Symposium on Non-Well-Posed Problems and Logarithmic Convexity, held in Heriot-Watt University, Edinburgh/Scotland March 22-24, 1972 (Lecture notes in Mathematics) 0387061592, 9780387061597

Citation preview

This series aims to report new developments in mathematical research and teaching - quickly, informally and at a high leveL The type of material considered for publication includes: 1. Preliminary drafts of original papers and monographs

2. Lectures on a new field, or presenting a new angle on a classical field 3. Seminar work-outs 4. Reports of meetings, provided they are a) of exceptional interest or b) devoted to a single topic. Texts which are out of print but still in demand may also be considered if they fall within these categories. The timeliness of a manuscript is more important than its form, which may be unfinished or tentative. Thus, in some instances, proofs may be merely outlined and results presented which have been or will later be published elsewhere. lt possible, a subject index should be included. Manuscripts should comprise not less than 100 pages. Publication of Lecture Notes is intended as a service to the international mathematical community, in that a commercial publisher, Springer-Verlag, can offer a wider distribution to documents which would otherwise have a restricted readership. Once published and copyrighted, they can be documented in the scientific literature. Manuscripts Manuscripts are reproduced by a photographic process; they must therefore be typed with extreme care. Symbols not on the typewriter should be inserted by hand in indelible black ink. Corrections to the typescript should be made by sticking the amended text over the old one, or by obliterating errors with white correcting fluid. Authors receive 75 free copies. The typescript is reduced slightly in size during reproduction; best results will not be obtained unless the text on any one page is kept within the overall limit of 18x26.5 em (7 x 10lj2 inches). The publishers will be pleased to supply on request special stationery with the typing area outlined. Manuscripts in English, German or French should be sent to Prof. Dr. A. Dold, Mathematisches Institut der U niversitat Heidelberg, 69 Heidelberg/Germany, Tiergartenstrafie or Prof. Dr. B. Eckmann, Eidgenossische Technische Hochschule, CH-8006 Zurich/Switzerland. Springer-Verlag, D-1000 Berlin 33, Heidelberger Platz 3 Springer-Verlag, D-6900 Heidelberg 1, Neuenheimer Landstrafie 28-30 Springer-Verlag, 175 Fifth Avenue, New York, NY 10010/USA

Lecture Notes in Physics:

Bisher erschienen/Aiready published

Vol. 1: J. C. Erdmann, Wiirmeleitung in Kristallen, theoretische Grund· lagen und fortgeschrittene experimentelle Methoden. II, 283 Seiten. 1969. DM 20,Vol. 2: K. Hepp, Theorie de Ia renormalisation. Ill, 215 pages. 1969. DM 18,Vol. 3: A. Martin, Scattering Theory: Unitarily, Analytic and Crossing. IV, 125 pages. 1969. DM 16,Vol. 4: G. Ludwig, Deutung des Begriffs physikalische Theorie und axiomatische Grundlegung der Hilbertraumstruktur der Ouantenmechanik durch Hauptsiitze des Messens. XI, 469 Seiten.1970. DM 28,Vol. 5: M. Schaaf, The Reduction of the Product of Two Irreducible Unitary Representations of the Proper Orthochronous Ouantummechanical Poincare Group. IV, 120 pages. 1970. DM 16,Vol. 6: Group Representations in Mathematics and Physics. Edited by V. Bargmann. V, 340 pages. 1970. DM 24,Vol. 7: R. Balescu, J. L. Lebowitz, I. Prigogine, P. Resibois, z. W. Salsburg, Lectures in Statistical Physics. V, 181 pages. 1971. DM 18,Vol. 8: Proceedings of the Second International Conference on Numerical Methods in Fluid Dynamics. Edited by M. Holt. IX, 462 pages. 1971. DM 28,-

Vol. 9: D. W. Robinson, The Thermodynamic Pressure in Quantum Statistical Mechanics. V, 115 pages. 1971. DM 16,Vol. 10: J. M. Stewart, Non-Equilibrium Relativistic Kinetic Theory. Ill, 113 pages.1971. DM 16,Vol. 11: 0. Steinmann, Perturbation Expansions in Axiomatic Field Theory, Ill, 126 pages. 1971. DM 16,Vol. 12: Statistical Models and Turbulence. Edited by M. Rosenblatt and C. Van Alta. VIII, 492 pages. 1972. DM 28,Vol. 13: M. Ryan, Hamiltonian Cosmology. V, 169 pages. 1972. DM 18,Vol. 14: Methods of Local and Global Differential Geometry in General Relativity. Edited by D. Farnsworth, J. Fink, J. Porter and A. Thompson. VI, 188 pages. 1972. DM 18,Vol. 15: M. Fierz, Vorlesungen zur Entwicklungsgeschichte der Mechanik. V, 97 Seiten. 1972. DM 16,Vol. 16: H.-0. Georgii, Phasenubergang 1. Art bei Gittergasmodellen. IX, 167 Seiten. 1972. DM 18,-

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dotd, Heidelberg and B. Eckmann, Z(Jrich

316 Symposium on Non-Well-Posed Problems and Logarithmic Convexity Held in Heriot-Watt University, Edinburgh/Scotland March 22-24, 1972

Edited by R. J. Knops Heriot-Watt University, Edinburgh/Scotland

Springer-Verlag Berlin-Heidelberg- New York 1973

A M S Subject Classifications (1970): 3 5 R 2 5 , 3 5 B 3 0

I S B N 3-540"06159"2 S p r i n g e r - V e r l a g B e r l i n • H e i d e l b e r g • N e w Y o r k I S B N 0-387-06159-2 S p r i n g e r - V e r l a g N e w Y o r k • H e i d e l b e r g . B e r l i n

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. (c.) by Springer-Verlag Berlin - Heidelberg 1973. Library of Congress Catalog Card Number 72-98023. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

PREFACE The following articles represent the invited lectures given at the Symposium on Non-Well-Posed Problems and Logarithmic Convexity held at Heriot-Watt University, Edinburgh, 1972.

Scotland from March 22-24,

The Symposium was organised jointly by the University of Dundee

and Heriot-Wati University, and for~ned part of the activities of the North British Differential Equations Symposium for the academic year 1971/72.

These activities were sponsored by the Science Research

Council of Great Britain. One of the main objects of the Heriot-Watt Symposium was To provide expository accounts of recent developments in the subject which would be of use not only to the specialist but also to someone wishing to become acquainted with this area of activity in partial differential equations.

The following collection of addresses

therefore contains both surveys and discussions of cut-rent problems and it is hoped they will further serve to fulfill the objects of the Symposium. The organisers take this opportunity of thanking Heriot-Watt University for the invitation that made the Symposium possible, and also for providing generous hospitality. thank the S,R.C. of Great Britain.

They would also like to

Finally, they gratefully

acknowledge the advice and assistance given by Professor L~E. Payne in the preparation of the programme,

and wish to express their appreciation

to Miss S. Corbey for her help in the prepamation and typing of the manuscripts. R.J. Knops Editor

CONTENTS

L.E. Pa~Tne: Some general remarks on improperly posed problems for partial differential equations . . . . . . . . . . . . . . . . . . . .

I

R.J. Knops: Logarithmic convexity and other techniques applied to problems in continuum mechanics ...................

31

D.L. Colton: Cauchy's problem and the analytic continuation of solutions to elliptic equations ........................

55

R.H. Dyer: Some properties of solutions of the Navier-Stokes equations

.....

67

K. Miller: Non-unique continuation for certain ODE's in Hilbert space and for uniformly parabolic and elliptic equations in selfadjoint divergence form . . . . . . . . . . . ". . . . . . . . . . . .

85

H.A. Levine: Logarithmic convexity and the Cauchy problem for P(t)utt + M(t)u t + N(t)u = 0 in Hilbert space ............

102

K. Miller: Stabilized quasi-reversibilite and other nearly-best-possible methods for non-well-posed problems .................

Lectures whose proceedinss do not appear here

R. Khosrovshahi:

Growth properties of solutions of Schrodinger type systems

161

SOME GENERAL REMARKS ON IMPROPERLY POSED PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS

L.E. PAYNE

INTRODUCTION

The aim of this paper is to present some historical remarks and a nottoo-technical introduction to the subject of improperly posed problems for partial differential equations.

We shall indicate various methods which

have been employed for handling such problems - methods whose power and applicability are discussed in the accompanying papers in this volume. A rather rough definition of a well posed problem for differential equations is as follows: ~ )

A problem is said to be well posed (or ~__roperl~ ....

if a unique solution exists which depends continuously on the data.

Of course, we must state precisely in what class the solution is to lie as well as the measure of continuous dependence.

A problem which is not well

posed is said to be i ~ ! p o s e d , non-well posed, or improperly posed.

Some

simple improperly posed problems are the final value problem for the heat equation, the Cauchy problem for the Laplace equation, and the Diriehlet problem for the wave equation.

It should perhaps be emphasised at this

point that in discussing the auestion of continuous dependence on the data we must consider as "data" any initial or boundary values, prescribed values of the operator, coefficients of the equation as well as the geometry of the domain of definition.

It is difficult to say when mathematicians first became concerned over the question of proper posedness of boundary and initial value problems. Most physically reasonable problems seem to lead to well posed problems for the corresponding mathematical models.

Equilibrium problems lead to

elliptic equations with boundary data required in both the physical and mathematical contexts. bolic equations, and by the physics.

Wave or vibration problems generally lead to hyperinitial or initial-boundary value problems are suggested

Diffusion or evolutionary processes are generally described

by parabolic equations, and it is again initial data that the physicist requires.

Until recently, the prevailing attitude toward the study of

classes of improperly posed problems, which may be found expressed, for instance, in Petrowsky's book

[121~, was that fortunately there is no need

to consider such problems since they are of no physical interest. Unfortunately, as we are now well aware, there are many physical situations which force us (if we are to get any answers at all) to consider problems that are improperly posed. Although it is clear from the work of Cauchy, Kowaleski, Holmgren and others that more than a century ago some consideration was being given to the question of proper posedness, the father of this study is generally considered to be Hadamard

~O].

It was he who, at the turn of the century, clearly

defined the question and illustrated by examples and counter examples the difficulties

involved.

Among other things, he derived necessary and

sufficient conditions for global existence of solutions of the Cauchy problem for the Laplace equation.

Unfortunately~

it is impossible to verify, in

general, whether the necessary conditions are or are not satisfied. Even today the question of global existence of solutions of the various classes of improperly posed problems is largely unsettled.

From a practical

point of view, one probably requires an existence theorem which is quite

different from the standard type of theorem required for properly posed problems.

Roughly speaking, what one really needs is a theorem which

asserts the existence of a solution whose data are close in some measure to the given data and which in addition satisfies some appropriate auxiliary stabilising condition.

Little has been done on the question of existence,

the major effort being directed rather toward the simpler question of uniqueness, where prominent among the earlier work is the fundamental theorem of Holmgren ~3] and some results due to Carleman [23] in the late thirties. However, a ~ r g e number of uniqueness proofs for improperly posed Cauchy problems began to appear in the literature of the 1950's and early 1960's, frequently as consequences of unique continuation theorems.

Among the

numerous mathematicians who have contributed in this area are Agmon ~,4], Agmon and Nirenberg ~], Aronszajn [7], Calderon [16], Cordes ~8], Douglis [34,35], Foias, Gussi and Poenaru [43], Heinz ~i], H~rmander ~4], John [60], Kumano-go [75], Landis [76], Lavrentiev

[79,80], Lions and

Malgrange [89], Lopatinskii [90], Mizohata [iO2,103], Miller [104], Nirenberg [107], Pederson [119], Plis [123], Protter [125], Pucci [129,130], Shirota [137] and many others (see e.g. Lattes and Lions [77], Lavrentiev [80,81], Payne [113] and the bibliographies contained in these references). These authors have dealt almost exclusively with the question of uniqueness.

Far more important, of course, are the questions of continuous

dependence (which implies uniqueness) and approximation of solutions.

In

this paper, we first make brief mention of some different classes of improperly posed problems, make a few brief remarks on the question of existence and then confine our discussion to classes of improperly posed Cauchy problems.

For such problems we mention various methods which have

been used to establish uniqueness and continuous dependence, and also to determine approximate solutions for the problems in question.

Finally, we

demonstrate, using a simple example, the principal features of several of these methods.

Mathematicians recognised very early that many problems of physical interest were completely intractable when tackled directly.

They

frequently adopted an inverse or semi-inverse method of trying to generate a reasonable approximate solution of the problem in question by combining simpler solutions of simpler problems.

The semi-inverse method was used

very successfully by St. Venant in his formulation of classical torsion and flexure problems, and fluid dynamicists through the years have tried to generate flows past obstacles by combining in an ingenious way the flows generated by various distributions of singularities.

In the late 1930's

mathematicians began to concentrate on the many mathematical questions raised in the treatment of these various types of inverse problems.

Since that time

much work has been done, particularly by the Russians, starting with Novikov

~08],

Rapaport

[133], Sretenskii [138], Tihonov

continuing with Lavrentiev Vasiliev

~i], Berzanskii

[139] and others and

[80], Lavrentiev, Romanov and ~i],

Mare~k [92,93] and many others.

(Reference [81]

provides a recent account of the Russian literature on this subject.) One of the most interesting classes of inverse problems is that in which coefficients are to be determined from a knowledge of certain functionals of the solution.

In general, such problems fall into the category of

improperly posed problems. As an example of this latter type of problem, suppose we are given an initial boundary value problem for the equation ~u a - ~ : Au where A is the Laplace operator and "a" is an unknown constant.

If, in

addition to appropriate initial and boundary data, the value of the solution at other suitable points in space-time is prescribed~ will it be possible to determine the coefficient "a"?

If so~ is it possible to

compute necessary and sufficient criteria?

Such problems have been studied

extensively in Russia but they have also attracted the attention of such authors as Cahen

[15], Cannon [17,19,20], Cannon and Dunninger

Jones ~ ,

and Jones

Vasiliev

[21], Douglas and

[63]. (See also e.g. Lavrentiev, Romanov and

~i], Lattes and Lions

[77], and the papers cited therein.)

The

work mentioned above deals mainly with stabilislng conditions and procedures for determining or approximating the unknown coefficients. Although we shall not be here concerned with boundary value problems for hyperbolic equations, it should at least be mentioned that until the past year or so very little work in this area has appeared in the literature. There was the early paper of Bourgin and Duffin by Abdul-Latif

~],

Dunninger and Zachmonoglou

[12] and subsequent papers ~6],

Fox and Pucei

~4],

John [58] and others, the investigations dealing primarily with questions of existence and uniqueness of classical solutions.

A number of papers

dealing with this class of problems are just now appearing in the literature and a summary of work in this area would seem appropriate at some later date. At this point let us briefly mention how other types of improperly posed problems might arise.

It often happens that we do not know precisely

the region of definition of the equation governing our problem (an elastic solid with internal cracks, inhomogeneities or holes, whose locations are not precisely known).

It also frequently happens that a portion of the

boundary of our region is inaccessible for measuring the desired data, as in the problem of locating oil deposits below the surface of the earth (see [131]).

It may also be the case that we need to locate certain internal

singularities which we know to exist but whose precise position is unknown or that we wish to place certain singularities a desired behaviour.

in positions which will guarantee

In this latter category we might mention the problem

of the electron focusing gun (see e.g. Hyman [55], Radley ~5a]).

One usually

tries to deal with the above mentioned types of problems by measuring an overabundance of data and then trying to compensate for the lack of knowledge of the geometry or location of singularities.

However, extreme care must

be employed since the resulting problems are almost invariably improperly posed.

For additional physically important examples see [80] and

~i].

As we mentioned earlier the existence question is an extremely difficult one, and we shall have very little to say about it. leaving the question we should, however,re-emphasise Fichera

Before

an observation of

[42] that the frequently held assumption that a partial differential

equation (even a linear one) with smooth coefficients possesses an infinite number of solutions is false. solutions at all.

Fichera

In fact, it may fail to have any non-trivial

~2] discusses the example of Lewy ~8] and an

earlier simple example due to Picone

[122] .

We mention also the example of

Plis [124]

Our subsequent remarks will in general be confined to improperly posed Cauchy problems or initial-boundary value problems.

Here it is known for

the typical problem that the solution will not exist globally unless strong compatibility relations hold among the data.

Even if for the particular

data of the problem the solution should exist, it will not depend continuously on the data in the sense of Hadamard

[50].

We note, for

instance, that Hadamard's condition for the existence of solution of the Cauchy problem for the Laplace equation was that a certain combination of the data be continuable as an analytic function throughout the domain of definition of the solution.

Extensions of Hadamard's method have been made

by Aziz, Gilbert and Howard [8], Colton

[26], Garabedian and Lieberstein

Ivanov [56], Payne and Sather [118], D. Sather and J. Sather [134] and others.

For additional discussion see the paper of Colton [25] in this

volume as well as the books of Bergman

[9], Gilbert

[49] and Vekna E143].

[48],

In practise, however, the type of information necessary for determining the continuability is usually not available.

The data are obtained by

measurement (the geometry, coefficients, boundary values of the solution, etc.), and hence are not known precisely. data which is not precise and

Thus one is forced to work with

the results must reflect and allow for

this possible error in data measurement.

As mentioned earlier, Cauchy

problems of this imprecise data type have been presented to mathematicians by petroleum engineers, meteorologists, physicists and scientists working in a wide variety of fields. ~0],

~i]

and [131~.)

(See for example the references cited in

In such problems it is frequently the case that the

physical problem which one really would like to solve is a standard well posed problem, but a portion of the boundary is either unknown, extremely irregular or inaccessible for measuring the desired data.

As we indicated

before, one hopes that by measuring an overabundance of data on an accessible portion of the boundary he can compensate somehow for the lack of data elsewhere. In [62] John pointed out that it is typical of such improperly posed problems that there exists at most one solution.

We would therefore expect

that by suitably restricting the class of solutions to be considered it shot£Ld be possible to bring about continuous data dependence.

As John emphasises,

the difference between properly and improperly posed problems is that the required restrictions for improperly posed problems cannot be inferred from the data alone or at least not from approximate knowledge of the data. In the middle 1950's, John [59-62] and Pucei E128-132]

published a series

of papers deriving continuous dependence results in suitably restricted classes. In particular, they obtained stability results for various problems in the class of uniformly bounded solutions. by Douglas and Gallie

[32J, Lavmentiev

Work along the same lines was continued ~8-81], and in the 1960's by a host

of authors (see e.g. the papers cited in Payne [113]).

It should be noted

that in a number of physically interesting problems this uniform bound (or appropriate norm bound) can be obtained by observation.

Since it need

not be sharp, any crude bound will suffice. It is perhaps fitting to recall at this point that this conference is devoted to the two topics - improperly posed problems and convexity methods. Although the tool of logarithmic convexity has had many important applications in the study of improperly posed problems (see e.g. Agmon [4], Conlan and Trytten [27], Crooke [29], Edelstein [39], Hills [52], Knops and Payne [67-72], Knops and Steele [73], Lavrentiev [78-79], Levine [83-86], Miller [94], Payne [112-114], Payne and Sather [115-116], Schaefer [135-136], Trytten [142] and others) it has also been used in the study of questions such as the growth properties of solutions as some variable tends to infinity (see Agmon [4,5], Edmunds and Dyer [38,40], Khosrovshahi E64], Ogawa [ii0-iii], and the references cited in the papers of Dyer [37] and Khosrovshahi which appear in this volume).

[65]

We should also mention that convexity argu-

ments have been used by Adelson [2] and by Payne and Sather [112] to study certain "singular perturbation" questions for classes of improperly posed problems.

At the same time it must be remarked that in addition to

logarithmic convexity methods, many other methods have been used in the study of various types of improperly posed problems.

The accompanying is a partial

list of tools used to tackle such problems. i.

Function theoretic methods ([8], [26], [48], [49], [56], [143])

2.

Eigenfunction methods ([17], [18], [95], [i01])

3.

Logarithmic convexity methods ([4], [27], [29], [39], [52], [67-72], [73],

[83-86], [i12-i14], ['115-i16], [135], [136], r142]) 4. 5.

Methods of

rotter (D25],

D26], D27], [lOS], [106])

,agra ge identity methods ([Z3], [14], DO6a])

I0

6.

Quasireversibility methods ([46], [77])

7.

Restriction of data to the class of band limit functions ([i00],

8.

Numerical and programming m e t h o d s

([31-33], [48],

[144])

[59], [61], [93],

[96-98] )

This is not intended to be an exhaustive list but rather an indication of the variety of tools that have been used. It is perhaps informative in an introductory paper such as this, to illustrate how several of these methods are applied, using as a model the initial-boundary value problem for the backward heat equation.

By

restricting attention to this simple model we will clearly be unable to indicate the power and wide applicability of the various methods.

We can

merely give some idea of the main arguments, and indicate the type of information on uniqueness, continuous dependence and growth of solution that can be obtained.

More will be said about several of these methods in the

other papers of this volume.

2.

EIGENFUNCTION METHODS The problem we shall consider is the following:

let D be a simply

connected closed bounded region in n-space with smooth boundary ~ D.

We

seek in D a C 2 solution u(x,t) of the following problem Lu~

3u ~-~ + Au = 0 in D (O,T)

u = 0 on ~ Dx[O,T]

(1)

u(x,O) = f(x) . We know that, in general, no solution exists and that if one does exist it will not, in general, depend continuously on the data.

We know too that the

stability problem can be overcome by suitably restricting the class of admissible data or the class of solutions to be considered. An eigenfunction method involves expansion of the solution in terms of the eigenfunctions of the Laplace operator, and it is easily seen that if u. l

11 and i. are the ith normalised eigenfunction and the corresponding i

eigenvalue

of Au + lu = 0 in D

(2)

u : 0 on SD , then the formal solution of the problem is given by

u(x,t) ~

where f

n

~

n:l

f e

In t

n

(3)

u (x) n

is the nth Fourier coefficient of f(x).

Obviously, for this series

to represent even an L 2 solution of (i) in Dx(O,T) it is necessary that the -I T n Fourier coefficients decay faster than e as n + ~. This is, of course, an extremely restrictive condition to impose on the initial data, particularly in view of the fact that f(x) was probably determined by measurement and hence subject to error. A restriction of the data to band-limited functions is essentially equivalent to replacing f(x) by fn(X), the first m terms of its Fourier series expansion. O(x,t) =

The corresponding solution 0(x,t) is then given by m I t n [ fn e Un(X) , n=l

(4)

a perfectly well defined solution for the approximate data. Now instead of replacing f(x) by fn(X), let us assume that we have made an error in measuring the initial data, calculating it to be f*(x) instead of the true value f(x).

Suppose further that we can compute a constant K

such that at some time T ID u2(x'T)dx

K2

(5)

This constant K can frequently be determined from the physics described by the problem.

f

D

Let

[f(x)-f~'~(x)]2dx

~ ~.

(6)

12 Thus our formal solution (3) now satisfies ~ 2 2 AnT K2 g f e .< n n=l

(7)

and eo

(8) n=l

Assuming K and a to be prescribed then any f(x) with Fourier coefficients satisfying

(7) and (8) could be regarded as giving an acceptable "solution"

of our problem.

Clearly such a solution need not exist for arbitrary K and

and if one does exist it will not, in general, be unique. 3.

QUASIREVERSIBILITY

METHODS

The idea of quasireversibility

is as follows:

one alters the operator

to make the problem a properly posed one, and using the solution of the altered problem as a guide one then constructs an approximate "solution" of the ill posed problem. For instance let u be a classical solution of ~--~ + Au + s2A2u = 0 in Dx(O,T) : 0, A~ = 0 o n ~(x,O)

= f(x)

~Dx~,T]

(9)

.

Under appropriate smoothness hypotheses the solution of this problem is known to exist and be unique. u(x,t) =

% f e n n=l

Formally the solution is expressible as

-Xn(S2Xn_l)t

If for fixed e 2, fsL 2 then ~sL 2.

u (x) . n

(i0)

The problem (9) is well posed and for

sufficiently small e the first several terms in the series (i0) are essentially the same as the corresponding terms in (3).

13 At t : T we find

u(x,t)

-~n(E2Xn-I)T :

~ %e n=l

Un(X)

Let us now define a new problem,

.

(ii)

i.e. we seek a solution U(x,t) of

~U 8-~ + AU : 0 in Dx(O,T) U : 0 on

Dx[0,T]

U(x,T) =

~ fn e n=l

Problem

(12)

-I (~21 -I)T n n Un(X) ~ u(x,T)

.

(12) is properly set and the solution is given by

E-in2~2T+in t] U(x,t) =

n=l

fn e

u n (x)

(13)

.

We wish now to see how close U(x,O) is to the initial data f(x).

l lU(x,O) - f(x)l 12 ~

D

[U(x,O) - f(x)]2d× =

In the quasireversibility "solution" of the problem

[

n:l

f2[l - e n

method the function U(x,t)

(3).

Thus

]

• (14)

is used as the

Note that as e ÷ 0 the right hand side of

(14) tends to zero, but the limit of the right hand side of (ii) may not exist.

There is clearly much non-uniqueness

in this procedure.

it has been pointed out by Gajewski and Zacharias advantages

In fact,

[46] that there are certain

to using instead of u, the solution u of the well posed problem

~---- [~ - e2AO] + AQ = O in Dx(O,T) ~t

G : o on ~Dx[O,J ~(x,O) : f(x)

(15)

.

The solution is then given by ~n ~ t i+~21 u(x,t) =

n=l

fn e

n

u n (x) .

(16)

14 As before we now define our approximating solution V(x,t) to be a solution of ~V 3-~ + AV : 0 in Dx(O,T) V : 0 on ~Dx[O,T]

(17)

I n - - T V(x,T) :

~ fn e n=l

i+e21n Un(X)

•

The solution of (17) may be expressed as

oo V(x,T) :

-~

-xn

[ f e n n=l

tl

u (x) . n

(18)

Thus in this case

llv(x,o)

- f(x)IJ

fn2

=

-e

~

.

(19)

n=l Clearly the right hand side of (19) is smaller than the right hand side of (14) for any positive E. This is a slight indication of the need for guidelines in the application of the quasireversibility

method.

It should be emphasised of course that

for a more complicated operator we might not be able to exhibit u(x,t) and U(x,t) explicitly in which case a bound for the left hand side of (14) might be more difficult to obtain. in the papers of Miller 4.

~9]

This method is discussed in fuller generality which appear in this volume.

LOGARITHMIC CONVEXITY METHODS The method of logarithmic convexity proceeds as follows:

we seek a

function F(t) defined on solutions of (3) which satisfies the following properties:(i)

F(t) >~ 0 for O .< t < T

15

(ii) (iii)

F(t) = O u(x,t) = O for O $ t < T FF'' - (F') 2 ~ O for O < t < T.

Here the prime denotes differentiation with respect to t.

It can be easily

shown that if F vanishes at any point tleEO,T ] then F(t) ~ 0 in ~,T]. Thus without loss one assumes that F(t) > O in [O,T].

Condition (iii) will

then yield the two inequalities F(t) $ [F(T~t/T[F(O)]I-t/T

(20)

and F(t) ~ F(O)exp(F'(O)/F(O))

.

(21)

Thus if one had the additional information that F(T) ~ K 2

22)

for some prescribed K~ (14) would lead to the H$1der stability result F(t) ~ K2t/T[F(O)] l-t/T

O ~ t ~ tI < T ,

23)

while (21) would give a lower bound for the growth (or decay) rate. For our example (3) we choose F(t) = (u,u) ~ [ u2dx JD

(24)

and compute F'(t) = 2(U,u t) F''(t) = 4(ut,u t)

(25) Q

Clearly then condition (iii) is satisfied (this follows from a simple application of Schwarz's inequality).

Thus assuming knowledge of the K in (5)

we see that (23) reduces to [

2t - - [I x] l-t/T u2(x,t)dx ~ K T f2 d , D D

(26)

while (21) leads to the result l I D [gradf 12dI

(27)

16

Since (26) represents an a priori upper bound for the L 2 integral of u it is not surprising that this method may be used to obtain explicit L 2 and pointwise bounds for the solution itself.

For more details see the

papers of Payne ~12], Schaefer [135-136], Trytten [142], etc. It is interesting that in this special example if one chooses

(28)

Fk(t) = ( U , U k ) 2 then FkF k ,,

o

_

(29)

and one obtains in fact the two identities (u(x,t),uk(x))2 = [fk]2(l-t/T)(u(x,T),Uk(X)) 2t/T 2 21kt (u(x,t),uk(x))2 = fk e

(30)

It should be noted that the inequality (29) holds in this case because of the special character of the example which we are considering. 5.

LAGRANGE IDENTITY METHODS The Lagrange identity method proceeds as follows:

suppose we are able

to construct a function V(x,t) which satisfies the adjoint equation (the forward heat equation) and homogeneous boundary conditions.

Then by the

Lagrange identity we have tr t 0 fJo;n =

L-~

,n) + Au(x,n)1 - u(x,n) ~ ~n(x,n) + Av(x,n ~ } dxdn

u(x,n)V(x,n)dx

.

D Thus,

for any such function

I

D

u(x,tlV(x,t)dx

(3l)

o

: I

V(x,n)

D

we o b t a i n

f(x)V(x,O)dx .

(32)

A particular choice of V(x,n) which leads to interesting results is V(x,n) = u(x,2t-n) .

(33)

17 Then clearly

V(x,t)

: u(x,t)

(34)

V(x,O) = u(x,2t) , and (32) becomes ;D u2(x,t)dx = ID f(x)u(x,2t)dx

(35)

.

In particular,

1

1 =

D

' p

q

IIf(x)llpllu(x,T)llq

(36)

and if we knew in addition that

....llu(x,T)llq ~ Kq

(37)

it would follow that

Ilu(x,~)ll 2 ~ Kqllf(x)ll r .

(3s)

One observes now the interesting fact that if the initial data is uniformly bounded and K I is prescribed then an application of (20) with T T T replaced by ~ establishes H~ider continuous dependence in L 2 for t $ ~ . On the other hand if feL 1 and lu(x,T)l

( K

continuous dependence in L 2 for 0 < t < T.

then one establishes H~Ider (The same result could have been

obtained in this special problem using eigenfunction expansion.) This method has been used by Brun [13,142 elastodynamics,

in treating problems in

viscoelasticity and thermoelasticity.

It can clearly be

used to obtain explicit bounds for the solution (which is assumed to lie in the appropriate class). 6.

METHODS OF PROTTER The method of Protter gives a weak result for the simple problem (3)

which we are considering.

However, it is applicable to a variety of problems

18

for which the other methods fail.

Protter's idea is to set

(39)

u(x,t) = e~tw(x,t) for an as yet undetermined constant ~. ~w ~-~ + ~w + Aw -- 0

Then w satisfies (40)

in Dx(O,T) .

We now form 0 = Itl ~ o D

+ ~W + AW] 2dxd~ >~ 2Ii I

D

~~,,~w(Aw+ew)dxd~ .

(41)

An integration of the right hand side yields the inequality 2 2 _ 2 allw(x,t)l122 - llgradw(x,t)ll 2 .< ~llf[l 2 llgradfll 2 •

(42)

Reinserting u(x,t) for w(x,t) one finds

IIu(x,t)II22 .< IIlgradu(x,t)II~ +

Ifll~

~

ec~t

(43)

We now assume that we know from observation that

f l gradu I 122 -~ ~2 for all ta(O,T).

i2

(u,u) . < -

(44)

It then follows that + (f,f)e ~t .

(45)

If one now chooses i

i

(46)

: ~T log (f,f) then (45) becomes (u,u) .
0 on tE(tl,t 2)

Then either tI = 0 or, by continuity, F(t I) = O.

If F(t I) = O

then using (2.6), it may be shown (Agmon [1] and Levine [26]) that F(t) = O, te[O,t2) , and hence by continuity that F(t) = O, ts[O,T].

When

t I = O, we have F(O) = O, and the same conclusion follows by repetition of the argument.

Uniqueness is therefore established.

As indicated in the introductory lecture, continuous dependence of the solution u(.,t) upon its (initial) data can be established only in a restricted class of solutions, namely those which at time T satisfy F(T) ~ M e

-a2T/al

(2.8)

for some positive constant M.

Initial data is measured by F(O), required

to be small, but not zero, and so by virtue of the uniqueness just proved, we may assume F(t) > 0, te[O,T].

Thus, (2.6) is valid on the interval [O,T],

and, on using (2.8), may be expressed in the form F(t) ~kM1-6[F(O)] ~ , for some constant k(T).

O ~(t)

~l

,

(2.9)

Inequality (2.9) clearly shows that the solution

depends H~ider continuously on its initial data in the measure F, on compact sub-intervals of ~,T].

(See Pueci [40], John [12] and [13].)

This type of

continuity must not be confused with continuous dependence in the usual sense nor with Liapounov stability.

(These concepts correspond respectively

with continuity on the closed finite interval ~,T] and on the semi-infinite interval [0,~).)

Thus, in the case of well-posed problems or for problems

with Liapounov stable solutions, the use of (2.9) and Holder continuity become superfluous.

On the other hand, in those non-well-posed problems or problems

with unstable solutions

which admit a function F(t) with a convex logarithm,

(2.8) and (2.9) demonstrate how the imposition of a priori bounds lead to the

36 recovery of continuous dependence,

in the modified sense of HSlder.

questions have been generally discussed by F. John

Such

~ 3 ] , who also proposed

the concept of logarithmic continuity which we shall introduce later in connexion with the Protter-Murray method. While the topic of continuous dependence is clearly very important,

it

is also of interest to learn how the solutions to non-well-posed and related problems evolve with time.

Information,

in the form of estimates, may be

obtained by integrating the fundamental inequality

(2.2) in the manner

analogous to the second geometrical property depicted in Figure 2. simplicity, we deal first with the case when a I = a 2 = O.

For

Then, from (2.2)

(under the assumption that F(O) 4 O) we obtain

r'(0)~

F(t) ~ F(O) exp t F - - ~ f

,

t ~ O ,

(2.10)

so that when F'(O) ~ O, we see that F(t) is bounded below by an increasing exponential function qf t~me.

When F'(O) = O, we have

r(t) ~ r(0) .

(2.11)

Let us return to (2.6) and write it as

g

F(T)~ '

F(t) $ F(O) exp t ~n F - - ~ f

0 ~ t $ T .

i If we assume that lim ~ ~nF(T) = 0 (or alternatively that F(T) T÷~

(2.12)

< 8eYT

1-~

TN~

where ~, y, c are positive constants and N is any positive integer) then from (2.19) we see that F(t) ~ r(o) ,

t ~ O .

(2.13)

Thus, by virtue of (2.ii) and (2.13), solutions with F'(O) = O and with the asymptotic behaviour just described, must satisfy F(t) : F(O) . Arguments of similar kind may be used to furnish other conclusions.

We

collect all the results in the following theorem (for proof, see for instance Knops and Payne

~9]

and

~ i ] ).

37 Theorem: (i)

Let the function F(t) have a convex logarithm.

Then

either (a) F(t) is bounded below by a time-increasing exponential function for sufficiently large time, or

(ii)

(b) F(t) sF(O);

when F'(O) > O, F(t) is bounded below by a time-increasing exponential function for t ~ O;

(ill)

when F'(O) = 0 either (c) F(t) is bounded below by a time-increasing exponential function for sufficiently large time, o_Er

(iv)

(d) F(t) : F(O);

when F'(O)

O. Generalisatlons and refinements of the above arguments are evidently possible, and details of some of these may be found in Agmon [i], Ogawa [36], [37], [38], Levine [27], Levine, Knops and Payne [32]; Khosrovshahi [14].

see also

38

It is worthwhile remarking that the above theorem and corresponding results for non-zero al, a 2 demonstrate the non-existence of the class of smooth solutions for which a possible F(t) may be defined.

In particular,

we see that when the convexity inequal~ty is satisfies, there are no solutions with a polynomial growth behaviour for large values of time. 3.

EXAMPLES Rather than use the theory of linearised anisotropic elasticity (1) to

illustrate the remarks of the preceding section, we shall use instead a more recent example from the director theory of rods (see Green, Knops and Laws

[9]).

Consider a curve embedded in euclidean three-space, which at each point P has associated with it two assigned vectors, called directors, and consider three configurations of the rod: directors are denoted b y ~

the initial configuration in which the

(8) (e : 1,2) and the position vector of P is

denoted by ~(8), where 8 is a convected coordinate;

a deformed equilibrium

configuration in which the directors are denoted by A (e) and the position vector of P by R(0){ denoted by a

and a final configuration in which the directors are

and the position vector of P by a (G,t).

~

Let

DR

us assume t h a t

the

a~ ~

= A- ~

+ ~b~ ~

,

r~

= R~

+ eu

,

final

We further define

~r

deformation

is

small

in

the

and orders of e higher than the first can be neglected.

sense

that

Let us define ~3 and

b.. through z] 9u ~3 = ~--~ '

(i)

b. i

= b..A.. i] 3

Logarithmic convexity has been applied to this theory by Knops and Payne [15], [16], [18], [19], [22].

39

It may be shown (see Green, Knops and Laws

[9]) that for a simply

extended director rod of length ~, in which ~i' Ai are orthonormal

sets, the

equations of motion for flexure in the plane normal to A I are (i_~12 )v Sb

~xx

+ b

~XX

- b - v

,X ,x

: mv" ,

(3.1)

: nb ,

(3.2)

where a dot now indicates differentiation 8 : ~x ,

~v : u 2 ,

w.r.t,

t,

b : b23 ,

and 6, m, n are certain material constants, The parameter 12 is proportional

which may be assumed positive.

to the end load, and is positive or negative

according as the rod is in compression or tension.

Finally,

for the purposes

of this example, we assume the rod has clamped ends so that v(t,x) : b(t,x)

= 0 ,

at x = 0,i .

The following conservation

(3.3)

law is an immediate consequence

E(t) : ½ Ii In ~ 2 + n b 2 + ( l ~ 1 2 ) ( v )2+2by x ~ ( b , x ) 2 + b ~ ,x o

of (3.1)-(3.3)

dx = E(O)

.

(3.4)

For the function F(t) we take F(t) = fl (mv2+nb2)dx o

+ B(t+to)2

,

(3.5)

where 8, t o are positive constants to be determined classical solution exists to (3.1)-(3.3) differentiability

requirements;

later.

We suppose a

and then (3.5) satisfies

the

clearly it is also positive-definite.

To show that F(t) given by (3.5) satisfies an inequality of the kind (2.2), we differentiate

(3.5) and use the equations of motion

stitute for the inertia terms arising in the expression differential. F'(t)

;

(3.2) to sub-

for the second time

Thus, 2

(mv~r+nbt;)dx + 2B(t+t "o

r,,(t)

(3.1),

= 2

v

~

"O : 4 Ii(mv2+nb2)dx ~o

-

o

)

,

2

O - 4E(O) + 2fl

)V2,x÷2bV,x+ b2,x+ 2]dxj + 28

40

where in the last line use has been made of (3.4).

It may now easily be

shown by means of Schwarz's inequality that FF''

(F') 2 ~ - 2(2E(O)+$)F(t)

which is of the desired form.

,

(3.6)

Uniqueness and continuous data dependence

then follow from the remarks of Section 2.

(Observe that for uniqueness,

the initial data is given by v(x,o) = ~(x,o) = b(x,o) = b(x,o) = 0 and therefore on setting B : O, we have F(O) = 0.) Growth of F(t) may be easily established in the case 2E(O) + ~ $ 0 (i.e. when the initial total energy E(O) is negative) as then the right-side of (3.6) is positive and can therefore be dropped.

Hence, we may obtain the

integrated form (2.10) in which, however, because of the present form (3.5) of F(t), we may always make F'(O) > 0 by choosing t conclude at once, therefore, G(t) z

O

suitably large.

We

that

(mv2+nb2)dx

(3.7)

O

has a time increasing exponential lower bound for sufficiently large time. When E(O) = O, we choose ~ = O, and (3.6) now corresponds to (2.2) with a I. = a 2 = O.

Discussion of growth estimates under these conditions has

already been given in Section 2. When E(O) > O, we cannot expect growth in all situations since it is known that when the strain-energy is positive-definite Liapounov stable.

the null solution is

However, provided this latter condition is violated, growth ]

of G(t) may be established if, for instance~ G'(O) ~ 212G(O)E(O)] ~. details may be found in the general treatment of Knops and Payne

Further

[21~ .

It must be remarked that growth, for this particular example, has been established only in terms of the measure (3.7), and therefore it is uncertain whether growth is due to an increase in either the v- or b-components of G(t), or indeed whether both components oscillate unboundedly and exactly

41

"out-of-phase" for large values of time.

This remains still largely an open

question, although E.W. Wilkes and the author have obtained some partial results.

The general method of analysis outlined above for the director rod theory holds generally for weak solutions to equations of the form TV

Mu = NU ,

(3.8)

where M, N are linear, time-independent, symmetric operators with values in a Hilbert space ~

, and M is also positive-definite.

An appropriate choice

for F(t) is F(t) = II(-)ll~ where ll(-)II~b

+ B(t+t )2 e

is the norm defined on ~

A general discussion is given

by Knops and Payne [21], but special cases of (3.8) appear for example in linearised theories of elasticity and multipolar elasticity (see Edelstein [7]), and also in the fields of plasma physics (Laval, Mercier and Pellat [24]) and stellar dynamics (Antonov [3]).

It is interesting to note that Laval,

Mercier and Pellat quite independently in 1955 developed the technique of logarithmic convexity to discuss certain instabilities arising in their problem. One possible generalisation of (3.8) is via the introduction of dissipation t~

Mu + n ~

= Nu

.

(3.9)

Levine [26] has shown that (3.9) specialises to the equations of linearised thermoelasticity (when u = (w,6)D where w is the three-dimensional vector displacement and 0 is the incremental scalar temperature).

Dafermos [6] has

studied the asymptotic behaviour of the solutions to equation (3.9) and has

42

shown that another specialisation of them corresponds to the linearised theory of two interacting homogeneous isotropic elastic materials (see Steel [~i]). It is possible to apply the technique of logarithmic convexity to equations of the form (3.9), and prove that, under suitable conditions, appropriately defined functions F satisfy inequalities of the type FF'' - (F') 2 ~ - alF2 - a2FF' ,

(3.10)

where al, a 2 are constants whose sign depends upon the precise problem under consideration.

For instance, in thermoelasticity (see Levine [26], Knops

and Payne ~i]), we may take F(t)

=

o B(n)

pw.w.dxdn + (T-t)

i a

I

B(o)

Ow.w.dx + y ,

(3.11)

i i

where w.± is the displacement, y is some computable positive constant, p is the density, and B(~) denotes integration over the volume B of the body at time t.

For the mixture of two linear elastic materials (see Knops and

Steel [22], ~3]), we may take F(t)

=

It;

o B(n)

(PlWiWid+o2vivi)dxdn

(3.12)

where wi, v.l are the displacements of the constituent materials, and PI' P2 their densities. As an example of a non-linear system of equations, we mention the incompressible Navier-Stokes equation backward in time in which logarithmic convexity arguments have been used to establish uniqueness and continuous dependence on the data (Knops and Payne [17]).

Similar arguments may be

applied to the Navier-Stokes equation forward in time.

HSlder continuous

dependence of the solution upon its initial data may be established by assuming that two solutions u~ I), u~ 2) exist for separate initial data

4]

{(i), f(2) i i , and that these solutions are restricted to satisfy

s~p u! l) u! i) ~ M2 x,t

1

sup x,t

(3o13)

,

1

.

i

+ ~ui'j-uj'i

for positive constants M, N.

[ui'j-uj'i] +

i

.< N 2

(3.14)

Then on setting v i = u~ I) - u(2)l" it may be

shown, with the help of equations governing vi, easily derived from (3.13), that the function F(t) =

[] B(t)

v.v.dx i 1

satisfies an inequality of the type (3.10) with a I < O, a 2 < O. and continuous dependence then follow as described in Section 2. F(t) =

for

t B(n)

v.v.dxd~ + (t +t) i ~ o

I

B(o)

Uniqueness By taking

v.v. + Y , z z

where y is a positive constant, uniqueness and continuous dependence may again be established by logarithmic convexity, but under assumptions weaker than those given by (3.14). Logarithmic convexity has also been applied by Hills ~0],

~i]

to the

linear dipolar fluid and linear micropolar fluid to establish uniqueness and continuous dependence.

The governing equations of motion are non-linear due

to the presence of convective terms. 4.

FURTHER CONVEXITY ARGUMENTS The method that will be described in this section, developed mainly by

Levine, Payne and Knops, is closely allied to the techniques of logarithmic convexity discussed previously but uses instead the polynomial functional relation between f and F (see Section 2) and moreover examines consequences of concavity inequalities of the kind (FY) '' ~ 0 .

44

Generally, this approach is applicable to non-linear theories of both parabolic and hyperbolic type (see for instance Levine Levine, Knops and Payne

[27], Levine and Payne [29], [30],

[32] ), but we shall content ourselves in this article

with treating the single example of isothermal (non-linear) elasticity with a strain-energy function, W. Thus, let ui(xi,t) be the displacement at time t of a point whose position vector in the reference configumation B is x.. i

Then the Piola-Kirchhoff

stress ~.. z] satisfies the equations of motion oij,j = PoUi ,

(4.1)

where Po' assumed positive, is the mass density in the reference configuration, and the constitutive relations ~W .... z] ~ u. . ' l,]

W

=

W(u..) z,]

(4.2)

where N is the strain energy function per unit volume of the elastic body in its reference configuration. u. i

Homogeneous data are assigned on the surface

= 0 on ~ B 1 (4.3)

n.o.. = 0 ] i]

on

~

B2

where ~B---IhJ~B2 = 9B, and 3B 2 # ~;

nj is the ~nit normal on ~B 2.

It easily follows from (4.1)-(4.3) that

f.-

E(t) ~ ½

PoUiUidx +

B

j

Wdx = E(0) .

(4.4)

B

We now postulate that there exists a constant ~ > 2 such that (1)

(4.5) B

z,]

and by means of (4.1)-(4.5) prove that F(t) = I

(1)

B

P°Uiuidx + $(t+t

)2 o

*

0

< t


O) and hence (4.10) proves that F(t)

becomes unbounded in finite time.

This result contrasts with linearised

elasticity where solutions, in general, become unbounded only after infinite time, but is in keeping with other non-linear hyperbolic theories (see

Levine [27] ). Further conclusions are possible when initial data and the form of W are such that E(O) > O, but they are somewhat too complicated to describe here. 5.

LAGRANGE IDENTITIES We illustrate this technique by applying it to the initial mixed boundary

value of linearised elasticity (see Brun [4], [5] ).

Other applications in

continuum mechanics are to linearised thermoelasticity (Brun [4], [5], Green [8] ), viscoelasticity (Brun [4]) [5]) and to the linearised theory of Cosserat surfaces (Naghdi and Trapp [35] ). The equations of linear elasticity are )!

(CijklUk)l) j = pu i in Bx(O)T) ,

(5.1)

in which u. is the small displacement, p = p(x) > 0 is the mass density, and i

Cijkl(X i) are the elasticities assumed to satisfy the symmetry conditions Cijkl = Cklij .

(5.2)

The last condition makes (5.1) formally self-adjoint. not essential to the success of the method;

(Self-adjointness is

see the opening lecture by Payne.)

As boundary conditions, we adopt the homogeneous data u. = 0 on~ BIZ(O,T) J-

njcijklUk, I = 0 on ~B2~(O,T) , where ~ B = $ B 1 U ~ B 2.

Initially, the values of ui(x,o) and ui(x,o) are

assigned and we suppose that a classical solution exists to the initial boundary value problem (5.1 )- (5.3 ).

(5.3)

47

Now, for any two sufficiently differentiable (distinct) functions v.(x,t),1 wi(x't)' the following identity holds. I B(t)

" Pvi(x,t)wi(x,t)dx = Itl ~$i(x,n)~i(x,n) + Pvi(x,n)wi(x,n)]dxdn o B(n) + I

B(O)

Pvi(x,o)wi(x,o)dx ,

(5.4)

where again B(t) denotes integration over the volume of the body at time t. Let us set wi(x,t) = ui(x,t) , v.(x,n)m = - ui(x,2t-n) . Then, by means of (5.1)-(5.3) and integration by parts, (5.4) becomes IB(t) Pui~idx = ½ JB [Pui(x'°)ui(x'2t)+ 0~i(x'°)ui(x'2t)]dx "

(5.5)

Uniqueness of the solution to the initial boundary value problem may be immediately deduced from (5.5) on setting ui(x,o) = ~i(x,o) = O.

We observe

that only the symmetry condition (5.2) is required on the elasticities, the present uniqueness proof not requiring definiteness conditions usually found in classical proofs.

This result and method of proof are due to Brun [4], [5],

but it is worth recording that the same result may be obtained either using logarithmic convexity (Knops and Payne [15]) or the method of Protter and Murray [33] (see also Murray [34] ).

In the next section, uniqueness in the

same problem will be established by the latter method for skew-symmetric elasticities. H~ider continuous dependence on the initial data in an L2-norm sense may be deduced from (5.5) in the manner described in the introductory lecture by Payne.

The fundamental identity (5.5) may also be treated slightly differently

48

when the initial data admits u.(x,o) = 0. l

An integration w.r.t, time and use

of the arithmetic-geometric mean inequality then leads to

G(2t) "< !4N i=l ~ i ~i l G(O) + iN 4 G(2Nt) ,

¢(t) "< ~G(o) + Ui 4

for any integral N, where G(t) = I see t h a t

solutions

with

B(t)

the asymptotic

(5.6)

Thus, on letting N ÷ ~, we

pu.u.dx. i l behaviour

G(t) = O(t 2) as t ÷ ~ , must r e m a i n l e s s

than or equal

G(t) ~ G(O) ,

(5.7) to their

initial

value

in the

t ~'0 .

sense that

(5.8)

An alternative choice of the functions v. and w. is i

i

w.(x,t)l = ui(x't)

~u. vi(x'n) =i'6-i (2t-n) .

(5.9)

Substitution into (5.4) and use of (5.1) then gives ]B(t) P~i~idx = I~IB(n)ICijkl~i'J(n)Uk'l(2t-n)-Cijkl~i'j(2t-~)Uk'l(~)]dxd~

+ IB ~.(2t)~.(0)dx 1 l = IB(t) CijklUi,j(t)Uk,l(t)dn - IB CijklUi,j(O)Uk,l (2t)dx

+ /B P~i(2t)~i(O)dx . Elimination of the strain energy by means of the energy conservation equation finally produces IB(t)Puluidx. = ½V(O) + ½T(O) - ½1Bei klUi,jj (O)uk,l(2t)dx + ½fBPUi(2t)ui(O)dx where V~ T are the strain and kinetic energies, respectively.

For simplicity

49

we suppose ui(x,o) : O, so that

T(t) : ½T(O) + ½ [ ~i(2t)~i(Oldx B

3 i ~ T(O) + ~ T(2t)

,

and in a manner analogous to that used in treating (5.6) we see that if T(t) = O(t 2) as t + ~, then T(t) $ T(O), t B O. G(t) = G(O) + tG'(O) + 8

i

By means of the identity

t (t-n)T(n)dn - 2E(O)t 2 O

this in turn implies G(t) $ t2T(O) and hence G(t) = O(t 2) as t ÷ ~. We remark that results of this section do not require the coefficients Cijkl to be sign-definite.

All the results, of course, carry over for

abstract operators of the form Mu = Nu mentioned in Section 3. 6.

THE PROTTER-MURRAY

TECHNIQUE

For purposes of illustration, we again consider the example of linearised elasticity introduced in the previous section, with, however, the elasticities now being skew-symmetric Cijkl = - Cklij

(6.1)

Murray [34] has considered elasticities more general than those satisfying either (5.2) or (5.1), while Levine has extended the treatment to certain abstract equations. To apply the Protter-Murray technique, we start by defining the function w(x,t) according to wi(x,t) = e-~tui(x,t)

,

(6.2)

where I (>0) is a parameter.

Substitution of (6.2) into the equations of

motion (5.1) yields the following equations for w T!

(Ci]klWk,l),j~

-

•

2

p(w.+2~w.+~ w.) 1 • l = O .

(6.3)

50

Because 2ab .< (a+b) 2, appropriate partition of terms in (6.3) leads at once to

IB PlWi+l wi~20Awi-(CijklWk'l)'J " 1dx

.< O ,

or i ~-~d B p (w.w.dx+ll i wiwi)d

- d'~ B P(CijklWk'l)'jwidx "< O ,

where (6.1) has been employed. Integration then gives ~ I

B(t)

p2w.w.dx - I .w.dx p2w'w'dx + A3 I z 1 B(t) P (CijklWk,l) ,] i i i B(t) ~< X fB(o )

p2wiwidx + A3 I

B(o)

P 2wiwidx + I

B(o)

" j dx . P c ij klWk, lWi, (6.4)

We now express (6.4) in terms of u i by means of (6.2) and obtain ~ I

B(t)

P2(~i- ui)(ui- ui)dx + A3 ]

B(t)

I

B(t) 0(CijklUk,l ) ,] 1

p2u.u.dx

l

1 i S

e2ltQ

'

(6.5)

where the initial data term Q is given by

q ~ ~[

JB(o)

S 2A3~1f

p2(~.-Au.)(~.-~u.)d~+~3[ 1

~B(o)

1

i

i

]B(o )

p2uiuidx+ I

B(o

)PCijklUk,l(~i,j-lui,j )dx

(C U ) (C u )pdX , uiuidx+(X+l)~lIB( ° ~.~.dx+e^[ ) i i Z]B(o ) ijkl k,l ,j ipqr q,r

(6.6) for computable constants ~i' ~2"

We now follow a development due to Levine,

and employ the inequality, for constant y > O, i P(CijklUk,l),j(~-Aui )dx . 0 such that a

~en

C~(O x [ O,T]) solution of the Navier-Stokes

equations exists which assumes the initial data on

smooth" ( s e e [ t ~ ] ) .

u0

and zero boundary data

aO × [O,T). The proof of such a theorem is not a little complicated and demands a fair

amount of preparation.

Historically, it has been fashioned through the efforts

of several mathematicians: Basic contributions concerning the existence of "weak" solutions of the initial value problem were made by Leray [18,19~20], Hopf [10], and I£iselev and Ladyzhenskaya [15], and subsequently, information about the regularity of weak solutions was obtained by and Kaniel and Shinbrot [ Ii~].

Ito [11], Serrin [27],

I must refer you to their work for the detail

and subtleties. Concerning uniqueness of solutions of the initial value problemj results establishing uniqueness as a weak solution were obtained by Serrin [28]:

provided

that the boundary is smooth, the proof of uniqueness, for~vard in time, in the clas~ of smooth solutions is an elementary matter. bounded domain in

If

O

denotes a fixed

R 3 with smooth boundary so that the application of Green's

theorem is justified then one can readily establish 1~neorem 2. Let

d

(Serrin [26] ) be the diameter of

Q.

Let

u (i) ¢ C2(O x [O,T)) and

p(i) ¢ CI(~ x [O,T)) (i = J,2) be solutions of the Navier-Stokes equations such

69

that

u (I) = u (2) on a Q ×

[O,T); let -m (~< O) be a lower bound for the charac-

teristic values of the d e f o m a t i o n tensor (Dij) = (½(ui, j + uj,i)) associated with u (I) in [to~T); let u = u (2) - u (I), an~ set K(%)

K(t 0) exp (2m-6~21d2)(t-t o )

K = ½~

0.< to-< t
0, what is kno~vn about the asymptotic behaviour of u as t ---~ + ~?

70 Concerning these questions let us begin by supplementing the result of ~ol~ollary 2 with one of the simpler versions of a theorem on lower bounds for the rate of decay.

If T = + ~, this theorem ~ i I also provide information

about asymptotic behaviour, and in par~its inspiration was sn attempt to i~rove an earlier result due to Edmunds [8] about such behaviour.

I quote his result

for comparison. ~leorem 3. Let

(Edmunds [8])

u c C2 (~ x [0,~)) be a classie~-~l solution of the Napier-Stokes equations

such t h a t u ( ~ , t )

= 0 if

(~,t)

Su_p I aui (x,t)I xcO ~-~j for every ~ > 0 (1 ~ i,j ~ 3).

~ a~ x [ O , A .

=

S~ppose t h a t as t

--->

+~

O(e-Wt)

Then u ( ~ , t )

= o for ( = , t )

~ o x [o,A.

In the statement and proof of the theorem on lower bolznds we make use of the following notation: Let O be a bounded domain in R 3 with smooth boundary a~ so that Green's theorem is valid in O.

For vector valued functions

g = (gl,g2,g3) and

h = (hl,h2,h3) in CI(O x [O,T)), 0 < T 4 + ~, set (g(t),h(t)) ~ /O gi(x't)hi(x~t)dx

and

Ig(t)I2 = (g(t)~g(t)) ,

so that (.,.) denotes the inner product in [L2(O)] 3, and set IIg(t)ll2=/olgrad g(x,t) I2 dx = ~ ax.

1

For brevity we also write Igl = Ig(t) l:

= gi

1

in referring to the Euclidean norm of

the vector g we indicate its arguments,

lg(x,t) l 2

~x.

( x , t ) gi(x,t)

In the above notation, the promised theorem is

71

Theorem ),.

(0gawa [ 24-] )

L et u ¢ C2(O x [ 0 , T ) )

Stoke~ (a)

equations

Zf

ruth

u = 0 on a O ×

lu(t) l > 0

lu(t)l

and p c C1(~ x [ O , T ) )

fo~

O-< t o-< t


[0,~).

-A(t-to)

be a s o l u t i o n

Suppose t h a t

(b)

~e~

T=+~.

U(t)

= sup~ l u ( x , t )

t

~, t h e n

for

t O . 0 m

o f the N a v i e r -

llL2.

[O,T) can b~ ~ l ~ e d . ]

If lu(t) l = O ( e ~ - ~ t )

ast---~+~

foreaeh

;~ > O, then u - O. Remark:

The proof of this theorem is based on convexity type inequalities

which are satisfied by solutions of cert~in differential Hilbert space.

inequalities in

To obtain a clear idea of the motive behind the operations

undertaken in the proof below it may be helpful to consult the abstract papers on differential inequalities by 0gamma [23], Agmon ~ud Nirenberg [2], and Cohen

and Lees [3]. Proof: We observe first that

lul2 dt

Using Green's theorem,

=

2 (u,ut)

=

2 (u,Au - u.grad u - grad p).

the fact that u has zero divergance~

and the boundary

conditions we find that (u,u.grad u) = 0,

(u, grad p) = 0

and so

-~-1~t 2 dt Similarly, we find that

= - 2 lbl 2

(1)

72

d I!u II2 d--~

T~t Q(t)

= I1 u(t)112/In(t)12

d lul~ ~ Q =

2

Au)

=

-

=

- 2 (ut, u t + u.grad u) :

(ut'

If

t ~ [to, T) then

- 21 lul2(ut , u t + u.grad u)

- (u,ut)(u,u t + u.grad u ) t

-

- 2Ilul 2 lu t+½u.sra~u

12 -~ 1 lu12 I~.~radul

2

+ J (u,u.grad u) 2 1

- ( u , u t + ½ u.grad u) 2

Thus, using Schwarz inequality and the hypotheses of the theorem,

a_.R.
lU(to) l

(exp

xpI-Q~t o) to

dr so

Since, by hypothesis, U ¢ L2([0, T)) we find

l u ( t ) l ~>

lU(to) l

expI-X(t-to)t,

t O ~< t < T

the desired inequality, where X = Q(t0) exp ½ IIU I122 is a positive constant. L To relax the condition that lu(t) I > 0 for

Co~llaG

I either u - 0 in [0,~) or lu(0) l > 0.

0 ~< t O ~< t < T note that by

Suppose that ru(0) l > 0 ~

also that there exists a t ~ [0, T) such that lu(t)I = o.

Clearly, i n . h a t

73

event there exists a least positive value of t, say t ~, such that lu(t~)l = O. However, since the conclusions of part (a) of the theorem hold in [O,t~), by the continuity of t ~

lu(t) l they must hold at t ~'~ also.

lu(t*)l ~ lu(0) lexp(-~t ~) > 0

which contradicts

m

lu(t*)l = O.

particular, It fomlows

then that either u =_ 0 or lu(t)I> 0 in [O,T). Lastly~ suppose the conclusion of part (b) is false.

If u ~ O, then

lu(O) l > 0 and from part (a) we obtain

~ t l ~ ( t ) l>~ lu(o) l ,

o-< t < + ~ .

But for this to be compatible ~ith the hypothesis of (b) it must be that lu(O) I = 0 giving a contradiction.

Thus u _ O.

O

Now, rather similar results concerning the exponential decay of solutions, which satisfy zero Diriehlet boundary data, of certain parabolic partial differential equations ~ t h Masuda

[21].

time independent coefficients have been obtained Ly

Loosely speaking, he proves that for solutions that are defined

for all real t and satisfy an exponential bound for all t, a maximum rate of decay as t ---->+ ~ can be found which depends on the exponential botund.

In

recent work, Ogawa [25], finds that a global theorem of this type is valid for parabolic differential inequalities in Hilbert space and that it is applicable to partial differential equations with time dependent coefficients.

Assuming

these results~ it is tempting to conjecture that a similar theorem on the maximum rate of decay of solutions of the Navier-Stokes equations may be obtainable;

such would be an interesting addition to the knowledge that~ within

a particular class, solutions of the Navier-Stokes equations are sandwiched between exponential bounds. Let us now turn to the matter of backward uniqueness. we have

As a basic result

74

Tj~eorem 5 (Serrin

[28])

Let Q be a bounded domain in R 3 with smooth boundary aO.

c2(F x [ o , T ) )

~d p

(i)

¢ cI(~ × [ o , T ) )

Navier-Stokes equations with t O c (O,T), Remark:

Let

u (i)

( i = 1.,2) be t~o solutions of the

u (I)= u (2) on aO x [O,T).

u ( 1 ) ( X , t o ) : u (2) (X, t o ) f o r a l l

If for some

x c Q, t h e n u ( 1 ) ~ u ( 2 ) .

Serrin~s proof was patterned after teclmiques developed in a paper

by Lees and Protter [17] on the unique continuation of solutions of parabolic differential equations and inequalities.

Howpver~ a proof may perhaps more

simply be obtained by using similar techniques to those adopted in establishing Theorem

4.

P r o o f (Ogawa [ 2 4 ] ) Let u = u (2)"" - u (I)" - and p = p(2)- _ p ( t ) ; u t - AU + u ( 1 ) . g r a d

then u has to satisfy

u + u. grad u (2) = - grad p

div u = 0 and u vsnishes on the boundary.

Just as in the proof of Theorem 4, we find

that if lul > 0 then d

!

!u(,,!!,,,,,,. grad u + u. ~rad U(2)12

d-~ Q "< 2

lul 2

. . . . . .

Setting Ui(t) = maxls~P

lu(i)(x't)[' suplgrad ' 0 u(i)(x't)Jl

we then have dQ O.

Plainly. we

may assume

tu(t) l > 0 for

Integrating (3) from tI to t, we find that Q(t) is bounded by

a constant depending on u(tl).

lu(t)l

From this fact together with (4) we obtain

>~ tu(tl)lexpl-

A (t - tl)t,

where A is a constant depending on t I. of t ~

that

t ~ [tl,to)

But then, because of the continuity

lu(t)i, we conclude tha{ lU(to) l > 0, contradicting the hypothesis

that u(.,t o) -- O.

~hus u -_-0 on [ 0 , % )

Foz~ard uniqueness follows from Corollary I of Theorem 2.

[3

Finally in connection with the problems that I raised associated with Theorem 2, in the class of C2(Q x [O,T)) solutions there is continuous dependence on the data at time t O for~ard in time.

Using methods of logarithmic convexity,

Knops and Payne [ 16] have determined a class of admissible solutions within which there is continuous dependence on data backward in time.

Professor Knops outlined

this result in his earlier talk at this conference. So far my remarks have been confined to solutions of the Navier-Stokes equations in a bounded domain in R 3, and it is natural to scan 'the unbounded scene' for similar results.

Concerning uniqueness~ various theorems are known, but possibly

the most interesting is that due to Graffi [9], the feature of his theorem being that it is proved without the strong assumptions about the convergence at infinity of the velocity field common in earlier papers on the topic.

To be specific, if

is the complement of a bounded domain QO c R 3 with smooth boundary @O, Graffi considers solutions of the Navier-Stokes equations satisfying the follo~wing: (i)

ui, u i,t , Ui, j (i,j = 1,2,3) are continuous functions of (x,t) defined

on O x ( - ~ )

which for every finite time interval T are bounded in Q x T by a

constant~pending upon T.

The second order derivatives of u. are continuous in 1

~ x (-~,oo). (ii)

The pressure p has continuous first derivatives with respect ~o x i in

O x (-~,~).

As r ---->+~(r2 = xi xi) , p tends to a limit PC in such a way that

76

P - PO = O(r-½- ¢)' ¢ > O, uniformly in every finite time interval. and he proves Theorem 6

( G r a f f i [9])

Let u (I) and u(2)be solutions of the Navier-Stokes equations satisfying

(i),

( i i ) (with the same PO in condition ( i i ) in both cases) and

u(1)(x,t) : u(2)(x,t) for all (x,t) ~ a O x

(-~,~).

If for some t o ~ (-~,~)

u(1)(X, to ) = u(2)(X, to ) for all x c O, then u(1)(x,t) = u(2)(x,t) for all

(x,t)

~ o × [to,~).

Actually, Graffi uses tile condition p - PC = O(r-1)' and the improvement above is due to Serrin and is incorporated in the supplementing baek~;ard uniqueness theorem due to Edmunds [7], whose proof is modelled on Serrffn's for the corresponding backward uniqueness theorem in the bounded case. Theorem 7

(Edmunds [ 7] )

Together with the hypothesis of Theorem 6, if u = u (2) - u (I) and

llu 1['2 =

(x,t)

~ o x

In passing,

0--~ ~

dx

exists f o r a l l

t C (-o%00) then

~_

for all

(-~,~). we note t h a t a corresponding

existence theorem i s highZy d e s i r a b l e .

The success of convexity methods in obtaining Io~er bounds and backward uniqueness in the case of bounded domains makes it natural to investigate by the same methods analogous postulates for the unbounded case. of ~ o t h e s e s ,

Under a variety

theorems concerning lower bounds and as~nptotic behaviour are

available, see [24] and [4].

However, concerning baclc,~ard uniqueness, only a

more restricted version of Theorem 7 has been found [ 2g], extra spatial growth conditions being required. For the remainder of t~is talk I would like to turn away from questions of evolution in time, and consider instead, how convexity methods may be used to

77

derive results concerning the behaviour of solutions of the Navier-Stokes equations at a point.

We shall consider first, a result pertaining to time

independent solutions of the Navier-Stokes equations and it is c~nvenient to preface its statement with two theorems, due to Agmon, which play a key role in its proof.

~t

~ = I~ ~ R3: l x i : ( ~

2

= Iv ~ C2(B) : v ( ~) = 0(I~1 n) Theorem 8

2 23)-~ * x2 + ~ .< r I and l e t ~s

i~1 - - > 0

for all

n ¢ IN]

(Agmon [ 1] )

If v ~ ~ and if there exists a constant K > 0 such that v satisfies the inequality

IAvl -< ~Ilvl + Ig~ad vll throughout B, then v = 0 in B. The proof of this theorem, on the behaviour of a solution of an elliptic inequality at a pointj is based upon an abstract theorem on differential inequalities in a Hilbert space, inequalities of the form

II U - A ( t ) u

11 -< k(t) Ill u 11? ~ ( A ( t ) u , u ) l

~ ,

where A(t) is a positive symmetric operator satisfying various conditions. For future needs, we note that Theorem 8 is also valid for vector-valued functions v = (vi) of x and, moreover,

that B can have its centre at any point in R 3 provided

that the conditions are modified appropriately.

~eorem 9

(~on

[1])

If v ¢ '~ then the first derivatives of v each have a zero of infinite order at the origin. For time independent one can prove ~ireetly, theorem,

(stationary)

solutions of the Navier-Stokes

equations

see [5], with the aid of •neorems 8 and 9 the follo~'Jing

78 Theorem 1 0 . Let O be a connected open set in ~3.

Let u c C3(Q) and p ¢ C2(Q)

be a classical solution of the time independent Navier-Stokes equations. If at a point of O, taken to be the origin of coordinates, u(x) = O(Ixl n) as

Ixl ---* 0 for every positive integer n, then u is identically zero throughout O. Now, of course, this kind of unique continuation property is immediately deducible from the spatial analyticity of solutions of the Navier-Stokes equations in a wlde class of circumstances:

Con~rning this, I refer you to

the important contributions of Masuda [22] and Kahane [ 12].

Nonetheless, I

think it of interest to see that such a unique continuation property can be obtained in an elementary manner, by appeal to the work of Agmon cited above. An outline of the proof of Theorem 10 will be given after a consideration of the development of a similar theorem for time-dependent solutions of the Navier-Stokes equations. Concerning time dependent solutions it burns out that by using methods developed by Lees and Protter [ 17] one can obtain a sequence of theorems comparable with Theorems 8, 9 and 10:

Let ~ = IU e C3(Bm) :

lira

j~J~o

e

tx1-13Ivl

=

o for eveG # > O,

uniformly in t for t ( [O,T]I , where BT = B × Theorem 11

[O,T].

(Lees and Protter [17])

If v ¢ ~ and if there exists a constant K > 0 such that v satisfies the inequality

IAv ~tl .< K[ Evl + Igrad vZl throughout ~ ,

then v _= 0 in ~ .

79 Theorem 12. If v ¢ @, then

lim

e

Ixl-~lav/axil

= 0 (i = 1,2,3) and

Ixl-~ o lira

elXi-~lav/atl

= 0 for every p > O, uniformly in [O~T].

Ixl-~0 Theorem 13.

L e t u ~ C4(0 x [ O , T ] ) the Navier-Stokes

equations.

the o~igin of ooor~nates for t ¢ [O,T],

and p E C2(Q x [ O , T ] )

be a c l a s s i c a l

solution

of

If at some point 0 of ~ which we select to be

lira e ixl Ixl-~O

lul : 0 for every p > 0 ~ i f o r ~ y

in t

then u is identically zero thz~ughout Q.

Below, we outline the proof of Theorem 13;

the proof of Theorem 10

from Theorems 8 and 9 exactly parallels that of Theorem 1 3 from Theorems

11 and

12. Proof of Theorem 13 (an outline). We use the equation for the convection of vorticity A~ - ~ t = u.grad ~ - ~.grad u, where ~ = curl u.

Let B have centre at 0 and let its radius r be so chosen

that B is contained in ~: we show first that in BT, ~ - 0. Because of the boundedness of u i and ~ui/ax k in ~

(i,k = 1,2,3), we have

throughout B T

IA~ - ~t i -< K(l~i + Igrad ~I), where K is a positive constant.

By hypothesis ~ ¢ C3(BT) , and Theorem 12

ensures that the components ~. (i = 1,2,3) of ~ belong to ¢. 1

By the obvious

generalisation of Theorem 11 for vector valued functions it then follows that ~ -z 0 in B T. Next, it can be sho~n that there exists ~ ¢ @ such that u = grad @ and A~ = O, from which it follows that @ - 0 in B T and so u = 0 in B T.

80

Lastly, using the cornaectedness of 0 it can be shown that u ~ 0 in

o x [O,T]. Naturally,

it is of interest to s upplemefit the information contained in

Theorems 10 and 13 about the behaviour of solutions at a point with information about the behaviour of solutions at infinity, and at least for stationary solutions of the Navier-Stokes equations

some progress has been made in this

direction making use of generalisations of Agmon's work~ see [6].

8t

References

I.

S. A~on,

Unlclte et convexlte dans les pI~bl~mes differenti¢Is'

University of Montreal Press, (1966).

2.

S. Agmon and L. Nirenberg, 'Properties of solutions of ordinary differential equations in Banach space'~ Cor~1. Pure Appl. Math. 16 (1963), 121-239.

3.

P. J. Cohen au@ M. Lees, 'Asymptotic decay of solutions of differential inequalities', Pacific J. Math. 11 (1961), 1235-1249.

4.

R. H. Dyer and D. E. Edmunds, 'Lower bounds for solutions of the NavierStokes equations', Prec. London Math. Soc. 18 (1968), 169-178.

5.

R. H. Dyer and D. E. E~lunds, 'On ~ e regularity of solutions of the Navier-Stokes equations', J. London Math. Soc. 44 (1969), 93-99.

.

R. H. Dyer and D. E. Edmunds, 'Asymptotic behaviour of solutions of the stationary Navier-Stokes equations', J. London Math° See. ~

(1969),

340-346.

7.

D. E. Edmunds, 'On the uniqueness of viscous flows', Arch. Rational Mech. Anal. 14 (1963), 171-176.

8.

D. E. Edmunds, 'Asymptotic behaviour of solutions of the Navier-Stokes equations', Arch. Rational Mech. Anal. 22 (1966), i5-21.

9.

D. graffi, 'Sul teorema di unicith nella dinamica dei f!uidi', Annali di Mat. 50 (1960), 379-388.

82

10. E. }~pf, 'Uber die Anfangswert&~gabe

fGr die hydrodynamischen

grundgleichungen', Math. Nachrichten g (1951), 213-231.

11. S. Ito~ 'The existence and uniqueness of regular solution of non stationary Navier-Stokes equation', J. Fac. Sci. Univ. Tokyo, Sec. Z, 9 (1961), I03-I~0.

12. C. Kahane, 'On the spatial analyticityof solutions of the Navier-Stokes equations'

Arch. Rational Mech. Anal. 33 (1969), 386-g05o

13. J. Kamp~ de Fermet, 'Sur la deerolssance de l'~nergie cin~tique d'un fluide visqueux incompressible occupant tun domaine born~ayant pour .4 frontlere des parois solides fixes', Ann. Soc. Sci. Bruxelles 63 (19J~9), 36-45.

I/+. S. Kaniel and M. Shinbrot,

'Smoothness of weak solutions of the Navier-

Stokes equations', Arch. Rational Mech. Anal. 2~ (1967), 302-32$.

15. A. A. Kiselev and O. A. Ladyzhenskaya, 'On the existence and uniqueness of the solution cf the nonstationary problem for a viscous incompressible fluid', izv. Akad. Nauk. SSSR 21 (1957), 655-680.

16. R. J. Knops and L. E. Payne, 'On the stability of solutions of the NavierStokes equations backward in time', Arch. Rational Mech. Anal. 29 (1968), 331-335.

17. M. Lees and M. H. Protter, 'Unique continuation for parabolic differential equations and inequalities'j Duke Math. J° 28 (1961), 369-382.

83

18. J. Leray, 'Etude de diverses ~quations integrales non llnealres et de quelques probl~mes que pose !'hydrodynamique', J. Math. Pares @ p l . 12 (1933), 1-82.

19. J. Leray, 'Essal sur les mouvements plans d'un liquide visque~x que

limitent des parers', J. Math. Pures Appl. 13 (1934), 331~18.

20. J. Leray, 'Sur le mouvement d'un liquide visqueux emplissant l'espace',

Acta Math. 63 (1934), 193-248.

21. K. Masuda, 'On the exponential decay of solutions for some partial differential equations', J. Math. Soc. Japan 19 (1967), 82-90.

22. K. Masuda, 'On the ~nalyticity and the unique continuation theorem for solutions of the Navier-Stokes equations'~ Prec. Japan Acad.

43 (1967), 827-832.

23. H. Ogawa, 'Lower bounds for solutions of differential inequalities in HAlbert space', Prec. ~ner. Math. Soc. 16 (1965), 1241-1243.

24. H. 0gawa, 'On lower bounds and uniqueness for solutions of the NavierStokes equations', J. Math. Mech. 18" (1968), 4/,5-452.

25. H. Ogawa, 'On the maximum rate of decay of solutions of parabolic differential 5_uequalities', Arch. Rational Mech. Anal. 38 (1970), !73-177.

26. J. Serrin, 'On the stability of viscous fluid motions', Arch. Rational

Meeh, Anal. 3 (1959), 4-13.

84

27. J. Serrin,

'On the interior regularity of weak solutions of the

Navier-Stokes equations', Arch. Rational Mech. Anal. 9 (1962),

187-155.

28. J. Serrin,

'The initial value problem for the Navier-S~okes equations',

Proc. Syrup. Non-linear Problems, Univ. of Wisconsin (1963), 69-/o8.

NON-UNIqUE CONTINUATION FOR CERTAIN ODE's IN HILBERT SPACE AND FOR UNIFORMLY PARABOLIC AND ELLIPTIC E~UATIONS IN SELF-ADJOINT DIVERGENCE FORM*

K. MILLER

I want to consider the problem of backward uniqueness for the uniformly parabolic equation: n ut = i,j=iZ (aij(x,t)Ux)x.]i e ~-a~u in ~×[0,~) ~Vu.v

= 0 on ~ x [ O , ~ )

,

,

(a)

(i)

(b)

and the problem of unique continuation

(and uniqueness for the Cauchy

problem) for the uniformly elliptic equation: n [ (aij(X)Ux )x. z ~ . ~ u i,j =i j 1

= 0 in ~ ,

(2)

where ~ is a bounded domain in Rn, ~ denotes the unit normal to 3 S, and the symmetric coefficient matrix a has its eigenvalues in [e, -i], with ellipticity constant 0 < e ~ i.

We construct examples of nonuniqueness

for (i) when n = 2 and for (2) when n = 3;

in each case e may be arbitrarily

close to i and the coefficients are also HSlder continuous. Equation (la) or (2) are the equations of time dependent or steady state heat flow, with variable nonisotropic conductivity matrix

C[, with constant

heat capacity, and with the conductivity in each direction bounded above and

Talk given at the Symposium on Logarithmic Convexity and Non-Well Posed Problems, 21-24 March 1972, Heriot-Watt University, Edinburgh. *Supported by a C.N.R. Visiting Professorship at Universita di Firenze and by N.S.F. grant

86

below.

The "no flow" condition (la) implies that ~

is totally insulated.

Sometimes we will want to replace the Neumann boundary condition (Ib) by the Dirichlet condition u = 0 on

~x[O,~)

,

(ib')

in which case we will speak of (la) and (ib') combined as (i'). The famous results of De Giorgi [5] and Nash [7] established ~ interior H~ider continuity of weak solutions of these equations with only measurable coefficients.

This led to a tremendous development of the theory

for second order elliptic and parabolic equations in divergence form, especially nonlinear equations.

In some sense ~ priori inequalities, those

that are independent of the smoothness of the coefficients, show the fundamental elliptic (or parabolic) behaviour of the equation.

I have been

attempting to prove backward uniqueness (and a conjectured ~ priori stability bound of logarithmic convexity type) for (i) for eight years. Backward uniqueness for (i) with C 1 coefficients was shown by Lions and Malgrange [6] in 1960.

However, probably the simplest proof is due to

Agmon and Nirenberg [i] in 1964, and to Agmon [2] in 1966, who used the general method of logarithmic convexity.

Carleman [4], long ago in 1939

established unique continuation for (2) with C 2 coefficients when n = 2.

(It

is possible that his proof can be carried through in this two-dimensional case with only measurable coefficients, but of this I am not certain.)

For

n ~ 3, unique continuation for (2) with C 2"I coefficients was proved by Aronszajn [3] in 1957, and more simply with C 1 coefficients by Agmon [2] in 1966, using logarithmic convexity techniques once again.

See [2] for

references to other results by Cordes, Holmgren, HSrmander, Landis, Lees and Protter, and others.

87

An example of nonunique continuation was constructed by Plis a uniformly elliptic equation in the nondivergence

[9] for

form:

3 3 [ aij(X)Ux.x. + [ b.(x)u + c(x)u : 0 , i,j=l i ] i=l l x.1 with H~ider continuous coefficients.

(3)

Despite this example, the question

of unique continuation for (2) has remained actively open.

In the first

place, it is well known that the solutions of the divergence equation (2) and the nondivergence equation (3) often exhibit extremely different p

behaviours when the coefficients become nonsmooth;

in fact, the Plis example

occurs at exactly that degree of smoothness at which (3) can no longer be changed into the form (2) (plus bounded lower order coefficients). whole ,

solutions

equation , while

of equation (2) tend solutions

On the

to behave like thoseof the Laplace

of (3) often exhibit

quite striking pathology.

In the second place, there is the strong physical interpretation of (i) and (2) in terms of heat flow (or diffusion, or electric flow).

It seemed

highly nonintuitive that an initially nonzero solution of (i) could manage to make itself vanish within a finite time T, or that a nontrivial solution of (2) could manage to wipe out all trace of itself on an open subset. Equation (3) on the other hand has no such physical interpretation. Before describing the nonunicity examples for (i) and (2), I would like to mention a few aspects of my previous attempts to prove backward uniqueness and ~ priqri stability for (i) and (i'). i.

BACKWARD STABILITY WITH C I COEFFICIENTS FOR (i) AND (i') The logarithmic convexity approach of Agmon and Nirenberg, when applied

to the simple equation (i) (or (i')), can be redescribed so simply that it is worthwhile to give a sketch of it here.

One writes (i) as an ODE in

the Hilbert space L2(~): u' = - A(t)u, u(t)aDA(t) , t > 0 ,

(4)

88 where A(t) is a symmetric operator with domain DA(t) consisting of all sufficiently smooth functions satisfying the boundary condition (ib) and with corresponding bilinear form: a(t)(v,w) ~ [ )

[ a..v w dx . i,j 13 x.] x.I

Now a(t)(v,w) is defined for all sufficiently smooth functions v and w, but integrating by parts we see that (A(t)v,w) : f J

[ a..v w dx - I i,j i] xj x i

w(~aij Vx.Vi )do ,

: a(t)(v,w) if VeDA(t) . Our first step is to differentiate loglIu(t)ll twice, as is usual in the logarithmic convexity method, and notice that a new term - a'(u,u)/(u,u) appears, due to the change in the bilinear form for fixed u. [loglIu(t)l I]'

:

-

~( u , u )

:

-

~( u , u )

Thus,

•

'

and hence, [logl[u(t)l]]'' = _

~u,u)[a(u'~u)+a(u,u')+a'(u,u)]-a(u,u)[2(Au,~,u!,,] (u,u) 2

~2(u;u)(Au,Au)-2(Au,u)2~ -

:

[-

(u,ui2

J

(u,u)

The first term is non-negative by the Cauchy inequality, as usual. we have:

Thus

suppose the A(t) are nonincreasing (in the sense that a(t)(v,v) is

nonincreas.ing for each fixed v), the n logIlu(t)II is convex for each sufficiently smooth solution of (4).

The same result holds even more simply

when (ib) is replaced by the Dirichlet condition (ib'), for then DA(t) is independent of t and u' will also be in DA(t) for sufficiently smooth solutions. Sather.

I understand that similar results have been proved by Payne and

89

The second step is to change the variable on the t axis; d~/dt = p(t) > 0 with ~(0) = O.

i.e., let

With respect to the new variable ~(t)

the equation (4) becomes du

d-~

-

(p(t))-lA(t)u

E B(T)U

(5)

.

Now B(~) has the bilinear form p-la(v,v) and clearly we can make this nonincreasing by making p increase quickly enough.

~

-la(v,v~'

_ pa'(v~v)-p'a(v~v) -

2

~

0

We must have

,

P

or

ogp(t

~'

a'(v,v) = P'/P ~ a(v,v)

'

which latter term is bounded in terms of the ellipticity constant ~ and uniform bounds on the time derivatives a!. of the coefficients.

z]

Thus by "distorting the t axis" by a sufficient amount we have forced logllu(t)ll

to be a convex function of the "distortion"

T(t), and we have

the following stability bound for backward solutions of (4): if

llu(T)lf

and ltu(O)tf

,

(6)

then

I1u(t)lt 2.

{ ~~(t)/~(T)EI-~(t)/T(T)

for 0 ~ t ~ T .

(7)

AN ABSTRACT EXAMPLE OF BACKWARD NONUNIqUENESS Our example of backward nonuniqueness for (i) closely resembles an

abstract example found in 1965 (but still unpublished)

for an ODE in Hilbert

space of the related form u' : - A(t)u, t > O, with

(8a)

0 < ~K ~ A(t) ~ e-iK, where

(Sb)

K and A(t) are self-adjoint operators all having

(8c)

the same domain.

90

Notice that equation (i') is in this form, with - K corresponding to the Laplacian operator A.

Since the abstract example is simpler, but also

shows certain limits to the logarithmic convexity approach, it is worthwhile to describe it here. I had conjectured at one time that conditions (4) should be sufficient to imply that logllu(t)II is an

t!~ 2

-convex ~! function of t.

We say that

2 the decreasing function % is ~ -convex on the interval I if for each triplet tI < t 2 < t 3 in I we have (%(t3)-%(t2))/(t3-t 2) ~ ~-2(%(t2)-~(tl))/(t2-tl). The case e = i of course corresponds to usual convexity.

One may show, by

a great variety of different proofs, that this conjecture is true provided that the A(t) and K all commute wit~ one another.

Without commutativity,

however, ~ priori convexity of logIIu(t)II is false, and in fact we now proceed to construct an example of backward nonuniqueness under conditions (8). Notice that the graph of log I lu(t)II has slope - (Au,u)/(u,u), which is comparable to - (Ku,u)/(u,u).

The size of (Ku,u)/(u,u), however, is

an indicator of whether u is mostly concentrated in low order eigenspaces of K, or in high order eigenspaces.

If logllu(t)II is to manage to plunge

to - ~ within finite time T, the solution u(t) must somehow manage to rotate itself so as to concentrate in higher and higher order eigenspaces. The significance of logarithmic convexity for the case A(t) z K ~ constant, however, is that this cannot then happen, for the high order eigencomponents of the solution eigencomponents;

then die out more quickly than the lower order hence u(t) must concentrate more and more in the lower

order eigenspaces and (Ku,u)/(u,u) must decrease. Within the constraints of (8), however, it turns out that it is still possible for a solution to start within one eigenspace of K and rotate itself completely into a higher order eigenspace.

Let us show that this

91

is possible in the two dimensional case. ~ = [~

ii, ~(t)=

RT(t)[~

We let

~] R(t) ,

(9)

and let ~(t) be the solution of

A 90 °.

the corners off in

Then we may vary a

parameter (such as the duration of the rotating stages) and by continuity select that value of it which gives a rotation of exactly 90 ° . without further explanation:

We state

there exists a C ~ choice of R(t) in (9)

such that ~(t) in (i0) rotates 90 ° from the x I axis into the x 2 axis in finite time To.

In an initial phase R(t) is ~ I and ~(t) = (e-t,o)T;

an intermediate phase R(t) varies and ~(t) rotates thro+u~h, g O°~ final phase R(t) { I and ~(t) is proportional to (O~e-lt) T.

in

and in a

The

magnitude of the solution decreases by a factor r z II~(To)II which can be made as small as we please merely by increasing the duration of the final phase. We now patch together an ~ of steps, each of which is essentially a carbon copy of the basic 2-dimensional example. matrix

Ill •

K =

We let K be the infinite

0

+o

considered as a self-adjoint operator on the space of 12 column vectors (Xo,Xl,...)T.

We adopt the notational convenience of starting time out

anew with t = 0 at the beginning of each step.

(12)

95 The Oth step has duration To, initial value u(O) = (I,0,...) T, matrix 'RT(t) (~ ~]R(t) 12 A(t) =

0

. T .

T - t ~ 1 -n in the nth interval,

I iu(t)II

= O(IT-tl m ) as t ÷ T.

u(1),u (2),...u (m) exist also Theorem equation

i.

There

of form

t ~ T but never made as large

exists

for t ~ T, but becomes

(17) implies

at t = T and are zero there.

(4) on the Hilbert

as we like.

condition

that

In this way we see that the strong

an example

O for t < T~

(18)

u(t)

of backward space

is strongly

The operator

discontinuous

12 .

A(t)

at t = T.

uniqueness The solution

C m on

derivatives

We summarise: for an ODE u(t)

[Oi~) ~ where

is z 0 for m can be

is C ~ (in any sense of the word)

95 In the above example the final phase of the nth step has duration proportional to l-n and the magnitude factor r in each step;

however, we can instead increase that duration,

to n -2 say; thus the magnitude in+In-2 rn < ein the nth step. with u(t) strongly ...... C ~ o n

IIu(t)II is reduced by the same

IIu(t)II is reduced by a factor In this way, we may construct our e x a ~ l e

[0,~).

It is easy to see that there is enough leeway here to make u(t) and all its derivatives + 0 without keeping the ratio between successive diagonal elements of K exactly a constant. where I

n

In fact we may even let Kn+l,n+i/Kn, n = In

is a sequence of numbers which converge downward to i, but not too

quickly.

Since the '!ellipticlty constant" ~ in the nth interval will equal

l-I in this way, we may construct our example with a variable "elli~tiqit~ n+l ' .......... constant" ~(t) in (8b) which tends to 1 as t ÷ T. Let A denote the unbounded self-adjoint operator on Lg(O,z) corresponding to the Laplacian with zero Dirichlet conditions at the end-points. With respect to the basis sinx,sin2x,...,sinkx,..,

for L 2,

A corresponds

to the infinite diagonal matrix

12 °1 22

- A ~

".

.

(19)

"k 2

We may now just pick a subsequence kl,k2,...,kn,.., k~+i/k ~ is always ~ I : - i

such that the ratio

Then using our previous construction, but

skipping over the intermediate eigenspaces, we can begin with a u(0) in the sinklX eigenspace and cause it to rotate successively into the sink2x,... , sinknx,.., eigenspaces. L2(0~)

with K b e i n g

In this way, we imay construct our example on

the negative Laplacian - A-

96 It can be shown immediately,

however, that the abstract operators

A(t) in the example just mentioned cannot be elliptic differential operators on ( 0 ~ ) .

It can be shown (due to variation diminishing properties

[8]

which follow directly from the maximum principle) that solutions of a parabolic differential equation on a rectangle (in one space dimension) with u = 0 on the lateral sides (or u changes of sign as time advances.

x

= O) cannot increase their number of

Since our abstract solution begins as

sinklX (with k I - i changes of sign) and ends the first step proportional to sink2x (with more changes of sign), it is impossible for a parabolic equation to copy the behaviour of even a single step of the abstract example (when n = i). All of the above results had been found by the summer of 1965.

From

that time until a short while ago I had hopes that the ellipticity of the operators

in (la) (i.e., the maximum principle) would somehow intervene and

permit one to prove that logIlu(t)II

2 is e -convex for solutions of (i').

There is no time here to describe some of the attempts to prove this ~njecture,

or the tens of thousands of computer, run tests for a discretised

version without finding counter examples (all unfortunately with n : i). (It still seems quite likely to me that eg-convexity for (i) and (i') does hold when n = I.)

It now turns out, however, that variation diminishing

properties fail drastically for n > i. 3.

STATEMENT OF NONUNICITY RESULTS FOR (i) AND (2) Tbeorem i.

~(ere exists an example of backward nonuniqueness on the

halfspace A : R2×[01 ~) for .a unif0rm! ~ parabolic e~uation u t = ((l+A+a)Ux) x + ( b uy)x (i)

+ ( b ux)y

+ ( ( l + C + c ) uy)y

in A .

(20)

The solution u(x,y,t) is C ~ on A, z 0 for t { a certain positive

value ~T, but never ~ 0 in an[ open subset of R2x[O,T).

97 (ii)

The coefficients a(x,v,t), b(x,y,t), c(x,y,t) are C ~ on

A,,,an_d

-= O for t >. T. (iii)

The coefficients A(t)~ C(t) are H~ider continuous (of order 1/6)

on [O,~), C ~ on !O,T), and ~ O for t ~ T. (iv) 2~;

All functions u. a, b, c are periodic in x and ~..with ~eriod

u is symmetric about x = j~ and y = j~, ~ integer. (v)

Moreover~ U satisfies the "no flow" condition (.la,) on the sides

$~×[0,~) of the ,c~lipder ~×[01~) I C = (0,~)×(0,~), since both b and the 3u normal derivative-~- are = 0 there. Theorem 2.

Th@re exists an example of nonunique cont.inuation on the

halfspace A = R2×!O,~) for a uniformly elliptic equation utt + ((l+A+a)Ux) x + (bUy) x + (bUx )y + ((l+C+C)Uy)y = 0 in A .

(21)

Conditions (i)-(v) on u~ a, b, c~ A~ C hold exactly as stated in Theorem i. 4.

OUTLINE OF THE PARABOLIC EXAMPLE Our construction proceeds with an ~ of steps of successively shorter

duration TI,T2,... whose sum T is finite.

Let us consider the nth step,

in which the solution begins proportional to cosl x.e n to cOSln+lY,e

-1~+it

, ends proportional

, and at intermediate times is always a linear combination

of cOSlnX and cOSln+lY. to be chosen later.

-1~t

Here 11,12,... is an increasing sequence of integers,

Each step will consist of three major phases (plus

four "transition" phases between for purposes of smoothness only).

These

seven phases have durations In 2, I-2n" I~ 2' Snln 2' I-2n' In 2' In2- respectively, where Sn, a certain sequence tending to ~, is also to be chosen later. 2 -(~[-l)s n then define ~n = In+i/A n and a n ~ e be ~ i and e

n

We

We point out that ~n will

will be ~ O, especially for large n.

Let n and B be two fixed C ~ functions on [0,I] with the following behaviour:

n(t) is ~ 0 near t = O, monotone on [0,i], and z i near t = i;

98

6(t) is ~ 0 near t = 0 and ~ t - i near t = I.

We now proceed to list the

duration, solution, and coefficients (in the order l+A+a, b, l+C+c) for each phase.

For notational convenience we start the time out anew with

t = 0 at the beginning of each phase and also employ the notation "~", "is proportional to". The "transition i" phase has duration 1-2, solution u ~ cosl x-e n n and coefficients i, O, @(t), where ~(t) = 1 + (~-l)q(l~t).

_~2t n

This phase

merely changes the coefficients smoothly from an initial i, O, 1 to a -2 final i, O, Pn The "seed" phase has duration 1-2 , solution n

_12t _12t -2 ~ u ~ cOSlnX.e n + n(l ~ t )EnCOSln+lY.e n , and coefficients 1 + a, b, Pn where the rather complicated a(x,y,t) and b(x,y,t) will be described later. This phase introduces a tiny cOSXn+lY component into the solution. The "transition 2" phase has duration X~2, solution -¢(l~t) u ~ cOSlnX,e

+ EnCOSXn+lY.e

where ¢(t) = t + (p~-l)B(t).

-X2t n , and coefficients ¢'(X~ t), O, Pn -2 "

This phase smoothly changes the decay rate

of the first component from an e

-~2tn

rate to an e

-X~+I t

rate.

It does not

change the i to ~n ratio of the two components. The "distorted decay" phase has duration ~ 2 S n ,

solution

-1~+it

-X2t u ~ cosl x°e + enCOSXn+lY.e n , and coefficients Pn' 2 0, Pn -2 " Since n -(X 2 .-X 2)X-2s 2 n+± n n n en = e , this phase reverses the initial 1 to en ratio of the two components to a final ~ to 1 ratio. n The "transition 3" phase has duration i~2, solution u ~ ~nCOSXnX.e

-X~+it

+ cOSXn+lY.e

where ~(t) = pit + (p~-l)B(l-t). of both components to the same e ratio of the two components.

-¢(l~t)

2 -2 , , and coefficients Pn' O, ~n ¢ (X~t),

This phase smoothly changes the decay rate -~+i t

rate.

It does not change the ~n to i

99

The "removal phase" has duration u ~ q(In2(l-t))enCOSXnX-e

-X2+lt

1-2, solution n

+ cOSXn+lY-e

-X2+lt

2

and coefficients ~n , b ,

I + c, where the b(x,y,t) and c(x,y,t) will be described later.

This phase

is quite similar in all respects to the previous "seed" phase, except that it removes a tiny component from the solution. The "transition 4" phase has duration Xn-2 ' solution u ~ cOSln+lY'e and coefficients ~(X~t), O, i, where ~(t) : p~ + ( l + ~ ) n ( t ) . merely changes the coefficients 5.

-l~+lt

This phase

smoothly from U~, O, I to i, O, i.

DERIVATION OF THE a; b It is convenient to normalise the geometry, expanding the x, y and t

scales by factors of In, In+l, and 12n respectively.

That is, we consider

defined by u(x,y,t) : u(X/ln,Y/ln+l,t/1~)

~ cosx-e -t + n(t)~ n cosy'e -t ,

(22)

~

which must be the solution of an equation with coefficients i + a, b, i. Since cosx°e -t and cosy°e -t are already solutions of the equation with coefficients i, O, i, the perturbations a and b need only take care of the q'(t) term in the equation and we are led to the following perturbation equation: (asinx)

x

= - nenSinyb

x

- sinxb

y

- q'e cosy n

.

(23)

By considering b of the form f(y) s(x,y), where s is a certain solution of the first order PDE ~s ~s q S n S i n Y ~ x + sinX-~y : 0 , one can construct

(for e

n

$ a certain

(24)

o

> O) a b which is : 0 on x : j~ and

y = j~ as desired and such that the right hand side of (23) has mean value zero across horizontal lines between x = j~ and x = (j+l)~. inserted in (23) then determines an a as desired.

This

Moreover, one may show

that each derivative of a(x,y,t) and b(x,y,t) is bounded by a constant (depending on the order of differentiation)

times E . n

100 Construction of the coefficients b and c for the "removal" phase is completely analogous. 6.

PUTTING THE PIECES TOGETHER One now stacks the solutions and coefficients for the various phases

and steps "end to end", after first multiplying the formulae (u ~ cOSlnX'e

_~2 t n etc.) in each phase by an appropriate magnitude constant

in order to maintain continuity.

Notice that the magnitude of u in each

phase of the nth step is decreasing at least at an e

_12 t n

rate.

Now with the proper choice of In and Sn we can make T z- ZTn = Z(6+Sn )I-2n finite, u and the a, b, c ÷ 0 (in C ~ fashion) and the A(t), C(t) + O (in H~ider continuous fashion) as t ÷ T.

It suffices for example to choose

In = (n+N) 3, sn = (n+N) 4, where the integer N is taken sufficiently large to keep en, a, b, c, A, C small also during the initial steps.

One then

sets u z a ~ b ~ c ~ A ~ C ~ 0 for t ~ T. 7.

THE ELLIPTIC EXAMPLE

The elliptic construction is extremely similar to the parabolic case. -I t n The solution in the nth step begins proportional to cOSlnX.e and ends proportional to cOS~n+lY-e -In+It

The perturbation equation for a and

in the "seed" phase turns out to be essentially in the same form as (23) and the "transition

2" and "transition 3" phases require a bit more attention.

101 REFERENCES

[1] [2] [3]

[4]

[5] [6] [7] [8] [9]

Agmon, S., Unicit6 et convexit6 dans les problhmes diff6rentiels, Seminar Univ. di Montr@al, Les Presses de l'Univer. Montr@al, 1966. Agmon, S., and Nirenberg, L., Properties of solutions of ordinary differential equations in Banach space, Comm. Pure and Applied Math., 16 (1963), pp. 121-239. Aronszajn, N., A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl., 36 (1957), pp. 235-249. Carleman, T., Sur un probl6me d'unicit@ pour les syst~mes d'@quations aux d@riv@es partielles ~ deux variables ind6pendentes, Ark. Mat. Astr. Fys. 26B (1939) No. 17, pp. 1-9. De Giorgi, E., Sulla differenziabilith e l'analiticit~ delle extremali degli integrali multipli, Mem. Acc. Sc. Torino, III, (1957), pp. 25-43. Lions, J., and Malgrange, B., Sur l'unicit6 r@trograde, Math. Scand., 8 (1960), pp. 277-286. Nash, J., Continuity of the solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), pp. 931-954. Nickel, K., Gestaltaussagen Uber L~sungen parabolischer Differentialgleichungen, J. Reine Angew. Math., 211 (1962), pp. 78-94. Plis, A., On nonuniqueness in the Cauchy problem for an elliptic second order differential equation, Bull. Acad. Polon. Sci., Set. Sci. Math. Astron. Phys., ii (1963), pp. 95-100.

Department of Mathematics University of California Berkeley, California

LOGARITHMIC

CONVEXITY AND TilE CAUCHY PROBLEM FOR

P(t)u~

+ M(t)u t + N(t)u = 0 IN HILBERT SPACE

Howard A. Levine

I.

duct

Let

H

be a Hilbert

(,)

and norm

t~[O,T), let D ( t ) ~ H and into

N(t)

II

INTRODUCTION space, real or complex, with inner pro-

II = (~,).

Let T >0

or

T = +~ and for

be a dense linear subspaee of

H . Let

M(t)

be linear operators (bounded or unbounded) mapping

H . In

[9]

for the following

and

D(t)

[i0], questions of uniqueness and stability

(generally nonwell posed) problems were dis-

cussed. Problem I .

M(t) dudt = N(t)u + fl(t,u)

; t ~ [0,T)

u(0) prescribed. Problem II.

M(t) d2u = N(t)u + f2(t,u,du/dt) dt 2 u(0)

Here

u : [O,T)

derivatives are

~H

and

; t e [0,T)

du/dt(0) prescribed.

is such that the required number of strong

D(t)

assumed to satisfy, for

valued. The "nonlinear" terms fl and f2 were Ul,U 2 : ~,T)----~D(t),

inequalities of the

form

This research was supported in part by N.S.F. Contract GP 7041X at the University of Minnesota and in part by the Battelle Institute, Advanced Studies Center of Geneva, Switzerland. This support is gratefully acknowledged. Permanent address: School of Mathematics, Institute of Technology, University of Minnesota, Minneapolis, Minnesota 55455.

103 (I)

Jjfl(t,ul) - fl(t,u2)ll (kl~(t,w)

or

(2)

llf2(t,Ul,dUl/dt)

- f2(t,u2,du2/dt)ll

kl~(t,w) + k2~(t,dw/dt) where

w = uI - u2

and t

(3)

~2(t,w)~

I(w(t),M(t)w(t)) I

+

~

0

and where

kl, k 2

and

urable functions of

~

are nonnegative

locally bounded, meas-

t . In these papers a number of examples

were cited from the applications. applications

[ I(w(n),MCn)w(n)) tdn

There, as well as in the

found in [5], [6], [12],

it was not generally assumed that

[13] and [14], for example,

N(t) (or at least its symmetric

part) was definite, thus leading to the (generally)

ill posed

nature of these problems. In [I] and [2] the authors were concerned with abstract differential

inequalities du

-

of the form

T B(t)ul I ~< }(t){iiulj2 + ~ ~(n)jlujl2dn}½ t

(8) and

d2u d--~- B(t)u

Idul2 ~< ¢(t){ ~

+

(u,Bu)

}½

104

Id2u 2L(t)d~+ B(t)uJ

(6)

idt 2 +

du 2 + Q t ( u , u ) ~ I + v o21lull .< v(t~ll~ll

where

~' ~' Y' Yo

Although in [i] and

are nonnegative,

scalar functions of

t E [0,T)

[2] the authors were primarily interested

questions of unique continuation

in

for solutions to elliptic equations,

their results also yield uniqueness and stability theorems for certain initial-boundary equations.

value problems in partial differential

(Note that questions of uniqueness and stability for

solutions to Problems I and II can be reduced to the corresponding questions for the null solution to the corresponding abstract evolutionary differential

inequalities

provided the upper bound

on the right hand side is of the form given by the right hand side of (i) or (2)). In [9], Problem I was studied and the results compared to those for inequality amined,

(~) . Likewise,

sufficient conditions on

in [i0], Problem II was ex-

M(t) and

N(t)

given in order

to insure uniqueness and stability and the results then compared with those for

(8)

in [i]. It is the purpose of this paper to

examine Problem Ill o

P(t) d2u + M ( t ) ~du + N(t)u = f(t,u,du/dt) dt 2 u(0)

and

du/dt (0)

prescribed.

t e [0,T)

105

This paper is divided into four sections.

In the first section we show

how Problem III can sometimes be reduced to Problem I or Problem !!.

In the

second section we treat the linear version of Problem III (f ~ O) in some detail, giving sufficient, but fairly general, conditions on the operator families P(.), M(') and N(.) in order to insure uniqueness of solutions and continuous dependence on the initial data as well as continuous dependence on the value of the operator;

X(t)u~

Putt + Mu t + Nu.

There we also indicate

how the corresponding results can be obtained for the nonlinear version of Problem III.

In the third section we mention some applications of the theory

to the theory of partial differential equations citing some interesting examples from the literature.

We also give two extensions of the results of

the second section which are needed to prove uniqueness

in certain problems

to which the results of Section II do not directly apply.

Finally, in the

fourth section we compare our theorems to those of Agmon [i] and Lions

[11].*

We wish to emphasise here that while we are only concerned with Problem III from an abstract point of view in this paper, we have in mind application of these results to initial and initial-bom~dary value problems for equations and systems of equations where the coefficients of the partial differential operator may not be analytic (in fact C 1 smoothness is all that we shall ever require on the principle part of the operator by way of regularity);

where the coefficients of the principle part do not satisfy any

conditions with respect to "type" so that the equation (5) may be of mixed type or even of no type.

In equations with non-constant coefficients of the form

in the title very little is known with regard to uniqueness and stability (and even less with regard to existence) called coersivity conditions.

if all the operators do not satisfy so

See Lions [ii], for example.

* We enphasise that we are concerned only with the uniqueness question here. As Lions [ii] remarks, the hypotheses needed for existence in such problems are usually different from those needed for uniqueness.

106 In a future work we hope to give some wider applications of the results presented herein.

However, we shall give a few simple, but non-trivial,

examples of Problem III to which our results can be applied and which do occur in the applications. II.

SOME SPECIAL CASES

In this section we shall assume that f satisfies the following Lipschitz condition. (4)

(Here u,t z du/dt in the strong sense):

llf(t,Ul,Ul, t) - f(t,u2,u2,t)II $ lle(t,w) + 12~(t,w,t )

where iI, ~2 and ~ (below) are nonnegative, locally bounded functions defined on [O,T), where (5)

a(t,w) ~

I(w(t),P(t)w(t))l + u

I(w(n),p(n)w(n))Idn 0

and w = uI - u 2.

We impose the following hypotheses, valid for each te[O,T),

on the operators P(t), N(t) and M(t); P-I.

P(t) is symmetric and there is a constant I ~ 0 (independent of te[O,T)) such that (P(t)x,x) ~ l(x,x) for all xeD(t).

M-I.

There is a nonnegative measurable function y e ~ ~ ([O,T)) such loc that for all tE[O,T) and xcD(t), [IM(t)x]I 2 ~ y(t)(x,P(t)x)

.

These are all the conditions needed to reduce Problem Ill to Problem II. One simply defines (6)

fl(t,u,u,t) z f(t,u,u,t) - M(t)u .

Then one obtains, upon application of the triangle inequality

(7)

IIfl(t,Ul,Ul, t) -fl(t,u2,u2,t)lJ llf II + lJMwll

where w = u I - u 2 and f* = f(t,Ul,Ul, t) - f(t,u2,u2,t) .

107

Squaring

both

on the cross more taking

sides of

(7)~ using the

t e r m on the right square

roots

and then

same form as on the right

replaced

P .

The q u e s t i o n

to the c o r r e s p o n d i n g

assumption

of

Assume

.

(4) and

P-II

question

(5) as f o l l o w s

The h y p o t h e s i s

in a d d i t i o n

hand

M-I

21abl~< a 2 + b 2

and

P-I

follows

side of

of u n i q u e n e s s of s o l u t i o n s

be r e d u c e d

N-I

M-I

, we see that t h e r e

bound of the by

inequality

to P r o b l e m

P-I

holds for

an upper

(2) with

for P r o b l e m

: Let

and once

N(')

M

III can

I under

hold and

the

suppose

.

that

. (u~P(t)u)

is d i f f e r e n t i a b l e

and t h e r e

is a n o n n e g a t i v e

for all

function

u e C 2([0,T),H) k(t) e ~ l o c ( [ 0 , T ) )

such that

l~t(u,P(t)u) Suppose, which

u I(0)

(8)

Let

moreover,

= u 2(0)

fl(t,u,du/dt)

i

fl

that

uI

- 2Re(u,P(t)ut) I < k(t)(UlP(t)u)

and

and let

are

solutions

of P r o b l e m

III

w = u I - u 2 . Let

= f(t,u,du/dt)

= fl(t,ui,dui/dt),i

u2

•

: 1,2

- N(t)u

and

f*

= P(t)utt

+ M(t)u t .

be as before,

obtain

llfll - f~ll-< IIf*ll + llN(t)wll

t h e n we

108

Whence it follows from

(9)

where

N-I

and

(4),

(5) that for suitable

llPwtt + Mwtl I ~ ¥1~(t,w)

~(t,w)

¥i' ¥2

+ y2e(t,wt )

is given by (5) . Now let

(lO)

v(t)

: ==(t)

dt

so that P

t

w(t) = ] v(n)dn 0

where the integral

(ii)

(w(O)

=

o)

J

is taken in the strong sense. Then (9) becomes

IIPvt + Mvl[ < yl~(t,w)

+ y2~(t,v)

The reduction to the desired differential

inequality will be com-

plete if we can show that there is an appropriate E~Io c ,

~(t) ~ 0,

such that

(12)

~(t,w) ~< y(t)~(t,v)

To do this, in view of

P-i

and the definition

of

~ , it suffices

only to show that t (13)

(w,P(t)w) ~ y(t) I (v(n),P(q)v(n))dq 0

109

For then, t 0

t

I (w,Pw)dn ~ t.ess.sup y(s) I (v(n),P(n)v(n))dn O~s 0

[ t o , t l ] C_ [O,T)

F(t) .< L [ F ( t o ) ] l - v ( t )

and

[2], on

[9],

[0,T).

[i0]

among

Moreover,

to < t < tI

then

[F(tl)]v(t)

where 2 kl(tl-to) In L = (t o - t I) kle

(24) and

(25)

v(t)

=

e e

The content of (22) is of course function

of

e

-klt

We wish to consider,

-klt o -klt o

- e - e

-kit -klt I

that

in F(t)

in this

section,

- k2t2

is a c o n v e x

the f o l l o w i n g

problem:

114 Problem IIl L

~(t)u~

P(t)utt + M(t)u + N(t)u = S ( t ) u(0), du/dt(0)

Here ~ ( - ) i s

0~ t < T ,

prescribed.

a prescribed function

(vector valued) which one may

suppose simply to be the prescribed value of the operator, Since

~(t)

is a linear operator and since the initial conditions u

are linear, we may suppose that lutions

Ul,U2~C2([0,T);D)

assume that D(t)cH

~ (t)u.

to

~(t)ui

P(t), M(t) and N(t)

and that

H

is the difference of two so:

~i

where

space.

(This last restriction

is made simply to avoid making already cumbersome cumbersome by being forced to write

symmetric operator for each

We

are defined on a dense domain

is a real Hilbert

We shall assume that whenever

~ ~ ~i~2"

Re(u,Av)

v,w E CI([0,T);D)

expressions more

in place of and

A(t)

(u~Av)). is a

t , the scalar valued function

f(t) ~ (v(t),A(t)w(t))

is continuously

differentiable.

f'(t) - (vt,Aw) - (v,Aw t) [i] and

by

We shall denote the quantity (v(t),A(t)w(t))

[2] for sufficient conditions

existence of

on

A(.)

See, for example, to insure the

f'(t)

We assume that the following hypotheses are satisfied by the the operator families

P(.), M(.), N(.)

:

115

P-I.

For all tE~,T) and some I > O, P(t) is symmetric and (x,P(t)x) ~ l(x,x) for all xcD(t).

P-II.

For all uECI(~,T);D) I), (u(t),P(t)u(t)) is continuously differentiable and there is a nonnegative function k(t)£

f"Ioc[O,T)

such that

l(u,Pu)l $ k(t)(u,Pu). M-I.

For each t, M(t) = Ml(t) + M2(t) where Ml(t) is symmetric and M2(t) is skew symmetric.

M-II.

MI(.) satisfies P-I for some ~ B O.

M-Ill.

The function (U,MlU) is continuously differentiable for all u~Cl( ~,T);D) and there are nonnegative kl(.),k2(.)e~oc([O,T)) such that l(U,MlU)l $ kl(t)(U,Ml(t)u) + k2(t)(u,P(t)u).

M-IV.

For each tEIO,T) and each xeD(t) llM2(t)xl] 2 ~ kl(t)(X,Ml(t)x) + k2(t)(x,P(t)x) for some kl(.),k2(.)~loc[O, T ) which are nonnegative.

l) Here ueCI(Io,T);D) means that u : [O,T) b H is differentiable in the strong sense and that for each te[O,T), u(t),du(t)/dteD(t).

116

One of the following sets of hypotheses hold for the operator family N('): N-I.

For each t, N(t) is symmetric.

N-If.

There is a real constant 6 (possibly $ O) and nonnegative yl (-)~Y2 (-)E~oe([O,T))

such that (u,Nu) is continuously

differentiable and (u,Nu) B - 6(u,Nu) - Yl(t)(U,MlU) - Y2(t)(u,Pu). N'-I.

For each t, N(t) = Nl(t) + N2(t) where N2(t) is skew symmetric and

N'-II.

For all te[O,T), (~)

(X,Nl(t)x)

~ O for all xcD(t)

or for all t~[O,T), (B) N'-III.

(X,Nl(t)x) ~ 0 for all x~D(t)

NI(-) satisfies N-If above with 8 9 0 in ease N'-II(~) holds or with 6 ~ O

N'-IV.

if N'-II(B) holds.

For each t~[O,T) some constant y ~ O and all xeD(t) IIN2xll 2 ~ yI(x~NlX)I + ~l(t)(X,Ml(t)x) + Y2(t)(x,P(t)x) for some nonnegative yl(-),y2(-)~[oc([O,T))

117 Theorem K

I .

contained

and

k2

Under the in

preceeding

[0,T)2),there

hypotheses,

on every

exist nonnegative

compact

computable

set

R2,k I

such that t

t

F(t) - J (u,P(n)u)dn + I (t-n)(u,Mt(n)u)dn + R2 0

0

(27) + (T'-t)(u,Pu) ° + ½[(T') 2 - t2](U,MlU)o

satisfies and

k2

depend

imposed R2

an inequality

bounds Proof

only upon bounds

conditions

depends

upon the initial

Let

F(t)

shall show how theorem.

R2

(22).

data

while u(0)

in the preceeding

be defined

Here the constants

for the functions

on the operators

on the operators :

of the form

given

kI

in the

T' = sup{t ~ K} < T and

ut(0)

and

as well as the

hypotheses.

by (27) for any constant

R 2 . We

may be chosen to satisfy the conditions

of the

We have t

F'(t)

: (u,P(t)u)

- (u,Pu) ° + ~ (U,MlU)dn

- t(U,MlU) o

0

(28a)

t 0

t 0

n 0

2)The interval [O,T) is chosen without loss of generality. If the problem had been formulated on [to,t I) with initial data at t o , then, since the translation t---t-t o does not affect the operator hypotheses, it is equivalent to the same problem formulated on [0,tl-t o) with initial data prescribed at 0

118

t

t

F'(t) : 2 ] (u,Pun)dq 0 t

(28b)

+

I (u,Pu)dn 0

+ 2

(t-n)(un,MlU)dq 0

t

+ [ (t-q)(U,MlU)dn J 0

211 + Jl where

Jl

is the sum of the last two integrals

in (28b) and

I I ~ ½(F' - Jl ) . Thus we find that t

r"(t)

t

= 2 I (u.'?un)d~ + 2 I (t-n)(%'MzUn)d~ 0

t (29)

+ 2(u'PUt)o +

0

t

2 I (u'Punq)dn 0

+ 2 j (t-n)(U,MlUnn)dq

+ 2t(ut,MlU)o +

0

t dJl + -~-+ 2

I

[(%,~) + (t-~)(un,~lu)]d~

0

Letting

(30)

L(t)

denote the last integral

F2(inF)"

= FF"

on the right of (29), we have

(F') 2 = 4S 2 + 4FQ 2 + 2FE(t) + FJ L + FDI - 4IiJl - Jl2 + 2FL(t)

where we have set (31)

Q2 ~ R 2 + (T,_t)(u,Pu) ° + ½((T,)2_t2)(u,Mu)o

,

t (32)

E(t) e

0

I [(u,Pu

nq

) - (uq,PUn)]dn t +

I (t-n)[(U,MlUqn) 0

- (Un,MlUn)]dn

119

(33)

D I ~ (u,Put) ° + t(ut,MlU) °

,

and

t

t

(34) S 2 -[ I [(u,Pu)+ (t-n)(U,MlU)]dn I [(un 'Pu + n) O 0 t -I

(t-n)(un,MlUn)]dn 1

I [(u,Pun) + (t-n)(U,MlUn)]dnl 2 0

The remainder of the proof will consist in showing that all of the terms following

4S 2

on the right of (30) can be bounded

below by an expression of the form kl,k 2

and

k3

-klF2 - k2FS - k3FF'

are computable, nonnegative constants,

where

("Computable"

is defined below). This leads to, upon completion of squares,

(22)

FF" - (F') 2 ~ - klFF' - k2F2

where

kI

and

k2

are computable and nonnegative.

We record, at this point, some inequalities which will be useful in the sequel. We shall denote generic, computable, nonnegative constants by subscripted

k's and

l's

Here, "computable" means

that the constants are known functions of the parameters appearing in the hypotheses on the operator families as well as any coefficients used in applications of the arithmetic-geometric equality

and (possibly)

T'

mean in-

Constants which depend upon these

120 computable constants and the initial data or the value of the operator

(~)

will be denoted by subscripted

d's .

We first observe that t Ir '(t)l .< (u,?(tlu) o + t(U,MlU) o + (u,P(t)u) + I (U,MlU)dn 0 However, from (28a) the sum of the last two tarms on the right is simply the sum of the first two and

(35)

so that

IF'(t)l 4 2d I + F'(t)

(Here

d I = (u,P(t)u) ° + T'(U,MlU) ° ~ 0)

Now we know that, from (34)

,

t

t

[ I [(u,Pu)+

(t-q)(U,MlU)]dq

0

I [(un, Pu ) + n

(t-n)(un'MlUq)]dnl½

0

2)½

=

(S 2 +

II

:

[s 2 + ¼(r'

- J1)2] ½

~ s + ½1r'l Now, a p p l y i n g that It

F'(t)

IJll

therefore

Hypotheses

< k l ( F - R 2) follows

for after

P-I,

+ ½1Jll

P-II,

M-i, M-II

some c o m p u t a b l e , using

(35)

that

and

positive, for

M-III

we s e e

constant

some n o n n e g a t i v e

k 1.

121

computable constants

( 0

(36)

( max

ki

t

k5 > 0 ,

t

(u,Pu)dq f (un,Pun)dn) ~ 0 t t (u,Pu)dn 0

0 t 0

with

f

!

( t - n ) ( u n 'Mlun)dn t

1~ z

(uq,Pun)dq I (t-n)(U,MlU)dnl 2 0 t t

(I ~t ~ u , ~ 0

0

I ~t ~

klF+k2F'+k3S + k4dl-k5R2

,~o~oI ~

Now we go to work on the right hand side of (30), term by term. Since [Jl ] ~ kI(F-R2)

Z klF

we have 0 ~ Jl2 ~ k~(F_R2)2 ~ klF2 - k2 FR2 as

0 .< F - R 2 % F . Now t

t

! I ~= f (u,Pu)d~ + I (t-n)(U,MlUn)dn 0

0

Therefore

t t f [(u,Pun)]dn + f (t-n)[(U,MlUn)~n

fill 0

0

122

Whence t

t

II

(38)

0

O

t

f

t

0

0

klF + k2F'

using

(36) and

constants

(37)

+ k3S + k4d I - ks R2

. Therefore,

k i, i = 1,..,4

there are computable

with

Jl2 + 1411Jl I $ klF2

(39)

i_

+

k4 > 0

nonnegative

such that

k2FS + k3FF , + k4Fd

_ ksFR2

Now t Jl' = (u,Pu)

+

•

so that

from

P-I, P-II

and

0

I

M-II

(U,MlU)dn we have

t

t

0

0

4 klF + k2F' + kad 1 Also,

it is easily We therefore

(38),

(39) and

seen that

IL(t)l

4

have the following

lll(t)l 3) estimate

(40) and use the facts that

if we combine

ILl 4

IIii

(37),

and that

123 d 2 ~ [(u,PUt)ol + T'](u t,M1u)oj is an upper bound for

DI

and is only data

:

(41) FJ~ + FD I - 411J 1 - Jl2 + 2FL >~F(13R2 where the i

k's

and

l's

lld 1

-

are n o n n e g a t i v e

12d 2) - klF2 computable

-

k2FS - k3FF'

constants

and

> 0 .

3

We now try to find a lower bound on the term form

- klF2 - k2FS - k3FF'

by o b s e r v i n g

FE(t)

of the

+ DF where D is a data term.

We begin

that

t (42)

E(t)

--- - I [(U,MlUn) 0

+ (u'M2un)

t

+ (u,Nu)

+ (Un,PUn)]dn

t

+ I (t-n)(U'MlUnT~)dn0

t

I (t-n)(u I ,MlUn)dn n 0

t

(U,MlUn)dn = 0

( t - n ) "~n (U'MlUn)dn + t ( u ' M l U t ) o 0

t

t

= I (t-n)(u'Mlunn)dn ÷ I (t-n)(un't~lun)dn 0

0

t . + I (U,MlUn)dn + t ( u ' M u t ) ° O

(u, ~ ) d n 0

However, t

+

•

124 It follows

that t

(43)

E(t)

: -

] [(u,Nu) 0

+ 2(t-n)(un,MlU n) + (un~Pur)]dn

t

t

t

+ I (u~M2uq)dn + I (u~'~)dq - f (U,MlUr)dn + D3 0

where

D3

the right

of (43) as follows

f (~.M2u.)~"

0

We treat

is a data term.

t

0

0

the last three

integrals

on

:

t

>._ I ll.',l tlM2..lld. 0 t

(.4)

Ilull

~--

[kl(Un,MlUq)~

+

k2(un,Pun)~]dn

(M-IV)

0

t >.- k I

t (u,Pu)~(u q , M l U n ) ~ d q - k

0

t

2 0

t

0

0

(u,Pu)~(u q ,Pu n )~dn !

f t

t

-k2[ I (u,Pu)dq I (un,PUn)dn] 0

>. -

f

(4S)

klF

-

t

k2S

-

Z3£'

+ k 4 d 1 - kSR2

O

[(36)]

t

I ]lull tl o~IId n

(u~ ~ )dn >~ -

0

8 t >"

h 0

t

[ (u.Pu)d, - k 2 J[ (~. ~)d,

J

>" - klF - k2d4

0

T'

" (d4 =- I ( ~ ' ~ ) d q ) 0

.

125

(6)

0

I t (u't~lUn)dn .< f t [(u'l~lUn)]dn 0 _1 .< kl [ it (U,MlU)dh

0

]t ( Un,MlUn)dn]2

0 ! 2

+

3)

(u,Pu)d n

0 0 k l r + k2r' + k3S + k5d 5 - k4R2

1(36)I.

(Here again, all the k's are generic, nonnegative computable constants.) Thus, for some constants k i with k i ~ O, we have t

E(t) ~ 0

I

l(un'Pun ) + 2(t-n)(un'Mlun) + (u'Nu)Idn

- klF - k2F' - k3S + k4R 2 - k5d 5 where

3) Here we have used the following fact which may be found in [I0] and [15]. If A is a symmetric operator on an inner product space V and if B is a positive definite operator on V (x,Bx) V > 0 unless x = O) such that

I(Ax,x)vt ~ [(x,B×)vl for all xeV then I(Ax,Y)v 12 ~ (x,Bx)v(Y,BY) V for all x,yeV.

126

(48)

d 6 = ll(U,Pu) ° + 12(ut,Put) ° + 13(U,MlU) ° + 14(ut,Mut) ° T !

+ 151(u'Nu)ol

16 I lJ ll2d

+

0

and where

the l's are nonnegative,

computable

constants.

Now we define t

(49)

G(t) 0 Assume

Lemma

:

G(t)

and

d5

the truth

the h y p o t h e s e s

constants

(5o)

where

(u ,Pu ) + 2 ( t - n ) ( u n , M , u n) + (u,Nu)]dn q n

for the moment

Under

nonnegative

I[

k.

i

of the theorem,

there

are c o m p u t a b l e

such that

{ klF + k2S + k3F'

is of the form given

in

- k4R2

+ ksd 8

(48) with the

l's n o n n e g a t i v e

computable. Combining

Lemma

(41),

(44),

(45),

(46) and the result

(50) of the

we obtain

F2(InF) '' ~ 482 _ klF2

for

of the f o l l o w i n g

some n o n n e g a t i v e

We now choose

R2

- k2FS

computable

k.z

to be this

last

- k3FF'

and a d5

+ k4(R2-d 5)

d5 and

of the form complete

the

(48). square

127

in

4S 2 - k2FS

satisfied by

to obtain an inequality of the form (22) which is F(t)

with

R2

of the form (48).

We now prove (50). We have t

t

2 ] (t-n)(un,Punn + Mun + Nu)dn : 2 I (t-n)(un, ~ ) d n 0

.

0

Which yields, after an integration by parts on the ].eft hand side, t G(t) : 2

t

(t-n)(un, ~ ) d n f 0

-

t

(t-n)(un,Pun)dn(t-n)(u,Nu)dn ] j 0 0

- t(ut,Put) ° - t(u,Nu) °

For the remainder of the argument we shall assume that satisfies

N-I

seen to hold if

and

N(-)

N-il . The corresponding argument is easily

N(')

satisfies the second set of conditions.

Using these conditions as well as Schwarz's inequality and P-!I

we find that t

(51a)

P-I ,

t

G(t) "< klu I ( t)- n ) ( u nd' P un) d nn + k2- I (t-n)(U'Ml 0 0 t t + 6 ] (t-n)(u,Nu)dn + k 3 I (t-n)(u,Pu)dn + k4d S 0

0

(51b) t

t

G(t) .< k I ] (t-n)(un'Pun)dn + ~ I (t-n)(u,Nu)dn 0 0

+ k2F + k3d 5

I28 where again the while

d5

inequality,

k's

are generic, computable and nonnegative

is of the form (47). We may take

kI > O

in this last

if it is not already positive.

We also have t 0

t

f (t-n)(un,Puq)dn

= 0

t

f (t-n)

(un,Pu)dn 0

t

f (t-n)(u,Punn)dn

0

t

t

= I (t-n)(u,Mtun)dn 0

f (t-n)(un,Pu)dn •

+ I (t-n)(u'M2un)dn 0

t

t

+ I (t-n)(u,Nu)dn - I (t-q)(u, ~ ) d n 0

0

t 0

f

(t-n)(u ,Pu)dn+ a data term of the n • form (~8).

In the second of these equations we integrate by parts in the first integral and then use the various hypotheses on

P(-)

and

M(')

to obtain an inequality of the following form : ¢t

t

t

t

I (t_n)(u ,pun)d n 0

0

0

t

(S2)

0

t

t 0

0

!

t

t 0

+ k4[ I (U,MlU)dn I (un,MlUn)dn] ½ + 0

0

a data term of the form (48).

129

For any integral

k m kI

in (51b),

where

kI

and for any

is the coefficient ~ e (½,1]

we have

t

(53)

of the first

t

G(t) .< (l-~)k I (t-q)(u [ 'Puq)dn n

+ ~k

0

(t-n)(uq'PUn)dn O

t

f ( t - n ) ( u , N u ) d q + k2F + k3d 5 .

+ 6 0

We now choose Using

k

so large that

(52) to estimate

the positive

the

definiteness

k ~ kI

second of

MI

P

and

integral

0 < (l-~)k < ~k + 6 ~ r.

on the right

of (53) and

we obtain,

t

G(t) .< r I G(~)d~

+ l(t)

0 where

l(t)

is a nonnegative

increasing

function

of

t . Thus,

after a quadrature,

(54)

G(t) $ ertl(t)

Therefore, linear F(t)

taking

combination

into consideration

. . . . Corollary Problem

i .

computable

Ill L with

and

and

. (It is a of (52) and

(36) we obtain

(54)

Vl,V 2 E C 2 ([0,T);D) ~(t)v.1 = ~ ( t )

l(t)

on the right

coefficients)

(50) of the Lemma from Let

.

the form of

of the last five terms

with nonnegative,

the statement

4 erT'l(t)

be solutions

vi(O)

on

= Uo, vi,t(0)

[O,T) to = Ul " If

130 the o p e r a t o r theorem Proof

coefficients

then

:

Thus,

vl(t)

Let

F(t)

satisfy

= v2(t)

the h y p o t h e s e s

for all

u = v I - v 2 . Then ~(t)u = 0 as g i v e n

by

(27) and

(48)

t F(t)

and

satisfies

F(0)

an i n e q u a l i t y

of the form

F(t)

2 .

solutions

to P r o b l e m

F. John

= 0

so that

using the p o s i t i v e

Coro!lar~

, u(t)

Proof

:

By (23) we

with

to = 0

= 0

or

and

I (u,Pu)dq + I

subsets

to

~ O

in this

and

of e i t h e r

= v2(t)

stable

case,

(U,MlU) P(t)

- O.

or

M(t)

. theorem,

the

in the sense of

0 4 t ~ T' - ~ < T' < T

we have,

t (t-q)(U'MlU)dn

"< L [ F ( 0 ) ] I - 6 E F ( T ' ) ] 6

0

u = vI - v 2

is the d i f f e r e n c e

blems

~(t)vi=

in

:

that

= 0 .

[0,T)

of

for

Since,

of the p r e c e e d i n g

where

(25).

= ut(0)

t I : T'

t

0

(u,Pu)

vl(t)

iII L are H S l d e r

see that

(22).

definiteness

Under the h y p o t h e s e s

[4] on c o m p a c t

u(0)

+ I (t-n)(u,Mlu)dn O

t c [0,T)

and

(R 2 =- d 5) r e d u c e s

8

for each

.

t

= I (u~Pu)dn

= O , we have

Therefore,

t e [0,T)

of the p r e v i o u s

-

Now

~i and

suppose

[0,T) vi(O),

0 < 6 ~ v(T'-~) , u(0)

and

of two vi,t(0) < i

ut(0)

solutions

prescribed.

where are

to the pro-

~

small

Here

is g i v e n in the

by

sense

131

T

0

I ll ~

l]2dn, (u,Pu) o , (u t ,Put) o , (U,MlU) o , (u t ,MlUt) o , I (u,NU)o 1

are all small. Then as

F(0) = T'(u,Pu) ° + ½T'2(u,MIU)o

where

R2

is of the form (48), we see that

(48) as well. Thus, if the above quantities prescribed and, for

(ss)

E

2

, then, F(0) ~ k~

t e [0,T'-p]

we have

t

t

2

F(0)

is of the form

are all less than some

for some computable

I (u,Pu)dn + I (t-q)(U,MlU)dn-< 0

+ R2

constant

k

Ke2-26[F(T')] ~

O

where T' F(T')

T'

= I (u,Pu)dn + I (T'-n)(U,MlU)dn 0

Thus if we consider bounded class,

+ R2

0

only those solutions which lay in some a priori

say T

c~

= {vg C2([0,T),D)~

I (v,Pv) + (T-n)(V,MlV)dn

< M 2}

0

where we have

M

is an a priori given constant.

Then for

t E [0,T'

p] ,

132 t

(s6)

t

J (u,Pu)dn

+

0 Thus

demonstrated

H61der

stability

in the

sense

of

[4].

The c o r r e s p o n d i n g

($6)

~(t)u

nonlinear

= Putt

is easily we have,

treated for

u = u

provided

i

prescribed

we assume

some n o n n e g a t i v e

llf(t,ul,utl)

problem

+ Mu t + Nu = f(t,u,u t) +

u(0),ut(0)

where

.< KH 28

0

we have

F. John

(t-n)(U,MlU)dn

that

2

ul,u 2

C 2 ([0,T);n)

li,~ i

_ f(t,u 2,ut2)I]2 4 lla(t,u)

- u

for

+ i2~(t,ut ) + X3B(t,u)

and t

~(t,u)

= (u,Pu)

+ ~i I (u,Pu)dn 0 t

B(t,u)

Let, ui

for

i = 1,2,

~ (0), u (0)

the p r o b l e m

and

ui ~i

= (U'MlU)

be a solution " Let

(U,MlU)dn

+ ~2 j 0

u = u

I

to - u

(56) c o r r e s p o n d i n g 2

and note that

to u

solves

133

(57)

~(t)u

= f* + ~ *

; u(0), ut(0) prescribed

where

f* = f(t,ul,u%)

- f(t,u2,u~)

But this can be viewed as just the linear problem with an inhomogeneous term, which namely

~ ~*

~ f* + ~ *

consisting

of two pieces, one of

, we treated as data in the preeeeding theorem.

Now inspection of the proof of the last theorem value of the operator

, ~,

entered into our

t

terms of the form t I

0

(t-~)Cu, ~ ) d n

J

shows that the

calculations

only as

t

(u,~)dn 0

f

(t-n)(u

n

,~)dn

and

as we see from (43) and the equations preceding

0 (51) and by

(52) respectively. f* , we can, using

form as in (51) and

If in these three integrals we replace (36), obtain inequalities

of the same

(52). We then complete our proofs as before.

The details are omitted.

IV.

As a first,

SOME EXAMPLES

simple example, we consider the following

boundary value problem for the equation

initial

134

~(t)U

:

i

~

~2U (~,t) -

i

0

such that

x eD(t)

N(-), N-I

such

t e [0,t)

JJM2(t)xJJ2 ~ kl(t)l(X,Ml(t)x)l Moreover,

and

+ k2(t)(x,P(t)x)

.

holds while the inequality

in

by

(u,N(t)u) > - 6 e pt (u,N(t)u) -Yl(t)J(U,MlU) j -Y2(t)(u,Pu).

Then this initial value problem has at most one solution. Proof

:

The proof of the theorem

~ ( t ) u : P(t)(utt Introducing ~*(r)u*

is very easy. We note that

- pu t ) + [M(t) + pP(t)]u t + N(t)u =

the change of variables, ~ ~(t)u

=

P*(T)U*

TT

~ = e pt

+ M*(T)U*

T

~(t)

we have,

+ N*(T)U*

=

$*(

T)

.

138

where P*(~)

z e2ptp(t)

M*(~)

~ ePt[M(t)

N*(~)

~ N(t)

+ pP(t)]

and u*(~) The initial

value problem

~*(T)U*

becomes,

: ~*

u*(1), We

shall

M*(.)

have our uniqueness

and

N*(.)

we have for

satisfy

x E D(t)

and

(x,P*(~)x) Using

the fact that

d/dT

(u*,P*(~)u*)

z u(t)

.

in this notation

on

u*(1) T

~,e pT)

prescribed

result

.

if we can show that

the hypotheses

of Theorem

P*(.),

I. For

x = e pt = e2pt(x,P(t)x) = (9~)-i d/dt

= p-l~(u,P(t)u)

> X(x,x)

.

we find + 2T--I(u*,P*(T)U *)

so that [(u*,P:'~(T)u*)[

< p -i T k(t)(u,P(t)u)

~. .,. + 2~-I(u",P"(~)u *)

k*(T)(u*,P*(T)U*) where k*(T) is locally

bounded

on

~ e-Pt(k(t)/p + 2)

[l,epT)

.

P*

,

139 For

we write

M*

M (t) = e

pt

M2(t)

M* : M1 +M 2

so that

M-I

(x,M~()x)

satisfies

(58)

MI(~) =ePt(Ml(t) +pP(t)) ,

holds. Since for

x ~D(t)

,

: ePt( x,[Ml(t) + pP(t)]x) >.

NI

where

~(x,x)

H-III . We observe

I (X,Ml(t)x)I ~< (x, [Ml(t) + pP(t)]x) + p(x,P(t)x) ~< e

Thus, for

~ = e pt

and

-pt

-' (x,MI(T)X)

+ pe -2

pt

(x,P*(T)X)

.

x ~D(t)

IIM2(~)x~ 2 = e2ptIIM2(t)xII2 ~< kl(T)(x,M

(T)x) + k (T)(x,P*(T)X)

where kl(T) = kl(t)ePt k2(~) - kl(t)p + k2(t) are both locally bounded and nonnegative (u'e,Ml(~)u*) = T-I(u*,M (T)U*) + " "

Whence, from

M'-!II

so that

M-IV

P-l(u,Ml )

holds. Now,

+ (u,Pu)

and (58)

I(u*,M (~)u*)I ~< kl(~)(u ,Nl(~)u ) + k2(~)(u ,P ( ~ ) u ) where kl(~) - e-Pt(l+ kl(t)/p) k2(~) -= e-2pt(kl(t)+ k(t) + k2(t)/p)

140

are both locally bounded andnonnegative, M*(.)

M-III

holds for

as well.

Finally, we must check

N-I

holds. We have, from

N"-II

N-I

so that

(u*,N*(T)U*)

and

N-Ii

for

N*(.)

. Clearly

and (58),

: p-l~-l(u,N(t)u)

-(6/p)(u*,N*(~)u*) - Yl(t)J(U,Ml(t)u) j - Y2(t)(u,P(t)u) >

-6"(u

,N (~)u*) - y (~)(u*,M~(~)U")

- Y2(U ,P"(~)u*) where 6* ~ ~/p

(constant)

* yl(~) ~ Yl(t

)e-Pt

* y2(~) ~ [pyl(t) + Y2(t)]e-2pt so that Remark : N'-I

N*(.)

satisfies

N-I

and

N-II.

There is an analogous result if the family

through

N(.)

satisfies

N'-IV .

Let us now return to our example. We see that

(S9)

I f(x)(Mlf)(x) dx = I a i j f , i f , j d x -

½ I aij,ij(x)f2(x) dx

where we have employed the summation convention as well as the notation f'i ~ sf/~x. . Let l

-~

be a uniform negative lower bound

141 for the in

~

smallest and

f2(x)

v'

eigenvalue

be a p o s i t i v e

in the i n t e g r a n d

(f,Mlf) Thus w i t h

of the m a t r i x upper

of the

>-~(f,Pf)

bound

second

A(x)

= (a..(x)) i]

for the c o e f f i c i e n t

integral

- ~'(f,f)

p = ~ + ~'/~ + i and

family

in

(59).

of

Then

>-(~

+ ~'/~)(f,Pf)

.

~= i , M ' - I I

is s a t i s f i e d

by

MI

M - I I I , M1 = 0 . S i n c e , f o r some n o n n e g a t i v e computable

As for

constants

k.

, we have

LIM2fl12

=

f (aij" ,j" f ' i

+ ½ a .ij . ,i~. f ) 2 d x

fl ~ kl I f , i f , i d x

k3 I f ' i f ' i d x

so that

M-IV

well as

N-I

is fulfilled.

The f a m i l y

if we note that

N = 0

In the e q u a t i o n that

MI

"bounded

and below"

P

Putt

had the by

P

= k3(f'Pf)

N(.)

and

satisfies

let

N-II

we had a s s u m e d

same

was at least

P

sign or that

MI

was n o n n e g a t i v e .

w h e r e this

as

6 = ~i = Y2 = 0 .

+ (M I + M 2 ) u t + Nu : 0

when

that we have an e q u a t i o n

+ k2 I f2dx

Suppose,

is not the case,

however,

as for e x a m p l e

utt + Uxx t + Nu = 0 where

N

M = d2/dx 2 much more

is some d i f f e r e n t i a l and

P : I

difficult

operator.

are of o p p o s i t e

but n e v e r t h e l e s s ,

Here we have that sign.

some

Then the

statement

situation

can be made.

is

142

Theorem

III.

following

Let

~(t~ e P ( t ) u t t

hypotheses

u ~ 0

We assume,

= 0 ,

u(O)

as a s o l u t i o n first

of all,

and that the o p e r a t o r

P

[O,T) D(t)

M"-I.

The C a u c h y dX(t) dt

of

P

Ml(t)

),

and

t

so that

>

0

> ~(x,x)

is d e n s e

I

in

I

H

~ M(t)X(t)

= I

X(-)

operators

: [O,T)

on

it p e r m u t e s

in the

strong

> B(H)

H ) such that

with

(the for each

contained

M(t)

in

. (Here

exist

in the

satisfies

M-III

. The d i f f e r e n t i a t i o n

for all

: M(t)v(t)

t ~ [O,T)

D X'(t)

sense.)

Ml(t)

d(M(t)v(t))/dt holds

is a c o n s t a n t

_ p-~M(t)P-~X(t)

such that

M(t)

t

is one to one w i t h r a n g e

is t a k e n M"-II.

of

of

problem

linear

t~ X(t) and

is i n d e p e n d e n t

of a s o l u t i o n

bounded

independent

x sD

X(O)

admits

= D

and t h e r e

(Px,x) The r a n g e

= 0

be fulfilled.

for all

P"-II.

= ut(O)

P(t)

is s y m m e t r i c

such that

in

that

family

P = 0 . Let the f o l l o w i n g P"-I.

. T h e n under the

the p r o b l e m

~(t)u has o n l y

+ M(t)u t + N(t)u

and

strong

sense and

+ M(t)vt(t) v e CI([o,T);D)

formula

143

M"-!II •

For each

t,M(t)

= -Ml(t)

skew symmetric

and

MI

for

and

xe D

t e [0,T)

M"-IV. We suppose, and

N-II

stringent

M2

satisfies

N'-I

C-I.

permutes

symmetric,

of this

we can,

be self adjoint. since

P > 11

where

and satisfies,

.

N'-IV

hypothesis

P

, the bounded and

N(t)

is very easy.

Since

somewhat

p-i

operator

P

P-½

and

Theoremjtake

it to

P½

is defined

self adjoint operators. Introducing the change ! ~ P2u(t) , the original equation becomes 1

1

1

M(t)

to verify that the operator ~ p-½M(t)P -½

satisfy the preceding

hypotheses

N(t) for

of variables,

1

vtt + P-~M(t)P-~v t + P-~N(t)P-~v It is not difficult

and

both exist and are

bounded v(t)

III below.)

is semibounded

theorem,

and

X(t)

. (See Remark

Extension

by the Spectral ~ > 0,

N-I

satisfies

will also be needed:

t e [0,T) with

N(-) -M(.)

. The following

by the Friedrichs

Thus,

is

> ~(x,x)

that the family

For each

The proof

M-IV

through

commutivity

M2

~ > 0 .

in addition or

where

is symmetric

(X,Ml(t)x) for some

+ M2(t)

= 0 . families

= p-½N(t)P -½

P = I . We thus only need to

144

consider

P = i . Writing

the case

M

and

N

for

M

and

and

letting

(60)

w(t)

we o b t a i n

~ X(t)v(t)

successively wt(t)

= X'(t)v(t)

+ X(t)vt(t)

= X(t){vt(t)

wtt(t)

+ M(t)v(t)}

= X'(t) [vt(t)

+ M(t)v(t)]

+ X(t)[vtt(t) Thus,

from

(61)

M"-I

with

wtt(t)

Therefore,

P = I

- M(t)wt(t)

from C-I and

(62)

wtt

with initial

data

(61)

w(0)

middle

Ml(t)

It t h e r e f o r e satisfy

Remark

I.

an o p e r a t o r

and

that,

Let the H i l b e r t

(62)

that

the

now,

= 0 is of the symmetric

semidefinite

by

since the c o e f f i c i e n t s

v ~ 0

I, we have and h e n c e

space

satisfying

+ M(t)v(t)}

that

- M(t))w

is p o s i t i v e

H

M"-I

.

, we have

= 0 . Now

of T h e o r e m

we o b t a i n

family

it f o l l o w s

except

follows

the h y p o t h e s e s

is one to one,

C-I

+ M(t)v(t)]

= X(t){-N(t)v(t)

= wt(0)

equation

is

and

- M ( t ) w t + (N(t)

as our o r i g i n a l term

+ M(t)vt(t)

same f o r m part of the M"-Ii

of

(62)

w ~ 0 . Since u z 0.

be complex.

.

X(t)

Q.E.D.

The e x i s t e n c e

is g u a r a n t e e d

by T h e o r e m s

of 3.1

145

and

3.2 of F r i e d m a n

[3] p r o v i d e d

I

P - ~ M ( t ) P -~ ~ M(t)

(ii)

(-~,0]

(Here

III

IN

lll[N(t)-

M"-III

is the o p e r a t o r

self a d j o i n t

of

t

for any

z

so that

(-®,O]

we have for

with

M

as well

e

and

e

extension

extension

norm and

or

R(z,M) =~I-R)-I).

(O < ~ < !)

are i n d e p e n d e n t M(t)

of

t,

is s y m m e t r i c

for

(which we know exists

Theorem

~53)

(which we take to be

Im z ~ 0

z = -p + in

of

with

l((-zI+M)x,x)1:1[+p(x,x)

is d e f i n e d

for each

each

in-

D ) then we k n o w

M(t)

t, s).

from

has d o m a i n

Re z 4 0, R ( z , M ( t ) )

is in the r e s o l v e n t and

for each

Re z 4 0)

~ elt-TI ~

I > 0 . Now if

and F r i e d r i c h ' s

dependent

M(t)

(z complex,

M(~)]M-I(s)III

hold w i t h

and if the

c Izl+m


.0 ,

therefore

Now Lattes and Lions seem to affirm (incorrectly) that

I

le.~A 2

-Ill .