Navier-Stokes Equations 0226115496, 9780226115498

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Navier-Stokes Equations

Chicago Lectures in Mathematics Series J. Peter May, Robert J. Zimmer, and Spencer J. Bloch, Editors

Other Chicago Lectures in Mathematics titles available from The University of Chicago Press: The Theory of Sheaves, by Richard G. Swan (1964) Topics in Ring Theory, by I. N. Herstein (1969) Fields and Rings, by Irving Kaplansky (1969; 2d ed. 1972) Infinite Abelian Group Theory, by Phillip A. Griffith (1970) Topics in Operator Theory, by Richard Beals (1971) Lie Algebras and Locally Compact Groups, by Irving Kaplansky (1971) Several Complex Variables, by Raghavan Narasimhan (1971) Torsion-Free Modules, by Eben Matlis (1973) The Theory of Bernoulli Shifts, by Paul C. Shields (1973) Stable Homotopy and Generalized Homology, by J. F. Adams (1974) Commutative Rings, by Irving Kaplansky (1974) Banach Algebras, by Richard Mosak (1975) Rings with Involution, by I. N. Herstein (1976) Theory of Unitary Group Representation, by George W. Mackey (1976) Infinite-Dimensional Optimization and Convexity, by Ivar Ekeland and Thomas Turnbull (1983) Commutative Semigroup Rings, by Robert Gilmer (1984)

Peter Constantin and Ciprian Foias

Navier-Stokes Equations

The University of Chicago Press Chicago and London

r PETER CONSTANTIN is professor of mathematics at the University of Chicago. CIPRIAN FOIAS is Distinguished Professor of Mathematics at Indiana University.

The University of Chicago Press, Chicago 60637 The University of Chicago Press, Ltd., London © 1988 by The University of Chicago All rights reserved. Published 1988 Printed in the United States of America 97 96 95 94 93 92 91 90 89

88

54321

Library of Congress Cataloging-in-Publication Data Constantin, P. (Peter), 1951 — Navier-Stokes equations / Peter Constantin and Ciprian Foias. p. cm.—(Chicago lectures in mathematics) Bibliography: p. Includes index. 1. Navier-Stokes equations. I. Foias, Ciprian. II. Title. III. Series. QA374.C655 1989 515.3’53—dc19 88-13122 ISBN 0-226-11548-8. ISBN 0-226-11549-6 (pbk.) CIP

CONTENTS

INTRODUCTION.vii CHAPTER 1.

Notation and Preliminary Material.

2.

The Stokes equations.

1

Existence and Uniqueness

of Weak Solutions.

11

3.

Regularity of Solutions of the Stokes Equations.

14

4.

The Stokes Operator.

31

5.

The Navier-Stokes Equations.

45

6.

Inequalities for the Nonlinear Term.

48

7.

Stationary solutions to the Navier-Stokes Equations ....

57

8.

Weak Solutions of the Navier-Stokes Equations.

63

9.

Strong Solutions.

74

10.

Further Results Concerning Weak and Strong Solutions ....

83

11.

Vanishing Viscosity Limits.

99

12.

Analyticity and Backward Uniqueness.

104

13.

Exponential Decay of Volume Elements.

110

14.

Global Lyapunov Exponents.

Hausdorff and Fractal

Dimension of the Universal Attractor . 133 15.

Inertial Manifolds.154

Bibliography.185 Index.189

v

INTRODUCTION

These constitute lecture notes of graduate courses given by the authors at Indiana University (1985-86) and the University of Chicago (1986-87), respectively. In recent years there has been considerable progress in some of the questions related to the Navier-Stokes equations and their relation to finite-dimensional phenomena.

For instance, the upper bound for the

dimension of the universal attractor for 2D Navier-Stokes equations has been lowered from an estimate of the type G2exp G4 to an estimate of the type G2^3log G1^2, where G is a nondimensional number, typically of the order of 1U0-1U00.

This most recent estimate can be understood in terms

of the Kraichnan length and seems to be optimal for general body forces. We try in these lecture notes to give an almost self-contained treatment of the topics we discuss.

These notes are by no means an

exhaustive treatise on the subject of Navier-Stokes equations.

It has

been our choice to present results using the most elementary techniques available.

Thus, for instance, the regularity theory for the Stokes

system (Chapter 3) is an adaptation of the classical L2 regularity theory for a single elliptic equation of [A 1]; our adaptation is inspired from [Gj.

Another example of our desire to illustrate the

general results, while avoiding excessive technicalities, is the way we describe the asymptotic behavior of the eigenvalues of the Stokes operator (Chapter 4).

For general bounded domains we provide a lower

vxi

Introduction

bound, usiny essentially elementary means. really need later on.

The lower bounds are all we

For completeness we give the elementary proof of

the exact asymptotic behavior in the periodic case.

The same asymptotic

behavior for general domains, while true, would have required considerably more effort to describe.

Questions regarding the notions

of weak and strong solutions and their relations to classical solutions are studied in some detail.

We prove that strong solutions are as

smooth as the data permit; thus, loss of regularity can only occur if the solution ceases to be strong.

We then show how, if there is an

initial datum leadiny to loss of regularity in infinite time, there exists another one which leads to loss of regularity in finite time.

We

give the argument of Scheffer and Leray estimating the Hausdorff dimen¬ sion of the singular times of a weak solution to be not more than 1/2. A simple argument is used to prove that, in the absence of boundaries, the vanishing viscosity limit of the Navier-Stokes equation is the Euler equation for incompressible fluids.

The same technique can be used to

show that as lony as the solution to the incompressible Euler equation is smooth, solutions to small viscosity Navi er-Stokes equations with the same initial data remain smooth.

We provide a proof of time analyticity

and consequent backward uniqueness for the initial value problem for the Navi er-Stokes equations. The importance of contact element transport is emphasized in the last chapter.

We study first (Chapter 13) the decay of volume elements

and give optimal starts.

lower bounds for the dimension at which this process

These bounds use inequalities of Li eb-Thi rri ng .

The construc¬

tion of the universal attractor for 2D Navier-Stokes is given in Chapter 14.

The fractal and Hausdorff dimensions of the universal attractor are

viii

Introduction

estimated making the connection with the Kaplan-Yorke formula involving ylobal Lyapunov exponents.

Upper bounds for the fractal dimension of

bounded invariant sets for 3D Navier-Stokes are given also.

The final

chapter deals with the concept of inertial manifolds for an artificial viscosity perturbation of the Navier-Stokes equation.

The spectral

blocking property and consequent cone invariance are illustrated in detail.

These are ideas of independent interest and were successfully

used to construct inertial manifolds for several physically significant equations.

As of this writing the question of the existence of inertial

manifolds for the Navier-Stokes equations remains open. We wish to thank E. Titi , who taught some of the classes at both Indiana and Chicago and assisted in the preparation of these notes. are indebted to Fred Flowers for his expert typing. performed while PC was a Sloan research fellow.

IX

This work was

We

1 NOTATION AND PRELIMINARY MATERIAL

Let $j c

be an open set. n is said to have the segment property

if the boundary of q, an, has a locally finite open cover (U^), i «I and for each i there exists a direction

e Sn_1 and ei > 0 such that, for

x € l)i r\ a, xt = x + tM. « S2 for 0 < t < ei. We denote by Lp(n) = {f|f:q ■> IR, measurable, J|f(x)|pdx < «}. shall use (...) for the scalar product in L2(a).

We

If a has the segment

property the notions of a weak derivative in the sense of distributions and in the Lp sense coincide. Daa =_ _2_ “1 3X,

We denote

i = l,...,n and

an *

“1 9X_

Wm,p(n) are the Sobolev spaces

Wm,p(q) = {f|D“fe Lp, |a| < mj. For

p = 2 we denote Hm(«) = Wm,2(n).

tug

= r m'p*°

When p = 2 we write

,

nun

The norm in Wm,p(q) is

y

1/2

nDau ii

|o| < m

) ' . Lp(n)

instead of

nun

«

.

Hm(q) is a Hilbert

i n»

space equipped with the scalar product (u,v)

n

=

II

(Dau)(x)(Dav)(x)dx.

|a| < m a The spaces Wm,p(q) are Banach spaces. Lp(n),

We shall use the same notation

Wm,p(q) for vectorial counterparts.

scalar product in (Hm(n))n will be denoted

1

For instance, the

Chapter One

2

where . signifies scalar product in IRn.

We shall use sometimes the

notation < , > for the scalar product in Rn. Wm,p(n) is denoted by

Proposition 1.1.

The closure of Cg( n) in

Wq’p(j}).

Let jj satisfy the segment property.

Then CQ( IR0) is

dense in Wnl,p(a), for 1 < p < ».

The idea of the proof is the following.

Let u

e

Wm,p( jj) .

We first

approximate u in Wm,p(n) by a sequence of elements in Wm,p(o) with com¬ pact support by considering um(x) =