Table Of ContentThe Linearization Method
in Hydrodynamical
Stability Theory
TRANSLATIONS OF MATHEMATICAL MONOGRAPHS
VOLUME 74
The Linearization Method
in Hydrodynamical
Stability Theory
V. I. YUDOVICH
American Mathematical Society Providence Rhode Island
B. H. I0, OBH4
MET0,1J JII4HEAPI43ALJI4I4
B i'I4AP0,IJI4HAMI4LIECKOVI
TEOPI4I4 YCT0ftLII4BOCTI4
I/I3JJATEJIbCTBO POCTOBCKOI'O
YHMBEPCMTETA, 1984
Translated from the Russian by J. R. Schulenberger
Translation edited by Ben Silver
1980 Mathematics Subject Classification (1985 Revision). Primary 76D05,
75E05; Secondary 35Q10, 35K22.
ABSTRACT. The theory of the linearization method in the problem of stability of steady-state
and periodic motions of continuous media is presented, and infinite-dimensional analogues of
Lyapunov's theorems on stability, instability and conditional stability are proved for a large
class of continuous media. Semigroup properties for the linearized Navier-Stokes equations
in the case of an incompressible fluid are studied, and coercivity inequalities and complete-
ness of a system of small oscillations are proved.
Library of Congress Cataloging-in-Publication Data
IUdovich, V. I. (Viktor losifovich)
[Metod linearizatsii v gidrodinamicheskoi teorii ustoichivosti, English]
The linearization method in hydrodynamical stability theory/V. I. Yudovich.
p. cm. - (Translations of mathematical monographs; v. 74)
Translation of: Metod linearizatsii v gidrodinamicheskoi teorii ustoichivosti.
Bibliography: p.
ISBN 0-8218-4528-4 (alk. paper)
1. Hydrodynamics. 2. Stability. 3. Navier-Stokes equations. I. Title. II. Series.
QA911.19313 1989 89-315
532'.5-dcl9 CIP
Copyright ©1989 American Mathematical Society. All rights reserved.
Translation authorized by the
All-Union Agency for Authors' Rights, Moscow
The American Mathematical Society retains all rights
except those granted to the United States Government.
Printed in the United States of America
Information on Copying and Reprinting can be found at the back of this volume.
This publication was typeset using AMS-TEX,
the American Mathematical Society's TEX macro system.
The paper used in this book is acid-fret and falls within the guidelines
established to ensure permanence and durability.
1098765432 9594939291
Contents
Introduction
1
Chapter I. Estimates of solutions of the linearized Navier-Stokes
equations 11
1. Estimates of integral operators in LP 11
2. Some estimates of solutions of evolution equations 20
3. Estimates of the "leading derivatives" of solutions of
evolution equations 32
4. Applications to parabolic equations and imbedding theorems 40
5. The linearized Navier-Stokes equations 47
Appendix to §5 65
6. An estimate of the resolvent of the linearized Navier-Stokes
operator 68
7. Estimates of the leading derivatives of a solution of the
linearized steady-state Navier-Stokes equations 91
Chapter II. Stability of fluid motion 99
1. Stability of the motion of infinite-dimensional systems 99
2. Conditions for stability 103
3. Conditions for instability. Conditional stability 115
Chapter III. Stability of periodic motions 121
1. Formulation of the problem 121
2. The problem with initial data 122
3. A condition for asymptotic stability 132
4. A condition for instability 135
5. Conditional stability 138
6. Stability of self-oscillatory regimes 147
7. Instability of cycles 154
8. Damping of the leading derivatives 161
Bibliography 163
v
Introduction
In this book the basis of the linearization method (Lyapunov's first
method) is expounded in application to the problem of steady-state and
periodic motions of a viscous incompressible fluid and also to the problem
of solutions of parabolic equations.
The first chapter is devoted to the Navier-Stokes equations linearized
near some steady flow. It contains estimates of various norms of solutions
of the homogeneous and inhomogeneous equations on an infinite time
interval. These estimates, in particular, show that a solution of the ho-
mogeneous equations decays (or increases) exponentially with time, while
the argument of the exponential function is related to the boundary of the
real part of the spectrum of the corresponding steady-state operator. Some
results of this chapter are subsequently applied in the proof of theorems of
Lyapunov's first method for the the nonlinear Navier-Stokes equations. It
is true that the first chapter contains much more material than is necessary
for the subsequent exposition. In particular, the estimates of the leading
derivatives of solutions of the Navier-Stokes system are almost never used
further on they can be applied to the investigation of stability in W (1 ;
corresponding results can easily be obtained by the methods of Chapter II.
We also remark that the proof of the solvability of boundary value prob-
lems on the basis of a priori estimates in this chapter is treated in a rather
cursory manner, since this question has been well studied in [43].
Chapter I consists of seven sections.
In § 1 a survey is given of some known methods of estimating integral
operators in Lp spaces, and the interpolation theorems of M. Riesz and
Marcinkiewicz are formulated, together with some simple corollaries of
them. Next in this section a vector-valued generalization of the so-called
extrapolation theorem of Simonenko is formulated, and it is shown how
the results of Calderon and Zygmund on singular integral operators can
be derived from it. The latter, as we know, are an appropriate tool for
i
2 INTRODUCTION
obtaining estimates in Lp of the leading derivatives of solutions of elliptic
equations. Some such estimates for the case of the Poisson equation are
presented at the end of § 1, and it is shown how these estimates behave as
p -+ 1, oo.
In §2 a number of estimates are established for solutions of the Cauchy
problem for a linear ordinary differential equation in a Banach space with
an operator A generating an analytic semigroup [37], [40]. Here the con-
cept of an operator having fractional degree relative to A is introduced,
and to estimate the values of such operators on solutions integral represen-
tations of the latter and interpolation theorems are used. The most precise
estimates are obtained in the case where the operator A generates an an-
alytic semigroup not in a single space but in an entire scale of Lp spaces.
At the end of this section it is shown how the method of M. G. Krein,
developed in [38] for the case of a bounded operator, in combination with
our method of deriving estimates, leads to theorems of the existence and
uniqueness of periodic and almost periodic solution which are bounded in
various norms.
The same class of evolution equations as in §2 are investigated in §3.
Here it is a question of so-called coerciveness inequalities estimates of
the leading derivatives of solutions. We show that for such equations co-
ercivity in Lp for some po > 1 implies coercivity in Lp for all p > 1 (The-
orem 3.1) . An analogous assertion (without investigation of the behavior
of the estimates for p -+ 1, oo) was formulated by Sobolevskii in [72].
We further show that coerciveness inequalities for solutions of the Cauchy
problem and for periodic solutions follow from one another (Theorem
3.2). Related to Theorem 3.2 is the assertion proved in the Appendix to
§5 that estimates of the "leading derivatives" on a semi-infinite time inter-
val in the Lp norm with a weight depending exponentially on time follow
from coerciveness inequalities on a finite time interval. Theorem 3.2 can
be applied to the Navier-Stokes equations in two ways. On the one hand,
according to this theorem the Lp-estimates of the leading derivatives for
the problem with initial data (Solonnikov [78], [79]) imply analogous es-
timates for the periodic problem. On the other hand, by applying Fourier
series and Marcinkiewicz's theorem on multipliers to prove coerciveness
inequalities in the case of the periodic problem, it is then possible with the
help of Theorem 3.2 to derive coerciveness inequalities for the problem
with initial data.
In the case where A is a selfadjoint operator in Hilbert space (or an op-
erator obtained from a self-adjoint operator by a weak perturbation of the
type of adding terms with lower order derivatives to an elliptic operator)
INTRODUCTION 3
coercivity for po = 2 can be obtained directly, and then by Theorem 3.1
it holds also for any p > 1 (see Theorem 3.3; a somewhat more special
assertion is formulated in [44]).
In §4 some applications of the results of §§ 1-3 to parabolic equations
are given. M. Z. Solomyak [74] for the first boundary value problem and
Agmon and Nirenberg [2] for a broad class of boundary conditions proved
that an elliptic operator A2,n of order 2m generates an analytic semigroup
in Lp. The fact that differential operators of order less than 2m have
fractional degree relative to A2m is derived from multiplicative inequalities.
Therefore, the results of §2 are made concrete in the case of parabolic
equations (in a bounded domain with sufficiently smooth boundary) in
the form of sharp estimates of solutions of the problem with initial data
in Lp and Lp,r = Lr (0, oo; Lp (92)).
Papers by Ladyzhenskaya, Solonnikov, and Sobolevskii have been de-
voted to estimates in L2 and Lp of solutions of parabolic equations (see
the bibliography in [44]). The method applied in §4 is apparently new and
well adapted to the needs of stability problems: the estimates are obtained
directly on an infinite time interval with indication of the correct order of
exponential growth (or decay) as t --; +oo. We further remark that this
method makes it possible to prove imbedding theorems for the function
classes Wp ,r,2mr in the cylinder 92 x [0, oo). It would be desirable also to
derive estimates of the leading derivatives of solutions of parabolic equa-
tions proceeding only from an estimate of the resolvent. So far no one has
been able to do this, but, using the estimates of Slobodetskii and Solon-
nikov in Lp, it is possible to estimate derivatives in Lp,r with the help of
Theorem 3.1.
The study of the Navier-Stokes equations linearized in a neighborhood
of some steady-state solution, which is henceforth called the basic solution,
is begun in §5 of Chapter 1. The flow region 92 is assumed to be bounded.
The subspaces Sp and Gp in the space of vector fields of class Lp on 92
obtained by closing the set of smooth solenoidal vector fields with zero
normal component, and hence the set of gradients of smooth functions,
are introduced at the beginning of this section.
It is proved that the operator 11 of orthogonal projection in L2 onto S2
admits continuation from M to a continuous projection onto Sp in Lp for
p > 1, and the space Lp is decomposed into a direct sum of the subspaces
Sp and Gp. Application of the operator IJ to the Navier-Stokes equations
makes it possible to eliminate the pressure and drop the condition that
the velocity field be solenoidal, which is a consequence of the equation ob-
tained in this manner. The operator 11 was used to this end by E. Hopf [24]
