The Mathematical Theory Of Viscous Incompressible Flow Ladyzhenskaya Pdf

Mathematical theory of viscous incompressible flow

Olga A. Ladyzhenskaya, R.A. Silverman

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In the three years since the Russian edition of this book was written, quite a few papers devoted to a mathematically rigorous analysis of nonstationary solutions of the Navier-Stokes equations have been published. These papers either pursue the investigation of differential properties of the solutions whose existence and uniqueness is proved in the present book, or else they give other methods for obtaining such solutions. However, the basic problem of the unique solvability "in the large" of the boundary-value problem for the general three-dimensional nonstationary Navier-Stokes equations (with no assumptions other than a certain smoothness of the initial field and of the external forces) remains as open as ever.

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The Mathematical Theory of Viscous Incompressible Flow byO. A. Ladyzhenskaya Revised Second Edition Translated from the Russian by Richard A. Silverman MATHEMATICS AND ITS APPLICATIONS A series of Monographs and Texts VOLUME 2 GORDON AND BREACH SCIENCE PUBLISHERS  Mathematics and Its Applications A Series of Monographs and Texts Volume 1 Jacob T. Schwartz lectures on the mathematical methods in analytical ECONOMICS Volume 2 O. A. Ladyzhenskaya the mathematical theory of viscous incompressible FLOW I Second English Edition. Revised and Enlarged) Volume 3 R. L. Slratonovich topics in ihe theory of random noise (in 2 volumes) Volume 4 S. Chowla the riemann hypothesis and hilbert's tenth problem Volume 5 Jacob T. Schwartz theory of money Volume 6 F. Treves linear partial difff.rential equations with constant coefficients Volume 7 Georgi E. Shilov generalized functions and partial differential equations Volume 8 /. /. Pyaletskii-Shapiro automorphic functions and the geometry of classical DOMAINS Volume 9 /. Colojord and С. Fioax theory of generalized spectral operators Volume 10 M. N. Barber and B. W. Niham random and restricted walks Volume 11 N. Nakanishi graph theory and feynman integrals Volume 12 Y. Wallach sti dy and compilation of computer languages Volume 13 T. S. Chihara an introduction to orthogonal polynomials Volume 14 J F Pommaret systems of partial differential equations and lie PSF.L DOGROl PS Volume 15 J F Pommaret differential galois theory This book iv part of a series. The publisher will accept continuation orders which may be cancelled at am time and which provide for automatic billing and shipping of each title in the >ene> upon publication. Please write for details.  The Mathematical Theory of Viscous Incompressible Flow O. A. LADYZHENSKAYA Second English Edition Revised and Enlarged Translated from the Russian by Richard A. Silverman and John Chu GORDON AND BREACH SCIENCE PUBLISHERS NEW YORK LONDON PARIS MONTREUX TOKYO MELBOURNE  © 1963, 1969 by Gordon and Breach, Science Publishers.;  Inc.. Post Office Box 786, Cooper Station, New York, New York 10276,U.S.A. All rights reserved. First edition published 1963 Second printing 1964 Second edition published 1969 Second printing 1987 Gordon and Breach Science Publishers Post Office Box 197 London WC2E 9PX England 58, rueLhomond 75005 Paris France Post Office Box 161 1820Montreux2 Switzerland 14-9Okubo3-chome Shinjuku-ku, Tokyo 160 Japan Camberwell Business Center Private Bag 30 Camberwell, Victoria 3124 Australia The Library of Congress has cataloged the first printing of this title as follows: Ladyzhenskaia, О А The mathematical theory of viscous incompressible flow. Translated from the Russian by Richard A. Silverman. Rev. English ed. New York, Gordon and Breach [1963] xiv, 184 p. 24 cm. (Mathematics and its applications, v. 2) Translation of Математические вопросы динамики вязкой несжи- маемой жидкости (transliterated: Matematicheskie voprosy dinamiki viazkoi neszhimaemoi zhidkosti) Bibliography: p. 177-180. 1. Hydrodynamics. 2. Boundary value problems. I. Title QA929.L313 532.58 63-16283 Library of Congress No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage and retrieval system, without permission in writing from the publishers. Printed in the United States of America.  This book is dedicated to three very different persons whom I hold in deep regard: my father Alexander Ivanovich Ladyzhenski, Vladimir Ivanovich Smirnov, and Jean Leray.  Author's Preface to the Second English Edition In the second English edition, more space has been given to the investigation of the smoothness of generalized solutions, and particularly, to the clarifica- tion of the conditions under which these solutions become classical solutions. In part, this is a concession to the tradition, of granting full rights only to classical solutions, a tradition not completely overcome even in mathematical circles. This tradition is exemplified in the first article in the recently published volume of the Handbuch der Physik [119]. A significant paragraph of this article presents results on nonstationary problems, emphasising classical solutions, and particularly results on the possible points of onset of turbulence and on bifurcation of the solutions in two-dimensional nonstationary problems. Only a casual reference to papers in which "various kinds of generalized solutions" are studied is made, among them [38], in which is shown that the two-dimensional nonstationary problem has a unique solution "in the large" and consequently in it there can be no onset of turbulence and bifurcation of solutions at all. Apparently the author of this highly systematic article is frightened by the term "generalized solution", as if it were synonymous to "unreal". In fact, as soon as the theorems on existence and uniqueness are proved for some class m, to which the classical solution (if it exists) also belongs, then the existence and uniqueness problem must be regarded as largely solved. The solution found in m is the only one possible. The problem of obtaining more detailed information concerning the solution although also interest- ing and possibly difficult; nevertheless, will occupy a secondary position and will not be involved in the questions of existence, uniqueness, and stability of the solution. In regard to boundary-value problems considered in this book, the determination of when the generalized solutions found are also classical solutions follows comparatively easily from the methods and theorems already given in detail in the first edition; this is shown in various paragraphs appended to the appropriate sections. I regret that this results in a certain complication of the exposition, thereby losing a definite virtue—that of being short and directed only toward the principal questions of solvability. This material has been added however, since I have wished to answer questions which readers have referred to me,  Viil AUTHOR'S PREFACE TO THE SECOND ENGLISH EDITION and to show the possibility of developing the ideas, methods, and results given in the first edition of this book. One of the main ideas of this book is that it is useful not to limit oneself to some one "class of solutions" selected a priori (for example, the class of classical solutions), but to use greater freedom in the choice of a class of solutions. This is particularly important since the question of the unique solvability in the large of the general three-dimensional nonstationary problem is still open. This problem will be solved if we succeed in finding some class m in which uniqueness holds, and a priori bounds simultaneously exist for all solutions of the problem. This requires that we obtain new a priori estimates valid for any interval of time, without smallness restrictions, for given data. It is possible that the following argument might permit us to by-pass these effective estimates. A nonstationary problem is stable for arbitrary finite intervals of time in all those classes in which we have succeeded in proving unique solvability. In view of this, it is sufficient to show unique solvability "in the large" only for some dense set of initial data and external forces (for a more precise discussion on this point, see chapter 6, section 6). To this latter end, good use might be made of a consideration of the entire set of possible solution-trajectories in the spirit of the ergodic theory of dynamical systems. Up to the present time, essentially only two cases of unique solvability of the general nonstationary problem have been proved: the first applies for arbitrary intervals of time, but only for small Reynolds number at the initial regime and for external forces f(x, t) derived from a potential (or for small departure of f(x, t) from a potential force). The second applies for arbitrary but not too bad initial regimes and external forces, but only for small intervals of time. Depending on the function space m in which the solution is to be found, the statements proved in these two cases have various analytical formulations. In this book we present in detail a version developed in [39]; for other versions, see references [12], [53], [68], [91-93], [96], [127], [128] and also chapter 6, section 6. In this edition, as well as in the first one, we restricted our considerations to the study of those cases where the region filled with fluid does not change with time, although the unique solvability of initial boundary-value problems is now established for the regions with changing boundaries and the methods we use to prove it are essentially the same as those described here. In a supplement, I propose alternate fundamental equations for fluid mechanics, whose mathematical character is advantageous relative to the Navier-Stokes equations, and which appear to me to be potentially useful in  AUTHOR'S PREFACE TO THE SECOND ENGLISH EDITION ix describing viscous fluid flows. For these equations, the initial-boundary- value problems are uniquely solvable in the large. Finally, I should like to draw the reader's attention to the following special feature of the book. Each chapter gives a distinct method of solution for the problem considered. This is done to acquaint the reader with as large a number of different methods of solution of boundary-value problems as possible, without significantly increasing the size of the book. However, each method might also have been used successfully (if appropriately modified) to solve the problems discussed in the other chapters. In this edition, aside from additions, improvements, and corrections of noticed misprints, we also make precise those statements which were subject to misinterpretation (particularly in the translation), and eliminate various errors and inaccuracies which crept into the first translation. Leningrad, Autumn 1968. O. A. L.  Author's Preface to the First English Edition In the three years since the Russian edition of this book was written, quite a few papers devoted to a mathematically rigorous analysis of nonstationary solutions of the Navier-Stokes equations have been published. These papers either pursue the investigation of differential properties of the solutions whose existence and uniqueness is proved in the present book, or else they give other methods for obtaining such solutions. However, the basic problem of the unique solvability "in the large" of the boundary-value problem for the general three-dimensional nonstationary Navier-Stokes equations (with no assump- tions other than a certain smoothness of the initial field and of the external forces) remains as open as ever. The most delicate results on the differentiability properties of generalized solutions are those due to К. К. Golovkin and V. A. Solonnikov, formulated in chapter 6, section 4. As for stationary problems, we call attention to the interesting papers by R. Finn, in which the behavior of solutions of the problem of stationary flow past obstacles is studied as | x | -> oo. In the analysis of stationary problems given here, we have directed our attention to problems involving flow past obstacles, or more exactly, problems in which the total flow through the boundary of an arbitrary obstacle in the flow is equal to zero. Of no less importance are problems involving sources, where this condition is not satisfied. The possibility is not precluded that such problems, unlike problems involving flow past objects, are not always solvable for large Reynolds numbers. In fact, in the case of an unbounded planar domain, the problem of flow with sources can have infinitely many solutions (so that extra conditions must be imposed to single out a unique solution). For example, the functions ¦ 11 - Г | __(c > иф — cil—r ° +Cl 2ClVr n = ° +l 2lV I l rB<7v) + 2 y 2r2 с Bc/v) + 2 where с and Cj are arbitrary constants, satisfy the equations of continuity and xi  Xii AUTHOR'S PREFACE TO THE FIRST ENGLISH EDITION the Navier-Stokes equations, written in polar coordinates r and ф. For fixed с < — 2v, these functions give infinitely many solutions in the domain r S: 1, which tend to zero sufficiently rapidly as r -* oo and satisfy the same boundary conditions «r|r=l=C, иф\г=1=0 at r = 1. In the present edition of the book, all detected misprints have been eliminated. Moreover, an extra section on effective estimates of solutions of the nonlinear stationary problem (chapter 5, section 4) has been added. Leningrad, January 7, 1963 O. A. L.  Author's Preface to the Russian Edition The aim of this book is to acquaint mathematicians and hydrodynamicists with the success which has been achieved so far in investigating the existence, uniqueness and solvability of boundary-value problems for both the linearized and the general nonlinear Navier-Stokes equations. Many of the fundamental results obtained are of such a simple and definitive form that it has been possible to present them in this small monograph. The reader is not required to know more than the elements of classical and functional analysis. The author is grateful to her young colleagues V. A. Solonnikov and К. К. Golovkin, and especially to A. P. Oskolkov and A. V. Ivanov, for their assistance in preparing the manuscript of this book. O. A. L.  Translator's Preface to the First Edition This book is a translation of O. A. Ladyzhenskaya's Matematicheskiye Voprosy Dinamiki Vyazkoi Neszhimayemoi Zhidkosti (literally, Mathematical Problems of the Dynamics of a Viscous Incompressible Liquid), which appeared in 1961 in the series Contemporary Problems of Mathematics, published under the auspices of the editorial board of the journal Uspekhi Matematicheskikh Nauk. The present edition has benefited greatly from the author's continued (and indefatigable) interest. Thus, it incorporates numerous corrections, additional references, further comments, and even an extra section. This "feed-back process" has been facilitated by Prof. Ladyzhenskaya's examina- tion of the translation in the galley proof stage. The subject index is a somewhat modified version of one proposed by the author. Of the various systems for transliterating the Cyrillic alphabet into the Latin alphabet, I prefer and have used that due to Prof. E. J. Simmons. I would like to take this opportunity to thank the author for her help, with the hope that I have acted as her faithful amanuensis, insofar as permitted by the divergence of stylistic and grammatical norms in our two languages. I would also like to thank Prof. L. Nirenberg of New York University for patiently assisting me in my quest for suitable terminological compromises. R. A. S.  Contents Introduction 1 Chapter 1—Preliminaries 7 1. Some Function Spaces and Inequalities 7 2. The Vector Space L2(ti) and its Decomposition into Orthogonal Subspaces 23 3. Riesz' Theorem and the Leray-Schauder Principle 31 Chapter 2—The Linearized Stationary Problem 33 1. The Case of a Bounded Domain in E3 35 2. The Exterior Three-Dimensional Problem 40 3. Plane-Parallel Flows 42 4. The Spectrum of Linear Problems 44 5. The Positivity of the Pressure 47 Chapter 3—The Theory of Hydrodynamical Potentials 49 1. The Volume Potential 49 2. Potentials of Single and Double Layers 53 3. Investigation of the Integral Equations 59 4. Green's Function 64 5. Investigation of Solutions in Wr{ii) 67 Chapter 4—The Linear Nonstationary Problem 81 1. Statement of the Problem. Existence and Uniqueness Theorems 82 2. Investigation of the Differentiability Properties of Generalized Solutions .. 89 3. Unbounded Domains and Behavior of Solutions as t^ + oo 101 4. Expansion in Fourier Series 104 5. The Vanishing Viscosity 105 6. The Cauchy Problem 106 Chapter 5—The Nonlinear Stationary Problem 113 1. The Case of Homogeneous Boundary Conditions 115 2. The Interior Problem with Nonhomogeneous Boundary Conditions 120 3. Flows in an Unbounded Domain 124 4. Effective Estimates of Solutions 127 5. The Differentiability Properties of Generalized Solutions 131 6. The Behavior of Solutions as | x | -» + oo 137 Chapter 6—The Nonlinear Nonstationary Problem 141 1. Statement of the Problem. The Uniqueness Theorem 141 2. A Priori Estimates 146 xvii  XV111 CONTENTS 3. Existence Theorems 156 4. Differentiability Properties of Generalized Solutions 162 5. The Continuous Dependence of the Solutions on the Data of the Problem, and Their Behavior as / -* + cc 168 6. Other Generalized Solutions of the Problem A) 173 7. Unbounded Domains and Vanishing Viscosity 184 8. The Cauchy Problem 188 Supplement I—New Equations for the Description of the Motion of Viscous Incompressible Fluids 193 Comments 203 Additional Comments 211 References 215 Name Index 221 Subject Index 223  Introduction Theoretical hydrodynamics has long attracted the attention of scientists working in a variety of specialized fields; the clear-cut nature of its experi- ments, the relative simplicity of its basic equations, and the clear-cut state- ment of its problems led to the hope of finding a complete quantitative description of the dynamical phenomena which takes place in a liquid medium. In reality, however, the seeming simplicity of these problems turned out to be deceptive, and so far, the effort expended in trying to answer the following two fundamental questions has not yet attained complete success: 1. Do the equations of hydrodynamics, together with suitable boundary and initial conditions, have a unique solution? 2. How satisfactory is the description of real flows given by the solutions of these equations? Apparently, as abundant as it is, accumulated hydrodynamical information, both theoretical and experimental, is still not adequate for a rigorous mathe- matical analysis of the phenomena occurring in fluids. Indeed, the numerous paradoxes of hydrodynamicsj serve as landmarks indicating the long and thorny path traversed since the beginnings of the subject. The first stage in the development of hydrodynamics, and one which ex- tended over a long period of time, involved the study of so-called potential flows of an ideal incompressible fluid. It was found that there is quite a large class of such flows, and that the means for investigating them (by using the theory of functions of a complex variable) are almost perfect. However, the famous Euler-D'Alembert paradox, according to which the total force acting on an object located in a potential flow is equal to zero, indicated that the theory of ideal fluids was not perfect. All attempts to eliminate this and a series of other paradoxes, within the framework of the theory of ideal fluids, turned out to be futile. This led to the creation of the mathematical model of a viscous fluid governed by the basic Navier-Stokes equations. This model had to serve as a scapegoat, answering for all the accumulated absurdities of the % A detailed analysis of these paradoxes is given in Birkhoff's book [1]. 1  2 MATHEMATICAL THEORY OF VISCOUS INCOMPRESSIBLE FLOW theory of ideal fluids, as well as accounting for the lifting force, the drag, the turbulent wake, and many other things. For a while, this scapegoat was silent and meek in face of the demands made on it; most of the time, it could neither answer yes or no with complete assurance, since in the case of the Navier- Stokes equations, it turned out to be impossible to solve the problem of flow past an obstacle, for even the simplest obstacles of finite size. Unlike the case of the ideal fluid, there are no potential flows satisfying the boundary con- ditions at the surface of the obstacle. Moreover, very few exact solutions of the Navier-Stokes equations were found, and almost all of these do not involve the specifically nonlinear aspects of the problem, since the corresponding non- linear terms in the Navier-Stokes equations vanish. However, in conjunction with a large number of experiments and approxi- mate calculations, even the meager information available on the Navier- Stokes equations made it possible to reveal various discrepancies between the mathematical model of a viscous fluid and actual phenomena occurring in such a fluid. Thus, paradoxes involving a viscous fluid came to light, of which only two will be discussed here. The first paradox is the following: It is well known that for any Reynolds number R, the only possible solutions of the Navier-Stokes equations in an infinitely long pipe which are symmetric with respect to its axis (directed along the x-axis, say) are given by where с is the radius of the pipe, and a is a free numerical parameter. How- ever, flows corresponding to these formulas (Poiseuille flows) are only observed for values of R which do not exceed a certain critical value, and the flows become turbulent when this critical value is exceeded. The second paradox was first observed in Couette flow, i.e., stationary flows between rotating coaxial cylinders which are invariant with respect to rotations about the axis of the cylinders and translations along it. Solutions possessing this same symmetry exist for all R, but in fact are observed only for small values of R; for large values of R, the flows are replaced by flows which are still laminar but no longer symmetric. This paradox leads to a contradiction with the deeply rooted belief that symmetric causes must produce symmetric effects. In both cases, it is not known whether the Navier- Stokes equations have solutions for large R which correspond to the observed flows; this would lead ipso facto to violation of the uniqueness theorem for stationary solutions of the Navier-Stokes equations. In connection with this second paradox, the following result proved by  INTRODUCTION 3 M. A. Goldshtik in [64] is of interest: In the problem of the interaction between an infinite vortex filament and a plane, there is a unique solution with the same symmetry as that of the problem itself, provided that R does not exceed a certain number Rt, but if R exceeds a certain number R2> Rt, there are no such solutions. Nevertheless, it might seem that this paradox and others involving viscous fluids can be quite satisfactorily explained within the framework of the mathematical model of a viscous fluid due to Stokes. Indeed, the Navier- Stokes equations are nonlinear, and it is well known that for nonlinear equations, a well-behaved solution of a nonstationary problem may not exist on the entire interval t S; 0; in a finite time interval, the solution may either "go to infinity" or else "split up", by losing its regularity, ceasing to satisfy the equations, and beginning to form branches. Moreover, even if a solution exists for all t ;> 0, it may not approach the solution of the stationary problem as the boundary conditions and the external forces are stabilized. In fact, depending on the values of the relevant parameters, a stationary boundary-value problem can have a unique solution, several solutions, or even no solutions at all (cf. the boundary-value problems for nonlinear elliptic equations, and the related problems of geometry and mechanics). Such comparisons of boundary-value problems for the Navier-Stokes equations with previously studied boundary-value problems quite naturally suggested the following conclusions: Because of the nonlinearity of the Navier-Stokes equations, the stationary problem has a unique solution for values of R less than a certain Rl, several solutions for R2 > R > Rl, and no solution at all for R > R2 .$ The above-mentioned result of Goldshtik might appear to confirm this point of view. (However, actually this result only shows that a solution with the symmetry prescribed by the author, starting from the corresponding symmetry of the data of the problem, ceases to exist. It is not known whether the problem has an asymmetric solution, but I suspect that it does.) On the other hand, even when the initial regime and the external forces are smooth, the solutions of the nonstationary problem may become progressively less regular as time increases, going over to "irregular", "turbulent" regimes and forming branches, where the particular branch which is actually "realized" depends on extraneous factors which are not taken into account by the Navier-Stokes equations. However, the only way to verify what the Navier-Stokes equations really X The inadequacy of this explanation of the paradoxes cited above may be seen by noting that the size of the critical value of R depends on the conditions of the experiment, and can be considerably increased by performing the experiment very carefully.  4 MATHEMATICAL THEORY OF VISCOUS INCOMPRESSIBLE FLOW have to say about the motion of actual fluids is first of all to carry out a rigorous mathematical analysis of the solution of boundary-value problems for the Navier-Stokes equations, corresponding to actual hydrodynamical situations. It turns out that incompressible fluids are the most suitable for such an analysis; in fact, for incompressible fluids, a whole series of results have been obtained which shed a great deal of light on the potentialities of the Stokes theory. The present book is devoted to a presentation of these results, and in it we have tried to touch upon everything of importance which has been discovered so far in this field. Without going into a detailed de- scription of the contents of the book, we shall now state in general terms the main results proved here. It is proved that stationary boundary-value problems have solutions v for any Reynolds number if i Y-ndS = Si for the boundary Sk of each obstacle. The boundaries of the obstacle past which the flow occurs and the external forces can be non-smooth. For bounded regions and small Reynolds numbers R the solutions are unique and stable. A nonstationary boundary-value problem for the Navier-Stokes equations has a unique solution for all instants of time if the data of the problem are independent of one of the Cartesian coordinates; the same is true for a problem with axial symmetry. In the general three-dimensional case, it is proved that the problem has a unique solution if the external forces can be derived from a potential and if the number R is small at the initial instant of time. In the general case, where these conditions are not satisfied, for all instants of time there exists at least one "weak solution" v(x, t) which belongs to L2(x) for all / ^ 0, and has \x. belonging to L2(x, t) and v(, yx.x. belonging to Ls/4(x, t), but its uniqueness cannot be asserted. If the initial conditions are not too bad (from the standpoint of their smoothness) then there is unique smooth solu- tion, at least during a certain time interval, whose size is determined by the data of the problem. As regards the stability of solutions of nonstationary problems for finite and infinite time intervals, the following results are proved: If in the course of time, the external forces die out, and if the boundary conditions corres- pond to a state of rest (i.e. v | s = 0), then the motion also dies out, regard- less of what the motion was at the initial instant of time. If as t -»• + oo, the values f(x, t) of the external forces approach stationary values fo(x), for  INTRODUCTION J which the corresponding boundary-value problem has a solution vo(x) with Reynolds number Ro small, then the solutions v(x, t) of the nonstationary problem corresponding to arbitrary initial regimes \(x, 0) approach vo(x) (and rather rapidly, at that) as t -* + oo. However, if the number Ro is large, then in general the solutions v(x, t) do not approach any definite limits as /-+ + 00. For a finite time interval, the solutions v(x, t) depend continuously on the initial values v(x, 0) and on the external forces f(jc, t). (This interval is arbitrary for plane-parallel flows, and small for arbitrary three-dimensional flows.) All these results are presented in the last two chapters. Before studying the nonlinear Navier-Stokes equations, we investigate various linearized versions of the equation. These studies show that the boundary-value problems for the linearized equations always have unique solutions, and that properties of the operators corresponding to stationary problems are very much like those of the Laplace operator, while the properties of the operators corresponding to nonstationary problems resemble those of the heat-conduction operator but have some distinctions. We call the reader's attention to the following three problems: 1. Whether there subsists unique solvability "in the large" for the general three-dimensional initial-boundary-value problem in some class of generalised solutions, if the smallness of given functions and regions where the problem is investigated is not supposed.f 2. Whether there exist solutions of general stationary boundary-value problems in multiply-connected regions if apart some smoothness conditions the boundary regimes satisfy only the necessary condition vndS= ? vndS = 3. Whether the solution of the boundary-value problem for the non- stationary Navier-Stokes equations approaches the solution of the boundary- value problem for an ideal fluid as v -»• 0$ The results given in this book support the belief that it is reasonable to use the Navier-Stokes equations to describe the motions of a viscous fluid in the case of Reynolds numbers which do not exceed certain limits. They partially refute the statements described above concerning the properties of solutions of the Navier-Stokes equations, and they force us to find other t Recently we have proved that the theorem of uniqueness of "weak solutions" described on p. 4 generally is not true. Jt This is true for the solution of the Cauchy problem in the planar case.  6 MATHEMATICAL THEORY OF VISCOUS INCOMPRESSIBLE FLOW explanations for observed phenomena in real fluids, in particular, for the familiar paradoxes involving viscous fluids. Apparently, in seeking these explanations, one must not ignore the fact that if a large force f acts on the fluid for an extended interval of time, then the quantities D™vk (where v = (vt, v2, v3) is the solution) can become so large that the assumption that they are comparatively small, made in deriving the Navier-Stokes equations from the statistical Maxwell-Boltzmann equations, will no longer be satisfied, just as other assumptions of the Stokes theory, i.e. the assump- tion that the kinematic viscosity and the thermal regime are constant, will be far from valid. Because of this, it is hardly possible to explain the transition from laminar to turbulent flows within the framework of the classical Navier- Stokes theory. The reader will find that the present book reflects the influence of Odqvist's work on linear stationary problems, Leray's results on nonlinear stationary problems, Hopf's investigations on the nonstationary problem, and finally investigations by the author and her colleagues and students A. A. Kiselev, V. A. Solonnikov and К. К. Golovkin. We have not dealt with the theory of nonstationary hydrodynamical potentials, developed by Leray for two space variables, and by К. К. Golov- kin and V. A. Solonnikov for three variables, partly because of its complexity and partly because the results enumerated above concerning the solution of the general nonlinear nonstationary problem were obtained by a different and simpler method. In the text and in the Comments (starting on p. 203), we give a more detailed description of what is done in various papers on the problems discussed in this book. Finally, we warn the reader who is accustomed to the classical methods of mathematical physics that the interpretations given here of what is under- stood by the solution of a problem and what it means to solve a problem differ from those with which he is familiar. To a large extent, a precise analysis of these matters is responsible for the success of the investigations reported here.  1 CHAPTER Preliminaries In this chapter, we present most of the auxiliary results from functional analysis which are used in this book. Since many of these results are well known, we only give proofs in cases where our proofs seem to be simpler than those available elsewhere. 1. Some Function Spaces and Inequalities 1.1. Throughout the entire book, we shall consider various functions of a point x = (*!, x2, x3) of three-dimensional Euclidean space E3; these functions may also depend on the time /, as well. The symbol П will denote a domain of the space E3 (i.e. a connected open set), П will denote the closure of Q. and S its boundary, so that Q = Q. + S. All our functions will be assumed to be real and locally summable in the sense of Lebesgue, while all derivatives will be interpreted in the generalized sense [6, 16]. A variety of Hilbert spaces will be used. For example, in the case of scalar functions, we shall consider the spaces W}(U) (/ = 0,1,2,...), introduced and studied in detail by S. L. Sobolev [6, 16]4 The Hilbert space W^Q) consists of all functions u(x) which are measur- able on П, have derivatives Dku with respect to x of all orders к _ /, and are such that both the function u(x) and all these derivatives are square- integrable over П. The scalar product in И^(П) is defined by the relation (u,v),= j X DkuDkvdx, and the norm is denned by } Numbers in brackets refer to items in the References, which begin on p. 215. 7  8 MATHEMATICAL THEORY OF VISCOUS INCOMPRESSIBLE FLOW CHAP. 1 The space W2(u) is complete. For / = 0, the space W2(Q) is usually denoted by L2(Q), and then the scalar product and norm are denoted simply by (,) and || ||, respectively. The Hilbert space ^(A) is the subspace of the space W2{Q.) which has as a dense subset the set of all infinitely differentiable functions which are of compact support in Q. A function is said to be of compact support in П if it is nonzero only on a bounded subdomain П' of the domain П, where П' lies at a positive distance from S, the boundary of Q. A whole series of integral inequalities and properties have been established for functions in W2\Q); it is customary to refer to these results briefly as imbedding theorems [6, 16]. We now prove several other inequalities which imply as simple consequences most of the imbedding theorems used in this book. The proofs given here are quite simple. In most cases, we shall be concerned with functions in W2(Q). Every such function can be regarded as a function of compact support defined on the whole space, if we extend the function by setting it equal to zero outside П. Because of this fact, the inequalities given below will be proved only for functions of compact support, although they can all be generalized to the case of functions defined on a domain П which are not of compact support, provided only that the boundary of П is subject to certain regularity con- ditions [6, 16]. Moreover, since the smooth functions are dense in W2{Q), all the inequalities given below are automatically valid for any function in W2(Q), although they are proved only for smooth functions. We begin by proving the following lemma: Lemma 1. For any smooth function u(xt, x2) of compact support in E2, the inequality Г Г и2 dxx dx2 grad2 udxxdx2 A) JJ Jim42 • holds. Proof: Because of the equality m(xi,x2) = 2 uuXkdxk (k=\,2), J -CO we have j* 00 maxu2(xbx2) g 2 |uuXk|Jxfc (k = 1,2). B) Xk J -00  SEC. 1.1 PRELIMINARIES 9 Then, using Schwarz' inequality, we obtain Я СО Л 00 Л 00 uxdxldx1 ^ maxu2 dx^ maxu2 dx2 -00 J-XX2 J - 00 JCi Я CO (* t* 00 |ии«|^1^2 \uuXl\dXldx2 Я со Л |* со .."'''HJ-.'11'"*1' which proves the lemma. For the case of three space variables, we have the following generalization of Lemma 1: Lemma 2. For any smooth function u(xt, x2, x3) of compact support in E3, the inequality m4dx1dx2dx3 u2 dx1dx2dx3) (Ml grad2 и dx i dx2dx3) C) holds. Proof. To estimate the integral in the left-hand side, we use A) and B). This gives m4 dxtdx2dx3 й2 Г ^зГ[П u2d ^2 max м2^Х!(/х2 ^4 | uuX3\ dxt dx2 dx3\ grad2 и dx1dx2dx3 h grad2 grad2M dxt dx2 dx3 и dx1dx2dx3) (III grad2иdxldx2dx3] , о which proves the inequality C).  10 MATHEMATICAL THEORY OF VISCOUS INCOMPRESSIBLE FLOW CHAP. 1 We can derive certain consequences from the inequalities A) and C) by using Young's inequality , a" bp' /1 1 , ab^-+— - + - = 1; P,p > P P \P P In fact, A) implies the inequality ff иЧх^х2йеП{ grad2udxldx2\ +-(\ f u2dXldxA, D) which is valid for any e > 0, and C) implies и4dx1dx2dx3 ^-d и2dx1dx2dx3\ -оо ? VJ J J -=о У ||T + 3e/||T grad2 и dx1dx2dx3) E) 2 for any e > 0. By using a method of proof similar to those given above, we can convince ourselves of the validity of the following lemma: Lemma 3. For any smooth function u(xt, x2, x3) of compact support, the inequality КГ u6dXldx2dx3S4%( I ff grad2 и dxldx2dx3] F) holds. Proof: It is easy to see that we can assume и ^ 0 without loss of generality. Then, setting dx = dxt dx2 dx3, we have u6dx= dxt u3u3dx2dx3 -=c J-x JJ-oo Лес ГЛх Лео -I ¦§, dxA maxu3Jx3 max«3rfx2 J-X |_J - °O *2 J-ХЛГз J u2uX31 *¦¦  SEC. 1.1 PRELIMINARIES 11 Next, we bring the first factor in the brackets outside the integral dxt..., replace it by its maximum, and use Schwarz' inequality to estimate the product of the last two factors. The result is ..... Dividing both sides of the inequality by yjj and replacing the geometric mean by the arithmetic mean in the right-hand side, we obtain which implies F). A remarkable feature of all the inequalities derived above is that the constants appearing in them do not depend on the size of the domain in which the function и is of compact support. However, in general, most of the inequalities appearing in the imbedding theorems do not have this property. Next, we exhibit a series of well-known inequalities which will be needed later. For any function u(x)eWi(Q), we have 1 Г u2dxu — \ graded*. G) Here, the number ц1 is the smallest eigenvalue of the operator-A in the domain Q with zero boundary conditions, i.e. the smallest number \i such that there exists a solution (which does not vanish identically) of the problem It is not hard to give an upper bound for \\\ix. Thus, for example, \\\it ^ d2, where d is the width of an «-dimensional strip containing the domain il. As the domain П is made larger, the constant ljnl may increase  12 MATHEMATICAL THEORY OF VISCOUS INCOMPRESSIBLE FLOW CHAP. 1 without limit, so that for unbounded domains, the inequality G) is in general not valid (it may turn out that /^ =0). The inequality G) with the constant d2 replacing 1//^ can easily be derived from the representation u(xl,...,xn) = u(a1,...,xn)+\ uxidxi, (8) by using Schwarz' inequality. It follows from this same formula that if u{x)bW\{Q), then J. (9) for any smooth (и— l)-dimensional surface S\ of finite size lying in П. It is also well known that the functions u(x) in PFf (П) are continuous functions of x if the dimension of the space of points x is no greater than 3; moreover, the functions u(x) obey the inequality ||i(O). A0) xeil If we restrict ourselves to functions t/(x,, x2, x3) of compact support, then it is easy to derive A0) by starting from the representation ^ If Au(y) u(x)=--r\ -dy, 4n)Ei\x-y\ which is familiar from the theory of the Newtonian potential. This implies the continuity of u(x) in the whole space, as well as the inequality 1 u(x)' = Щ,гЛ\Ж1'A"|2 dyJ - C(fi)""" w'm' where П denotes the domain in which и is of compact support, and the constant С obviously depends on the size of the domain Q. 1.2. We now give some compactness criteria for families of functions in W^Q.). In the first place, any bounded set in WJ(Q.) is weakly compact, since W2'(Q) is a Hilbert space (see e.g. [16]). Moreover, if П is a bounded domain, then any bounded set {un(x)} in Wi{Q) is compact in L2(Q). This is Rellich's theorem (see [3]) and is most easily proved as follows: Extend each и„(х) onto the whole space by setting it equal to zero outside П, and then use formula (8) and Schwarz' inequality to see that the family of functions is equicontinuous in the norm of L2(O-). However, as is well known, a uniformly  SEC. 1.2 PRELIMINARIES 13 bounded, equicontinuous family in L2{?1) is compact in Z.2(Q). Moreover, this theorem and the inequality C) imply the following lemma: Lemma 4. A weakly convergent sequence of functions in W2(Q) converges strongly in the space ?4(П). In fact, by Rellich's theorem such a sequence converges strongly in L2(Q) so that by inequality C) it will also converge strongly in the norm of L4(Q). To study the differentiability properties of the solutions to the linear and nonlinear problems in which we shall be interested (these questions will be dealt with in special sections in each chapter), it is necessary to use more general imbedding theorems than those just stated. We give these without proof. Let Lm(Q), m^l, denote the Banach space of functions u(x), xeQ, with the norm / Г \u\mdx И^П), m ^ 1, is the Banach space consisting of the elements of Lm(u) having generalized derivatives up to order / (inclusive) which belong to Lm(Cl). In this space, the norm is defined as: t = 0 (k) Lemma 5. Let u(x) be an integral "of potential type", i.e. /GO , ,dy. \x-y\* * Let f(y) e Lp(Q), p > 1, and let П be a bounded domain in the n-dimensional Euclidean space. Then for any bounded domain Cl l, the function u(x) is continuous for X < n(l — 1/p) and || и 1 „>ni = max | u(x) | + max ' "-^^ й С \\f\\ tp(n,, A1) where h = n(l — llp) — X. For X ^ n(l — \jp), the function и is summable with any finite exponent 1 and Hk<n,)^C||/lMn). A2) The constants С in A1) and A2) depend only on n, X, p, q, Q., andQl and not on f.  14 MATHEMATICAL THEORY OF VISCOUS INCOMPRESSIBLE FLOW CHAP. 1 Lemma 6. Suppose that u(x)e WlJQ), m > 1, / ^ 1, where Q. is a bounded domain in the n-dimensional Euclidean space, and Sr is some r-dimensional plane region contained in Q (in particular, we may have Sr = Q). Then for n^ml and r > n — ml, the function u(x) belongs to Lq(Sr) for any finite q ^ mr/(n — ml), and U Lq(Sr) й С || U I Wmi(a) ¦ For n < ml, the function u(x) is continuous in Q and *,n h<\. The constants С in these inequalities depend only on n, m, I, r, q, П, and Sr, but not on u(x). Lemma 6 holds under the condition that the boundary of fi possesses some regularity property (for example, when Q is the union of a finite number of domains, each of which is star-shaped with respect to some w-dimensional sphere contained in it). The reader may find the proofs of Lemmas 5 and 6 in [6] and [16]. In the sections devoted to the differential properties of solutions, we also use spaces W&tt(.QT),, CUh(U) and C*^hl-l+hl(QT). We shall now give the definitions of these spaces for the case when the boundary of Cl is smooth. We shall say that a function u(x), defined in Q, satisfies a Holder condition with exponent h, he@, 1), and Holder constant |f|(fc)>n in the region П if, u(x)-u(x')\ max COh(U) is the Banach space whose elements are all the continuous functions u(x) in Q having finite values of |u|(A)_n. The norm in COh(U) is defined as I и ||„,п = max | и | +|u|Win. xeCl A function u(x) belongs to Co Л(П) if it belongs to Co h(Q') for every П' с П. С, А(П) is the Banach space of /-times continuously differentiable functions with finite norm II « II /.*д = I Z max I D^u(x) | + ? | D<"n(x) | (Jk)>o, * = 0 (k) xeCl (i) where^ denotes summation over all possible derivatives of order к. (к) _ Clih(Q) is the set of functions belonging to Clh(u') for all W a Q.  SEC. 1.3 PRELIMINARIES 15 Ckjhl-l+h2(QT), 0 < *! < 1, 0 < h2 < 1, is the Banach space of functions u(x, t), continuous in the cylinder QT = {xeil, te[0, Г]}, having continuous derivatives with respect to x up to order k, with respect to t up to order /, and possessing a finite norm к I I u || Ct+Ai.i+*2(QT) = X Z max I D<xm)u(x, 0 I + X max I D™u(x, f) I *•' ra = O (m) Qt m = 0 QT Dxk)u(x,t)-Dxk)u{x',t)\ " "In + max It X~X D\u{x,t)-Dl,u{x,t') (x,t),(x,t')sQT I ' 'I Ckx+thul+hl(Q.T) is the set of functions belonging to Ck+thl'l+h2(U' x [е,Г-е]) for all W а П and ? > 0. The Banach space Wk'lxt(QT) consists of all the elements of Lm(QT), which possess generalized derivatives with respect to x up to order k, and with respect to t up to order / (inclusive) in Lm(QT). The norm in this space is defined as i = 0 (i) i = 0 Finally, let us agree that the notation for the spaces С and W with the different subscripts and superscripts introduced above will be used for spaces of vector functions u = (ul, u2, и3), the components of which belong to С or W. 1.3. In investigating the differentiability properties of generalized solutions, the averaging operation is used [6, 16]. Here, we define the averaging operation and list only its basic properties. As an averaging kernel we take a function which depends only on \x\. In fact, let ш(?) be a nonnegative, infinitely differentiable function which is not identically zero, but vanishes identically for ? ^ 1. The function co(\ x \jp) obviously vanishes for | x | ^ p, and its integral over the whole space equals some constant x multiplied by I'  16 MATHEMATICAL THEORY OF VISCOUS INCOMPRESSIBLE FLOW CHAP. Then we choose the function cop(x) =—„col— XP \ P as the averaging kernel. For an arbitrary summable function f(x), the averaging operation takes the form fP(x)={a>P(\x-y\)f(y)dy, where the integration is nominally over the whole space, but effectively over the ball \x — y\ ? p. lff(y) is specified only in the domain Q, then fp(y) is denned in the smaller domain Qp <= П whose boundary lies at the distance p from the boundary of П. We now enumerate some properties of the averaging operator: 1. The averaging operator commutes with the differentiation operator, i.e., dxk)p wherever dffdxk and fp(x) exist. 2. Suppose that/(j)eLp(Q), p ^ 1, and let f(y) = 0 outside П. Then /,(v) is defined on the whole domain П, is infinitely differentiable in il, and converges as p -> 0 to f(y) in the Lp(Q) norm. 3. Suppose that/and geLp(il), p ^ I, and let/and g vanish outside Q. Then, we have J. '*= f9Pdx. n 1.4. We now derive some other inequalities which will be used in studying stationary problems in unbounded domains. First we show that the inequality : uidx en) x — y\ holds for any smooth function u(x) of compact support, where the integral is carried out over the whole space E3, and у is an arbitrary point of E3.  SEC. 1.4 PRELIMINARIES 17 To prove this, consider the equality Z Uxk{x)u(x)^Ldx k=i \X — y Su2 xk-yk Г и )\х- —.-2dx, У\ obtained by making a single integration by parts. Using the Schwarz inequality to estimate the left-hand side, we obtain u\x) = c-y\ which implies A3), since > (xk-ykJ \2h\\2 h\*-y\ It is easy to see that inequalities of the type A3) are valid for functions of compact support in any number of variables greater than 2, except that the factor 4 in A3) is replaced by [2/(n —2)]2. For n = 2, instead of A3), certain other relations hold, from which we choose the following result: Let u(x) be an arbitrary smooth function of the variable x = (xlt x2), of compact support inQ = {l^|x|<oo}. Then, u(x) satisfies the inequality u\x) _, ^ Jl 2 In2 I I x |g fc = 1 A4) In fact, integration by parts gives 2 Г У их и fi dx = I Z-i xk | 2 i I J |x|2 1 k=l X | 1П Л | 2 du2 _ t=idxk |; U2{X) dx !ln2| ,dx, and from this we obtain A4), just as before. A noteworthy and useful feature of the inequalities A3) and A4) is that they involve constants which do not depend on the size of the domain in which u(x) is of compact support. There exist more complicated inequalities where the constants have the same property, but they will not be discussed here. Using A3) and A4), we now construct Hilbert spaces 6(Q) for the case of two and three space variables. In fact, let П be any domain (bounded or  18 MATHEMATICAL THEORY OF VISCOUS INCOMPRESSIBLE FLOW CHAP. 1 unbounded) in one of these Euclidean spaces, and let D(Q) be the set of all smooth functions of compact support in П. We introduce the scalar productf [u,v-]=[ uXkvXkdx A5) in t>(Q) It is clear from the symmetry of A5) with respect to и and v, and from the inequalities A3) and A4), that A5) actually defines a scalar product in D{C1). The completion of D(Q) in the norm corresponding to this scalar product gives just the Hilbert space which we denote by D(Q). It is not hard to prove that D(Q) consists of all locally square-summable functions u(x) which vanish on S, have square-summable first-order derivatives over all П, and obey the inequality A3) or A4), as the case may be. For n = 3, the functions u(x) also satisfy the inequality F). When Q is unbounded, there is an important difference between D(Q.) for the cases of two and three space variables: When n = 2, 6(Q) contains functions which do not go to zero as | x | -> oo. It can be shown that if П is the exterior of any bounded domain, then the smooth function which equals a constant for large \x\ belongs to D(D). However, this is impossible if /2 = 3, as is at once apparent from F). Roughly speaking, the inequality F) implies that the functions in D(u) "go to zero" as | x | -> oo. 1.5. Finally, we give some further inequalities, which are special cases of inequalities we have derived for elliptic operators [2, 17]. If the domain Q is bounded and if its boundary S has bounded first and second derivatives, then the inequality || " || w2hq) й С j| Am Ц i2(n) A6) holds for any function u(x)e WlCDi) r\W\(Q). We now give a short derivation of A6). As before, all the arguments can be carried out for sufficiently smooth functions. Let u(x) be a function which is continuously differentiable three times and vanishes on S. Integration by parts gives Г , Г дАиаи С ди , (АиJ dx = - —dx + Au--ds ]п JadXidXi Js дп дп vXj дп дх{ i Here and below, unless the contrary is explicitly stated, pairs of identical indices imply summation from 1 to 3.  SEC. 1.5 PRELIMINARIES 19 Take any point ?eS, and introduce local Cartesian coordinates у = (Уi > У2 ¦> Уз) at ?> '-e- 'et tne >'i and ^2 axes ue m tne tangent plane to 5 at <!;, and let j>3 be directed along the exterior normal to S at ?. The expression ди д2и ди is invariant with respect to rotations of the coordinate system, and hence д2и ди д2и du\ 2 /д2и ди д2и ди\ dyf dy3 dvi dy3 dyj ^ \dyf dy3 dytdy3 dyj' The derivates dujdyt (/=1,2) vanish, since и s = 0. Moreover, the derivatives d2ujdyf (i = 1,2) can be expressed in terms of the derivative cu/dn. In fact, let y3 = со (yx, y2) be the equation of the piece of the surface S in the neighborhood of the point ? = @, 0, 0). Differentiating the identity и(уиу2,со(уиу2)) = 0 twice with respect to y\ and y2, we obtain ди ди дса <5)'/ ду3 dvi д2и д2и ды д2и/дш\2 ди д2ш -2 + 2 + -(--)+ - =0 (/=1,2). dyf дУ1оу3ду\ cyWcyJ dy3dyf At the point ?, the last equality gives d2u ди д2оз t yj о у з о у i since at t; ^ = 0 (/ = 1,2). It follows that yen/ i = , and A7) can be written in the form (ДИ)>*с=Г itXlbx-H^Kds, A9) Jq i.U\oxdxJ J\cnJ  20 MATHEMATICAL THEORY OF VISCOUS INCOMPRESSIBLE FLOW CHAP. 1 where If the surface 5 is convex, it is not hard to see that K(S) 5^ 0, and hence 3 ' д2и ^2 J. dxu\ №2dx B0) J for such 5. Moreover, in the case of an arbitrary surface S with bounded first and second derivatives, we have the following estimate of the surface integral, where e is an arbitrary positive number: To see this, it is sufficient to reduce the surface integral to a volume integral by using Gauss' formula (after first extending cos (и, хк) from S to all П) and then use the inequality We now substitute B1) with e =\/2C1 into A9). After simply reducing similar terms, we obtain k) dx = The term in grad2 и can be eliminated from the right-hand side, since in view of the inequality G) we have grad2udx=-l Auudx-u-\ u2dx + —\ (АиJdx то In 2 I о 2fi I n ^ — grad2 и dx H— (AmJ dx ^ljn 2eJn for any г > 0. Setting ? = /^ , we see that Г 1 Г grad2 u Jx ^ — (AuJ Jx. B3) Jn /^ijn  SEC. 1.5 PRELIMINARIES 21 Together with G) and B2), this inequality gives A6), as required. It also follows from this derivation of the inequality A6) that the estimate II « || жЛе„) = К || и 1 l2(En) + II А» II м?„>) B4) holds for any twice continuously differentiable function u(x) of compact support in Е„, and since the set of such functions is dense in W\(?„), B4) also holds for any u(x)e W%(En). The inequality B4) is also valid for any unbounded domain Q. whose boundary has bounded first and second derivatives, more precisely, for any we have II ii II 2 , < C( II и II 2 -I- i! An I 2 ^ СУ^Л || u || Wz2({i) = *^\ jl u || L2(O) ' || Liu I L2(O.)>' \*-J) Moreover, the inequality f t <*jd* ^ c\ [ i«i dx+1д" ii Mi»! B6) JO i,j=l LJn *=1 J also holds. Particularly simple estimates of the type A6) and B4) can be established for the Newtonian potential u(x) = - — 4л In fact, first \etf(y) be a twice continuously differentiable function of compact support. Then, u(x) will be a function of x which is continuously differenti- able three times and satisfies Poisson's equation Am =/. As x|->oo, the functions u, ux. and ux.Xj tend to zero as *|~\ x ~2 and x ~3, respec- tively. Consider the equality f2 dx = I AmAu dx. Integration by parts transforms the right-hand side into з f where all the surface integrals vanish because of the above-mentioned behavior of ux. and uXiX. for large \x\. The last equality gives the desired estimate of the second derivatives and shows that if/is of compact support  22 MATHEMATICAL THEORY OF VISCOUS INCOMPRESSIBLE FLOW CHAP. 1 and belongs to L2(E3), then и has generalized second-order derivatives which are square-summable over E3. Moverover, the estimates of и and ux. follow from Lemma 5 on integrals "of potential type". Thus, if/vanishes outside a finite domain, the corresponding Newtonian potential u(x)= --- - - dy 4J|| satisfies the inequality И ,)||/||мп,>. B7) with a constant С which is finite for any bounded D and Qx. Inequalities A6) and B7) also hold in Lp for arbitrary p > 1, i.e. when Wj norms are used in the left-hand side and Lp norms in the right-hand side of the inequalities (cf. [82], [85], [86]). In addition to the Newtonian potential, we shall also encounter the volume potential 1 r = ~o\ \x-y\f(y)dy, x = (xl,x2,x3), 8?tJ which is a solution of the nonhomogeneous biharmonic equation A2v=f. If/is a function of compact support which is square-summable over E3, then Lemma 5 enables us to assert that v has derivatives up to order 3, inclusively, which are summable with exponents greater than 2 over any bounded domain. Moreover, estimates of the fourth derivatives are obtained as follows: Let/be of compact support and twice continuously differentiable. Then v has continuous derivatives up to order 5 and satisfies A2v =/, while f2dx=[ A2vA2vdx=[ ? vliX]XkXidx. B8) 3 jEi jEi i,j,M=l The surface integrals vanish in this case too, since as x\ -> со, D3v and D4v fall off like \x \~2 and | x ~3, respectively. The equation B8), which remains valid for any feL2(E3), gives the desired estimate of D4v. Because of B8) and Lemma 5, the inequality holds for the biharmonic volume potential v, when / vanishes outside a bounded domain Qt; here the constant С is finite for any bounded Q and Ql.  SEC. 2 PRELIMINARIES 23 Here, we have given estimates for the potentials и and v and their deriva- tives in the L2 norm. Estimates of these same quantities in the norms of the spaces C,,, are more familiar [18, 19, 82-84, 106, 107, etc.], i.e. H|2,*^CJ|/|j0,» C0) and Hk*^c 11/11 <,.*• CD If/ is a function of compact support in E3, which satisfies a Holder condition with exponent h in ?3, then the norms || ||2/, and || ||4/, in C0) and C1) can be taken over any bounded domain. However, if the integrals и and v do not extend over all of ?3, but just over a bounded domain Q, then the norms || ||2Й, || ||4;, and || ||0!l in C0) and C1) can be taken over the domain Q, provided that its boundary is sufficiently smooth. We shall say that the boundary S of the domain П belongs to C{ h if it is a Lyapunov surface of index h (see e.g. [18, 19]), i.e. if it can be decomposed into a finite number of overlapping pieces each of which has an equation of the form where феСг h. Moreover, if ф e C2h, we shall write Se C2h. 2. The Vector Space L2(L1) and its Decomposition into Orthogonal Subspaces Let П be a domain of Ei (or E2), and let L2(Q) be the Hilbert space of vector functions u(x) = (m,(x), u2(x), m3(.v)), xeQ (or u = (m,,m2)) with components щ in L2(?l). The scaiar product in as L2{Q) is defined by the relation (u, v)= u-vdx= ukvkdx, and the length of the vector u is denoted by = V- The basic problem studied in this section is the decomposition of the space L2(Q) into two orthogonal subspaces B(Д) and J(Q). The first of these sub- spaces contains the gradients of all functions which are single-valued in Q. (and only gradients of such functions). The second subspace contains the  24 MATHEMATICAL THEORY OF VISCOUS INCOMPRESSIBLE FLOW CHAP. 1 set of all smooth solenoidal vectors of compact support in Q. as a dense subset. To solve this problem (and for use in subsequent sections), we must first consider the following auxiliary problem: 2.1. Problem.J Construct a solenoidal vector field a(x) in П which takes specified values a |s = a on the boundary S. Since a(x) is solenoidal, i.e., since diva = 0, the field a must satisfy the condition Г <x-ndS = 0, C2) s where n denotes the exterior (with respect to П) normal to S, because divadx = x-ndS. n J s Thus, suppose C2) holds. This problem has an infinite set of solutions. Construct one of these solutions, which will be used in what follows. The smoothness requirements on S and a will vary, depending on how smooth a must be for various purposes. First, we consider the more complicated case where the domain Q is three-dimensional. We decompose a into normal and tangential components with respect to S, i.e. a = а„ п + ar, an = a • n, and we use an to construct a solenoidal vector field of the form b = grad ф with bn |s = а„. This reduces to the Neumann problem . , = «, C3) on s in the domain П. It is well known that because of C2), this problem can be solved to within an additive constant, which we fix by requiring that ф(х0) = 0, x0 e 5*. We now set Then, we have to find c(x) from the conditions divc = 0, c|s = («-b)|s = fc where (/J-n)|s = O. J In chapter 3, we give another method for solving this problem, which uses the theory of hydrodynamic potentials.  SEC. 2.1 PRELIMINARIES 25 Next, we represent the function identically equal to 1 in П as a sum of sufficiently smooth functions of compact support in E3, i.e., Moreover, we choose the (t(x) such that we can introduce smooth curvi- linear coordinates (y\ , y\, y\) in terms of which the intersection Sk of the surface S with the domain where (*(*) ф 0 (if this domain has a nonempty intersection with S) has the equation y\ = 0 and such that the curvilinear net (y\, y\, y\) is orthogonal on the surface Sk. Writing /f* = (t/f, we construct a vector dk(x) in Q such that curl d* = ck(x) is equal to /J* on S. Then gives us the desired vector c(x). We now show how to choose d^x) on S so as to satisfy the condition curld*(x)js = /f*. C4) If the point Me 5- Sk, then fik = 0 at M, and we can take d* and d^ to be zero at M. If MeSk, then in a neighborhood of M, we introduce local Cartesian coordinates (zl, z2, z3) such that all the zk vanish at M and such that the axes are directed along the coordinate lines (y\ , y\, y\). In the (z) coordinate system, equation C4) takes the form dd$ dd\_ dz3 We satisfy these equations at the point M by setting all the ddfjdzm equal to zero except for We also set d$, — 0 (m = 1, 2, 3) at the point M, and then return to the (x) coordinates. The values dk(z) = 0 and d&k{z)jdzm at the point M uniquely  26 MATHEMATICAL THEORY OY VISCOUS INCOMPRESSIBLE FLOW CHAP. 1 determine d*(.v) and ddk(x)jdxm at M; since dk = 0, only /}k and cos(.vt, zm), but not derivatives of cos(.vt, zm), appear in the expression for rdk(x)jcxm. It only remains to show that the values of d* and cdk!,cxm calculated in this way at every point MeS are compatible. The only condition relating these quantities is the fact that the derivatives of dk(x) with respect to the tangent directions to 5 must vanish (since dk = 0). But it is easy to see that this con- dition is satisfied by our choice of ddk(z)jczm (m = 1, 2). The smoothness of dk(x) and of ddkjdxm on S is guaranteed by the smooth- ness of the system of (y) coordinates and of the fields j$k, which amounts to the smoothness of 5 and of the field /J. The vector dk vanishes everywhere on 5, and ddk(x)jdxm vanishes everywhere on S—Sk. From the values of these quantities, we can construct the field dk(x) in Q. In so doing, we can assume that dk(x) is very smooth inside D and vanishes for points .v at a fixed distance from S. The sum curldk(x), k= 1 as already noted, gives the desired vector c(.v), which in turn determines a(x) =c(x) + gradc6. If 5 is a Lyapunov surface and a is a continuous field on S, then the above method allows us to construct a field a(x) which is continuous on Q and is as smooth as we please inside П. If 5 is a surface with bounded first and second derivatives, and if a sel?j(S) [20,21,22], i.e. if each component of a can be continued inside S onto П in such a way that the continuation belongs to W\{0), then the above construction gives a vector field а(л) in W\{Q). Moreover, this field can be represented in the form [17] a(x) = gra where Д0 = О, 0eW22(fi) and deWi(Q). If 5 and a are smoother, we can take a(.v) to be smoother in П. Below, we shall be interested in the solution of the problem for a domain П with a surface S which has "edges", specifically, for a tubular domain П whose ends are right cylinders with bases 5, and S2 ¦ Thus, the whole surface S will consist of three pieces, two planes 5\ and S2, and a third piece 53, which is the lateral surface of the tube. On S3, the vector a = 0, while on St and S2, the vector a is smooth and vanishes on the intersections Sx, 22 of the bases S^, S2 with 53 and near them. Then, concerning the solution ф  SEC. 2.2 PRELIMINARIES 27 of the problem C3), we can say that it is continuous in fi — Ej — ?2, and its first derivatives are continuous in fi — Ej — X2 and bounded in fi. Moreover, the vector c(*) can be constructed to be continuous in fi — Et — ?2, bounded in fi, and infinitely differentiable inside fi, by using the construction given above. Then the ddf/dxj will be bounded on S and continuous everywhere on S except on Zt +Z2. In the case where the domain ?1 is planar, the solution of the problem is very simple. In fact, if Q is a simply connected domain, the field a can be found in the form /дф Вф \дх2 dx The condition a |s = a gives the values of дф/дп and дф\дх on S. From the values of дф/дт, we find ф on S (to within an arbitrary constant), where ф is a single-valued continuous function, since °^- dx = a • n d~ = 0. Is* Js Then, from the functions ф\8 and дф/дп\8, we construct a smooth function i/л If the domain Q. is multiply connected, then we look for я(х) in the form a = gradp + curl^, where p is a solution of the problem C3), and ф is defined just as before. 2.2. We now turn to the decomposition of the space L2(u), discussed at the beginning of this section. Let J(Ci) denote the set of infinitely differentiable solenoidal vectors of compact support in fi, and let J(fi) denote its closure in the L2(Q) norm. The set of elements of L2(Q) which are orthogonal to J(fi) form a subspace which we denote by G(fi), so that L2(fi) = G(fi) 0 J(Ll). C5) We now prove the following theorem: Theorem 1. G(Q) consists of elements дгайф, where ф is a single-valued function on Q, which is locally square-summable and has first derivatives in L2(Ci). Proof: LetueG(fi), i.e. let Г u-\dx = 0 C6) n  28 MATHEMATICAL THEORY OF VISCOUS INCOMPRESSIBLE FLOW CHAP. 1 for all vej(fi). Choose curlwp as the vector v, where w is a smooth vector of compact support in il, and wp is its average: w,(x) = Г ojp(\x-y\ My) dy. C7) J\x-y\Sp We chose the number p to be smaller than the distance from the domain Q' where w ф 0 to the boundary S, so that wp(x) is defined over all Q and is of compact support in fi, if we set w equal to zero outside O. Substituting curl wp into C6) and bearing in mind that curl wp = (curl w)p and that the functions w and wp vanish outside fi, we obtain 0= и(л) «Д | x-y |) curl My) dy dx = u,O0- curl v/dy. C8) Jn J\x-y\Sp Jn Here, the function up(y) is infinitely differentiable and is given by formula C7) in fi' с Q. Integrating C8) by parts, we obtain J, curl Up • w dy = 0. n It follows from this identity that curlup = 0, since w is sufficiently arbitrary. The function u,, is defined for all xe?lp, where Qp is the subdomain of fi at the distance p from S, and curl up = 0 in пр. Next, we make suitable cuts in fi, so that fi becomes simply connected, and we construct the function ф(х, р) = J хо к = 1 in Qp, choosing a fixed point x0. Since curlup = 0, the function ф(х, р) is defined by the given integral and u,, = grad ф(х, р). We now let p -> 0. It is well known (see [6] or [16]) that for any fixed interior subdomain П' of the domain D, up will converge to u in L2(Cl'), and then, as is easily verified, ф(х, р) will converge to a function ф(х) in W\{Q!) (if fi' is bounded), and grad ф = u. Since fi' is an arbitrary subdomain of fi, the function ф(х) is defined on all П and grad ф = u. If the domain fi is bounded, then ф eL2(Q). However, the domain Q was just assumed to have cuts, and if we want to remove these cuts, we have to verify that ф is continuous in Q without cuts, or, more precisely, that <j>, dф = 0 for almost all closed paths in fi. We now take a smooth tube ГсП and draw a transverse planar cross- section S1! in T. We choose the tube as in the preceding problem, except that  SEC. 2.2 PRELIMINARIES 29 in this case Sl and S2 coincide. On S^ we specify an arbitrary smooth field of vectors a, which have directions orthogonal to 5, and equal zero near the boundary 5t. In T, we construct a solenoidal field a(x) which is smooth inside T, vanishes on the lateral surface of T, and equals a on St, S2. The field a(x) is bounded in T and continuous in the tube and on its boundary, with the possible exception of the curve X, in which S1, intersects the surface T. It was shown in studying the auxiliary problem (section 2.1) that such a construction is possible. We now extend a(x) onto all ?3 by setting a(x) = 0 outside T, and we then average a(x) by using a kernel top(| x — у |), where p is smaller than the distance from T to S1. If we let it is easy to see that veJ(fi). In fact, v is of compact support in fi, v is infi- nitely differentiable, and д f . _-" cop(\x-y\)ak(y)dy Cxkj x-y\Sp С д = - , -vP(\x-y\)ak(y)dy J \х~у\^рС1Ук = a>0( \x — у ) div a dy = 0. I \x-f I < Here we have used the fact that шр(| x—^ |) vanishes for | x—y | ^ p, and the fact that integration by parts is permissible for our a(x). We substitute this v into C6) and integrate the resulting equality by parts, obtaining 0= Г gradtf>-v<bc= - 0divvJx+ [»ndS= [(j>]vndS, C9) J J J J where Sx is the planar cross-section of fi containing St, and [0] is the jump of the function ф on this cross-section. We take the number p to be so small that the domain T, outside which v vanishes, differs by very little from T, and the cross-section 5t differs only slightly from S^ . As p -»¦ 0, the field a() remains uniformly bounded and approaches a uniformly in fi-St.  30 MATHEMATICAL THEORY OF VISCOUS INCOMPRESSIBLE FLOW CHAP. 1 Therefore, we have "»b, =Op),,!s1->a|Sl (a = 0 on Si-Si). Taking the limit as p -* 0 in C9), we obtain from which, since a = а„п is arbitrary on St, it follows that [$] = 0, i.e., ф is continuous as we pass through the cross-section Si. This proves the theorem. Theorem 1 is also valid for planar domains. Remark. It is not hard to show that for wide classes of domains fi, the inequality Г Г ф2 dx йСА u2 dx, D0) Jo Jo holds, where f* ф(х) = J^ukdxk, x0, xeCl, Jxo к if fi is bounded, and the inequality <p2(.x)dx \2dx, D1) holds if Q. is unbounded. The constants Ct and C2 are determined by fi and do not depend on u. The inequality D0) is certainly valid if fi is the sum of a finite number of star-shaped domains. The inequality D1) is valid, for example, for a domain fi which is the sum of a finite number of star-shaped domains and the exterior of a sphere. The proofs of the inequalities D0) and D1) will not be given here, but the inequalities themselves will be used later. 2.3. We now consider the set j(ii) of all sufficiently smooth solenoidal vectors of compact support in fi, and in j(fi) we introduce the scalar product Г [u, v] = uXk ¦ \Xk dx Jo and the norm  SEC. 3 PRELIMINARIES 31 The completion of J(?l) in the metric corresponding to this scalar product leads us to a complete Hilbert space, which we denote by H(Q). What was said about the elements of D(Q) in section 1.4. is certainly true for the elements of tf(fi). Finally, we denote byJ0A(Q) the Hilbert space of vector functions, obtained by completing j(ii) in the norm || • ||, corresponding to the scalar product f (u, yI = (u-v + uv, •\Xk)dx. J« In the case where the domain Q. is bounded, the spaces H(Q) and Jo ^D) coincide, and the corresponding norms are equivalent. However, if Q. is the exterior of a bounded domain (for example), then the norm in Jo ,(fi) is stronger than the norm in H{il) and H(Q) is a larger set than /0 ,(Q). 3. Riesz' Theorem and the Leray-Schauder Principle We now state two theorems which will be used later to prove existence theorems for stationary problems. The solution of linear problems will be based on Riesz' theorem (see e.g. [16]): Riesz' Theorem. A linear functional% l(u) on a Hilbert space H can be expressed as a scalar product of a fixed element aeH with the element usH, i.e. The element a is uniquely determined by the functional I. As for the solution of nonlinear stationary problems, we shall use one of the "fixed-point theorems", i.e. the so-called Leray-Schauder principle [23]. We shall not need this principle in its full generality, and therefore here we only state one of its implications. Suppose that we are given an equation x = Ax D2) in a separable Hilbert space, where A is a completely continuous and, in general, nonlinear operator. We recall that an operator A is said to be completely continuous in H if it maps any weakly convergent sequence {xt, x2, ¦ ..} in H into a strongly convergent sequence {Axl, Ax2, . • •} in H. The existence of solutions for equation D2) is guaranteed by the following result: % In this book, all linear functionals are assumed to be bounded (and hence continuous).  32 MATHEMATICAL THEORY OF VISCOUS INCOMPRESSIBLE FLOW CHAP. 1 Leray-Schauder Principle. If all possible solutions of the equations x = lAx for Ae[0,1] lie within some ball \x\ rg p, then the equation D2) has at least one solution inside this ball. This principle is particularly remarkable in that it can even be used to investigate problems for whose solution there is no uniqueness theorem.  CHAPTER 2 The Linearized Stationary Problem The basic problem investigated in this book is that of determining the motion of a viscous incompressible fluid, when we know the volume forces acting on the fluid, the boundary regime, and, in the case of nonstationary flows, the initial velocity field. In all cases considered here, the only important assumption is that a system of coordinates can be chosen in which the domain Q filled by the fluid does not change. This assumption is satisfied in the following important practical problems, and in many others: 1. The problem of the motion of a rigid body in an infinite flow, or equivalently, the problem of an infinite flow past a rigid body immersed in the flow; 2. The problem of the motion of a fluid acted upon by volume forces in a vessel with rigid walls, whose spatial position is varied in a known way; 3. The problem of the motion of a fluid between two coaxial cylinders, or two concentric spheres, rotating with different velocities. In an inertial Cartesian coordinate system, the characteristics of the motion of the fluid which can be determined, i.e. the velocity field v and the pressure p, satisfy the system consisting of the Navier-Stokes equations and the equation of incompressibility: v, - vAv + vk yXk = - grad p + f(x, tf\ divv = 0 J Here, and henceforth, we set the density of the fluid equal to 1, and we assume that the kinematic viscosity v is constant. In any other Cartesian coordinate system which moves with respect to the given inertial system, the second equation A) has the same form, but new linear terms in v and \Xk can appear in the first equation. The methods pre- sented here are such that if we include such terms in the Navier-Stokes 33  34 MATHEMATICAL THFORY OY VISCOUS INCOMPRESSIBLE FLOW CHAP. 2 equations, with coefficients which are not too bad, no basic technical diffi- culties are introduced. Because of this, we can confine ourselves to the case of Navier-Stokes equations in inertial systems, and to linearizations of the Navier-Stokes equations in which all nonlinear terms are discarded. We reiterate that the investigation of the problem in other, noninertial coordinate systems and the investigation of other linearizations of the system A) can be carried out in an analogous fashion. The system A) has to be supplemented by boundary conditions. In the case of rigid walls, we obtain the "adhesion condition", according to which the velocity v of the fluid at points next to the wall coincides with the velocity of motion of the corresponding points of the wall. In the general case, this condition takes the form v|s = «, B) where a is a specified velocity field on 5". It follows from the equation divv = 0 that 0. C) Except for the case of exterior three-dimensional problems, it can be assumed that a always satisfies this condition. In the present chapter, we establish our first basic result, i.e. we shall prove that when they are linearized, the above-mentioned stationary problems have unique solutions. This fact is most easily established in the Hilbert space L2(Q) of vector functions, after we have made a certain well-defined extension of the concept of a solution, to be described below. The comparative simplicity of investigations in L2(Q) is largely explained by the fact that in this space it is easy to separate the problem of finding v from that of finding/). In fact, we can obtain a closed system of equations for v from which v can be determined uniquely, and then p can be found either directly from the Navier-Stokes equations or from a corresponding integral identity. Because of this, in denning the "generalized solution of the problem", we shall discuss only the function v, and not the pair v, p. The considerations given in this chapter allow us to assert not only that the problems in question have unique solutions but also that various approxi- mation methods, e.g. Galerkin's method, can be used to find these solutions. The reader who is familiar with approximate methods for solving the Dirichlet problem for the Laplace operator will see in reading sections 1 and 2 of this chapter that these methods carry over to hydrodynamical problems, except that here the basic functions must satisfy the solenoidality condition.  SET. 1 THE LINEARIZED STATIONARY PROBLEM 35 1. The Case of a Bounded Domain in ?3 In this section, we consider the so-called Stokes problem, i.e. the problem of determining v and p in a domain fi from the conditions vAv = grad p — i, 1 } D) divv = 0, J = a. Da) Concerning a and S, we require that a can be extended inside Q as a solenoidal field a(x) with a{x)eW\(Q>); sufficient conditions for this are given in chapter 1, section 2. In this section, we assume that the domain fi is bounded. By a generalized solution of the problem D), Da), we mean a function v(.v) which satisfies the identity f ¦Фх dx=\ f-OJ.v E) for any ФеЯ(О), such that v — ae#(fi). It is easy to see that the classical solution of the problem is a generalized solution. In fact, if we multiply the first of the equations D) by ФеЯ(П), integrate over fi, and carry out an integration by parts in the first term, we obtain E) as a result. The term containing p drops out, due to the orthogonality of grad/>eG(fi) and OeJ(fi), Conversely, if it is known that a generalized solution v belongs to Wl{?l'), where fi' is any interior subdomain of fi, and if feL2(fi), tnen E) can be transformed into Г = 0 F) for OeJ(fi')- Since J(fi') is dense in J(fi') (see chapter 1, section 2), since Ф is an arbitrary element of J{&¦'), and since vAv+/eL2(n'), it follows from F) that vAv + f is the gradient of some function p(x). Since fi' с fi is arbitrary, we find that vAy + f = grad p . inside fi, i.e. v(x) actually satisfies the Navier-Stokes system. This extended notion of a solution is also justified from another point of view, i.e. the uniqueness theorem is preserved. Thus, if we find a generalized solution, it will also be the classical solution, if the latter exists. However,  36 MATHEMATICAL THEORY OF VISCOUS INCOMPRESSIBLE FLOW CHAP. 2 with the weak restrictions on the data of the problem for which a generalized solution can be found, there may not be a classical solution, whereas the existence of a generalized solution follows from very general and very simple considerations. All this shows how reasonable it is to go from classical solutions to generalized solutions. The generalized solution of the problem D), Da) and of the boundary-value problems to be considered below, can be found for a large class of functions f describing the external forces. In fact, the only restriction on f in this chapter and in chapter 5, unless the contrary is explicitly stated, is that the integral should define a linear functional for Ф in the space #(fi). This in turn will be the case if and only if the inequality I ^ с И ф||и holds. The following are among a variety of conditions which imply the validity of this inequality: 1. If fi is an arbitrary domain and if f e L6/5(fi), then according to Holder's inequality and the inequality F) of chapter 1, section 1, 2. If fi is an arbitrary domain and if converges for some y, then I n > -vl2 x-y because of the inequality A3) of chapter 1, section 1.  SEC. 1 THE LINEARIZED STATIONARY PROBLEM 37 3. Let/, have the form f - dJA^l and ktfik(x)eL2(n) (see [24]). Then f f <bdx ^ -f i/^ J П J n l,к for any Oej(fi). Since J(Q) is dense in H(Q), this inequality will hold for any Ф in #(fi). Here, the domain fi can be arbitrary. 4. The vector f need not be a function in the usual sense. It can also be a so-called "generalized function" (see [25, 26] and elsewhere), e.g. a Dirac delta function 5Ej)e concentrated on some smooth surface St lying in a bounded region of fi. For such f, the integral Jnf-O<?x is interpreted as the integral of Фе over St, i.e. f<t>dx= Ф edS. This integral actually defines a linear functional on //(fi), because of the familiar inequality (9) of chapter 1, section 1, which is valid for any function Ф in ^(fi'). S1! с Q' с Q. in the third case listed above, the /, can also be generalized functions. Of course, the cases just enumerated do not exhaust all possible situations in which the integral |пГФ^х defines a linear functional of Фе#(П). However, there is no need to explore all these possibilities, since in all the theorems on the existence of a generalized solution, proved in chapters 2 and 5, we shall not use concrete properties of f, but only the fact that defines a linear functional on H(Q). Theorem 1. There exists no more than one generalized solution of the problem D), Da). Proof: According to E), if u were the differe.n.ce_ between two possible solutions, we would have u e H(Q) and v [u, Ф] = 0. j Setting Ф = u and recalling that [ , ] is the scalar product in H(Q), we find that u = 0.  38 MATHEMATICAL THEORY OF VISCOUS INCOMPRESSIBLE FLOW CHAP. 2 Theorem 2. The problem D), Da) has a generalized solution if for the given f, the integral Jnf-O^x defines a linear functional of ФеН(СТ) and Proof. We rewrite the identity E) in the form v[v - а, Ф] = - v[a, Ф] + (f, Ф) G) and note that the right-hand side defines a linear functional of Ф According to Riesz' theorem, this functional can be represented in the form [u, Ф], where u is a well-defined element of H(Ci) which is uniquely specified by f, a and v. Obviously, the function v = a + u is the solution we are looking for. Theorem 3. IffeL2(Cl') and Cl' <= fi, then the generalized solution v found in Theorem 2 belongs to W\{?1") for Q." <= fi' and satisfies the system D) almost everywhere in Q", with grad peL2{0."). Proof: Here fi" is any subdomain which lies strictly inside Q!'. We choose a fixed Q". Without loss of generality, we can regard the function a(x) in fi' as being as smooth as we please. In E), we choose Ф of the form Ф = curl [C2 curl у pip, where the index p denotes averaging with the kernel cop(\x — y |), and ((x) is a twice continuously differentiable non-negative function of compact support in fi', which equals 1 in fi" <= fi' and does not exceed 1 anywhere in fi'. We shall assume that the width of the boundary strip in fi' where С = 0 is greater than p. Then, we substitute our Ф into E) and carry out a series of transformations, noting that the averaging operation commutes with the differentiation operation. The result is f • Ф dx = v vXk ¦ ФХк dx = v \Xk [curl (C2 curl yp)\Xk dx Jf!' J«' J"' = v vpXk [curl (C2 curl \p)~]Xk dx Jn- = -v A\p curl (C2 curl vp)dx. (8) Jn' But curl (?2 curl \p) = 4'2 curl curl yp + grad ?2 x curl v,, = - C2Avp + grad ?2 x curl vp,  SEC. 1 THE LINEARIZED STATIONARY PROBLEM 39 since divv,, = 0. Therefore, from (8) we obtain Г Г v i'2(AvpJ dx = [fp • curl (C2 curl vp) + vA\p ¦ grad C2 x curl vp~\ dx. Jo' Jo' We estimate the right-hand side by using the inequality 2b2 with arbitrary e > 0. It is not hard to see that this leads to the inequality v | B(Av/ dx й ? f vC2(AvpJ dx + Cl\ (f2 + ? v2X(, J Же, (9) n- Jn- ? Jn-\ k=i / with a constant Cj which depends only on the choice of the function l,(x). We choose e < 1 in (9) and use the fact that the estimate 3 ? vlkdx g const, A0) holds for v, and hence for \p also, as follows easily from E) if we set Ф = v-a. From (9) and A0), we see that the inequality (AvpJ dxu\ C2(AvpJ dx g const holds for any p > 0, with one and the same constant. This in turn implies the following estimate for the second-order derivatives of vp (see chapter 1, section 1): (D2xypJdxu const. A1) Jn" Since the constant in A1) does not depend on p, the function v which is the limit of yp as p -> 0 has second-order derivatives, which also obey the inequality A1) (see [16]). Gathering together all the estimates for v, we obtain II v || wwiu const. A2)  40 MATHEMATICAL THEORY OF VISCOUS INCOMPRESSIBLE FLOW CHAP. 2 We can now transform E) into assuming that <S>eJ(Q"). Since J(Q") is dense in J(?l"), and since vAv + feL2(fi"), it follows that vAv + f is the gradient of a function pe W\(Q."), so that vAv + f = grad/э and p is the pressure we are looking for. This completes the proof of Theorem 3. To investigate the behavior of v near S, as will be done in chapter 3, section 5, more complicated calculations are needed. In all the above theorems, the requirements on the smoothness of a and of the boundary S1 reduce to just the fact that it should be possible to continue a inside the domain as a solenoidal field a(x) with a,e W\(fi). If a = 0, then no smooth- ness requirements at all are imposed on S1. By using the method of Theorem 3, we can show that if f e PF2m(Q')» then 2. The Exterior Three-Dimensional Problem In this section, we consider linearized problems for unbounded domains fi. If we have the homogeneous boundary conditions v|s = 0, v»=0, A3) both on S and at со, then the proof that the problem D), A3) has a unique solution is identical, word for word, with the proofs of Theorems 1 and 2 of the preceding section (here a(x) = a = 0). The boundary conditions are satisfied in the sense that the solution v belongs to the space H(Q). Thus, we have the following theorem: Theorem 4. //Jnf -<f>dx defines a linear functional o/Oe#(fi), then there exists a unique generalized solution of the problem D), A3), i.e. there exists a function \(x) belonging to H(Q) which satisfies the identity v[v,O] = f-Ф^л- E) for any ФеН(СТ). If, in addition, f is locally square-summable, then v has locally square-summable second-order derivatives and satisfies the system D) almost everywhere, with a pressure p which has a locally square-summable gradient. Finally, ifQ. contains a complete neighborhood of the point at infinity,  SEC. 2 THE LINEARIZED STATIONARY PROBLEM 41 i.e. a domain {\x\ ^ R} and if fe L2{\x\ 2; R}, then vXiX. and px. are square- summable over the domain {\x\^ R + г}, e > 0. The last statement may be proved in just the same way as Theorem 3, if we take into account the inequalities of chapter 1, section 1.5. We now assume that the boundary conditions at oo are nonhomogeneous. In fact, suppose we have n immovable objects of finite size, bounded by surfaces Slt ..., Sn, past which there occurs a flow v that approaches a given vector v" = const as \x\ -* со. The problem consists in determining v and p from the equations D) and the conditions . =0, v||x| = o0=v°°. A4) S= E Sk k=l We construct a smooth solenoidal field a(jt), which equals zero on S = *?i S" and equals v" for large x I. For example, we can take a(x) to be where Цх) = curl (C(x) d(x)), d(x) = (v?x3 ,vfXl, vTx2), and C(x) is a smooth "cutoff" function, equal to 1 on S1 and near S, and equal to 0 for large |jc|. We call the generalized solution of the problem D), A4) the function v such that v — a e ЩС1), which satisfies the integral identity E) for all Ф e #(fi). Then the proof of the following theorem is similar to the proofs of Theorems 1 to 4: Theorem 5. All the assertions of Theorem 4 are valid for the problem D), (И). To prove that the problem D), A4) has a solution, it is enough to verify (see the proof of Theorem 2) that the expression defines a linear functional of ФеЯ(П). But this is certainly the case, since яХк = 0 for large | x |, and hence v | ar, • ФХк dx Ф n  42 MATHEMATICAL THEORY OF VISCOUS INCOMPRESSIBLE FLOW CHAP. 2 The differentiability properties of the solution v, p are improved to the extent that one improves the differentiability properties of f; in particular, if f = 0, then v and/? are infinitely differentiable. The boundary conditions A4) are understood "in the mean square" [6] on S, and in the sense that v(.r)-a(x)|2 2— ax < oo x-y at infinity. Using the fundamental singular solution for the Navier-Stokes equations, it is easy to ascertain when the generalized solutions obtained above belong to one or another Holder space, and at what rate they approach their limits at oo. The final results are the same as in the Dirichlet problem for the Laplace operator. The dependence of the differentiability properties of the generalized solutions on the differentiability properties of the problem data described above is also valid for nonlinear equations; this will be shown in chapter 4. The case of boundary conditions which are nonhomogeneous both at oo and on S may be studied in the same way as the case considered above. 3. Plane-Parallel Flows For the case of two space variables, the problem D), Da) reduces by a familiar argument to the first boundary-value problem for the biharmonic equation. In fact, because of the equation dv-i cv2 dxt ox2 there exist a "stream function" ф(х1, x2) defined by the equations дф дф cx2 ox1 Taking the curl of both sides of the Navier-Stokes system and replacing vt and v2 by their expressions in terms of ф, we obtain the following equation for ф: As is easily seen, the boundary condition v|s = a determines the values of ф and сф\сп on S (the first to within a constant which can be chosen arbi- trarily). Thus, for plane-parallel flows, the problem D), Da) actually reduces  SEC. 3 THE LINEARIZED STATIONARY PROBLEM 43 to the well-studied problem of determining \\i. Here, we shall not give the results pertaining to this problem, and we only remark that the methods of the preceding section are of course applicable to the present special case. For bounded domains, these methods lead to the same results as in three- dimensional problems. The situation is otherwise for the problem of flow past an object, i.e. for the problem D), A4). In fact, in the case of two space variables, it is impossible to satisfy the preassigned conditions A4) at infinity. By analogy with the basic electrostatic problem, the problem of plane- parallel flows past an object takes the following form: Find a solution of the system D) satisfying a boundary condition which for simplicity is taken to be homogeneous v|s = 0, Da) and which is bounded at infinity. Moreover, it is natural to state the following generalized formulation of this problem: Find a function v(x) belonging to #(fi) which satisfies the identity E) for all Ф in Н(п). Theorem 4 guarantees that this problem has a unique solution in H(Q) for any linear functional f on H(Q). In particular, if f = 0, then the solution is v=0, despite the fact that the condition Vго = 0 is not assumed to hold at infinity. We note that the fact that v belongs to H(Q) does not compel v to converge to zero as \x -> oo (for example, v may be constant for large x |), but it does exclude the possibility that v grows logarithmically as | x —> oo. Using the funda- mental singular solution of the Navier-Stokes equation, one readily shows that if f(x) tends to zero sufficiently rapidly at infinity, the generalized solution \eH(Q) has a fully defined limit уж = const as x | -> со. In its classical formulation, the problem of plane flow past an object was discussed by various authors in connection with an analysis of the familiar "Stokes paradox". This paradox consisted in the fact that a solution of the homogeneous system D) which is equal to 0 on 5 and to a given vx at infinity had not been found. It follows from what has been said above that such a solution generally does not exist. In the paper by B. V. Rusanov [27], dealing with the case where fi is the exterior of a circle, it is shown that the solution \(x, t) of the nonstationary problem corresponding to a zero force f, a homogeneous boundary condition on S and a nonhomogeneous boundary condition v 1*1 = 0;= (Cl, 0) at infinity, converges to zero as t->¦ +00, for any fixed x. The same is also true for the exterior of an arbitrary bounded domain. Another result pertaining to the Stokes paradox is due to Finn and Noll [28], who proved that the homogeneous system D) with a zero boundary  44 MATHEMATICAL THEORY OF VISCOUS INCOMPRESSIBLE FLOW CHAP. 2 condition on S1 has only a zero solution in the class of twice continuously differentiable functions which are bounded at infinity. 4. The Spectrum of Linear Problems Let fi be a bounded domain in the Euclidean space of points x = (Xj, x2, x3). To the linear problem D), Da) studied in this chapter corresponds a linear operator in a Hilbert space whose properties we now intend to study. We introduce the space J(fi) as the basic Hilbert space, and we introduce the operator A in J(Ci), which establishes a correspondence between the solutions \(x) of the linear problems vAv + gradp = ф(х), "! divv = 0, v|s = and the corresponding external force ф(х), i.e. Ay = ф. In section 1, we proved that to any ф in J(?l), or even in L2(?i), there corresponds a unique solution (v, p), where \ e Wl( In order to justify introducing the operator A, we have to show that different functions v satisfying A5) correspond to different ф in J(Ci), or, equivalently, that if the solution of the problem A5) is identically zero, then ф = 0 also. But this is actually so, since for v = 0, from E) it follows that (ф, Ф) = 0 for arbitrary Ф e #(fi); but #(fi) is dense in J(?l) and ф s J(Q), hence ф = 0. Let D(A) denote the set of all solutions of the problem A5), corresponding to all elements ф е J(il). The set D(A) is the domain of definition of the operator A, and A establishes a one-to-one correspondence between D(A) and J(fi). We note that the operator A can be regarded as an extension of the operator vPjA, where Pj is the operator projecting L2(fi) onto ^(^)> defined on Wl{Q)n H(Q). Then we have the following theorem: Theorem 6. The operator A is self-adjoint and negative-definite on D(A). Its inverse operator A'1 is completely continuous. Proof: Suppose that yeD(A) and Ay = ф. Then, by the definition of A the identity 4 yxk-4>Xkdx= - Jn Jn A6)  SEC. 4 THE LINEARIZED STATIONARY PROBLEM 45 holds for any ФеЯ(П). If we set Ф = v, A6) implies the inequality ф • у dx ^ I ф || || v || ^ С | i// | Jf and also the inequality hLucUL A7) because of the equivalence of the H and W\ norms. We now show that A is closed on D(A). Let y"eD(A), v"=>v and Ay" = ij/" => ф in J(fi) (i.e. in L2(H)). By A7), v" converges to v in the #(fi) norm, and A6) holds for v". Letting n approach oo in this identity, we arrive at A6) for v and v|/, so that v actually belongs to D(A) and Av = ф. Next, we verify that A is symmetric on D(A). Let u and v belong to D(A) (and, a fortiori, to H(Q)). Then A6) will hold for u, with any Фе//(О), and in particular, with Ф = v, i.e., v[u, v] = -( similarly, A6) holds for v, with Ф = u, i.e., v[v,u] = -(A\, u). Comparing these equalities, and recalling that we are considering only real spaces, we find that A is symmetric on D(A) and negative-definite. Thus, the operator A is closed and symmetric, and its range fills the entire space J(Cl). Therefore, A is self-adjoint (see e.g. [16]). The fact that A'1 is completely continuous follows from the inequality A7) and the fact that a set of functions which is bounded in ^(fi) 's compact in L2(?l) (see chapter 1, section 1.2). This proves Theorem 6. The properties just established for the operator A imply a whole series of properties for the eigenfunctions and eigenvalues of A [3, 29], such as the following: The spectrum Я = kl, X2, ¦ ¦ ¦ is discrete, negative and of finite multiplicity, kk converges to - со, the eigenfunctions are orthogonal and complete in the metrics of L2(O) and H(Q), etc. We have the following theorem on the convergence of orthogonal series expansions a(x)= ? (а,фк)фк(х) A8) k= 1 for arbitrary functions a(jc), in terms of the eigenfunctions фк of the operator A.  46 MATHEMATICAL THEORY OP VISCOUS INCOMPRESSIBLE FLOW CHAP. 2 Theorem 7. The series A8) converges in the norm Wf"(fi), n = 0, 1, ..., if a(x) belongs to J(d) n WJ"(Ci) and satisfies the boundary conditions a|s=...= l"-)a|s = 0, and SeC2n. If a(x) belongs to J(C1) n W%"+ '(Cl), n = 0, 1, ..., and satisfies the con- ditions a |s = ... =zf"a |s = 0, and Se C2n+1, then the series A8) converges in the norm of Win+\u). The symbol A1 denotes the /-th iteration of the operator A. This theorem will be proved on the basis of the properties of the operator A established above and inequality G7), chapter 3, section 5, for a = 0 and r = 2. The proof is similar to that given in our book [2] (chapter 2) for expansions in eigenfunctions of elliptic operators. First note that the following relation holds for the coefficients of expansion A8) under the conditions of the theorem (а, ф„) = лЛа, Л"фк) = /ЛЛ "а, фк) = Aj where and correspondingly that (Л,фк) = Ак"~1(л,1 where A9) B0) k= I On the other hand, from inequality G7) of chapter 3, section 5, it follows that the norm | а 2n=||/d""a|| is equivalent to the norm ||а||^,2П(П), and the norm |a|2n+i = \\Д"Л н(П) 's equivalent to the norm ||aЦи^п + цп) , for the set of vector-functions a possessing the properties listed in the conditions of Theorem 7. Thus = С = С In B1) and B2) i.e. the assertions of Theorem 7 are indeed true.  SEC. 5 THE LINEARIZED STATIONARY PROBLEM 47 The assertions of Theorem 7 are sharp in the limit, in the sense that if the series A8) converges in the norm W'2(Cl) (/= 2л and 2л + 1), then its sum possesses the properties stated in the conditions of the theorem. We recall that from the convergence in the norm W'2{Ql) follows convergence in the norm С,_2^(П) (cf. chapter 1, section 1.2). For domains containing a complete neighborhood of the point at infinity, the spectrum of the operator A is continuous and fills the entire negative semi-axis. This is proved in approximately the same way as the analogous fact for the Laplace operator [16, 30]. 5. The Positivity of the Pressure The system D) determines the pressure p(x) to within an arbitrary additive constant. If we knew that the function p(x) which is obtained had a bounded absolute value, then by adding a sufficiently large, positive constant to p(x), we could see to it that the pressure is positive. However, from Theorem 3 of section 1, it is only known that grad/> is summable with exponent 2 over any interior subdomain Q' of the domain ?1 (if feL2(Q)). Moreover, for arbitrary feL2(Q), the function p(x) + const will in fact neither be bounded in absolute value nor have constant sign. To see this, we can choose p(x) + const to be any function in W\(?$), and we can choose \(x) to be any solenoidal vector in Wl(Cl) which vanishes on 5; then, the sum — Av + grad/? gives the value of the force f which corresponds to the chosen values of p and v. Thus, it is reasonable to relinquish the requirement that p{x) (or, more exactly, p(x) + const) be positive at every point; instead, we replace the physical requirement that the pressure be non-negative by the requirement that the integrals p | dS be bounded over two-dimensional surfaces I. This weakened non-negativity condition is more natural than the condition f that p(x) + const be non-negative for all x. Actually, the integrals p dS only have physical meaning for areas Z whose sizes are not less than a certain positive number (stipulated by the limits of accuracy of measurement and by the discreteness of the liquid medium). If we know that these integrals do not exceed a certain constant in absolute value, then we can add a constant f С to p{x) such that the integrals (p + C) dS, giving the pressure on the h Г Г areas E, are non-negative. Moreover, the finiteness of pdS and |^|u?5  48 MATHEMATICAL THEORY OF VISCOUS INCOMPRESSIBLE FLOW CHAP. 2 for all planar bounded E and all I obtained from such I by making con- tinuously differentiable transformations у = y(x) with bounded [ dykjdxm |, dxjdyk | and d(xltx2,x3) > Q I follows from the finiteness of | grad2pdx. Later, in chapter 3, section 5, we shall prove that the estimate II v I w2hh) + I SradP II ыа) й С || f || ш holds for the whole domain Q.  CHAPTER 3 The Theory of Hydrodynamical Potentials The linear stationary problem considered in the preceding chapter was originally solved by the methods of potential theory. In fact, Odqvist and Lichtenstein independently constructed hydrodynamical potentials, investi- gated their properties, and used them to solve the problem D), Da). In the present chapter, we present this classical method. The method has many advantages over the functional method presented earlier. For example, it allows us to study the differential properties of solutions in the "Holder norms" C, h and in the Lp norms, not only inside the domain, but also near its boundary. The weakness of the method is its great complexity as compared to the functional method, and the requirement that the boundary of the domain be sufficiently smooth. The present theory differs essentially from the widely known theory of electrostatic potentials only in the concrete analytical form of its potentials. However, the properties of these potentials, due to the polarity (singular character) of the kernels, are completely analogous to the properties of electro- static volume potentials and potentials of single and double layers. Therefore, we shall not give a detailed analysis of the convergence of various improper or singular integrals, and we shall also not give a careful derivation of the integral equations which are satisfied by the hydrodynamical potentials of single and double layers. Moreover, everything which is proved for hydro- dynamical potentials in the same way as for ordinary potentials, and is therefore familiar, will be given without proof. Thus, we now present the formal theory of hydrodynamical potentials, mainly for the case of three-dimensional space. 1. The Volume Potential First of all, we have to determine the fundamental singular solution of the linearized Navier-Stokes system, or, more exactly, the tensor made up 49  50 MATHEMATICAL THEORY OF VISCOUS INCOMPRESSIBLE FLOW CHAP. 3 of the solutions corresponding to concentrated forces directed along the various coordinate axes. Thus, we consider the problem vAak(x,y)-gT&dqk(x,y) = S(x-y)ek,') divu* = 0, J . where к = 1, 2, 3. Here, e* is a unit vector directed along the kth coordinate axis, and 5(x — y) is the Dirac delta function. All differentiations are carried out with respect to the variable x, and the pointy plays the role of a parameter (the applied force is concentrated at y). The system is supplemented by the requirement that u* and qk approach zero as | x | -> oo. To find uk and qk, we use Fourier transforms, recalling that the familiar relations and imply that An x-y = -8(x-y), A2l-~-y-±=-5(x-y) on 1 4n\x-y \x-y B) oo eitz(x-y) doc, D) wherej Let v(x) denote the Fourier transform of the function v(x) then t All expressions written here are understood to be generalized functions. The reader can acquaint himself with the theory of these functions in the books [25] and [26]. We shall use such functions formally only fo find the concrete form of the basic tensor. After the tensor has been found, we can immediately verify that it has all the required properties.  SEC. 1 THE THEORY OF HYDRODYNAMICAL POTENTIALS 51 Going over to Fourier transforms in equation A), we obtain (k,j= 1,2,3), where 5k is the Kronecker symbol. From this system we can uniquely deter- mine й) and qk: г*- l Г #4-а'аЛ y"vB7t)V[ J a2 J' * "B7t)V The inverse Fourier transform and formulas B), C) and D) give ir г* | аг 1х-уП v [_ An | x — у | 8xj дхк 8л J' Bл)^_жа2 ахк4л|х These representations also imply the Lorentz formulas 4n\x-v\3' E) It is clear from the formulas E) and the equations A) tha

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