiplying (2.1.6b) by an arbitrary function η ∈ C0∞ (Ω) and integrating by parts we obtain ∂η ∂η + ui dxdt = 0. ∂t ∂xi D

410

Chapter 13. Appendix

Noting that ∂η ∂η ∂η = − Wi , ∂t x=const ∂t y=const ∂yi

ui

∂η ∂η ∂η = ui Uip = wp , ∂xi ∂yp ∂yp

we obtain for ζ(y, t) = η(x(y, t), t), ∂ζ ∂ζ + (wi − Wi ) dydt = 0. ρ ∂t ∂yi Ω0 ×R It now follows from the equality v = w − W that for any ζ ∈ C0∞ (Ω0 × R), ∂ζ ∂ζ dydt = 0, (13.1.10) ρ + vi ∂t ∂yi Ω0 ×R which is equivalent to the equation ∂t ρ + div(ρv) = 0.

(13.1.11)

In particular, we have ∂ψi ∂Wi ∂ψi ∂Wi + vj + vj dydt = − dydt. ρWi ρψi ∂t ∂yj ∂t ∂yj Ω0 ×R Ω0 ×R From this and (13.1.9) we obtain the integral identity "

Ω0 ×R

ρvi

∂ψi ∂ψi 1 ∂ψi 1 ∂ψi + ρ vi vj Sij (v) − + p ∂t ∂yj Re ∂yj Ma2 ∂yi # ∂Wj ∂Wi dydt + ρ Fi ψi + ρvj ψi − ∂yi ∂yj " # 1 ∂|W|2 ∂Wi + ρψi − dydt = 0, 2 ∂yi ∂t Ω0 ×R

which is equivalent to the equation ∂t (ρv) + div(ρv ⊗ v) +

1 1 ∇p − div S(v) + ρ Cv Re Ma2 ρ = ρF + ∇|W|2 − ρ ∂t W. (13.1.12) 2

Combining this with (13.1.11) we obtain (2.4.5). Let us calculate the expression for the hydrodynamic force, the power devel˜ (x, t) oped by this force and the work of these forces in the moving frame. Let n be the outward normal vector to ∂Ωt at x and n(y) be the outward normal vector to ∂Ω0 at y. It is easy to see that ˜ |x=x(y,t) = U(t) n(y). n

13.2. Normal coordinates

411

Thus we get the following expression for the components Rf i of the hydrodynamic force Rf at x = x(y, t): Re Rf i = σpUiα nα − Sij (u)Ujα nα . Next, note that ∂wn ∂wn ∂wn + Uik Ujn + (λ − 1)Upk Upn ∂yk ∂yk ∂yk ∂wn ∂wn = Ujk Uin + Uik Ujn + (λ − 1) div w δij . ∂yk ∂yk

Sij (u) = Ujk Uin

Thus we get Sij (u)Ujα nα = Uin

∂wn ∂wα nα + (λ − 1) div w Uiα nα , + ∂yα ∂yn

which gives an expression for Rf in the new variables, Re Rf = U(−S(w) + σp I)n.

(13.1.13)

Noting that the dimensionless velocity of ∂Ω is V = UW we obtain Re V · Rf = W · (−S(w) + σp I)n,

(13.1.14)

which leads to equalities (2.4.6) and (2.4.7).

13.2 Normal coordinates Let Σ be a 2-dimensional closed C 2 surface in the Euclidean space R3 of points x = (x1 , x2 , x3 ) such that Σ = ∂Ω ⊂ R3 for a domain Ω. For any point ω ∈ ∂Σ denote by n(ω) the unit outward normal vector to ∂Σ at a point ω and by ν(ω) = −n(ω) the inward normal vector to Σ. It is well known that there exist a neighborhood O of Σ and a number μ > 0 so that the mapping (ω, y3 ) → ω + y3 ν(ω) takes diﬀeomorphically the set Σ × [−μ, μ] onto O. Let us evaluate the derivatives and the Jacobian of this mapping. To this end, we ﬁx ω0 ∈ Σ. In some neighborhood O0 of ω0 , the surface Σ admits a parametric representation Σ ∩ O0 :

ω = r(y1 , y2 ),

|yi | ≤ μ,

r(0, 0) = ω0 ,

where the vector-valued function r(y1 , y2 ) is C 2 in the rectangle Q = {(y1 , y2 ) : |yi | ≤ μ, i = 1, 2}, for an appropriate choice of μ independent of ω0 . We write y = (¯ y , y3 ),

y¯ = (y1 , y2 ),

(13.2.1)

412

Chapter 13. Appendix

and introduce the moving frame y ) = ∂yi r(¯ y ), ei (¯

i = 1, 2,

e3 (¯ y ) = ν(r(¯ y )) =

1 e1 × e2 . |e1 × e2 |

(13.2.2)

This means that the orientation of Σ is compatible with the direction of the inward normal vector. Thus, the mapping X : y → r(¯ y ) + y3 e3 (¯ y ) ≡ ω + y3 ν(ω) maps diﬀeomorphically Q × [−μ, μ] → O0

and

Q × [0, μ] → O0 ∩ Ω.

Recall the notation y ) · ej (¯ y ), gij = ei (¯

bij = e3 (¯ y ) · ∂y2i yj r(¯ y ),

for the coeﬃcients of the ﬁrst and second fundamental form of the surface Σ and set g 2 = det (gij ), (g αβ ) = (gαβ )−1 . Note that gdy1 dy2 = dΣ is the surface element of Σ. In this notation, the mean and Gauss curvatures of Σ are deﬁned by H=

1 1 (b + b22 ), 2 1

K = b11 b22 − b12 b21 ,

where

bij = g iα bαj .

Let us consider the structure of the Jacobi matrix X (y) deﬁned by $ . M(y) ≡ X (y) := ∂y1 X, ∂y2 X, ∂y3 X ,

(13.2.3)

where the notation [a, b, c] stands for the (3 × 3)-matrix with columns a, b, and c. Using the Weingarten equations ∂yj e3 = −bij ei with i, j = 1, 2, we obtain the columns of the matrix function M: ∂yj X = ej − y3 (b1j e1 + b2j e2 ),

j = 1, 2,

∂y3 X = e3 .

(13.2.4)

In order to derive an expression for the inverse M−1 we use the identity $ . M− ≡ ∇x y1 (X(y)), ∇x y2 (X(y)), ∇x y3 (X(y)) $ . (13.2.5) = (det X )−1 ∂y2 X × ∂y3 X, ∂y3 X × ∂y1 X, ∂y1 X × ∂y2 X .

13.3. Geometric results. Approximation of unity

413

Next, note that for j = 1, 2, ej × e3 = g −1 ej × (e1 × e2 ) = g −1 (ej · e2 )e1 − (ej · e1 )e2 ) = g −1 (gj2 e1 − gj1 e2 ) and e1 × e2 = ge3 . From this and (13.2.4), (13.2.5) we obtain ∂yj X × ∂y3 X = g −1 (gj2 e1 − gj1 e2 ) − y3 g −1 (bj2 e1 − bj1 e2 ), and ∂y1 X × ∂y2 X = g(1 − 2Hy3 + Ky32 ) e3 . In particular we have the following expression for the Jacobian: |M| := det M ≡ (∂y1 X × ∂y2 X) · ∂y3 X = g(1 − 2Hy3 + Ky32 ).

(13.2.6)

We also get the following expression for the rows of M−1 (columns of M− ): (∇x y1 )(X(y)) = g −1 (1 − 2Hy3 + Ky32 )−1 g22 e1 − g21 e2 − y3 (b22 e1 − b21 e2 ) , (∇x y2 )(X(y)) = −g −1 (1 − 2Hy3 + Ky32 )−1 (g12 e1 − g11 e2 ) − y3 (b12 e1 − b11 e2 ) , (∇x y3 )(X(y)) = e3 . Normal conformal coordinates. The formulae can be essentially simpliﬁed if the parameterization x = r(¯ y ) is conformal. This means that g12 = 0,

g11 = g22 = g.

In this case the expressions for the rows of the inverse Jacobi matrix become (∇x y1 )(X(y)) = g −1 (1 − 2Hy3 + Ky32 )−1 ge1 − y3 (b22 e1 − b21 e2 ) , (∇x y2 )(X(y)) = g −1 (1 − 2Hy3 + Ky32 )−1 ge2 − y3 (b11 e2 − b12 e1 ) , (∇x y3 )(X(y)) = e3 . In particular, we have the following ⎛ 1/2 g 0 M ≡ X (y) = ⎝ 0 g 1/2 0 0

expression for the Jacobi matrix on Σ ∩ O0 : ⎞ 0 (13.2.7) 0 ⎠ , |M| = g for y3 = 0. 1

13.3 Geometric results. Approximation of unity In this section we prove an important technical result which shows that in a cylinder Q = Ω × (0, T ) the constant function 1 can be approximated in a special way by a sequence of Lipschitz functions whose traces on the lateral side ST = ∂Ω × (0, T ) vanish outside of the inlet. It is assumed that the ﬂow domain Ω and a given vector ﬁeld U : Rd × [0, T ] → Rd satisfy the following structural conditions:

414

Chapter 13. Appendix

Assume that Ω = B \ S, where B is a hold-all domain and a compact set S is an obstacle placed in B. Hence ∂Ω is a union of two disjoint compact sets: a surface ∂B and a compact set ∂S. Denote by Σ the cylindrical surface ∂B × (0, T ). By Σin we denote, as usual, the inlet, that is, the open subset of Σ which consists of all points (x, t) ∈ Σ at which U · n < 0. Recall that n stands for the outward unit normal vector to ∂Ω. Furthermore, we assume that ∂B and U satisfy the following conditions: Condition 13.3.1. • The vector ﬁeld U ∈ C 1 (Rd ×[0, T ]) vanishes in the vicinity of the compact set S × [0, T ]. • The surface ∂B is of class C 2 . Let Γ = Σ \ Σin ∩ cl Σin ⊂ Σ be the interface between the inlet set and the outgoing subset of Σ. Then lim sup σ→0

1 meas Oσ < ∞, σd

(13.3.1)

where Oσ is a tubular neighborhood of Γ, Oσ = {(x, t) ∈ Rd+1 : dist((x, t), Γ) ≤ σ}. Remark 13.3.2. We do not impose any restrictions on the obstacle S. Theorem 13.3.3. Let ∂Ω and U satisfy Condition 13.3.1 and suppose u − U ∈ L2 (0, T ; W 1,2 (Ω). Then there is a sequence of Lipschitz functions ψn : Q → [0, 1], n ≥ 1, with the following properties: ψn vanishes in some neighborhood of Σ \ Σin , ψn (x, t) 1 as n → ∞ everywhere in Q, and Φ(∂t ψn + ∇ψn · u) dxdt ≤ 0 (13.3.2) lim inf n→∞

Q

for any nonnegative Φ ∈ Lq (Q) with 2 < q ≤ ∞. Proof. Deﬁne M (x) := dist(x, ∂B),

N (x, t) := dist((x, t), Σ \ Σin ),

(x, t) ∈ Rd+1 , (13.3.3)

and choose η ∈ C ∞ (R) such that η(s) = 0 for s ≤ 1/2 and

η(s) = 1

for s ≥ 1.

Next, set ψn (x, t) = η(nN (x, t)),

(x, t) ∈ Q.

Obviously M and N satisfy the Lipschitz condition. Moreover, |∇M | ≤ 1,

|∇N | ≤ 1.

(13.3.4)

13.3. Geometric results. Approximation of unity

415

Since M (x) ≤ N (x, t), it follows that |∇ψn | ≤ cn in B1/n × (0, T ),

|∇ψn | = 0 in (B \ B1/n ) × (0, T ),

(13.3.5)

where B1/n ⊂ B is the set of all points x ∈ B with dist(x, ∂B) < 1/n, in other words M (x) < 1/n. Hence the functions ψn , n ≥ 1, also satisfy the Lipschitz condition. Obviously they vanish in the 1/(2n)-neighborhood of Σ \ Σin and are equal to 1 outside the 1/n-neighborhood of Σ \ Σin . Hence ψn 1 in Q as n → ∞. Let us rewrite the integral in (13.3.2) as Φ(∂t ψn + ∇ψn · u) dxdt = Φ(∂t ψn + ∇ψn · U) dxdt + Φ ∇ψn · v dxdt, Q

Q

Q

(13.3.6) where v = u − U. Since v ∈ L2 (0, T ; W01,2 (Ω)), in particular we can extend it by zero onto S × [0, T ] and assume that v ∈ L2 (0, T ; W01,2 (B)). By the general property of the space W01,2 (B), we have

|v|2 dx = (dist(x, ∂B))2

B

which gives

B×[0,T ]

|v|2 dx ≤ c(B)v2W 1,2 (B) , 0 M2

B

|v|2 dxdt ≤ cv2L2 (0,T ;W 1,2 (B)) ≤ C, 0 M2

(13.3.7)

where C is independent of n. It now follows from (13.3.5) that Φ∇ψn · v dxdt = Q

B1/n ×[0,T ]

T

≤ cn

Φ∇ψn · v dxdt

T

M ΦM −1 |v| dxdt

Φ|v| dxdt = cn

0 T

B1/n

B1/n

0

T

(M

−1

|v|)

q/(q−1)

dxdt

B1/n

0

≤c

B1/n

ΦM −1 |v| dxdt

≤c ≤c

0

T

B1/n

0

≤c 0

(M −1 |v|)q/(q−1) dxdt

T

dxdt B1/n

≤ cn−(q−2)/(2(q−1))

(q−1)/q

0

T

0

0

(q−1)/q

(q−2)/(2(q−1))

T

T

Φq dxdt

B1/n

1/q

B1/n

M −2 |v|2 dxdt

(M −2 |v|)2 dxdt B\B1/n

1/2

.

q/2(q−1) (q−1)/q

416

Chapter 13. Appendix

Combining this with (13.3.7) we obtain the following estimate for the second integral on the right hand side of (13.3.6): Φ∇ψn · v dxdt ≤ Cn−(q−2)/(2(q−1)) → 0 as n → ∞. (13.3.8) Q

To estimate the ﬁrst integral on the right of (13.3.6) we split the cylinder Q into three disjoint parts, Q = Zn ∪ Gn ∪ On , where the closed set Zn and the open set Gn are deﬁned by Zn = (x, t) ∈ Q : dist((x, t), Γ) ≤ 2/n , Gn = (x, t) ∈ Q : dist((x, t), Γ) > 2/n and dist((x, t), Σ \ Σin ) < 1/n . Notice that dist((x, t), Σ \ Σin ) ≥ 1/n, hence ψn (x, t) = 1 on On = Q \ (Gn ∪ Zn ). It follows that the derivatives of ψn vanish a.e. on On , which yields Φ(∂t ψn + ∇ψn · U) dxdt = Φ(∂t ψn + ∇ψn · U) dxdt Q Zn + Φ(∂t ψn + ∇ψn · U) dxdt. (13.3.9) Gn

The remaining part of the proof is based on the following lemma. Lemma 13.3.4. Under the assumptions of Theorem 13.3.4, there is an n0 , depending only on B, with the following property. If n > n0 and (x, t) ∈ Gn then ∇ψn (x, t) · U(x, t) ≤ c

and

∂t ψn (x, t) = 0,

(13.3.10)

where c is independent of (x, t) and n. Proof. Since ∂B is a C 2 surface, there is an n0 such that the normal coordinate x ↔ ω + τ n(ω), ω ∈ ∂B, |τ | ≤ 1/n0 , is well deﬁned and takes diﬀeomorphically the 1/n0 -neighborhood of ∂B in Rd onto the cylinder ∂B × (−1/n0 , 1/n0 ). By construction we have |τ | = dist(x, ∂B) for all x ∈ B \ B1/n0 . Let (x, t) ∈ Gn with n > n0 . Then dist(x, ∂B) = dist((x, t), Σ) ≤ dist((x, t), Σ \ Σin ) ≤ 1/n < 1/n0 . Hence x ∈ B1/n0 and x = ω + τ n(ω),

ω ∈ ∂B,

−1/n < τ ≤ 0.

13.3. Geometric results. Approximation of unity

417

Let (ω ∗ , t∗ ) ∈ Σ \ Σin be at minimal distance to (x, t), i.e. |(ω ∗ , t∗ ) − (x, t)| = dist((x, t), Σ \ Σin ). For any (ω , t ) ∈ Γ we have |(ω ∗ , t∗ ) − (ω , t )| ≥ |(ω , t ) − (x, t)| − |(ω ∗ , t∗ ) − (x, t)| ≥ 2/n − 1/n = 1/n > 0. Hence (ω ∗ , t∗ ) does not belong to Γ, i.e., (ω ∗ , t∗ ) ∈ Σ \ (Σin ∪ Γ). Since Σ \ (Σin ∪ Γ) is an open subset of ∂B ×(0, T ) it is a classical result that the vector (ω ∗ , t∗ )−(x, t) is parallel to (n∗ , 0), a normal vector to ∂B ×(0, T ) at (ω ∗ , t∗ ). It follows that (x, t) admits the representation x = ω ∗ + τ ∗ n(ω ∗ ) ∈ B1/n0 , hence

ω ∗ ∈ ∂B,

t∗ = t,

−τ ∗ = dist((x, t), Σ \ Σin ).

Because of uniqueness of the normal coordinates we have ω = ω ∗ , τ = τ ∗ and, as noted earlier, t = t∗ . Thus for every (x, t) ∈ Gn with normal coordinates (ω, τ ) we have N (x, t) := dist((x, t), Σ \ Σin ) = −τ in Gn . Notice that τ is a C 1 function of the variable x in B \ Bn0 . Hence N (x, t) is independent of t on the open set Gn , which yields ∂t ψn = 0 on Gn . Next, recalling the formula ∇τ (x) = n(ω) for x = ω + τ n(ω) we obtain for x = ω + τ n(ω), and (x, t) ∈ Gn , U(x, t) · ∇N (x, t) = −U(x, t) · ∇τ (x) = −U(ω + τ n(ω), t) · n(ω). Since U(ω, t)n(ω) ≥ 0 for (ω, t) ∈ Σ \ Σin we conclude that U(x, t)∇N (x, t) ≤ (U(ω, t) − U(ω + τ, t))n(ω) ≤ c|τ | ≤ c/n in Gn . This obviously yields the inequality U(x, t) · ∇ψn (x, t) ≤ c in Gn .

Continuing with the proof of Theorem 13.3.3, it follows from Lemma 13.3.4 that Φ(∂t ψn + U · ψn ) ≤ cΦ. Since Gn ⊂ B1/n × (0, T ) we conclude that

T

Φ(∂t ψn + U · ψn ) dxdt ≤ c Gn

≤c

T

≤ cn

dxdt

0

Φ dxdt B1/n

0

B1/n

−(1−1/q)

.

1−1/q

T

Φq dxdt 0

B1/n

1/q

≤c 0

T

dxdt B1/n

1−1/q

418

Chapter 13. Appendix

It follows that

Φ(∂t ψn + U · ψn ) dxdt ≤ 0.

lim sup n→∞

Next, since |∂t ψn | + |∇ψn | ≤ cn we have Φ(∂t ψn + U · ψn ) dxdt ≤ cn Zn

(13.3.11)

Gn

Φ dxdt

Zn

≤ cn

dxdt

1−1/q

Zn

Φq dxdt

1/q

Zn

≤ cn meas(Zn )

1−1/q

.

Since Zn ⊂ O2/n , it follows from Condition 13.3.1 that meas(Zn ) ≤ cn−2 . Thus we get Φ(∂t ψn + U · ψn ) dxdt ≤ lim cn2/q−1 = 0. (13.3.12) lim sup n→∞

n→∞

Zn

Combining (13.3.11) and (13.3.12) with (13.3.9) we obtain (13.3.2), which completes the proof of Theorem 13.3.3.

13.4 Singular limits for normal derivatives of solutions to diﬀusion equations This section is devoted to the proof of Theorem 5.3.13. We derive estimates near the boundary for solutions to a linear parabolic problem depending on a small parameter ε > 0. Namely, we consider the ﬁrst boundary value problem for the second order parabolic equation ∂t + div(u) − εΔ = 0 in Q, = ∞ on ST , (x, 0) = 0 (x) in Ω.

(13.4.1a) (13.4.1b) (13.4.1c)

Here, Q = Ω × (0, T ),

ST = ∂Ω × (0, T ),

Ω ⊂ R , d = 2, 3, is a bounded domain with C 3 boundary, and u is a given vector ﬁeld. In this framework, (13.4.1a) can be regarded as a singular perturbation of the linear transport equation. Our main goal is to estimate the normal derivative of a solution to problem (13.4.1). The following arguments show that this question is important for the theory of zero viscosity limits of solutions to the diﬀusion equation. First of all, recall the deﬁnition of a weak solution to the ﬁrst boundary value problem for the nonperturbed transport equation d

∂t + div(u) = 0 in Q, = ∞ on Σin , (x, 0) = 0 (x) in Ω,

13.4. Singular limits of normal derivatives

419

where the inlet is deﬁned by Σin = {x ∈ ∂Ω : U∞ (x) · n(x) < 0}. Assume for a moment that u, ∞ and 0 are continuous. Then an integrable function is a weak solution to this problem if the integral identity (∂t ψ + u∇ψ) dxdt + ψ(x, 0)0 dx = ψ∞ u · n dSdt Q

ST

Ω

holds for all smooth functions ψ vanishing at ST \ Σin and for t = T . On the other hand, multiplying both sides of equation (13.4.1a) by ψ and integrating by parts we obtain, at least formally,

(∂t ψ + u · ∇ψ) dxdt + ψ(x, 0)0 dx − ψ∞ u · n dSdt Q Ω ST =ε ∞ ∂n ψ dSdt − ε Δψ dxdt − ε ψ ∂n dSdt. ST

Q

ST

If solutions to problem (13.4.1) are uniformly integrable with respect to ε, the integrals on the right hand side, except the last one, tend to 0 as ε → 0. Convergence to zero of ε∂n is not obvious. What is more, in the general case such a statement is false. Indeed, if ε∂n converges to zero even in the sense of distributions on the whole surface ST = ∂Ω × (0, T ), then the limit transport equation has a solution which takes the prescribed boundary data on the whole surface ST , which is impossible. Hence we can expect that ε∂n converges to zero only on the inlet. For an elliptic regularization, this result was proved by Ole˘ınik and Radkevich under the assumption that the set of solutions to the regularized problem is bounded in the L∞ -norm, i.e., that they are uniformly bounded. In fact, Ole˘ınik and Radkevich proved that ∂n ∼ 1 in the interior of the inlet. With applications to Navier-Stokes equations in mind, we extend this result to the case of parabolic equations and unbounded solutions. The result is purely local and it is independent of initial data provided that solutions are uniformly bounded in Lγ (Q). Throughout this section we assume that ∞ belongs to C 2 (Q), and the vector ﬁeld U satisﬁes the following smoothness and positivity conditions. Condition 13.4.1. For any (z, t∗ ) ∈ Σin there exist positive constants c0 , a, b with the following properties. Denote by Qa the cylinder B(z, a) × (t∗ − a, t∗ + a), where B(z, a) ⊂ Rd is the ball of radius a centered at z. • In Qa ∩ Q the velocity ﬁeld u admits the representation u = ∇H + v in Q ∩ Qa , H = 0, v · n = 0 on ST ∩ Qa .

(13.4.2)

420

Chapter 13. Appendix Moreover, the function H and the vector ﬁeld v satisfy the inequalities HC 2 (Q∩Qa ) + vC 1 (Q∩Qa ) ≤ c0 , sup t∈(t∗ −a,t∗ +a)

v(·, t)C 1 (B(z,a)∩Ω) ≤ c0 .

(13.4.3)

• The potential H satisﬁes 1 1 ∂t H(x, t) + |∇H(x, t)|2 > b > 0 in Qa ∩ Q, 2 4

(13.4.4)

−∇H(x, t) · n(x) > b > 0 on Qa ∩ ST . It follows from these conditions that diminishing a if necessary, we can assume that ∈ C 1,2 (Qa ∩ Q), but smoothness properties of depend on ε and global estimates of in the cylinder Q. Our task is to prove the following theorem which is equivalent to Theorem 5.3.13. Theorem 13.4.2. Assume that is a solution to problem (13.4.1) that belongs to L2 (0, T ; W 2,2 (Ω)) ∩ C(0, T ; W 1,2 (Ω)) ∩ W 1,2 (0, T ; L2 (Ω)) and that γ > d + 2. Then there are positive constants C0 , a0 , ε0 , depending only on d, γ, Ω, and on the constants in Condition 13.4.1, such that for all ε ∈ (0, ε0 ), ∂ on ST ∩ Qa0 , where σ = 1/2 − d/(2γ). ε ≤ C0 εσ Lγ (Qa ) + ∞ C 2 (Qa ) ∂n We split the proof into six steps. Furthermore, we denote by c, ci generic constants depending only on c0 , a, b, z, and ∂Ω. To avoid repetitions the proof is given for d = 3 only. Step 1. Normal coordinates. The ﬁrst step is standard: we make a change of the independent variables in order to rectify the boundary of Ω and reduce an inhomogeneous boundary problem for a homogeneous parabolic equation to a homogeneous boundary problem for an inhomogeneous equation. Fix (z, t∗ ) ∈ Σin . As mentioned in Section 13.2, there exist constants c2 > 0 and μ > 0 with the following properties. There is a mapping y → X(y) with X(0) = z which takes diﬀeomorphically the cube Dμ = {y ∈ Rd : |yi | < μ, i = 1, 2, 3}

(13.4.5)

onto some neighborhood G of z, the ﬂat square Dμ ∩ {y3 = 0} onto the surface ∂Ω ∩ G, and the slab (13.4.6) Dμ+ = {y ∈ Dμ : y3 > 0} onto G ∩ Ω. Moreover, this mapping and its inverse satisfy the estimate XC 2 (Dμ ) + X −1 C 2 (G) ≤ c(z, Ω)

(13.4.7)

13.4. Singular limits of normal derivatives

421

and we have the representation X(Y ) = r(y1 , y2 ) + y3 ν(y1 , y2 )

(13.4.8)

where x = r(y1 , y2 ) is a conformal parameterization of ∂Ω in the vicinity of z and ν = −n(r(y1 , y2 )) is the inward normal vector to ∂Ω. Notice that y3 (x) = dist(x, ∂Ω) in G. Further, consider the cylinders Qμ = Dμ × (t∗ − μ, t∗ + μ),

+ ∗ ∗ Q+ μ = Dμ × (t − μ, t + μ).

(13.4.9)

Notice that the mapping (X, I) : (y, t) → (X(y), t) takes diﬀeomorphically the cylinder Qμ onto some neighborhood of (z, t∗ ). By (13.4.7) we have Qμ/c ⊂ (X, I)(Qμ ) ⊂ Qcμ ,

(13.4.10)

where c = c(z, Ω) is the constant in (13.4.7). Without loss of generality we can assume that 0 < μ < c(z, Ω)a, 0 < μ < a, (13.4.11) and the mapping (X, I) takes the cylinder Qμ into Qa . Let us recalculate the normal derivative of in the normal coordinates. To this end notice that (13.2.2) and (13.2.5) imply 1 e1 × e2 1 = ∂y1 X × ∂y2 X = |M|M− k, |e1 × e2 | g g

k = (0, 0, 1),

|M| = det M.

Since by (13.2.7), M− ∇y = ∇x and |M| = g for y3 = 0, we obtain the following expression for the normal derivatives of : ∂ = −M−1 M− k · ∇y for y3 = 0. ∂n Again, using (13.2.7) for the restriction of M to {y3 = 0} we ﬁnally obtain ∂ ∂ =− ∂n ∂y3

for y3 = 0.

For technical reasons it is convenient to introduce a new unknown function ρ(y, t) deﬁned by (13.4.12) X(y), t = |M(y)|−1 ρ(y, t) + ∞ (X(y), t). It is clear that ρ vanishes at y3 = 0. Thus we get ∂ρ ∂∞ ∂ = −|M|−1 − . ∂n ∂y3 ∂y3 Consequently, the proposition will be proved if we show that ∂ρ ≤ cεσ γ + + ∞ 2 + ε for (y, t) ∈ {y3 = 0} ∩ Qμ/4 , (13.4.13) L (Qμ ) C (Qμ ) ∂y3 where c depends only on γ, d, z, Ω, and the constants in Condition 13.4.1; and ∞ are considered as functions of y, t, e.g. = (X(y), t).

422

Chapter 13. Appendix

Step 2. Change of independent variables. Choose ζ ∈ C0∞ (G × (t∗ − μ, t∗ + μ))). Multiplying (13.4.1) by ζ and integrating by parts we obtain

t∗ +μ

t∗ −μ

Recall the notation

(∂t ζ + u · ∇ζ − ε∇ · ∇ζ ) dxdt = 0. G

M := X (y),

|M| := det M,

and notice that ∇x ζ = M− ∇y ζ. This gives (∂t ζ + M−1 u · ∇y ζ − εM−1 M− ∇y · ∇y ζ)|M| dydt = 0, Q+ μ

which yields the following equation for (X(y), t) in the cylinder Q+ μ: |M|∂t + div(|M|M−1 u) − ε div(|M|M−1 M− ∇) = 0.

(13.4.14)

Next, let ρ(y, t) deﬁned by (13.4.12) be the new unknown function. Substituting (13.4.12) into (13.4.14) we obtain the following equation and boundary conditions for ρ: ∂t ρ + div(ρW) − ε div(A∇ρ) = f

in Q+ μ,

(13.4.15a) (13.4.15b)

ρ(y, t) = 0 for y3 = 0, where A = M−1 M− ,

W = M−1 u +

ε A∇|M|, |M|

f = −|M|∂t ∞ − div(∞ |M|M−1 u) + ε div(|M|M−1 M− ∇∞ ).

(13.4.16)

Here, ∞ is regarded as a function of y, t, i.e., ∞ = ∞ (X(y), t), and similarly for u. It follows from Condition 13.4.1 that sup t∈(t∗ −μ,t∗ +μ)

WC 1 (Dμ+ ) ≤ c1 , (13.4.17)

AC 1 (Dμ+ ) ≤ c1 , 2 2 c−1 1 |ξ| ≤ Aξ · ξ ≤ c1 |ξ| ,

where the positive constant c1 depends only on the constant c0 in Condition 13.4.1 and the constant c in (13.4.7). Using representation (13.4.2) and the identity ∇x H = M− ∇y H we rewrite the expression for W in the form W = A∇y H + V, where

V(y, t) = M−1 v(X(y), t) +

ε A∇|M|. |M|

(13.4.18)

13.4. Singular limits of normal derivatives

423

It follows from Condition 13.4.1 that sup t∈(t∗ −μ,t∗ +μ)

VC 1 (Dμ+ ) ≤ c2 ,

HC 2 (Q+ ≤ c2 /2, μ)

(13.4.19)

where c2 depends only on the constant c0 in Condition 13.4.1 and the constant c in (13.4.7). Moreover, since the vector M− k, k = (0, 0, 1), is collinear with the normal vector ν, we have M−1 v(X(y), t) · k = 0 for y3 = 0.

(13.4.20)

On the other hand, using formula (13.2.6), |M| = g(1 − 2H(y1 , y2 )y3 + K(y1 , y2 )y32 ), where H and K are the mean and Gaussian curvatures of ∂Ω, respectively, we obtain ε A∇|M| · k = −2εH(y1 , y2 ) for y3 = 0. |M| Combining this equality with (13.4.20) and using (13.4.19) we conclude that |V(y, t) · k| ≤ c2 (ε + y3 ) in Q+ μ.

(13.4.21)

It follows from (13.4.11) and Condition 13.4.1 that ∇y H · k > b for (y, t) ∈ Qμ ∩ {y3 = 0}. From this and (13.4.19) it follows that for a suitable choice of μ, depending only on the constants c2 and b, ∇y H · k > b/2 in Q+ μ,

(13.4.22)

which along with (13.4.21) yields for a suitable choice of μ and ε0 , depending only on c2 and b, W · k > b/2 − c2 (ε + y3 ) > b/3 in Q+ μ for all ε ∈ (0, ε0 ).

(13.4.23)

Step 3. Change of unknown function. This step is crucial. We eliminate the potential part of the transport term in equation (13.4.15) using an appropriate change of unknown function. To this end, set " # H(y, t) ρ(y, t) = exp w(y, t). (13.4.24) 2ε Calculations show that w satisﬁes ∂t w − ε div(A∇w) + V · ∇w + w = 0 for y3 = 0.

C w=F ε

in Q+ μ,

(13.4.25)

424

Chapter 13. Appendix

Here, the coeﬃcient C and the function F are deﬁned by 1 ε 1 1 ∂t H + A∇H · ∇H + V · ∇H + ε div V + div(A∇H), 2 " 4 2 2 # H(y, t) F = exp − f. 2ε

C=

(13.4.26)

Let us show that C is bounded from below and above by constants independent of ε. First we observe that 1 1 ∂H 1 1 ∂H 2 |∇ + H| = + |A∇y H · ∇y H|2 . x 2 ∂t x=const 4 2 ∂t y=const 4 It now follows from Condition 13.4.1 that 1 1 ∂t H + A∇H · ∇H ≥ b in Q+ μ. 2 4 On the other hand, since ∇y H is collinear with k, it follows from (13.4.21) that ∇H · V = (∇H · k)(V · k) ≤ c22 ε

for y3 = 0.

From this and estimates (13.4.19) we conclude that |∇H · V| ≤ c22 (ε + y3 ) ≤ c22 (ε + μ), hence C ≥ b − c22 (y3 + μ) − 2εc22 . Therefore, we can choose μ and ε0 , depending only on the constants in Condition 13.4.1 and the constant c in (13.4.7), such that b/2 ≤ C ≤ c22

in Q+ μ for all ε ∈ (0, ε0 ).

(13.4.27)

Note that for such a choice of μ we have H > 0,

|w| ≤ |ρ| ≤ c(|| + |∞ |),

In particular,

|F | ≤ |f | ≤ c∞ C 2 (Q+ μ)

Q+ μ

|w|γ dydt ≤ c

Estimates for truncated functions.

Q+ μ

(||γ + |∞ |γ ) dydt.

Now choose η1 ∈ C0∞ (R) with

η1 (s) = 1 for s ∈ [−μ/2, μ/2] and and set

in Q+ μ . (13.4.28)

η = η1 (y1 )η1 (y2 )η1 (y3 )η(t − t∗ ),

η1 (s) = 0

for |s| ≥ μ

ϕ(y, t) = ηw(y, t).

(13.4.29)

13.4. Singular limits of normal derivatives

425

Multiplying (13.4.25) by η we arrive at C ϕ ε = div Φ0 + Φ1 + ηF

∂t ϕ − ε div(A∇ϕ) + V · ∇ϕ +

in Dμ+ × (0, T ),

(13.4.30)

ϕ = 0 for y ∈ ∂Dμ+ , ϕ(y, 0) = 0, where Φ0 = −2εwA∇η, Φ1 = V · ∇η + ε div(A∇η) + ∂t η w

in Q+ μ,

(13.4.31)

and Φ0 = 0, Φ1 = 0 otherwise. Energy estimates. Multiplying (13.4.30) by |ϕ|γ−2 ϕ and integrating by parts we arrive at 1 γ

|ϕ| dy + ε(γ − 1) γ

+ Dμ ×{T }

+ Dμ ×(0,T )

ε |ϕ| C − div V dydt + γ Dμ ×(0,T ) γ−2 |ϕ| ϕ(Φ1 + ηF ) − (γ − 1)|ϕ|γ−2 ∇ϕ · Φ0 dydt. 1 + ε

=

|ϕ|γ−2 A∇ϕ · ∇ϕ dydt

+ Dμ ×(0,T )

γ

Here, we use the identity ∇(|ϕ|γ−2 ϕ) = (γ − 1)|ϕ|γ−1 ∇ϕ. Noting that for suﬃciently small ε we have C−

ε b b ε div V ≥ − c2 ≥ γ 2 γ 3

we obtain, for suﬃciently small ε, b 3ε

+ Dμ ×(0,T )

|ϕ|γ dydt ≤

+ Dμ ×(0,T )

|ϕ|γ−2 ϕ(Φ1 + ηF ) − (γ − 1)|ϕ|γ−2 ∇ϕ · Φ0 dydt.

426

Chapter 13. Appendix

Next, notice that (with all integrals over Dμ+ × (0, T )) −

|ϕ|γ−2 ∇ϕ · Φ0 dydt = 2ε η γ−2 |w|γ−2 wA∇(ηw) · ∇η dydt γ−2 γ = 2ε η |w| A∇η · ∇η dydt + 2ε η γ−1 |w|γ−2 wA∇η · ∇w dydt 2ε = 2ε η γ−2 |w|γ A∇η · ∇η dydt + η γ−1 ∇(|w|γ ) · A∇η dydt γ 2ε γ−2 γ = 2ε η |w| A∇η · ∇η dydt − (γ − 1) η γ−2 |w|γ A∇η · ∇η dydt γ 2ε η γ−1 |w|γ div(A∇η) dydt. − γ

Thus we get |ϕ|γ−2 ∇ϕ · Φ0 dydt − + Dμ ×(0,T )

= = It follows that (γ −1)

+ Dμ ×(0,T )

2ε γ 2ε γ

+ Dμ ×(0,T )

η γ−2 |w|γ A∇η · ∇η − η γ−1 |w|γ div(A∇η) dydt

|ϕ|γ−2 |w|2 A∇η · ∇η − |ϕ|γ−1 |w| div(A∇η) dydt.

+ Dμ ×(0,T )

|ϕ|γ−2 ∇ϕ·Φ0 dydt ≤ cε

+ Dμ ×(0,T )

(|ϕ|γ−2 |w|2 +|ϕ|γ−1 |w|) dydt,

where c depends only on γ, μ, and the constant c1 in (13.4.17). Using the representation for Φ1 and inequalities (13.4.19) we obtain |ϕ|γ−2 ϕ(Φ1 +ηF ) dydt ≤ c |ϕ|γ−1 (|w|+|F |) dydt, (13.4.32) + Dμ ×(0,T )

+ Dμ ×(0,T )

where c depends only on the constants c1 and c2 in (13.4.17) and (13.4.19). Combining the results obtained and noting that ϕ is supported in Q+ μ , we arrive at b b |ϕ|γ dydt = |ϕ|γ dydt 3ε Dμ+ ×(0,T ) 3ε Q+ μ γ−1 ≤c |ϕ| (|w| + |F |) + ε|ϕ|γ−2 |w|2 dydt. (13.4.33) Q+ μ

It follows from the Young inequality that for any positive σ and κ < 1, γ/(γ−2) γ/2 γ−2 2 γ κ γ σ |w| ≤ cε|ϕ| + εc|w| . ε|ϕ| σ κ

13.4. Singular limits of normal derivatives

427

Choosing σ = ε2(γ−2)/γ we obtain 1 ε|ϕ|γ−2 |w|2 ≤ cκγ/(γ−2) |ϕ|γ + cκ−γ/2 εγ−1 |w|γ . ε Next, we have |ϕ|

γ−1

γ/(γ−1) γ κ γ σ |w| ≤ c|ϕ| + c|w| . σ κ γ

With σ = ε(γ−1)/γ we obtain 1 |ϕ|γ−1 |w| ≤ cκγ/(γ−1) |ϕ|γ + cκ−γ εγ−1 |w|γ . ε Similar arguments give 1 |ϕ|γ−1 |F | ≤ cκγ/(γ−1) |ϕ|γ + cκ−γ εγ−1 |F |γ . ε Inserting these inequalities into (13.4.33) and noting that κ < 1 we obtain b γ/(γ−1) |ϕ|γ dydt ≤ cεγ κ−γ (|w|γ + |F |γ ) dydt. − cκ + 3 Dμ ×(0,T ) Q+ μ Choosing κ depending only on γ and c suﬃciently small and recalling that |w| ≤ c|| + |∞ | we obtain + F Lγ (Q+ ). ϕLγ (Dμ+ ×(0,T )) ≤ cε(Lγ (Q+ μ) μ) Recalling that |F | ≤ c∞ C 2 (Q+ we conclude that μ) + ∞ C 2 (Q+ ). ϕLγ (Dμ+ ×(0,T )) ≤ cε(Lγ (Q+ μ) μ)

(13.4.34)

Step 4. Extension. Since the solution ϕ to problem (13.4.30) and the function w vanish for y3 = 0, we can extend them over the cylinder Dμ × (t∗ − μ, t∗ + μ) by setting ϕ(y, t) = −ϕ(¯ y , t), y¯ = (y1 , y2 , −y3 ) ∈

w(y, t) = −w(¯ y , t) for y ∈ Dμ− , Dμ+ .

where

(13.4.35)

The extended ϕ is Lipschitz and compactly supported in Dμ × (t∗ − μ, t∗ + μ). Next, deﬁne extensions of the matrix A, the vector ﬁelds V, Φ0 , and the functions C, Φ1 , ηF on Dμ by the formulae ⎞ ⎛ ⎞ ⎛ A12 −A13 1 0 0 A11 A22 −A23 ⎠ y¯ , V(y, t) = ⎝ 0 1 0 ⎠ V(¯ y , t), A(y) = ⎝ A21 −A31 −A32 A33 0 0 −1 ⎛ ⎞ y , t), Φ1 (y, t) = −Φ1 (¯ −1 0 0 ⎝ ⎠ C(y) = C(¯ y ), 0 −1 0 Φ0 (¯ y , t), Φ0 (y, t) = 0 0 1 ηF (y) = −ηF (¯ y , t).

428

Chapter 13. Appendix

We keep the same notation for the extended quantities. By (13.2.7) we have ⎞ ⎛ −1 0 0 g g −1 0 ⎠ for y3 = 0. A=⎝ 0 (13.4.36) 0 0 1 Hence the extended matrix A and the function C are Lipschitz in Dμ . It follows from (13.4.17) that λ ≤ C ≤ c21 ,

where λ = b/2 > 0,

|A(y ) − A(y )| ≤ c1 |y − y | for all y , y ∈ Rd , 2 c−1 1 |ξ|

(13.4.37)

≤ Aξ · ξ ≤ c1 |ξ| . 2

Since w vanishes for y3 = 0, the extended vector ﬁeld Φ0 is Lipschitz in Dμ . In contrast, the extensions of V, Φ1 and F are discontinuous and have jumps at y3 = 0. Nevertheless, since ϕ vanishes for y3 = 0, we have the representation V · ∇ϕ = div Ψ0 + Ψ1 , where Ψ0 = ϕV,

Ψ1 = − div V ϕ in Dμ+ ∪ Dμ− .

(13.4.38)

It follows from (13.4.19) and (13.4.34) that Ψ0 Lγ (Qμ ) + Ψ1 Lγ (Qμ ) ≤ cϕLγ (Qμ ) ≤ cε(Lγ (Q+ + ∞ C 2 (Q+ ). μ) μ)

(13.4.39)

Notice that the functions ϕ, Φ1 , Ψ1 , ηF and the vector ﬁelds Ψ0 , Φ0 are compactly supported in the cylinder Qμ deﬁned by (13.4.9). We assume that all these functions and vector ﬁelds are extended by 0 over R3 × (0, ∞). We also extend A(y) over R3 . By the sharp form of Whitney’s extension theorem (see [31]), we can assume that the extended matrix satisﬁes (13.4.37) in the whole R3 . Finally, we extend C over R3 × (0, ∞), simply setting C = λ outside of Qμ . Thus, we arrive at the Cauchy problem for the extended function ϕ in the whole space Ω: C ϕ ε = div(Φ0 − Ψ0 ) + Φ1 − Ψ1 + ηF ϕ = 0 for t = 0. ∂t ϕ − ε div(A∇ϕ) +

in Rd × (0, ∞),

(13.4.40)

Step 5. Heat kernel. Our next task is to estimate the L∞ -norm of the solution ϕ to problem (13.4.40) by using the representation of ϕ via the heat kernel associated with the parabolic diﬀerential operator in (13.4.40). We start by estimating this 1 = A 1 and a 1 : Rd → Rd2 with A kernel. Assume that a matrix-valued function A

13.4. Singular limits of normal derivatives

429

bounded function C˜ : Rd → R satisfy the inequalities (13.4.37) with Dμ replaced by the whole space Rd . For simplicity, we omit the tildes. Let us consider the Cauchy problem ∂t ϕ − ε div(A∇ϕ) +

C ϕ=f ε

in Rd × (0, ∞),

ϕ = 0 for t = 0,

(13.4.41)

where f is a bounded function. It is well known that the unique strong solution to this problem admits a representation via the heat kernel Gε , t ϕ(y, t) = 0

Rd

Gε (y, t; z, s)f (z, s) dzds.

(13.4.42)

The next proposition gives Aronson-type estimates for Gε . Proposition 13.4.3. Suppose the matrix A and the coeﬃcient C satisfy (13.4.37). Then there are positive constants α and β, depending only on d and the constants c1 , λ in (13.4.37), such that α |y − z|2 λ(t − s) exp −β − , ε(t − s) ε (ε(t − s))d/2 α |y − z|2 . exp −β |∇z Gε (y, t; z, s)| ≤ ε(t − s) (ε(t − s))(d+1)/2 0 ≤ Gε (y, t; z, s) ≤

(13.4.43)

The proof is given at the end of this section. The following lemma is the main result of this subsection. Lemma 13.4.4. Under the assumption of Theorem 13.4.2, there exists c > 0, depending only on the constants in Condition 13.4.1, ∂Ω and μ, such that the solution ϕ to problem (13.4.30) satisﬁes sup (y,t)∈Q+ μ/3

|ϕ(y, t)| ≤ cεσ

1 −κ/ε e + εσ (Lγ (Q+ + ∞ C 2 (Q+ ). μ) μ) ε

Here,

" κ = min

βμ2 μλ , 144T 12

# ,

σ=

(13.4.44)

d 1 − , 2 2γ

and the constant c depends only on d, γ > d+2, ∂Ω and the constants in Condition 13.4.1. Proof. Choose (x, t) ∈ Q+ μ/3 and write the solution to problem (13.4.40) for the extended function ϕ in the form ϕ(x, t) = I1 (x, t) + I2 (x, t) + I3 (x, t),

430

Chapter 13. Appendix

where

t I1 = − Gε (x, t; y, s)(div Ψ0 + Ψ1 )(y, s) dyds, Rd 0 t Gε (x, t; y, s)(div Φ0 + Φ1 )(y, s) dyds, I2 = Rd 0 t I3 = Gε (x, t; y, s)(ηF )(y, s) dyds. Rd

0

To estimate I1 , since Ψ0 , Ψ1 are compactly supported we can write t t ∇y Gε (x, t; y, s) · Ψ0 (y, s)dyds − Gε (x, t; y, s)Ψ1 (y, s)dyds. I1 (x, t) = Rd

0

0

Rd

Recalling that, by (13.4.38), |Ψ0 |, |Ψ1 | ≤ cϕ and applying Proposition 13.4.3 we obtain t |y − x|2 1 exp −β |ϕ(y, s)| dyds. |I1 (x, t)| ≤ cα (d+1)/2 ε(t − s) Rd (ε(t − s)) 0 The Hölder inequality yields |I1 (x, t)| t ≤c 0

Rd

1 (d+1)γ

(ε(t − s)) 2(γ−1)

exp −

γβ|y − x|2 (γ − 1)ε(t − s)

# γ−1 γ dyds

ϕLγ (Rd ×(0,T )) .

γβ|y − x|2 dy = cεd/2 (t − s)d/2 , (γ − 1)ε(t − s) Rd where c depends only on γ, β, d, i.e., only on γ, Ω and the constants in Condition 13.4.1. Thus we get t γ−1 d+γ d γ − 2(γ−1) − 12 − 2γ (t − s) ds ϕLγ (Rd ×(0,T )) . |I1 (x, t)| ≤ cε

Note that

exp −

0

Since γ > 2 + d, we have (d + γ)/(2(γ − 1)) < 1, which along with (13.4.39) yields 1 d . + varrho∞ C 2 (Q+ |I1 (x, t)| ≤ cε− 2 − 2γ ϕLγ (Rd ×(0,T )) ≤ cεσ Lγ (Q+ μ) μ) (13.4.45) Estimating I2 is similar to those for I1 . Arguing as before we obtain t 1 |y − x|2 |Φ0 (y, s)| dyds exp −β |I2 (x, t)| ≤ cα (d+1)/2 ε(t − s) Rd (ε(t − s)) 0 t 1 |y − z|2 λ(t − s) exp −β − |Φ1 | dyds + cα d/2 ε(t − s) ε 0 Rd (ε(t − s)) = I20 + I21 .

13.4. Singular limits of normal derivatives

431

To estimate the ﬁrst integral notice that, by (13.4.31), Φ0 (y, t) is compactly supported in Qμ , where it satisﬁes |Φ0 (y, t)| ≤ c|w(y, t)|. Moreover, it vanishes for |yi | ≤ μ/2, i = 1, 2. Recall that w is given by (13.4.24). It follows that t 1 |y − z|2 |Φ0 (y, s)| dyds exp −β (d+1)/2 ε(t − s) Rd (ε(t − s)) 0 1 |y − z|2 exp −β ≤c |w| dyds, (d+1)/2 ε(t − s) Et (ε(t − s)) where Et = {(y, s) ∈ Qμ : 0 < s < t, |yi | ≥ μ/2, i = 1, 2}. For every (x, t) ∈ Qμ/3 and (y, s) ∈ Et , we have |y − x| ≥ μ/6, which yields |y − x|2 |y − x|2 −κ/ε ≤e . exp −β exp −β ε(t − s) 2ε(t − s) Thus we get −κ/ε

I20 ≤ ce

Et

1 |y − x|2 exp −β |w(y, s)| dyds. 2ε(t − s) (ε(t − s))(d+1)/2

Applying the Hölder inequality we obtain I20 ≤ ce−κ/ε

"

exp −

1

γβ|y−x|2 2(γ−1)ε(t−s)

# γ−1 γ

wLγ (Qμ ) (d+1)γ (ε(t−s)) 2(γ−1) # γ−1 " t γ 1 γβ|y−x|2 ≤ ce−κ/ε exp − wLγ (Qμ ) . dyds (d+1)γ 2(γ−1)ε(t−s) Rd (ε(t−s)) 2(γ−1) 0 Et

Recalling that Rd

exp −

γβ|y − x|2 2(γ − 1)ε(t − s)

we obtain 1

I20 ≤ ce−κ/ε ε− 2 − 2γ d

t

dy = cεd/2 (t − s)d/2

d+γ

(t − s)− 2(γ−1) ds

γ−1 γ

0 1

dyds

≤ ce−κ/ε ε− 2 − 2γ wLγ (Qμ ) . d

Next, note that wLγ (Qμ ) = 21/γ wLγ (Q+ , μ)

wLγ (Dμ ×(0,T )

(13.4.46)

432

Chapter 13. Appendix

hence from the equality

1 2

+

= 1 − σ we obtain

d 2γ

I20 ≤ cεσ−1 e−κ/ε wLγ (Q+ . μ) It follows from the expression (13.4.24) for w that |w| = |e−H/ε ρ| ≤ |ρ| ≤ c|| + |∞ | in Q+ μ,

(13.4.47)

I20 ≤ cεσ−1 e−κ/ε (Lγ (Q+ + ∞ C 2 (Q+ ). μ) μ)

(13.4.48)

which leads to

Next, we estimate I21 . Again note that, by (13.4.31), Φ1 is compactly supported in Qμ , where it does not exceed c|w|. Moreover, it vanishes in Qμ/2 . Thus we get λ(t − s) 1 |y − x|2 I21 ≤ c − |w| dyds, exp −β d/2 ε(t − s) ε Ft (ε(t − s)) where

Ft = (y, s) ∈ Qμ \ Qμ/2 : 0 < s < t .

Notice that for (y, s) ∈ Ft and (x, t) ∈ Qμ/3 we have either |x − y| ≥ μ/6 or |t − s| ≥ μ/6, which gives |y − x|2 λ(t − s) λ(t − s) |y − x|2 exp −β − ≤ e−κ/ε exp −β − . ε(t − s) ε 2ε(t − s) 2ε Using this result we arrive at λ(t − s) 1 |y − x|2 −κ/ε − |w| dyds. I21 ≤ ce exp −β d/2 2ε(t − s) 2ε Ft (ε(t − s)) Combining this with the Hölder inequality leads to I21 ≤ ce−κ/ε wLγ (Qμ ) " t # γ−1 γ 1 γ λ(t − s) |y − x|2 × exp − . + β dyds γd γ−1 2ε(t − s) 2ε 0 Rd (ε(t − s)) 2(γ−1) Recalling again (13.4.46) we get −κ/ε

I21 ≤ ce

" wLγ (Qμ )

t

1 d

0

(ε(t − s)) 2(γ−1)

# γ−1 γ γλ(t − s) ds exp − . 2(γ − 1)ε

Since for γ > d + 2, "

t

1 d

0

(ε(t − s)) 2(γ−1)

# γ−1 γ γ−1−d γλ(t − s) exp − ≤ cε γ ≤ c, ds 2(γ − 1)ε

13.4. Singular limits of normal derivatives

433

we ﬁnally obtain from (13.4.47) + ∞ C 2 (Q+ ). I21 ≤ ce−κ/ε wLγ (Qμ ) ≤ ce−κ/ε (Lγ (Q+ μ) μ) Combining this estimate with (13.4.48) yields |I2 (x, t)| ≤ cεσ−1 e−κ/ε (Lγ (Q+ + ∞ C 2 (Q+ ). μ) μ)

(13.4.49)

It remains to estimate I3 . Recall that F is given by (13.4.26). Since the extended function ηF is compactly supported in Qμ we have |I3 (x, t)| t ≤ 0

t

Dμ

Gε (x, t; y, s)|ηF (y, s)| dyds ≤ ηF C(Qμ )

0

Rd

Gε (x, t; y, s) dyds.

It follows from (13.4.43) and the equality 1 |y − x|2 dy = exp −β exp(−β|y|2 ) dy = c d/2 ε(t − s) Rd (ε(t − s)) Rd that t 0

Rd

Gε (x, t; y, s) dyds t ≤c

λ 1 |y − x|2 − (t − s) dyds exp −β d/2 ε(t − s) ε 0 Rd (ε(t − s)) t λ =c exp − (t − s) ds ≤ cε. ε 0

Since, by (13.4.28), ηF C(Q+ ≤ c∞ C 2 (Q+ , μ) μ) we obtain . |I3 (x, t)| ≤ cε∞ C 2 (Q+ μ) Combining this with (13.4.45)–(13.4.48) and noting that 0 < σ ≤ 1/2 we obtain (13.4.44). Corollary 13.4.5. There are constants c3 ≥ 1 and ε0 > 0, depending only on ∂Ω, T , d, γ, and the constants in Condition 13.4.1, such that the solution ϕ to problem (13.4.30) satisﬁes sup (y,t)∈Q+ μ/3

|ϕ(y, t)| ≤ c2 εσ (Lγ (Q+ + ∞ C 2 (Q+ ). μ) μ)

(13.4.50)

Proof. It suﬃces to note that for a suitable choice of ε0 , 1 −κ/ε e ≤ 1 for all ε ∈ (0, ε0 ). ε

434

Chapter 13. Appendix

Step 6. Majorants. We are now in a position to complete the proof of Theorem 13.4.2. First of all we use Lemma 13.4.4 to obtain the L∞ estimate for ρ connected with w via (13.4.24). To this end, note that w coincides with ϕ in the cube Qμ/2 . From this and (13.4.24) we obtain 2 ∗ ∗ |ρ| ≤ eH/ε |ϕ| in Q+ μ/3 = [−μ/3, μ/3] × [0, μ/3] × [t − μ/3, t + μ/3].

On the other hand, as H vanishes at y3 = 0, estimates (13.4.19) yield H ≤ c2 y3 in Q+ μ , which leads to (13.4.51) |ρ| ≤ ec2 y3 /ε |ϕ| in Q+ μ/3 , where c2 is the constant in (13.4.19). Denote Q = {(y, t) ∈ Qμ/3 : 0 ≤ y3 ≤ ε} = [−μ/3, μ/3]2 × [0, ε] × [t∗ − μ/3, t∗ + μ/3]. It follows from (13.4.51) that |ρ| ≤ ec2 |ϕ| in Q. Combining this inequality with (13.4.50) we obtain |ρ| ≤ ψ(ε) in Q,

(13.4.52)

where + ∞ C 2 (Q+ ). ψ(ε) = c3 ec2 εσ (Lγ (Q+ μ) μ) Next, ﬁx a nonnegative ζ ∈ C ∞ (R3 ) such that ζ(y1 , y2 , t) = 0 for |y1 |, |y2 |, |t − t∗ | ≤ μ/4, ζ(y1 , y2 , t) = 1 for |y1 |, |y2 |, |t − t∗ | ≥ μ/3 and

|∂t ζ| + |∇ζ| + |∇2 ζ| ≤ 200μ−2 .

We set P (y, t) = ψ(ε)ζ(y1 , y2 , t) + Ky3 . Let us show that for a suitable choice of the positive constant K, the function P can be regarded as a majorant for |ρ|. First, observe that P (y, t) = ψ(ε) + Ky3

for |y1 |, |y2 |, |t − t∗ | ≥ μ/3.

It follows from this and (13.4.52) the functions P ± ρ are nonnegative on the bottom and the lateral sides of the slab Q for any nonnegative K. On the top y3 = ε we have P (y, t) − |ρ(y, t)| = ψ(ε)ζ(y1 , y2 , t) + Kε − |ρ(y1 , y2 , ε, t)| ≥ Kε − ψ(ε).

13.4. Singular limits of normal derivatives

435

Hence for K = ψ(ε)/ε, the functions P ± ρ are nonnegative on ∂Q. To prove that they are nonnegative in the whole cylinder Q we calculate ∂t (P ± ρ) + div((P ± ρ)W) − ε(A(P ± ρ)) =: L(P ± ρ). It follows from (13.4.15) that " # 1 L(P ± ρ) = ψ(ε) W · k + y3 div W − ε div(Ak) + L(ζ) ± f. ε Here k = (0, 0, 1), f is deﬁned by (13.4.16) and satisﬁes |f | ≤ 2c1 ∞ C 2 (Q+ μ)

in Qμ+ ,

where c1 is the constant in (13.4.17). It follows from (13.4.17) and (13.4.23) that W · k ≥ b/3,

| div W| + | div(Ak)| ≤ 2c1 ,

|L(ζ)| ≤ 400(c1 + 1)μ−2 .

Thus we get L(P ± ρ) ≥ ψ(ε)(b/ε − 2c1 − 400(c1 + 1)μ−2 ) − 2c1 ∞ C 2 (Qμ+ ) in Q. Recalling the expression for ψ(ε) and the inequality c3 ≥ 1, we obtain, for all ε ∈ (0, ε0 ), L(P ± ρ) ≥ {εσ−1 (b/3 − ε2c1 − ε400(c1 + 1)μ−2 ) − 2c1 }(Lγ (Q+ + ∞ C 2 (Q+ ). μ) μ) Recall that μ and c1 depend only on ∂Ω and the constants in Condition 13.4.1. Hence for a suitable choice of ε0 we obtain, for all ε ∈ (0, ε0 ), L(P ± ρ) ≥ 0 in Q,

P ± ρ ≥ 0 in ∂Q.

It follows that P ± ρ ≥ 0 in Q. Since ζ vanishes for |y1 |, |y2 |, |t − t∗ | ≤ μ/4, taking into account the expression for P , we conclude that + ∞ C 2 (Q+ )y3 |ρ| ≤ Ky3 = ψ(ε)ε−1 y3 ≤ c3 ec2 εσ−1 (Lγ (Q+ μ) μ) for |y1 |, |y2 |, |t − t∗ | < μ/4 and 0 < y3 < ε < ε0 . Hence the inequality ∂ρ c2 σ ε + ∞ C 2 (Q+ ) ∂y3 ≤ c3 e ε (Lγ (Q+ μ) μ)

(13.4.53)

holds for all y = (y1 , y2 , 0) and

t with |y1 |, |y2 |, |t − t∗ | ≤ μ/4 and

which yields (13.4.13), and the theorem follows.

0 < ε < ε0 ,

436

Chapter 13. Appendix

Proof of Proposition 13.4.3. The proof is based on the following lemma which is a simpliﬁcation of the more general Lemma 3.1 in [132]. For any positive c > 0 set # " |x − y|2 , x, y ∈ R2 , t > s. Gc (x, t; y, s) = (t − s)−d/2 exp − c t−s Lemma 13.4.6. For any 0 < a < b and λ > 0, t e−λ(τ −s) c(a, b, λ, d) Ga (x, t; z, τ )Gb (z, τ ; y, s) dzdτ ≤ Ga (x, t; y, s). 1/2 (t − s)1/2 s Rd (t − τ ) Proof. We can assume that s = 0, so the desired inequality reads t e−λτ c(a, b, λ, d) G (x, t; z, τ )Gb (z, τ ; y, 0) dzdτ ≤ Ga (x, t; y, 0). 1/2 a (t − τ ) t1/2 d 0 R Split the integral into t ρt + Rd

0

ρt

Rd

e−λτ Ga (x, t; z, τ )Gb (z, τ ; y, 0) dzdτ = J1 + J2 , (t − τ )1/2

where ρ ∈ (0, 1) is speciﬁed below. Let us estimate J1 . We have

ρt

J1 =

−λτ

e 0

Rd

2 2 |z−y|2 exp −a |x−z| exp −(b − a) |z−y| t−τ − a τ τ dzdτ. (t − τ )(d+1)/2 τ d/2

Next we have τ = νt, t − τ = (1 − ν)t, where 0 < ν ≤ ρ. It follows that |z − y|2 |x − y|2 1 |x − z|2 |z − y|2 |x − z|2 + − = + − |x − y|2 t−τ τ t t 1−ν ν 1 − ν 1 ν = |x − z|2 + |z − y|2 + 2(x − z) · (y − z) t 1−ν ν 1 ν 1 − ν ≥ |x − z|2 + |z − y|2 − 2|x − z| |y − z| ≥ 0, t 1−ν ν and hence

|z − y|2 1 1 |x − z|2 −a ≤ 1/2 exp −a Ga (x, t; y, 0). (d+1)/2 t−τ τ (t − τ ) t (1 − ρ)(d+1)/2

Thus we get ρt |z−y|2 1 −λτ exp −(b − a) τ Ga (x, t; y, 0) e dzdτ J1 ≤ 1/2 t (1 − ρ)(d+1)/2 τ d/2 0 Rd ∞ 1 ≤ c(a, b, ρ) 1/2 Ga (x, t; y, 0) e−λτ dτ ≤ c(a, b, λ, ρ)t−1/2 Ga (x, t; y, 0). t 0

13.4. Singular limits of normal derivatives

437

It remains to estimate J2 . To this end, split Rd into Ω+ = {z : |z − y| ≥ |x − y|(a/b)1/2 } and

Ω− = {z : |z − y| < |x − y|(a/b)1/2 }.

Notice that for all z ∈ Ω+ and τ ∈ (ρt, t), |z − y|2 |x − y|2 τ −d/2 exp −b ≤ (ρt)−d/2 exp −a . τ t It follows that t e−λτ G (x, t; z, τ ) Gb (z, τ ; y, 0) dzdτ 1/2 a ρt Ω+ (t − τ ) t 2 λ exp −a |x−y| − λρt e− 2 τ |x − z|2 t 2 ≤ exp −a dzdτ (d+1)/2 t−τ (ρt)d/2 ρt Rd (t − τ ) |x−y|2 2 λ t exp −a |x−y| λρt exp −a e− 2 τ − λρt − t t dτ ≤ ce 2 ≤ ce 2 1/2 (ρt)d/2 td/2 0 (t − τ ) 2 exp −a |x−y| t ≤c = ct−1/2 Ga (x, t; y, 0). (13.4.54) td+1/2 On the other hand, for z ∈ Ω− , we have |x − z| ≥ |x − y|(1 − (a/b)1/2 ) and t − τ ≤ (1 − ρ)t, which yields a

a |x − z|2 a |x − z|2 |x − y|2 ≥ + (1 − (a/b)1/2 )2 . t−τ 2 t−τ 2 (1 − ρ)t

Choose ρ so that 1 1 (1 − (a/b)1/2 )2 = 1. 2 1−ρ Thus we get a

|x − z|2 a |x − z|2 |x − y|2 ≥ +a . t−τ 2 t−τ t

Hence t

e−λτ G (x, t; z, τ ) Gb (z, τ ; y, 0) dzdτ 1/2 a ρt Ω− (t − τ ) t λ e− 2 τ c |x − y|2 |x − z|2 λρt ≤ d/2 exp −a exp −a − dzdτ (d+1)/2 t 2 2(t − τ ) t ρt Ω− (t − τ ) t λ λρt e− 2 τ |x − y|2 c |x − y|2 c dτ ≤ exp −a ≤ d/2 e− 2 exp −a . 1/2 t t t t(d+1)/2 0 (t − τ )

Combining this with (13.4.54) we obtain J2 ≤

c t(d+1)/2

|x − y|2 exp −a . t

438

Chapter 13. Appendix

Corollary 13.4.7. Under the assumptions of Lemma 13.4.6, for any 0 < a < b, t e−λ(τ −s) c(a, b, λ, d) G (x, t; z, τ )Gb (z, τ ; y, s) dzdτ ≤ Ga (x, t; y, s). 1/2 b (t − τ ) (t − s)(1/2 d R s Let us turn to the proof of Proposition 13.4.3. First of all we use scaling in order to eliminate ε from the equations. To this end, we set ϕε (x, t) = ϕ(εx, εt), fε (x, t) = f (εx, εt), Aε (x) = A(εx), Cε (x, t) = C(εx, εt). A change of variables in (13.4.41) leads to ∂t ϕε − div(Aε ∇ϕε ) + Cε ϕε = εfε

in Rd × (0, ∞),

ϕε = 0 for t = 0. (13.4.55)

It is well known that the solution to this problem has the representation t ϕε (x, t) = ε G(x, t; y, s)fε (y, s) dyds. (13.4.56) 0

Rd

Along with equation (13.4.55) we consider the truncated problem ∂t ϕε − div(Aε ∇ϕε ) = εfε

in Rd × (0, ∞),

ϕε = 0 for t = 0.

(13.4.57)

with heat kernel G0 (x, t; y, s), and the auxiliary problem ∂t ϕε − div(Aε ∇ϕε ) + λϕε = εfε

in Rd × (0, ∞),

ϕε = 0 for t = 0, (13.4.58)

with heat kernel (fundamental solution) Gλ (x, t; y, s). Since 0 < λ ≤ C it follows from the general theory of parabolic PDE’s that G(x, t; y, s) ≤ Gλ (x, t; y, s) ≤ G0 (x, t; y, s).

(13.4.59)

Moreover, since A is independent of t, we have Gλ (x, t; y, s) = e−λ(t−s) G0 (x, t; y, s).

(13.4.60)

Since the operator div(Aε ∇·) is symmetric, the heat kernels are symmetric with respect to x and y. Estimates for Green functions of second order parabolic equations with bounded coeﬃcients were derived in [7] (see also [132] for further discussion). A feature of these estimates is that they are sensitive to lower order terms and in general nonuniform in time. Nevertheless, as shown in [7] and [132], global in time estimates do hold true for the fundamental solution G0 : |y − z|2 α0 exp −β0 G0 (x, t; y, s) ≤ , t−s (t − s)d/2 |y − z|2 α0 exp −β0 . |∇x G0 (x, t; y, s)| ≤ t−s (t − s)(d+1)/2

13.4. Singular limits of normal derivatives

439

It follows that G0 (x, t; y, s) ≤ α0 Gβ0 (x, t; y, s), α0 Gβ (x, t; y, s). |∇x G0 (x, t; y, s)| ≤ (t − s)1/2 0

(13.4.61)

From this we obtain an estimate for G: G(x, t; y, s) ≤ e−λ(t−s) G0 (x, t; y, s) ≤ α0 e−λ(t−s) Gβ0 (x, t; y, s).

(13.4.62)

We take advantage of strong positivity of C in order to prove that ∇G admits the same estimate as ∇G0 . To this end, we write (13.4.57) in the form ∂t ϕε − div(Aε ∇ϕε ) = −Cε ϕε + εfε , which leads to t ϕε (x, t) =

Rd

0

G0 (x, t; y, s)(−Cε ϕε (y, s) + εfε (y, s)) dyds.

Next, substitute (13.4.56) into the right hand side to obtain t ϕε (x, t) = −ε

0

Rd

0

Rd

G0 (x, t; z, τ ) 0

t

τ

Rd

G(z, τ ; y, s)Cε (z, τ )fε (y, s) dyds dzdτ

G0 (x, t; y, s)fε (y, s) dyds.

+ε

Recall that ϕε also admits representation (13.4.56). Since fε is an arbitrary smooth function, after changing the order of integration, we arrive at t G(x, t; y, s) = −

Rd

s

G0 (x, t; z, τ )G(z, τ ; y, s)Cε (z, τ ) dzdτ + G0 (x, t; y, s).

Hence ∇x G(x, t; y, s) t =− s

Rd

∇x G0 (x, t; z, τ )G(z, τ ; y, s)Cε (y, s) dzdτ + ∇x G0 (x, t; y, s).

It now follows from inequalities (13.4.61)–(13.4.62) that t |∇x G(x, t; y, s)| ≤

c1 α02 +

s

Rd

1 Gβ (x, t; z, τ )e−λ(τ −s) Gβ0 (z, τ ; y, s) dzdτ (t − τ )1/2 0

α0 Gβ (x, t; y, s). (t − s)1/2 0

440

Chapter 13. Appendix

Next, applying Corollary 13.4.7 we obtain, for any β < β0 and α = α02 c1 + α0 , |∇x G(x, t; y, s)| ≤ α

1 Gβ (x, t; y, s). (t − s)1/2

Combining this inequality with (13.4.62) we ﬁnally obtain α |x − y|2 − λ(t − s) , G(x, t; y, s) ≤ exp −β t−s (t − s)d/2 α |x − y|2 exp −β |∇x G(x, t; y, s)| ≤ . t−s (t − s)(d+1)/2

(13.4.63)

Next, a change of variables in representation (13.4.56) gives t x t y s −d G ϕ(x, t) = ε , ; , f (y, s) dyds, ε ε ε ε 0 Rd t x t y s −d−1 , ; , f (y, s) dyds. [∇x G] ∇ϕ(x, t) = ε ε ε ε ε 0 Rd Thus we get

x t y s , ; , , ε ε ε ε x t y s , ; , . ∇x Gε (x, t; y, s) = ε−d−1 [∇x G] ε ε ε ε Gε (x, t; y, s) = ε−d G

Combining these identities with (13.4.63) we obtain the desired estimates for the heat kernel Gε . This completes the proof of Proposition 13.4.3.

Bibliography [1] D. Adams, L. Hedberg, Function Spaces and Potential Theory, Springer, Berlin, 1996. [2] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [3] C. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial diﬀerential equations satisfying general boundary conditions. I. , Comm. Pure Appl. Math. 12 (1959), 623–727. [4] H. Amann, Linear and Quasilinear Parabolic Problems, Birkhäuser, Boston, MA, 1995. [5] A. A. Amosov, A. A. Zlotnik, Properties in the large of quasi-averaged equations of the one dimensional motion of a viscous heat conducting gas, Dokl. Akad. Nauk 346 (1996), 151–154 (in Russian). [6] A. A. Amosov, A. A. Zlotnik, On quasi-averaged equations of the onedimensional motion of a viscous barotropic medium with rapidly oscillating data, Comput. Math. Math. Phys. 36 (1996), 203–220. [7] D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa 22 (1968), 607–694. [8] N. S. Bakhvalov, M. E. Eglit, Processes in periodic media not describable in terms of mean characteristics, Dokl. Akad. Nauk SSSR 268 (1983), 836–840 (in Russian). [9] J. M. Ball, A version of the fundamental theorem for Young measures, in: PDEs and Continuum Models of Phase Transitions, Lecture Notes in Phys. 344, Springer, Berlin, 1989, 207–215. [10] V. Barbu, I. Lasiecka, R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Mem. Amer. Math. Soc. 181 (2006), no. 852. [11] H. Beirão da Veiga, Stationary motions and the incompressible limit for compressible viscous ﬂuids, Houston J. Math. 13 (1987), 527–544.

P. Plotnikov and J. Sokoáowski, Compressible Navier–Stokes Equations: Theory and Shape Optimization, Monografie Matematyczne 73, DOI 10.1007/978-3-0348-0367-0, © Springer Basel 2012

441

442

Bibliography

[12] H. Beirão da Veiga, An Lp -theory for the n-dimensional, stationary, compressible Navier-Stokes equations, and the incompressible limit for compressible ﬂuids. The equilibrium solutions, Comm. Math. Phys. 109 (1987), 229– 248. [13] H. Beirão da Veiga, Existence results in Sobolev spaces for a transport equation, Ricerche Mat. 36 (1987), suppl., 173–184. [14] J. A. Bello, E. Fernández-Cara, J. Lemoine, J. Simon, The diﬀerentiability of the drag with respect to the variations of a Lipschitz domain in a NavierStokes ﬂow, SIAM J. Control. Optim. 35 (1997), 626–640. [15] J. A. Bello, E. Fernández-Cara, J. Simon, Optimal shape design for NavierStokes ﬂow, in: System Modelling and Optimization, P. Kall (ed.), Lecture Notes in Control and Inform. Sci. 180, Springer, Berlin, 1992, 481–489. [16] J. Bergh, J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin, 1976. [17] Y. Brenier, Résolution d’équations quasilinéaires en dimension N d’espace à l’aide d’équations linéaires en dimension N + 1, J. Diﬀerential Equations 50 (1983), 375–390. [18] D. Bucur, G. Buttazzo, Variational Methods in Shape Optimization Problems, Birkhäuser, Boston, 2005. [19] X. F. Chen, W. Q. Xie, Discontinuous solutions of steady state, viscous compressible Navier-Stokes equations, J. Diﬀerential Equations 115 (1995), 99–119. [20] R. Courant, D. Hilbert, Methods of Mathematical Physics. Vol. II: Partial Diﬀerential Equations, Interscience Publ., New York, 1962. [21] M. C. Delfour, J.-P. Zolésio, Shapes and Geometries, Adv. Design Control 22, SIAM, Philadelphia, PA, 2001, 2nd augmented ed., 2011. [22] B. Desjardins, E. Grenier, P.-L. Lions, N. Masmoudi, Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions, J. Math. Pures Appl. 78 (1999), 461–471. [23] R. J. DiPerna, P.-L. Lions, Ordinary diﬀerential equations, transport theory and Sobolev spaces, Invent. Math. 48 (1989), 511–547. [24] N. Dunford, J. T. Schwartz, Linear Operators. Part 1: General Theory, Interscience Publ., New York, 1958. [25] H. Ebert, Physikalisches Taschenbuch, Vieweg, Braunschweig, 1957. [26] R. E. Edwards, Functional Analysis. Theory and Applications, Holt, Rinehart, and Winston, New York, 1965.

Bibliography

443

[27] L. C. Evans, Partial Diﬀerential Equations, Amer. Math. Soc., Providence, RI, 1998. [28] L. C. Evans, R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992. [29] R. Farwig, Stationary solutions to compressible Navier-Stokes equations, Comm. Partial Diﬀerential Equations 14 (1989), 1579–1506. [30] R. Farwig, G. Galdi, H. Sohr, A new class of weak solutions of the NavierStokes equations, J. Math. Fluid Mech. 8 (2006), 423–443. [31] C. L. Feﬀerman, A sharp form of Whitney’s extension theorem, Ann. of Math. 161 (2006), 509-577. [32] E. Feireisl, On compactness of solutions to the compressible isentropic Navier-Stokes equations when the density is not square integrable, Comment. Math. Univ. Carolin. 42 (2001), 83–98. [33] E. Feireisl, Shape optimization in viscous compressible ﬂuids, Appl. Math. Optim. 47 (2003), 59–78. [34] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Univ. Press, Oxford, 2004. [35] E. Feireisl, Asymptotic analysis of the full Navier-Stokes-Fourier system: from compressible to incompressible ﬂuid ﬂows, Uspekhi Mat. Nauk 62 (2007), no. 3, 169–192 (in Russian). [36] E. Feireisl, Š. Matuš-Nečasová, H. Petzeltová, I. Straškraba, On the motion of a viscous compressible ﬂuid driven by a time-periodic external force, Arch. Ration. Mech. Anal. 149 (1999), 69–96. [37] E. Feireisl, A. H. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Birkhäuser, Basel, 2009. [38] E. Feireisl, A. H. Novotný, H. Petzeltová, On the existence of globally deﬁned weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech. 3 (2001), 358–392. [39] E. Feireisl, A. H. Novotný, H. Petzeltová, On the domain dependence of solutions to the compressible Navier-Stokes equations of a barotropic ﬂuid, Math. Methods Appl. Sci. 25 (2002), 1045–1073. [40] W. Feller, An Introduction to the Probability Theory and its Applications, 3rd ed., Wiley, New York, 1968. [41] G. Fichera, Sulle equazioni diﬀerenziali lineari ellittico-paraboliche del secondo ordine, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Nat. Sez. I (8) 5 (1956), 1–30.

444

Bibliography

[42] I. Fonseca, G. Leoni, Modern Methods in the Calculus of Variations: Lp Spaces, Springer, New York, 2007. [43] J. Frehse, S. Goj, M. Steinhauer, Lp -estimates for the Navier-Stokes equations for steady compressible ﬂow, Manuscripta Math. 116 (2005), 265–275. [44] J. Frehse, M. Steinhauer, W. Weigant, The Dirichlet problem for steady viscous compressible ﬂow in three dimensions, J. Math. Pures Appl. 97 (2012), 85–97. [45] J. Frehse, M. Steinhauer, W. Weigant, The Dirichlet problem for viscous compressible isothermal Navier-Stokes equations in two dimensions, Arch. Ration. Mech. Anal. 198 (2010), 1–12. [46] G. Frémiot, W. Horn, A. Laurain, M. Rao, J. Sokołowski, On the analysis of boundary value problems in nonsmooth domains, Dissertationes Math. 462 (2009), 149 pp. [47] P. Fulmański, A. Laurain, J.-F. Scheid, J. Sokołowski, Level set method with topological derivatives in shape optimization, Int. J. Computer Math. 85 (2008), 1491–1514. [48] A. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, Transl. Math. Monogr. 187, Amer. Math. Soc., 2000. [49] H. Gajewski, K. Gröger, K. Zakharias, Nightlineare Operatorgleichungen und Operatordiﬀerentialgleichungen, Akademie-Verlag, Berlin, 1974. [50] G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearized Steady Problems, Springer, Berlin, 1994; 2nd ed., 2011. [51] D. Gilbarg, N. S. Trudinger, Elliptic Partial Diﬀerential Equations of Second Order, Springer, Berlin, 1983. [52] V. Girinon, Navier-Stokes equations with nonhomogeneous boundary conditions in a bounded three-dimensional domain, J. Math. Fluid Mech. 13 (2011), 309–339. [53] L. Hedberg, Spectral synthesis in Sobolev spaces, and uniqueness of solutions of the Dirichlet problem, Acta Math. 147 (1981), 237–264. [54] A. Henrot, M. Pierre, Variation et optimisation de formes: une analyse géométrique, Springer, Paris, 2005. [55] J. G. Heywood, M. Padula, On the existence and uniqueness theory for steady compressible viscous ﬂow, in: Fundamental Directions in Mathematical Fluid Mechanics, Adv. Math. Fluid Mech., Birkhäuser, Basel, 2000, 171–189. [56] D. Hoﬀ, Strong convergence to global solutions for multidimensional ﬂows of compressible, viscous ﬂuids with polytropic equations of state and discontinuous initial data, Arch. Ration. Mech. Anal. 132 (1995), 1–14.

Bibliography

445

[57] D. Hoﬀ, Discontinuous solutions of the Navier-Stokes equations for multidimensional ﬂows of heat-conducting ﬂuids, Arch. Ration. Mech. Anal. 139 (1997), 303–354. [58] L. Hörmander, Pseudo-diﬀerential operators and non-elliptic boundary problems, Ann. of Math. 83 (1966), 129–209. [59] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147–150. [60] N. Itaya, On the initial value problem of the motion of compressible viscous ﬂuid, especially on the problem of uniqueness, J. Math. Kyoto Univ. 16 (1976), 413–427. [61] W. Jäger, A. Mikelić, Couette ﬂows over a rough boundary and drag reduction, Comm. Math. Phys. 232 (2003), 429–455. [62] B. Kawohl, O. Pironneau, L. Tartar, J.-P. Zolésio, Optimal Shape Design, Lecture Notes in Math. 1740, Springer, Berlin, 2000. [63] A. V. Kazhikhov, V. A. Solonnikov, Existence theorems for the equations of motion of a compressible viscous-ﬂuid, Ann. Rev. Fluid Mechanics 13 (1981), 79–95. [64] A. Kaźmierczak, P. I. Plotnikov, J. Sokołowski, A. Żochowski, Numerical method for drag minimization in compressible ﬂows, in: 15th International Conference on Methods and Models in Automation and Robotics (MMAR), 2010. 97–101; DOI: 10.1109/MMAR.2010.5587258. [65] R. B. Kellogg, Discontinuous solutions of the linearized, steady state, compressible, viscous, Navier-Stokes equations, SIAM J. Math. Anal. 19 (1988), 567–579. [66] J. J. Kohn, L. Nirenberg, Degenerate elliptic-parabolic equations of second order, Comm. Pure Appl. Math. 20 (1967), 797–872. [67] A. N. Kolmogorov, S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Mir, Moscow, 1989 (in Russian). [68] M. A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations, Macmillan, New York, 1964. [69] J. R. Kweon, R. B. Kellogg, Compressible Navier-Stokes equations in a bounded domain with inﬂow boundary condition, SIAM J. Math. Anal. 28 (1997), 94–108. [70] J. R. Kweon, R. B. Kellogg, Regularity of solutions to the Navier-Stokes equations for compressible barotropic ﬂows on a polygon, Arch. Ration. Mech. Anal. 163 (2000), 35–64. [71] J. R. Kweon, R. B. Kellogg, Singularities in the density gradient, J. Math. Fluid Mech. 8 (2006), 445–454.

446

Bibliography

[72] J. R. Kweon, R. B. Kellogg, Regularity of solutions to the Navier–Stokes system for compressible ﬂows on a polygon, SIAM J. Math. Anal. 35 (2004), 1451–1485. [73] J. R. Kweon, R. B. Kellogg, Compressible Stokes problem on nonconvex polygonal domains, J. Diﬀerential Equations 176 (2001), 290–314. [74] O. A. Ladyženskaja, V. A. Solonnikov, N. N. Ural’ceva, Linear and Quasiliniear Equations of Parabolic Type, Transl. Math. Monogr. 23, Amer. Math. Soc., Providence, RI, 1967. [75] L. D. Landau, E. M. Lifshitz, Course of Theoretical Physics. Vol. 6. Fluid Mechanics, Pergamon Press, Oxford, 1987. [76] N. S. Landkof, Foundations of Modern Potential Theory, Springer, Berlin, 1972. [77] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969. [78] J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications, Vols. 1, 2, Dunod, Paris, 1968. [79] P.-L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 1, Incompressible Models, Clarendon Press, Oxford, 1996. [80] P.-L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible Models, Clarendon Press, Oxford, 1998. [81] P.-L. Lions, Compacité des solutions des équations de Navier-Stokes compressibles isentropiques, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), 115–120. [82] P.-L. Lions, Bornes sur la densité pour les équations de Navier-Stokes compressibles isentropiques avec conditions aux limites de Dirichlet, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), 659–662. [83] P.-L. Lions, On some challenging problems in nonlinear partial diﬀerential equations, in: Mathematics: Frontiers and Perspectives, V. Arnold et al. (eds.), Amer. Math. Soc., Providence, RI, 2000, 121–135. [84] P.-L. Lions, B. Perthame, P. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Comm. Pure Appl. Math. 49 (1996), 599–638. [85] P.-L. Lions, B. Perthame, E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7 (1994), 169–191. [86] J. Málek, J. Nečas, M. Rokyta, M. Růžička, Weak and Measure-Valued Solutions to Evolutionary PDE, Chapman and Hall, London, 1996.

Bibliography

447

[87] A. Matsumura, T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive gases, J. Math. Kyoto Univ. 20 (1980), 67–104. [88] V. G. Maz’ya, Sobolev Spaces, Springer, Berlin, 1985. [89] B. Mohammadi, O. Pironneau, Shape optimization in ﬂuid mechanics, Ann. Rev. Fluid Mech. 36 (2004), 255–279. [90] B. Mohammadi, O. Pironneau, Applied Shape Optimization for Fluids, 2nd ed., Oxford Univ. Press, Oxford, 2010. [91] M. Moubachir, J.-P. Zolésio, Moving Shape Analysis and Control: Applications to Fluid Structure Interactions, Chapman & Hall/CRC, Boca Raton, FL, 2006. [92] J. Nash, Le problème de Cauchy pour les équations diﬀérentielles d’un ﬂuide général, Bull. Soc. Math. France 90 (1962), 487–497. [93] S. Novo, Compressible Navier-Stokes model with inﬂow-outﬂow boundary conditions, J. Math. Fluid Mech. 7 (2005), 485–514. [94] S. Novo, A. Novotný, On the existence of weak solutions to the steady compressible Navier-Stokes equations when the density is not square integrable, J. Math. Kyoto Univ. 42 (2002), 531–550. [95] S. Novo, A. Novotný, On the existence of weak solutions to the steady compressible Navier-Stokes equations in domains with conical outlets, J. Math. Fluid Mech. 8 (2006), 187–210. [96] A. Novotný, About steady transport equation. I. Lp -approach in domains with smooth boundaries, Comment. Math. Univ. Carolin. 37 (1996), 43–89. [97] A. Novotný, About steady transport equation. II. Schauder estimates in domains with smooth boundaries, Portugal. Math. 54 (1997), 317–333. [98] A. Novotný, M. Padula, Existence and uniqueness of stationary solutions for viscous compressible heat conductive ﬂuid with large potential and small non-potential external forces, Sibirsk. Mat. Zh. 34 (1993), no. 5, 120–146 (in Russian). [99] A. Novotný, M. Padula, Lp -approach to steady ﬂows of viscous compressible ﬂuids in exterior domains, Arch. Ration. Mech. Anal. 126 (1994), 243–297. [100] A. Novotný, M. Padula, Physically reasonable solutions to steady compressible Navier-Stokes equations in 3D-exterior domains (v∞ = 0), Math. Ann. 308 (1997), 439–489. [101] A. Novotný, I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford Univ. Press, Oxford, 2004.

448

Bibliography

[102] O. A. Ole˘ınik, E. V. Radkevič, Second Order Equations with Nonnegative Characteristic Form, Plenum Press, New York, 1973. [103] M. Padula, Existence and uniqueness for viscous steady compressible motions, Arch. Ration. Mech. Anal. 97 (1987), 89–102. [104] M. Padula, Steady ﬂows of barotropic viscous ﬂuids, in: Classical Problems in Mechanics, Quad. Mat. 1, Dept. Math., Seconda Univ. Napoli, Caserta, 1997, 253–345. [105] P. Pedregal, Parametrized Measures and Variational Principles, Progr. Nonlinear Diﬀerential Equations Appl. 30, Birkhäuser, Basel, 1997. [106] B. Perthame, Kinetic Formulation of Conservation Laws, Oxford Lecture Ser. Math. Appl. 21, Oxford Univ. Press, Oxford, 2002. [107] P. I. Plotnikov, E. V. Ruban, J. Sokołowski, Inhomogeneous boundary value problems for compressible Navier-Stokes equations, well-posedness and sensitivity analysis, SIAM J. Math. Anal. 40 (2008), 1152–1200. [108] P. I. Plotnikov, E. V. Ruban, J. Sokołowski, Inhomogeneous boundary value problems for compressible Navier-Stokes and transport equations, J. Math. Pures Appl. 92 (2009), 113–162. [109] P. I. Plotnikov, J. Sokołowski, Stationary boundary value problems for Navier-Stokes equations with adiabatic exponent γ < 3/2, Dokl. Akad. Nauk 397 (2004), 166–169 (in Russian) [110] P. I. Plotnikov, J. Sokołowski, On compactness, domain dependence and existence of steady state solutions to compressible isothermal Navier-Stokes equations, J. Math. Fluid Mech. 7 (2005), 529–573. [111] P. I. Plotnikov, J. Sokołowski, Concentrations of solutions to time-discretized compressible Navier -Stokes equations, Comm. Math. Phys. 258 (2005), 567– 608. [112] P. I. Plotnikov, J. Sokołowski, Domain dependence of solutions to compressible Navier-Stokes equations, SIAM J. Control Optim. 45 (2006), 1165–1197. [113] P. I. Plotnikov, J. Sokołowski, Stationary solutions of Navier-Stokes equations for diatomic gases, Uspekhi Mat. Nauk 62 (2007), no. 3, 117–148 (in Russian). [114] P. I. Plotnikov, J. Sokołowski, Stationary boundary value problems for compressible Navier-Stokes equations, in: Handbook of Diﬀerential Equations: Stationary Partial Diﬀerential Equations, Vol. VI, Elsevier/North-Holland, Amsterdam, 2008, 313–410. [115] P. I. Plotnikov, J. Sokołowski, Shape derivative of drag functional, SIAM J. Control Optim. 48 (2010), 4680–4706.

Bibliography

449

[116] P. I. Plotnikov, J. Sokołowski, Inhomogeneous boundary value problem for nonstationary compressible Navier-Stokes equations, J. Math. Sci. 170 (2010), 34–130. [117] P. Plotnikov, J. Sokołowski, A. Żochowski, Numerical experiments in drag minimization for compressible Navier-Stokes ﬂows in bounded domains, in: Proc. 14th International IEEE/IFAC Conference on Methods and Models in Automation and Robotics MMAR’09, 2009, 4 pp. [118] A. Sard, The measure of the critical values of diﬀerentiable maps, Bull. Amer. Math. Soc. 48 (1942), 883–890. [119] H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York, 1955. [120] L. Schwartz, Analyse Mathématique I, Hermann, Paris, 1967. [121] J. Serrin, On the uniqueness of compressible ﬂuid motions, Arch. Ration. Mech. Anal. 3 (1959), 271–288. [122] J. Serrin, Mathematical principles of classical ﬂuid mechanics, in: Handbuch der Physik VIII/I, Springer, Berlin, 1959, 125–263. [123] J. Simon, Domain variation for drag in Stokes ﬂow, in: Control Theory of Distributed Parameter Systems and Applications (Shanghai, 1990), Lecture Notes in Control and Inform. Sci. 159, Springer, Berlin, 1991, 28–42. [124] T. Slawig, A formula for the derivative with respect to domain variations in Navier-Stokes ﬂow based on an embedding domain method, SIAM J. Control Optim. 42 (2003), 495–512. [125] T. Slawig, An explicit formula for the derivative of a class of cost functionals with respect to domain variations in Stokes ﬂow, SIAM J. Control Optim. 39 (2000), 141–158. [126] J. Sokołowski, J.-P. Zolésio, Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer Ser. Comput. Math. 16, Springer, Berlin, 1992. [127] E. Stein, Singular Integrals and Diﬀerentiability Properties of Functions, Princeton Univ. Press, Princeton, NJ, 1970. [128] L. Tartar, Compensated compactness and applications to partial diﬀerential equations, in: Nonlinear Analysis and Mechanics, Heriot-Watt Symposium, Vol. IV, Pitman Res. Notes in Math. 39, Pitman, Boston, MA, 1979, 136– 212. [129] L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lecture Notes of the Unione Matematica Italiana 3, Springer, Berlin, and UMI, Bologna, 2007. [130] A. Valli, W. M. Zajączkowski, Navier-Stokes equations for compressible ﬂuids: global existence and qualitative properties of the solutions in the general case, Comm. Math. Phys. 103 (1986), 259–296.

450

Bibliography

[131] K. Yosida, Functional Analysis, Springer, Berlin, 1965. [132] Q. S. Zhang, Gaussian bounds for the fundamental solutions of ∇(A∇u) + B∇u − ut = 0, Manuscripta Math. 93 (1997), 381–390. [133] W. P. Ziemer, Weakly Diﬀerentiable Functions, Springer, New York, 1989.

Notation The Banach space of bounded continuous functions on cl Ω, p. 9

C(Ω)

The linear space of continuous functions compactly supported on Ω, p. 9

Cc (Ω)

C0 (Ω) The space of bounded continuous functions vanishing at ∂Ω for bounded or unbounded domains, p. 16 C m+β (Ω) The Banach space of functions on cl Ω which are continuous and bounded along with all derivatives of order ≤ m, and whose derivatives of order m are Hölder continuous with exponent β ∈ [0, 1), p. 9 C m+β (0, T ; X) p. 11 f (s) dF (s) I

The space of C m+β functions with values in a Banach space, Lebesgue-Stieltjes integral, p. 21

L∞ w (Ω; M(R)) [f ],k , [f ]m, , [f ]m,k

The space of Young measures, dual to L1 (Ω; C0 (R)), p. 29 Molliﬁers of f , p. 35

A bounded domain in Rd

Ω

Q = Ω × (0, T ) ST = ∂Ω × (0, T ) T

= ST ∪ (cl Ω × {t = 0})

S(u)

The viscous stress tensor, p. 47

C[f ]

The nonlinear operator of the kinetic equation, p. 182

J(Ω)

The drag functional for stationary ﬂow, p. 300

dJ(Ω)[T] The shape derivative of the drag functional in the direction of the ﬁeld T, p. 372 W l,r (Ω)

A Sobolev space, p. 32

W s,r (Ω)

A fractional Sobolev space, p. 32

W0s,r (Ω)

The interpolation space [Lr (Ω), W01,r (Ω)]s,r , p. 33

P. Plotnikov and J. Sokoáowski, Compressible Navier–Stokes Equations: Theory and Shape Optimization, Monografie Matematyczne 73, DOI 10.1007/978-3-0348-0367-0, © Springer Basel 2012

451

452

Notation

W −s,r (Ω)

The dual of W0s,r (Ω), p. 34

W−s,r (Ω)

The dual of W s,r (Ω), p. 347

X s,r = W s,r (Ω) ∩ W 1,2 (Ω) Y s,r = W s+1,r (Ω) ∩ W 2,2 (Ω) Z s,r = W s−1,r (Ω) ∩ L2 (Ω)

p. 308 p. 308 p. 314

U s,r = (W s−1,r (Ω))3 × (W s,r (Ω))3 × R

p. 333

V s,r = (W s+1,r (Ω))3 × (W s,r (Ω))3 × R

p. 333

E

s,r

= (Z

) × (X

s,r 3

) ×R

s,r 3

F s,r = (Y s,r )3 × (X s,r )3 × R

p. 334 p. 334

Index absolute variation, 15 accretivity defect, 316, 382 adiabatic exponent, 48, 64 adiabatic ﬂow, 64 adjoint state, 376 admissible obstacles, family of, 244 admissible set, 297 a.e. convergence, 24 Alaoglu theorem, 5 angular velocity, 52 anisotropic Lebesgue space, 28 dual of, 29 Aronson-type estimates, 429 Arzelà-Ascoli theorem, 11 autonomous system, 53

ball, 3, 231 Banach space, 3 barotropic ﬂow, 47 Bessel capacity, 229 Bessel kernel, 230 bi-Lipschitz map, 230 Bochner integrable, 24 Bochner integral, 24 Bochner theorem, 25 Bochner-Lebesgue theorem, 26 Borel function, 14 Borel map, 180 Borel measure, 14 Borel set, 12 Borel σ-algebra, 12

boundary condition ﬁrst, 48 no-slip, 48 no-stick, 48 second, 48 boundary of class C m+β , 10 bounded energy functions, 64 bounded operator, 3 bounded variation, 20 capacity, 229 Bessel, 229 Carathéodory function, 142 Carathéodory integrand, 31 characteristic set, 49 combined S-convergence, 236 compact embedding, 33 compact family of obstacles, 244 compact mapping, 106 compactly supported, 9 compactness in Lp , 20 compensated compactness, 71 complete space, 3 compressible Navier-Stokes equations, xiv nonstationary, 48 nonstationary problem, 54, 58, 101, 250 regularized, 104 renormalized solution, 60 weak renormalized solution, 101, 102, 251, 253, 255, 281

P. Plotnikov and J. Sokoáowski, Compressible Navier–Stokes Equations: Theory and Shape Optimization, Monografie Matematyczne 73, DOI 10.1007/978-3-0348-0367-0, © Springer Basel 2012

453

454 compressible Navier-Stokes equations nonstationary problem weak solution, 59 parabolic regularization, 103 regularized problem strong solution, 106 stationary problem, 58, 299 algebraic scheme, 305 material derivative, 301, 359 shape derivative, 301, 371 small perturbation, 307, 311 thermodynamical inconsistency, 48 compressible Navier-Stokes-Fourier equations, 46 condenser type domain, 235 conformal parameterization, 413 constitutive equations, 46 constitutive law, 46 continuity of work functional, 284 convection term, 68 convergence a.e., 24 in measure, 20 strong, 3, 19 weak, 4 weak in Lp , 18 weak , 4 weak in L∞ , 19 Coriolis force, 52 critical exponent, 64 cylinder, 28 lateral surface, 99 parabolic boundary, 99 dense set, 3 density, 45 deterministic random function, 184 deviation parameter, 64 diﬀerentiable mapping, 11 DiPerna and Lions lemma, 36 Dirichlet problem, 231 Dirichlet-type condition, 48 distribution function, 22, 181 distributional derivative, 32

Index divergence, 2 of matrix, 333 domain Lipschitz, 10 of class C m+β , 10 dominated convergence, 13, 25, 26 drag, 300 drag functional, 300 dual space, 3 duality pairing, 3 Eberlein-Šmulian theorem, 4, 5 eﬀective viscous ﬂux, 78 eﬀective viscous pressure, 78, 303, 305 Egoroﬀ theorem, 24 elliptic BVP, 40 ellipticity, 41 embedding, 3 emergent ﬁeld condition, 381 energy conservation law, 45 energy density, 45 energy estimates, 64 energy inequalities, 70 entropy density, 46 equicontinuous, 11 equi-integrable, 19 equivalent functions, 17 existence for compact obstacles, 102, 253, 255, 281 extension to obstacle, 252 external energy ﬂux, 45 external mass force, 45 fast oscillation data, 176 solution, 176 Fatou theorem, 14 ﬁctitious domain method, 102 ﬁne neighborhood, 233 ﬁne properties, 229 ﬁne topology, 233 ﬁnely closed set, 233 ﬁnely continuous function, 233 ﬁnely open set, 233

Index ﬁrst boundary condition, 48 ﬁxed point theorems, 8 ﬂat spearhead, 233 ﬂow adiabatic, 64 around moving body, 53 around obstacle, 250 barotropic, 47 isentropic, 47 stationary, 52 ﬂow domain, 45 fundamental theorem on Young measures, 31 Galileo transformation, 52 generalized derivative, 32 generalized viscous ﬂux, 88 Gibbs equation, 46 governing equations, 45 gradient, 2 of vector, 333 Hahn-Banach theorem, 4 Hausdorﬀ distance, 225 heat capacity, 46 heat conduction coeﬃcient, 45 heat kernel, 429 Hedberg theorem, 234 Helly theorem, 21 hold-all domain, xv, 54 Hölder domain, 10 Hölder inequality, 18, 28 Hölder space, 10 vector-valued, 11 hydrodynamic force, 50 hydrodynamic power, 50 hydrodynamic work, 50 in/out ﬂow problem, xiv inlet, 49 integrable function, 13, 17 internal energy, 46

455 interpolation, 6 of dual spaces, 7 of subspaces, 7 interpolation space, 6, 7 invertibility, 8 isentropic ﬂow, 47 Jacobi matrix, 412 Jacobian, 1 J-interpolation space, 7 Jordan decomposition, 15 kinetic energy tensor, 64 kinetic equation deterministic solution, 185 weak solution, 183 kinetic problem, 181 K-interpolation space, 6 Kuratowski-Mosco convergence, 226, 235 lateral surface, 99 Lebesgue dominated convergence theorem, 13 Lebesgue integral, 17 Lebesgue measurable, 17 Lebesgue measure, 17 Lebesgue measure zero, 16 Lebesgue space, 17 anisotropic, 28 dual of, 18 vector-valued, 25 Lebesgue-Stieltjes integral, 21 integration by parts, 22 Lebesgue-Stieltjes measure, 21 Leray-Schauder theorem, 8 Levi theorem, 14 linear operator, 3 Lipschitz domain, 10 local solutions of stationary problem, 311, 354 locally bounded, 15 Lusin theorem, 24 Mach number, 46

456 mass balance, 45 mass balance equation generalized, 78 relaxed, 377 renormalized, 78 steady, 377 mass conservation, 63 mass diﬀusion coeﬃcient, 68 mass dissipation rate, 68 material derivative, 301, 357 Mazur theorem, 238 measurable function, 13, 17 vector-valued, 23 weakly, 23 weakly , 23 measure, 12, 17 Borel, 14 ﬁnite, 12 probability, 12 Radon, 14 σ-ﬁnite, 12 Young, 30 measure space, 12 method of characteristics, 378 metrizability of weak topology, 5 Minkowski inequality, 17 molliﬁer, 35 mollifying kernel, 35 momentum, 64 momentum balance equation, 45 generalized, 78 momentum conservation, 63 monotone convergence, 14 multiplicators in Sobolev spaces, 334 mutually singular, 15 Navier-Stokes-Fourier equations, 46 Neumann-type condition, 48 nonpermeability, 380 nonstationary problem, 58 normal coordinates, 384, 411 conformal, 413 normal integrand, 31 normed space, 2

Index no-slip boundary condition, 48 no-stick boundary condition, 48 obstacle, 235, 250 Ole˘ınik theorem, 379 optimal obstacle, 297 outgoing set, 49 parabolic boundary, 99 parabolic regularization, 103 Pecle number, 46 perfect polytropic gas, 46 Pettis theorem, 24 Piola transformation, 301 Poisson equation, 39 power of hydrodynamic force, 50 pressure, 46 probability distribution, 180 probability measure, 12 product of tensors, 2 q.e., 234 quasicontinuous function, 234 quasieverywhere, 234 Radon measure, 14 signed, 15 space, 16 random variable, 180 real interpolation method, 6 reference domain, 301 reﬂexive space, 4 regular solution, 182 regularized problem, 104 renormalization, 36, 60 renormalized equation, 60 renormalized solution, 60 weak, 101 representative of function, 234 Reynolds number, 46 Riesz operator, 40 Riesz theorem, 15 rigid body motion, 51 scalar product, 1

Index Schauder theorem, 8 S-convergence, 236 second boundary condition, 48 separable set, 23 separable space, 5 sequentially weakly compact, 5 sequentially weakly continuous, 4 shape derivative, 301 shape gradient of drag functional, 301 shape sensitivity analysis, 300 σ-algebra, 12 signed measure, 15 simple function, 13 vector-valued, 23 S-limit, 236 Sobolev embedding theorem, 82 Sobolev space, 32 dual of, 33 embedding, 33, 66 extension operators, 35 fractional, 32 dual of, 347 solution regular, 182 renormalized, 60 very weak, 332 weak, 59 weak renormalized, 101 special orthogonal matrix, 51 stability of solutions for compact obstacles, 255 standard parabolic neighborhood, 383 state equations, 375 state variables, 45, 375 stationary ﬂow, 52 stationary problem, 58 in perturbed domain, 300 Stokes equations, 314 Stokes problem, 42 stress tensor, 50 viscous, 45, 47, 64 strong convergence, 3, 19

457 strongly measurable, 23 symmetric diﬀerence, 286 temperature, 45 tensor product, 1 thick set, 231 thin set, 231 Tikhonov ﬁxed point theorem, 9 total variation, 20 trace of function, 234 transport equation, 377 weak solution, 379 velocity, 45 very weak solution, 332, 349 viscosity coeﬃcients, 45 viscosity ratio, 46 viscous ﬂux, 95 generalized, 88 viscous stress tensor, 45, 47, 64 weak compactness, 19 weak convergence, 4 in anisotropic space, 29 in Lp , 18, 31 weak convergence, 4 in anisotropic space, 29 in L∞ , 19 weak renormalized solution, 101, 251 weak solution, 59 weak topology, 4 weakly compact set, 5 weakly complete space, 4 weakly continuous map, 4 weakly measurable, 23 weakly measurable, 23 Whitney theorem, 218 Wiener regularity condition, 231 work of hydrodynamic force, 50 Young measure, 30 Zygmund-Calderón theorem, 40