Contemporary Developments in Continuum Mechanics and Partial Differential Equations 1977: International Symposium Proceedings

C'ON T E M P O R A R Y D E V E L O P M E N T S IN CONTINUUM MECHANICS A N D PARTIAL DIFFERENTIAL EQUATIONS

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N0 RTH-H0 LLAND MATHEMATICS STUDIES

30

Contemporary Developments in Continuum Mechanics and Partial Differential Equations Proceedings of the International Symposium o n Continuum Mechanics and Partial Differential Eyuations Rio de Janeiro, August 1977 Edited b y

GUILHERME M . DE LA PENHA Universidade Federal do Rio de Janeiro and Financiadora de Estudos e Projetos (FINEP/SEPLAN) and

LU lZ A D A U T O J. M E D E I R O S Universidade Federal do Rio de Janeiro

1978

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM - NEW YORK- OXFORD

0

North-Holland Puhlishing Compan,~ 1978

All rights reserved. No part of this publication may he reproduced, stored in a retrieval s.vstem, or transmitted in any form or h.v any means, electronic, mechanical, photocopying, recording or otherwise, without the prior pcrmission of the copvright owner.

ISBN 0 444 85166 6

PUBLISHERS

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM. NEW YORK,OXFORD

SOLE D I S T R I B U T O R S F O R T H E U . S . A . A N D C A N A D A :

ELSEVIER/ N O R T H - H O L L A N D , INC. 5 2 V A N D E R B l L T A V E N U E . NEW Y O R K . N.Y. 10017

Library of Congress Cataloging in Publication Data

International Symposium on Continuum Mechanics a d Partial Differential Equations, Rio de Janeiro, Brazil, 1977. Contemporary developments in continuum mechanics and partial differential equations. (North-Holland mathematics studies ; 30) Includes index. 1. Continuum mechanics--Congresses. 2. Differential equations, Partial--Congresses. I. Penha, Guilherme M. de la. 11. Medeiros, Luis Adauto da Justa. 111. Title.

~~808.2.1585 1977 ISBN 0-444-851666

531

78-5270

P R I N T E D I N THE N E T H E R L A N D S

FOREWORD

An I n t e r n a t i o n a l Symposium on Continuum Mechanics and P a r t i a l D i f f e r e n t i a l E q u a t i o n s was h e l d a t t h e I n s t i t u t o de Matemztica, U n i v e r s i d a d e F e d e r a l do Rio d e . J a n e i r o , Rio de J a n e i r o , R . J , 1977.

Brasil, from August 1 - 5 ,

T h i s meeting had an o b j e c t i v e o f a c q u a i n t i n g m a t h e m a t i c i a n s and

o t h e r s c i e n t i s t s i n p h y s i c s and e n g i n e e r i n g , w i t h t h e p r i n c i p a l l i n e s o f modern r e s e a r c h i n such i n t e r r e l a t e d s u b j e c t s so t o a f f o r d s t i m u l a t i n g d i s c u s s i o n s l e a d i n g t o t h e i n v e s t i g a t i o n o f new problems o r r e v i v i n g o l d o n e s s t i l l open.

Another aim o f t h e Symposium was t o a t t e m p t t o a s s e s s

what t h e r e c e n t r a p i d growth o f i n t e r a c t i o n s between t h e two mathematical s u b j e c t s h a s accomplished and may accomplish i n t h e n e a r f u t u r e . Wide-spread a c t i v e i n t e r e s t was shown i n t h e Symposium, and twelve European, f i f t e e n North-American and n i n e t y - o n e B r a z i l i a n s c i e n t i s t s registered, papers. by J . L .

The c h i e f a c t i v i t y o f t h e c o n f e r e n c e was t h e p r e s e n t a t i o n o f

Two s h o r t c o u r s e s were g i v e n , Lions.

one by M . E .

G u r t i n and t h e o t h e r

C . T r u e s d e l l p r e s e n t e d a c r i t i c a l a p p r a i s a l o f continuum

therniomechanics and t h e k i n e t i c t h e o r y o f g a s e s , a l l o f t h e s e b e i n g reproduced h e r e .

S e v e r a l p a r t i c i p a n t s c o n t r i b u t e d p a p e r s , and s h o r t

communications were p r e s e n t e d .

Duplicated a b s t r a c t s of presented papers

were a v a i l a b l e a t t h e meeting and a l m o s t a l l p a p e r s were followed by a q u e s t i o n and d i s c u s s i o n p e r i o d .

I t i s n a t u r a l l y n o t p o s s i b l e t o reproduce

a l l t h e s e d i s c u s s i o n s , b u t most of t h e p a p e r s which were p r e s e n t e d are included here.

Nearly a l l t h e p a p e r s p r i n t e d were r e t y p e d from t h e o r i g i n a l s

s u p p l i e d by t h e a u t h o r s and s u b s e q u e n t l y e d i t e d .

To speed t h e a v a i l a b i l i t y

o f t h i s volume, t h e f i n a l typed v e r s i o n was n o t s u b m i t t e d t o t h e r e v i s i o n of t h e authors, thus t h e e d i t o r s bear a l l t h e r e s p o n s i b i l i t y f o r anything t h a t may n o t p l e a s e t h e a u t h o r s o r r e a d e r s o f t h e c o n t e n t s o f t h i s volume, I t i s hoped t h a t t h e s e p r o c e e d i n g s w i l l s e r v e b o t h t h o s e who a t t e n d e d t h e Symposium and t h o s e u n a b l e t o a t t e n d , and t h a t t h i s permanent r e c o r d w i l l add t o t h e s t i m u l a t i o n and d i a l o g u e g e n e r a t e d a t t h e m e e t i n g . We s h a l l be most p l e a s e d i f , b e s i d e s t h i s , it w i l l h e l p t o some e x t e n t i n e n c o u r a g i n g m u t u a l l y advantageous c o n t a c t s between t h e p r o d u c e r s and u s e r s o f mathematics.

Many t h a n k s a r e due t o t h e B r a z i l i a n a g e n c i e s f o r r e s e a r c ] ] ,

F i n a n c i a d o r a de Estudos e P r o j e t o s (FINEP) and Conselho Nacional de Desenvolvimento C i e n t f f i c o e Tecnol6gico (CNPq) f o r t h e i r generous f i n a n c i a l V

vi

FOREWORD

s u p p o r t which made t h e Symposium p o s s i b l e .

The Universidade F e d e r a l do

Rio de J a n e i r o , t h r o u g h t h e Centro de C i e n c i a s Matematicas e da Natureza and t h e I n s t i t u t o de Matematica p r o v i d e d f a c i l i t i e s for t h e c o n f e r e n c e , and many p e o p l e o f t h e s e two u n i v e r s i t y i n s t i t u t i o n s gave f r e e l y o f t h e i r h e l p . I t remains t o e x p r e s s a p p r e c i a t i o n t o t h o s e p e r s o n s who have h e l p e d i n v a r i o u s ways t o b r i n g t h e s e p r o c e e d i n g s i n t o b e i n g :

.

Our c o l l e a g u e s on t h e O r g a n i z a t i o n Commission, Gustavo P e r l a Menzala, Luiz C a r l o s M a r t i n s and Rubens Sampaio,

.

The Symposium s p e a k e r s f o r t h e i r c o o p e r a t i o n i n s u b m i t t i n g manuscripts (mostly i n t i m e ) ,

.

Wilson Goes for t h e n e a t n e s s o f t h e t y p i n g ,

.

t o t h e p u b l i s h e r s , North-Holland,

and i n t h e person o f D r . E . Fredriksson.

Rio d e J a n e i r o , December 1977 G.M.

d e La Penha

L.A.

Medeiros

CONTENTS

Foreword

S.S. Antman

V

A family of semi-inverse problems of nonlinear elasticity

1

G.S.S. Avila and D.G. Costa Asymptotic properties of general symmetric hyperbolic systems

25

J.B. Baillon and J.M. Chadam The Cauchy problem for the coupled SchroedingerKlein-Gordon equations

37

T.B. Benjamin

H. Brezis

Applications of generic bifurcation theory in fluid mechanics

45

The Hamilton-Jacobi-Bellman equation for two operators via variational inequalities

74

Nonlinear problems related to the Thomas-Fermi equation

81

H. Brezis

F. Cardoso and J . Hounie Global solvability and hypoellipticity of abstract complexes and equations

90

Michael O'Carroll On the inverse scattering problem for linear evolution equations

102

Nonlinear bifurcation problem and buckling of an elastic plate subjected to unilateral conditions in its plane

112

B.D. Coleman

On the thermodynamics of non-classical systems

135

R.L. Fosdick

On an inequality in thermodynamic stability

143

Claude Do

R. Glowinski and 0. Pironneau Least square solution of non linear problems in fluid dynamics

171

P. Podio Guidugli Elastic bodies in a Signorini-type environment

225

M.E. Gurtin

On the nonlinear theory of elasticity

237

D.D. Joseph

Constitutive equations and free surfaces

254

J.L. Lions

On some questions in boundary value problems of mathematical physics

284

vii

viii

R.C. Maccamy

CONTENTS Memory effects in one-dimensional problems of continuum mechanics

34 7

A general framework for problems in the statics of finite elasticity

363

P. Nowosad

Elliptic metrics on Lorentz manifolds

388

S. Osher

Boundary value problems for equations of mixed type

396

J.P. Puel

A free boundary, nonlinear eigenvalue problem

400

James Serrin

The concepts of thermodynamics

411

W.A. Strauss

The nonlinear Schrddinger equation

452

Walter No11

G. Svetlichny and P. Otterson Derivative dependent infinitesimal deformations of differentiable functions L. Tartar

466

Non linear constitutive relations and homogenization

472

R. Temam

Qualitative properties of Navier-Stokes equations

485

G. Truesdell

Some challenges offered to analysis by rational thermodynamics

495

Some simplified equations from the theory of mixtures

604

W.O. Williams Author Index

613

G.M. de La Penha, L.A. Medeiros ( e d s . ) Contemporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations Worth-Holland Publishing Company (1978)

A FAMILY O F SEM I-INVERSE

PROBLEMS OF N O N L I N E A R ELASTICITY

STUART

s.

ANTMAN*

D e p a r t m e n t o f Ma t,herna t i c s and I n s t i t u t e f o r P h y s i c a l S c i e n c e and T e c h n o l o g y U n i v e r s i t y of Maryland,

C o l l e g e P a r k , Maryland

20742

f ' o r t h e development of

the

1. I n t r o d u c t-i _on Much o f t h e i m p e t u s

theory

o f n o n l i n e a r e l a s t i c i t y f o l l o w i n e ; t h e Second World War was s u p p l i e d by the s t u d y of

c o n c r e t e problems

that

exhibit

i n t e r e s t i n g e f f e c t s due t o n o n l i n e a r m a t e r i a l r e s p o n s e . the study of

i n v e r s e p r o b l e m s R i v l i n f s work

[

111 was

In

paramoiint.

The n a t u r e o f s u c h p r o b l e m s was e x p o s e d by E r i c k s e n l s p r o o f

61 t h a t t h e o n l y d e f o r m a t i o n s p o s s i b l e i n e v e r y homogeneous ( c o m p r e s s i b l e ) e l a s t i c m a t e r i a l a t r e s t u n d e r z e r o body f o r c e a r e a f f i n e and by h i s s t u d y

[51

p o s s i b l e i n e v e r y homogeneous

of n o n - a f f i n e

deformations

incompressible e l a s t i c material

a t r e s t u n d e r z e r o body f o r c e .

(Ericksenls analysis

spawned a s m a l l l i t e r a t u r e d e v o t e d

[51

t o those aspects o f

the

problem t h a t h e l e f t u n r e s o l v e d . )

Semi-inverse

problems,

studied i n t h e 1950's, lead t o

?+

The r e s e a r c h r e p o r t e d h e r e was s u p p o r t e d b y N a t i o n a l S c i e n c e Foundation Grant

MCS77-03760.

2

STUART S.

a r i c h e r c l a s s of d e f o r m a t i o n s , by q u a s i l i n e a r s y s t e m s of (Cf.

G r e e n & Adkins [

for references.)

7,

Most

of solutions, A notable

many of w h i c h a r e d e s c r i b e d

ordinary d i f f e r e n t i a l equations.

Ch 111 a n d T r u e s d c l l & N o 1 1 [ t r e a t m e n t s of

y i e l d i n g a number o f r e s u l t s avoided the questions

ANTMAN

of

t h e s e problems,

o€ p h y s i c a l

interest,

1 3 , Sec.591 while

either

e x i s t e n c e and q u a l i t a t i v e b e h a v i o r

or e l s e t r e a t e d them o n l y f o r s p e c i a l m a t e r i a l s . to this i s the work

exception

radial oscillations elastic material.

of

of Knowles

a c y l i n d r i c a l t u b e of

181 on t h e

incompressible

The manner i n which Knowles employed

constitutive inequalities i s the natural precursor of

the

method u s e d h e r e . I n t h i s p r e s e n t p a p e r we examine a f a m i l y o f s e m i i n v e r s e problems

f o r compressible e l a s t i c m a t e r i a l s s a t i s f y -

ing the strong e L l i p t i c i t y condition.

( F o r s i m p l i c i t y we

assume t h a t t h e m a t e r i a l i s homogeneous and i s o t r o p i c . ) By f u l l y e x p l o i t i n g t h e s t r o n g e l l i p t i c i t y c o n d i t i o n w e r e a d i l y show t h a t

" r e a s o n a b l e " s e m i - i n v e r s e boundary v a l u e problems

a l w a y s h a v e s o l u t i o n s a n d t h a t a number of

their qualitative

f e a t u r e s can be determined. N o-n . - o t a t i.

E',

V e c t o r s , which a r e h e r e d e f i n e d t o b e e l e m e n t s o f

and v e c t o r - v a l u e d

l e t t e r s over t i l d e s .

f u n c t i o n s a r e denoted by lower-case S _e _cond-order

t e n s o r s , which a r e t a k e n

~

t o be elements o f

L ( E 3 ,(E 3) ,

and f u n c t i o n s w i t h v a l u e s

in

L(E3,E3),

a r e d e n o t e d by u p p e r c a s e l e t t e r s o v e r t i l d e s .

The s u b s e t

o f second-order

i s denoted tensors

L+(E3,S3)

i s denoted

tensors with p o s i t i v e determinant

and s u b s p a c e o f symmetric second-order

S(E3,E3).

The d _ o. . t product

of

2

and

b

SEMI- I N V E R S E PROBLEMS O F NONLINEAR ELASTICITY

a.!. -A*

is written The a d j o i n t

-

all

a,?.

-

A

of

-

B,

and

- --

= A.(B-A).

(ab)-c

=

(b*c)a 2

d i f f e r e n t i a l o f t h e mapping

-

i s denoted

t h e mapping

- -

U - I - z F- ( U-)

3.

-2

[af(a)/ay]

- --

b . ( A * a ) = a*(;*.!)

= bu . A- .

The p r o d u c t of

(fi-n)-,a = ah --

The dyad

2.

for all

k+,f(u)

3

at

We s e t

a (fl,.

a (f19f2,f3)

= A

a(Av+Bw,u 2

is

the

The FrGchet

i n the direction

i n the direction

-

B

i s denoted

.. , f n ) ,11

n

3(fl

a

afi

= d e t (-),

) I

a U .

3

f2

9

f3)

(v,u2,u3)

a ( f l 3 f -~ *,f3)

+ B-

a(w,u2,u3) *

The E q u a t i o n s o f E l a s t o s t a t i c s .

' (21,i2,i-3) =

Let

domain

n

in

E3

( i1 ,,I.z ,& 3 )

B3.

orthonormal b a s i s f o r that

( -i , J , E )

I

he a f i x e d

We i d e n t i f y a body ~ i t h the

i t occupies i n i t s r e f e r e n c e

c o n f i g u r a t i o n and w e i d e n t i f y a m a t e r i a l p a r t i c l e

-z

with i t s p o s i t i o n

- z a I. -

-a

-z

for

and t h e F r G c h e t d i f f e r e n t i a l o f

at

a (u,, . . .

2.

Asa.

D i a g o n a l l y r e p e a t e d i n d i c e s a r e summed f r o m

[aF(A)/$U]:B*. 1 to

A*?.

- -

is written

i s d e f i n e d by

As?,

= trace

A::

t e n s o r d e f i n e d by

h

- -

A*.b

-

denoted

-

We s e t

-a

at

i s d e f i n e d by

We a c c o r d i n g l y w r i t e

-A

tensors

A

The v a l u e of

3

= xi + y j

-

-

+

i n this configuration.

-

zk.

Let

~ ( 5 )h e

i n a deformed c o n f i g u r a t i o n .

The d e f o r m a t i o n

F

of

We s e t

t h e p o s i t i o n of

t h e body

-z

particle

W e set

preserves orientation i f

=

and o n l y i f

4

STUART S. ANTMAN

Let at

5.

z(z)

be the first Piola-Kirchhoff stress tensor

Then the equilibrium equations f o r a body subject to

zero body force are

(2.3)

Div

n

The material of

I

-

ia.-

a

aza

AT*= 0,

-

-

is h_omogeneously elastic if there are

T: L + ( E 3 , E 3 )

functions

?-

4

L(!E3,E3)

2:

and

S(IE3,E3)

+ S(E3,E3)

such that

We assume that

T

and

5

are continuously differentiable.

This representation ensures Kirchhoff stress tensor.

_ _ _ _ _ _ . _ _ _ _ ~ -

-S

only if

where

-

I

(2.4).

,S

is the second Piola-

The material is isotropic if and

has the form

is the identity on

E3

and where

depend on the principal invariants of

c.

ao, a l , a 2

Assuming that

T

is continuously differentiable, we require it t o satisfy the strong ellipticity condition

Below we shall impose specific conditions ensuring that large stresses accompany large strains and strains for which det is small.

F

SEMI-INVERSE PROBLEMS OF NONLINEAR ELASTICITY

5

3. Formulation of the Semi-Inverse Problem. We consider deformations defined by

p(z) =

(3.1)

f(x)_el(z)

+

[h(x)+C~+Dz]k,

where

--

(3.2) ~ ~ ( =2 C)O S e(z)i

e(.)

+ sin e ( ? ) i ,

(This is ”Family 1 ” of [ 13, Sec. 591 . )

= g(x)

+ AY + BZ.

O u r problem will be

to determine the existence and properties o f functions and numbers

A, 8 , C, D

f,g,h

such that (3.1) satisfies the

equations (2.3) and (2.5) and certain subsidiary conditions. We set

(3.3)

-

e ( z ) = &~e_~(z), e = k , -2 -3

c?g

x

Denoting derivatives with respect to

where

f’el + fg’e -2

+

h‘

(3.5)

/

f’

0

0

(3.6)

Condition (2.2) reduces to

(3.7)

(AD-BC)ff’ > 0 .

e,c .

by primes, we have

(3.4)

aE2 ax =

=

,e3,

6

ANTMAN

STUART S.

(3.7) as

By making o b v i o u s s i g n c o n v e n t i o n s , we may i n t e r p r e t being e q u i v a l e n t t o t h e requirement be p o s i t i v e .

l i-a i-' j

C

The components o f

that

(3.7)

e a c h f a c t o r of

with r e s p e c t t o t h e b a s i s

are

+

,/(f')2

(cap) =

(3.8)

I\

+

(fg')2

(h')2

A f 2g ' + C h '

Bf 2 g ' + D h ' \

A2f2

2 ABf +CD

+C2

'> 'a

,/

B2f2+D2 /

T

We decompose

as

-

T = ~a L - eL- ia

(3.9) s o that

( 2 . 3 ) and

( 2 . 5 ) imply t h a t

(3.10)

Relations

on

G

( 3 . 5 ) and ( 3 . 8 ) t h u s i m p l y t h a t

- -

{ F a = e L . F . -a i ]

(3.12)

In p a r t i c u l a r , that

h',

g',

f , A,

B,

C,

D.

--

. + * i a ]a r e i n d e p e n d e n t

of

y

and

(3.10) has the componential f o r m 1

(3.13)

(3.14)

From

depend o n l y

and t h e r e f o r e o n l y o n

f',

(:&

(Tta]

(T2')'

+

-

g'Tll

AT2

+

( 3 . 1 5 ) we o b t a i n

(3.16)

T~~ = H ( c o n s t ) .

2

-

AT12

BT2

2

-

+ BT13

BT2'

= 0,

= 0,

z

so

SEMI-INVERSE

Among o t h e r r e l a t i o n s

f[ g ' T l l

(3.17)

implies that

(2.14)

+

+ BT13] =

ATI'

t h e s u b s t i t u t i o n o f which i n t o

D

f'T2

1

y

produces

the integral

(const).

I

Without loss of

C,

(3.14)

T, 1 = G

f

(3.18)

domain

7

PROBLEMS O F NONLINEAR E L A S T I C I T Y

R

i s the unit

t h e reference

g e n e r a l i t y we s u p p o s e t h a t cube

( 0 ~ ) ~ W e .s u p p o s e t h a t AD-BC

are prescribed with

>

0.

I n Section

7

By

A,

we d i s c u s s

how t o r e l a t e t h e s e c o n s t a n t s t o v a r i o u s r e s u l t a n t s a c t i n g over t he m a t e r i a l f a c e s o f any t r a c t i o n s p r e s c r i b e d with the i n t e g r a l s

H

and

(3.16), (3.18).

of

a non-zero

W e accordingly regard

corresponds

d e a d s h e a r l o a d s on t h e f a c e s

p r e s c r i p t i o n of

be c o m p a t i b l e G

e n t e r i n t o a d i s c u s s i o n of

€1

the prescription of

x = O,1

( 3 . 1 6 ) and (3.18), b u t m e r e l y n o t e

c o n d i t i o n s compatible b i t h that

the faces

011

W e do not

as g i v e n .

W e a l s o require that

t h e cube.

G

t o the p r e s c r i p t i o n

x = 0 , 1 , whereas

the

c o r r e s p o n d s n e i t h e r t o a dead

l o a d n o r t o a c o n s t a n t Cauchy s t r e s s .

The __ g o -. v_ e -r.n i n g e q u a t i o n s f o r o u r problem c o n s i s t of

(3.13), ( 3 . 1 6 ) laws

4.

(3.18),

which i n c o r p o r a t e

the s t r e s s - s t r a i n

(2.5).

Conseauences o f C o n s t i t u t i v e R e s t r i c t i o n s .

\\ie now t r a n s f o r m o u r e q u a t i o n s f u r t h e r by b r i n g i n g the strong e l l i p t i c i t y condition choosing

-a

-

= i

in

(2.7) we get

( 2 . 7 ) t o b e a r on them.

By

8

ANTMAN

STUART S .

b = b'

for

e

-L

f u n c t i o n of' Note t h a t

2.

Thus

T *-i

i s a s t r i c t l y monotone

f o r f i x e d v a l u e s of i t s o t h e r a r g u m e n t s .

ag/ax

t h e remarks p r e c e d i n g ( 3 . 1 2 )

i n d e p e n d e n t of g, y,

#

t h e b a s i s u s e d h e r e and t h e r e b y i n d e p e n d e n t of

I n consonance with

Z .

(TLa] are

imply t h a t

(4.1),

we impose t h e growth

conditions

(4.2)

TZ1

4

as

*m

for f i x e d values

of

fg'

+

+m,

T

+

3

t h e o t h e r arguments.

r e s t r i c t t h e arguments o f

-

T

as

fa

h'

+

fm,

Here and below we

t o b e d e f o r m a t i o n s of

t h e form

( 3 . 1 ) ; we t h e r e b y a v o i d t h e i n t e r e s t i n g q u e s t i o n o f p o s i n g r e a l i s t i c growth c o n d i t i o n s f o r e l a s t i c m a t e r i a l s u n d e r a r b i t r a r y deformation.

( T h i s g e n e r a l q u e s t i o n might b e hand-

l e d by combining i d e a s of B a l l Brezis

[4]

31 . ) The m o n o t o n i c i t y c o n d i t i o n

conditions

(4.2)

ensuring t h a t

(4.1)

and t h e growth

j u s t i f y a g l o b a l i m p l i c i t f u n c t i o n theorem

( 3 . 1 6 ) and (3.18), r e g a r d e d as a l g e b r a i c

e q u a t i o n s , can b e uniquely of

w i t h t h o s e of Antman &

t h e o t h e r v a r i a b l e s of

solved

for

fg'

t h e problem.

and

h'

i n terms

W e represent these

s o l u t i o n s by

(4.3)

(4.4)

fg' h'

= y(f',

G/f,

= n(f', G/f,

A,

B,

C, D ) ,

€1, A ,

B,

C,

H,

A l o c a l i m p l i c i t f u n c t i o n theorem e n s u r e s t h a t a r e c o n t i n u o u s l y d i f f e r e n t i a b l e because

T

is.

D).

y

and

q

SEMI - I N V E R S E

We

=ubstitute

a u t on ornous,

9

PROBLEMS O F N O N L I N E A R E L A S T I C I T Y

( b . 3 ) , ( 4 . 4 ) i n t o (3.13) t o get the

q u a s i 1i n e a r ,

s e c ontf

- ord e r

ord inary differentia 1

equation

h = (A,B,C,D,G,H)

where

T2' + A T 2 2 + BTz3 F21

-

[P(f',f,X)I'

(4.5)

and

F31

a ( f ' , f , h ) = 0,

aiid \ % h e r e p

evaluated

at

and

The m o n o t o n i c i t y c o n d i t i o n system

(3.l3),

semi-monotone

(fT2 (for

1

)

1

are

3

of

(3.6)

by d e f i n i t i o n o f

with

for

ensures t h a t the f,

g, h

i s formally

~t c o m e s a s n o s u r p r i s e t h a t

( 4 . 5 ) i s i t s e l f a f o r m a l l y semi-monotone e q u a t i o n f o r Indeed,

and

T1

('t.h).

(4.1)

(3.75)

= 0,

f > 0).

4,

{F a]

the

(4.3)

r e p l a c e d by

and

y, q

and

p,

f.

we f i n d

(4.6)

w h i c h i s s t r i c t l y p o s i t i v e by

a l s o compute

(4.7)

(4.8)

(4.1).

I n a l i k e manner we

10

STUART S.

ANTMAN

(4.9)

The s t r o n g e l l i p t i c i t y c o n d i t i o n i s i n c a p a b l e that the first

t e r m on t h e r i g h t s i d e d o f

s o t h e mapping monotone.

(p(f’,f,k),

(f’ ,f)-

of ensuring

(4.9) i s positive,

~ ( f, f‘ , x ) )

need n o t be

( 4 . 5 ) need n o t be f o r m a l l y

Therefore the equation

i t i s l i k e l y t h a t b o u n d a r y v a l u e problems

monotone

and h e n c e

for ( 4 . 5 )

f a i l t o have unique s o l u t i o n s .

T o e x a m i n e t h i s and r e l a t e d q u e s t i o n s f u r t h e r we G = €1 = 0.

c o n s i d e r t h e s p e c i a l c a s e i n which 1

(2.6) imply t h a t

1

= T3

T2

=

F

when

t h e monotonicity c o n d i t i o n ( k . 1 ) alone, that

2

Now ( 2 . 5 ) and

= F31 = 0.

without

Thus

( 4 . 2 ) , implies

( 3 . 1 6 ) and ( 3 . 1 8 ) h a v e t h e u n i q u e s o l u t i o n = 0,

g’

(4.10)

h’

= 0;

t h i s i n t u r n i m p l i e s t h a t t h e d e f o r m e d c o n f i g u r a t i o n s of t h e faces

y = O,l,

z = 0,1

(2.6)

imply t h a t

(4.11) so that (4.12)

T2

1

= T3

a r e planes.

1

= T1

2

I n t h i s case

= TI3 = 0

(3.13) o r ( 4 . 5 ) r e d u c e s t o p’-a

E

w h e r e t h e a r g u m e n t s of

-

(Tll)’

T1

1

,

T2

AT2 2

2

,

-

T2’

BT23 = 0

are

(2.5) and

SEMI-INVERSE

PRODLEMS O F NONLINEAR E L A S T I C I T Y

O

0

Af

BY

C

D

11

‘.

I n t h i s case

1

-Pa

(4.14)

af7 =-I

a T1

aFll

(4.15)

(4.16)

2

as

--

(4.17)

af

T+AB---aF 2

-

(4.1)

Inequality but

A

2

(p(-,-,h),

a T2 +

aF22

n o w implies that U(-,-,h))

3 AB -+ B

ap/af‘

>

2 aT2 __ 2 ‘ aF

and

0

as/af

> 0,

may n e v e r t h e l e s s f a i l t o be

monotone.

5 . E x i s t e n c e Theory f o r Boundary Value P r o b l e m s . W e impose b o u n d a r y c o n d i t i o n s

1 T1

(5.1 a,b)

I x=o

- qo

or

f(0) = fo

More g e n e r a l c o n d i t i o n s a r e p o s s i b l e . that

A,

y = 0,1,

B, z

C,

D

= 0,1

are prescribed, can e n s u r e t h a t

moment o n t h e body v a n i s h when

> O,

S i n c e we a r e a s s u m i n g

the reactions

on t h e f a c e s

t h e r e s u l t a n t f o r c e and

( 5 . l a ) and ( 5 . 2 a ) a r e prescribed.

STUART S. ANTMAN

12

There are several effective ways to attack the boundary value problem (4.5), ( 5 . 1 ) ,

(5.2).

The first method

is to give it a weak formulation i n a reflexive Sobolev space, the choice o f which is dictated by sharper growth conditions that must be imposed. In this setting the problem can be cast as an equation involving a pseudo-monotone operator.

The difficulty here

lies in the treatment of the strict inequality (3.7) and the

T

growth o f

where

(3.7) is small.

of Antman [2] and Antman & Brezis

Methods similar to those

133

can be used to handle

this difficulty and to yield a full regularity theory.

(The

latter work describes a useful set of growth conditions in Section

4.) This approach shows that there is a weak solution for

each

h

satisfying (3.7) and that each such weak solution is

classical.

Instead of carrying out the details of this theory,

we turn to another that provides somewhat more information about classical solutions. We impose the growth condition

(5.3) for fixed values of

f

and

1.

This condition and the positivity of

(4.6) imply that the

algebraic equation

(5.4)

P

has a unique solution for

(5.5)

(f’,f

1) = 9

f‘, which we denote by

f’ = cp

s,f,X).

13

SEMI-INVERSE PROBLEMS O F N O N L I N E A R ELASTICITY

W e c a n r e w r i t e t h e boundary v a l u e problem

(4.5), ( 5 . 1 ) ,

(5.2)

a s t h e system

(5.9 a , b )

q(l) =

q1

or

f ( l ) = fl.

T o b e s p e c i f i c and t o a v o i d minor problems w i t h Neumann conditions w e r e s t r i c t o u r attention t o conditions (5.8a),

(5.9b). f

>

0

where

a

Let

( T h e s e c o n d i t i o n s do n o t r e n d e r t h e r e q u i r e m e n t of

( 3 . 7 ) c o m p l e t e l y i n n o c u o u s a s would ( 5 . 8 b ) . )

@

i s a c o m p l e t e l y c o n t i n u o u s mapping f r o m

reduces

a

=

;(p),

0 E IR,

a

so that

A

(5.12a)

to

A

If

ExA i n t o E.

be c o n f i n e d t o l i e o n a s i m p l e smooth c u r v e i n

g i v e n p a r a m e t r i c a l l y by

Then

i s continuously d i f f e r e n t i a b l e ,

then

Y

i s completely

14

STUART S. ANTMAN

continuous.

W e assume t h i s .

F o r s u c h problems we h a v e R a b i n o w i t z [ 101

Theorem ( L e r a y & S c h a u d e r [ 9 ] ,

cf.

(uo,e0)

(5.12b).

b e a s o l u t i o n p a i r of

u = Y(u,po)

equation

t h e component of

E x

Suppose t h a t t h e

t h e s e t of

i s e i t h e r unbounded i n

To e x p l o i t t h i s

of

Then

s o l u t i o n p a i r s that E x

and i n

[po,m)

o r e l s e a p p r o a c h e s t h e boundary of

(-m,po]

(u0,$,)

t h e c l o s u r e of

(uo,p0)

contains

Let

as i t s u n i q u e s o l u t i o n .

uo

has

-

)

these s e t s .

theorem we must f i r s t f i n d a s o l u t i o n p a i r One way t o g e t t h i s i s t o u s e t h e weak

(5.12b).

But i t i s more i l l u m i n a t i n g t o

f o r m u l a t i o n mentioned a b o v e .

u s e a d i f f e r e n t approach m o r e i n keeping with t h e continuation methods d e v e l o p e d by R a b i n o w i t z [ 103

.

I n m o s t problems t h e n a t u r a l c h o i c e f o r t r i v i a l s o l u t i o n , which would c o r r e s p o n d s t a t e f o r o u r e l a s t i c i t y problem. s t a t e i s n o t i n t h e form

(uo,Bo)

is a

t o the reference

Unfortunately the t r i v i a l

( 3 . 1 ) of a d m i s s i b l e s o l u t i o n s ;

i s rather a singular l i m i t of

such s o l u t i o n s .

it

The continuation

of s o l u t i o n s from t h i s t r i v a l s o l u t i o n i s o f t e n termed " b i f u r c a t i o n from i n f i n i t y " .

We c a n u s e t h e p h y s i c a l

structure

of o u r problem t o r e n d e r t h i s t e c h n i c a l d i f f i c u l t y i n n o c u o u s . W e s t a r t t h i s p r o c e s s by c o n s i d e r i n g t h e problem i n which

B = C = G = H = 0, case (3.6)

D = 1.

(Cf.

(4.10)-(4.17).)

In this

i s d i a g o n a l and i s t h e s q u a r e r o o t of t h e correspond-

i n g v e r s i o n of

(3.8).

L e t u s assume t h a t t h e r e f e r e n c e s t a t e

is a natural stress-free

state, i.e.,

:(I)

=

2.

We assume

that

(5.13)

-

H:

-

-

aT(I)/aF : H > 0 -

I

V diagonal

I;! f

2.

(This i s a milder r e s t r i c t i o n linear elasticity.) then says t h a t

F‘s

c l a s s of

t h a n t h a t commonly made i n

The c l a s s i c a l i m p l i c i t f u n c t i o n t h e o r e m

the constitutive

relations f o r our special

a r e e q u i v a l e n t t o e q u a t i o n s o f t h e form

(5.14)

f’

1 = V ( T ~,

(5.15)

Af

= V(T1

where

(P,v)

15

PROBLEMS OF NONLINEAR E L A S T I C I T Y

SEMI-INVERSE

i s monotone,

1

,

AT,^), AT2

w(O,O)

2

),

= 1,

v(0,O)

= 1.

Note t h a t t h e c u b i c a l r e f e r e n c e s h a p e i s a t t a i n e d i n t h e l i m i t

+

that

A + 0,

while

( 4 . 12) g i v e s

f

Af

+

1.

(Tll)’

-

m ,

(5.17) The s y s t e m ( 5 . 1 6 ) ,

From ( 5 . 1 4 ) ,

AT2‘

(5.17) replaces

(5.15)

we get

= 0. Equation (5.16)

(4.12).

i s a compatibility condition.

Now t h e b o u n d a r y c o n d i t i o n s (5.18)

T1 1( 0 ) = q 0 ,

(5.8a), ( 5 . 9 b ) become v ( T 1 1(I), A T 2 2 ( 1 ) ) = Afl.

In v i e w of o u r m o n o t o n i c i t y c o n d i t i o n s , t h e u n i q u e s o l u t i o n of

(5.16)-(5.18)

I f we v a r y

when

(A,Afl,qO)

9,

= 0,

Afl

from (O,l,O)

= 1

is

T1

along a curve i n

t h e n a s t a n d a r d i m p l i c i t f u n c t i o n theorem e n s u r e s (5.16)-(5.18) enough t o

= T 2 2 = 0. R3,

that

has a unique s o l u t i o n f o r t h e s e parameters n e a r

(0,1,0). B y o u r c o n s t r u c t i o n ,

such a s o l u t i o n

would c o r r e s p o n d t o a d e f o r m a t i o n i n w h i c h t h e f a c e s

x = 0,l

l i e on c o n c e n t r i c c i r c u l a r c y l i n d e r s o f f i n i t e r a d i u s . A n y such s o l u t i o n i s a s o l u t i o n of

(5.10), (5.11) f o r

16

STUART S .

the se parameters.

L e t one s u c h s o l u t i o n p a i r be d e n o t e d

A condition sufficient

(uo,ao).

u = Y(u,@,), that

ANTMAN

a. = & ( P o ) ,

where

t h e mapping

t o ensure that

the equation

h a s a t m o s t one s o l u t i o n i s

1 2 1 1 2 ( F 1,F * ) A T 1 ( F 1 ,O , O , 0 , F ~,0 , O , O , D )

2 1 2 T2 ( F l , O , O , O , F 2,0,0,0,D)

a.

o u r c o n s t r u c t i o n of

be s t r i c t l y monotone.

,

(Note t h a t

F

e n t a i l s t h a t t h e components o f

have t h e form i n d i c a t e d i n t h e a r g u m e n t s o f

TI1

and

T2

2

.

T h i s r e s t r i c t e d m o n o t o n i c i t y c o n d i t i o n i s i m p l i e d by t h e

[ I 3 1 b u t n o t by t h e s t r o n g e l l i p t i c i t y

Coleman-No11

inequality

condition.)

The c o n t i n u a t i o n theorem o f Leray & S c h a u d e r

a p p l i e s t o parameters

a =

a^(@)

a

confined t o c u r v e s of

t h e form

with The c o n t i n u a t i o n method o f Leray & S c h a u d e r i m p l i e s

that solutions of

(5.10), ( 5 . 1 1 ) a r e c l a s s i c a l u n t i l the

continuum of s o l u t i o n p a i r s becomes unbounded o r e l s e approaches everywhere.

~ ( E x A ) . Such a c l a s s i c a l s o l u t i o n s a t i s f i e s

(3.7)

One c a n g e t a somewhat s t r o n g e r r e s u l t by look-

ing f o r solutions

(5.10),

( 5 . 1 ~ )i n a s m a l l e r c l a s s of

f u n c t i o n s , say L i p s c h i t z continuous f u n c t i o n s .

By s t r e n g -

t h e n i n g o u r growth c o n d i t i o n s o n t h e c o n s t i t u t i v e f u n c t i o n s we c a n show t h a t any- L i p s c h i t z c o n t i n u o u s s o l u t i o n must satisfy

( 3 . 7 ) everywhere provided t h i s

with t h e c h o i c e of

a.

The p r o o f

i s not

incompatible

r e l i e s on t h e o b s e r v a t i o n

t h a t a L i p s c h i t z c o n t i n u o u s f u n c t i o n whose r e c i p r o c a l i s i n t e g r a b l e o n a n i n t e r v a l c a n n o t v a n i s h on t h a t i n t e r v a l . (Cf.

C11.)

SEMI - I N V E R S E

PROBLEMS O F N O N L I N E A R E L A S T I C I T Y

17

6. Qualitative Behavior o f Solutions.

Since

(4.5)

o r the equivalent system ( 5 . 6 ) ,

(5.7) is

autonomous, we can readily determine the qualitative behavior of all solutions of these equations by studying their phase-

plane trajectories.

For simplicity we fix the parameter

at a value for which

aJ/af

> 0

(cf. ( 4 . 1 0 ) - ( 4 . 1 7 ) )

1

and we

assume that

(6.1)

D ( f / , f , ~ )

as

+

f +

{+om].

Then the algebraic equation

o(f',f,h) =

(6.2)

0

has a unique solution

We sketch the curve defined by (6.3) in Fig. 1.

Figure 1 We are now ready to s t u d y the

(5.6),

(5.7).

j-rnplies that

(f,q)

phase-plane diagram o f

W e first note that our construction o f

f'

cp

> 0. Thus there are no singularities f o r

18

Y

STUART S. ANTMAN

>

0.

left f

of

Moreover

(5.7)

t h e image o f

i s t o the right.

says t h a t

the curve

q‘

f =

< 0 (f’

when

,x)

the rurve ( 6 . 2 )

and

i s to the

q’

> 0

when

U s i n g t h e s e i d e a s we s e e t h a t t h e p h a s e -

p l a i i e d i a g r a m h a s t h e c h a r a c t e r shown i n F i g . of

f

or

2.

The image

( 6 . 3 ) , which i s g i v e n by all t h e

(f,q)

satisfying

(6.4a) or

(6.’kb) i s i n d i c a t e d b y t h e dashed l i n e . ‘1

= o

Figure 2

SEMI-INVERSE PROBLEMS O F WONLI'JEAR

Suppose t h a t

t h e boundary ( - 0 n c 1 i t i o n s a r e

c a n d i d a t e s for t h e s o l u t i o n s of a r e the t r a j e c t o r i e s

123, 2 3 , h i 6 ,

a segment o f u n i t

( J. 8 a ) , ( 5.9a)

. Then

t h i s b o u n d a r y v a l u e problem

c a n d i d a t e would be a s o l u t i o n i f traverses

19

ELASTICITY

5 6 , c t c . of F i g . 2 .

A

the independent v a r i a b l e

length as the point

x

(f,y)

traverses the indicaked t r a j e c t o r y . (From t h e pseudo-monotone

operator analysis described i n the

l a s t s e c t i o n w e know t h a t t h e r e i s a t l e a s t one s o l u t i o n f o r

X .)

each

Fig.

2 t e l l s us that

different solutions that

1 2 3 , has t h e s t r e s s

q

there a r e t w o q u a l i t a t i v e l y

arc possible: d e c r e a s e from

The f i r s t , of

x = 0

t o an i n t e r i o r

x = 1.

minimum and t h e n i n c r e a s e t o i t s v a l u e a t of

the f o r m 23, has the s t r e s s increase with

7.

F u r t h e r R e s u l t s and Comments.

t h e form

The second,

x.

-

A l l our r e s u l t s a r e v a l i d f o r non-homogeneous, materials

aeolotropic

with t h e property t h a t t h e r e s u l t i n g equations a r e still

ordinary d i f f e r e n t i a l equations with

independent variable x .

S u i t a b l e a e o l o t r o p i c m a t e r i a l s would even y i e l d autonomous e q u a t i o n s ( c f . [ 7 ] ) . I n place o f prescribing some r e s u l t a n t

A,

B,

C,

D

w e c o u l d prescribe

f o r c e s and moments on t h e f a c e s

y = 0,1, z=O,l.

Then t h e u n s p e c i f i e d c o n s t a n t s a r e t o be d e t e r m i n e d f r o m a system o f e q u a t i o n s o f

(7.1)

t h e form

'I

T ( f ' , f , h ) d x = const.

By u s ng t h e s t r o n g e l l i p t i c i t y c o n d i t i o n or

-

k

( 2 . 7 ) with

2 = J

we o b t a i n a m o n o t o r i i c i t y c o n d i t i o n , which w h e n coupLd

STUART S. ANTMAN

20

with mild growth conditions, enables us to solve appropriate parameters as functionals of ing parameters.

f

(7.1) for

and the remain-

These functionals may be substituted into

(4.5) to convert it into a functional-differential equation. A preliminary analysis indicates that the resulting equakion

together with reasonable boundary conditions generates both a pseudo-monotone operator equation on a suitable Sobolev space and an equation on

C1([O,l])

involving the sum of the

identity plus a compact operator.

Thus these problems can be

readily treated by the methods of analysis described in Section

5. The difficulty with the singular character of the

cubical reference state disappears in "Family 2 " of semi-inverse problems (as categorized in [ 1 3 ] ) .

Here the reference

configuration may be taken as a body described in cylindrical polar coordinates z1 < z < z 2 .

(r,B,z)

by

r

1

< r < r2,

8 , < 8 < €I2,

The deformation is defined by a representation

obtained from (3.1) by replacing

(x,y)

by

(r,B).

Because

the independent variables are polar coordinates, the resulting equations will be singular at

r = 0.

This singularity

is manifested i n a semi-inverse problem when

rl = 0 ,

i.e.,

when the reference configuration contains the material line r = 0.

When

E (0,n)

U (n,m)

this singularity might

cause serious analytical difficulties because the domain has a corner.

n

It is not so obvious that tliis singularity can

cause serious difficulty when a segment o f a solid cylinder.

B2-01 = m , i.e.,

L-2

is

I n this case the analogs

of

when

the various completely continuous operators used i n the

21

SEMI-INVERSE PROBLEMS O F N O N L I N E A R ELASTICITY

analysis o f Section

6 may f a i l t o he c o m p l e t e l y c o n t i n u o u s r = 0.

when t h e m a t e r i a l i s a e o l o t r o p i c n e a r i s o i r o p y f o r s u c h problems t r e a t s t h e h u c k l i n g of When

rl

The r o l e o f

i s examined i n d e t a i l i n

[ 7 1 , which

a circular plate.

> 0 , t h i s problem d o e s n o t a r i s e .

The

5 c a n he a d a p t e d w i t h o u t change t o t r e a t

methods o f S e c t i o n

I n particular,

t h e c o r r e s p o n d i n g s e m i - i n v e r s e problems.

the

p r o c e d u r e b e g i n n i n g w i t h t h e weak e q u a t i o n s p r o d u c e s a soluticn

t o t h e e v e r s i o n problem,

d i s c u s s e d by T r u e s d e l l [ 1 2 ]

elsewhe=

i n t h i s volume.

T h i s problem c a n h e s o l v e d u n d e r t h e

requirement

that

there he zero r e s u l t a n t

t h e ends of

the tube.

equations of t h e problem.)

t h e form

f o r c e and moment a t

(This r e s t r i c t i o n yields

(7.1)

I n general,

by v i r t u e o f

.just t w o

t h e symmetries o f

t h e r e w i l l be n o n - z e r o

tractions

r e q u i r e d o v e r t h e end f a c e s i n o r d e r t o e n s u r e t h a t of

the c i r c u l a r c y l i n d e r s

This semi-inverse

z e r o end t r a c t i o n s .

remain c i r c u l a r cylinders.

r = r 1,r2

p r o c e s s would n o t

t h e images

treat

t h e problem w i t h

The f l a r i n g t h a t T r u e s d e l l d e s c r i b e s ,

which o c c u r s u n d e r z e r o end t r a c t i o n s , singular perturbation of

seems t o r e p r e s e n t a

t h e s e m i - i n v e r s e s o l u t i o n and t o

o f f e r a n e x p e r i m e n t a l v e r i f i c a t i o n of a S t .

Venant P r i n c i p l e

f o r t h i s problem o f n o n - l i n e a r

This e v e r s i o n

problem and o t h e r s o f

elasticity.

Family 2 w i l l b e d i s c u s s e d elsewhere.

( A l t h o u g h t h e g o v e r n i n g e q u a t i o n s of a r e n o t autonomous,

t h e problems

t h e q u a l i t a t i v e b e h a v i o r of

of F a m i l y 2

t h e i r solutions

may b e s t u d i e d by means of P r f l f e r t r a n s f o r m a t i o n s and o t h e r a p p a r a t u s u s e d i n t h e s t u d y of S t u r m - L i o u v i l l e

systems.

STUART S.

22

ANTMAN

The r e s u l t s r e p o r t e d h e r e s u g g e s t t h a t a l l r e a s o n a b l e boundary v a l u e problems f o r q u a s i l i n e a r systems of o r d i n a r y d i f f e r e n t i a l equations describing semi-inverse

problems of

compressible e l a s t i c bodies ha>e c l a s s i c a l solutions.

o n a f o r m u l a t i o n i n m a t e r i a l c o o r d i n a t e s and

analysis relies

on t h e e x p l o i t a t i o n o f Many of

the strong e l l i p t i c i t y condition,

t h e s t u d i e s of semi-inverse

employed s p a t i a l d e s c r i p t i o n s , equations

(cf.

s i m p l i c i t y of

Our

[7,l3]).

problems i n t h e 1950's

which y i e l d o s t e n s i b l y s i m p l e r

Bu1. t h e s e f o r m u l a t i o n s o b s c u r e t h e

the strong e l l i p t i c i t y condition.

At

one s t a g e

o f o u r a n a l y s i s w e made a c o n s t i t u t i v e a s s u m p t i o n t h a t i s i m p l i e d b y t h e Coleman-No11

inequality.

I t s use could be avoided

but a p e r i p h e r a l r o l e i n our bork: by u s i n g a c o n t i n u a t i o n t h e o r e m b a s e d (cf.

[41)

This i n e q u a l i t y plays

on weak er h y p o t h e s e s

o r by u s i n g t h e pseudo-monotone

operator theory.

8. R e f e r e n c e s [l]

S.S.

Antman,

O r d i n a r y D i f f e r e n t i a l E q u a t i o n s of

Non-linear

E l a s t i c i t y 11: E x i s t e n c e a n d R e g u l a r i t y T h e o r y f o r C o n s e r v a t i v e Boundary V a l u e Problems, Mech. [2]

S.A.

Antman, Arch.

[ 3 ] S.S.

Anal.

Arch. R a t i o n a l

6 1 ( 1 9 7 6 ) 353-393.

B u c k l e d S t a t e s o f N o n l i n e a r l y E l a s t i c Plates,

R a t i o n a l Mech.

Antman & H .

BrGzis,

Anal.,

67(1978)

111-149.

The E x i s t e n c e o f O r i e n t a t i o n -

Preserving Deformations i n Nonlinear E l a s t i c i t y , Research Notes i n Mathematics, ed. R. Knops, Pitman, London,

t o appear.

SEMI-INVERSE PROBLEMS O F NONLINEAR ELASTICITY

B a l l , Convexity Conditions

J.M.

i n Non-linear

-63 _ ( 1977) \J.L. E r i c k s e n ,

and E x i s t e n r e Theorems

E l a s t i c i t y , Arch.

337-'ho'3

R a t i o n a l Mech.

Anal.,

-

Deformations P o s s i b l e i n Every I s o t r o p i c ,

Incompressible,

P e r f e c t l y E l a s t i c Body.

Z.A.M.P.

5

( 1 9 5 4 ) , ,466-486. E r i c k s e n , D e f o r m a t i o n s P o s s i b l e i n E v e r y Compressible,

J.L.

Isotropic, Perfectly Elastic Material,

J . Math.

Phys.

-qb _ (195'1)~ 126-128.

A.E.

G r e e n & \J.E.

Non-linear Oxford,

L a r g e E l a s t i c D e f o r m a t i o n s and

Continuum M e c h a n i c s , C l a r e n d o n P r e s s ,

1960.

Knowles,

J.A.

Adkins,

L a r g e AmpLitude O s c i l l a t i o n s

Incompressible E l a s t i c Material, (1960, J.

o f a Tube of

Appl.

Q.

Math.

18 -

71-77.

Leray & J.

Schautler, Topologie e t Gquations

fonctionelles,

Ann.

S c i . E c o l e Norm.

Sup ( 3 ) j l

-

(1934), 45-78. Rabinowitz,

[lo] P . H .

v a l u e Problems

A G l o b a l Theorem f o r N o n l i n e a r E i g e n -

and A p p l i c a t i o n s ,

l i n e a r Functional Analysis, Academic P r e s s , New Y o r k ,

[ll] R . S .

e d . E.H.

Zarantonello,

1971.

R i v l i n , Large E l a s t i c Deformations of I s o t r o p i c

Materials,

Parts I V , V , V I , P h i l . T r a n s . Roy.

London A 2 4 1 (1948) A

C o n t r i b u t i o n t o Non-

379-397,

Proc.

Roy.

SOC.

S O C . London

195 ( 1 9 4 9 ) 4 6 9 - 4 7 3 , P h i l . T r a n s . Roy. S o c . London ( 1 9 4 9 ) 173-195.

A 242

24

[l21

STUART S. ANTMAN

C. Truesdell, Comments on Rational Continuum Mechanics, i n this volume.

[13l

C. Truesdell & W.

Noll, T h e Non-Linear Field Theories o f

Mechanics, Handbuch d e r Physik, III/3, Springer-Verlag, Berlin, 1965.

G.M.

de La Penha. L.A. Medeiros (eds.) Contemporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations QNorth-Holland P u b l i s h i n g Company (1978)

ASYMPTOTIC PROPERTIES OF GENERAL SYMMETRIC HYPERBOLIC SYSTEMS

L V I L A and D.G. COSTA

G.S.S.

Departamento de Matemgtica Universidade de Brasilia Brasilia, DF

-

Brasil

1. Introduction

We consider the Cauchy problem

-a +U at

n C

A . aU --= .I a x . j=1 J

where the A .Is are constant J

u

and

f

x = (xl,

are column vectors o f

... ,xn) E

the time.

n,

R

Hermitian symmetric matrices, k

components,

is tlie space variable and

t E IR

is

We use the Radon transform to study the behavior

of the solution times.

kxk

0,

u

for large (positive as well as negative)

We obtain results on the energy distribution o f

according to the several components of

f

u

(Theorem 3.1 and

Corollary 3 . 2 ) , o n energy decay (Theorem 4.1 through b . h ) , and on the existence o f wave operators, which are defined by comparing equation (1.1) (regarded as unperturbed) with a perturbed equation au

+ at

E(x) -

11

aU C A. j ax = O. j=l j

26

(Theorem 5 . 1 ) .

COSTA

A V I L A and D . G .

G.S.S.

These a r e e x t e n s i o n s of r e s u l t s p r e v i o u s l y

o b t a i n e d by t h e a u t h o r s [

11,

[2] and by Wilcox [ 91.

Proofs

w i l l appear elsewhere. Systems

(1.1) have b e e n d i s c u s s e d i n t h e l i t e r a t u r e

X

u n d e r c e r t a i n a s s u m p t i o n s on t h e r o o t s

= h(p)

o f the

c h a r a c t e r i s t i c equation P ( h , p ) = det(X1

n

-

C

p.A.) = 0

j=1

(see

[43, [ 51, [ 9 ] ) .

symmetric,

A

J

I t should be n o t e d t h a t o u r r e s u l t s a r e

v a l i d f o r g e n e r a l s y s t e m s of on t h e m a t r i c e s

3

type

( l . l ) , w i t h no r e s t r i c t i o n s

o t h e r than t h a t

j’

they a r e Hermitian

t h e r e f o r e w i t h no r e s t r i c t i o n s on t h e r o o t s

X(p).

Of c o u r s e , t h i s i s i m p o r t a n t n o t o n l y from a p u r e l y t h e o r e t i c a l p o i n t of view, but a l s o f o r a p p l i c a t i o n s , s i n c e t h e r e a r e c a s e s , s u c h a s i n magnetogasdynamics the r o o t s

X(p)

[ 6 1 , [S]

,

where

d o n o t s a t i s f y t h e r e s t r i c t i o n s u s u a l l y made

i n the l i t e r a t u r e . I n what f o l l o w s of

i s g i v e n by

f E Ho,

u(t,*) = Uo(t)f,

u n i t a r y group i n

w i l l i n g e n e r a l d e n o t e an e l e m e n t

k H o = L2(Rn)

t h e H i l b e r t space

It i s known t h a t i f

f

Ho

,

w i t h norm d e f i n e d b y

the solution where

(see [9]).

Uo(t)

u

of

(1.1)-(1.2)

i s a one-param*

Such a s o l u t i o n i s c a l l e d a

s o l u t i o n w i t h f i n i t e energy, s i n c e i t s energy, defined as 2

Ilu(t,.)llo,

is finite:

27

SYMMETRIC HYPERBOLIC SYSTEMS

The group

is generated by the operator

Uo(t)

-iA,

where k A = -i"L

A .

j=1

D(A) = { f € H o :

a ~

J ax. J

Af E H o ] .

We observe that under the Fourier transformation,

A

is unitarily equivalent to the operator of multiplication

by the matrix

.

k

A(p) =

Z j=1

p.A. J J'

that is,

Af = 5-l [A( .)5f]

D(A) =

{r

E H ~ :A ( . ) ? : E H ~ ] .

2. The Radon transform and solutions of the Cauchy problem. -.

In this section we shall use the Radon transform in order to derive an expiicity formula for the solution of The Radon transform provides a translation

(1.1)-(1.2).

representation for the free solution group

Uo(t),

and this

is a key fact in establishing the results anounced i n the following sections.

If

f

is a function i n the Schwartz space of rapidly

decreasing functions,

8(Rn),

n

2

3,

its Radon transform

= @ f is defined by the formula (see [ S l y [ 5 1 y

?(S,uJ)

=

f

(xIdsx 7

The following properties are valid:

x E IR,

*i

c71)

E Sn-l.

28

A V I L A and D.G.

G.S.S.

COSTA

Also, we point out the following relation between the A

Radon transform

where

5,

and the Fourier transform

f

-f:

denotes Fourier transformation i n the variable Now, for

n

2

3

odd, let

operators

a

K = a

K

L

and

s.

denote the

n-1

(-)

n as

where

a > 0 n

is a suitable constant.

Also,

for

n > 3

even, let

where

bn

shown ([ 31

is another suitable constant.

, [ 51 ,

[ 71 )

Then, it can be

that the following Inversion Theorem

and Parseval Theorem hold: Theorem 2.1 (2.3)

-

Let

f

E 8 (IRn),

f(x) =

I

Ilu

n

2

3.

Then

(K;)(x-~,a)dw,

I =I

x E Rn.

29

SYMMETRIC HYPERSOLIC SYSTEMS

-

Theorem 2.2

@ = Lm.

Let f

I f(x) I

(2.14)

Then, f o r any

' Rn @

In fact, onto

that

-

K = L

'l*rl=l

2

n

3,

5

l@f(s,J;)I2 dsds.

I-m

extends to a unitary mapping from all of

L2(Rn)

.

L~( [ ~ s xn-')

Remark 2.3

f"

dx =

f E 8(Rn),

I t follows from the definitions (2.2)0 and (2.2)e

.

Ipln-'

Also, since

the definition of

K

= sgn(p) p

in this case differs -1

ki = 5,

o d d by the Hilbert transform

n-1

for

even,

n

from that for

sgn(*)Zl

n

(up to a

constant). Now, let datum

f

u(t,*) 8(Rn)

belongs to

C(t,.,-)

=

?.

k

.

Then, i n view of (2.1)

a; t aC = a + A(&) as

0

n C

with corresponding eigenvalues

A(&)

and setting

we diagonalize (2.5) into the scalar equations

a;.

&+ &.(O,s,u))

J

a;.

x J. ( W ) A as

= gj(s,w),

= 0,

J

J

I S j S k.

w e obtain

C.(t,S,W) = P.(s-x.(W)t,&), hence

(iii),

w .A The initial datum is n o w 6 ( 0 , - , * )= j=l J j' S o , denoting b y [ej(u)] a complete set of orthonormal A(w) =

eigenvectors o f

Since

-

satisfies the eyuation

(2.5) where

be a solution of (1.1) whose initial

J

1 5 j S k ,

xj(d),

30

AVILA

G.S.S.

and D.G.

COSTA

(2.7)

As we see, the Radon transform provides a translation representation f o r the solutions of (1.1). the eigenvalues order.

and eigenvectors

Xj(;u)

First, we are enumerating the

A few words about

e.(a)

are now in

J

h j ( a ) I s

which are not

identically zero in decreasing order (counting multiplicity): X,(m)

(2.8)

hence setting

Xr+l(~)

...

2

... =

=

2

Xr(JJ),

= 0

A,(*)

this way, it is a known fact that the functions (see

[ l O l , Theorem

i n case

1.Is

r < k.

In

are continuous

.J

O n the other hand, it has

1).

been proved by Wilcox ([ 101, Theorem 2) that the corresponding (orthonormalized) eigenvectors measurable functions,

u(t,x) =

Remark

-

2.4

J

can be chosen to be

Assuming that this choice has been made,

(2.7) and obtain

we use Theorem 2.1 to invert

12-91

e.(a)

“i

z

j=1

~

~

( Ki‘ ) ( x - w - X . (a) t ,a ) e . (3) d W ,

l

=

~J

J

The non-zero eigenvalues i n (2.8) can be re-

numbered as w,(m)

...

5

2

pL(w),

L

5

r,

where all the p . ( u J ) ! s are distinct, except f o r J

No

of measure zero i n

1 < j

S

L ,

projection o f =

[ 101

.

We denote by

the orthogonal projection of

space associated with

Xr+l(w)

Sn-1

Ck

pj(w),

and by

iu

Ck

in a set

Pj ( a ) ,

onto the eigen-

P0(w)

the orthogonal

onto the eigenspace associated with

... = Xk(;u)

E

0,

make the convention that

i n case

P0 ( w )

=

0,

r < k Cr0(w)

(if

=

0).

r = k

we

Then, we

SYMMETRIC HYPE RRO L I C SYSTEMS

31

(2.10) and

respectively.

3.

Partition of

energy.

F o r any g i v e n

matrix

kxk

M,

we c o n s i d e r t h e

bounded o p e r a t o r s

Ho = L2(Rn)

mapping

k

we d e f i n e t h e M-energy

where

into f

of

i s t h e norm i n

-

If

u(t,.)

L2(IR~Sn-l)

t h e n t h e e n e r g y of

of

as

It]

E

S

t Ho,

Then, we 'nave: (1.1) w i t h i n i t i a l

t o t h e M-energy

tends

Mu(t,*)

m:

In f a c t , there exists a s e t f

.

i s a s o l u t i o n of

f,

+

f-

by

datum f

Also for

L2(iRxSn-1)k.

k

11) *II[

Theorem 3 . 1

each

1 s j s k,

M . = MP.(*)@, J J

(3.1)

we h a v e

S,

dense i n

2 l l M u ( t , * ) ( j o= EM[f]

HO

,

such t h a t f o r

for a l l

It[

sufficiently large. C r o_ l l a_r y_ 3 .~2 _ o_

-

F o r any f i n i t e e n e r g y s o l u t i o n

(1.1) t h e r e e x i s t s

E . = E .[u(O, *)] J J

,

j=l,

... , k ,

u(t,x)

of

such t h a t

4.

A V I L A and D.G.

G.S.S.

32

COSTA

Energy decav.

A solution

non-static

if

with f i n i t e energy i s c a l l e d

Uo(t)f

E ( k e r A)'.

f

R e c a l l i n g (2.l)(iv) a n d Remark

2.4, i t i s n o t h a r d t o s e e t h a t t h i s is t h e c a s e i f P o ( ~ ) ? ( s , , u )E

0.

i s non-static

if

Thus, i n view o f and o n l y i f

(2.10),

has

u(t,*) = U(t)f

the representation

.e. G(t,s,s) =

(4.1)

T j=1

Theorem 4__ .1 ( k e r A)', (i) (ii)

cone

-

P.(JJ)P(S-V . ( o ) t , w ) . J J So

There e x i s t s a s e t

C

(ker A ) I ,

dense i n

such t h a t Uo(t)So =

so

f o r each

f

1x1 5 a l t l - R ,

for all

E So, It1

t E R;

Uo(t)f

R/a,

2

v a n i s h e s i n some d o u b l e -

where

a = a ( f ) > 0,

R = R ( f ) > 0. T h i s r e s u l t is u s e d t o p r o v e t h e e n e r g y d e c a y s t a t e d i n the following: Theorem 4 . 2

-

G i v e n a bounded m e a s u r a b l e s e t

for any n o n - s t a t i c

solution

More g e n e r a l l y , i f increase "too fast" as any n o n - s t a t i c

t

-b

u.

B(t) m,

i s a s e t t h a t does not

then t h e energy i n

s o l u t i o n also d e c a y s t o z e r o ,

p r e c i s e l y i n the following:

B c Rn,

B(t)

as stated

of

SYMMETRIC 1IYPERBOLTC SYSTEMS

Theorem

'1.3

-

Let

t > O}

(B(t):

e(t)

m e a s u r a b l e s e t s such t h a t

8 ( t ) = sup{ 1x1: x E B ( t ) ) .

37

be a f a m i l y o f hounded = o(t)

t

as

-+

dx = O,

t-+ m

t-+ m f o r any n o n - s t a t i c

sol.utio11

u.

F o r t h e n e x t r e s u l t we n o t e t h a t

g = (I-P

and

)f E

( k e r A)'.

f E Ho

any

f = g+h,

decomposed u n i q u e l y i n t h e form

B

If

C

a s t h e energy o f

On t h e o t h e r h a n d ,

in

h

B,

0

Rn

measurable s e t s converges s l o w l y t o

where t o t h e c h a r a c t e r i s t i c i n this situation, tends t o the value Theorem

4.4

-

If

Eg[ f ]

B(t)

B

B(t)

as

t +

B(t) m,

{B(t): if

i n_ _

B,

t > 0)

e(t)

= o(t)

of and

converges almost cvery-

f u n c t i o n of

t h e energy i n

i s any

that i s ,

we s a y t h a t a f a m i l y

the c h a r a c t e r i s t i c function o f

can b e

h = P f' €

where

m e a s u r a b l e s e t , we d e f i n e t h e _ s t a_t i c. _ e n_ e r_ g y_of f _ Ei[f],

bhere

Then,

l i m Ilu(t,

E ker A

m,

B, of

as

t +

m .

any s o l u t i o n Uo(t)f

that is:

converges slowly t o

Then,

B,

then

5 . The p e r t u r b e d e q u a t i o n and t h e wave o p e r a t o r s . W e now c o n s i d e r t h e p e r t u r b e d Cauchy problem

A V I L A and D . G .

G.S.S.

34

COSTA

u(0,x) = f ( x ) ,

(5.2)

i s a positive definite

E(x)

where

kxk

H e r m i t i a n symmetric

matrix s a t i s f y i n g the following properties: t h e r e e x i s t p o s i t i v e c o n s t a n t s c and c’ s u c h t h a t

i)

(5.3)

CI

ii)

s ~ ( x s) e

there exists

for a l l

>

such t h a t

0

1x1 +

= ~(~xl-l-‘) as I n view o f c o n d i t i o n

11 * ] l o

Letting

(5.3) ( i )i t i s c l e a r that the

/I - 1 1 ,

d e f i n e d by

E(x)f(x)-f(x)dx.

€lo = L2(Rn)

denote the H i l b e r t space

H

1) * I ) ,

norm

b;.

=

IE(x)-II =

m.

norm i s e q u i v a l e n t t o t h e norm

II f l l

x;

C’I

k

we d e f i n e t h e o p e r a t o r A~ = E ( X ) -1 A ,

It follows that

-iAE

= D(A).

D(A,)

i s a self-adjoint

o p e r a t o r on

t h e f i n i t e energy s o l u t i o n o f

(5.1)-(5.2)

u(t,-) = U(t)f,

i s t h e one-parameter

group i n

where

U(t)

generated by

H

-iAE

The wave o p e r a t o r s groups

with t h e

Uo(t)

and

U(t)

W+,

H

and

i s g i v e n by

unitary

(see [9]).

W-

associated w i t h the

a r e d e f i n e d by

(5.4) Po = I - P

where

Theorem 5.1 exist

.

-

i s the projection of

Under t h e h y p o t h e s i s

Ho

onto

(ker A I L .

( 5 . 3 ) , t h e wave o p e r a t o r s

35

SYMMETRIC HYPERBOLIC SYSTEMS

Remark 5 . 2

(5.4)

-

Letting

€*

and the u n i t a r i t y of

l i m t+*m

f E Ho,

= W*f,

that

U(t)

lIU(t)f*

i t f o l l o w s €rom

- U ~ ( ~ ) P ~ ~ =/ I 0.

T h e r e f o r e , by c o n s i d e r i n g s o l u t i o n s o f

(5.1)

of

t h e form

i t i s now e a s y t o s e e t h a t m o s t o f o u r p r e v i o u s

U(t)f*,

results

( n a m e l y Theorem 3 . 1 ,

through

4.4)

C o r o l l a r y 3.2

a n d T h e o r e m s 4.2

are s t i l l valid f o r the perturbed equation

(5.1)

(5.3).

under t h e hypothesis

References -

111

G.S.S.

A v i l a and D.G.

symmetric hyperbolic equations

[z]

D.G.

costa,

Decay o f

Costa,

systems of p a r t i a l d i f f e r e n t i a l

( t o a p p e a r i n Rocky M o u n t a i n J .

of Math.).

O n p a r t i t i o n of energy f o r uniformly

propagative systems, J. M a t h . pp.

solutions of

Anal.

Appl.

52 (1977),

56-62.

[ 3 ] S. H e l g a s o n ,

T h e Radon t r a n s f o r m o n E u c l i d e a n s p a c e s . . . ,

A c t a Math.

113 ( 1 9 6 5 ) , p p . 93-106.

[ b ] T. I k e b e , S c a t t e r i n g f o r u n i f o r m l y p r o p a g a t i v e s y s t e m s ,

[5]

P.D.

Proc.

Int.

Tokio

( 1 9 6 9 ) , p p . 225-230.

Lax a n d R . S . Press,

C6] D .

Conf'.

Ludwig,

on Func.

Phillips,

Anal.

and Related T o p i c s ,

S c a t t e r i n g T h e o r y , Academic

1967. The S i n g u l a r i t i e s o f

NYU R e s a r c h R e p o r t

(1961).

t h e Riemann F u n c t i o n ,

G.S.S.

36

AVILA and D.G.

[71 D. Ludwig, The Radon transform Cornm.

[8] S.I. Pai,

COSTA

on Euclidean spaces,

Pure Appl. Math. 19 (1966), pp. 49-81. Magnetogasdynamics and Plasma Dynamics,

Springer, 1962.

[9]

C.H.

Wilcox,

Wave operators and asymptotic solutions o f

wave propagation problems of classical physics, Arch. Rat. Mech. Anal. 22 (1966) PP. 37-78. [lo] C.H. Wilcox, Measurable eigenvectors f o r Hermitian matrix-valued polynomials, J. Ma h. Anal. Appl. 4 0

( 1 9 7 2 1 , ~p.12-19.

G.M. de La Penha, L.A. Medeiros (eds.) Contemporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations @North-Holland Publishing Company (1978)

THE CAUCHY PROBLEM FOR THE COUPLED

SCHROEDINGE R-KLE IN- GORDON E QUATI0N S

and

J.B. Baillon

John M .

Department of Mathematics

Dept. de Math6matiques &ole

Chadam*f

Pontificia Universidade

Polytechnique

Cat6lica do Rio de Janeiro Paris, France

Rio de Janeiro, Brasil

Introduction. The classical versions of the non-linear equations o f relativistic quantum physics have for a long time been a subject of great interest.

One o f the major unsolved problems

in the area has been the global existence o f the solution t h e Cauchy problem in

R3

to

o f certain important equations

-

having quadratic non-linearities

the coupled Dirac-Klein-

Gordon equations,

P (-iY au + ~ ) =t g $ @ ,

-

( A - a tt

2

m ) @=

6t1

and the coupled Maxwell-Dirac (-i?

)r

+

a,,

+

O n leave from Mathematics Department, Indiana University, Bloomington, Indiana 47401.

Supported in part by NSF grant MPS 73-08567

J.B. BAILLON and J.M. CHADAM

38

Here we consider the Cauchy problem for the closely related coupled Schoredinger-Klein-Gordon ( S K G ) equations which are a semi-relativistic version o f equations (1):

with nucleon field

8 : R 3X R

-b

C

and meson field

@:

R'xR

4

R

I n previous works the Cauchy problem has been solved for equations (1) and (2) i n one space dimension [l] and for special cases in higher dimensions [2,31.

Equations ( 3 ) have

recently been treated i n bounded regions of

R3[

4,5]

.

The Results. We begin by writing equations ( 3 ) i n vector form,

from

R

JI

(g

) are considered as maps t to complex and R2-valued functions respectively. The

where the components

and

conventional solution spaces in which one wishes to solve equations

(4) for the components

"escalated energy" spaces

C m = Hm

( $ y(:t)) @

(HmCBHm'')

are the so-called [l].

Indeed,

following the example of quantum mechanics, it is really the exponentiated o r , in the non-linear case, the integrated form of equation

(k),

-

SC HR O E D IN GE R K LE IPJ - GO R D 0N E QUA T I0 N S

which a r e of

interest.

S(t) = e

Here

i A t

and

a r e t h e f r e e S c h r o e d i n g e r and Klein-Gordon respectively.

(+ ,(:

Specifically then,

39

propagators

))

is a

Cm solution

t of

t h e SKG e q u a t i o n s o v e r t h e i n t e r v a l

($,(:

)):

(0,T)

4

Cm

i s continuous

(0,T)

i f t h e map

and s a t i s f i e s e q u a t i o n s

t

( 5 ) where t h e i n t e g r a l i s i n t e r p r e t e d i n t h e s t r o n g Riemann

zm.

sense i n

-

Theorem

solution i n Proof:

zm

if

problem.

of

+

S ( t ) : tim

a r e continuous, uniformly

( a , (m 0 12 ) )

rn 2 2 .

-+

boundedness,

[ 6 1 c a n be a p p l i e d t o this

and

-+

K( t ):

bounded o n e - p a r a m e t e r

, ( O

IJI

I

2)): Zm + Cm

H

~

~

g r o u p s and

i s l o c a l l y Lipschitz

t h i s f o l l o w s from t h e l o c a l

o f t h e map v i a some t r i v i a l a l g e b r a .

The l o c a l

on t h e o t h e r h a n d , f o l l o w s f r o m t h e f a c t t h a t

i s an a l g e b r a i f

dimension

(@,$

H~

One c a n s h o w t h a t

boundedness

Hm

Segal

The l o c a l e x i s t e n c e f o l l o w s i n t h i s way b e c a u s e t h e

f r e e propagators

if

m)

2.

m 2

The a b s t r a c t r e s u l t

t h e map

T =

The SKG e q u a t i o n s h a v e a u n i q u e g l o b a l ( i . e .

m > d/2

where

d

i s the s p a t i a l

"71.

The main c o n t r i b u t i o n o f t h i s n o t e i s t o show t h a t this s o l u t i o n can b e e x t e n d e d t o

T =

p r e v i o u s l y mentioned r e s u l t

of

+m.

This o b t a i n s f r o m t h e

Segal i f t h e

Zm

-

norm of

s o l u t i o n c a n be s h o w n t o be f i n i t e f o r e a c h f i n i t e t i m e .

the We

H

~

-

J.B.

40

B A I L L O N and J . M .

CHADAM

b e g i n with t h e p h y s i c a l l y r e l e v a n t conserved q u a n t i t i e s o f c h a r g e and e n e r g y and t h e n s y s t e m a t i c a l l y b o o t - s t r a p o u r way

t o the xm-norm.

Specifically

[4,

p.4031

one h a s

4

3x = c o n s t a n t . I $ ( x , t ) 1 2 @ -b( x , t ) d F r o m t h i s one h a s

t h e second l i n e f o l l o w i n g f r o m H s l d e r ' s

i n e q u a l i t y and t h e

l a s t l i n e from S o b o l e v i n e q u a l i t i e s and t h e f a c t t h a t li$(t)ii 2 i s conserved.

Now

Combining t h e above t w o e s t i m a t e s , one o b t a i n s

Adding e q u a t i o n ( 6 a ) t o i n e q u a l i t y

(7),

one o b t a i n s on t h e

l e f t h a n d s i d e t h e s q u a r e o f a norm which i s e q u i v a l e n t t o t h e

Cl-norm

t h u s p r o v i n g i t i s u n i f o r m l y bounded i n s p i t e o f

non-definiteness

of

t h e i n t e r a c t i o n energy.

the

SCHROEDINGER-K LE I N - GORDON EQUATIONS

and m o s t c r u c i a l ,

The n e x t ,

Proof:

s t e p i s t o show

Here we f o l l o w c l o s e l y t h e t e c h n i q u e of B a i l l o n e t .

C 8 , 9 I by showing t h a t

a1

~ ~ $ ( t ) ~i s~ fl i,n4i t e and t h e n u s i n g

111

t h e Sobolev i n e q u a l i t y

( t ) l l mS

( t ) l ( z / 4 119 (t)ll

C11 vp

to

To b e g i n

establish the desired result.

A l l t h a t r e m a i n s t h e n i s t o e s t a b l i s h t h e f i n i t e n e s s of

IlVh(t)114.

Let

D

d e n o t e a n a r b i t r a r y weak s p a t i a l d e r i v a t i v e .

D i f f e r e n t i a t i n g e q u a t i o n ( 5 a ) , u s i n g t h e L e i b n i t z f o r m u l a and t a k i n g norms

D

since

one o b t a i n s

commutes w i t h

S(t).

U s i n g t h e well-known

e s t i m a t e f o r t h e Schroedinger p r o p a g a t o r [lo,

decay

p.601

1 1 -d(-- -) IlS(t)fIlp s

for a l l

f

E Lq(Rd)

where

C't

+

l/p

l/q

= 1

and

1

2;

q S

2,

one o b t a i n s f r o m i n e q u a l i t y ( 8 )

rt IIS(t)$0112,2 +

IID$(t)l14 s

Igl C i ,

( t - s ) - 3 / 4 I I D 0 ' ~ + @ D $ l l ~ / 3d s

t s

co

+

lglc

S

Co

+

Igl

(t-s)-3/4 c I I D 0 ~ s ~ l 1 2 1 1 ~ ~ s ~ l 1 4 + l l s ~ s ~ l l , I l ~ d s~ ~ s ~ l l , 1 C{C,

c

(t-s)-3/4 ds

+

C2

c

( t - ~ ) - ' / I~I D + ( S ) ( ~d~s ] .

The r e s u l t n o w f o l l o w s f r o m a v e r s i o n o f G r o n w a l l ' s sketched i n reference Lemma

3

-

[91.

The C2-norm of

f i n i t e f o r each

t <

lemma as

m.

the solution

($(t),(st(t)

42

J.B.

Proof:

But,

B A I L L O N and J . M .

CHADAM

From e q u a t i o n ( 5 b ) one h a s

o f Lemma 2 ,

f r o m t h e proof

one h a s t h a t

/IIII(S)//~,~ is

a

l o c a l l y i n t e g r a b l e f u n c t i o n g i v i n g t h e f i n i t e n e s s o f t h e meson p a r t of

t h e norm.

The r e s u l t now f o l l o w s from t h e s t a n d a r d v e r s i o n o f Gronwall’s lemma b e c a u s e e v e r y t h i n g e x c e p t

/ID2$

(.)I\

has previously

b e e n shown t o b e l o c a l l y i n t e g r a b l e .

To t h i s p o i n t we h a v e shown t h e e x i s t e n c e of a u n i q u e g l o b a l s o l u t i o n t o t h e Cauchy problem f o r t h e S K G e q u a t i o n i n t h e space

Z2.

A l l t h a t remains i s t o prove t h e r e g u l a r i t y

p a r t of t h e r e s u l t .

-

Lemma

4

than

2,

have

114 ( t ) l l m , 2 11@(t)11m,2 and

Suppose

m

i s an a r b i t r a r y p o s i t i v e i n t e g e r l a r g e r

t h e n t h e a b o v e , g l o b a l s o l u t i o n of e q u a t i o n s

II@,(t)ll

m-l,2

locally

(5))

-

S C H R O E D I N G E R - K LE I N GORDON E Q U A T I O N S

f o l l o w s f r o m L e m m a 3 by g i v i n g t h e i n d u c t i v e

Proof:

The p r o o f

step.

Suppose then that

some

But

m 2

H

3,

m- 1

the r e s u l t

is known in

Then

m 2

for

the left-hand

47

~

m- 1

for

i

3

i s an algebra g i v i n g t h e f i n i t e n e s s of

side f r o m the inductive hypothesis.

and t h e r e s u l t f o l l o w s

from Gronwall's

Similarly

l e m m a b e c a u s e of t h e

11 a ( s)II m , 2 .

l o c a l i n t e g r a b i l i t y of

References

[l] C h a d a m , J . M .

-

G l o b a l Solutions of

the ( C l a s s i c a l )

Coupled Maxwell-Dirac

Space D i m e n s i o n ,

[2]

Chadam,

J.M.

J. Functional A n a l . ,

and G l a s s e y ,

Solutions of

-

R.T.

Equations

3,1 7 3

in One

(1973).

On C e r t a i n G l o b a l

the C a u c h y P r o b l e m for the ( C l a s s i c a l ) E q u a t i o n s i n O n e and T h r e e

Coupled Klein-Gordon-Dirac Space D i m e n s i o n s ,

[ 3 ] C h a d a m , J.M.

the C a u c h y P r o b l e m f o r

Arch.

and G l a s s e y ,

Rat. R.T.

Mech.

-

Anal.,

54

(1974).

O n the M a x w e l l - D i r a c

E q u a t i o n s w i t h Z e r o M a g n e t i c F i e l d and T h e i r S o l u t i o n

i n T w o Space D i m e n s i o n s ,

53,

495 ( 1 9 7 6 ) .

J. M a t h .

Anal.

and A p p l i c .

44

C 41

J.B. BAILLON and J.M.

Fukuda, I. and Tsutsumi, M.

-

CHADAM

On the Yukawa-Coupled

Klein-Gordon-Schroedinger Equations i n Three Space

Dimensions, Proc. Japan Acad., 5 1 ,

C 51

Fukuda, I. and Tsutsumi, M.

-

Schr8dinger Equations, 11,

C 61

Segal, I . E .

-

402, (1975).

O n Coupled Klein-Gordonto appear.

Non-linear Semi-groups, Ann. Math. 7 7 ,

339 (1963).

C 73

Grisvard P,

-

Contribution i n Proceedings o f Evolution

Equation Seminar, Nice, France

1974-5.

[ 81 Baillon, J.B., Cazenave, T. and Figueira, M.

-

Equation

de Schr8dinger Non-lingaire I, t o appear C.R. Acad. Sci

[ 91

.

Baillon, J.B., Cazenave, T. and Figueira, M.

-

Equations

de Schradinger Non-lin&aire, 11, to appear CR. Acad. Sci. [lo] Reed, M. and Simon, B.

-

Methods o f Modern Mathematical

Physics 11, Academic Press,

1975.

G.M. de La Penha, L . A . Medeiros (eds.) Contemporary Developwnts i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations @North-Holland Pub1i s h i n g Company (1978)

APPLICATIONS OF GENERIC BIFURCATION THl3ORY IN FLUID MECHANICS

T.B. BENJAMIN Fluid Mechanics Research Institute University of Essex, England

1. Introduction

In thiq lecture

some

abstract theoretical results

are reviewed with the particular aim of illustrating their application to practical problems in the mechanics of fluids. From the standpoint of the practical-minded investigator into flow problems the modern theory o f partial differential equations seemingly offers much relevant information, and the complexities of the problems in question give weight to the seeming generality of this iiifsrmation.

Its precise bearing

on observed phenomena is sometimes elusive, however, and

physical insights guided by experimental information may then appear to be the more dependable, albeit limited, means of understanding.

Thus, in this subject, there are distinct and

sometime,: ill-matched stvles o f scientific inquiry, and their more effective alliance is still an outstanding objective. T h e present discussion begins by recalling J.Leray’s classic demonstration, elaborated by E . Hopf, O.A.

Ladyzhens-

kaya and others, that the boundary-value problem determining steady motions of a viscous incompressible fluid in any finite domain has at least one solution f o r all value o f the Reynolds number

R

(the dimensionless velocity parameter).

46

T.B.

By i t s e l f

this result

p r e t a t i o n s of

BENJAMIN

i s hardly helpful as regards i n t e r large R

observed s t e a d y f l o w s , p a r t i c u l a r l y a t

when t h e problem can h a v e m u l t i p l e s o l u t i o n s , may r e p r e s e n t

some o f which

s t a b l e f l o w s and o t h e r s r e p r e s e n t u n s t a b l e f l o w s .

O n t h e same b a s i s , however,

a corpus o f

additional inferences

can h e b u i l t up which p r o v i d e s f o r c o m p l e t e i n t e r p r e t a t i o n s of

v a r i o u s c o m p l i c a t e d phenomena.

To t h i s e n d , that are typical of

i t i s essential t o identify properties

e x p e r i m e n t a l l y observed f l o w s , e x c e p t i n g

s i n g u l a r b e h a v i o u r t h a t m a y h e i n c l u d e d w i t h i n t h e s c o p e of the

Comments a b o u t t h i s r e q u i r e m e n t

abstract theory.

made i n

$3.

Then,

$4,

in

a s e t of

are

typical properties

a s s o c i a t e d w i t h hounded s t e a d y f l o w s i s p r e s e n t e d .

In

$5

a

concomitant i d e a with p a r t i c u l a r p r a c t i c a l i n t e r e s t i s introduced, being explained i n t h e f i r s t p l a c e with r e f e r e n c e

t o a problem i n e l a s t i c i t y .

Finally,

in

$ 6 , some r e l e v a n t

experimental o b s e r v a t i o n s a r e reviewed.

2.

The t h e o r y of Leray-Hopf-Ladyzhenskaya

a c c o u n t of

The best-known

L a d y z h e n s k a y a ' s monograph ( 1 9 6 9 ) . recalled.

Here t h e g i s t i s m e r e l y

W e c o n s i d e r s t e a d y motions o f

a v i s c o u s incompres-

s i b l e f l u i d t h a t f i l l s a bounded smooth domain

aD

d e n o t e s t h e boundary,

i s prescribed,

and

Eulerian velocity,

5 and

D

on which t h e v e l o c i t y of

= D U aD. P:

fi +

IR

pressure corrected f o r the effect conservative force f i e l d s ) .

5 of

t h e t h e o r y i s Ch.

Let

U:

5

+ R3

in

the f l u i d he t h e

t h e gauge p r e s s u r e of

El3.

(i.e.

g r a v i t y and o t h e r

The N a v i e r - S t o k e s

equation of

47

GENERIC BIFURCATION THEORY I N FLUID MECIIANICS

motion i n dimensionless f o r m ,

t h e c o n d i t i o n of

incompressibil-

i t y and t h e boundary c o n d i t i o n a r e

-nu

+

+ R u.vU

v-U = 0

VP = 0 U = q

D,

i n

R,

The p o s i t i v e p a r a m e t e r

that

q-n = 0

aD.

on

the Rey n o ld s number,

p r o p o r t i o n a l t o t h e v i s c o s i t y of assume

D,

in

all,

U = W

i s t o put

which i s so i n t h e

later.

s p e c i f i c applications t o be considered The f i r s t

of t h e problem (2.1)

s t e p i n the treatment

-

+ q,

g

%-here

i s a solenoidal extension of

t h e boundary data t o t h e i n t e r i o r of n o i d a l and v a n i s h e s on

aD.

It i s h e l p f u l t o

the fluid.

everywhere on

is inversely

Hence,

Thus

D.

assuming

i s sole-

W

W,

;,

P

and

t h e boundary t o be s u f f i c i e n t l y smooth, m u l t i p l y i n g t h e f i r s t o f

(2.1)

compact

Let

H

by any smooth s o l e n o i d a l f u n c t i o n

D,

support i n

I

[v$:vW

-

D + kJ

with

a n d i n t e g r a t i n g b y p a r t s , we o b t a i n

+

R{{*(W*Vt) =

W-(~*V$) + W.(W*W)]ldx

-

b.(Ag W

d e n o t e t h e subspace of

192

i n t h e Hilbert space

H

(u,v) =

I

(2.2)

Rq-Vq)dx.

(D)3

c o n s i s t i n g of

aD.

s o l e n o i d a l e l e m e n t s w i t h z e r o t r a c e on product

@:

the

The s c a l a r

i s taken t o be VU:VV

dx,

D

and i n view of

t h e Poincar6 i n e q u a l i t y ,

c l o s e d u n d e r t h e norm

I/uII = ( u , u )

for arbitrary

i s evidently

A weak s o l u t i o n o f

t h e hydrodynamic p r o b l e m i s d e f i n e d as a (2.2)

H

W E H

satisfying

$ E H.

The f i r s t f u n c t i o n a l of

(I

i n (2.2)

is just

(W,a),

48

T.B.

and a n i m p o r t a n t

BENJAMIN

s t e p i n t h e t h e o r y i s t o show t h a t

remaining l i n e a r f u n c t i o n a l s a r e d e f i n e d i n

H

the

and have

needed p r o p e r t i e s a c c o r d i n g t o t h e r e p r e s e n t a t i o n theorem of Riesz.

-

q

A c h o i c e of

i s shown t o b e p o s s i b l e s u c h t h a t t h e

f u n c t i o n a l on t h e right-hand t w o involving

s i d e i s d e f i n e d , and t h a t t h e

l i n e a r l y on t h e l e f t - h a n d

W

completely continuous o p e r a t o r s

-+

H

H.

side define

It also appears that

a c o m p l e t e l y c o n t i n u o u s q u a d r a t i c o p e r a t o r i s d e f i n e d by t h e

l a s t term on t h e l e f t - h a n d

c(R) =

where

@:

H

+

H

+

side.

Thus

(2.2)

i s equivalent t o

i s a p r e s c r i b e d e l e m e n t of

RC2

H

and

i s completely continuous. The most t a x i n g s t e p i n t h e t h e o r y i s t o e s t a b l i s h

t h e f o l l o w i n g a p r i o r i e s t i m a t e depending on a s u i t a b l e c h o i c e of

p

(p >

G.

\lclll)

s E [0,1]

F o r any f i n i t e

R,

t h e r e e x i s t f i n i t e numbers

s u c h t h a t t h e r e i s no

w E H,

IIwII

= p ,

and

satisfying

w = sR@w

+

C(sR).

A c c o r d i n g t o t h e homotopy i n v a r i a n c e of

L e r a y - S c h a u d e r degree,

i t f o l l o w s that deg(1 where

S = ( u E H: P

p r o p e r t y of d e g r e e ,

-

RH,c(R),S

IIu/l < p } .

P

) = 1,

(2.41

Hence, by t h e f i x e d - p o i n t

( 2 . 3 ) h a s a t l e a s t one s o l u t i o n i n

The a v a i l a b l e r e g u l a r i t y t h e o r y shows t h a t , j e c t t o m o d e r a t e c o n d i t i o n s on t h e smoothness of equivalence c l a s s o f

aD,

S

P

.

subthe

t h i s s o l u t i o n h a s a member c o n s t i t u t i n g

a classical solution of

(2.1).

GENERIC BIFURCATION THEORY IN FLUID MECHANICS

3.

The n o t i o n of ~

"typical" o r

~

~

49

'generic' ~

With r e g a r d t o s p e c i f i c p r a c t i c a l a p p l i c a t i o n s ,

the

a n a l y s i s o u t l i n e d above i s i n c o n v e n i e n t l y s w e e p i n g i n i t s g e n e r a l i t y , and t h e p o s s i b i l i t i e s i n a b s t r a c t need rowed i n o r d e r t h a t solution set

a u s e f u l account

{W(R)]

c l a s s i f i c a t i o n s of

in

R+xH.

If

t o be n a r -

c a n b e made of

the

one a c c e p t s p l a u s i b l e

the r e a l p o s s i b i l i t i e s f o r steady flows,

however, v a r i o u s p r a c t i c a l l y s i g n i f i c a n t p r o p e r t i e s may be deduced from t h e c e n t r a l r e s u l t framework.

I n o t h e r words,

(2.4)

and i c s t h e o r e t i c a l

s i m p l i f y i n g h y p o t h e s e s a r e needed

t o d i s c r i m i n a t e between t y p i c a l b e h a v i o u r o f

t h e system ( 2 . 1 )

and e x t r a o r d i n a r y b e h a v i o u r t h a t may b e i n c l u d e d

i n the

i n f i n i t e s e t s of a d m i s s i b l e domains and boundary d a t a

q

but

i s impossible t o capture experimentally.

I t i s p e r t i n e n t h e r e t o acknowledge t h e r e c e n t s t u d y by F o i a s & Temam

( 1 9 7 7 ) , i n which p r o g r e s s h a s b e e n

made t o w a r d s a r a t i o n a l c l a s s i f i c a t i o n of s t e a d y Navier-Stokes

flows.

problem w i d e r t h a n ( 2 . 1 ) , non-conservative

field of

the properties o f

T h i s s t u d y was c o n c e r n e d w i t h a

which i n c l u d e s t h e e f f e c t s o f a e x t e r n a l f o r c e s a c t i n g on t h e f l u i d .

Considering a c e r t a i n space

F1

boundary f u n c t i o n s , F o i a s & Temam showed by means o f Sard-Smale value o f

theorem t h a t , f o r a g i v e n domain

R,

F 2 of

of f o r c e s and a s p a c e

D

t h e r e i s a n open d e n s e s u b s e t of

the

and a g i v e n FlxF2

for

which t h e number o f

solutions i s f i n i t e .

number i s c o n s t a n t ,

and e a c h s o l u t i o n i s a f u n c t i o n of p o s i -

tion,

i n e v e r y c o n n e c t e d component

Moreover,

o f t h i s subset.

this

This

50

BENJAMIN

T.B.

p r o p e r t y , which i s g e n e r i c i n t h e s p e c i f i e d s e n s e , h a s sub-~ s e q u e n t l y b e e n shown by Tromba & Marsden

(1977) t o e x t e n d t o

a s p a c e o f domains. While t h e s e r e s u l t s g o some w a y t o w a r d s d e t e r m i n i n g t h e o v e r a l l c h a r a c t e r of

t h e s o l u t i o n s e t , many i n t e r e s t i n g

q u e s t i o n s r e m a i n which s o f a r h a v e o n l y b e e n answered speculatively.

For instance, i s the finiteness property nearly

universal i n the sense t h a t

i t e x h a u s t s some a p p r o p r i a t e

measure p u t on t h e s p a c e s o f p o s s i b i l i t i e s ( s e e n o t e b e l o w ) ? Are t h e e x c e p t i o n s c o n f i n e d t o a s e t of t h e f i r s t B a i r e category?

I n view of

t h e e m p i r i c a l f a c t t h a t t h e number of

s o l u t i o n s c a n change w i t h c h a n g e s of

D

(cf.

$ 6 ) , what i s

t h e t y p i c a l form of t h e b i f u r c a t i o n s from which t h e new solutions arise?

What p r o p e r t i e s

of

such b i f u r c a t i o n s and o f

o t h e r s t e a d y - f l o w phenomena a r e s t r u c t u r a-~ l l y s t_a_b le, so that t h e y may b e o b s e r v a b l e i n p r a c t i c e ? Adopting t h e p r a c t i c a l s t a n d p o i n t a l r e a d y proposed, we may c l a s s i f y p r o p e r t i e s as t y p i c a l when t h e weight a v a i l a b l e evidence and i n t u i t i v e

-

-

mathematical,

suggests that

of

such a s i t i s , e m p i r i c a l

t h e s e p r o p e r t i e s a r e predominant

and t h e r e i s no o b v i o u s r e a s o n t o t h e c o n t r a r y .

Such p r o -

p e r t i e s may be u s e f u l l y j u x t a p o s e d w i t h o t h e r p r o p e r t i e s t h a t have been p r o v e d t o h o l d i n g e n e r a l ( e . g . P r o p e r t y 3 i n A s regards bifurcations,

$4).

we s h a l l i d e n t i f y a s t y p i c a l t h e t w o

s i m p l e s t s t r u c t u r a l l y s t a b l e p r o c e s s e s t h a t comply w i t h d e g r e e t h e o r y and a c c o u n t p l a u s i b l y f o r known phenomena.

In

f a c t , we t h u s r e c o v e r t h e t w o s i m p l e s t c a t a s t r o p h e s a c c o r d i n g t o R.

Thorn's t h e o r y o f

c a t a s t r o p h e s , even t h o u g h t h e p r e s e n t

problem d o e s n o t have a ' g e n e r a t i n g p o t e n t i a l '

and s o i s n o t

G E N E R I C BIFURCATION T H E O R Y I N F L U I D M E C H A N I C S

d i r e c t l y amenable t o t h e s t a n d a r d v e r s i o n of

that

theory.

(Note:

It i s w o r t h remembering i n t h e p r e s e n t c o n n e x i o n t h a t

a

S

set

that

i s open and d e n s e i n a s p a c e

n e c e s s a r i l y h a v e t h e p r o p e r t y of

covering

X

does n o t

'almost all'

of

X,

a p r o p e r t y t h a t may be d e f i n e d i n t e r m s o f a measure p u t on X

For instance,

xn

c o n s i d e r t h e d e n u m e r a b l e s e t o f r a t i o n a l numbers

belonging t o

a n open i n t e r v a l x

n

]O,l[,

and f o r e a c h

Sn c ] O , l [

and which h a s l e n g t h

n

= 1,2

,...

define

which i n c l u d e s t h e r e s p e c t i v e

6/2n,

where

0 < 6 < 1.

]O,l[,

t h e s e t o f r a t i o n a l numbers i s d e n s e i n

Because

so also is

m

t h e open s e t

S =

not g r e a t e r than

u

Sn b u t t h e m e a s u r e of S i s evidently n=1 6 , which we can make a s s m a l l as w e p l e a s e

without invalidating the property t h a t

4.

S

i s open and d e n s e . )

T y p i c a l p r o p e r t i e s o f bounded s t e-~ a d y f_ l.-. ows -. ~ ~ ~

_

The f i v e p r o p e r t i e s l i s t e d a s f o l l o w s h a v e b e e n d i s c u s s e d i n previous papers

(Benjamin

1976,

1977a), i n the

l a t t e r of which t h e i r p r a c t i c a l i m p l i c a t i o n s were emphasized. Here v a r i o u s p o i n t s o f of

and i n

remarks,

$5

i n t e r p r e t a t i o n a r e covered i n a s e r i e s

a r e l a t e d i d e a w i l l be d e v e l o p e d which

has p a r t i c u l a r i m p o r t a n c e r e g a r d i n g t h e e x p e r i m e n t s r e p o r t e d

in

96. P r o p e r t i e s 1 and 2 a r e g e n e r i c i n t h e d e f i n i t e sense

e s t a b l i s h e d by Foias-Temam

and Tromba-Marsden,

l i k e l y t o be v i r t u a l l y u n i v e r s a l t h e c l a s s of Property

4

observable f l o w s .

and t h e y seem

('measure-exhausting') i n Property

3 i n p a r t and

p e r t a i n t o t h e c o r r e s p o n d i n g time-dependent

problem.

T.B. BENJAMIN

52

Property

3 i s c e r t a i n , h a v i n g b e e n p r o v e d q u i t e g e n e r a l l y by

S e r r i n ( 1 9 5 9 ) , and P r o p e r t y

Thm 3 ) .

Property

4

i s a l s o c e r t a i n (Benjamin

1976,

5 i s t h e m o s t d e p e n d e n t on p l a u s i b l e reason-

ing f o r i t s c l a s s i f i c a t i o n , f i e d by t h e d i s c u s s i o n i n

and i t s meaning w i l l be e x e m p l i -

56.

(2.3) are isolated.

1. The s o l u t i o n s of

Being t h e f i x e d

p o i n t s o f a completely continuous o p e r a t o r , the s o l u t i o n s e t i s therefore f i n i t e . (i,,

m = 1,2,.

degree,

An i n d e x i s d e f i n a b l e f o r e a c h s o l u t i o n

.., k <

a n d , by t h e a d d i t i v e p r o p e r t y o f

a);

(2.4) implies that k

C i =I. m m= 1 Remark .( i f :The t y p i c a l p r o p e r t y t h a t s o l u t i o n s a r e i s o l a t e d

i s n o t u n i v e r s a l , a s t h e f o l l o w i n g couter-example s h o w s . C o n s i d e r t h e T a y l o r e x p e r i m e n t on f l o w b e t w e e n c o n c e n t r i c rotating cylinders,

and s u p p o s e t h a t t h e l e n g t h o f t h e f l u i d -

f i l l e d a n n u l u s i s s m a l l enough f o r a f l o w i n t h e form of j u s t t w o a x i s y m m e t r i c c e l l s t o b e p o s s i b l e a t m o d e r a t e v a l u e s of

R

(se

56).

I t i s w e l l known t h a t a l l s u c h c e l l u l a r f l o w s

i n t h e T a y l o r e x p e r i m e n t d e v e l o p a z i m u t h a l waves when

R

is

r a i s e d above a c r i t i c a l v a l u e , and i t a p p e a r s c e r t a i n t h a t

if

t h e i n n e r and o u t e r c y l i n d e r s a r e r o t a t e d i n o p p o s i t e d i r e c t i o n s , t h e i r v e l o c i t i e s c a n b e a d j u s t e d t o make t h e wave motions s t e a d y ( i . e .

t h e p h a s e of

t h e waves, which i s u s u a l l y

moving a z i m u t h a l l y , i s b r o u g h t t o r e s t a t p a r t i c u l a r combinations o f t h e boundary v e l o c i t i e s ) . flow i s p o s s i b l e t h a t i s n o n - t r i v i a l l y muthal c o - o r d i n a t e .

If

Thus, a s t e a d y

periodic i n the azi-

t h e a p p a r a t u s were p e r f e c t l y symmetric,

GENERIC B I F U R C A T I O N THEORY IN F L U I D MECHANICS

53

all translations of this flow would be equally possible. Therefore the solution of the hydrodynamic problem is not isolated.

It is likely, however, that such exceptions to

Property 1 are extremely rare. 2. The solutions

of (2.3)

Wm(R)

possibly intersecting) curves i n

Cm

are

IR+xH,

(though i = rtl m

and

except

at points o f intersection (two-sided bifurcation points) and turning points (one-sided bifurcation points). Remark (ii): In view of (4.1) this obviously implies that, except at isolated critic-a2values of

1

2

solutions is odd and

3. F o r

R

(k-1)

Rc

of

R,

the number

of them have index

sufficiently small, the solution

W1

k

-1.

is unique

and the f l o w that it represents is uncondition lly stable (i.e.

W1

is an attractor for the whole of

H

Remark .. - - flow may accordingly be defined. . I n - _ _ _ (iii): A primary R+XH,

the corresponding solution curve

one extending to

R = 0.

W1(R)

Uniqueness for small

is the only R

is obvious-

ly to be expected, because (2.1) becomes a linear elliptic problem in the limit

R

4

0.

Stability is to be expected

because the time-dependent, parabolic problem also becomes linear i n this limit.

4. At any non-critical value of by a solution with index

i = -1

R,

the flow represented

is unstable.

Remark (iv): Combined with Property 2, this implies that

i=l

is a necessary condition for stability of a steady flow away

from a bifurcation point.

But it is generally not a

sufficient condition for stability.

The most significant

54

T.B. BENJAMIN

p r a c t i c a l i n f e r e n c e t o be drawn i s t h a t any o b s e.r ~ v eId_ s t e a d y . f l o w must have i n d e x 1.

5. A t b i f u r c a t i o n p o i n t s , t h e d i m e n s i o n o f t h e c e n t r e manifold i n

R+xH

is

2,

and t h e i n d e x of t h e r e s p e c t i v e

solution i s zero. Remark-

(v):

This property i s a t t r i b u t a b l e t o bifurcations

that are realizable i n practice,

f o r the exceptions t o i t

depend on s i n g u l a r c o n d i t i o n s of

symmetry or o t h e r s p e c i a l

f a c t o r s which c a n n o t b e produced e x a c t l y i n a n y e x p e r i m e n t , b e i n g a n n u l l e d by any p e r t u r b a t i o n ( e x p e r i m e n t a l ' n o i s e ' however small. Property

Thus,

5 a r e not

f o r t h e problem (l), t h e e x c e p t i o n s t o

' s t r u c t u r a l l y s t a b l e ' , a l t h o u g h t h e y may

sometimes need c o n s i d e r a t i o n a s f e a t u r e s o f i d e a l i z e d t h e o r e t i c a l models.

< i= 0

r

)

R

f

Cr --iIO

- --

55

G E N E R I C BIFURCATION THEORY I N FLUID M E C H A N I C S

Consider the forms o f b i f u r c a t i o n i l l u s t r a t e d i n Fig.

1, where

mernhers of ( b ) and

r

i s any l i n e a r f u n c t i o n a l d i s c r i m i n a t i n g

the solution s e t

[Wn(R)].

I n t h e diagrams

( c ) t h e a x i s r e p r e s e n t s t h e p r i m a r y s o l u t i o n , and

a d d i t i o n a l solutions a r e represented off c r i t i c a l form o f

Property

Fig.

c’

il = 1

The s u p e p

l ( a ) , being

i s e x c l u d e d by

a t the bifurc-

But t h e t r a n s c r i t i c a l ( a s y m m e t r i c ) form i n

l ( b ) i s admissible,

p a i r of

R = R

5 s i n c e (4.1) r e q u i r e s t h a t

ation point.

the axis.

bifurcation i l l u s t r a t e d i n Fig.

symmetric a t t h e b i f h r c a t i o n p o i n t

Fig.

as a l s o i s t h a t i n Fig.

l ( c ) where a

s o l u t i o n s a r i s e s s e p a r a t e l y from t h e p r i m a r y s o l u t i o n .

l ( d ) shes a n o t h e r c a s e c o n s i s t e i i t w i t h P r o p e r t y

primary where

5: t h e

s o l u t i o n curve i s f o l d e d , having t w o t u r n i n g p o i n t s il = 0 .

single-valued

The t y p i c a l p r o c e s s whereby a n o r i g i n a l l y s o l u t i o n curve i n

IR+

x H

becomes f o l d e d a s a

s u p p l e m e n t a r y p a r a m e t e r i s v a r i e d w i l l be d i s c u s s e d i n Remark

(a),

$5.

( v i ) : I n t h e i d e a l i z e d t h e o r e t i c a l m o d e l for t h e Taylor

experiment,

t o infinity.

t h e e n d s of

t h e f l u i d - f i l l e d a n n u l u s a r e removed

A c c o r d i n g l y , a n a r b i t r a r y c o n d i t i o n of p e r i o d -

i c i t y i s imposed i n p l a c e of

t h e r e a l i s t i c end c o n d i t i o n s

i n c l u d e d i n t h e problem ( 2 . 1 ) . p r o v i d e s examples of

A s i s w e l l known,

t h i s model

s u p e r c r i t i c a l b i f u r c a t i o n , but

no r e a s o n t o e x p e c t P r o p e r t y

there is

5 t o be evaded by a bounded

model o f t h e T a y l o r f l o w . P r o p e r t y 5 does not

carry

over w i t h t h e s a m e f o r c e

t o t h e i n t e r p r e t a t i o n o f t h e Bdnard e x p e r i m e n t on i n c i p i e n t c o n v e c t i v e m o t i o n , f o r which t h e t h e o r e t i c a l problem i s a k i n

to

( 2 . 1 ) b u t somewhat more c o m p l i c a t e d .

Among v a r i o u s bounded __

56

T.B.

BENJAMIN

models that may be considered, the simplest o f them features supercritical bifurcations and s o Property 5 does not apply.

In this model the thermal boundary conditions are taken to be commensurate with a motionless state of the fluid for all values of the temperature parameter (the Rayleigh number), the dependence of viscosity on temperature is ignored, and

If any one of these three

internal heating is also ignored.

simplifications is relaxed, however, Property 5 becomes typical.

For a discussion o f these facts, see the Appendix to the paper by Benjamin (1977a).

5.

Morphogenesis T o introduce a far-reaching idea whose importance

in the hydrodynamic problem will be demonstrated presently we

turn to a familiar problem i n elasticity.

Its qualitative

aspects being intuitively obvious, this problem serves to secure the idea i n question, whose generality then becomes plain i n terms of Leray-Schauder degree.

The problem is

considered i n three progressive stages of elaboration, the first two of which are well known but the last of which presents a new aspect. (i) We first consider Euler's elementary model of buckling (cf. Love 1927, $263; Reiss 1969).

A thin inextensible but

flexible wire (elastica), constrained to lie i n a plane and straight when unstressed, is subjected to a compressive end load

P.

The bending stiffness

only of arc length s = 0

and

s = ,f.

s

$

of the wire is a function

along its centreline, and its ends at

are free to turn.

If

y

is the displace-

57

G E N E R I C B I F U R C A T I O N THEORY I N F L U I D M E C H A N I C S

ment of then

t h e c e n t r e l i n e from t h e s t r a i g h t l i n e b e t w e e n t h e ends,

dy/ds

= sin

where

@,

i s angle t h a t

In t e r m s of

makes w i t h t h i s line. of

0

t h e tangent

= @(sf,

@

the equation

e q u i l i b r i u m and t h e end conditions a r e

(5.1) and i n p h y s i c a l r e s p e c t s n o g e n e r a l i t y i s l o s t by t h e

< n.

assumption t h a t

P,

For a l l

( j . 1 ) has the

r e p r e s e n t i n g t h e unflexed

s t a t e of

additional solutions a r i s e , exceeds t h e

@

the wire.

0+

and

t h e l i n e a r i z e d form of

i s a constant).

@-

@ z 0

But t w o

= -@+,

bhen

P

which i s t h e f i r s t

Pc

‘Euler buckling load’

eigenvalue o f if

say

t r i v i a l solution

( 5 . 1 ) thus

The m u l t i p l i c i t y o f

Pc = O n 2 / L

2

solutions further

i n c r e a s e s by t w o a t e a c h of a n i n f i n i t e s e q u e n c e o f h i g h e r c r i t i c a l l o a d s , b u t we c o n f i n e a t t e n t i o n t o t h e f i r 5 t p a i r o f non-trivial

solutions.

f a c t a case o f

The g e n e s i s of

these solutions i s i n

s u p e r c r i t i c a l b i f u r c a t i o n from t h e n u l l

solution, as i l l u s t r a t e d by Fig.

l(a).

t h e f i g u r e c o u l d b e , f o r example

0(0)

The f u n c t i o n a l

or

$I(&),

f

in

although

f o r analysis the preferable choice i s f

where

C(s)

value

Pc

=

i s t h e e i g e n f u n c t i o n corresponding t o the eigen (thus

5 =

Questions

cos(l~s/t) is

i s a constant).

o€ s t a b i l i t y c a n b e answered by s t u d y i n g

t h e p o t e n t i a l energy o f

P < Pc

@

t h e system.

the n u l l function realizes

It a p p e a r s t h a t when a minimum o f

t h e energy

58

T.B.

BENJAMIN

and so represents a stable state, but when

P

7

Pc

the un-

flexed state is unstable and then the non-trivial solutions @+

and

represent stablc states.

@-

The problem (5.1) can

be reformulated so that degree theory is applicable (rf. Krasnosel’skii 1964, pp. 181-183), and the disposition o f indices f o r the first bifurcation is found to b e a s shown in Fig. l(a). A perturbation analysis shows that, respective to

p(s),

every given positive function

(P,f)-plane is locally parabolic.

the branching in the

Thus the bifurcation is

described by the cubic equation

-

f3

(P-P )f = 0,

the real roots of which are f = &(P-P

)

f = 0

P > Pc.

for

(5.2) P

for all

and

O n the understanding that the

appropriate normalizing factor is included i n the definition of

f,

this representation holds for all choices of the

linear functional such that

f(@+) = -f(@-)

#

0.

(ii) W e next consider the more general problem where the unstressed wire need not b e perfectly straight, that is, where

Q, = @ o ( s ) f o r

letting the number flexure.

G

P = 0.

We write

(p@g)’ =

EF(s),

represent the magnitude of the residual

While the end conditions remain the same as before,

the equation o f equilibrium becomes

+ P sin

(PO’)‘

@ = SF(S).

(5.3)

The solution set is now qualitatively different from the previous case

E

= 0.

T h e symmetric bifurcation of

before is unfolded i n the manner that was first made clear

59

G E N E R I C B I F U R C A T I O N T W O R Y IU F L U I D M E C H A N I C S

i n the hritings of W.T.

Koiter about

It can be shown that provided

problems of elastic stability. Q0(s)

‘imperfection-sensitive’

satisfies the undemanding condition

‘0

I€\,

the situation for small

P-P

with

positive and small

o r negative, is described by the cubic equation

f7

-

(l’-Pc)f

-

6

= 0

(cf. Chillingworth 1 9 7 5 , Thompson & Hunt

(5.k)

1975).

P

FIGURE 2.

FIGURE 3 .

This situation is illustrated in Fig. 2 , where the real roots of

(5.4)

are plotted against

curves are for

E

= 0.05,

P

with

P c = 1.

The continurns

and the dashed curves f o r

E

= 0.

Note also the disposition in indicesshownin the figure. The discriminant of the cubic values o f

P

and

E

satisfying

(5.4)

vanishes for

60

T.B.

BENJAMIN

which d e s c r i b e s a c u s p i n t h e

(P,E)-,~lane(Fig.

7).

For

p o i n t s i n s i d e t h e c u s p t h e c u b i c has t h r e e r e a l r o o t s , f o r p o i n t s o u t s i d e i t h a s only one. d e s c r i b e d by

(5.4) i s a

and

The u n f o l d i n g p r o c e s s

p a r t i c u l a r i n s t a n c e of t h e a l g e b r a i c

c h a r a c t e r i z a t i o n commonly known by t h e t e r m

'cusp c a t a s t r o p h e ' .

The s i t u a t i o n i s i n t u i t i v e l y o b v i o u s

f r o m Fig.

4.

The p r i m a r y mode of

the

f l e x i n g upwards i n t h e f i g u r e ,

-system,

P

P d e v e l o p s s m o o t h l y and r e m a i n s

i- L(;LRF: ' t .

with increasing

P.

stable

P

But f o r

s u f f i c i e n t l y l a r g e , a n o t h e r s t a b l e mode becomes p o s s i b l e , f l e x i n g downwards evident that

there i s an intervening s t a t e o f

which i s u n s t a b l e .

and i t i s

i n the figure,

equilibrium

Having b e e n r e a l i z e d by p u s h i n g t h e column

dohnwards u n d e r a s u f f i c i e n t l y l a r g e mode w i l l a b r u p t l y d i s a p p e a r when

P,

t h e s t a b l e secondary

P i s reduced g r a d u a l l y t o

the c r i t i c a l v a l u e corresponding t o t h e one-sided p o i n t i n t h e lower h a l f mined a p p r o x i m a t e l y by

of F i g .

(5.5)

( i i i ) Suppose now t h a t

2

when

(i.e.

c

bifurcation

t o the value deter-

i s small).

t h e p e r t u r b e d system j u s t c o n s i d e r -

ed i s s u b j e c t e d t o a n a d d i t i o n a l p h y s i c a l f a c t o r t e n d i n g t o c a n c e l t h e p r i o r i t y o f t h e p r i m a r y mode.

F o r example, a

b e n d i n g moment may b e a p p l i e d a t t h e end

s = 0

column,

s o t h a t t h e end c o n d i t i o n becomes

It i s c l e a r from F i g . the

left-hand

4

of

the

Q ' ( 0 ) = m.

t h a t a c l o c k w i s e moment a p p l i e d a t

end w i l l have t h e r e q u i r e d e f f e c t , and i f

large

61

G E N E R I C BIFURCATION TIIEORY I N FLUID M E C € I A N I C S

e n o u g h i t w i l l c a u s e t h e column t o b u c k l e downwards r a t h e r t h a n upwards u n d e r a g r a d u a l l y i n c r e a s e d l o a d

P.

mode i s t h u s made t o u n d e r g o a m o r p h o g e n e s i s :

t h e continuum

~

The primary

~~~

s t a t e s g e n e r a t e d by i n c r e a s i n g t h e l o a d g r a d u a l l y f r o m z e r o

of

c h a n g e s f r o m d e f i n i t e l y u p w a r d s - t o d e f i n i t e l y downwards, l e a v i n g t h e u p w a r d s c o n t i n u u m a s a s e c o n d a r y mode r e a l i z a b l e

It i s c l e a r , moreover, that

only a t s u f f i c i e n t l y l a r g e loads. if

i s small,

E

then a comparatively s m a l l

m

w i l l

switch

Clie p r i m a r y mode f r o m u p w a r d s t o downwards. I n view of p r o p e r t i e s a p p l i c a b i l i t y of §I+),

degree theory

the nature of

that are inherent i n the (like the properties

t h e morphogenesis can be p r e d i c t e d i n

geometrical terms without

explicit

analysis.

d i a g r a m s u c h a s i s m o d e l l e d by F i g .

a

P = 0

to the f a r

t h e d i a g r a m , and a s e c c i n d a r y mode by o n e b r a n c h o f

l o o p s u c h as i n t h e l o w e r h a l f

of' F i g .

i n d i c e s i s a l w a y s 1, t h e c u r v e s a r e points,

In a state

2 , a p r i m a r y mode i s

r e p r e s e n t e d by a c u r v e e x t e n d i n g from right of

listed in

Cm

2.

The sum of' t h e

except a t bifurcatim

and d i s c o n n e c t e d s e c o n d a r y l o o p s a r e s m o o t h a t t h e i r

extremities

(i.e.

locally p a r a b o l i c ) .

W e seek a process

whereby a n o r i g i n a l p r i m a r y modc a n d a s e c o n d a r y mode e x change r o l e s as a supplementary parameter continuously.

is varied

Hence w e e x p e r t t h e p r i m a r y and s e c o n d a r y

c u r v e s t o b e deformed c o n t i n u o u s l y s o t h a t i n t o c o n t a c t a t one p o i n t . order,

m

they a r e brought

They t h e n d i v i d e i n a d i f f e r e n t

f o r m i n g a new p r i m a r y c u r v e a n d new s e c o n d a r y l o o p . One p o s s i b i l i t y f o r s u c h a p r o c e s s i s t h e u n f o l d i n g

o f a symmetric s u p e r c r i t i c a l b i f u r c a t i o n ,

a s d e s c r i b e d by

62

T.B.

(5.4),

but

BENJAMIN

t h i s i s e v i d e n t l y not ge n e r i c i n t h e considered

c l a s s of p h y s i c a l p o s s i b i l i t i e s . singular specifications

(i.e.

w i l l e n f o r c e symmetry o f

of

I n o t h e r words,

~ ( s )i n ( 5 . 2 ) )

the function

t h e m u t a t i o n when t h e p r i m a r y and

secondary c u r v e s i n a s t a t e diagram a r e brought The s i n g u l a r e v e n t u a l i t y would c o r r e s p o n d the generic Property

(c F I G U R E 5.

only

into contact.

t o a n e v a s i o n of

5 l i s t e d i n 93.

1

(a)

S u c c e s s i v e f o r m s of

l o c i i n a s t a t e diagram f v s P.

T h e r e r e m a i n s o n l y one p o s s i b l e form o f e v e n t s , which i s i l l u s t r a t e d i n F i g . sense (a)

( d ) of

4

as t he parameter

5.

The p r o c e s s m a y g o i n t h e

t h e f o u r diagrams,

m

i s i n c r e a s e d , but

o r i n t h e r e v e r s e sense, i n e i t h e r case the

p r i m a r y mode m u t a t e s a t a t r a n s c r i t i c a l b i f u r c a t i o n p o i n t a s shown i n t h e d i a g r a m ( c ) . by t h e f i g u r e i s t h a t

An i m p o r t a n t c o n c l u s i o n made c l e a r

the continuous process

through t h i s b i -

f u r c a t i o n n e c e s s a r i l y causes t h e primary curve t o be f o l d e d (i.e.

a s i l l u s t r a t e d by F i g .

parameter

m

(P,m)-plane,

1 ( b ) ) when t h e s u p p l e m e n t a r y

l i e s i n some i n t e r v a l .

P r o j e c t e d onto t h e

t h e f o l d a p p e a r s a s a c u s p , whose a p e x

63

G E N E R I C BIFURCATLON THEORY I N FLUID M E C H A N I C S

f = f(P)

c o r r e s p o n d s t o t h e p o i n t where t h e p r i m a r y c u r v e

T h i s i s a n o t h e r cusp c a t a s t r o -

f i r s t has a v e r t i c a l tangent.

p h e , b u t w i t h o r i e n t a t i o n d i f f e r e n t from t h e one c o n s i d e r e d previously.

The f o l d i n g of

i n practice,

a h y s t e r e s i s w i l l be m a n i f e s t e d i n t h e dependence

of

t h e p r i m a r y s t a t e on

t h e primary curve i m p l i e s t h a t ,

P.

A p p r e c i a t i n g t h e g e n e r a l i t y of

t h i s argument,

we

may p r e s e n t t h e c o n c l u s i o n a s f o l l o w s , where no p a r t i c u l a r p h y s i c a l meaning i s a t t a c h e d t o

and

P

m:

P R I N C I P L E O F MORPHOGENESIS W I T H RESPECT T O TWO PARAMETERS.

u E H

Suppose t h a t t h e s t a t e s a primary

'loading'

parameter

m. ~

-a b_-l e- (A.e.--a

P E R+

parameter

F o r every

s. u f f i c i e n t l y s mall,

o f a s y s t e m a r e d e t e r m i n e d by

m,

u(P)

and a s u p p l e m e n t a r y

i s u n i q u e when -

s o t h -~ a t - a p r i. m a r y s e t o f s t a t e s i s d e fin~

R+XH).

curve i n

~

-

Suppose ___ _. t h a t a s

c o n t i n u o u s l y t h r o - u g p z c_r_i t i.___ c a l -~ value u n d e r g-o__ e s a m o r p h o g e n e s i__ s (i.e. ~

mc,

m

i s _va r ied _

t h e p r i m a r y ___ set ~

a r a d i c a l q.u a l i t a t i v e change -

i n t.. h e__ s.-t . a. t e .s d.e v e l o p e d a t s u f f-i c i e n t l y _l ~a r_g_e_ . -- -

TbE,

P).

~

g_e_n e r i c a l l y~, t!le P

when - __

m

is

P

~s y s. t e_ m-

man -i f e s t s h y s t e r-____ e s i s w i t h res-pect

l_i_e-s -i n a n i n t e r v a l above - o- r below -

B "c

C

FIGURE 6 .

m

.

to

64

T.B.

BENJAMIN

This principle is illustrated in Fig. 6, which shows the typical form of the 'catastrophe set' i n the (P,m)-plane. Except f o r

B

and

C,

every point i n the set is the project-

ion o f a one-sided bifurcation point, the extremity of a secondary loop, i n R+xH

.

T h e turning point

B

corresponds

to the transcritical bifurcation point shown i n F i g . and the cusp between

C

and

fold in the primary curve

m = m

C

u = u(P).

5(c),

is the projection of the

T h e generic attribute

being asserted is simply that the cusp points either upwards or downwards relative to the P-axis.

The singular case where

it is aligned with the P-axis recovers the case o f mutation through a symmetric bifurcation, which we have earlier dismissed as a practical possibility. It is noteworthy that closely parallel reasoning has been used by Zeeman (1975) a s the basis of a speculation about biological processes.

6.

Experimeptal results The qiialitative predictions reviewed in 994 and

5

have been borne out by some recent experiments on steady flows.

In particular, morphogenetic processes have been studied, and they have been found to accord with the general principle explained above.

We summarize as follows the outcome o f one

set of experiments which has already been reported (Benjamin 1977b). Fig. 7 is a diagram o f the apparatus, which is basically the same as that used i n the famous experiments o f Taylor (1923) mentioned earlier.

The length

4,

of the fluid

G E N E R I C R I F U R C A T L O N F - I E O R Y I N FLUID M E C H A N I C S

f i l l e d annulus i s comparatively continuously adjustable.

s m a l l , however,

The o u t e r c y l i n d e r made of a

62

i s r o t a t e d a t c o n s t a n t speed

a continuous range o f speeds. arid o u t e r r a d i i i s 0 . 6 2 ,

( r a d / s ) by a s e r v o m o t o r w i t h

2

R = 62r,/V,

the fluid.

hhere

R.

wide r a n g e s o f

R

cells.

(mm2/s)

R,

but

a l l of

i t was

? t a b l e s t e a d y f l o w s e x i s t s over

The p r i m a r y f l o w was p r o d u c e d by g r a d u a l -

R

f r o m s m a l l v a l u e s , and f o r

a b o u t 100 i t s s t r u c t u r e became q u i t e c l e a r , number of

V

flow v i s u a l i z a t i o n .

which became u n s t e a d y a t s u f f i c i e n t l y l a r g e of

- 2 3 mm. -

a pearly

V a r i o u s a x i s y m m e t r i c f l o w s were o b s e r v e d ,

found t h a t a m u l t i p l i c i t y

the inner

An aqueous s o l u t i o n

i n which a s m a l l amount o f

s u b s t a n c e w a s mixed f o r t h e p u r p o s e o f

of

d = r '-rl

and t l i c . gap w i d t h

i s t h e k i n e m a t i c v i s c o s i t y of

g l y c e r i n bas u s e d ,

rl/r2

The r a t i o

The Reynolds number i s g i v e n b y

l y increasing

trans-

( P e r s p e x ) i s f i x e d , and t h e i n n e r c y l i n d e r

parent material

of

and i s

g r e a t e r than

f e a t u r i n g a n even

c e l l s w i t h c o n t r a r y s p i r a l l i n g motions i n a d j a c e n t

S e c o n d a r y modes were p r o d u c i b l e by s e v e r a l means,

s u c h a s by s t a r t i n g t h e m o t i o n of

L

f r o m r e s t , or by v a r y i n g

t h e i n n e r c y l i n d e r suddenly

g r a d u a l l y t o change a w e l l -

d e v e l o p e d p r i m a r y f l o w i n t o a s t a t e where i t i s no l o n g e r primary. A n example of

t h e p o s s i b i l i t i e s i s g i v e n with photo-

graphs i n t h e c i t e d paper. demonstrated a t

L/d

= 3.3,

t w o c e l l s f o r t h i s v a l u e of modes h a v e f o u r c e l l s .

F i v e d i s t i n c t s t e a d y flows a r e

R = 650. L/d,

I n the

The p r i m a r y f l o i q h a s

and t w o of

'normal'

t h e secondary

f o u r - c e l l mode,

c a l l e d because it occurs a s t h e primary flow f o r l a r g e r

so

66

T.B.

values o f

L/d,

BENJAMIN

the motion is inwards near the end walls;

and i n the other, abnormal four-cell mode, the motion is outwards there.

The remaining two secondary modes have three

cells, with a n abnormal cell at the t o p of the annulus in one case and at the bottom in the other. 2 and

4 explained i n $4, this evidence

Because of Properties o f five different

stable flows implies the existence o f four other flows which, having index

-1, are necessarily unstable.

FIGUIILI 7 .

-dh 1.01

FIGURE 8.

1.

a:

0

I

400

I

600

R*

G E N E R r C n L F U R C A T I O N TYEORY I N F L U I D M E C H A N I C S

4.01-

-1

d -

3.8

-

3.6 -

3.4 -

> -

ioo

160 IZ

?GO

140

120

FIGURE 9. The s t r u c t u r e o f

t h e primary f l o w ,

d i s t i n c t l y a t moderately l a r g e

L/d,

and i t i s o f

R,

as manifest

d e p e n d s d i s c o n t i n u o u s l y on

obvious i n t e r e s t t o i n q u i r e i n t o t h e

n a t u r e of' t h e m u t a t i o n t h a t

takes place a t c r i t i c a l values of

k/d.

i n v e s t i g a t i o n was t o measure

The a d o p t e d method of

properties

of

t h e s e c o n d a r y modes t h a t e v o l v e i n t o t h e

p r i m a r y flow a s t h e c r i t i c a l v a l u e i s c r o s s e d r e s p e c t i v e l y from a b o v e or below. Fig.

A s a n example o f s u c h m e a s u r e m e n t s ,

8 shows e x p e r i m e n t a l v a l u e s of t h e h e i g h t

c e l l i n t h e normal f o u r - c e l l

d e c r e a s i n g f u n c t i o n of

R

significant feature of Fig.

s i d e of

R

end,

and h

is a

The most

8 i s t h a t t h e curve

has a v e r t i c a l tangent a t i t s left-hand

t h e end

where t h e

the f i g u r e ,

up t o t h i s l i m i t .

of

L/d = 3 . 3 .

s e c o n d a r y mode a t

T h i s f l o w becomes u n s t e a d y a t t h e v a l u e of c u r v e e n d s on t h e r i g h t - h a n d

h

h = h(R)

indicating the

68

T.B.

c r i t i c a l v a l u e of (with

R

f o r t h e one-sided

bifurcation point

from which t h e s e c o n d a r y mode a r i s e s .

i=O)

t o the p r o p e r t i e s explained

i s a view of

R xII.

BENJAMIN

According

e a r l i e r , t h e experimental curve

one b r a n c h ( w i t h

i = l ) of a s e c o n d a r y l o o p i n

The complementary b r a n c h ( w i t h i = - l )r e p r e s e n t s a n

u n s t a b l e f l o w and t h e r e f o r e c a n n o t be r e a l i z e d e x p e r i m e n t a l l y . The r e s u l t s of many s u c h measurements a r e shown i n

9.

Fig.

For

L/d

B

below

i n the figure,

h a s t w o c e l l s , and t h e l i n e generating the four-cell t h e l o c u s of flow.

two-cell

R

flow i s primary,

is

b u t below

B

a new e f f e c t

i s gradually increased f r o m small values, a

f l o w w i t h embryonic f o u r - c e l l

i s then g r a d u a l l y reduced,

f e a t u r e s i s evolved

a t which a n a b r u p t

CB,

t r a n s i t i o n t o a c l e a r two-cell

DC,

DC

T h i s f l o w i s a s e c o n d a r y mode

B,

c o n t i n u o u s l y up t o t h e l i n e

line

The l i n e

c o r r e s p o n d i n g b i f u r c a t i o n s i n v o l v i n g t h e two-cell

and t h e f o u r - c e l l If

i s the locus o f bifurcations

s e c o n d a r y mode.

Above t h e l e v e l of

appears.

RC

t h e primary f l o w

s t r u c t u r e takes place.

If

R

t h i s s t r u c t u r e remains u n t i l t h e

a t which t h e f l o w r e t u r n s a b r u p t l y t o i t s o r i g i n a l .

form. The i n t e r p r e t a t i o n o f l y a s e x p l a i n e d a t t h e end of

ween F i g .

t h e s e o b s e r v a t i o n s i s precise-

$ 5 , and t h e e q u i v a l e n c e b e t -

6 and t h e e x p e r i m e n t a l F i g . 9 c a n b e a p p r e c i a t e d .

H y s t e r e s i s of t h e p r i m a r y t w o - c e l l

CB

flow t a k e s p l a c e between

and t h e o p p o s i t e s i d e o f t h e c u s p , and

B

corresponds

t o t h e t r a n s c r i t i c a l b i f u r c a t i o n p o i n t a t which t h e p r i m a r y f l o w m u t a t e s b e t w e e n t h e t w o - c e l l and f o u r - c e l l forms. O t h e r e x p e r i m e n t s have b e e n made o n s t e a d y f l o w s i n

GENERIC B I F U R C A T I O N THEORY I N F L U I D YECIIANICS

a domain w i t h s e m i - c i r c u l a r

c y l i n d r i c a l boundary,

69

t h e diametric

f a c e and e n d s o f w h i c h a r e s t a t i o n a r y a n d t h e c i r c u l a r portion of w h i c h i s r o t a t e d a t c o n 5 t a r r t s p e e d . llake c o m p l i c a t e d c e l l u l a r

f’orms, a n d ,

of‘ t h e p r i m a r y f l o w e v o l v e d a t liigh

l y o n t h e l e n g t h o f t h e domain. and t h e m u t a t i o n s o f

T h e o b s e r v e d €lobs

as before, R

the structure

depends d is c o n tin u o u s -

P r o p e r t i e s of

s e c o n d a r y modes

t h e p r i m a r y f l o w ha\-e a p p e a r e d t o be t h e

5ame q u a l i t a t i v e l y as i n t l i e f i r s t e x p e r i m e n t s ,

thus a g a i n

confirming t h e t h e o r e t i c a l p r e d i c t i o n s .

It should be noted f i n a l l y t h a t t h e n o n l i n e a r boundary-value

problems r e - p e c t i v e

t o t h e s e two s e t s o f

e x p e r i i n e n t s a r e beyond t l i e r e a c h o f Moreover, no numerical r c . + u l t

15

constructive theories.

yet

available.

The

clualitative theory i s therefore particularly useful i n c l a r i f y i n g s u c h experitrierital

fxndings.

REFERENCE S B e n j a m i n , T.R.

1976

Applications

o f Leray-Schauder d e g r e e

t h e o r y t o problems o f hydrodynamic s t a b i l i t y .

Benjamin, T.B. of

1977a

Benjamin, T.B.

I. Theory.

Proc.

Roy.

S O C . Land. -.A ~

~

[ F l u i d Meclianics R e s e a r c h I n s t i t u t e , U n i v .

of E s s e x , Rep.

of

B i f u r c a t i o n phenomena i n s t e a d y f l o w s

a viscous f l u i d .

359, 1-26.

Math. __ __

no 83.1

197713

a viscous

B i f u r c a t i o n phenomena i n s t e a d y f l o w s

fluid.

11. E x p e r i m e n t s .

~

~ Roy-._Soc. ~ c A.

359, 2 7 - 4 3 . Chillingworth,

D.

1975

The c a t a s t r o p h e o f a b u c k l e d beam.

I n Dyriamical S y s t e m s Mathematics v o l . -~

468,

-

Warwick pp.

1974.

86-91.

L e c t u r e s N o t e s i -_n

Berlin:

Springer-Verlag.

70

T.B. BENJAMIN

Foias, C.

Temam, R .

&

1977

Structure of the set of station-

ary solutions of the Navier-Stokes equations.

Communs -

Pure Appl. Math., 30, 149-164.

1964

Krasnosel'skii, M.A.

Topological Methods i n the Theory -

o f Nonlinear Integral Equations.

1929

Love, A.E.H.

London: Pergammon.

TA-Mathema&icaLTheor_y of Elasticity,

4th ed. Cambridge U n i J ersity Press. (Dover edition l9hh.)

1969

Reiss, E.L.

Column buckling

Bifurcation.

Xar-ue

-

A n elementary example o f

In Bifurcation Theory and - -Nonlinear _____ Eigen_ _ __ Problems (ed. J.B. Keller & S. Antman). New Y o r k : ~-

B en,jam in. Serrin, J.

1959

O n the stability o f viscous fluid motions.

Arch. Ration. Mech.- _ Analysis _ --

1923

Taylor, G.I.

1,1-17.

Stability o f a viscous liquid contained

between rotating cylinders.

Phil. Trans. Roy.~SOC. A-

2213, 2 8 9 - 3 4 3 . Thompson, J.M.T.

& Junt, G.W.

furcation theory. Tromba, A . J .

1975

Towards a unified bi-

z.

Math. Phys. 26, _. angew. ~- 581-604.

& Marsden, J.E.

1977 Generic finiteness of

the stationary solutions of the Navier-Stokes equations. (Preprint.) Zeeman, E.C.

1975

Levels of structure i n catastrophe theory

illustrated by applications in the social and biological sciences.

Proc. Int. Cong. Math., Vancouver 1974,

pp. 533-546.

Canadian Mathematical Congress.

G E N E R I C BIFURCATION THEORY I N F L U I D MECHANICS

L e c t u r e hy T . R .

71

Benjamin

S U P P LEME NT

A t the conclusion of

the lecture,

a practical

d e m o n s t r a t i o n w a s made i l l u s t r a t i n g t h e p o s s i b i l i t y of m u l t i p l e s t a b l e s t a t e s i n a s y s t e m w i t h i n f i n i t e freedom. A c o i l e d wire capable o€ l a r g e f l e x u r e s without

permanent s t r a i n ( ' c u r t a i n w i r e ' ) i s f i x e d i n a h o r i z o n t a l board, forming a n a r c h i n a v e r t i c a l plane.

The w i r e p a s s e s

t h r o u g h t h e b o a r d a t one e n d , g i v i n g a s p a r e l e n g t h b e n e a t h , and s o t h e l e n g t h varied.

L

of w i r e i n t h e a r c h c a n r e a d i l y be

[ T h i s kind o f wire roughly s i m u l a t e s E u l e r ' s e l a s t i c a

w i t h c o n s t a n t bending s t i f f n e s s

8,

moment t o c u r v a t u r e i n a p l a n e ( c f .

the r a t i o of bending

$5(i)).

But i n f a c t

p

i s a m i l d l y d e c r e a s i n g f u n c t i o n of c u r v a t u r e . ]

The p r o p e r t i e s s t a t e d i a g r a m , where

f

p o s i t i o n o f equilibrium. about t he L-axis,

of

t h i s system a r e i n d i c a t e d i n t h e

i s d e f l e x i o n from t h e u p r i g h t Ideally,

t h e diagram i s symmetrical

and t h e u n f o l d i n g d u e t o r e s i d u a l i m p e r f e c t -

72

T.B.

BENJAMIN

i o n s i s shown by t h e d a s h e d l i n e s .

The f o l l o w i n g p r o p e r t i e s

a r e noteworthy:

( i )F o r stable.

L < L’

,

the upright position i s unconditionally

T h i s i s d e m o n s t r a t e d by d e f o r m i n g t h e w i r e g r o s s l y

and t h e n r e l e a s i n g i t , whereupon t h e e x t r a e n e r g y i s q u i c k l y d i s s i p a t e d and t h e w i r e comes t o r e s t u p r i g h t .

( i i )F o r

L

L’,

i n a n i n t e r v a l above

position i s s t i l l stable

( d e m o n s t r a t e d by g i v i n g t h e w i r e a

s m a l l p u s h from t h i s p o s i t i o n ,

t o which i t t h e n r e t u r n s ) , b u t

t w o other s t a b l e e q u i l i b r i a e x i s t .

Given a s u f f i c i e n t l y

vigorous push f r o m t h e u p r i g h t p o s i t i o n , one o r o t h e r o f

the upright

t h e wire f a l l s i n t o

t h e s e lower p o s i t i o n s .

( i i i ) The e x i s t e n c e of

three stable

e q u i l i b r i a implies the existence of

t w o a d d i t i o n a l e q u i l i b r i a which a r e unstable. theory,

T h i s f o l l o w s by d e g r e e

or by t h e Morse i n e q u a l i t i e s

r e f e r r e d t o t h e energy f u n c t i o n a l ( e x p r e s s i n g s t r a i n energy p l u s g r a v i t y p o t e n t i a l ) , common s e n s e i n t h i s example.

E i t h e r one of

or by

these unstable

e q u i l i b r i a i s d e m o n s t r a t e d by g e n t l y g u i d i n g t h e w i r e i n t o t h i s position.

On b e i n g r e l e a s e d ,

t h e w i r e t h e n s l o w l y moves

away f r o m t h e p o s i t i o n e i t h e r upwards or downwards. ( i v ) The t u r n i n g p o i n t

P

i n t h e s t a t e diagram i s

d e m o n s t r a t e d by p u t t i n g t h e w i r e i n t o one o f positions of

s t a b l e equilibrium with

gradually decreasing

L

towards

L’.

L

i t s lower

> L’ ,

and t h e n

Thus t h e a r c

QP

is

G E N E R I C D I F U R C A T L O N THEORY I N F L U I D M E C H A N I C S

f o l l o w e d , and a s t h e upright

(v) For unstable.

P

i s a p p r o a c h e d t h e w i r e a b r u p t l y jumps t o

position.

L

sufficiently large,

t h e uprigllt

position is

The w i r e t h e n falls s p o n t a n e o u s l y if r e l e a s e d i n

t h i s position.

71

de La Penha, L.A. Medeiros (eds.) Contenporary Developments in Continuum Mechanics and Partial Differential Equations @North-Holland Publishing Company (1978) G.M.

THE HAMILTON-JACOBI-BELLMAN EQUATION FOR TWO OPERATORS VIA VARIATIONAL INEQUALITIES

HAIM BREZIS DGpartement de Mathgmatiques Paris VI

Universit;

4 pl. Jussieu, 75230 Paris Cedex 0 5

Let

(Au]

denote a family of second order elliptic

0c

operators on a bounded domain

where aa. 1J

p > 0

!,isj

f'(x),

is a constant, 2

2

N

R :

aa

(a > 0) 4

and

:a

RN,

+a.

Ij

5 E

are smooth and Given functions

we consider the problem: find

(1)

u(x)

n

on

and

such that u = 0

on

Sup {A'u(x) U

-

fa(x)]

= 0

an.

Problem (1) occurs in the theory of optimal stochastic control (see e.g.

In case

C21).

N 62 = R ,

assumptions on $L

p <

m.

difficult

Krylov [ 3 ] proves

f&)

His proof

-

(under reasonable

that (1) has a unique solution

-

u E

which is extremely technical and

relies on an explicit construction of

u

by use

of probability tools.

In what follows, I report on a joint work with L.C. Evans [l]. O u r results concern o n l y the case where the family

of two operators.

A'

consists

It is not clear how to extendour proofs to

the case of more than two operators.

O n the other hand our

THE H A M I L T O N - J A C O B I - B E L L M A N

EQUATION

75

method h a s some a d v a n t a g e s : a ) i t i s q u i t e s i m p l e and c o n s t r u c t i v e , b) i t leads t o c l a s s i c a l c)

solutions

(u E C2’a(n)),

i t i s v e r y f l e x i b l e and c a n be a d a p t e d : ( i ) t o t h e c a s e where

A1

and

( i i ) t o t h e c a s e where

A1

i s e l l i p t i c and

parabolic

are parabolic

A*

( t h i s h a s been d o n e by P . L .

is

A2

-

Lions)

s u c h a problem o c c u r s i n a w o r k of B e n s o u s s a n and L e s ourne

.

Our main r e s u l t s a r e t h e f o l l o w i n g Theorem 1 then

V

f

1

Theorem 2 of

Po > 0

There i s a c o n s t a n t

,f

2

E L‘(n)

1 Max { A u

(2)

u

-

-

If

3

-

11

E H2((n)

2 fl, A u

Theorem 3

-

When

solution

u

of

to

f ]

H:(n) = 0

> p0,

@

unique s o l v i n g a.e.

f 1 , f 2 E H1(n)

i n addition

( 2 ) belongs

2

-

n

such t h a t i f

in

R.

then the s o l u t i o n

H3(n).

f1,f2

E

W1

”(n)

p

>

N,

then the

~ ~ , ~ f( o nr some )

o

( 2 ) belongs

with

o < a < 1.

Remarks and open p r o b l e m s . f1,f2 E C”(n),

1) I n g e n e r a l , e v e n f o r

( 2 ) does n o t belong t o = A,

(2)

continuous

reduces

prove a t b e s t t h a t a s k i s whether

C1

i n general.

u E

C2’l.

a

A’

of

2 = A =

which i s o n l y l i p s c h i t z T h e r e f o r e we may hope t o

A more r e a s o n a b l e q u e s t i o n t o

v a < 1

u E C29a

s h a l l see l a t e r the

F o r example when

Au = M i n ( f 1 , f 2 }

to

and n o t

C3(n).

u

the s o l u t i o n

when

f 1 , f 2 E Cm.

A s we

which o c c u r s i n Theorem 3 i s found by

u s i n g D e G i o r g i s theorem,

t h e r e f o r e i t might be very s m a l l .

76

H A I M BREZIS

A related question i s :

( o r f o r what

p ’ s ?)

does

belong t o

u

W39p(n)

that

f1,f2 E Lm(n)

3

m

f1,f2 E LP(n),

2 ) It would b e o f i n t e r e s t t o p r o v e t h a t i f

u

p <

f1,f2 E Cm.

when

then the solution

+

of

u E W2”

(2) satisfies

u E W2”

p > N).

for some

( o r just A positive

answer would be v e r y u s e f u l i n d e a l i n g w i t h t h e problem: find

u

such t h a t

1 Max(A u

-

which o c c u r s i n q u e s t i o n s o f s t o c h a s t i c systems ( J . L .

f

1

,

2 A u

-

f

2

,

u]

= 0

on

R,

o p t i m a l s t o p p i n g t i m e for

L i o n s , p r i v a t e communication).

3 ) Many q u e s t i o n s c a n be a s k e d c o n c e r n i n g f o r example t h e ” f r e e boundary” s e p a r a t i n g the s e t s [A’u

= f

1

3,

1 [A1u < f ]

i t s s h a p e , smoothness e t c . . .

and

Many r e s u l t s h a v e

been o b t a i n e d r e c e n t l y o n v a r i o u s o t h e r f r e e boundary v a l u e problems

( s e e t h e works

Nirenberg,

o f C a f f a r e l l i , Friedman, K i n d e r l e h r e r ,

Riviere, e t c ...)

Sketch o f proof

f o r Theorem 1

-

B y an e a s y change o f unknown

I

we c a n a l w a y s assume t h a t 1 Max ( A u

(3) Set

1 2 -1 T = A (A ) ,

Dirichlet condition. operator i n Set

2 cp = A u

(A2)

-1

f2 = 0

that is

2 f , A u]

= 0

in

0.

being understood with zero

Clearly,

T

i s a l i n e a r and bounded

~ ~ ( 0 ) . a s new unknown,

By a w e l l known theorem of

and s o

( 3 ) becomes

Stampacchia ( s e e

141)

problem

(5)

THE H A M I L T O N - J A C O R I - B E L L M A N

tlas a u n i q u e s o l u t i o n p r o v i d e d '

'n

T W ' Y dx

2

alp)

T

i s coercive i n

a > 0.

E L2,

V $

1 \\here

=

v

-1

fI

Tq * $ d x =

n

A1(A2)

w*$

P

=

allvl

'hl

'

-1 (A')

$

2

T y * v dx

Therefore

al$l2

2

l7

'.

< Cllvll

i.e.

A v - A v dx

( b y a n i n e q u a l i t y o€ S o b o l e v s k i , H2 Appendix i n [ 11 ) .

3

L'

But

L2

f

77

EQUATION

see e.g.

2

u; = A v

since

the

I(! 1

and

S

L

L2

I1

o f p r o o f f o r Theorem 2

Sketch

-

We a r i t c ( 3 ) a s a m u l t i v a l u e d

equation

(6)

A

P

where

(6)

1 A u

(7)

P,

+ p(a2u)

U

3 f

=

r

y

by

+ 6 ( A 2u e )

E

= f

0

> 0

for

r > 0.

'2

in

y : R + IR

( f o r example) and

nondecreasing f u n c t i o n siirh t h a t

Y(r)

R,

in

i s t h e maximal monotone g r a p h d e f i n e d a s

a n d we a p p r o x i m a t e

where

~

Y(r)

= 0

i s a n y smooth for

r < 0,

It f o l l o w s f r o m t h e s t a n d a r d t h e o r y o f

monotone o p e r a t o r s ( a n d f r o m S o h o l e v s k i Is i n e q u a l i t y )

u

h a s a s o l u t i o n and t h a t

(7) is

G

+ u

s t r o n g l y nonlinear, but

smooth p r o v i d e d

f

as

0

in

11'.

F o r s i m p l i c i t y we d r o p now

is.

1 (A u

+

)

+~ 8 '

x

2 2 ( A u ) ( A u),

we find

= fx.

(7)

Equation

e l l i p t i c and t h e r e f o r e

Differentiating (7) with respect t o

(8)

E

that

uF 6 .

i s

78 Let

HAIM B R E Z I S

;

b e a smooth f u n c t i o n w i t h compact s u p p o r t i n

M u l t i p l y i n g ( 8 ) by

2

C2(A u

)

a.

~y i e l d s

(9) Note t h a t

R1,

where

R2

so that (10) 0 ) and S o b o l e v s k i ‘ s i n e q u a l i t y t h a t

I n e q u a l i t y (11) l e a d s independent of

c

To e s t a b l i s h t h a t

E H’

-

in

Hloc

Hloc’

u p t o t h e b o u n d a r y one u s e s f i r s t

l o c a l c h a r t s t o s t r a i g h t e n t h e boundary. a s above

3

c

3

and t h e r e f o r e LI

u

t o an estimate f o r

used i n t a n g e n t i a l d i r e c t i o n s

t h i r d order d e r i v a t i v e s , except

u

The same a r g u m e n t

-

x x x ’

shows t h a t a l l belong t o

N N N

(xN

d e n o t e s t h e normal v a r i a b l e ) .

F i n a l l y we g o b a c k t o e q u a t i o n

with

S1,S2

E

H1,

But U

x x

( 1 2 ) says t h a t

= min(-

N N

and t h e r e f o r e

1 E H .

u X~ X~

( 3 ) w h i c h we w r i t e as

s1 , 7s2 1 1

a

NN

L2

THE HANILTON- JACOBI-BELLMAN E Q U A T I O N

-

Sketch of proof for Theorem 3 ~

(13)

M E A1

Set -

> 0

6

79

s o small that

A2

E

is still uniformly elliptic with top order coefficients

ak d *

We may rewrite ( 3 ) as

(14)

+

max{Mu

E A

2

-

u

f, A ~ U ]= 0

0.

in

Set v = M u - f

(15) and s o

(14) becomes ~2 u

E

(16)

We already know that

v

+

v

+

= i n~

6 M

2

A u

M

let us apply

HI-;

resulting expressions makes sense i n

(17)

a.

+

H")

to (16) (the

:

M v+ = 0.

The commutator 2 2 K u z M A u - A M u

involves at most third order derivates from (15) and

(18)

of

u.

(17) that C[A

2

(v+f)

+

Ku]

+

M V+ = 0.

We rewrite (18) as

(gkL

vXk ) x

= Rl + R 2

4,

where

(XE

denotes the characteristic function of

E)

It follows

80

HAIM B R E Z I S

where

a

ap

E

(note f o r example that

Lm

can be included i n

R2

The coefficients

a

term like

by ( 1 5 ) ) . are bounded measurable and uniformly

elliptic; thus the interior estimates o f De Giorgi-MoserStampacchia apply to (19). u E H3,

Since

- -1

it f o l l o w s that

u

I

(and the usual modification when

therefore apply Theorem

,E ~L2” ,

~ J

N=l

w2 y 2 * * ( n 2 )

and so

J

A n easy bootstrap argument shows that

some u E

Lp

ux.x . E L2**(n2) 1

p > N.

a < 1.

1

F -

W e may

u E

estimates with

:w:

Then Theorem 6 . 2 i n [ 4 1 implies that 0 <

2=

5.4 i n [41 to ( 1 9 ) to conclude that

Then by equation (15) and the standard

u E

1

where or 2).

n2 r c

R,.

f o r some

v E CoYa f o r

Schauder estimates applied to (15) lead to

c2ya. References

[l] H . Brezis [2]

W.H.

-

L.C.

Fleming

-

Evans,

to appear.

R.W. Rishel, Deterministic and stochastic

optimal control, Springer ( 1 9 7 5 ) .

[33 N.V. Krylov, Control of a solution o f a stochastic integral equation, Th. Proba. Appl. 1_7 (1972) p.114-13L [4]

G. Stampacchia, Equations elliptiques d u second ordre & coefficients discontinus, Presses Univ. Montreal(l966).

G.M. de La Penha, L.A. Medeiros (eds.) Contemporary Developments in Continuum Mechanics and Partial Differential Equations @North-Holl and Pub1 ishing Company (1978)

N O N L I N E A R PROBLEMS R E L A T E D T O T I I E THOMAS-FERMI

EQUATION

HAIM H R E Z I S

D 6 p a r t e m e n t d e MathGmatiques U n i v e r s i t G P a r i s VI pl.

’I

Jussieu

75230 P a r i s Cedex 05

Consider t h e following f u n c t i o n a l

p : a Z 3 + R+

where

and

a n i m p o r t a n t example i s

V:

R3

V(X)

+ R.

V(x)

I<

m.

c

=

i s a g i v e n function:

m

where

i=t Ix-ail ai

E ~3

e(p)

where

> 0

and

a r e given.

i s c a l l e d t h e Thomas-Fermi

f u n c t i o n a l and t h e f o l l o w -

i n g p r o b l e m o c c u r s i n quantum m e c h a n i c s (1)

i

Min e ( p ) w h e r e K = ( p PEK I > 0

[41 )

E L 1 (IR 3) , p 2 O a n d

i s a f i x e d number.

The f u n c t i o n electrons.

(see

p

t o be determined r e p r e s e n t s a d e n s i t y of

The s y s t e m c o n s i s t s o f e l e c t r o n s and o f

positive nuclei of The f u n c t i o n a l

e

t h e k i n e t i c energy,

charge

mi

has 3 terms,

placed a t p o i n t s

a.

k i n space.

corresponding r e s p e c t i v e l y to

t h e a t t r a c t i v e p o t e n t i a l energy

( i n t e r a c t i o n b e t w e e n e l e c t r o n s and n u c l e i ) and t h e r e p u l s i v e p o t e n t i a l energy ( i n t e r a c t i o n between e l e c t r o n s ) . Even t h o u g h p r o b l e m (1) seems t o b e a s i m p l e c o n v e x m i n i m i z a t i o n problem, i t need n o t have a s o l u t i o n .

The

82

HAIM BREZIS

pn

difficulty lies in the fact that if sequence for (l), then 1< p

5 / 3 and

that /p(x)dx K

S

converges weakly to

pn

e(pn)

is a minimizing

7

e(p),

in Lp,

but we can only assert

In fact if we replace in (1) the convex

I.

by = (p E Ll,

and

p 2 0

/p(x)dx

then the problem becomes much simpler.

< I]

To convince the reader

that there is a serious difficulty in solving (l), we begin with a non-existence result. 1

Proposition 1

-

as

Then, n o solution of problem (1) exists (no

1x1 -+ m.

matter what Proof:

I

Clearly

Assume

V(x)

E Lloc,

V

S

0

a.e. and V ( x) -+ 0

is). e(p) 1 0

p

for

E K.

On the otherhand we

have

Indeed, we choose ball of radius

R,

I ' = ' R = -

lBRl

where

BR

denotes the

XBR

centered at

0

in

R3

and

X

is the BR

characteristic function.

It is not hard to check (using

properties of the Marcinkiewicz spaces, see e.g. 131 Appendix)

and therefore

C (p,)

-+ 0

as

R

.$

-.

Thus (1) can not have a solution since it would have to be p

E

0.

We recall first remarkable resultsdue to E. Lieb and

B. Simon [41.

NONLINEAR PROBLEMS RELATED T O THE T H O M A S - F E R M I

Theorem 2

-

k m C -i=l lx-ail

V(x) =

Assume

83

EQUATION

k

=

I.

and s e t

C mi. i=l

Then ( a ) If

0 < I < Io,

(b) If

I > Io,

( c ) If

I < I .

problem (1) h a s a unique s o l u t i o n .

( 1 ) h a s no s o l u t i o n .

problem

( 1 ) h a s compact

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

support.

The proof

-

then

in

-

[ 4 ] i s i n d i r e c t : f i r s t one s o l v e s

Min E ( p ) K one shows t h a t i f 0 < I S

-

by a c l e v e r a r g u m e n t

the solution l i e s i n f a c t i n

K.

a j o i n t w o r k w i t h Ph.

( t o a p p e a r ) which e x t e n d s

Renilan

I n what f o l l o w s

I

and

Io,

describe

Theorem 2 i n v a r i o u s d i r e c t i o n s : 1)

p5”

i s r e p l a c e d by

j(p)

where

i s any

j

C1

convex

j ( 0 ) = j ’ ( 0 ) = 0.

f u n c t i o n such t h a t

m. 2)

general function d e t e r m i n e what

--I- i s r e p l a c e d by a

V(x) =

The s p e c i a l p o t e n t i a l

V(x).

I

I

x-ai O f course w e w i l l have t o

plays t h e r o l e of

Io.

3 ) The v i e w p o i n t i n s o l v i n g ( 1 ) i s t o t a l l y d i f f e r e n t f r o m t h e one i n

C41.

d e r i v e d f r o m (1) directly. ,j(p)

= pp,

some

p’s

Inf E K

=

One of

W e c o n s i d e r the Euler-Lagrange

-

which i s a n o n l i n e a r p . d . e . t h e advantages i s

that,

equation

-

and s o l v e i t

for exemple when

we c a n h a n d l e a l a r g e r r a n g e o f

p’s,

including

f o r which t h e v a r i a t i o n a l a p p r o a c h f a i l s s i n c e

-m.

Consider now

where

c

i s a normalization c o n s t a n t such t h a t

f u n d a m e n t a l s o l u t i o n of

-A

i.e.

IXI

is the

84

HAIM B R E Z I S

T h e Euler-Lagrange

e q u a t i o n d e r i v e d ( f o r m a l l y ) from Min K

(3)

e(p)

is

Here

E

p

Find

and f i n d a c o n s t a n t

K

an

[p >

03

j’(p)

-v+

an

[p

=

01

j’(p)

-

h

v +

p l a y s t h e r o l e of

with t he c o n s t r a i n t

I

B

B

1

so that

=

P

A.

2

P

a Lagrange m u l t i p l i e r a s s o c i a t e d

pdx = I .

Our n e x t r e s u l t d e s c r i b e s t h e

p r e c i s e l i n k between ( 3 ) and ( i t ) . Proposition 3

-

Assume t h a t

there i s a constant

j*(v+c)

(5)

d e n o t e s t h e c o n j u g a t e convex f u n c t i o n o f

-

j*(r) = s u p ( r s

such t h a t

E L ~ ( R ~ ) .

( 3 ) and ( 4 ) a r e e q u i v a l e n t .

Then t h e problems

C

j

Here

j*

i.e.

j(s)].

s2 0

The p r o o f t h a t when

j(r) =

o n l y when

p >

of P r o p o s i t i o n 3 w i l l be found i n [ 2 ] . and

3

V(x) = C

(in particular

mi

F5q ,

p =

-3

is

then

Note

(5) holds

O.K.)

From now o n , we w i l l c o n c e n t r a t e on ( 4 ) .

Onemain

r e s u l t i s the following Theorem k

-

Assume

(6)

V

1 E * L1

(7)

V(X) >

1.1

o

a.e.

(i.e.

AV E L1

and

V(m)

= 0)

on some s e t o f p o s i t i v e measure.

85

NOWLINEAR PROBLEMS R ELATED T O THE THOMAS-FERMI E Q U A T I O N

A)

€3)

C)

(a)

If

0 < I

(h)

If

I > Io,

When

I < I .

then

p

such t h a t

t h e r e e x i s t s a u n i q u e s o l u t i o n o€ ('t)

Io,

t h e r e e x i s t s no s o l u t i o n o f

V ( x ) -+ 0

and

as

1x1 -+

(4).

(uniformly),

a

h a s compact s u p p o r t .

S(r) = rp

If

> 0

I .

Then t h e r e e x i s t s

p 2

and

4/3

( t h i s i s j u s t a n example

-

t h e g e n e r a l assumption a p p e a r s l a t e r ) , t h e n

s I .

\(-Av)~x

r0

In p a r t i c u l a r D)

If

V E

I d e a of A)

1

dx.

z 0.

4

p > -

3

B)

= bounded m e a s u r e s o n

and C ) hold.

the proof.

(4)

we f i n d

-

v +

13

j'(p)

1x1 -+

t o transform

2

A

a.e.

on

we s e e t h a t n e c e s s a r i l y

m

(4)

P

IR

u = V - B

P

s o that

o

= u + 2

. I n order

i n t o a nonlinear p a r t i a l d i f f e r e n t i a l

(9)

j'(p)

3

A < O.

e q u a t i o n we i n t r o d u c e t h e new unknown

and

R3)

( f o r the general conditions see

F i r s t n o t e t h a t by

Letting

(-Av)' -:iV

when

(h

1.1

A),

l a t e r ) then

SVdx

-* h

j ( r ) = rP,

and

-

J

( 6 ) we assume t h a t

i n s t e a d of

(8)

=

<

u

X

on the s e t

[p >

+ X

on the s e t

[P

=

01

01.

86

HAIM B R E Z I S

In other words we find

-

u + x < o

o

i u + h >

p = o

=1

p = (j')

-1

(u+X).

We can summarize (11) i n a single equation

Y: IR

where

-b

[R

is defined by (j/)-'(r)

(13)

Problem

r >

for

r < 0

(4) becomes equivalent to Finding a function

-nu +

and a constant

u

1

=

Y(u+X)dx

the following.

(4).

First we

0 such that

o

= =

I.

(14) has been solved, the function

provides a solution of

X s

-zv

y(u+A) =

u(=

Once

o

for

~ ( r=)

p(x) = y(u(x)+X)

O u r approach i n solving

fix x

5;

0

and find

ux

(14) is

unique

solution of: -nux

+ y(ux+X) = - i v

(15)

such that

Y(ux+h) E L1.

The existence and uniqueness of

ux

follow f r o m the next

Lemma Lemma 1 ( [ 3 1 )

-

Let

function such that unique

u

8 : R + IR e(0)

solution of

= -3u

0.

+

b e a continuous nondecreasing Given

f

@ ( u )= f

E

Ll,

in

R3

there exists a and

u(=) =O.

87

N O N L I N E A R PROBLEMS RELATED T O THE TIIOMAS-FERMI E Q U A T I O N

X

Next we s e t f o r e a c h

I(X)

=

0

5;

[ Y ( u , ( x ) + X ) dx. i

I(X)

The f o l l o w i n g p r o p e r t i e s of

-

Lemma 2

I(h):

The f u n c t i o n

I ( , ) = 0.

l i m

n o n d e c r e a s i n g and

+

(-m,O]

play a c r u c i a l role: i s continuous,

[O,+m)

I ( X )

I n addition

is

X+-m

s t r i c t l y i n c r e a s i n g a s s o o n a s i t becomes p o s i t i v e and

>

I(0)

0.

I t f o l l o w s immediately t h a t w i t h

< I0

0 < I

h a s a unique s o l u t i o n f o r

= I(0)

I.

(14)

problem

and n o s o l u t i o n when

I > Io.

The f a c t t h a t c o n s e q u e n c e of

X +

as

I

i s nondecreasing i s an easy

t h e maximum p r i n c i p l e .

I ( X )

To s e e t h a t

+

0

i t i s s u f f i c i e n t t o n o t e t h a t ( b y maximum

-m

principle )

(17) and s o Next

y(uX+X) < y(V+X) + 0 I(0)

solution

> u

satisfies

X +

as

-03.

would imply t h a t t h e

of

y(uo)

E

0

+ Y (u,)

i.e.

u

S

= -AV

0

and

= V

u

-

acontradih

(7). I(x)

I n proving t h a t becomes p o s i t i v e , B) For a given

1

a.e.

I(0) = 0

otherwise

0;

I -nuo with

v

ux s

such t h a t

I

i s s t r i c t l y i n c r e a s i n g as s o o n a s i t

one u s e s Lemma such t h a t

I(X)

= I

3.5 f r o m [ 3 ] .

0 < I < Io,

satisfies

the corresponding

X < 0.

Since

88

HAIM B R E Z I S

p ( x ) = Y(u,(x)+X)

Y(V(x)+x)

p(x) = 0

follows that

when

Under t h e a s s u m p t i o n s of

C)

<

5

,/

/ p(u)' (-All)+

dx <

if'

dx.

and 1x1

V(x)

4

0

as

1x1

it

m

4

is sufficiently large.

Lemma 1 , o n e a l w a y s h a s

= /Y(uo)dx =/Y(uo)+dx

I .

Therefore

[31)

(see

dx.

The i n e q u a l i t y -AV d x < I .

(18)

i s more d e l i c a t e t o p r o v e .

I n f a c t one uses t h e following:

3 - Under t h e a s s u m p t i o n s of Lemma 1 a n d i n a d d i t i o n

Lemma

dx =

S k e t c h of

t h e proof

assume t h a t = 0

-

-

f a r away.

By a d e n s i t y a r g u m e n t w e c a n a l w a y s

E (r)

4

0

r

as

4

m

-

which shows t h a t

u(x) < c ( l x l ) 1x1 and t h e c o n c l u s i o n f o l l o w s .

-

When

holds provided

1

j(r) = - r p D p 2

1x1 F i n a l l y one

Y(u) =

2 + P-1 (u ) and

so

(19)

4 -

3 .

hold for a g e n e r a l

proof

u)

t h e n we must h a v e

then

D ) The a n a l o g u e o f Lemma 1 when

Ournext r e s u l t

u(x) S

(we u s e h e r e (19)).

proves t h a t i f

Remark

+ p(u)=

-Au

i t i s easy t o construct a supersolution

Next

h a v i n g s p h e r i c a l symmetry

where

m .

h a s compact s u p p o r t a n d t h e r e f o r e

f

if

f

i s a measure does

not

@.

r e p l a c e s Lemma 1 when

i s e s s e n t i a l l y unchanged.

f

E h,

the r e s t o f the

NONLThmAR PROBLEMS RELATED TO THE TI-IO!IAS-FER?II

Lemma

-

4

p:

Assume

R + R !

b,

f E

Then f o r e v e r y

i s a continuous nondecreasirig

= 0

@(O)

function such t h a t

89

EQUATION

and s u c h t h a t

t h e r e e x i s t s a unique f u n c t i o n

u

such t h a t

-nu + p ( u )

= f

(21)

=

u ( m )

p(u) E

and i n a d d i t i o n

o

L1.

1) A r e s u l t s i m i l a r t o Lemma h

Remarks:

(when

was p r o v e d i n d e p e n d e n t l y by B a m b e r g e r [ 11

p(u) E

such t h a t

L1

f =

and

f

6

i s a power)

.

2 ) The a s s u m p t i o n (20) i s a l s o n e c e s s a r y ; i f

u

@

( 2 1 ) h a s a solutica?

( D i r a c mass a t t h e

o r i g i n ) , then (20) holds.

1 + P-1 Y(u) = ( u ) ~

3) I n c a s e

1 j ( r ) = - rp, P p >

provided

4 3

then

( 2 0 ) holds w i t h

.

Ll i b 1i o g r a phy

[11 A .

B a m b e r g e r , E t u d e tle d e u x e q u a t i o n s n o n l i n 6 a i r e s a v e c u n e m a s s e d e D i r a c a u s e c o n d membre Benilan

[ 2 ] Ph.

-

H.

B r e z i s , D e t a i l e d p a p e r on t h e Thomas

Fermi e q u a t i o n ,

[3I

Benilan

l'h.

-

equation i n

[41

H.

Brezis

L1(IRN),

-

M. Ann.

Crandall, A semilinear Sc.

Norm.

P.

523-555.

-

B.

Simon, The Thomas-Fermi

Lieb

M o l e c u l e s and S o l i d s , Adv.

p.

22-116.

-

( t o appear).

2 (1975) E.H.

( t o appear).

i n Math.

Sup.

Pisa,

T h e o r y o f Atoms,

29 ( 1 9 7 7 )

G.M. de La Penha, L.A. Medeiros (eds.) Contemporary Developments in Continuum Mechanics and Partial Differential Equations Worth-Holland Publishing Company (1978)

GLOBAL SOLVABILITY AND HYPOELLIPTICITY

OF ABSTRACT COMPLEXES AND EQUATIONS

FERNANDO CARDOSO

and

JORGE HOUNIE

Universidade Federal de Pernambuco

1. Introduction_. -.

n

Let

be a V-aimeiisiorial

ed, orientable by

A

Cm

l),

compact, connectWe will denote

a linear selfadjoint operator, densely defined in a

has bounded inverse instance,

IDXI ) .

H,

Am'. on

(l-AX)'

which is Enbounded, positivo and (We may think of

Rn,

A

defined by

elements

u

)I UII

H

of

= ))ASullo ,

s < 0, HS

A:

if

such that where

s

2

0, HS

ASu E H,

1) 1)

s E

R,

m E R,

( f o r the Hilbert space structures) of

By

H+m

HS

is the space of equipped with the

H Am HS

is a n isomorphism onto

HSmm.

we denote the intersection of the spaces

union, with the inductive limit topology. H'and

so can

H+=

H = Ho; IIulIs =

for the norm

equipped with the projective limit topology, and by

sER,

(for

denotes the n o r m in

is the completion of

= )IASullo. Whatever

as being, for

o r a selfadjoint extension of

We will use the scale of Sobolev spaces

s E R),

if

2

manifold without boundary.

complex Hilbert space

norm

(V

H-'

and

HS,

H-m their

Since for each

can be regarded as the dual of each other, Hmm:

with their topologies, they are the

91

GLOBAL SOLVABIL I T Y AND H Y P O E L L I P T I C ITY

s t r o n g d u a l o f each o t h e r . role of

space o f

I n concrete cases

H+m

plays

the

H - ~ of distributions.

test-functions,

When s t u d y i n g a n a l y t i c i t y p r o p e r t i e s i t i s advantageous

t o introduce another s c a l e of spaces: let

denote t h e subspace o f

ES

functions

u

ES

let

the

Eo = H

s'

< s, I

embedded, w i t h a g r e a t e r norm,

s , ES

i n the

i s a Hilbert space.

ES

and f o r any t w o r e a l numbers

r e a l number

H

denote t h e completion of

Equipped w i t h t h i s norm, have

consisting of

satis€ying

s < 0,

f o r any

H

s 2 0,

f o r any r e a l

ES

arid d e n s e i n

ES

.

We

is

Given a n y

E-'.

i s t h e d u a l of

A s w i t h S o b o l e v s p a c e s we f o r m t h e u n i o n and i n t e r s e c t i o n of

t h e spaces

than going t o (1.2)

or

-m

E+O

=

ES, +m):

u

ES,

E-O

s> 0 The s p a c e s

E+O

and

E-O

> 0,

=

n S<

E ~ .

0

w i l l be e q u i p p e d w i t h t h e i r n a t u -

r a l l o c a l l y convex t o p o l o g i e s , s

going t o z e r o ( r a t h e r

s

b u t for

t h e i n d u c t i v e l i m i t of

on t h e f i r s t , t h e p r o j e c t i v e l i m i t

of the

on t h e s e c o n d , which i s t h e n a FrGchet s p a c e .

of

each o t h e r .

t h e r o l e of funct io n s

From o u r p r e s e n t s t a n d p o i n t

"analytic functions1',

62

CJ(62;Hk)

that

s < 0,

They a r e d u a l s

Efo

w i l l play

of t h e

"hyper-

.

We d e n o t e by in

E-O

ES,

t h e ES,

valued i n H+m.

Cm(n;Hfm)

t h e space of

Cm

functions

It i s t h e i n t e r s e c t i o n o f t h e spaces

(of the j-continuously

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

92

FERNANDO CARDOSO and JORGE H O U N I E

R

defined i n

j, k ,

tend t o

Hmm.

a s t h e nonnegative i n t e g e r s with i t s n a t u r a l

W e equip Cm(R;H+")

+m.

8' ( R ; H - m )

W e w i l l d e n o t e by

topology. and r e f e r

Hk)

and v a l u e d in

Cm(R;H+T,

t h e d u a l of

t o i t as t h e space o f d i s t r i b u t i o n s i n

R

Cm

valuedin

A c t u a l l y , we s h a l l need f o r m s and c u r r e n t s w i t h v a l u e s

E

i n some t o p o l o g i c a l v e c t o r s p a c e

I?"

either

E*O).

or

t h e space o f E-valued

(@; tl,

I;

p G V ,

p-currents

simply distributions i n charts

0

If

...,t v )

n

in

R,6

C

0

w i l l always be

we d e n o t e by

n

p=O

(if

valued i n

of

I?

(here

A'

fl'(n;E)

they a r e

I n terms o f l o c a l

E).

a n open s e t , s u c h a

c u r r e n t i s a l i n e a r combination

(1.3) where

J

i s an ordered m u l t i - i n d e x

such t h a t

to

p,

and

1S

<.

j,

. .<

j P

dtJ = d t .

5

A...A

AP

Cm(n;E)

J

-b

of E - v a l u e d

P

)

of i n t e g e r s

i t s l e n g t h , here equal The c o e f f i c i e n t s

uJ

in

P

distributions i n

Cz(8)

l i n e a r mappings

.

dt.

1 (1.1) a r e E - v a l u e d

[ J[

V ,

(jl,. . . , j

6,

i.e.,

continuous

6.

A n a l o g o u s l y one d e f i n e s t h e space

Cm

p-forms

in

0.

( 1 . 3 ) c a n he made g l o b a l by i n t r o -

Of c o u r s e , f o r m u l a

d u c i n g a p a r t i t i o n of u n i t y s u b o r d i n a t e d t o a l o c a l l y f i n i t e covering o f

R

by l o c a l c h a r t s

s h a l l always assume t h a t in

2.

(in fact,

from now o n , w e

such formulas a r e g l o b a l l y d e f i n e d

0). G lobal S o l v a b i l i t y . ____

O u r b a s i c datum i s a on t he parameter

h E o(A)

Cm

one-form i n

( t h e s p e c t r u m of

R, A):

depending

GLOBAL SOLVAHI L I T Y A N D F I Y P O E L L I PTICITY

V

b(t,k) =

(3.1)

b.(t,h)dt

J

j=1

We s h a l l assume t h a t t h e 12

.(t,A),

jzl,..

J

folloLbs:

.,v,

of

%A(")

A-I,

iihich converge i n

x(EI,H)

o p e r a t o r s on

a s w e l l a s e a c h one o f

t

iD = d

on

t h e i r t-derivatives,

n.

" d i f f e r e n t i a l operator" i n

n

+ h(t,h)AA.

t

p = O,...,v-l,

F o r each

C"(n),

in

( t h e Banach s p a c e o € bounded l i n e a r

We s h a l l t h e n f o r m t h e (2.2)

defined as

w i t h complex c o e f f i c i e n t s

to

uniformly w i t h r e s p e c t

%,(n)

a r e t h e s e r i e s i n t h e non-

n e g a t i v e powers of

H),

..

J

"convolution operators"

belong t o t h e r i n g

t h e elements

93

i t defines a l i n e a r operator

(2.3) D"

W e set

= 0.

Our f i r s t r e q u i r e m e n t i s ( t h e complex g e n e r a t e d by t h e

= 1,.

..

i.e.

,V),

that

( 2 . 3 ) form a complex

operators

V

I m IDp c Ker

__ a

atj

+

b.(t,A)A,j= J

rihich means t h a t

the

s t a n d a r d "Frobenius i n t e g r a b i l i t y c o n d i t i o n s " a r e s a t i s f i e d , and t h i s i s w e l l known t o be e q u i v a l e n t w i t h t h e r e q u i r e m e n t t h a t t h e 1-form We

be c l u s d e d f o r a . e .

f o r a.e.

where

a.e.

XEu(A),

t h e one-form b ( t , X ) i_____ s exact

i s t o b e u n d e r s t o o d i n t h e s e n s e of

A.

We t h e n i n t r o d u c e a p r i m i t i v e of

t (2.5)

1.

s h a l l assume more:

(2.4)

measure o f

b(t,X)

B(t,X)

=

b:

(in

n),

the spectral

FERNANDO CARDOSO and J O R G E H O U N I E

94

O u r purpose is to study the equation (2.6)

IDU = f,

where form

is a E-valued p-current,

u

(p s

V

-

l),

n.

in

f

a H+m-valued, Cm (p+1)-

Let us consider a spectral

resolution of the operator

A

(considered i n

(2.7)

jX

dE(X).

A =

H):

F r o m it one easily derives that a compatibility c o r r l i t h

for

(2.6) is

Eq

the - (p+l)-form ___- e B(t,A)AE f I -

(2.8) for a.e. XEu(A), T h e set o f forms will be denoted by

f

E'+'A

is exact.

- -

satisfying (2.8)

Cm(n;H+")

gp+I Cm(h2;H+m). D

I t is our goal to complement (2.8) with conditions

B

bearing on the function

so as to obtain conditions which

a r e necessary as well sufficient for the Ttglobalsolvabilitytt of ( 2 . 6 ) . We state now o u r fundamental hypothesis about the primitive

B( t

,X)

or rather aoout its "principal part".

Let

(2.9)

B(t,X)

= Bo(t)

F o r any.real number

(2.10)

nr

=

The natural injection of

nr

+

B,(t)X-'

... .

w e write

r

t E

+

n;

Re Bo(t) < r]

into

h2,

.

gives rise to homo-

morphisms of the De Rham homology spaces:

apPr) -

(2.11)

i

P.'DP(").

We also have the transpose homomorphisms (restriction of cohornology classes):

GLOBAL SOLVABILITY AND HYPOELLIPTICITY

95

i*

4 " @ ) P4 P ( n r ) .

(2.12)

We d e f i n e t h e f o l l o w i n g " g l o b a l " ( Q )

-

Definition 2 . 1 i n d .i m e nsio - _ . __n

p

(0 5

Ker i

(2.13)

D

W e say t h a t t h e system

p

F

s

= {O],

i.e. _-

(see

has Property

(O),

0 , ______ i f f o r every r e a l

in

V - l ) l

condition

i

is injective

P

--

o r equivalently I m i* = @ P ( n r ) , P

(2.14)

i* p

i.e. ~

i s s ur j e______ ctive.

($)

We o b s e r v e t h a t a l t h o u g h P r o p e r t y i n t e r m s of on

the particular primitive

(2.5),

i s formulated

i t depends only

b.

We s h a l l s t a t e now a r e s u l t p r o v e d i n

-

Theorem 2 . 1

The f o l l o w i n g c o n d i t i o n s

( I ) : T h e system

ID

(11): G i v e n a n y e l e m e n t ..-

element -~

u

of

A'

f

are equivalent: ~

(e),

has Property

of

121 :

i n dimension

C"(O;H+")

- OaD

p,

in

t h e r.e _.___ i s an

c ~ ( ~ ; H + " ) ______ s o l u t i o n of

~u = f

in

n. (111): G i v e n a n y e l e m e n t element

u

of

f

of -

6dP+l

AP 8' (o;H-")

C"(n

ID

H'")

s o l u t on of

t h e r e i s an

iDu = f

in ~~

Remark

2.1

-

When

V

= 1,

deals with a single operator o n l y one c a s e :

p = 0 = V-1.

= S1 L =

(the unit circle),

a + at

b(t,A)A

one

and t h e r e i s

I t i s s e e n at once t h a t t h e

96

FERNANDO CARDOSO and JORGE H O U N I E

(,I,)

v a l i d i t y of

in

( p e r f o r c e i n dimension z e r o ) i s

S1

equivalent t o the following property: Re b o

-

Remark 2 . 2 model.

vanishes -identically i n

i t i s worth t o mention t h a t t h e r e i s o n e c a s e

where t h i s model i s m o r e t h a n t h a t :

t h i s i s when t h e Ley&

f o r m of t h e complex i s n o n d e g e n e r a t e .

m a t r i x of

.

1

T h e complex u n d e r s t u d y p r o v i d e s a v e r y s i m p l e

However,

means t h a t ,

S

t

whatever

B(t,h)

n

X

and

A =

(1-n X )l!'

on

E a(A),

notation i t t h e Hessian

a '€3

t,

with respect t o

I t i s shown i n [l]

i s nonsingular. situation that

in

our

In

...,

( -a-t-j-a b . ] ~. , k = 1 ,

( i n the

"concrete"

H = L2(Rn))

IRn,

V '

that

u n d e r s u c h a h y p o t h e s i s , o u r complex i s t h e t r u e m i c r o l o c a l f o r m ( p o s s i b l y a f t e r a s i m i l i t u d e by a n e l l i p t i c F o u r i e r i n t e g r a l o p e r a t o r ) o f a n y g e n e r a l complex w i t h t h e same property,

and i n f a c t c a n be f u r t h e r s i m p l i f i e d ( l o c a l l y or

m i c r o l o c a l l y ) i n t o t h e complex g e n e r a t e d o p e r a t o r s ) p r e c i s e l y b y o p e r a t o r s of

a a t 3. +

L . =-J

Remark 2 . 3

-

c . tJA, J

the type:

j=l,.

..,

V

E . = kl. J

A n i n t e r e s t i n g a l t h o u g h more d i f f i c u l t q u e s t i o n

t o investigate i s the study of if

(modulo r e g u l a r i z i n g

one r e p l a c e s

the global solvability of

(2.6)

( 2 . 4 ) by t h e following l e s s r e s t r i c t i v e

condition:

(2.15) f o r a.e.

XEa(A),

t h e one-form b ( t , X ) i s c l o s e d

(in 0).

3. G l o b a l H y p o e l l i p t i c i t y a n d H y p o a n a l y t i c i t y I n t h i s s e c t i o n , we o n l y c o n s i d e r t h e c a s e

V

= 1,

GLOBAL SOLVABILITY A K D HPPOELLIPTICITY

L-2

= Sp

T1 = [R/2nr&).

( w h i c h we i d e n t i f y w i t h Let

a

(3.1) IQhere

at

thought

of

at +

s1

b ( t , ~ ) ~ ,t E

gt

- and t h e c o e f f i c i e n t

means

aA(S1)

the r i n g

=

Sectioii 2 .

defined i n

as an oper ator oil

b(t,A) Thus,

belongs

!L

to

may be

with periodic coe€ficients

lR

and we o n l y l o o k a t p e r i o d i c s o l u t i o n s , when p e r i o d i c d a t a are prescribed. We o b s e r v e t h a t

(2.15)

holds automatically, although

i n general

( 2 . 4 ) may b e f a l s e ( s e e C o r o l l a r y 3 . 3 I s e l o h ) .

Definition

3.1

s1

We s a y t h a t

i s globally hypoelliptic i n

[L,

u E c"(s';H-")

_______ i f given

Lu E C"(Sl;H+")

(3.2) D e f i n i t i o n 3.2

s1

-

if

given

-

W e say that

u E

C"(SI;E-O)

Remark

3.1

-

Cm(S1;H+m).

i~-s g l o b a l l y h y p.o a n a l y t i-~ c in

L

~

LU E Cm(S1;E+o)

(3.3)

u E

=

u E Cm(S1;E+O).

A c c o r d i n g t o P r o p o s i t i o n 111.2.2 i r i

i s no g a i n i n g e n e r a l i t y i n D e f i n i t i o n

3.1,

o u r " s p a c e of

Crn(S1;E-O)

solutions"

C"(S';H-"),

bigger spaces o f d i s t r i b u t i o n s Remark 3 . 2

-

to

!Lu

with r e s p e c t

the coefficients i n

t)

b(t,A)

to

t

,

if

we replace

there

by t h e

Q' ( S1;E-").

Q' ( S 1 ; H - " ) ,

Traditionally i n Definition 3.2,

a n a l y t i c i t y of that

3.2,

[ 61

one assumes also

(under the hypothesis

a r e analytic with respect

and d e r i v e s t h e same p r o p e r t y f o r

u.

But

in fact

t h e r e i s no g a i n i n g e n e r a l i t y i n d o i n g t h i s h e r e :

if

the

coefficients i n

to

t,

b(t,A)

are a n a l y t i c with respect

98

FERNANDO CARDOSO and JORGE HOUNIE

u E 8' ( S 1 ; E ) ,

and if in

1 S ,

and

Lu

E,

with values in

Proposition 111.2.2 i n [ 6 ] , of

t

S1

in

t

is an analytic function of

then by virtue of a variant of will be an analytic function

u

with values in

(E

E

is one of the four

dm, E*').

spaces

In [ 5 ] , a theorem characterizing globally hypoelliptic operators in of

u(A)

of the form (3.1), w a s proved; the behavior

S1,

(the spectrum of

A)

in a neighborhood of infinity

played an important role.

In this section we shall present a n analogous theorem (see

[31) characterizing globally hypoanalytic operators in

S1,

of the form (3.1). According to the hypotheses made on

C

IR, for certain positive

(3.4)

r(z) =

m liQ m

which is analytic on

p.

> p,

= r z+rl+r2z

A

is clear that constant.

C

where

-1

+...,

z

E

C

and either has a pole of order

one or a removable singularity at

be the subset of

C

We introduce a function

zb(t,z)dt IzI

A,u(A) c [p,+-)

r(z)

z

=

m.

Let

takes integral values.

is a discrete set when

r(z)

It

is not a

We denote b y

Theorem 3.1

-

defined on

S1,

The evolution operator

L,

given by (3.1)

is globally hypoanalytic i n

if the following conditions hold:

S1

and

if and only

GLOBAL SOLVABILITY AND HYPOELLIPTICITY

(p1

Re bo

(GG)

If

Corollary 3.1

-

S1;

~

Re b o

5 E a(A)

for a l l

does -__ not change sign i n

99

0, for any positive real number

I

k

sufficiently large. ~

If -

L

~~~

is _ _ globally hypoelliptic, then it is

globally hypoanalytic. Proof:

Condition

Remark 3.3 -

-

(6)

(see [ 51 ) implies (Cis).

The converse of Corollary 3.1 is not true in

general, as follows from Corollary 3.2, b e l o w , and We recall that if

P

s m o o t h orientable manifold

hypoanalytic if

Pu E A(M) u E

is a differential operator on a M,

we say that

P

is globally

(the distributions on

(the real analytic functions on

M)

and

M)

imply that

A(M).

Theorem 3.2 on -

u E r9'(M)

[4].

S1

-

Let

b(t)

be an analytic complex valued function

and consider the vector field

(3.8) defined on the 2-torus

T'

= S1xS1 = {eit]x{eix].

Thep

P

is globally hypoanalytic if and only if the following

conditions hold:

(e ) (GLJJ)

I m b(t)

If

Im b(t)

t

d o e s n o t change sign

0, y =

211

("

Re b(t)dt

is an

10

irrational number satisfying: (A)

Given

K > 0, there is

Q > 0

s o that for

q

2.

Q,

FERNANDO CARDOSO and JORGE H O U N I E

100

lp-yql

e-Kq,

2

When

p, q

b(t)

Lntegers.

is constant in ( 7 . 8 1 ,

Theorem 3.2 yields a

result due t o Greenfield (see [ 4 ] ) namely: Corollary 3.2

-

The complex vector field

is globally hypoanalytic in Re b

if and only if

T2

-

hax, b E C,

Im b f 0

pr

is an irrational number satisfying (A).

Corollary 3.3 L

P = at

-

If -

(2.4) holds, that is if

b(t,A)

is exact,

is hypoanalytic (in fact the same is true - not globally --_ ~~

concerning global hypoellipticity). This Corollary is also a consequence of the proof of the following necessary and sufficient condition for hypoellipticity at the first step 1 E U(A),

p = 0

B(t,X)

the primitive

(see [ 6 ] ) :

for each

should not have any local

t.

minimum with respect to

References [l] Boutet de Monvel, L.

-

Hypoelliptic operators with double

characteristics and related pseudodifferential operators, Comm. Pure Applied Math., vol. 2 7 ( 1 9 7 4 ) . [2]

Cardoso, F. and Hounie, J.

-

Global solvability of an

abstract complex, to appear in the Proc. Amer. Math. SOC.

[3] Cardoso, F . and Hounie, J.

-

Global hypoanalytic first

order evolution equations,

[4]

Greenfield, S . J .

-

to appear.

Hypoelliptic vector fields and

continued fractions, Proc. Amer. Math. SOC., vol. 3 1

( 1 9 7 2 ) , 115-118.

GLOBAL SOLVABILITY A N D HYPOELLIPTICITY

[ 5 ] H o u n i e , J.

-

equations,

[ 6 ] Treves, F.

-

101

Global hypoelliptic first order evolution t o appear.

Study of a m o d e l in the theory of complexes

o f pseudodifferential operators, Ann. Math.,

vol.

( 1 9 7 6 ) , 269-324.

T h i s w o r k o f F. Cardoso w a s partially supported b y CNPq (Brasil).

104

G.M. de La Penha, L.A. Medeiros (eds.) Contemporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations @North-Holland Pub1 i s h i n g Company (1978)

ON THE INVERSE SCATTERING PROBLEM

FOR L I N E A R EVOLUTION E Q U A T I O N S

M I C H A E L 0 1 CARROLL

bepartment

of M a t h e m a t i c s

P o n t i f i c i a Universidade C a t 6 l i c a do R i o d e J a n e i r o , B r a s i l

I t i s o u r aim t o p r e s e n t r e s u l t s w i t h s i m p l e p r o o f s o f some a s p e c t s o f dimension

> 1.

n

t h e i n v e r s e s c a t t e r i n g problem i n s p a c e T h e s e r e s u l t s were o b t a i n e d j o i n t l y w i t h

Paul O t t e r s o n o f PUC/RJ. s c a t t e r i n g problem.

Let us f i r s t r e c a l l t h e d i r e c t

W e c o n s i d e r a system undergoing l i n e a r

e v o l u t i o n i n a H i l b e r t space U(t)

= e

-it H

evolution.

61

g i v e n by t h e u n i t a r y g r o u p

and want t o a n a l y s e t h e o r b i t s of

The e v o l u t i o n may r e p r e s e n t t h a t

this

o f an e l e c t r o -

m a g n e t i c f i e l d i n a d i e l e c t r i c and m a g n e t i c medium, a c o u s t i c wave i n a l i q u i d o r s o l i d , mechanical system o f p a r t i c l e s

(take

o f an

o f a quantum or c l a s s i c a l

W

t o be

L2

o f phase

space i n t h e c l a s s i c a l mechanical c a s e ) i n t e r a c t i n g through p a i r p o t e n t i a l s and w i t h a s t a t i c e l e c t r o m a g n e t i c f i e l d , e t c . One can a l s o c o n s i d e r o b s t a c l e s i n t h i s framework by d e f i n i n g H

w i t h a p p r o p r i a t e b o u n d a r y c o n d i t i o n s , f o r example, Dirichlet,

Neuman or mixed.

I n many c a s e s f o r l a r g e p o s i t i v e and n e g a t i v e t i m e s and s p a t i a l d i s t a n c e s t h e e v o l u t i o n of n o n - s p a t i a l l y ed o r b i t s of

U(t)

localiz-

may b e a p p r o x i m a t e d by a s i m p l e r e v o l u t i o n

O N THE INVERSE SCATTERING

u

(t) = e

-itHo

where s a y

Ho

103

represents a self-adjoint

p a r t i a l d i f f e r e n t i a l operator with constant c o e f f i c i e n t s , T h i s can be made p r e c i s e by r e q u i r i n g t h a t

s-lim t-b*

e

i H t e-iHot

= w*

(la)

m

e x i s t s and

R ( Y ) = Pc# where

R = r a n g e and

denotes t h e orthogonal p r o j e c t i o n

Pc

on t h e c o n t i n u o u s s p e c t r u m of f o r every

f E

If

H.

Pc#

there exist

l i m

-iHot (e

fit

( l a ) and ( l b ) h o l d t h e n

f* E #

-

e

-iHt

such t h a t

fJ = 0

t+*m

and i t i s i n t h i s s e n s e t h a t e v e r y o r b i t

(t +

asymptotically Also if

Uo(t).

H

f

behaves

( l a ) and ( l b ) h o l d t h e n t h e l i n e a r o p e r a t o r

f

ween

-iHt

l i k e t h e o r b i t s under t h e e v o l u t i o n

km)

S : # + # ,

i s u n i t a r y and

e

S-I

-

s = w ?++ w -

-+ f +

g i v e s a measure o f t h e d i f f e r e n c e b e t -

Ho.

and

Now l e t u s c o n s i d e r t h e i n v e r s e s c a t t e r i n g problem. W e consider the non-linear

function

s: c c C

w i t h domain

-b

L(#)

-t

s(c)

some c l a s s of i n t e r a c t i o n s and r a n g e t h e

unitary operators i n

#.

We p o s e t h e f o l l o w i n g q u e s t i o n s :

1.

I s t h e r e an a l g o r i t h m f o r f i n d i n g

2.

Is

S

injective?

3.

If

S

i s i n j e c t i v e and we i n t r o d u c e a Banach s p a c e

C

from

S(C)?

104

MICHAEL O'CARROLL

topology on

@,

such as

(Clg

S-I

is

continuous and

differentiable?

!I.

Can we completely characterize the image

S(@)

C

of

L(#)?

in

We now respond to questions 1-4 for some systems of interest in mathematical physics: Case I:

C =

{V:

id = L 2(R),

n=l, R + R

1

2

H = -d /dx

+

2 2 H o = -d /dx ,

V,

Faddeev El1 has given a

E L1].

(l+lxl)V(x)

2

solution to 1 by generalizing the Gelfand-Levitan equation

S

and to b e

is not 1-1. However if the scattering data is taken

and the normalization constants of' bound states

S

C

then there is a 1-1 correspondence between

and the

scattering data and 4 has a n affirmative answer. Case 11:

@ =

In V: R

-A,

8 = L ~ ( R ~ ) , H~ =

n=I,

{vE

L'

n LP

n

H = H +V

~ q , p < n/2 < q j , 3 L3j2) n L1],

+ R: {VER(Rollnik class

{v

-1/2-0

= (1+x2)

W for some $ > O

n > 3 n = 3 1

and W € L m n L

1,

n = 2

Mochizuki [ 2 ] has answered 1 and 2 affirmatively for a smaller class t h a n

@

for

n=3.

We now give a simple proof showing

that the response to 1, 2 and 3 is yes. space

S

has the kernel

-

S(k/ ,k) = 6(k/-k) where

k,k'

E Rn

lim Oig-bO

f

e

2

m i 6(kf2-k )T(k' ,k)

and for large

T(k/ ,k) =

-

In Fourier transform

- ik' - x

k2

~ ( x eikSx ) dx

-ik/.x V ( x ) [ (H-k2-io)

-1

Ve

ik.

'1

(x) d x

(2.1)

O N THE I N V E R S l i : S C A T T E R I N G

x E Rn,

where

i n t e g r a l of

k2 = kj2

V1j2

(2.1)

-1

V1’*(H-k2-is)

and

L2 -+ L

2

.

( s e e Simon

[31).

= M(II,c ,k2)

( n > 1) and G i n i b r e and Moulin [

k

31 ( n = 3 ) ,

Agmon [

41

51

(M(Ho,k

SUP

L2

as a n o p e r a t o r f r o m

From t h e e s t i m a t e s of Simon [

2

I n t,he second

i s c o n s i d e r e d a s a n e l e m e n t of

V1/’

l i m

105

2

,c)l =

0

O
-+m

s o using the resolvent equation l i m sup /M(H,k 2 k -+a W c S 1 The i d e a of

2

k‘-k = q

T(k’ ,k)

each

we i n t r o d u c e a u n i t v e c t o r

cl

and w i t h

so that

keeping

0 < a E

k(a)’

R

= k’(a)

To do t h i s f o r

fixed. n(q)

perpendicular t o k ‘ ( a ) = an+q/2

k ( a ) = an-q/2,

write 2

0.

1 and 2 i s t o i s o l a t e t h e f i r s t

the proofs o f

term o f q

, ~ ) j=

2

+

= q /I1

a

2

,

k‘(a)-k(a) = q.

For

n o t a t i o n a l c o n v e n i e n c e we d r o p t h e a d e p e n d e n c e i n what We g i v e t h e p r o o f s t o

follows.

1) W e find

1 , 2 and

( A denotes the F o u r i e r transform)

l i m IT(k‘,k)-G(q)\

l i m IVIL/(

a3 m

a+ a

2 2 IM(h,a cq / 4 , 6 ) (

sup O<ES

= ?2(q)

T ( k ‘ ,k,V1)

so that

= T ( k / ,k,V2) V1

= V2

+ lvlLP

for

n >

3

T

on

as

lvlg

B,

=

T(k’,k,V) G1(q)

=

#.

as operators i n

or

V

t h e n a s i n 1 above

3 ) I n t r o d u c i n g t h e Banach s p a c e

= 0.

1

2 ) W r i t i n g t h e e x p l i c i t dependence of if

3:

= (VILl

l v l R + lvlLl

+ for

lVILq

f

n = 3

and u s i n g t h e f a c t t h a t

s UP

SUP

2 k E[O,m) by t h e e s t i m a t e s of [

31

2

lM(Ho,k

,El

<

m

0<~<1

and [ 51 f o r

IVIB

small t h e Fr6chet

106

MICHAEL O'CARROLL

derivative of

I

e-ik'.x

T(V)

is

-1

( ( I-V(H-k2-is )

V, W

and is continuous where a +

rn

fixed

W ( I-( H-k2-ia

belong to

V) e

B.

ik..

) (x) d x

Taking the

limit gives

q

w

dV Therefore

)-'

dT(V)/dV

= ij(q).

is injective.

inverse function theorem

B

sional subspaces of

T(V) is a

By a version of the

.restricted to finite dimenC1

diffeomorphism.

Case 111: The same results as in Case I1 hold for the wave equation with a linear multiplicative perturbation, i.e.

Ott + (For

n=3

and small

V

o

(3.1)

see also Perla [ 6 ] ) .

A s in Birman [ 7 ]

the free equation with

2

( - ~ + m+v)@ =

we transform equation (3.1) and

V = 0

to the Schroedinger form

(3.2) where =

r

x = (xl,x2),

(au(O),

v(0)).

HPac(H = -A+V)

x1 = q B u, B

x2 = du/dt = V

x(0) =

and

is the positive self-adjoint operator

for equation (3.1) where

Pac

is the ortho-

gonal projection on the subspace of absolute continuity of H; for the free equation we use the operator place of

B.

A = -A

= H

in 0

T h w n i t a r p group which gives the time evolution

of ( 3 . 2 ) i n the Hilbert space

L2@L2

is

ON THE I N V E R S E SCATTERING

107

The wave operators associated with the unitary group e -iat

and the free particle unitary group = s-lim

W,(b,a)

e

ibt

e

-ibt

are

-iat e

t + f m

Combining the results o f A g m o n [ b ] on the existence and completeness o f the wave operator

W*(B,A)

= s-lim e

iBt .-iAt

t+*m and the invariance principle of Birman 161 i.e.

shows that

W*(b,a) =

W*(b,a)

where

exist and are complete.

/I (w* + w*)t)/2

(w*

-

w*)/2i

(-W* + W*)/2i

(w*

+

W*)/Z

W* = W*(B,A).

Also

s = W (b,a)*W

- (b,a),

scattering operator associated with (3.2),

where

S = W+(B,A)*

and

S*-I;

,’

the unitary

is

-

W (B,A).

From Equation (3.3) we s e e that

S-I

Furthermore

s-I

only involves

the restriction to the energy shell of

S-I,

as we have seen, is given by ( 2 . 1 ) . Rather than transforming (3.1) to (3.2) we could take

T

as given by the time independent approach to

scattering where we let

@(x,t)

= $(x,k) e -iwt

By imposing the outgoing wave condition o n

(w=(k 2+m 2 )1/2),

$(x,k)

-

ik-x e 9

108

MICHAEL 0 ' C A R R O L L

$(x,k)

e i k - x -[ ( H - k 2 - i C ) - l

i s g i v e n by

i s g i v e n by 2 . 1 .

T(k',k)

dependence

x ( x , k ) eiwt

x

condition

= e

i k -x

-ik. x

(x)

and

We a l s o have a s o l u t i o n w i t h t i m e

and by i m p o s i n g t h e o u t g o i n g wave

-

[ ( H-k2+ie )

V ( s ) x(x,k)ds.

-1

Ve

ik.

T-(k' , k )

i n g t o t h i s t i m e dependence f e

Veik"]

Both

T

'1 ( x )

.

Correspond-

i s g i v e n by

and

T

a r e considered as

-

g i v e n s c a t t e r i n g d a t a b u t a s we h a v e s e e n i n Case I1 i t i s

T

enough t o c o n s i d e r Case I V :

3.

i n responding t o q u e s t i o n s 1, 2 ,

We c o n s i d e r t h e S c h r o e d i n g e r e q u a t i o n i n a n e x t e r n a l

s t a t i c electromagnetic f i e l d , i . e . i

where

a@/at

= [ ( - i a/axi

2 A ~ )+

-

i s a s t a t i c r e a l p o t e n t i a l and

V

vlo

A = (A1,A2,

...,An)

i s t h e r e a l m a g n e t i c v e c t o r p o t e n t i a l i n t h e Coulonb gauge which we r e q u i r e t o s a t i s f y a r e t o b e summed o v e r i = 1 , 2 , We assume

V,

2 Ai

and

Ai

C = (Q:

and

A

t o be s m a l l .

U(x) = 2 i A

i

lim a+ m and a s mines

i3Ai/axi A.

= 0

Knowing

or A

i n the distributionalsense.

2 -1/2-c

Q = (l+x

=

e

+

A:

)

W E Lm

and

For large

a/axi

(repeated indices

t o belong t o t h e c l a s s

c > 0

T(k',k) where

...,n )

Rn + R :

some

= 0

i)Ai/axi

n

W

for

L']

k2

- i k ' ax

+

T(k',k)/a

V.

We f i n d t h a t

= i(q)-n(q)

q-i(q) = 0 we o b t a i n

u(x) $ (x,k)dx

2 Ai

equation

+

(4.1)

(4.1)

V and h e n c e

s u b t r a c t i n g from t h e s e r i e s e x p a n s i o n of

T

deter-

?(q)

by

a l l ternis involv-

ON T H E I N V E R S E SCATTERING

A

ing

a +

and t a k i n g t h e

limit.

m

/Q1"

have a l s o used t h e e s t i m a t e where

i s a c o n s t a n t for a l l

C

b u t d e p e n d i n g on

2 ko.

(4.1)

(Ho-k

.

2

-1

-1s)

c

2

[ko,m)

only t h e f i r s t

remains i n t h e l i m i t o f

T

we a p p r o x i m a t e d a f i n i t e number o f t e r m s i n and

V

b y f i n i t e sums o f

Ai

a r b i t r a r y mean and v a r i a n c e , r e p r e s e n t e d i ( I I o - k 2- i s ) t by i e dt f o r E > 0 and u s e d

the e x p l i c i t expression f o r the kernel

Case V:

2 - i.c ) Q1 / 2 / <

i n the interval

A l s o i n showing t h a t

t h e s e r i e s by a p p r o x i m a t i n g G a u s s i a n s of

I n t h i s a n a l y s i s we

a/axi(-4-k k2

term i n t h e s e r i e s expansion o f equation

109

e

iHot

Here we c o n s i d e r t h e K l e i n - G o r d o n e q u a t i o n m i n i m a l l y

coupled w i t h t h e s t a t i c e l e c t r o m a g n e t i c f i e l d , i . e .

-a2Q/at2 V

and

Ai

Letting

2

= (-k+m - 2 i A i

@ ( x , t )= $ ( x , k ) e

U = V2

-

2 A.

-

-iwt

ik-x

-

2Vw

-

i

T(k',k) = The s o l u t i o n o f e q u a t i o n n o t needed t o r e c o v e r -2w

2 2 A ~ - V

2iAi

and t a k i n g t h e

V

e

,

V

+

2vi

and

Ai

w = ( k +m

)

2

-1

[ (-A+U-k - i e )

s c a t t e r i n g d a t a corresponding

by

+

a r e a s i n Case I V w i t h

$(x,k) = e with

a/axi

a/axi

t o the

-ik' .x

aG/at. small. where

U eik"]

a 4

m

Ai.

(x)

and we t a k e as o u r e

-iwt

t i m e dependence

U(x) 1 ( x , k ) d x .

( 5 . 1 ) w i t h time dependence and

(5.1)

(5.2) e

iwt

Dividing equation (5.2)

l i m i t we f i n d

is

MICHAEL O'CARROLL

110

Witti

fixed take normal directions

q

determine

?(q)

and

i(q)

n(q)

and

-n(q)

to

separately.

Case VI: We consider the Dirac equation minimally coupled to a static electromagnetic field, i.e. (-iyo a/at where

yo, y

spinor.

i

+

iyi a/axi

+

m

+ vy0

are Dirac matrices and

V,

We assume

av/aXi,

and

satisfy the conditions of Case IV.

-

aiyi)@

=

o

is a multi-component

@

2 Ai

are small and

We omit the details but

by assuming a solution of equation (6.1) with time dependence -iwt and

this case is similar to Case V and we can determine

A

V

separately. In conclusion we think i t would be interesting to

extend these results to a wider class o f interaction (i.e. the

L1

condition can probably b e dropped and a distribution

type p r o o f given) and to find applications of the inverse method to solve non-linear evolution equations i n more than one space dimension.

References

111 L.D. Faddeev,

AMS Translations, Series 2, Vol.

65,

139-166 (1967).

[2] Mochizuki,

[3] B. Simon,

Proc. Japan Acad.

43 (1967).

Quantum___.___ Mechanics for Hamiltonians Defined as

Quadratic Forms, Princeton University Press, 1971.

[ k ] Agmon, S.. Ann. Scu. Norm. Sup. Pisa 2 (1975), 151-218.

ON THE INVERSE SCATTERING

[51

J . G i n i b r e a n d M.

Moulin,

Ann.

111

Inst. Henri Poincar6 21,

8 (1974).

[6]

G.

Perla,

J o u r n a l o f D i f f e r e n t i a l E q u a t i o n s 20 ( 1 9 7 6 )

NO 1, 2 3 3 - 2 4 7 .

[ 7 ] B i r m a n , M.S., 91-117

Am.

Math.

(1966).

SOC.

Transl.,

Ser.

2, Vol.

54,

G.M. de La Penha. L.A. Medeiros (eds.) Contenporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations @North-Holland Publishing Conpany (1978)

NONLINEAR BIFURCATION PROBLEM AND BUCKLING OF AN ELASTIC PLATE SUBJECTED TO UNILATERAL CONDITIONS IN ITS PLANE

DO

CLAUDE

Universit6 de Nantes (ENSM) et

Laboratoire de Mbcanique ThGorique associ6 au

CNRS, Universitg Paris VI

I. --Introduction. Many concrete situations of buckling o f elastic structures arise in engineering Sciences. The corregondingproblems have been studied since a long time, when the prescribed conditions are bilateral; they are described by an eigenvalue problem for a system of partial differential equations and boundary conditions.

One can find a general discussion i n

[161 and the corresponding mathematical analysis i n [121. However, in many situations, the structures are subjected to unilateral conditions; in these cases, we still have an eigenvalue problem, but there are some inequalities amongst the relations which describe the mechanical system. The case of elastic plates, when the vertical displacement is subjected to unilateral conditions, has been recently studied. It has been formulated as an eigenvalue problem for a variational inequality on a Hilbert space (Cf. [ 2 ]

[41 ,

[ 181

,

[ 201

,

[ 211

,

[ 221 )

.

,

[3I,

The corresponding bending

problems have been studied i n [ 61 , [ 111

, [ 171 .

A new situation with unilateral conditions, this time

113

NONLINEAR r3IFURCATION

i m p o s e d on t h e h o r i z o n t a l d i s p l a c e m e n t ,

h a s b e e n s u g g e s t e d by

e n g i n e e r s , b h o h a v e t o coiisicler a c r a c k i n s i d e t h e p l a t e . The p r e s e n t p a p e r i s r e l a t e d elastic plate, of

t o d i s c u s s t h e b u c k l i n g of

an

when a c r a c k i s e x p a n d e d a t t h e b o u n d a r y ( c r a c k

t h e clamp, f o r example); a l o n g t h e c r a c k ,

d i s p l a c e m e n t i s l i m i t e d by t h e s u p p o r t .

the horizontal

Under some assumptions,

we c a n d e s c r i b e t h e c r i t i c a l l o a d a n d e x h i b i t b i f u r c a t e d solutions; but

t h e number of

t h e s e s o l u t i o n s r e m a i n s unknown.

U s i n g v o n Karman's

theory,

we h a v e t o b e g i n t o

s o l v e a problem o f l i n e a r e l a s t i c i t y i n t h e p la n e o f plate,

w i t h u n i l a t e r a l boiindary c o n d i t i o n s

so

t h e horizontal

d i s p l a c e m e n t i s g i v e n b)- a v a r i a t i o n a l i n e q u a l i t y . ing t h i s displacement,

where

1: + c(1,;)

(I-XL), I

Eliminat-

obtain a functional equation f o r

hnic

the v e r t i c a l displacement

(1.1)

the

i s the identity,

L

i z i n g the given l a t e r a l load,

= 0,

i s a l i n e a r o p e r a t o r character-

X

i s a p o s i t i v e r e a l parameter

measuring t h e i n t e n s i t y of

t h e s e l o a d s and G i s a n o n l i n e a r

o p e r a t o r i n t r o d u c e d by

von K a r m a n ' s

The r e l a t i o n

(1.1) h a s

p r o b l e m s a n d we c a n hope

that

theory.

t h e u s u a l f o r m f o r t h i s k i n d of t h e l i n e a r i z e d problem

(1-XL)L

w i l l s t i l l g i v e t h e c r i t i c a l l o a d and t h e b u c k l i n g mode.

The

f o l l o w i n g s t u d y only g i v e s a p a r t i a l answer t o t h i s q u e s t i o n . We h a v e n o t situations

the very p r e c i s e r e s u l t s as obtained i n c l a s s i c a l

([ 1 2 1 , [ 131

of

C

i s v e r y weak:

to

I?

and

z

, C

[: 141 , [ 151 )

,

because t h e r e g u l a r i t y

i s o n l y Lipschitxian with respect

( t h a t comes f r o m t h e f a c t t h e r e i s a

v a r i a t i o n a l i n e q u a l i t y i n our problem).

Then we c a n n o t u s e

CLAUDE DO

114

methods requiring differentiability of

C

(implicit function

theorem, for example). A s s u m i n g that the first characteristic value o f

L

is

simple, we still use the classical Lyapunov-Schmidt method (Cf. [lg]),

but only

Brouwer's fixed point theorem allows us

to solve the bifurcation equation and from this alone we cannot know the number o f solutions.

We give another example (without

physical application) for which it is possible to use Picard's fixed point theorem in place of Brouwer's theorem, and so, to recover the fact that there exists only one bifurcating

branch of solutions. The plan is as follows 2. The Physical Problem.

Some results are summarized at the

end of this section. 7 . Variational formulation, and functional formulation for

the vertical displacement.

4. Bifurcation.

The Theorem 4.2 is the main result.

11. The Physical Problem.

I n its natural state, the plate occupies a bounded

open

0 in

R2,

generic point of

with sufficiently smooth boundary

n is

denoted by

the outward unitary normal and deduced from

n

by a

n/2

x = (x )

a

T=(T

n = (n )

a

A

is

) is the tangent vector,

a

rotation.

characterized by the thickness

,

r.

h,

The plate is

the flexural rigidity

D,

Greek indices take values 1 and 2; we use the summation

af

convention on repeated indices, and the notation-=

axa

-

f ,a

N O N L I N E A R BIFURCATION

v

the Poisson's r a t i o n

aa$Y6

and

115

LamGIs e l a s t i c i t y c o n s t a n t s

with

'

1

a

(2.1)

a

agY6

-

a B ~ 6- a a p b ~

A

aPAY6

' a.

--

a

~6aS

*aBAaO'

symmetric t e n s o r ;

(ao

-U A

= (A

a$

),

' 0).

H o r i z o n t a l and v e r t i c a l d i s p l a c e m e n t s a r e r e s p e c t i v e l y c a l l e d u = (u

a

)

and

C.

E q u i l i b r i u m s t a t e s , when body f o r c e s a r e

z e r o , a r e d e s c r i b e d by t h e f o l l o w i n g von Karman's r e l a t i o n s ( s e e [11):

U

Y6

i s t h e n o n l i n e a r t e n s o r of deformations used i n

von Karman's t h e o r y ; stresses

0

aP

i s the tensor of

tension

( i n the plane o f the p l a t e ) .

Boundary c o n d i t i o n s . The theorem o f v i r t u a l w o r k s p r o v i d e s t h e v a r i o u s b o u n d a r y problems a s s o c i a t e d w i t h t h e t h e o r y o f plates.

L e t us i n t r o d u c e some n o t a t i o n s :

116

CLAUDE DO

a bs

i s the tangent d e r i v a t i v e ;

a an

i s t h e normal d e r i v a t i v e .

T h e n , when body f o r c e s a r e z e r o ,

[5 ] ) ,

works can be w r i t t e n ( s e e [ 3 ] , displacement

( v ) dx =

R

-

From

a t t h e boundary i s

(2.9).

for a l l virtual

(v,z) as:

(2.11)

Remark 2 . 1

t h e theorem o f v i r t u a l

1r

aUp nB vu du

.

( 2 . 1 0 ) , w e observe t h a t the v e r t i c a l load

,a

hUuD

na

+

F,

where

F

i s g i v e n by

T h i s p r o p e r t y a l s o r e s u l t s from t h e s t u d y i n

[SI.

The boundary conditions u s e d i v the f o l l o w i n g a r e w r i t t e n i n (2.12),

On

rl

. . . , (2 . 2 0 1 .

(crack of

the clamp), the v e r t i c a l

a n d t h e r o t a t i o n a r e f r e e ( ( 2 . 1 6 ) and ( 2 . 1 7 ) ;

(2.8)

i s t h e b e n d i n g moment;

by t h e p r e v i o u s Remark 2 . 1 ) ; l i m i t e d o n a s i d e ((2.12),

t h e m e a n i n g of

displacement

M,

g i v e n by

( 2 . 1 7 ) i s provided

the h o r i z o n t a l displacement i s

(2.13),

( 2 . 1 4 ) ) and t h e p o s s i b l e

contact with t h e support i s without f r i c t i o n ( ( 2 . 1 5 ) ) .

On

r2

and

r3,

the p l a t e i s clamped

the h o r i z o n t a l displacment i s f i x e d ( ( 2 . 2 0 ) ) . the plate

( ( 2 . 1 8 ) ) ; on Finally,

i s s u b j e c t e d t o g i v e n boundary t r a c t i o n :

i s a given l i n e a r d e n s i t y of f o r c e s i n t h e p la n e o f and

a r e a l parameter

on

r3 r2,

t = (t,) the plate,

( p o s i t i v e i n t h e f o l l o w i n g ) measur-

NONLINEAR BIFURCATlON

i n g the intensity of the boundary traction ((2.20)). u

(2.12)

n

= u

a

S O ;

a

n n

R = u

(2.13)

n

S O ;

a B

R un = 0 ;

(2.1k)

,

(2.15)

0

i-1

UP

a

T

on

rl.

= 0;

13

M = 0;

(2.16)

h o

(2.17)

c

(2.18)

aB

0

aP

(2.20)

'la

+

F = 0.

? L =0 an

= 0,

(2.19)

,p

n

B

= Xt,.

r2 u r q ;

on

r2;

on

r

on

U = O

3'

In the following, we assume (2.21) and (2.22).

Then

rigid displacements of the plate are n o t possible; meas

(2.21)

r2 u

(2.22)

Remark 2.2 (2.12),

- 5

= 0

. . . , (2 . 2 0 ) ,

r3

r3

> 0;

is nonrectilinear.

,...,( 2 . 6 ) ,

is a solution of the problem (2.2)

for all real

X,

i.e. for all intensity

o f the lateral load; the horizontal displacement is the

solution of a classical elasticity problem (with unilateral boundary conditions), The question is to k n o w i f , for some real

h ,

load

X.

other solutions

exist

and to find the critical

I t is natural to introduce the Problem 2.3 (linearized problem) _ _ ~

_

-

To find

5

and

uo

such

CLAUDE DO

118

in

with the boundary conditions (for

<

and

on

r2.

n,

uo) (2.12), ...,(2.18),

(2.20) and a,;

(2.26)

nB = t,

It is an eigenvalue problem. The r e s u l t s obtained in this paper are as follows: Results -

2.4

-

Let ?,

be the first positive characteristic

value of Problem 2.3.

Then, for the problem (2.2),,..,(2.6),

(2.12), ...,(2.20), (i) (ii)

5 =

0

if

l o is simple,

lo

is the critical load).

is the unique solution;

l o is a bifurcation point (i.e.

111. Variational formulation and functional formulation.

Using the theorem of virtual works, it is easy t o obtain the variational formulation of the previous problem (2.2), ...,(2 . 6 ) ,

(2.12),...,( 2.20).

First, we have to fix

the admissible displacements, the selection of functional spaces being clear and classic.

(3.1)

V =

{V

=

(v,) E

1

(H

Let us introduce

( n ) )2 1

v = 0

on

r33 ,

119

NONLINEAR B I F U R C A T I O N

By Korn inequality,

V

is a Hilbert space.

Now, the set of

kinematically admissible horizontal displacements is the closed convex cone of K = (v

(3.3)

V:

E v

I

o

vn = v n s a a

r,}.

on

The set o f vertical admissible displacements is defined by (3.4)

2 Z = ( z E H ( n )

I

Because of(2.21) and ( 2 . 2 2 ) , a(z1,z2)

z = O ,

az-

an -

o

on

r2 u r33.

the symmetric bilinear form

introduced by ( 2 . 7 )

is a scalar product on

Z,

equivalent to the natural one; in the following, we will use this scalar product, and we will write, more simply:

(3.5) Now, we assume that

(3.6) The variational formulation is obtained from the theorem o f virtual works, using admissible virtualCKSplacemen0j; then, we obtain Problem 3.1

-

T o find

u E K

(3.7)

and z

5

E Z

d x = O

such that ' & z € Z ,

(3.9) Elimination _ ___

of

u.

We have already observed that

a solution f o r all positive real solutions, we eliminate

u

1.

5 =

0

is

To find other possible

between the relations

(3.7), ( 3 . 8 ) ,

120

CLAUDE DO

(3.9):

i t remains a f u n c t i o n a l f o r m u l a t i o n o n l y f o r

t h i s , we f i r s t s o l v c Lemma 3 . 2

(3.8), ( 3 . 9 ) ,

when

gives the existence o f

<

and

1

j.

For

a r e given.

u = ~ ( 1 , s )and u = ~(1,;).

I n o r d e r t o e x h i b i t a l i n e a r i z e d p r o b l e m , we may i n t r o d u c e the following n o t a t i o n s

(3.10)

II

(because

li

i s a cone):

u ( h , < ) = xuo

+

..(A,<),

where

u

= xuo

+

s(x,c),

where

0

O ( h , ; )

0

= u(1,O);

0

= u(1,O).

2

0.

O bser v e t h a t

Lemma 3 . 2

-

exists a

unique

(L2(62))‘,

Let

z

be given i n

u = u(X,z)

in

with t h e decomposition

Z, V,

and

h

10) T h e r e

u = a(1,z)

and

in

(3.10), s u c h t h a t

c

2 0 ) W e have t h e f o l l o w i n g e s t i m a t e s C )

(with various constants

:

(3.14)

(3.16) 30)

I n addition,

and compact f r o m

the a p p l i c a t i o n

Z

into

L2(61).

z+sag

(X,z) i s c o n t i n u o u s

NONT,TUFAR H I F ' T J R C A T I O N

Proof: results

about v a r i a t i o n a l i n e q u a l i t i e s

particular,

results

e l a s t i c i t y (cf. (and in

(7.13)

Tlle problem ( 3 . 1 2 ) )

1 0 )

L2("),

i/-

7

theorem ( c f . [

i s coercive.

( c f . [ 71

General

) axid, i n

a b o u t \ - a r i a t i o n a l i n e c l u a l i t i e s of

C 51 ) by

Uao(h,<)

121

give

(3.13)),

E Z.

u(X ,:)

e x i s t e n c e arid u n i c i t y of

i f t$e o b s e r v e t h a t

7

,Y

3 8

T h i s p r o p e r t y comes f r o m t h c imbedding

91 )

(3.17)

H1(n)

t

12"(0),

tL.ith a compact i n j e c t i o n , v e r y u s e d i n what f o l l o w s . particular,

is

z

In

we have

w i t h a compact i n , j e c t i o n . 2 0 )

U s i n g t h e e s t i m a t e , for all

- -

f , g, f, g

in

we d e d u c e from t h e g e n e r a l r e s u l t 5 c o n c e r n i n g t h e

of

L

4( n ) :

continuity

the solution o f a v a r i a t i o n a l i n e q u a l i t y with respect t o

the datas

(cf.

101 )

the rollowing, with various constants:

3 g ) F r o m t h e s e e s t i m a t e s , and (3.18), we d e d u c e t h e compactness o r the statement. Remark and

z

3.3

-

s

i s L i p s c h i t x continuous b-ith r e s p e c t t o

(at least locally i n

z ) .

W e cannot h o p e f o r a b e t t e r

result and, i n p a r t i c u l a r , h c c a n n o t hope f o r any d i f f e r e n t i a t l i l i t y properties,

because

s

comes f r o m a v a r i a t i o n a l i n e q u a l i t > - .

122

CLAUDE DO

Now,

because

3.2) a r e i n

L2(n),

uo

and

( g i v e n by ( 3 . 1 0 )

s

and because ( 3 . 1 8 ) , we have

( L

Then, by R i e s z theorem, t h e r e e x i s t in

Z

and Lemma

and

C(k,<)

such t h a t

Theorem 3.4

(Properties of

L

-

L

i s a compact s e l f a d j o i n t

Z.

l i n e a r o p e r a t o r on Proof:

L)

i s s e l f a d j o i n t because

' 0

is symmetric.

c o m p a c t n e s s comes from t h e f i r s t i n e q u a l i t y ( 3 . 1 9 )

The

and

(3.17),

(3.18). Theorem 3.5

(Properties of

C)

i s a n o n l i n e a r o p e r a t o r on

Z,

(i)

c(X,O)

(ii) z

c

5

clIzI13,

z bC(k,z)

0,

w i t h the f o l l o w i n g p r o p e r t i e s .

c

1

(c(h,z),z)

2

and 0

( i i )L e t u s w r i t e

1;

c1

constant with

z;

v z E Z; w E W

Z ;

c o n s t a n t , independent of

~ ~ C ( ~ , z ) - C ( ~5, zc)l II1~- i I I I z l l ,

there exists Proof:

2

i s c o n t i n u o u s and compact on

C(1,z)

respect t o (V)

1

For a l l

= 0;

(iii) IlC(x,z)II (iv)

-

when t h e e q u a l i t y h o l d s ,

such t h a t

e

UB

(w)

+

1 2

-z

z

,Y ,6

= 0.

N O N L I N E A R BIFURACTION

123

and use the decomposition, with clear notations:

1I;I

then, i f

S

1,

we have

Lemma 3 . 2 ,

Now,

2 0 ) , gives

and the conclusion foflows,xqith (3.18), (iii)

Comes from ( 3 . 2 3 ) and (ii).

(iv)

We write

-

IlC(h,z)

like in ( 3 . 2 2 )

C(5,z)II

and tho

conclusion comes from ( 3 . 1 6 ) . (v)

First, we know solution, f o r

(3.12),

(3.13)

v = Xuo

+ w

that ?,

(with

0,

5

z=O),

Xu

0

,

defined in ( 3 . 1 0 ) , is

of the variational inequality and we choose the test function

to obtain the inequality:

Afterwards, we choose the test function

(3.12),

(3.13)

0

to obtain + s

a@) cap

Using

v = Xu

(w) d x s

[

t w

n a a

dU.

( 3 . 2 4 ) , we deduce: /

(3.25) Computing obtain:

(C(X,z),z)

with ( 3 . 2 1 ) , and using ( 3 . 2 5 ) ,

we

in

CLAUDE DO

124

Now,

we u s e t h e c o n s i t i t u t i v e r e l a t i o n s , i n e q u a l i t i e s

and ( 3 . 1 1 ) .

(2.1),

Then, /

(C(X,z),z)

c

5

(3.26)

( C ( x , z ) , z ) 5 0;

W e get

f o r a suitable

€

ap

a@

aB

i n addition, if

then there

z,

IJ

LJ

n

(w)

+

is

w

1 5 Z,aZ

,P

dx,

the equality i s true

V such t h a t

in

= 0.

T h i s c o n d i t i o n shows t h a t t h e p l a t e can bend w i t h o u t s t r e t c h i n g d i t w a s a l r e a d y found,

M.

POTIER-FERRY

[ 113

.

S o , t h e Theorem Remark

-

3.6

i n c l a s s i c a l s i t u a t i o n s , by

3.5 i s proved.

The above d i s c u s s i o n shows t h a t ,

i s s u f f i c i e n t l y f i x e d a t i t s boundary,

if

the p l a t e

s o t h a t i t c a n n o t bend

w i t h o u t s t r e t c h i n g , t h e n we have

Functional formulation. problem f o r

5

3.7

-

Problem

U s i n g t h e above n o t a t i o n s , we g e t a

only:

Let

1 2 0

be g i v e n .

Find

5 E

Z

such

that

(3.27) where

(I-XL)< I

Remark 3 . 8

i s t h e i d e n t i t y on

-

+ C(X,C) 2.

5

A s a l r e a d y observed,

t h e previous problem,

for all

= 0,

X

2

0.

= 0

i s a solution of

Now,

we have t o look

12 5

NONLINEAR BIFURCATION

h.

€or nontrivial solutions with suitable of

Existence results

nontrivial solutions describe the buckling.

By (iii), Theorem 3.5, it is clear that the linearized problem is

We may hope that this linearized problem gives (as in classical situations) the critical load and the buckling mode. while, the regularity of

C

C

with respect to

is o n l y Lipschitz-continuous ((iv),

1

Mean-

is very weak,

Theorem 3 . 5 )

S O

we

cannot hops f o r a better resu1t.C comes f r o m the solution of a variational inequality (see Remark 3 . 3 ) . techniques of H . B . FERRY [ 1 2 1

,

The results and

KELLER and W.F. LANGFORD [ls], M. P O T I E R -

D.H. SATTINGER [14] cannot be used here, because

they require the differentiability of

C.

Meanwhile, the

classical Lyapunov-Schmidt method (already used i n these papers) will be used i n Section I V to study the Problem 3 . 7 . Let of

L,

Eo

Lo

be the smallest positive characteristic value

the associated proper subspace (of finite

dimension, because the Theorem 3 . 9 ) .

Concerning the unique-

ness, we have the Theorem 3.8

-

If

o f the Problem

h = h

0

0

S

h < h

3.7 and

c=O

tknc=0

is the unique solution

still is the unique solution f o r

if we have the following condition.

126

CLAUDE DO

I f we h a v e

5

e q u a l i t y , then

E Eo,

equality i s

but

n o t p o s s i b l e when ( 3 . 2 9 ) h o l d s .

IV.

B i f u r c a t i o n a t a Simple C h a r a c t e r i s t i c Value. An a b s t r a c t v e r s i o n of

t h e problem d e s c r i b e d i n t h e

above s e c t i o n c a n b e f o r m u l a t e d

w i t h p o s s i b l e new applicaticns.

I n t h e f o l l o w i n g , we c o n s i d e r t h i s

abstract

version,

and t h e

a s s u m p t i o n s which a r e done a r e t h e p r o p e r t i e s o b t a i n e d i n t h e concrete case o f

the previous sections.

i s a r e a l H i l b e r t space, w i t h s c a l a r product

Z

11 1 1 .

and r e l a t e d norm

(4.1)

(4.2)

(4.3)

r

L

simple;

C

L,

Eo

is

t h e a s s o c i a t e d p r o p e r subspace

Go,

with

i s d e f i n e d on

R+XZ,

(4.4)

c(x,0) = 0,

(4.5)

zctC(h,z)

c

w

x

2

Il$oll

,

)

a s follows:

the f i r s t characteristic value o f

g e n e r a t e d by

and

and

L

i s a compact s e l f a d j o i n t l i n e a r o p e r a t o r o n

l o > 0,

C

The d a t a s a r e

(

Z; is

= 1.

w i t h t h e f o l l o w i n g properties:

0;

i s c o n t i n u o u s and compact on

constant with respect t o

z,;

and

Z;

x;

N O N L I N E A R BIFURCATION

4.1

Problem

-

x

Find

(4.11)

2

and

0

<

E Z

+ ~(x,;)

(I-xL),:

127

such t h a t

= 0.

We a r e l o o k i n g f o r n o n t r i v i a l s o l u t i o n s o f problem,

t h e above

using the

Lyapunov-Schmidt

method.

It c o n s i s t s i n looking f o r

s o l u t i o n s with the f o r m

<

(4.12)

I

+ w,

= EOo

I h - X o l < rl.

w E Eo'

I

Let

P

Eo.

be t h e o r t h o g o n a l p r o j e c t i o n on

Then

( 4 . 1 1 ) i s e q u i v a l e n t t o t h e s y s t e m ( 4 . 1 3 ) , (4.14), i f sufficiently s m a l l so that

w

+

(I-IL)-l

F i r s t , we s o l v e

w e have t o f i x ; f o r this.

(4.14)

e x i s t s on

Pc(x,e@o+ w) = for a l l

is

*O'

0.

and

i n a domain

E

P i c a r d ' s f i x e d p o i n t theorem can b e used

Afterwards,

of bifurcation)

(I-XL)-'

q

we p u t t h i s s o l u t i o n i n

(4.13)

wich we h a v e t o s o l v e w i t h r e s p e c t t o

Implicit function

theorem c a n n o t b e used

sufficiently regular;

Picard's fixed point

because

C

(equation

X. i s not

theorem c a n n o t b e

u s e d b e c a u s e we h a v e n o t a s t r i c t c o n t r a c t i o n .

Only

R r o u w e r ' s f i x e d p o i n t theorem a l l o w s u s k s o l v e t h e e q u a t i o n o f b i f u r c a t i o n ; f r o m them, i n f o r m a t i o n

s o l u t i o n s c a n n o t be o b t a i n e d .

on t h e number of

128

CLAUDE DO

The m a i n

-

Theorem b . 2

result i s the

eo >

There e x i s t s

such t h a t ,

0

h,

t h e Problem 11.1 p o s s e s s e s a s o l u t i o n

all

for

; (,; #

leI<EOY

if

0

6

#

0)

with t he e s t i m a t e s :

1 = A0(1

€ao

j = 0 s p

<

C l E l

+

Then,

u), w E El,,

W)

P-1 I'WII

+

s

(and

p

clc

I P.

> 0

if

E

f 0),

t h i s s o l u t i o n b i f u r c a t e s from t h e t r i v i a l one a t t h e

lo.

point

-

Remark j1.3 By ('1.9) ~-

,

t h e u n i q u e n e s s Theorem 3 . 8 h o l d s i n

i t s s t r o n g e s t form.

-

4.4

Remark

l o a d if it is

Theorem k . 2 shows t h a t

-

Lemma j1.5

IIwI1 5

There e x i s t s and

<

IE

w = w(1,S)

solution

4.5,

14.6,

4.7

L.

s e t up t h e p r o o f

of

4.2.

< no

Ih-kol

i s the c r i t i c a l

a s i m p l e c h a r a c t e r i s t i c v a l u e of

The f o l l o w i n g Lemmas Theorem

lo

clgl',

qo > 0,

where

(4.14)

C o y

with

Eo

IIwI S

> 0

such t h a t ,

possesses a IC

I.

if

unique

I n a d d i t i o n , w e have

c

i s a c o n s t a n t with r e s p e c t

A = A(h,€)

be t h e o p e r a t o r d e f i n e d on

to

X

and

E. Proof:

Let

2

if

IEl

(4.15)

and A

Ix-hol

*(I-lL)-l

PC(h,E~o+z);

are s u f f i c i e n t l y small,

then:

i s a s t r i c t contraction on the b a l l

R(0,IeI).

NONLINEAR B I F U R C A T I O N

T o s e e t h i s , we c h o o s e

(4.16) so,

when

(4.6), we

IG

if

Ih-hol

I I ~ z l ls

cIc

EL

a n i n v e r s e on

1

< Eo,

there c

are

d

for a l l

i s proved.

c:

IS\

6

<

if

0'

z

1.

and

tliat i s necessary,

we o b t a i n

the statement

w(X,E)

c

I G I P. of

(4.14),

IX-X~

(c

16

1

we h a v e t h e

< Co):

constant i n

But

so

and t h e r e s u l t f o l l o w s f o r

c0

Then,

we h a v e

= IIAwll S

C ~ E I

Now, b y

B(O,ISl),

L i p s c h i t z c o n d i t i o n (uniform with r e s p e c t t o l l w ( ~ , ~ ) - w ( i , ~ ) I 5I

I).

11 z - ~ l l;

c c p-l

(4.16),

For the solution

1;

IS

The f i r s t p a r t o f

Eo.

But now, by

-

in

1 1

IE

B(0,

a constants, ~ independent of

l!lyll

Lemma 4.6

5

possesses

i t i s e a s y toohtajn

Then, IIzII 5

if

and

z

AZ-AZII

a f t e r a reduction o f

(4.15)

no.

o p e r a t e s on

A

obtain,for a l l

I)

Ip,

(I-hL)

such t h a t

qo

c

129

sufficiently s m a l l .

x

and c ) .

CLAUDE DO

130

Now, let us study the bifurcation equation (4.13) after w = w(X,E),

substitution of Lemma

4.7

-

I

IE

For all

<

C

solution of o,

(4.14). = X (E )

there exists

such

that:

o

(4.18) (4.19)

(1

x -

< -

x

-

1 s

*

if

CJEIP-',

E

# o

( c ( 1 , E $ o + w(X,E)),Qo)

= 0-

0

Proof: We use Brouwer's

(4.7) and

have by

Lemma

I~C(X,E:G~

fixed point theorem.

First, we

4.5

+ w(a,e)>ll

4

~ I I +~ wG( x~, d P

c ~ ( E ( ~ ;

and it follows that P- 1

I n addition, (4.18) and Lemma I I C ( X , E ~+ ~

-

w(~,c))

That is enough to use

4.6 give + ~(x,E)II

c(X,CQ,

Brouwer's

c3lc

I 1x-X I

fixed point theorem.

The

conclusion follows. Remark 4.8 Lemma

-

Theorem

4.2 (and

4.7) shows that the solutionsof

(4.19) are inside the dark region the annexed picture.

of

But, without

uniqueness, a local representation

X = X(P)

is not possible.

I n fact,

when the assumptions o f the implicit function

theorem are not valid, all the dark region can be

h

131

NONLINEAR BIFURCATION

f ( h ,F:)= 0 .

s o l u t i o n o f a problem l i k e i n g c l a s s i c a l e x a m p l e , where

f(A,e) =

r

I

f

0

-E -h -2E

2

2

W e r e c a l l t h e follow-

i s d e f i n e d by: 2

,

if

h

,

if

-E2

,

if

0 s h s

,

if

E 2 S

,

if

h

-E

5

s

< 0 6

S

29

2

x

2

29

2

2

ExoEpt f o r t h e c o n t i n u i t y of p a r t i a l d e r i v a t i v e s a t

the origin,

e v e r y a s s u m p t i o n o f t h e i m p l i c i t f u n c t i o n theorem h o l d s . we check t h a t

is Lipschitzian, but

f

t h e c o n c l u s i o n s of

i m p l i c i t f u n c t i o n s theorem a r e n o t v a l i d : representation

-

Remark '+.9

member o f

Picard's

On t h e c o n t r a r y , i f

replaces

r e p l a c e d by

a local

f i x e d p o i n t theorem t o

b e c a u s e we h a v e no i n f o r m a t i o n on t h e c o n s t a n t

i n (4.20).

below,

(4.10),

I

cl(e 9

( 4 . 2 0 ) by

1h-A

t h e assumption ( l k . 1 0

t h e r i g h t member i n

I +

c4l?

1'

clP

I

IX-il,

1) W-GI~ with

4 . 7 and we o b t a i n T h i s assumption

C ( h , z ) = XC(2).

l o c a l e x i s t e n c e of a

bis),

( 4 . 1 7 ) can be and t h e r i g h t

a >

Then,

0.

Picard's f i x e d p o i n t t h e o r e m c a n b e u s e d i n t h e p r o o f

this case.

the

i s not p o s s i b l e for t h e s e t

X(C)

We c a n n o t u s e

solve (4.19) c4

h =

Also

unique

of Lemma

h = X(o).

( 4 . 1 0 b i s ) i s , f o r example, s a t i s f i e d i f Theorem 4 . 1 0

But, p r o b a b l y ,

is the result

obtained i n

(4.10 b i s ) i s not true f o r the

problem d e s c r i b e d i n s e c t i o n s I1 and 111.

CLATJDE DO

132

Then, there exists one, unique, family o f nontrivial solutions

4.1 and i t possesses the representation

for the Problem

Besides, the estimates given i n the statement of Theorem 4.2 hold good.

References [11 L. Landau et E. Lifschitz, ThGorie de l'Glasticit6, Physique th6orique , t. V I I , Editions M i r y lloscou (1967). [2] C1. D o ,

~'robli?mes de valeurs propres pour une in6quation

variationnelle sur un ccne et application auflambement unilateral d'une plaque mince, C.R. Acad.

Sc.,

A , 280,

P. 45-48 (1975). c3]

C1. Do, The Buckling Plate Subjected ~- o f a Thin Elastic . __ ~~~~

~

to Unilateral Conditions; Proceedings of the IUTAM/IMU Symposium -

on Applications of Methods of Functional

~

Analysis to Problems in Mechanics, Lecture Notes i n Math.

[4] C1.

Do,

,

5 0 3 , p. 307-316, Springer ( 1 9 7 6 ) .

Bifurcation Theory for Elastic Plates Subjected

to Unilateral Conditions, -~ 6 0 , no 2 ,

[51

J. M a t h . A n a l . A p p l . ,

p. 435. 448, 1977.

G. Duvaut et J . L .

Lions, Les in6quations - - . _en _ MGcanique - . et _ _

en Physique, Dunod, Paris (1972).

133

N O N L I N E A R BIFURCATION

Duvaut e t J . L .

G.

Lions,

Prob1;rnes --

u n i l a t 6 r a u x d a n s ~la

t h e o r i e d e l a f l e x i o n f o r t e d e s p l a q u e s , J . MGca., no 1 J.L.

(1974).

Lions,

Q u e l q u e s rn6thodes d e - r 6 s o l u__t i o n .d-e s p r o b l & m e s ~~~

aux -~ A.

~

Necas,

Sur- l e f l a m b e m e n t d e s p l a q u e s

rniE"s,

Cycle, U n i v e r s i t g de P o i t i e r s

(1976).

Les m6thodes d i r e c t e s d a n s l a t h e o r i e d e s __.

6 q u a t i o n s e l l i p t i q u _e s_, ~

Prague,

(1969).

Dunod, P a r i s

~

Cirnetiere,

G.

~

l i m i t e s rion l i n 6 a i r e s ,

T h e s e d e 3;me J.

13,

e t Masson,

Acad.

T c h g c o s l o v a q u e d e s Science,

(1967).

Paris

Stampacchia, V a r i a t__ i on a l I n e q u a l i t i e s ; T-heory t i o n s o f Monotone O p e r a t o r s , _A -p p l i c a.~

and

Proceedings of a

NATO Advanced S t u d y I n s t i t u t e , V e n i c e ( 1 9 6 8 ) . M.

Potier-Ferry,

C harge s -l _i r_ n-i t e s - e t Th6orerne d ' e x i s t e n c e ~~

~

e n t h 6 o r i e d e s plaqles 6 l a s t i q u e s , C . R .

280, p. M.

Sc.,

A,

1317-1319 e t 1385-1387 ( 1 9 7 5 ) .

Potier-Ferry,

B -i f u r c a t i o n e t s t a b i l i t 6 p o u r d e s

s-y s t 6 m e s d 6 r i v a n t d ' u n . p o.t_ entiel, ~

H.B.

Acad.

K e l l e r e t W.F.

& paraqtre.

L a n g f o r d , I--___ terations , -P e r_ t u_ rb_ a t i_ o n_ s ~

and M u l t i p o r N o n l i__ n e ar.-B _ l_i c -i t.i.e s f-~ - i f u r c-a t i o n P r o b l e m s , Arch. D.H.

Rat.

Mech.

Anal.,

48,2, p . 83-108 ( 1 9 7 2 ) .

S a t t i n g e r , S t a_b_i l i t y o f B i f u r c a t i n g S o l u t i o n s b y

L e r a y - S c h a.___--uder Degree, A r c h .

- .

no 2 ,

D.H.

p.

Rat.

Mech.

Anal.,

43,

154-166 (1971).

S a t t i n g e r , Topics i n S t a b i l i t y and B i f u r c a t i o n

T h e o r y , L e c t u r e Notes i n Math.,

309, Springer ( 1 9 7 3 ) .

CLAUDE D O

134

[16] B. Budiansky, Theory _ _ _ of _ ~Buckling and Post-Buckling Behavior ____-- of Elastic Structures, Advance in Appl. Mech.,

14, Chia Shyn Yih ed., Academic Press (1974). El71 J. Naumann, On Value Problems __Some Unilateral - -__- Boundary _in Nonlinear Plate Theory, A paraztre dans Beitrdge zur Analysis.

[I81 J. Naumann et H.U.

Wenk, On Problems for ._ _Eigenvalue _ _ . -.. ______

Variational Inequalities; an Application to Nonlinear __.___ ___---__--___ -Plate Buckling, Rendiconti di Matematica ( 3 ) , Vol. 9, s6rie VI, p. [l9] M.S.

439-463 (1976).

Berger, A Bifurcation Theory for Nonlinear Elliptic

Partial Differential Equations and Related Systems; Bifurcation Theory and Nonlinear Eigenvalue Problems;

J.B. Keller et S. Antman ed., Benjamin (1969). [ Z O ] E.

Miersemann, Uber Nichtlineare .~ Eigenwertaufgaben in konvexen Mengen, Preprint, Karl Marx Universit3t, Leipzig (1976).

1211

R.C. Riddell, Eigenvalue Problems for Nonlinear ___ - - Elliptic Variational on a Cone, J. Functional Anal., . -. Inequalities - - ..._._ to appear.

[22]

R.C. Riddell, Eigenvalue Problems for ___Nonlinear Elliptic ~

-_

Variational Inequalities, to appear. -._

G.M. de La Penha. L.A. Medeiros (eds.) Contemporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations @North-Holland Publishing Company (1978)

OY Tflk’

THERMODYNAMICS OF NON-CLASSICAL SYSTEMS

BERNARD D. COLEMAN Department of Mathematics Carnegie-Mellon University Pittsburgh, Pennsylvania 15213

In recent years David R. Owen and I have been developing a mathematical theory of thermodynamics in which the existence and regularity of caloric functions o f state, such as entropy and free energy, are deduced, rather than assumed [ 11 -[ e.g.

51 .

When applied to

several classes of materials,

elastic and viscous materials, materials with internal

state variables, and materials with fading memory, our new formulation o f thermodynamics yields restrictions on the constitutive relations for the stress, temperature, and heat flux agreeing with those obtained i n treatments [ 61 -[ 103 which start with a differentiable entropy function and employ the Clausius-Duhem inequality. to hypoelastic materials [ 3 ]

The theory has been applied also and elastic-plastic materials

[Z] [4] for which the thermodynamical restrictions had not previously been fully understood.

The new formulation o f the

principles of thermodynamics has two advantages:

(1) It

permits one to determine the class of entropy functions compatible with given constitutive relations for stress, temperature, and heat flux.

(2) It permits one, in principle,

to find thermodynamical restrictions on such constitutive

136

BERNARD D.

r r l - i t i o n s without

e v e r m e n t i o n i n g e n t r o p y or f r e e e n e r g y .

T h i s s e c o n d a s p e c t of

t h e t h e o r y was i l l u s t r a t e d i n a n o t e

i n t h e Reridiconti d e l l ’ b c c a d e m i a d e i Lincei

r e c e n t l y published

[41.

COLEXAN

I t i s t h e m a t t e r of t h e e x i s t e n c e of e n t r o p y f u n c t i o n s

which I s h o u l d l i k e t o d i s c u s s t o d a y .

L 51

Owen and I g i v e p r o o f s

of

I n r e f e r e n c e s [l]

t h e theorems s t a t e d below.

(C , V ) i n

I n the present theory, a system i s a p a i r

Z

which

i s a t o p o l o g i c a l space shose elements

II

s t a t es , and

a n open s u b s e t called form

g(P)

Pp:

C + R(P) ’6

P-valued f u n c t i o n t

Z

of

Z;

pp

P

)

H

#

Q Y

P”P‘

i s called

pP’,

ll &(P”)),

Pp

is

of

R(P).

Thus

I t i s assumed

I”’

processes

and

there i s specified a with t h e p r o p e r t y t h a t

PNP’,

i.e.,

is the

for e a c h

t h e r e holds

J

pPNplD

in

= pp,,p’a;

t h e p r o c e s s r e s u l t i n g from t h e s u c c e s s i v e

P‘

a plication of __

with

of

and i s w r i t t e n i n t h e

i s d e n o t e d by

(P” yP‘)

pairs

With

Pp

t h i s function

and i s c o n t i n u o u s .

(I)” , P ’

pPN

-1 f i ( p ” p ’ ) = pp’(6?(P’)

if

into

t h e transformation induced b y

“composition” of

P”P’

C

R(P’) n a S ( P ” )

f o r which

P P// p‘

Z

of

The r a n g e o f

on t l i e s e t

that

I”

Q(P)

p,a. C

a r e called

i s a s s o c i a t e d a c o n t i n u o u s mapping

t h e t r a n s f o r m a t i o n inducg-d by 0 I--,

u

i s a s e t o f e l e m e n t s c a l l e d __ p r o-c___ esses.

each process t h e r e

and

and P”. -

(z,n)

The p a i r

f o r m s a system

t h e s t r u c t u r e ,just described has t h e following p r o p e r t y ,

which w e c a l l p r o p e r t y

I:

F o r eacli

states accessible f r o m

uY

i.e.,

i.s d e n s e i n

0

in

c,

the set of

t h e set

C.

We have r e c e n t l y observed [ 5 ] t h a t s e v e r a l o€ our r e s u l t s f o r s y s t e m s h o l d a l s o f o r more g e n e r a l o b j e c t s c a l l e d

ON THE THERMODYNAMICS O F N O N - C L A S S I C A L

F o r a semi-system,

semi-systems.

n

the s e t

assumed t o h a v e p r o p e r t y

-

I;

o f s t a t e s and

a s t r u c t u r e of' t h e t y p e

o f processes possess

employed t o d e f i n e s y s t p m s , b u t

c

the s e t

137

SYSTEYS

t h i s s t r u c t u r e 1s no l o n g e r

w e assume i n s t e a d t h a t t h e

s t r u c t u r e h a s a l e s s r e s t r ' i c t l v e p r o p e r t y , which I s h a l l h e r e

I t :T h e r e

c a l l property

I

n u o := { p p a o i s dense i n

C

in

C.

A state

such t h a t

the

set

E l9(P)3

Do

such t h a t t h e s e t

Do

1%

E n,

p

is c a l l e d a b a s e s t a t e .

b a s e s t a t e , and a s y s t e m

oo

is a state

70

i s dense

A semi-system h a s a t

l e a s t one

a s e m i - s y s t e m f o r which e v e r y

s t a t e i s a base s t a t e . Let

noc

fi

An action

nOc

-4

IR

(i)

19 (P"P' )

:=

n ~ cI

u E ~J(P)}.

(C,F)

i s a function

with the following two properties: Additivi-ty

,

(ii)

(X,y).

E

f o r a semi-system

-

if

Continuity IR

(P",P')

is in '6

and

C7

is in

then ,cL(l-"'P'

$(P) +

((r.,~)

-

+

,a) = p(P' ,a) f o r each

p(p" , p p ' " ) ; 9,

in

P

the function

&(P,.):

i s continuous.

Let

A

b e a n a c t i o n and

If

@

i s a subset of

s t a t e o f a semi-system

0

C,

.A{O + S]

we w r i t e

t h e s e t o f numbers o b t a i n e d by l e t t i n g

P

vary over t h e

p r o c e s s e s whose i n d u c e d t r a n s f o r m a t i o n s t a k e A{.

If € o r e a c h

-4

c > 0

(43

:=

(,A(P,U)

1

P

E

n,

for

u

into

@,

i.e.

of

a

p p a E 81.

t h e r e i s a n open n e i g h b o r h o o d

8

U

BERNARD D.

138

COLEMAN

such t h a t

u.

t h e n P i s s a i d t o have t h e C -l a u s i u s p r o p e r t y a t

( C ,ll)

i s a b a s e s t a t e of

8

neighborhood

and i f

each s t a t e

u

uo

If

h a s an open

such t h a t

D

t h e n we s a y t h a t

i s u p p e r l i m i t e d from

,&

F o r systems,

uo.

t h e f o l l o w i n g t w o theorems a r e e a s i l y

p r o v e d [ 11 : Theorem 1

-

a system,

then

from

i.e.,at

0,

I f A -

,a, h a s t h a t p r o p e r t y a t a l l s t a t e s a c c e s s i b l e a l l s t a t e s i n the s e t

h y p o t h e s i s , dense i n

-

Theorem 2

uo

which i s , by

An a c t i o n kL h a s t h e C l a u s i u s p r o p e r t y a t a s t a t e

F o r g e n e r a l semi-systems, t o base s t a t e s ,

i s upper l i m i t e d f r o m

Do.

when a t t e n t i o n i s c o n f i n e d

t h e p r o p e r t y of b e i n g u p p e r l i m i t e d i s a t

l e a s t as strong as the Clausius property,

-

nu,

C.

of a s y s t e m i f a n d o n l y i f ,&

Theorem 3

of

u

has t h e C l a u s i u s property a t a s t a t e

f o r we have C51:

If a n a c t i o n ,A f o r a s e m i - s y s t e m i s u p p e r

from a base s t a t e . -

00'

thenp -

limited

has t h e C l a u s i u s p r o p e r t y a t

Do* There a r e semi-systems

( b u t n o t s y s t e m s ) , f o r which

a n a c t i o n which h a s t h e C l a u s i u s p r o p e r t y a t a b a s e s t a t e n e e d ' n o t be upper l i m i t e d f r o m t h e base s t a t e . I f p. i s a n a c t i o n f o r a s e m i - s y s t e m A:

G

C

C

4

lR

i s c a l l e d an upper p o t e n t i a l f o r

(C

,n), ,&

i f

a functim

O N THE THERMODYNAMICS O F NON-CLASSICAL SYSTEMS

(1)

t h e domain

(2)

whenever

E > 0,

n

Note: -

If

Pa

of

u1

i s dense i n

A

a2

and

are i n

8

a n open n e i g h b o r h o o d

in

P

G

with

Pal

8,

in

G ,

u2

of

and

t h e r e i s , f o r each such t h a t ,

f o r every

there holds

i s a n u p p e r p o t e n t i a l f o r p-

A

C,

are b o t h i n t h e domain of

,

u

and i f

4

-

If

f o r a semi-system h a s an upper

then p has the C l a u s i u s p r o p e r t y a t every

s. t a t e i n t h e domain ~

[1][5]: ~~

A,

potential

an a c t i o n A ,

and

then

A,

The f o l l o w i n g theorem h a s a v e r y s h o r t p r o o f Theorem

1 39

G

f A.

~~~~~~

A far less t r i v i a l result

i s t h e f o l l o w i n g theoremC51,

w h i c h , w i t h i t s c o r o l l a r y , Theorem 6 ,

i s the principal result

[l] of t h e p r e s e n t t h e o r y : Theorem 5 if

-

I f p. i s a n a c t i o n f o r a s e m i - s y s t e m -

t h e r e i s a base s t a t e

limited,

then

L%

uo

in Z

(C,n)

from & , ____ which --

has an upper p o t e n t i a l

A

and

i s upper

which i s upper __

semicontinuous. I n v i e w of Theorem 2 ,

t h e theorem j u s t s t a t e d has t h e

following corollary: Theorem 6

-

I f p. i s a n a c t i o n f o r a s y s t e m -

there is a s t a t e i n th e n L?,

C

a t which U ,

(C,n),

and i f

h a s t he.-~ Clausius property, ___

has an upper semicontinuous upper p o t e n t i a l .

~

BERNARD D .

140

COLEMAN

Without w r i t i n g o u t h e r e o u r p r o o f s h o u l d l i k e t o show you t h e form of t h e proof y i e l d s .

(C,n),

l e t ,&

Let

oo

5, I

of Theorem

t h e u p p e r p o t e n t i a l which

be a base s t a t e o f

b e u p p e r l i m i t e d from

no,

t h e semi-system

and c o n s i d e r t h e

(oo,o):

f o l l o w i n g f a m i l y o f open s e t s a s s o c i a t e d w i t h t h e p a i r

6 ( a o , u ) := { Q c C

I

O o p e n , UEO, s u p p / { o o + S] <

m } .

Let

Here t h e infimum may o r m a y n o t b e f i n i t e ; set states

0

We s h o w t h a t in C.

for which i t i s f i n i t e , i . e .

nuo

i s a s u b s e t of

be t h e

let

C o and h e n c e

We a l s o s h o w t h a t t h e f u n c t i o n A o :

Co

let

C o + R,

co

i s dense

d e f i n e d by

i s u p p e r s e m i c o n t i n u o u s and i s a n u p p e r p o t e n t i a l f o r A . If

the s e t

m(oo,uo) = 0 P

Ao,

in

n

Co

uo,

contains the base s t a t e

and i f

( f o r t h i s t o b e s o i t s u f f i c e s t h a t t h e r e be a

and s u c h t h a t

Pao =

uo

and

,&(P,ff

) =

0),

then

d e f i n e d a b o v e , i s t h e minimum e l e m e n t o f

t h e s e t of a l l

Co

and v a n i s h a t

u p p e r p o t e n t i a l s f o r ,& which have domain 00

T h i s r e s e a r c h w a s s u p p o r t e d by t h e U . S . Science Foundation.

I am v e r y g r a t e f u l

National

t o our B r a z i l i a n

c o l l e a g u e s w h o have s p e n t time a t Carnegie-Mellon namely P r o f e s s o r s Guilherme M . F i l h o , and L u i z C a r l o s M a r t i n s ,

University,

d e L a P e n h a , Rubens Sampaio f o r making t h i s v i s i t t o B r a d

ON THE THERMODYNAYICS OF NON-CLASSICAL SYSTEMS

141

possible and f o r their friendship and kindness to me here.

References [l] Coleman, B . D . ,

D.R. Owen:

&

Arch. Rational Mech. Anal. 54,

€or thermodynamics.

1-104

A mathematical foundation

(1974).

[ 2 1 Coleman, B.D.,

D.R. Owen:

&

On thermodynamics and e l a s 6

59, Arch. Rational Mech. Anal. -

plastic materials.

25-51 ( 1 9 7 5 ) .

[3] Coleman, B.D.,

& D.R,

Owen:

On thermodynamics and

intrinsically equilibrated materials.

108,189-199

Pura Applicata (LV),

[4] Coleman, B.D.,

&

D.R. Owen:

plastic materials.

&

(1976).

Thermodynamics of elastic-

Rend. Accad. Lincei (Cl. Scienze

fisiche, Serie VIII)

[ 5 ] Coleman, B.D.,

Annali Mat.

61,

D.R. O w e n :

77-81 ( 1 9 7 7 ) . On the thermodynamics of

semi-systems with restrictions on the accessibility 6 6 , 173-181 ( 1 9 7 7 ) . of states. Arch. R a t i o n a l blech.Ana1. -

[6] Coleman, B.D.,

& W. N o l l :

The thermodynamics o f elastic

materials with heat conductionand viscosity. Rational Mech. Anal.

[ 7 1 Coleman, B.D.,

&

3,167-178

V.J. Mizel:

(1963).

Existence of caloric

equations of state i n thermodynamics.

9,1116-1125 [8] Coleman, B.D.:

Arch.

J. Chem. Phys.

(1964).

Thermodynamics o f materials with memory.

Arch. Rational Mech. 1 7 , 1-46 (1964).

BERNARD D .

142

[9]

Coleman, B.D.:

O n thermodynamics, s t r a i n impulses, and

viscoelasticity.

230-254

COLEMAN

Arch.

R a t i o n a l Mech. Anal. 17, -

(1964).

[lo] Coleman, B.D.,

& M.E.

Gurtin:

internal state variables,

(1967).

Thermodynamics with

J. Chem. Phys.

47, -

597-613

G.M. de La Penha, L.A. Medeiros (eds.) Contemporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations @North-Holland P u b l i s h i n g Company (1978)

OK AN INEQUALITY IN THERMODYNAMIC STABILITY

ROGER L. FOSDICK

1

Department of Aerospace Engineering and Mechanics University o f Minnesota Minneapolis, Minnesota 55455

1. Introduction. -.

In recent years significant progress has been reported on the relationship between the thermodynamics and

A key step in much of

the stability of a general continua.

this work is the demonstration that a functional o f the form

is, i n a broad class of processes

P,

a non-increasing

function of time, i.e.,

61dt for all

t > 0. Here,

Be >

@(t) s 0 0

is the prescribed constant

equilibrium absolute temperature;

and

E(.,.)

n ( - , * ) are,

respectively, response functions f o r the densities of internal energy and entropy;

e(X,t) > 0

and

K(X,t)

are,

w

respectively, the fields o f absolute temperature and a deformation tensor (e.g., an appropriate strain tensor for an 'This article is based o n work that was done jointly with J. Ernest Dunn, a more detailed version o f which will appear shortly in the Archive for Rational Mechanics and Analysis. See, f o r example, [: 1-71 and, for a general theory [: 81

.

144

ROGER L. F O S D I C K

elastic material,

rl-31,

c.f.

or t h e s t r e t c h i n g t e n s o r i n

c e r t a i n incompressible f l u i d s , c . f . velocity field. d e n o t e s t i m e and

the processes

course, i n the foregoingdescription,

t

B

i s t h e body m a n i f o l d w i t h p a r t i c l e s

XEB

1

$ ( t ) is, for

t h e i n e q u a l i t y ( 2 ) shows t h a t

IP,

o f

bounded ____ above by i t s i n i t i a l v a l u e @ ( O ) ,

a f i n i t e lower bound f o r

t h e e x i s t e n c e of

i s the

v(X,t)

Of

and mass measure m ( . ) . While

[9]);

@ ( t ) i s provided

by t h e i n e q u a l i t y

(e,E) f ( B e , K e ) ,

for all

justified [1,2] w i t h ( 2 ) and result

under

an i n e q u a l i t y which E r i c k s e n h a s

f a i r l y natural conditions.

Moreover,

( 3 ) t h e f o l l o w i n g r a t h e r weak i m p l i c i t

stability

i s immediate:

i

B

f(e,K)dm

@ ( o )-

where t h e p o s i t i v e d e f i n i t e f u n c t i o n

f(.,-)

C+,+eed-,.>l

me,

f(.,.)

(4) i s d e f i n e d by

- C ~ ( e e l ~ e ) - e e ~ ( e e , ~, e ) l( 5 )

The f o l l o w i n g q u e s t i o n may b e r a i s e d :

Since i n

1 F o r s i m p l i c i t y , we assume t h r o u g h o u t t h i s w o r k t h a t B is regular a t a l l X E B. T h i s n o t i o n o f r e g u l a r i t y i s made p r e c i s e i n t h e forthcoming paper r e f e r r e d t o i n f o o t n o t e 1 on page 1. I n words, i t means r o u g h l y t h a t B h a s a smooth c o n n e c t i n g s t r u c t u r e s o t h a t a t any X E B any n e i g h b o r h o o d can b e decomposed i n t o d i s j o i n t s u b b o d i e s e a c h h a v i n g a p r e s c r i b e d f r a c t i o n a l mass t o t a l i n g t h e mass of t h e neighborhood.

ON AN INEQUALITY

practice neither known,

nor

€ ( a , * )

the function

I N THERMODYNAMIC

I](.,-)

i s ever e x p l i c i t l y

-

i s a l s o n o t known

f(.,.)

145

STABILITY

u n d e r what

c o n d i t i o n s i s i t n e v e r t h e l e s s p o s s i b l e t o c o n c l u d e from

more e x p l i c i t bounds f o r

8(

a

,

t)

c o n n e c t i o n Coleman and G r e e n b e r g

-

K(

and

[4]

a

,

(4)

t)? In this

and Coleman

~ s t a b l i s h e dt h e i n t e r e s t i n g r e s u l t t h a t i f

[5]

f(.,-)

llave

i s not

o n l y p o s i t i v e d e f i n i t e b u t a l s o convex o u t s i d e a compact s e t

>

then f o r every

where

II*II

points

0

there exists a

6 > 0

d e n o t e s a norm on Lhe s p a c e o f n - t u p l e s

(e,g).

The a p p l i c a t i o n of

we s e e t h a t i f

Ie

enough" t o LI

sense

P

by

( 4 ) and ( 7 )

i s convex o u t s i d e a compact s e t

f(-,.)

f o r any p r o c e s s i n

containing

s u c h a lemma t o t h e

s t a b i l i t y a n a l y s i s o u t l i n e d above i s c l e a r :

an

such that

with the i n i t i a l value

it follows that close" t o

Be

e(.,t)

and

Ke.

and

I(0)

hut broadly generalized, i s the subject o f

"close

K(-,t)

A r e s u l t of

then

are i n t h i s type,

this work.

In

f a c t , i n o r d e r t o b e a p p l i c a b l e t o a w i d e r c l a s s of m a t e r i a l s , t h e requirement w i l l b e dropped,

that

f(.,.)

b e convex o u t s i d e a compact s e t

and t h e s p e c i f i c norm

11 - 1 1 ,

as introduced

a b o v e , w i l l n o t b e assumed. F o l l o w i n g a d i s c u s s i o n of problem and r e l a t e d results,

t h e a b s t r a c t fundamental

theorems i n S e c t i o n 2 ,

i n S e c t i o n 3,

we t h e n a p p l y o u r

t o show how c e r t a i n m i l d c o n d i t i o n s on

t h e s t r a i n energy f u n c t i o n i n h y p e r e l a s t i c i t y t h e o r y can l e a d to a result

similar t o

(7),

and t h e r e b y b r o a d l y g e n e r a l i z e a

s t a b i l i t y t h e o r e m o f G u r t i r i [ 101.

146

ROGER L. F O S D I C K

2. The Fundamental Problem and Related Theorems.

Following upon the question that was raised in the introduction, we state, somewhat more precisely, the associated fundamental problem:

Given a continuous function _____.

on a connected subset

functions

R2,

on D -

N(*)

of a complete metric space

D

the non-negative reals

f(.) defined

into

V

into

determine those continuous

R2

which share with

f(-) 2

common non-empty compact proper subset of zero's, i.e., 1 = f- (0) E

N-'(O)

for every -

E 7 0

P, C

D,

an? which have the ;>roperty that

there is a

wise continuous function

6 > 0

g(*) on B

such that for any pieceinto - D

with ___

one has

More generally, for

where

C

P

A > 0 and for the set of functions given

denotes the class of piecewise continuous €unctions,

the fundamental problem is to determine, in terms of the general structure of N(.)

f(.),

those conditions which any function

must satisfy i n order to ensure that the number

is finite and tends to zero with main point, if

f(.)

A.

Thus, to emphasize the

is known only in a vague, qualitative

way, then it is not likely that a bound on

( B

f(g)dm

will

ON AN INEQUALITY I N THERMODYNAMIC STABILITY

g ( * ) . Never-

be at once tractable regarding information about f(.)

theless, qualitative information on construction o f an explicit

may suffice for the

and "nice" function

which

N(.)

satisfies the conditions of the fundamental problem statement

g(.).

and so yield concrete information on t o this

proceQure as

f-N

sbitching.

associated with the possibility of when is

A?

G(A)

We shall refer

Thus the basic questims

f-N

switching are, then,

finite valued, and how does its value depend on

i(* E S,)

and when does there exist G(A)

=

i

N(g)dm

such that

?

B

We shall refer to any such function

gc.t

E

s,

as a

Clearly the study o f C; -maximizers is relevant A to the questions associated with the possibility of f-N

% -maximizer. switching.

As we shall now see, the local characterization,

at its points of continuity in

B,

of any

A

-maximizer

requires that a certain relative concavity property must hold between

f(*)

and

N(.).

This, in turn, will be given a

useful geometric interpretation in terms of the notion of f - N touch points.

First, then, we begin with a theorem which is

concerned with necessary conditions for the existence o f a

GA -maximizer. Theorem 1

-

Let

-g ( . )

be a SA-maximizer and let

a point of continuity of and g 2 -

in D -

E- ( - ) .

are such that

Suppose

x o E B

a E (O,l),

and

__

be El

148

ROGER L.

g(Xo)

Proof: Clearly, if trivial.

is connected and

gl

and

g2

f(-)

not in

E E

"X0)

Thus, take

Z

FOSDICK

then both (10) and (11) are

E D 4

D

and observe that since

is continuous there are many points which satisfy (10) for any

a E (Oil).

Now, with these three quantities fixed, suppose initially that

where

f(gl) f

(S,]

f ( _ g 2 ) and define

is a monotone decreasing sequence of open

spheres about

Xo

of continiii.ty o f

with limit

-(.)

Xo.

a E (O,l),

neighborhood of for large erioiigh

subbodies,

Sn 1

= (l-an)m(Sn)

an

the value of

ri

n

and

is a point

it follows that the numerator in the

above fraction vanishes in the limit as large eiioiigh

Xo

Because

so that also

1

with

a.

Thus, for

is within any prescribed

Sn

we m a y decompose

Sn,

n +

m(Sn)

=

an E (0,l). Hence,

into two disjoint

a n m(Sn)

and

2

m(S,)

and define the following function: X E B X €

sIl

1

sn

1

This function has the property that piecewise continuous and

gn(*)

E

S,

since it is

=

ON AN I N E Q U A L I T Y I N T H E R M O D Y N A M I C S T A B I L I T Y

of a Q - m a x i m i z e r ,

A

or,

we s e e t h a t

f o r l a r g e enough

n

gn(-),

w i t h the d e f i n i t i o n of

T h u s , by d i v i d i n g t h r u b y

n

and l e t t i n g

m(Sn)

4

m

we

o b t a i n (11) t o c o m p l e t e t h e p r o o f u n d e r t h e a s s u m p t i o n t h a t

f(gJ

f f(g2)To remove t h i s

f(g(Xo))

that since

f(g)

such t h a t Then,

S

>

f(g(XO))

and by what

q u a l i t y must h o l d f o r see that

then there e x i s t s

0

c l e a r l y for e v e r y

< f(g(XO)),

l a t t e r restriction, first notice

f(2)

a E (0,l)

f f(g') <

we hove

in

D

f(i(XO)).

af(g)

+

-

(1-a)f(g')

we h a v e a l r e a d y p r o v e n t h e same i n -

N(.).

N(g) S N(g(XO))

and

g'

and

Thus,

for a l l

letting

E

D

a +

1

such t h a t

we r e a d i l y

150

f(g)

ROGER L. F O S D I C K

f(C(XO)).

5

Now,

returning to the problem of removing

the restriction noted above, we suppose that

gl

g in D -2 it s o happens that

and

that

a E

and

(0,l)

have been chosen to satisfy (lo), and that

f(gl) = f(g2).

fki) s f(g(Xo))

as we have just noted.

Then, from (10) we have and s o

(i=1,-2,),

Nki)

I:

N(.'(XO)),

I

But then clearly (11) holds.

Those points

E D

which satisfy

are called points of f-concavity f o r

N.

By Theorem 1, it is

in this set that the value of a 6 -maximizer, g ( X ) , 3 whenever X is a point of continuity for g ( * ) .

must lie

-

G

While every point in €-concavity for

N,

is trivially a point of

it is perhaps worth noting that

for GA-maximizer could have all o f its points o f f-concavity __

N

& L,

since

?Z

is a proper subset o f

D.

A s a first step toward a geometric characterization

o f the points of f-concavity for

N

through the notion of

touch points we have the Theorem 2

-

A point

g E

w

D-C

if and_ only ~ _ _ _ i f there exists

g E D.

for all D

then

Proof:

-

If

-X

is a point of f-concavity for N

> 0

such that

is not a global maximum for

N(.)

on -

X > 0.

To seethat (13) with

-

1

2

0

is sufficient for

6

to

151

ON A N I N E Q U A L I T Y I N TKERMODYNAMIC S T A B I L I T Y

be a p o i n t

of

f-concavity

g

respectively, f o r

l-a,

g2

and

b e any

(13) is written, and

g2,

a E (0,l)

and when t h e

and t h e s e c o n d

and t h e n a d d e d , we o b t a i n

a E (O,l),

T h u s , if holds,

gl

r e p l a c e d by

f i r s t i n e q u a l i t y i s multiplied by by

g1

let

N,

and o b s e r v e t h a t when

D

two p o i n t s i n

for

-

1 z

and i f

0,

gl

and

we l e t

a r e chosen such t h a t

(12)2

$

t h e n (12)1 h o l d s which i m p l i e s t h a t

i s a point o f f-concavity f o r Now,

g2

g

N.

be a p o i n t o f f - c o n c a v i t y f o r

N

a E (0,l) by

and d e f i n e

where

p E (0,l)

where

-

g2 E D

gl E

and where

meets

f(g2) <

D

meets

f(g).

Thus,

f(gl) since

2

f(g)

a

is

and

s t r i c t l y l e s s t h a n t h e r a t i o i n b r a c e s and s i n c e t h e denominator i s s t r i c t l y p o s i t i v e , f o r a l l such

N

for of

a

and

g2.

Since

we may t h e n i n v o k e ( 1 2 ) 1 ,

-g

-

i s a point

of f-concavity

which w i t h t h e d e f i n i t i o n

may b e w r i t t e n as

f o r a l l such ting

g1

(12)2 h o l d s

then c e r t a i n l y

p

+

0

g1

and

E2

and any

E

inf N ( E )

g

-

f(E) -

N(g)

Clearly, l e t -

we s e e t h a t

2_

0,

-g

E D

(15)

f(g)

where t h e infimum i s t a k e n o v e r a l l

f(g),

(0,l).

i n t h i s i n e q u a l i t y and d e f i n i n g

x’ f(g) <

p E

(13) holds f o r a l l

which a l s o meet

g E

D

of

this

FOSDICK

ROGER L.

152

kind. F o r all

g E

D

such that

f(g)

= f(E)

we showed

N(g)

i n the closing remarks of the proof of Theorem 1 that

< N(g).

Thus, ( 1 3 ) holds for such

g

establish (13) for all

E D

Toward this end, n o w , define

gl E

f(_g2)

again meets and

g2

f(g) >

such that a E (0,l)

f(gJ f(&) - f(gJ

is such that

< f(E).

’

f ( g l ) > f(E)

Then clearly ( 1 2 )

is a point of f-concavity for

gl

with

g2

and

f(g).

by

interchanged, holds f o r all such

since (12)1,

D

and we have left to

f(g) -

a s where, here,

g,

N

51

2’

and

-g2

with and

El

g21

and

we may invoke a

interchanged, and for this

write

Finally, since choices o f

g1

(14) and (15) are still valid under the present and

g2 *

it follows that by letting

p

-t

1

i n (14), by using this result to continue the latter inequality and then by applying (15) we reach

for any such

g

E D

gl.

such that

Thus, we find that ( 1 3 ) holds for all

f(g) >

f(Z).

M

The condition ( 1 3 ) has an interesting and helpful geometric interpretation which I shall now describe.

To do

so, requires that the set of values o f the functions

f(.)

and

N(.)

defined on

D

be interpreted as surfaces, as is

illustrated, for example, i n Figure 1 .

(Here, 8:

is the

ON A N INEQUALITY I N THERMODYNAMIC STABILITY

-

singleton set

the origin,

a simple "cone".)

(.)

a ,b

5

-

(:,6)

b 2

0,

s u r f ac es

happens t h a t

,.,

but

g E D - i s a l w a y s a t l e a s t a s l o w as

(16)

0.

F- - ( * ) ,just a,b otherwise f o r a l l other

j t

Y

such s u r f a c e i s i l l u s t r a t e d .

>

a

13,

a t some p o i n t

f ( * )

downward t r a ns 1a t e d

and

M

aN(*)

Suppose t h a t f o r some touches

h a s been c h o s e n t o h e

N(.):

associated with

F

N(.)

C o n s i d e r t h e f o l l o w i n g two p a r a m e t e r famiLy

u n i f o r m l y c omp r e s s ed

of

and

153

f(g).

I n F i g u r e 1, o n e

Thus,

which g i v e s

-

- -

h = aN(z)

f(5) 2

0,

and

Accordingly, .g . E D. ( 1 7 ) h o l d s f o r some a E

for all

we c a l l a n y p o i n t

that

[O,m]

f-N

a proper

is clear that

(17) i s

touch point

-

E D

if

impossible.

N

on

D

E D

then -

if and o n l y if

iZ

f-N

i s a touch point

8

5

then

g l o b a l maximum on

N

in

D,

= 0

in

touch p o i n t s ,

i s n o t a g l o b a l maximum f o r

- ,Z

i s a point o f f-concavity

i s a proper

f-N

touch p o i n t .

A

for

It

(0,m).

l i n g t h a t n o ( $ -maximizer can have a l l o f i t s p o i n t s f-concavity

such

touch point

a t

t r u e f o r some

Thus, i n t h e terms of

Theorem 2 a s s e r t s t h a t if

N( . )

if

an

E D

Z,

we s e e t h a t

if N(-)

-

Recalof

has no

t h e n t h e e x i s t e n c e of p r o p e r

for

f-N

154

ROGER L. FOSDICK

touch points is necessary of a QA -maximizer. __ -~ for the existence . -. - One could now ask, to what extent is the existence of proper touch points -~ sufficient for the existence o f a maximizer?

SA -

Before turning to this important question let us

introduce the notion of a touch manifold associated with -~ -

g,

i.e.,

(in Figure 1, 3(g)

has two elements as shown) and consider

the following independent criterion concerning the existence

of touch points.

-

3 Theorem -

M

_Let _

E

D

a compact .set containing -be -~

C

such __

that

f@some~-2 > 0.

Then _ _ _ -proper __

- _.

in particular,

5(2)

point then ^a __ - - ____

f(i)/N(g)

where

q

E

Eg E

I

touch points exist and, ____

is not null.

I

D

f-N

> 0

f(g) s

_If _ ..2 E D

D

C

is a touch

and

f(g)3

*

It is worth noting that if of

-

is a compact subset

M;'

then the necessary condition (18)* is a complete

converse to the sufficient condition (18)1.

However, recal-

ling that a central motive in much of this work is that in

f(.) is

not usually known explicitly, we see

that neither is

M;.

A l l is not s o bad though in that for

any touch point

g

applications

I

{g E D

I

N(g)

S

known, s o that if

(17) requires that

N(g)],

M;

and since

Mh

N(-)

E

MG

E

is known

is a compact subset of

D

M i

is

then s o too

155

O N AN I N E Q U A L I T Y I N THERMODYNAMIC STABILITY

w i l l be

M.:

T h i s theorem w i l l be u s e d l a t e r i n t h e d e v e l o p -

ment of m i l d c o n d i t i o n s f o r non-linear merely

It

elasticity.

f ( * )

i s compact and

-

f(g) 5

$N(g)

t o u c h p o i n t s when

t h i s theorem,

-

ZN(g)

f(g)

f ( g ) < $N(g)

g E

a proper

t o u c h p o i n t and

D

on

0

2

h o l d s f o r all f-N

outside

N(.)

-

5

observe t h a t s i n c e

E M

there e x i s t s

- f(g)

GN(g)

-

of

Z E M

(18)1 t e l l s u s t h a t we have

guarantees

D.

Proof: F o r t h e f i r s t p a r t -__

ZN(g)

switching i n the theory o f

l i e above some c o m p r e s s i o n of

some compact s e t i n -

M

f-N

M.

On t h e o t h e r h a n d ,

D-M.

on

0

on

f($)

such t h a t

D-M,

and f o r t h e g i v e n number

5(2)

A fortiori,

(17)

and t h u s

2 >

-

0

g*

is

i s n o t empty.

F o r t h e s e c o n d p a r t of t h i s t h e o r e m , r e c a l l t h a t since

2

E

E

impossible i n -

8

.

g = g E

D

-

E

D

-

(17).

Further, if

M;

(where

(17)

then

N(.)

g l o b a l l y maximizes

-

f(g) >

which s h o w s t h a t

aN(g)

on

f(8))

D.

5

0

and

= 0

is

for t h e t o u c h p o i n t

i be

a p o i n t which

I n t h i s c a s e f o r any

g E

D-

we may w r i t e

(18)* h o l d s .

- f(8)

5 =

requires that

i n ( 1 7 ) and n o t e t h a t s i n c e then

-

a

i s a touch p o i n t then

We n o w assume t h a t E

iE

D

-

g

a

E

( 0 , m )

i s a touch p o i n t

156

ROGER L.

-

g E D

for any

MP.

FOSDICK

Thus, appropriately defining

2

we

establish (18)

2'

N(.)

We saw earlier that if on

D, then the existence of proper

has n o global maximum f-N

touch points is

necessary for the existence of a Q -maximizer. i\

is a proper

touch point s o that (17) holds for a l l

f-N

and for some

a

E

Then, if

(0,m).

g ( * )E

s,

ED

Suppose

g E D

(recall (8)),

it follows from (17) that

\

s m(B)N(t.)

N(_g(X))dm

+

[!ho)f(ij)l/a

B G(A) <

and we see that

m

exists, even though it is not at

In fact, if

all certain that a Q*-maximizer exists. L!

.-- E

then not only is the constant function with

f-'(A/m(B))

value

in

but it is clearly a @, -maximizer, i.e.,

s,

D

3

f

/

N(g(X))dm

f o r all

g(.)

E 9,.

given

>

the set

*

C,

0

m(B)N(g)

= G(A)

B

While it is hardly likely that for a f-'(A/m(B))

will contain a proper

touch point, o u r observations here indicate that the mere existence

of a proper touch point may be the single most

important sufficient requirement for the existence of a

A

-

maximizer even in this more general situation. The here',

following theorem, which we shall not prove

gives a sufficient condition for both the existence of

a QA-maximizer as well as for the validity o f the procedure

of

f-N

switching.

'The proof o f a similar b u t stronger result w i l l appear shortly in a paper by Dsnn and myself i n the Archive for Rational Mechanics and Analysis.

ON AN I N E Q U A L I T Y

Theorem

4 -

Suppose t h e r e ...e-x.i s t s

mizer e x i s t s . -i

nto

and

D

Moreover, --______ E

> 0,

g i v e n any

> 0

a*

-

such t h a t

g(-)E

Cp

3(a*)$ G

such t h a t

F i g u r e l g i v e s a n example i l l u s t r a t i o n of

a*

f o r a s i n g l e c h o i c e of

-

t h e r e a r e many o t h e r s ) aiid f o r

For this particular

"cone".

t h e touch manifold consists of x(a*)

$

a

(taken equal t o

a*,

-)

P(a*)

i n the f i g u r e

assumed t o b e a sirnple

there are t w o points i n

3(a)).

(=

5(a*)

N(

Clearly, since, here,

the s i n g l e point a t the coordinate o r i g i n ,

E.

Since

i s compact,

P(a*)

B

mapping

6(c) > 0

t h e r e e_. xists

~~

157

I N THERMODYNAMIC STABILITY

iZ

wehave

i t i s b o t h c l o s e d and

t o t a l l y bounded i n t h e c o m p l e t e m e t r i c s p a c e

V

3

D.

Roughly,

t h i s c o n d i t i o n avoids c e r t a i n d i f f i c u l t i e s t h a t could g e n e r a l l y e x i s t a t t h e boundary p o i n t s o f

i t requires that

for f i n i t e dimensional V)

a*N(*),

o s c i l l a t e f o r e v e r about above i t .

I t i s worth n o t i n g t h a t

o u t s i d e of

a compact s e t ,

f-N

f o r the

switching

D,

f(-)

( a t least not

but rather ultimately r i s e need n o t be convex

f(.)

a s Coleman [ 5 ] procedure

and

required,

i n order

t o be p o s s i b l e .

B e f o r e l e a v i n g t h i s s e c t i o n we r e c o r d a s u f f i c i e n t condition o f practical v a l i d i t y of

t h e procedure

g e n e r a l l y the

5

-

{g E

D

Theorem

M;

P

s i g n i f i c a n c e which g u a r a n t e e s t h e

of

existence o f a

Suppose

N(g) <

E D

N(i)]

f-N

S

s w i t c h i n g , though n o t

a -maximizer.

i s a proper i s compact i n

f-N

D.

t o u c h p o i n t and Then,

the

ROGER L.

158

procedure Proof:

of

FOSDICK

s w i t c h i n g-.i s v a l i d .

f-N

-

E D

Supposing f i r s t t h a t

C

i s a p r o p e r touch

p o i n t , we may conclude from t h e second p a r t of Theorem

;E

where

f(g)/N(g) > 0

MA f

B u t , a s noted e a r l i e r ,

M;'

Thus, s i n c e

and where C

G(A) <

M;

f(i)].

i s compact i n D.

3(;)

i s n o t n u l l which, a s r e -

o f Theorem 3 , i s s u f f i c i e n t f o r t h e

marked a f t e r t h e proof e x i s t e n c e of

and s o

f(g) <

we may a p p l y t h e f i r s t p a r t o f

C,

contains

3 t.o conclude t h a t

Theorem

Mi,

= {g€D I

MA f

3 that

m.

Moreover,

f o r any

a

E

we s e e

(0,;)

that

which g u a r a n t e e s n o t o n l y t h a t t h e s e t s t a t e m e n t of Theorem

4 ) satisfies

P(a)

MT,

P(a) G

(defined i n the and i s t h e r e -

f o r e compact, b u t a l s o , f r o m t h e f i r s t p a r t of Theorem 3 , that one

i s not n u l l .

5(a) a

E

3(a)

$

Z

f-N

f(g) for

all

g

E D.

a E

for a l l

5 ( a ) E L!

t h e n f o r any one of t h e s e v a l u e s we s e e from I;

the

switching i s v a l i d .

On t h e o t h e r hand, i f

aN(g)

for a t least

we may a p p l y Theorem 4 t o s e e t h a t

(0,;)

procedure of

Thus, i f

Thus, f o r any

(17)

(0,;)

that

g ( * )E

s,

we

obtain

'B which shows t h a t procedure

of

f-N

Finally,

'B

G(A) I

A/a

and, t h e r e f o r e ,

t h a t again the

switching i s v a l i d . supposing t h a t

E

Z

i s a proper touch

159

O N AN I N E Q U A L I T Y I N THERMODYNAMIC STABILITY

p o i n t , we s e e f r o m ( 1 7 ) t h a t

a

and f o r some

E

a N(g)

f(g)

5

D

t h e l a s t paragraph

The argument of

(0,m).

g E

for all

1

may t h e n be r e p e a t e d t o c o m p l e t e t h i s p r o o f .

T h i s theorem w i l l b e u s e f u l i n t h e n e x t s e c t i o n where we e s t a b l i s h a c e r t a i n s w i t c h i n g specific function

3.

Application:

Let of

procedure

for a

i n t h e t h e o r y of n o n - l i n e a r elasticiiy.

N(.)

Hyperelastic Materials.

Lin

d e n o t e t h e s e t of

l i n e a r transformation

a t h r e e dimensional Euclidean v e c t o r s p a c e , i n t o i t -

E’,

self-the

elements o f

two v e c t o r s

2,

2

in

and

F

are tensors.

E Lin

The ____ i n n e r product o f

i s d e n o t e d by

E’

a 4 2,

t e n s o r p r o d u c t by any e l e m e n t

Lin

2-2

and t h e i r

which i s a n e l e m e n t of

-

we u s e t h e n o t a t i o n s

det F,

t o denote the determinant, t r a c e ,

For

Lin.

F, FT

tr

transpose,

and

inner t r a c e norm, t h e l a t t e r of which i s b a s e d upon t h e ____ product

E

--

t r UWT

f o r any

denote t h e u n i t tensor .

u,

W

in

Lin.

F i n a l l y , we s h a l l u s e

and O r t h t o d e n o t e , r e s p e c t i v e l y ,

We l e t

1

E Lin

I n v , Syrn, Sym+

the s e t o f i n v e r t i b l e ,

s y m m e t r i c , p o s i t i v e d e f i n i t e s y m m e t r i c , and orthoe;orial t e n s o r s .

It i s clearthat I n v , and

Sym+

Inv

i s open i n

i s open i n

Sym

L i n , Sym

i s a s u b s p a c e of

and convex.

A s t r a i n energy f u n c t i o n f o r a h y p e r e l a s t i c material

i s a smooth ( i . e . ,

1 c l a s s C ) mapping

a(.)

of

Inv

into

IR

w h i c h , b e c a u s e of t h e p r i n c i p l e o f m a t e r i a l f r a m e i n d i f f e r e n c e

[ll],

s a t i s f i e s the invariance condition

160

ROGER L.

F E

for all

Inv

FOSDICK

E Orth.

and f o r a l l

I n addition, we

(19) holds i f

know by t h e p o l a r d e c o m p o s i t i o n theorem t h a t and o n l y i f

dF) where,

-

--

F = RU ed on

8(*):

F

f o r any with

;E

E Inv,

f? E O r t h

=

a(;)

i s u n i y u e and i s s u c h t h a t

Sym'

and u n i q u e .

Sym+ + !R

a(*)

Thus,

by i t s v a l u e s o n l y on

Inv

(20)

Sym+ C Inv

u(.)

be t h e r e s t r i c t i o n of

i s determin-

and i f we l e t

to Sym'

we

have

-

a ( F ) = a(!)

= a(;).

(21)

I t i s straightforward t o s h o w t h a t the gradients

- -)

and

E Sym

;E

F E

Inv

1

=

-

c!

T

'F(F) -

+

F

i s a r b i t r a r y and

r a F ( E ) I T 133

-

(22)

9

f? E O r t h

= f?; with

and

Sym+. If

i n a d d i t i o n t o ( 2 0 ) i t a l s o happens t h a t e i t h e r

f o r some s m o o t h symmetric f u n c t i o n of positive reals

R> x

IR'

>

x IR

d e n o t e s t h e p r o p e r numbers o f

a(?) f o r some smooth f u n c t i o n that

E Lin

a r e r e l a t e d by

):(,' where

a,(.)

a(*)

6(.,.,.)

into

u

IR,

E Sym+,

of

where

the t r i p l e (u,}

or

-

(24)

= E(det U)

;(*):

R'

+ R,

t h e n we s h a l l s a y

i s t h e s t r a i n e n e r g y f u n c t i o n f o r a i s o t r o p i c hy-

p e_ r e_ la_ s t i~ c material or a e l a s t i c f l u i d , respectively.

.

T h e r e a r e c e r t a i n i n e q u a l i t i e s t h a t w i l l be mentioned i n t h e r e m a i n d e r

of

t h i s s e c t i o n i n connection with a

O N AY I V E Q U A L I T Y I N TTII~R"40DYNAMIC! S T A B I L I T Y

specific notion of For t h i s reason,

f-N

s h i t r h i n g i n h y p e r e l a s t i c i t y theory.

i t i s c o n v e n i e n t t o f i r s t i n t r o d u c e aird t o

discuss the various relations

t h a t e x i s t hetween t h e s e i n -

and t h e n t u r n t o t h e q u e s t i o n of

equalities,

W e say t h a t

obeys the

U ( * )

-

--

u ( F ) + (GF-F)-UF(E) <

E

for a l l

Inv

-

I

-

G E

and f o r a l l

Sym+

C-N

f-N

switching.

inequality

if

u(GF)

(25)

{i}.

-

In

contrast

,

f o r a much weaker i n e q u a l i t y o f t h e same k i n d , w e sa3 t h a t

G

E

Inv

i s a point of

C-N

u(*)

growth for

I

-

f or all

-

E

G

Sym+.

--

+

a(?)

(G6

-

Clearly,

-

f).OF(E)

U(

i f and o n l y i f e v e r y member o f

for

.)

S

--

obeys t h e

Inv

(26)

o(G$) C-N

i s a point

of

(26) i s a s t r i c t inequality T o r

a t which

U ( . )

O-

We s a y t h a t

ti(-)

i f merely ~

inequality

C-N

growth

2 # 3.

E ~ y m + i__s a p o i-n t o f c o n v e x i t y f o r ~

if

-

i(ir)+ ( -u --G ) G u ( ~ )s ; ( u )

-

E Syrn'.

for a l l

a x i a l convexity f o r that,

b(*)

if

g

Sym+

i___._s a p o i n t of co-

(27) holds f o r a l l

E Sym'

s h a r e a c o m m o n orthonormal b a s i s o € p r o p e r

i n addition,

-

$

A n clerneilt

(27)

A

vectors ~ t i t h U,

i.e.,

g ( g ) + ( c & - ~ ,-. q i js) 8(.,&) for a l l

2

E

Sym+

such t h a t

@

E Sym+. C l e a r l y , c o a x i a l

c o n v e x i t y i s weaker t h a n c o n v e x i t y . p o i n t of

C-N

growth f o r

a x i a l convexity f o r generally true.

; ( a ) ;

U(.)

(28)

Also,

if

E Sym+

is a

then i t i s a p o i n t o f c v -

however,

t h e converse i s not

To emphasize t h e d i f f e r e n c e between t h e s e

ROGER L. F O S D I C K

162

E Sym+,

two notions of convexity at

5 2

common, that any two tensors

-A

stretch if

E

is a point of Inv

-_

GB C-N

for some

growth for

B

E Sym'. a(*)

differ by a pure Then, if

fi

the graph of

E Sym+

a(*)

on

$

is restricted at all those tensors that differ from

by a pure stretch.

Sym'

iE

On the other hand, if

point of coaxial convexity for on

and

let us say, as is

;( * )

Sym+

is a

then the graph of

;(

)

has the same formal restriction but only at those A

tensors that in addition to differing from

';I

by a pure

1

by a pure

n

stretch also differ, like

U

itself, from

w

stretch

I

A

-

for

C-N

growth

U

is composed with any pure

stretch, while for coaxial convexity

.-.

is composed only with

those pure stretches that yield a pure stretch composition. 1 For isotropic elastic materials we have the following Theorem 6

2

-

a(.)

be the strain energy function -______-- for an

isotropic elastic material.

~~

Then the following three state-

are equivalent: -ments -. - _____ ~

-1

fi

One argument [121 for motivating that E S y m + be a point o f C-N growth for a ( . ) is based on the requirement that the actual wor$ done at a particle in a deformation which differs from by a pure stretch be greater than or equal to the work that would be done if the same deformation was executed under locally dead loading. It would seem at least as reasonable to require, additionally, that the deformation a pure stretch. The restriction of coaxial itself convexity would then follow. 2

The proof of this theorem is given by Dunn and myself in an article to appear shortly in the Archive for Rational Mechanics and Analysis,

ON AN INEQUALITY IN THERMODYNAMIC STABILITY

;(.).

(i)

E Sym+

is a point of convexity . for

(ii)

E Sym+

is a point of coaxial convexity ~- for

(iii)

(G1,G2,G3) of -

The __ proper values ___

E(*,.,.),

point of convexity for

q,G2,G3)

;(*I.

~

3

,.

+ c

p

E Sym+

is a

i.e., -

- (Gl,G2,G3)

(ui-u.) 1 aui

+l1,U2,U3)

i= 1 __ and

ul, u 2 ,

for all

u

R'.

in -

3

Now, since, f o r such materials, Coleman and No11

have

shown that if E Sym+

(iv)

C-N

is a point o f

growth for

a ( .)

,

then (iii) must hold, and since it is known [ll, p.323, and

141 that there exists a symmetric smooth function E Sym+

which satisfies (iii) for all all

p

in a non-empty subset of

Sym',

; ( - , a , - )

while (iv) fails for we see that within

the class of isotropic elastic materials any one of (i), (ii)

or (iii) is generally milder than (iv).

However, for elastic

fluids, where ( 2 4 ) holds, this difference breaks down and all four statements are, indeed, equivalent.

We turn now to the subject of touch points and switching within the context of hyperelasticity theory. Here, Gurtin 1101 has noted, essentially, that if

a(-),

the

strain energy function for an isotropic elastic material, has a strict global minimum at E Sym'

for all

-

then the procedure E

;(-)

ed o n

-

a(;)

Sym'

and into

{_1) ) , of N(.) R.'

-1

on

Syh+

and obeys the

f-N

(i.e.

a ( ? ) < a(!)

C-N

inequality,

switching is valid for

1-

- -11,

f(*) c

both of which are defin-

Our purpose in the remainder o f this

164

ROGER L.

FOSDICK

article is to show that the same conclusion holds for any

,1

hyperelastic material, not necessarily isotropic, provided a(-)

is a strict global minimum of

on

Sym'

and provided

there is a "properly distributed" set of points o f coaxial convexity for

6(')

"arouiid" 1.

In order to more clearly

state the precise result, it is convenient to have available

-

the set theoretic notions of a star effectively containing 1 ~ _ _ _ _ _ and a radiant shell of the star. We say that

--

containing 1 if

S E Sym

-

R

-1 is

not a limit point o f

-

the line

S

S,

is a star effectively

is compact and if there exists some sub-

S

set s -

E

+

called a radiant shell about 1, such that (i)

-

UU

+ ( 1 - a )-l

empty intersection with effectively containing

f o r all

RS.

-1

-

and (ii) -~ for any E Sym'

RS,

u E [ 0,1] has a non-

Clearly,

since

Sym'

admits of stars

is open in

Sym'

Sym.

For

example, any small enough closed ball in Sym centered at

,1

will suffice if its boundary is chosen as a radiant shell. This brings us to the following main Theorem

7

-

a(.)

Let

be the strain energy function for an

~

elastic material and suppose that minimum- at

1 c

ly containing -_

____.

RS,

on --

Sym'.

Let __

O(*)

S E Sym

f

has a strict global be a star effective-

,1 and suppose that one of its radiant shells,

is composed entirely of points of coaxial convexity for

;(-).

Then, f o r

Sym+,

proper

-

f(*) E ;(.)

f-N

U(;)

__ and

N(-)

T

1.

-

,I[ on

touch points exist.

Before proving this theorem, it is perhaps worth noting that it's conclusion procedure

of

f-N

guarantees the validity of the

switching.

That is, since at least one

165

OK AN I N E Q U A L I T Y I N TFTERMODYNANIC STABILITY

f-N

proper

touch point exists,

we see that the set

compact in

then f o r it

i n the hypotheses of Theorem 5 , i.e.,

M;

in the present context

E Sym+,

say

(u E

I ];-?I

Sym+

s

1$-11}

,

is

Whence, Theorem 5 is applicable. Of course,

Sym'.

as a prerequisite consideration it is clear that the and the

of the present

N(-)

f(.)

heorem satisfy all of the

conditions that we required of such functions throughout this work. Proof:

A s a first step we shal

point o f coaxial convexity for ;( .. -

for all

> 1

CI

a

Y

s a

Sym'

then ___

)

-1 + I J ( ~ - 1E ) Sym'.

such that

E

show that if

To see this,

-

n

let and

G1

C2

and

G2i

be two tensors in

are in

G1 and G2. containing El by a E G2

containing

by

G 1;

and note that (28) can be written

Sym+,

f o r each

such that

Sym'

Multiplying the resulting inequality

1-a

and the second inequality

(O,l),

and adding the two together, we find

that

f o r all

a

E

G1

and

(O,l),

and

G -2

as described above

such that

+ Now,

let

LI

> 1 be such that n

since

U

-CI

and

-

E

6-l

-Up -

and

-CI

E

-1 + p ( i - & ) E

Syrn'.

Then,

share a common orthonormal basis of proper

U

vectors it follows that G1

U

G2

s

U

-I-I

fi-' -

E Sym+.

Finally, if we set

it follows that

El, G 2 ,

6

G -1- = -Up '

166

ROGER L.

G26 =

and

a

,1

are a l l i n

FOSDICK

and

a € (0,l) p r o v i d e d we s e t

from

(30). Now, A

-

that

U

E

-

aU

RS

convexity f o r f(-)

This,

aGl + (l-a)G2 = 1,

u

Thus,

RS

that

l/p.

E

E Sym+

-

E RS.

T h a t s u c h a number

6 E

and l e t

S

b e i n g a r a d i a n t s h e l l of

,1 + 'E(g-,1),

=

C l e a r l y , we have

and s i n c e

of

u

let

+ (1-E):

a p r o p e r t y of

-1.

and t h a t

Sym',

-

( 2 9 ) follows

be s u c h

(0,l)

6

exists is

the s t a r

where

I

l/a' >

1,

i s composed e n t i r e l y o f p o i n t s o f c o a x i a l

i t f o l l o w s f r o m ( 2 9 ) and t h e d e f i n i t i o n

; ( a )

-

fi f ( 6 )

s f(;),

so that

-

E Sym'

of c o u r s e , h o l d s f o r e v e r y c h o i c e o f

and f o r e a c h s u c h c h o i c e t h e r e e x i s t s a c o r r e s p o n d i n g

N(*),

Thus, u s i n g t h e d e f i n i t i o n of

f!~)

2

2

inf

E

-

where t h e s t r i c t i n e q u a l i t y

be a l i m i t p o i n t of

.fLY)2 -

0,

2 >

0

h o l d s s i n c e a p r o p e r t y of

RS.

S

be compact and t h a t

To c o m p l e t e t h i s p r o o f ,

we f i n a l l y a p p e a l t o t h e f i r s t p a r t of Theorem g u a r a n t e e s t h e e x i s t e n c e of

proper

A s mentioned p r e v i o u s l y , theorem e n s u r e s t h e v a l i d i t y of

hyperelasticity theory, t h a t the p o i n t s of

0 E RS.

we have

s t a r s and r a d i a n t s h e l l s demands t h a t

switching, f o r a s p e c i f i c

S,

UERS N(U)

N@)

-1 n o t

about

S

f-N

this

touch points.

p r i o r t o i t s proof,

t h e procedure

f ( * ) and

3;

then,

N(.),

of

this

f-N

i n non-linear

It i s , I t h i n k , worth ernphasising

c o a x i a l convexity i n the hypotheses a r e

n o t n e c e s s a r i l y t h e same a s t h e p r o p e r t o u c h p o i n t s which a r e

167

ON AN INEQUALITY IN THERMODYNAMIC STABILITY

shown to exist.

However, since it, indeed, can be shown that

any proper touch point which is a point of convexity for N(.) is also a point of convexity f o r has the special form Sym+,

1. -

f ( * ) , and since here

N(.)

,I1 which is convex everywhere on

then in the present context we see that every proper

touch point must be a point of convexity for

b(-)

on

Sym'.

Thus, the existence of a "properly distributed" set of points of coaxial convexity for

'Taround'T ,1,

; ( a )

its point o f

strict minimum, implies, through "touching", the existence o f points of convexity for

;(-)

somewhere on

Syrn'.

The results of this and the previous section suggest that the condition o f having proper

f-N

touch points is a

regularity condition that may well be intimately connected to the development of mild a priori inequalities in the theory of hyperelasticity.

Acknowledgement.

Support o f the U . S .

tion is gratefully acknowledged.

National Science Founda-

168

\

--

A

h

la

v

u \ \

ROGER L.

\

I

FOSDICK

. .. WI

3

a

\

-1-

1

I-

I

Yr i

‘1 N

”

169

O N A N INEQUALITY I N THERMODYNAXIC STABILITY

R e f erenc es -

[ 11 E r i c k s e n , J . L . , theory.

A thermo-kinetic

J. S o l i d s S t r u c t u r e s

Int.

[Z] E r i c k s e n , J . L ,

2,

Appl.

(1966).

573-580

Thermoelastic s t a b i l i t y , Proc.

N a t i o n a l Congr.

[ 3 ] K o i t c r , W.T.,

view o f e l a s t i c s t a b i l i t y

5 t h U.S.

187-193 (1966).

Mech.,

Thermodynamics o f e l a s t i c s t a b i l i t y .

of A p p l .

3 r d C a n a d i a n Congr.

Mech.,

Calgary,

Proc.

29-37

(1971).

[ b 1 Coleman, B . D .

Bc J . M .

G r e e n b e r g , Thermodynamics and t h e

s t a b i l i t y o f f l u i d motion. Anal.

25, 3 2 1 - 3 4 1

[ 5 1 Coleman, B . D . ,

[61

R a t i o n a l Mech.

(1967).

O n the s t a b i l i t y of equilibrium s t a t e s o f

general fluids.

1-32

Arch.

Arch.

R a t i o n a l Mech.

36,

Anal.

(1970).

Coleman, B . D .

& E.H.

D i l l ,

O n thermodynamics

and t h e

s t a b i l i t y o f m o t i o n s o f m a t e r i a l s w i t h memory. Arch.

R a t i o n a l Mech.

,

[ 71 G u r t i n , M.E.

Anal.

51, 1-53 ( 1 9 7 3 ) .

Thermodynamics a n d t h e e n e r g y c r i t e r i o n f o r

stability.

Arch.

R a t i o n a l Mech.

Anal.

52,

93-10?

(1973). 181 G u r t i n , M . E . ,

Thermodynamics a n d s t a b i l i t y .

R a t i o n a l Mech.

[ 9 ] Dunn, J . E .

& R.L.

Anal.

Fosdick,

and boundedness o f

o f second grade. 191-252

(1974).

2,63-96

Arch.

(1975).

Thermodynamics, s t a b i l i t y ,

f l u i d s o f c o m p l e x i t y 2 and f l u i d s Arch.

R a t i o n a l Mech.

Anal.

56,

ROGER L. FOSDICK

[lo] Gurtin, M.E., Moden continuum thermodynamics. Mechanics Today.

New York: Pergamon 1973.

[ll] Truesdell, C. & W. Noll, The Non-Linear Field Theories Flbgge's Handbuch der Physik, III/'j,

o f Mechanics.

Berlin-Heidelberg-New York: Springer 1965. [12] Coleman, B.D., Mechanical and thermodynamical admissibility of stress-strain functions. Mech. Anal.

4, 172-186

Arch. Rational

(1962).

[l3] Coleman, D.B. & W. Noll, On the thermostatics tinuous media,

Arch. Rational Mech. Anal.

of con-

k,

97-128 (1959)*

[14] Bragg, L.E.

& B.D.

Coleman, On strain energy functions

for isotropic elastic materials.

4, 424-426

(1963).

J. Math. Physics

G.M. de La Penha. L.A. Medeiros (eds.) Contemporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations @North-Holland P u b l i s h i n g Company (1978)

LEA.I;T SQUARE SOLUTION O F NON L I N E A R PROBLEMS

I N F L U I D DYNAMICS

R.

GLOWINSKI* and

Universitb de P a r i s Iria-Laboria,

0.

PIRONNEAU

6 and

Iria-Laboria, 78150 le Chesnay

France

France

1. I n t r o d u c t i o n .

Many problems i n Mathematical P h y s i c s a r e of

the

following forms, e i t h e r A(u)u = f,

+ B(u)

Au

In

(l.l), A ( v )

i s in (1.2)

v

i s for

=

f.

g i v e n a l_i _ n e_a_ r . o p e r a t o r and

B

a non l i n___ e a r operator.

This suggests obviously the following algorithms: F o r s o l v i n g (1.1)

(1.3)

uo

then f o r

n

2

0

we o b t a i n

u n+l

given from

u

n

by

I

A(un)un+' = f ,

For s o l v i n g ( 1 . 2 )

we may use an a l g o r i t h m s i m i l a r t o

(1.5)

(1.4)1 i s r e p l a c e d by

*

except t h a t

(1.3)-

~.

V i s i t i n g P r o f e s s o r IJF'RJ,

R i o d e J a n e i r o , July-August

1977.

R. GLOWINSKI and 0. P I R O N N E A U

172

There are some applications in which (l.3), (resp. ( 1 . 3 ) , ( 1 . 4 ) 2 , (resp. (1.2)).

-

(1.5))

( 1 . 4 ) 1’ (1.5)

converges to a solution o f (1.1)

Unfortunately these are important applications

mathematically modelled by equations like (1.1) or (1.2)

-

f o r which the above algorithms can not be used, therefore one

has to use more sophisticated methods.

In some cases (cf., e.g., Glowinski-Marocco Ell) these problems may be reduced to a saddle-point problem by introduc-. ~

ing an artificia-1 constraint and the corresponding Lagran* multiplier. - __ -..~

In some other situation such an approach is not suitable (by lack o f monotonicity usually) and it may be convenient to use aleast square method.

A typical least square formulation o f problem (1.1) is Min J(v)

(1.6)

V

where

(1.7) In

(1.7),

]I*]]

denotes a suitable norm (for more details

about this last point, see Remark 1.2 below) and

J(v)

the

solution o f the linear problem

We obtain similarly a least square formulation of (1.2) by replacing (1.8)1 by

(1*8)2

Ay = f-B(v).

LEAST SQUARE S O L U T I O N O F NON L I N E A R PROBLEMS

Remark _ _ 1~ .1:

I t a p p e a r s c l e a r l y from t h e above r e l a t i o n s t h a t

(1.6), (l.7), s t r u c t u r e of

(l.7),

( 1 . 8 ) l and ( 1 . 6 ) ,

( l . 8 ) 2 have t h e

an o p t i m a l c o n t r o l problem i n which t h e c ontrol ._ ~~

variable

(i.e.

t h e independent v a r i a b l e i n t h e problem) i s

the s t a t e variable i s

y

and t h e c o-s t-f-u n-c.t_ i -o. n is -

v,

J.

T h i s p a r t i c u l a r s t r u c t u r e l e a d s v e r y n a t u r a l l y t o compute t h e derivative o f

111 , [ 2 ] )

J

v i a an a d j o i n t s t a t e e q u a t i o n ( s e e Lions

f o r t h e o p t i m a l c o n t r o l o f s y s t e m s m o d e l l e d by

p a r t i a l d i f f e r e n t i a l equations). T h i s k i n d o f methodology h a s b e e n a d v o c a t e d by Cea-Geymonat

[l] f o r s o l v i n g non monotone, non l i n e a r t w o - p o i n t b o u n d a r y v a l u e problems l i k e

1

(1.9)

+

-u”

cp(u) = f

10,1[

over

u ( 0 ) = u ( 1 ) = 0.

I n B e g i s - G l o w i n s k i [1] a s i m i l a r a p p r o a c h h a s b e e n u s e d f o r s o l v i n g f r e e boundary problems

l i k e those r e l o t e d t o flows o f

l i q u i d s i n porous media. I n a n u n p u b l i s h e d work Derviauc-Glowinski h a v e a l s o u s e d t h i s t y p e o f methods t o compute t h e maximal s o l u t i o n of

I

(1.10)

-u

= Xe

U

over

10,1[

”

u(0) = u ( 1 ) = 0 where

>

0.

In t h e above r e f e r e n c e s t h e o p t i m i z a t i o n i s a c h i e v e d u s i n g e s s e n t i a l l y -~ descent o r steepest descent algorithms. ~

~~

~~~

More

~~~~

r e c e n t l y G l o w i n s k i and P i r o n n e a u o b s e r v e d t h a t t h e u s e of a conjugate gradient algorithm,

t o achieve the minimization,

c a n d r a s t i c a l l y improve t h e s p e e d o f c o n v e r g e n c e ,

specially

17 4

R. GLOWINSKI and 0. PIRONNEAU

in the solution of some hard problems like those to be considered in the following sections.

J

Remark 1.2: The choice of a convenient cost function

is a

very important matter if one wishes to use the above rnethodology.

Let us mention some possibilities we have found very

attractive since they simplify the adjoint state equation and lead to good convergence properties.

In the case of the problem (1.1) assume that for A(v)

E e(V,V’)

dual space. if

A(v)

where

(

where

is an Hilbert space and

V

Assume also thak

A(v)

v

given, V’

is self-adjoint. .

is _ strongly - a natural choice for _ _ _ V-elliptic

,*)

denotes the cu-ality -~_ pairing ~ _ . _o _ f _V’

Similarly i f in (1.2),

A

and

its Then

J

is

V.

is strongly V-elliptic it is

convenient to use

In the following sections we shall see how the above principles can be applied to the numerical solution of two important problems i n Fluid Mechanics, namely: *

The transonic flow of an isentropic, compressible perfect ~

fluid. ’ Navier-Stokes equations for incompressible fluids.

Numerical results will be presented in order to illustrate the possibilities of the above methods.

To conclude this report some indication will be given on the minimization of non quadratic functionals by conjugate gradient algorithms.

LEAST S Q U A R E S O L U T I O N OF NON L I N E A R PRODLEMS

175

2. Numerical solution-of transonic flow problems. . __ ~

2.1

Generalities on transonic flows.

The theoretical and numerical studies of transonic flows for -_ perfect fluids have always been very important,

But these

problems have become even more important these last years in relation with the design and development of large subsonic aircrafts. These transonic problem are very difficult, theoretically and mumerically, for the following reasons: They are n o n linear. Shocks may exist in the flow. An Entropy Condition is needed, in a way o r another, to avoid non physical solutions. They are mixed problems i n the sense that they are el_l.ipt-icin the Eubsgsic_ part of the flow, LyEerbolic i n the supersonic part of the flow. From a theoretical point of view let us mention the work of Morawetz [l].

At the present moment the numerical methods

which are the more commonly used have originated from Murman-Cole [l]

and we shall mention B a u e r - G a r a b e d i a n - K o r n r l ] ,

Bauer-Garabedian-Korn,Jameson

11

,

Jameson [ 11 ,[ 21 ,[ 31 ,[ 41

and the bibliographies there i n (see also Hewitt-Illingworth and Co-editors [ 13 ) . These above numerical methods use the key idea of Murman and Cole which consists to use a finite difference scheme, centered i n the subsonic-part of the flow, kackward (in the direction of the flow) in the supersonic part.

The switching

1 76

R.

GLOWINSRI a n d 0 .

between t h e s e two-schemes

PIRONNEAU

i s a u t o m a t i c a l l y done v i a a

t r u n c a t i o n o p e r a t o r only a c t i v e i n the supersanic p a r t of flow ( s e e Jameson,

loc.

cit.,

f o r more d e t a i l s ) .

We s h a l l d e s c r i b e h e r e a d i f f e r e n t a p p r o a c h ,

which seems v e r y

c o n v e n i e n t f o r comput i n g f l o w s p a s t p r o f i l e s , infinity,

or f l o w s i n n o z z l e s .

the

subsonic a t

I n o u r method t h e t r a n s o n i c

f l o w problem i s f o r m u l a t e d a s a non l i n e a r l e a s t s q u a r e problem,

w h i c h i s i n t u r n i n t e r p r e t e d a s a n o p t i m a l~control

p r o b l -e m .

Then f i n i t e e l e m e n t s a r e u s e d t o a p p r o x i m a t e t h e S i n c e t h e .____ e n t r o p y~-c o n d i t i o n i s f o r m u l a t e d by

above problems.

~

a l i_ n e_ ar _ ine q u a l i t y c~-o n s t . ra i n t , a c o n v e n i e n t method t o h a n d l e .

i t i s t o u s e p e n a l t y a n d / o r d u a l i t y methods ( u s i n g a n augmented l a g r a n g i a n i f p e n a l t y and d u a l i t y a r e co mb in ed ) . Duality techniques a r e not

c o n s i d e r e d i n t h i s r e p o r t and w e

r e f e r t o Glowinski-Pironneau [ d e s c r i p t i o n of

2.2 2.2.1

11,

Glowinski

[13 for a

them.

M a t h e m a t i c a l m o d e l s f o r t r a n s__ o n __ i c f.l_ o ws. . .. B..-a s i c a s s u m p t i o n s .

We a s s u m e t h a t t h e f l u i d u n d e r c o n s i d e r a t i o n i s p e r f e c t a n d compressible,

and t h a t t h e flow of

a n d i r r o t a t i o-n a l ( i . e . ,

potential).

-

such a f l u i d is isentropic These assumptions a r e n o t

t r u e i n g e n e r a l s i n c e t h r o u g h a s h o c k wave t h e r e i s a v a r i a t i o n of e n t r o p y , and a n i r r o t a t i o n a l f l o w becomes rotational

(i.e.

t h e v a l i d i t y of i n t h e c a s e of

t h e r e i s c r e a t i o n o f v o_ r t_ i c i_ ty).

Therefore

t h e model t o f o l l o w i s assumed t o b e c o r r e c t "weak s h o cks". -.

177

LEAST SQUARE SOLUTION O F N O N LINEAR PROBLEMS

I n t h e c a s e o f a f l o w s p a s t a s- h a_ r p~ p_r o_f ._ i l e w e s h a l l suppose t h a t t h e r e i s no wake b e h i n d

2.2.2

E--q u a -~ t i o n s of

the tr a- _ i l ~i n g edge.

t h e flow.

~

n

Let

be t h e domain o f

r

t h e f l o w and

i t s boundary;

then

we s h a l l assume t h a t t h e f l o w i s m o d e l l e d by 1

,

-

(2.1)

, 2

y-1

v-((1 -

where i n ( 2 . 1 ) .

- 0 -

09

i s the flow p otential, _____

the flow v e l o c i t y ,

i s t h e c.r i t i c a l v e l o c i t y ,

C,

~~

-

y

i s t h e r. __ a t i o of

W e have *

(y = 1 - 4 f o r

specific heats

4.

_ _ I

t o add t o ( 2 . 1 )

Boundary c o n d i t i o n s ( o f D i r i c h l e t a n d / o r

Neumann t y p e ,

for

instance), *

A K u t t a - J o u k o v-k y

condition i n the case of

a l i f t i n g body ( s e e L a n d a u - L i f c h i t z

-

the flow past

[ l , Sec.

46]),

An e n t r o p y c o n d i t i o_ n . i n o r d e r t o e l i m i n a t e t h e non p h y s i c a l .s o l u t i o n s of

(1.1); t h i s p o i n t

w i l l be d i s c u s s e d i n S e c .

2.2.3. Remark 2 . 1 :

I t c a n h a p p e n t h a t on some p a r t of

0

have

:

and

ness; -

t o be g i v e n s _ i m_ u l t a n e o u s- __ ly t o i n s u r e unique-

i t is, f o r i n s t a n c e ,

of F i g .

2.1 i f

~~

t h e c a s e of

'4

g i v e n on

t h e divergent nozzle

the velocity a t the entrance i s supersonic.

T y p i c a l boundary c o n d i t i o n s a r e 5-n

t h e boundary,

r4, rl, r 2 .

If

@

g i v e n on

r l , Fg

and

the velocity a t the entrance

178

(i.e.

R.

I-,)

GLOWINSKI and 0. P I R O N N E A U

is subsonic we require ___ less boundary conditions.

Figure 2.1

Figure 2.2

Remark 2.2:

In the case o f t h e f l o w past a multibody profile

(like i n Fig. 2.2)

each

body requires a Kutta-Jourkowsky

condition.

2.2.3

Formulation of the entropy condition.

It follows f r o m Landau-Lifchitz [ l , ch.91 condition c a n be formulated as follows:

that the entropy

1 79

LEAST SQUARE SOLUTION OF NON LINEAR PROBLEMS

1 In the direction

o f the flow, One- cannot have a - ___ ~~

-!

I

(2.2)

subsonic-supersonic transition through

a_;t?ck.

F o r a one-dimensional . . flow, (2.2) implies ~

*<

2 d

(2.3)

dx

3-

2

+m,

is a measure bounded from_ above; _ ~ _ _weak (and more dx rigorous) formulation of (2.3) are: i.e.

-~

There exists a constant ~~

~

M,

such that-either -

where

(We recall that fJ

(R) =

{cp E Cm ( f i )

,

cp

has a compact support in

n] . )

In the case o f a t ~ oo r three dimensional flow we shall suppose that (2.2) can be formulated by

(2.9)

190

R.

GLOWINSKI a n d 0 . P I R O N N E A U

T h e n u m e r i c a l r e s u l t s o b t a i n e d f o r t__ wo and _ t r e_e -_ d i_ m e~n s_ i o_ n a_ l flows u s i n g d i s c r e t e analogous of above formulation Remark 2 2 :

(2.7) seanto justify the

of t h e entropy condition.

The c o n d i t i o n ( 2 . 7 )

i s a c t u a l l y a p a r t i c u l a r case

O f

(w)+ E LPP)

(2.10) where,

assuming t h a t

is a m a s_ u_ re, _e_

( ~ c p ) + = positive p a r t of

(2.11) Then

Acp

(2.7)

corresponds t o

f r o m a computational point

p =

+m;

~cp.

another useful choice i s ,

of view,

p=2

i.c.

(2.12)

To c o n c l u d e t h i s S e c .

2.2

a t t h e moment,

l e t u s mention t h a t ,

t h e t r a n s o n i c f l o w problem i s s t i l l open from a t h e o r e t i c a l ~

~~

of view, even i n t h e simpler c a s e of t h e so-called

point

s m a l l d i s t u r b a n c_e _ c c l u a t i o n ~~

details).

( s e e J a m e s o n [ 11 ,[

41

f o r more

T h i s l a c k o f e x i s t e n c e and u n i q u e n e s s r e s u l t s h a s

v e r y l i k e l y i t s c o u n t e r p a r t i n t h e n u m e r i c a l methodology.

2.3

L - east

s q u a r e f o r m u l a t i on. ~

~~

R edu5;ion _ _t o~ a n o p t i m a l -~ ~~

~

~

~~

c o n t r o l problem.

If

w e s u p p o s e t h a t t h e d e n s i t y o f t h e f l u i d i s o n e when t h e

velocity is zero,

then the coefficient

a s the density of

the

fluid.

of

i n (2.1)

We s h a l l u s e t h e n o t a t i o n 1

2 Y - l (2.13)

V@

lvcpl

)

.

appm

181

LEAST SQUARE SOLUTION OF NON L I N E A R PROBLEXS

The i d e a o f t h e method t o f o l l o h i s t o d ecoup_ le t h e _ density ~ and t h e p o t e n t i a l those described

4

potential

0

and

-

via a _ l e_a s -~ t s_ q u_ a r_ e~ f o_ rm u_ l a.t_____ i o_ n like _ _

@

To d o s o we i n t r o d u c e a new

1.

i n Sec.

-

the c o. n t r o l p o t e n t i a l ~

by m i n i m i z i n g a c o n v e n i e n t

least

square c o s t function.

W e may u s e f o r i n s t a n c e t h e f o l l o w i n g f o r m u l a t i o n formulations s e e Glowinski-Pironneau pironneau

5

and t r y t o r e c o u p l e

(for other

[ 2 1 , Gowinski-Periaux-

[ 11 )

(2.14) where

5

i s a f u n c t i o n of

in

via the state

(2.15) p l u s boundary c o n d i t i o n s f o r

I-.

on

d,

i

In

(2.14

,

a convex s e t Since

a

t h e parameter

'' c o n v e n i e n t l y ''

p ep(x)) = 0

is either cho s e n

or

0

1

.

1 Y+1 2

I Vcp(x) 1

i f and o n l y i f

X

and

= (Fl)

is

and

C,

1

y =

that for a i r , appears t h a t

i n the

1.4

implies

y+1 2

(--)

Y-1

t r a n s o n i c range

=

--

,,/6 = 2 . 4 5 , =

(say

it

lVCpl

1.5

c*)

w e have

(2.16)

o

< 6 s p(cp(x)) s 1

I t f o l l o w s from ( 2 . 1 6 )

that

a.e.

on

0.

( 2 . 1 5 ) i s an e l l i p t i c p r o b l e m f o r

a p p r o p r i a t e boundary c o n d i t i o n s . l i f t i n g b o d i e s , Kutta-Joukowsky

I n t h e case of flows p a s t conditions are a l s o required

t o h a v e , w i t h t h e o t h e r b o u n d a r y c o n d i t i o n s , a -__ physical s o l u t i o n o f problem ( 2 . 1 5 )

(modulo a c o n s t a n t i f

Neumann c o n d i t i o n s on t h e b o u n d a r y ) .

one o n l y h a s

R.

182

2.4:

Remark

If

GLOWINSKI and 0 . P I R O N N E A U

i n t h e o r i g i n a l problem,

0

r,

be s i m u l t a n e o u s l y p r e s c r i b e d on some p a r t of approach,

5

w i t h t h e two p o t e n t i a l

and

'@ have t o an

and

a,

t h e above

i s very

convenient s i n c e t h e boundary c o n d i t i o n s can be s p l i t t e d b e t ween

@

4.

and

However i f

boundary c o n d i t i o n s f o r t a k e a c c o u n t of

0

one w i s h e s t o u s e t h e same and

4

i t i s always p o s s i b l e t o

t h e e x t r a boundary c o n d i t i o n s

(assumed t o be

(2.14) a

o f D i r i c h l e t t y p e ) by a d d i n g t o t h e c o s t f u n c t i o n I

quantity proportional t o e i t h e r

( I

15-@,1

(9-0,1

2

br,

or

'd (or t o a l i n e a r c o m b i n a t i o n o f b o t h ) , where

2

' rd

i s the part

of

on which one r e q u i r e s

A s i m i l a r i d e a i s used i n Begis-Glowinski

@Ird

=

rd

Od.

[l] t o s o l v e some

f r e e boundary p r o b l e m s . Remark 2.5: variant

To s t a t e t h e e n t r o p y c__ o n d-i t_i_ o n ( 2 . 7 )

4

( 2 . 1 2 ) ) we h a v e t h e c h o i c e between

(or its

and

F

(actud-

l y we can a l s o u s e t h e s e two p o t e n t i a l s ~ i m y l t a n e o u ~ l y ) I. f one u s e s

E

( r e s p . @ ) we have a c o n t r o l c o-n s t r a_i_n t

(resp.

a s t a t e c-onstr-int). Remark 2 . 6 :

W e observe t h a t for t h e c l a s s o f flows under

c o n s i d e r a t i o n we h a v e

T h i s o b s e r v a t i o n s u g g e s t s t h a t t h e convex

X

( 2 . 1 4 ) h a s t o be t a k e n a s a convex s u b s e t of

occuring i n

Wl'"(S2).

We

o b s e r v e a l s o t h a t t o s t a y i n t h e t r a n s o n i c r a n g e i t may b e convenient t o intr oduce t h e following c o n s t r a i n t s

or

183

LEAST SQUARE S O L U T I O N OF NON LINEAR PROBLEMS

(2.18) Actually the computations we have done prove that for a physically well-posed transonic problem it is not necessary ~~

to introduce

Remark

~

(2.17), (2.18).

2.7: If the transonic problem has a solution and if

X

is "large enough" then the least square problem has a solution such that the cost function is equal to zero.

This last

property gives us means to check the quality of the computed solution.

2.4

Finite element approximations. __

We assume that

0c

_. Three-dimensional calculations will

2

!R

be reparted i n a forthcoming paper.

2.4.1

Generalities.

The above control problem will be approximated by finite elements.

Compared to finite differences, finite elements

give us the possibility o f handling problems posed on rather complicated geometrical domains.

Moreover the variational . .~

framework of finite element formulations is very appropriate to the problem under consideration and to the methodology we use to solve it.

It will be in particular fairly easy to

approximate the weak formulations of the entropy condition (2.7) (and also the alternate entropy condition (2.12)).

If

R

is unbounded, it is replaced by a bounded domain

still denoted by

-

as large as possible.

-

To approximate

the above continuous problem, we introduce a standard tri-

184

R. GLOWINSKI and 0 . PIRONNEAU

Zh

angulat ion elements).

of

61

(we can also use quadrilateral finite -

7

Then we approximate the functions

and

$I

by

piecewise polynomial functions belonging to the following subspace of

with

H1(n)

(and

Wl’”(n))

Pk = the space o f polynomials of degree

Sk.

Remark 2.8: In the case of a lifting body, to take into account the Kutta-Joukowsky condition, one usually introduces -~ .. (see Fig. 2.3) an arc

y

between the trailing edge o f the

body and the external boundary.

Thus arc

constant jump, a priori unknown of

a@

requires the continuity of

Y.

Since

@

-

an

@

(and

is discontinuous along

and

y

supports a

<,

ae z) at y,

but one the crossing o f

Hl(61)

c a n not be

used any more, but introducing

.

.

0

\

n = n - y one can use

A.

HL(n)

and define

Vh

over a triangulation of

Results of computations for such profiles are given in

Sec. 2.6;

however the method f o r t r e a t i n g t h e Kutta-Joukowsky

condition will not be treated in this report since they are not specific of transonic flows. In order to simplify the presentation o f our method we shall assume i n the sequel that one works directly over

61.

LEAST SQUARF S O L U T I O \

@rU O U

Figure

Remark 2 . 9 :

If

@

and

?

2.3

have t o s a t i s f y o n l y N e u m-a r i r i

b-o u n d a r y_ _c o n d i t -~ i o n s , t h e s u b s p a c e of ~

Vh

itself.

If

on t h e c o n t r a r )

i$

Vh

t o h e u s e d w i l l be

and/or

haxe to satisfi

F

D i r i c h l e t boundary c o n d i t i o n s over s o m e p a r t of s h a l l have t o use subspaccs o f Remark 2 . 1 0 : domain o f

R

Vh,

,

",

t h e n we

s t r i c t l y included _-___

We have t a c i t l y assumed t h a t 2

185

LIhFAR l'ROI3LEVS

R

in V

h'

i s a polygonal

or has b e e n a p p r o x i m a t e d b y s u c h a d o m a i n .

Hoizever i n t h e c a s e o f a curx-ed b o u n d a r y i t i s a l w a y s r > o s s i b l e t o use

( a t s o m e e x t r a computational c o s t ) curved f i n i t e

elements

t-11).

(see,

for i n s t a n c e , S t r a r i g - F i x [I], C i a r l e t - R a v i a r t

R. GLOWINSKI and 0. PIRONNEAU

185

2.4.2

o f the state equation . and . . o f_ _the cost Approximation __. ~~

function. T o simplify the presention, we shall assume that we only have

Neumann .boundary cond>t-ions, i.e.

aa@ n

(2.20)

= g

on

7

(it is the case for the important application of flows subsonic at infinity, past profiles). F

sections that

and

0

It follows from the above

will satisfy the same boundary ~

conditions.

We shall also assume that if

then the corresponding value of (2.21)

p g

p

dr =

g(x)

f 0, x E

r,

is known, and that

0.

'iThen the state equation (2.15) h a s the following variational

(2.22)

which is approximated by

r

i

@

and

f

(pg)h q h

dr

qhevh,

'I-

[

Oh vh' ( ~ g )is~ a Convenient approximation of @h

pg.

Since

are only defined modulo a constant we shall

prescribe the values of some point of ed by

dx =

4

(2.23)

where

P(ch)v@h'v"h

r.

@

and

cph

(and

5

and

5,)

at

The cost function in (2.14) is approximat-

187

LEAST SQUARE S Q L U T I O N O F PJON L I N E A R PROULEFIS

If one

R e m a r k 2.11:

(k=l),

uses

the piecewise linear approximation

then the integrals occuring i n ( 2 . 2 3 ) ,

to compute since property for

05,

p(ch)

(2.24)

piecewise constant implies the same (from (2.1'3)).

If

k=2

a ~ numerical _

integration procedure has to be used in ( 2 . 2 3 ) ,

a

are easy

_

and also in

= 1.

(2.24)

if

2.4.3

Approximation of .the entropy condition. -~ _. ~~

~

~~~

~

T o avoid non physical shocks (i.e. shocks for which the

thermodynamical entropy condition is n o t satisfied) we use the formulations o f Sec. 2 . 2 . 3 .

We still assume that w e o n l y

have Neumanii b o u n d a r y conditions.

We use the notation of the continuous oroblem. the condition ( 2 . 1 2 ) (2.14)

If

one uses

we may "regularize" the cost function

by adding to it, with

C

> 0, either

(2.25)

or (2.26)

E

[ '*

l ( h E ; ) + 1 2 dx

( o r a linear combination of both).

In

(2.25),

(2.26),

is

a "small" parameter.

We can make this approach more sophisticated by using instead of ( 2 . 2 5 ) ,

(2.26),

regularization functionals like

188

R.

GLOWINSKI and 0 . PIRONNEAU

0.

p o s s i b l y e q u a l t o z e r o o v e r some p a r t of

To u s e t h e above methodology f o r t h e a p p r o x i m a t e problem i t i s n e c e s s a r y t o have a n a p p r o x i m a t i o n o f W e may u s e a n a p p r o x i m a t i o n a p p r o x i m a t i o n of

C i a r l e t - R a v i a r t [ 21

,

[

(2.28)

73 ) :

i s s u f f i c i e n t l y smooth, t h e n from

we h a v e

Ai/lcpdx =

'n

(2

i

epdP

-

(2.29)

I

{R

an a p p r o x i m a t i o n of

E H1(n).

-

i

'R

Ah

$I,

0Bh'VCPh dx

A@

of

as

cph E Vh,

bL

I n (2.29),

t o define

the function

g

of

gh

is

(2.20).

we a l s o h a v e D i r i c h l e t boundary c o n d i t i o n s

If

o v e r some p a r t

cp

E Vh*

We u s e t h e same method

Remark 2 . 1 2 :

ghCPhc

=

'h'hcphdx

AhOh

+

V w - v c p dx

F

U s i n g t h i s i d e a we d e f i n e a n a p p r o x i m a t i o n €allows

[21,

e q u a t i o n ( s e e Glowinski

-

Glowinski-Pironneau [

$

L e t u s assume t h a t Grgeq-for5qJ-a-

s u g g e s t e d by F i x e d -f i n i t e e_l e_m_e n t

t h e b i h a r m o n~ic

~

AS).

(resp.

of

r,

t h e above i d e a i s s l i g h t l y more

c o m p l i c a t e d t o a p p l y , b u t t h i s can be d o n e . Remark 2.13: To o b t a i n

from ( 2 . 2 9 ) we have t o s o l v e a

Ah@,

l-~ i n e a r s ys t e m t h e m a t r i x o f which i s _s_yrnmetric, p o s i t i v e ~~

definite,

sparse, but p o t d i a g o n a l

approximation of t h e o p e r a t o r I

approximate

n

4 0 cp

h h h

dx

I).

( t h i s m a t r i x i s an If

k=l

one c a n

u s i n g t h e two-dimensional,

189

LEAST SQUARE SOLUTION O F N O N L I N E A R I'RODLEMS

t____ r a p e-z o i d a l n u m e r i c a l i n t e g r a t i o - n m e t h o d .

D o i n g s o we o b t a i n

~

s o l v i n g a l i n e a r s y s t e m w i t h a _d_i a g o n a l m a t r i x .

by

ahQh

If

t h e a b o v e r e g u l a r i z a t i o n method i s t e c h n i c a l l y more

k=2

complicated t o use.

A

Once <Jh

h a s b e e n a p p r o x i m a t e d we add t o t h e c o s t f u n c t i o n

(see ( 2 . 2 4 ) ) the functional

I

2

I (Ahah)+]

('. 3')

dx

(rcsp'

Eh(x)

I ('h
2

dx)

'R

ch

with (2.30)

" s m a l l " and

0.

2

W e use i n f a c t approximations o f

o b t a i n e d v i a a n u m e r i c a l i n t e g r a t i o n p r o c e.d u r e .

W e have t o m e n t i o n t h a t t h e o p t i m a l c h o i c e f o r

is still

Fh

a n open q u e s t i o n , n e v e r t h e l e s s t h e r e e x i s t s o m e e m p i r i c a l methods t h a t w e d o n ' t d i s c u s s h e r e .

Numerical r e s u l t s

ed w i t h t h e p i e c e w i s e l i n e a r a p p r o x i m a t i o n i n Sec. Eemark

(k=l)

obtairr

a r e given

2.6.

2.14:

T h e r e g u l a r i z a t i o n by

€

(or i t s

l ( h @ ) + 1 2 dx

v a r i a n t s ) m a y be s e e n a s a t e c h n i q u e f o r i n t r o d u c i n g s o m e pseudo-viscosity

B

-

i n t h e problem under c o n s i d e r a t i o n .

A method u s i n g ( 2 . 7 ) .

L e t u s d e s c r i b e t h i s method f o r t h e c o n t i n u o u s p r o b l e m ,

first:

~~

We assume t h a t t h e e n t r o p y c o n d i t i o n i s f o r m u l a t e d by ( 2 . 7 ) . Let

M(x)

b e a s u f f i c i e n t l y s m o o t h u p p e r bound o f

e s t i m a t e d or g u e s s e d ) .

a weak f o r m u l a t i o n o f

R e p l a c i n g ( 2 . 7 ) by

(2.31)

is

hb,

(M i s

GLOWINSKI and 0. P I R O N N E A U

R.

190

-

(2.32)

I

1

[email protected] dx

S

I

n

M ( x ) c p dx

V

cp E 8 + ( n ) .

R i t i s v e r y convenient t o i n t r o d u c e t h e

M

I n s t e a d of u s i n g

s o l u t i o n ( d e f i n e d modulo a c o n s t a n t ) o f

Ago = M

over

0,

= g

over

r,

(2.33)

then

i""'o.vv

(2.33) has t h e f ollow ing v a r i a t i o n a l formulation dx =

-

(

M(x)cp dx +

f

v cp t H 1 ( n ) ,

mdr

r

R

(2.34)

I t f o l l o w s from ( 2 . 3 4 )

that

-i

(2.35)

W e observe t h a t

(2.32)

v ( O - @ O ) ' ~ c p dx

5

a (@-a0) = an

0.

c a n a l s o be w r i t t e n

v cp E

0

rs+(n).

F o r t h e d i s c r e t e problem t h e o b v i o u s s t r a t e g y seems t o be t h e following: F i r s t we a p p r o x i m a t e

vaoh-vvh

dx =

Q0

-

by

\ R

where

Mh

Go,,

solution of

Mh(x)vhdx +

f ghcphdr r

i s a c o n v e n i e n t a p p r o x i m a t i o n of

M.

+ e p , c vh,

191

LEAST SQUARE SOLUTION O F N O N L I N E A R PROBLEMS

( 2 . 3 5 ) by

F i n a l l y we a p p r o x i m a t e

1

-

(2.38)

'

v(@h-@o,)'vqh dx

qh

n

I n f a c t t h e above " o b v i o u s ' l

s t r a t e g y f o r t h e d i s c r e t e problem

h a s t o be m o d i f i e d f o r t h e f o l l o w i n g r e a s o n s : 1)

V:h

k=l,

If

can be g e n e r a t e d by t h e c a n o n i c a l b a s i s

2)

B u t i t i s n o t the c a s e f o r

Voh.

f u n c t i o n s of

k=l

C o m p u t a t i o n s u s i n g ( 2 . 3 8 ) and done w i t h t h a t t h e approximation o f

close t o

k=2. have shown

t h e s o l u t i o n i s n o t good

t h e p o i n t s a t which s o n i c l i n e s t o u c h

r.

To overcome t h e s e d i f f i c u l t i e s we may p r o c e e d a s f o l l o w s : If

k=l:

ed from 1

C,

Let

be t h e s e t o f

Nh,

t o

where

Vh,

c a n o n i c a l b a s i s of

Nh

t h e v e r t i c e s of

= dim(Vh).

Sh,number-

Wh

Let

be t h e

i.e.

(2.39) with

(2.40)

wi

E Vh,

W e observe t h a t

po - .s -i t i v e Z of

e

w

VL

i

2

of

Wi(Pj)

0

onver

Vh

v P j E c.,

= 6 1. J.

R,

+k

i,

i s g e n e r a t e d by

and t h a t

Bh.

the

Then i n s t e a d

t o formulate t h e d i s c r e t e entropy c o n d i t i o n

using (2.38)

one t a k e s

(2.41)

-

V(@h-@oh)*vp dh x

which i s e q u i v a l e n t t o t h e

s e t of

the

'

' Nh

ph

Vi

following l i n e a r

192

R.

GLOWINSKI a n d 0.

T-'IRO"EALl

i z c i u a 1i t y c o n s t r a i n t s

f

( 2 .I42 ) 5,

If

v(ql-cboh)

-1

-vwi

V

dx < 0

i = 1,

...,N h .

i s u s e d , i n s t e a d o€ ( 2 . 4 2 ) x e h a v e I

-

(2.'43)

v(~tl-@oll)'v\\idx

0

4

V

...,N h .

i = 1,

62 Computations done with

and u s i n g ( 2 . 4 2 )

k=l

have p r o d u c e d

good r e s u l t s . k=2: The t e c h n i q u e s

I.f ~

~~

t o be u s e d a r e m o r e c o m p l i c a t e d a n d

a r e d e t a i l e d i n G l o ~ i n s k i - P i r o n n c a u[ 1 1 ,

-Remark ..2.15:

(It holds f o r

If

k=1,2).

some D i r i c h l e t

r,

b o u n d a r y c o n d i t i o n s a r c ' p r e s c r i h e d somewhere o v e r p o s i t i v e cone used

the

t o de€irie t h e d i s c r e t e entropy c o n d i t i o n

w i l l b e r e l a t e d t o t h e subspace of

Vtl

c o n s i s t i n g of

those

f u n c t i o n s v a n i s h i n g a t t h e boundary nodes corresponding t o the

(discrete) Dirichlet condition.

2.26:

Remark

(or

Mh)

may b e a n u n e a s y

computations. p a r t A,

Houever,

t~e c h n i q_u_e i s u s e d

e s p e c i a l l y for p r o f i l e

l i k e i n t h e r e g u l a r i z a t i o n method o f

t o handle

2.'+./+ A p p r o x.i m~-~ ation of ~

(2.'+2) ( o r

if

Part A,

~_

X.

take i n t o account

(2.17),

(2.18),

Xh

then

Xh

is

the d i s c r e t e entropy conditions.

2.4.3 e i t h e r

one u s e s t h e r e g u l a r i z a t i o n methods

or

a penalty

(2.43)).

one u s e s t h e methods d e s c r i b e d i n S ec.

= Vh

if

~~

e s s e n t i a l l y determined b y If

task,

some e m p i r i c a l r u l e s h a v e b e e n o b t a i n e d ,

If we d o n ' t

M

The o p t i m a l c h o i c e f o r the hound f u n c t i o n

of

Sec.

xh =

2.4.3,

i s d e f i n e d by t h e l i n e a r i n e q u a l i t y constraints

LEAST S Q U A R E SOLUTIOV O F NON L I N E A R PKOI)LE,PIS

(2.42)

2.5

( o r (2.43))

ICe-rative

~-

2.5.1

so

i f w e proceed

lution of

l i k e i n Sec.

193

2.4.3,

P a r t B.

t h e a p p~r o x i m a t e p r~. oblems.

-~

~~~

Preliminary statements.

G eneralitie - s.

We s h a l l r e s t r i c t o u r a t t e n t i o n t o t h e f o l l o w i n g s i t u a t i o n : *

R

i s hounded anti w e o n l y have Neumann boiindary

conditions.

~

We assume a l s o t h a t K u t t a - J o u k o w s k y required

i s not

since t h e i r treatment

conditions are not s p e c i f i c of

transonic

flows. '

We d o n ' t

take

i n t o account

*

We s u p p o s e t h a t

a

= 1

in

(2.17), ( 2 . 1 8 ) .

(2.14)

and t h a t an e n t r o p y

c o n d i t i o n i s f o r m u l a t e d u s i n g t h e method o f Part B,

with the formulation (2.43)

Sec.

2.4.3,

(control's constraints).

The c a s e where t h e r e g u l a r i z a t i o n method o f P a r t A i s u s e d i s s i m p l e r , and i n f a c t may b e s e e n a s a p a r t i c u l a r c a s e o f other ( a t least i f '

one u s e s p e n a l t y t o h a n d l e

the

(2.43)).

We f i n a l l y assume t h a t we u s e p i e c e w i s e l i n e a r f i n i t e elements

(k=l).

S i n c e we d o n ' t closed

take i n t o account

convex s u b s e t of

Vh

(2.17), ( 2 . 1 8 ) ,

d e f i n e d by

t h e a p p r o x i m a t e c o n t r o l problem

(2.43).

xh

i s the

Therefore

i s a non l i n e a r ( a n d non ~

c o n v e x ) programming p r o b l e m i n w h i c h t h e i n d e p e n d e n t v a r i a b l e

is

qh.

method.

~~

We s h a l l s o l v e t h i s p r o b l e m by a c o n j u g a t e g r a d i e n t ~~~

T o take i n t o account t h e l i n e a r c o n s t r a i n t s ( 2 . 4 3 )

we h a v e u s e d a p e n a l t y method

( f o r t h e s o l u t i o n by p e n a l t y -

d u a l i t y m e t h o d s v i a ____ augmented ._______ Lagrangians see GlowinskiP i r o n n e a u [ 11 )

.

194

R.

2.5.2

GLOWINSKI and 0 . PIRONNEAU

Penalty approximation of the discrete ____ problem. ~

~

We use the notation o f Sec. 2.4.3.

(Uh9Vh)h

-- _

vh

2

L (n)-scalar product: ~-

the following _ approximate (2.44)

Let us define over

Nh

c

miuh(Pi)vh(Pi),

pi E

ch,

tf

i,

3 i=l where

i :f

m

-

6. =

measure union of

(nil, ni TE

0

=

a.

1'

6, such that Pi is a vertex of T;

the corresponding norm is denoted by

\*\,.

Then we define

It follows therefore from (2.45) that (2.43) can also be written

which is equivalent to

where in

cph

(2.48)'

(cp,)'

does not denote the positive part of

but the approximation of it defined by

LEAST SQUARE

In

(2.48)s

@Il

195

SOLUTION O F N O N LINEAR PROULEPIS

Eh

is a function of

t h-r~ ou _ g_h_ ( 2 . 2 3 )

and

.

r

i s a p o s i t i v e p a r a m e t er. Then w e c a n p r o v e t h e f o l l o w i n g :

- -~ P rop-osition

2 .1

-

If

t h e approximate c o n t r o l problem ~~

Min

(2.50) has a s o l u t i o n ,

Jh(Fh)

'h

I-1'

t h e n t h e s o l u t i o n o f - t h e p e n a l i z e d problem

(2.51) converges

Remark 2.17: ______

If

Ooh

= 0

notation) a variant of i n Sec.

2.4.3,

in

~~

the r e g u l a r i z e d f u n c t i o n a l i n t r o d u c e d

(2.51) i s a f i n i t e dimensional c o n t r o l

constraint.

We h a v e s o l v e d t h i s p r o b l e m

the n - on l i n e a r c o n j u g a t e

_g _r _ a d i e n t method

( s e e P o l a k [ 1, C h . 2 ,

more e f f e c t i v e ,

f o r our problem,

pp.

53-551)

w h i c h seems

than the F l e_ tche r.- R e e v e s _

The s c a l a r p r o d u c t u s e d i n t h i s a l g o r i t h m i s t h e

s c a l a r product important

+m.

(2.48) w e recover ( w i t h d i f f e r e~nt

u s i n g t h e P o l a k - R i b i & ~ ev e r s i o n o f

version.

+

P a r t A.

The p e n a l i z e d problem problem without

r

(2.50)

toasolutioLo-f

i n d u c e d by

H1(n)

over

s t e p i n t h e s o l u t i o n of

algorithm i s t h e computation of

(2.51)

Vh.

Therefore a very

by the ab o v e

the gradient

r _J h_

d5 h

~ (5,)

; this

196

R.

point i s discussed

GLOWINSKI and 0.

PIRONNEAU

i n t h e f o l l o w i n g s e c t i - o n 2.5.4.

A d e s c r i p t i o n o f t h e m o s t i m p o r t a n t non l i n e a r c o n j u g a t e g r a d i e n t a l g o r i t h m s i s g i v e n i n Sec.

2.5.4

Computation of

-- J h r

4.

( F ~ ) .

d5 h Owing t o t h e p r a c t i c a l i m p o r t a n c e of

.

Jhr

"h

we s h a l l d i s c u s s

i t s c o m p u t a t i o n w i t h some d e t a i l s : We have

I

We c a n e a s i l y compute t h e l a s t t e r m of

(2.53). of

(2.23)

About

t h e right-hand

s i d e of

t h e s e c o n d t e r m , w e o b t a i n by d i f f e r e n t i a t i o n

LEAST SQUARE S O L U T I O N O F N O N L I N E A R PROBLEMS

197

Since

(2.57) !

t h e second term o f with t h e l a s t

Jhr ( 5 h )

' 6!3

( 2 . 5 6 ) i s e a s i l y computed and by a d d i t i o n

t e r m of

t h e r i g h t hand s i d e o f

( 2 . 5 3 ) we o b t a i n

h'

Remark 2.18: I f -~ ..-.

i n s t e a d of t a k i n g

U.=l

i n (2.24),

one t a k e s

~

a

= 0

w i l l r e q u i r e the use o f

t h e n t h e c o m p u t a t i o n of

an adjoint s t a t e equ ation. -~

2.5.5

Computational c o n s i d e r a t i o n s . __

When s o l v i n g ( 2 . 5 1 )

by t h e non l i n e a r c o n j u g a t e g r a d i e n t

method d i s c u s s e d a b o v e , we h a v e t o s o l v e a t e a c h i t e r a t i o n the state equation

(2.23).

S i n c e t h e b i l i n e a r form o c c u r i n g

i n (2.23)

is positive de€inite

p o i n t of

6

( o n c e t h e v a l u e of'

h a s b e e n p r e s c r i b e d ) we c a n u s e t o s o l v e

e i t h e r i-t e r a t i v e m e t h o d s l i k e c o n j u g a t e g r a d_--.I ient, I_

S.S.O.R.

etc...

A x e l s s o n [ 13

qjh

,

(cf.

e.g.,

Young [ 11 )

P o l a k [ 1 1 , Concus-Golub

i n one

(2.23)

S.O.R., __

[l]

,

o r d i r e c t m e-. thods l i k e G a u s s ' s o r

Choleskyls.

2.6

Numerical e x p e r i me n t s .

The f o l l o w i n g n u m e r i c a l t e s t s by Mrs. B r i s t e a u a n d MM.

( i n two d i m e n s i o n s ) were done

P e r i a u x , P o i r i e r on I B M 3 7 0 - 1 6 8 ,

198

R.

G L O W I N S K I and 0.

PIRONNEAU

o t h e r t e s t s and r e s u l t s a r e r e p o r t e d

C 11 ,

21

,

i n Glowinski-Pironneau

~ l o w i n s ~ < i - ~ e r i a u x - ~ i r o n n[c11 a u,[: 21

Concerning 3-dimensional

problems,

.

some c a l c u l a t i o n s h a v e been

d o n e on t h e e f f e c t s o f o b l i q u e wind on t h e f l o w a t t h e i n t a k e (see Fig.

2.4)

of

a j e t e n g i n e a t l o w mach number ; t h e s e

-

c a l c u l a t i o n s w i l l be r e p o r t e d

elsewhere.

Aircraft

Velocity

\

J e t engine

F i g u r e 2.4 The f o l l o w i n g r e s u l t s c o n c e r n

-

A Naca 0012 p r o f i l e ,

A Kornls p r o f i l e , An O n e r a ' s p r o f i l e ,

A two-piece

2.6.1

airfoil.

Naca 0012 p r o f i l e .

The t e s t d e s c r i b e d b e l o w c o r r e s p o n d s i n f i n i t y of

.85

(M,

= .85)

t o a mach number a t

and i s w i t h o u t _ i n~c_i _ d -e n c e . ~

We

199

L E A S T SQUARE SOLUTION OF NON LINEAR PROCLEMS

have used __piecewise linear finite elements ing to a triangulation %h

correspond-

(k=l)

of 600 triangles and 341 nodes.

The entropy condition (2.12) was required via regularization. ~

~~

The convergence was obtained in 80 iterations for a CPU time of

4

mn.

O n Figure 2.5 we have shown the mach numbers at the midpoint

of the edges, on the skin of the airfoil, extrapolated from their values at the centroid o f the triangles. ed by Bauer-Garabedian-Korn's

Results obtain-

code are shown also.

We observe

the good agreement between the results, except for the amplitude of the shock jump,

This seems to be related to the

fact that in the B-G-K code we have used the discretization -. finite difference - scheme is achieved by a Eon conservative -

and that the small disturbance model is used.

(The schemes

we use are fully conservative). O n Figure 2 . 6 we show the results obtained for the same problem, using piecewise quadratic finite elements 2.6.2

(k=2).

Korn's profile.

The Korn's profile, which is not symmetric (see Fig. 2.7) has the property of giving a shock free flow if

for s o m e specific incidence angle).

Mm = 0.75

(and

On Figure 2.8 we have

shown the mach numbers on the skin of the airfoil;

we observe

that the computed flow is also shock free.

2.6.3

ONERA profile.

Wind tunnel measurements have been made by ONERA (Office Nationale d'Etudes et de Recherches AGrospatiales) for the

R.

200

GLOWINSKI and 0 . P I R O N N E A U

profile shown o n Fig. 2.9.

O n Fig.

2.9 is also indicated the

mach distribution on the skin o f the profile.

O n Fig. 2.10

the computed results are shown and the agreement with the measured results is very good, including the shock location and amplitude.

The agreement is less good after the shock

since the physical flow is n o longer potential.

We observe

that the computed shock is sharper than the measured shock. This fact is due, may b e , to the fact the actual fluid is slightly viscous, but very likely the main reason of this difference is the fact that the velocity fluctuations at the intrance o f the wind-tunnel induce fluctuations of the shock wave location; therefore the measurement taking same average value of the velocity has a smoothing effect,

2.6.4

Two-piece airfoil. ____~

The geometry shown on Fig. 2.11 corresponds to a &)ractical situation.

Kutta-Joukowsky conditions have to be imposed o n

both profiles.

T h e position o f the front piece makes the

funnel slightly convergent. incidence and (k=l)

Mm = . 5 5 .

The main airfoil is at 5 O of

Lagrangian elements of order 1

were used with 2936 triangles and 1 5 5 5 nodes.

Regularization i s used and convergence is obtained in iterations for a C P U time o f 1 6 mn. guessed by measuring

3.5

X

from

lo3

and at

IlV(@-C)Il

n=50

2;

L it is 2.

T h e precision can be at

n=O

its value is

O n each triangle it varies

in the subsonic region to

sonic region.

50

lom2

i n the super-

li

LEAST SQUARE S O L U T I O N OF N O N L I N E A R PROBLEMS

i

0

20 1

202

0

*a

R. G L O W I N S K I and 0 . P I R O N N E A U

W

m

N

F

LEAST SQUARE SOLUTION OF NON L I N E A R I’R0DLJ)MS

203

204

a U I

R.

h

x

G L O W I N S K I and 0 . P I R O N N E A U

h

h

4

LEAST SQUARE SOLUTION O F N O N L I N E A R PROBLEMS

'I;

205

R.

206

G L O W I N S K I and 0 .

PIRONNEAU

M

X 0.5

Figure 2.10

1.

L E A S T SQUARF: S O L U T I O N O F NON L I N E A R PROBLEMS

R. GLOWINSKI and 0. PIRONNEAU

208

3.

Numerical solutions of Navier-Stokes equations. ___ __

3.1

-

Generalities.

Or$entati&n.

In this Section 3 we should like to briefly introduce the reader to some aspects o f the numerical analysis o f the NavierStokes equations.

This is a very large subject in constant

evolution; for information prior to 1 9 7 6 see Temarn [l] and the bibliography therein.

In the following subsections we s h a l l see how the non linear least square methodology can be applied for solving the continuous Navier-Stokes equations in the -~ pressure-velocity ~~

formulation.

Similar methods can also be used for the stream

function-vorticity __ formulation; these methods will bedescribed in a forthcoming report. In practice the above problems have to-be approximated;

we

have used some mixed finite element methods which are described in the report mentioned above.

However we shall only

consider the continuous case whose formalism is simpler.

3.2

The - _N _a v i e r - S t o k e s - e - q u a t i o n s .

Let

0

be a domain o f

iRN

(N = 2 , 3

in practice) and

we consider the

stationary Navier-Stokes equations

(3.2)

uIT. = g

where

(

g'n dr = 0 ,

Jl-

r

= 30;

2 01'

LLAST S Q U A R E S U 1 , I J T I O N O F N O N L I K E A R I'ROB1,CMS

F o r e x i s t e n c e anti u r i i q u c i i e s s see, e.g.,

7.3

Lions

I31 ,

(3.I)-( 3 . 3 )

tlieorems concerning

Tctnam [ 11

A l e a s t s q u.a.r e f o r m u l a t i o n

,

L a d y z e n s k a y a [ 11

.

o f~-t h e Y a v i e r - S t o k e s

equatio _ n_s .

It can b c

Miri VEU

(3.4)

where i n

v 5 :6

i

-

I v(y-v) I

dx,

'R

(3.4)

and where

i s a function of

v

through

(3.5)

problcrn i n w h i c h t h e s t a t e e q u a t i o n

W e h a v e o b t a i n e d a c-ontrol

~

~

~~

i s a Stokcs problem.

If t h e o r i g i n a l p r o b l e m

(3.1)-(3.3) has a s o l u t i o n then the

l e a s t s q u a r e p r o b l e m has a s o l u t i o n f o r w h i c h t h e c o s t function takes

the value

0.

Remayk 3 .-1-.: O t h e r f o r m u l a t i o n s c a n b e u s e d ; f o r i n s t a n c e we can r e q u i r e f o r

etc..

.

v

the condition

v-v

= 0,

o r t a k e as c o s t

R. GLOWINSKI and 0. PIRONNEAU

2 10

3.4

Calculation o f t h e gradient of the cost function. ___ - ._ ~. ~

~

~~

In practical applications, an essential step, in view of using descent or conjugate gradient methods, is the calculation o f the d e r i v a t i v e , o f the cost function. Let

J

J(v) = ; 2

where

r,

the functional defined by

y

is a function of

v

lv(y-v)l

via ( 3 . 5 ) .

2

dx,

Then we f o r m a l l y

have =

V(y-v).V&(y-v)dx

(3.6) = -V

[

v(y-v)*06v

dx +

V

i

~ ( y - v *V6ydx )

W~V€(H’,(~))~

R

where

6y

is solution o f

I

+

-VA6y

v6p = - ( & v . v ) v

-

(v-V)bv,

At that stage it is convenient to introduce the following S t o k e s equation (which is the adjoint state equation i n o u r

problem)

-van + m (3.8)

y,v E

(H;(anN

on

a,

= 0,

nlr

van = If

= -v~(y-v)

0.

(H1(n))N,

then (3.8) has a unique solution in

x (L2(W/R).

Taking variational formulations of ( 3 . 7 ) , (3.8) we have ~

LEAST SQUARE

where

Wo

Taking

z

SOLUTTON

= { z

E

= 7

i n ( 3 . 9 ) and

I n t e g r a t i n g by- p a r t hand s i d e o f

7 - 2 = 03

(Hi(n))N,

(i.e.

z

= 6y

211

L I N E A R PROBLEMS

O F NON

*

i n (3.10)

we o b t a i n

using Green's formula) the right-

( 3 . 1 1 ) c a n a l s o be w r i t t e n

I t f o l l o w s t h e n from ( 3 . 6 ) , ( 3 . 1 1 ) , ( 3 . 1 2 ) t h a t w 6 v E (H:(R))~

which c a n also be w r i t t e i i

(3.14) (

where and

(J'(V),~V)=(-VA(Y-V)

- ,*)

(H:(n))N.

(3.15)

3.5

+

(V*V)n

+

(V.V)q

-

(V~)q,q)

d e n o t e s t h e d u a l i t y p a i r i n g between

(H-l(n))N

Therefore

J' (V) =

-vA(V-y)

+

(V*V)

n

-k

(v'V:Y

-

(vV)n

-

Numerical e x p e r i m e n t s .

I n p r a c t i c e we have a p p r o x i m a t e d t h e N a v i e r - S t o k e s by a mixed

f i n i t e e l e m e n t method.

equations

Then u s i n g a d i s c r e t e

212

R.

v a r i a n t of

GLOWINSKI and 0 . P I R O N N E A U

the l e a s t square formulation

( 3 . ) + ) , ( 3 . 5 ) we have

minimized t h e c o s t f u n c t i o n by a c o n j u g a t e g r a d i e n t a l g o r i t h m _ _ ~ . ~ . _ ( o f R i b i & r e - P o l a k t s t y p e ) . I n t h e conjugate g r a d i e n t algorithm 1 N

we used as a s c a l a r P r o d u c t a d i s c r e t e f o r m of t h e ( ~ ~ ( 0 ) ) s c a l a r product. The n u m e r i c a l t e s t p r e s e n t e d h e r e i s t h e t w o - d i m e n s i o n a l , time d e p e n d e n t f l o w p a s t a d i s k , o f a v i s c o u s i n c o m p r e s s i b l e f l u i d a t a Reynolds number o f 2 0 0 . f o r t h e time d i s c r e t i z a t i o n ,

U s i n g a n i_ m - p l i c i t scheme .__ ~

t h e numerical i n t e g r a t i o n i s

r e d u c e d t o a s e q u e n c e of s t a t i o n a r y a p p l y t h e t e c h n i q u e s . d e s c r i b e d above.

problems

t o which we

T h e r e f o r e a t each time

s t e p we h a v e t o s o l v e a l e a s t s q u a r e p r o b l e m ; however s i n c e

t o compute t h e f l o w

at

(n+l)At

we s t a r t o u r i t e r a t i v e

procedure u s i n g t h e informations a t of

nAt,

t h e convergence

the conjugate gradient algorithm i s f a i r l y f a s t .

S i n c e t h e domain o f t h e f l o w i s unbounded we have imbedded t h e d i s k i n a much l a r g e r d i s k on which a u n i f o r m boundary velocity i s prescribed. l y perturbated Fig.

3.1

S t a r t i n g from a u n i f o r m f l o w s l i g h t -

t o 3.4

show t h e e v o l u t i o n o f t h e f l o w

d u r i n g some t i m e i n t e r v a l , and p a r t i c u l a r l y t h e b e h a v i o u r of t h e f l o w behind t h e o b s t a c l e .

LEAST SQUARE

SOLUTION O F NOK L I N E A R PROBLEMS

F i g . 3.1

CYCLE DE TEMPS REYNOLDS

10

200.

214

R.

G L O W I N S K I and 0. P I R O N N E A U

Fig. 3 . 2

CYCLE DE TEMPS REY NClLDS

20

200.

LEAST S Q U A R E S O L U T I O N O F NON L I N E A R I’ROULEIIS

Fig. 3 . 3

CYCLE DE TEMPS REYNOLDS

40 200.

216

R.

G L O W I N S K I and 0 .

PIRONhTAU

Fig. 3 . 4

CYCLE DE TEMPS REYNOLDS

GO 200.

LEAST SQUARE S O L U T I O N OF NON L I N E A R PRODLEMS

4.

On c o n j u g a t e g r a d i e n t m e t h o d s .

An e s s e n t i a l t o o l i n t h e s o l u t i o n o f non l i n e a r p r o b l e m s by l e a s t s q u a r e methods i s t h e c_o_ n j-u.g_ a t. e_ - g r a d i e n t method t o b e d e s c r i b e d below:

N

be a s u f f i c i e n t l y smooth f u n c t i o n ,

Let

f: R

R

vf

af be {a fK ... , ,axN]

4

let

i t s g r a d i e n t , and l e t S b e a N x N 1 s y m m e t r i c , p o s i t i v e- d e_ f-i n i t e m a t r i x . Then t o s o l v e t h e m i n i =

~

~

r n i z a t i o n problem

I

x

Find

E RN

such t h a t

(4.1)

we c a n u s e t h e t w o f o l l o w i n g c o n j u g a t e g r a d i e n t a l g o r i t h m s t h e c o n v e r g e n c e o f which is s t u d i e d i n P o l a k [I]

(forS=I).

F i r s t method ( F l e t c h e r - R e e v e s ) :

(4.2)

xo

(4.3)

go =

s-l

(4.4)

w

-

-

0

E I R ~ given, Vf("),

0

= g .

Then a s s u m i n g t h a t

-x n

wn

-

and

a r e known we compute

x n+1

-

by

(4.5)

X n + l= I

where problem

p,

-n

n -p,y9

i s t h e s o l u t i o n f o r t h e one-dimensional minimization

R.

218

GLOWINSKI a n d 0 . PTRONNEAU

Then w e compute

-gn+'

(4.7)

gn+l =

(4.8)

wn+1 = -gn+1 + x n w n

wn+ 1

and

s-1

by

Vf(?n+l),

where

(4.9) i n (4.9),

denotes t h e u s u a l inner-product

( a , . )

Second method

of

N

R

.

(Polak-Ribisre).

T h i s method i s l i k e t h e p r e v i o u s o n e e x c e p t t h a t

(4.9) i s

r e p l a c e d by

-

( sgn+l, _g"+l-gn) (4.10)

Remark 4.1:

If

E RN

where

and where

A

is a

NxN

symmetric p o s i t i v e

d e f i n i t e m a t r i x , t h e n t h e two a b o v e a l g o r i t h m s a r e i n f a c t

similar,

s i n c e i t c a n be proved t h a t i n t h i s c a s e

-

( S g n + l , g") Remark 4 . 2 :

A t e a c h i t e r a t i o n of

= 0.

t h e two a b o v e a l o r i t h m s w e

have t o s o l v e a one d i m e n s i o n a l problem t o o b t a i n

pn.

For

t h e s o l u t i o n o f t h i s k i n d o f one d i m e n s i o n a l problem s e e , e.g.,

P o l a k [: 1 3

Remark 4.3: round-off

,

Brent [

13 .

The above a l g o r i t h m s a r e f a i r l y s e n s i t i v e t o

e r r o r s ; h e n c e d o u b l e p r e c i s io n may be r e q u i r e d o n

some c o m p u t e r s .

M o r e o v e r i t may b e c o n v e n i e n t t o r e s t a r t

LEAST SQUARE SOLUTION OF MON LINEAR PROBLEMS

periodically with

wn =

-g

n

.

219

Actually Powell [l] has recently

introduced a new type of restarting procedure which increases the efficiency of the algorithm, but which requires more storage.

The Powell's analysis shows also that the Polak-

RibiBre's algorithm may b e more convenient than the FletcherReeves' algorithm for some problems.

In fact f o r the type of

problems we are solving a similar observation has been done, from our computational experiments. Remark 4.4: F o r the problems described i n this paper a convenient choice f o r the matrix form of

S

is to take a discrete

-A.

5 . Conclusions, The methods presented in this paper may appear as very heuristical, however when convexity o r monotonicity are lacking they seem one o f the few possibilities for computing the solutions o f some difficult non linear problems, taking into account their special structure. Applications to non linear problems i n elasticity are to be done. Acknowledgments: Many thanks are due to Mrs. Bri,steau and MM. Perrier, Periaux, Poirier for their help at various stages of this report.

R. GLOWINSKI and 0. PIRONNEAU

220

Bibliography Axelsson, 0.

-

[l]

A class of iterative methods for finite

element equations. Ccmp.

Math. . Applied _ ~ _ .. Mech. Eng.,

Bauer, F., Garabedian, P., Korn, D.

~~

-

[l]

Supercritical _____ wing

sections, . -~ Lecture Notes in Economics and Math.

.

Systems, Vol. 66, Springer-Verlag, 1972. Bauer, F., Garabedian, P., Korn, D., Jameson, A. [l]

Supercritical wing sections (11), Lecture Notes

in Economics and Math. Systems, Vol. 108, SpringerVerlag, 1975. Begis, D., Glowinski, R.-[l] 616ments finis

Application de la m6thode des la r6solution d'un problbme de

domaine optimal. Vol. 2, NO 2, Brent, R.

-

Applied Math. and Optimization,

(1975), pp. 130-169.

[l] Algorithms for minimization without derivat~

~~~

~

ives, __ Prentice Hall, 1973. Cea, J., Geymonat, G.

-

[l] Une methode de lin6arisation via

l'optimisation.

Instituto d i Alta Mat. Symp. Mat.,

Vol. X, (1972), Bologna, pp. Ciarlet, P.G., Raviart, P.A.

-

431-451.

111 Interpolation theory over

curved element with applications to finite elements methods. Comp. Meth. Applied Mech. Eng., Vol. 1, (1972) , PP. 217-249.

[2] A mixed finite element method for the biharmonic equation, i n Mathematical aspects of the

LEAST SQUARE SOLUTION OF NON LINEAR PROBLEMS

finite elements in partial di-fferen&ial C. de B o o r Ed., Acad. Press, Concus, P., Golub, G.H.

-

221

equations,

1974, pp. 125-145.

[l] A generalized conjugate gradient

method for non symmetric systems of linear equations, in Computing Methods in Applied Sciences and Engineering, R. Glowinski and J.L. Lions Ed., -~ Lecturca Notes i n Ecsiiomics and Math. Systems,

VOI. 134, Springer-Verlag, 1976, pp. 56-65. Glowinski, R.

-

[ 13 Numerical methods f o r non linear

variatipnal problems, Lecture Notes at the Tata Institute, Bombay, to appear in 1977-1978. [2] Approximations externes par 616ments finis

d'ordre un et deux d u probl&me de Dirichlet pour

A 2 , in Topics i n Numerical Analysis , J. J.H. Miller ed., Acad. Press, (1973)) pp. 123-171. Glowinski, R., Marroco, A .

-

[l] Sur lfapproximation par

616ments finis dfordre un, et la r6solution par p6nalisation-dualit6 d'une classe de probl;?mes de Dirichlet non linGaires, Revue Franqaise d'dutoma___ ____ tique, d'Informatique et de Recherche Opgr-ationnelle, e

9

ann&e, aojt 1975, R-2, pp.

41-76.

Glowinski, R., Periaux, J., Pironneau, 0.

-

[l] Transonic

flow simulation by the finite element method via optimal control.

Proceedings of the Second Inter-

national Symposium on Finite Elements i n Flow Problems, Santa Margherita, Italy, June 1976,

PP. 249-259.

R. GLOWINSKI and 0 . PIRONNEAU

222

[2] Use o f optimal control theory for the numerical

simulation of transonic flow by the method of finite lements.

Proceedings o f the

5th Conference _____

on Numerical Methods i n Fluid Dynamics.

Enschede,

Netherlands, July 1976. Glowinski, R., Pironneau, 0.

-

[l] On the computation of

transonic flows, Proceed+-ngs o f the first FrancoJ_po&Ese

Co-lloquium on Functional and Numerical

__ Analysis, Tokyo, Kyoto, September 1976. [2] Calcul d'ecoulements transoniques par des

methodes dlel6ments finis et de controle optimal, in Computing Methods in Applied Sciences and E n- -g i n e e r i x ,

R. Glowinski and J.L.

Lions Ed.,

Lectures Notes i n Economics and Math. Systems, vol. 134, Springer-Verlag,

1976, pp. 276-296.

[ 3 ] Numerical methods f o r the first biharmonic equation and for the two-dimensional- Stokes problem, C.S. Report, University of Stanford, Computer Science Dpt., Hewitt, B.L.;

1977.

Illingworth, C.R.;

Lock, R.C.;

Mc Donnel, J.H.; Richards, C.; Walkden, F . ;

Mangler, K.W.; Ed.

[l] Computational method2 and problems i n aeronau-

tical fluid Acad. Press, 1976. ~ - dynamics, -___-Jameson, A.

- [l]

Transonic flow calculations, in Proceedings

of Conference o n Computational Fluid Dynamics, - Von .

Karman Institute, March 1976, Brussels (Belgium). [2] Three dimensional flows around airfoils with

LEAST SQUARE SOLUTION OF NON LINEAR PROBLEMS

223

shoks , in Computing Methods _in ApplA-e-d-Sci-e-Eces_and Engineering, Part 2 , R . -~

Glowinski and J.L. Lions

_ I

Ed., Lecture Notes in Comp. Sciences, V o l . Springer-Verlag,

131

11,

(1974), pp. 185-212.

Iterative solution of transonic flows over

airfoils and wings, including flows at Mach 1,

Comm. Pure Appl. Math., ~ _ _Vol. 27, (1974), pp.283-309.

[4] Numerical solution of non linear partial differential equations of mixed type, in Numerical Solution of Equations, 111, Synspade - Partial Differential - __. ~~~

~~

1975, B. ___

~

Hubbard Ed., Acad. Press, (1976),

PP. 275-320. Landau, L.; Lifchitz, F.

-

[l] M6canique des - Fluides, Editions ~

MIR, MOSCOU, 1953. Ladyzenskaka, O . A .

-

[l] The ___. mathematical theory of viscous

incompressibA-e fluids, Gordon Breach, New York, 1963. Lions, J.L.

-

[l] Gontrole optimal des _ systemes _ ~ gouvern6s par ~

des 6quations aux d6rivGes partielles.

Dunod, 1968.

[ 2 ] Some aspects of the optimal control of

distributed systems, Regional Conference Series in Applied Math., Siam Publication NO 6, 1972.

[ 3 ] Quelques m6thodes de rdsolution des problhmes aux limites non lineaires, Dunod, 1969. Morawetz, C.S.

-

[l] O n the non existence of continuous

transonic flows past profiles, C o m m . Pure Applied Math., V o l .

9, 1956, pp. 45-68.

224

R. GLOWINSKI and 0. PIRONNEAU

Murman, E.M.;

Cole, J . D .

-

[11 Calculation of plane steady

transonic flows, AIAA Journal, V o l .

9, (1971),

pp. 114-121. Polak, E.

-

[l]

Computational methods in optimization, Acad.

Press, 1971. Powell, M . J . D .

-

[l] Restart procedures f o r the conjugate

gradient method, Math. Programming 12,

(1977),

pp. 241-254. Strang, G.; Fix, G.

-

[l] &n--analysis of the finite element -

method, Prentice Hall, 1973. Temam, R.

-

[l] Navier-Stokes Equations, North-Holland,

1977 Young, D . M .

- I: 11

Iterative s-ol~~ix-of__large linear systems,

Acad. Press, 1971.

G.M. de La Penha, L.A. Medeiros ( e d s . ) Contemporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations @North-Holland P u b l i s h i n g Company (1978)

E L A S T I C BODIES I N A SIGNORTNI-TYPE ENVIRONMENT

P.

POD10 G U I D U G L I

%

Dipartimento d i Costruzioni Facol

ta

d i Ingegneria

U n i v e r s i t ' a d i Aiicona,

Noll's formulation of

Italy

t h e t r a c t i o n boundary-value

p r o b l e m o f e l a s t o s t a t i c s i s s p e c i a l i ~ e dt h r o u g h a s i m p l e

I t s dependence on a para-

c h o i c e of

environmental operator.

meter

l e a d s t o a r e p r e s e n t a t i o n o f s o l u t i o n s a s power

C

series i n G

E ;

t h e c o e f f i c i e n t s of

t h e s u c c e s s i v e powers

a r e s o l u t i o n s o f c e r t a i n Linear d i f f e r e n t i a l problems,

of

each

of w h i c h l e a v e s a c e r t a i n i n d e t e r m i n a c y i n t h e s o l u t i o n w h i c h

i s g e n e r a l l y removed by c o m p a t i b i l i t y c o n d i t i o n s d e r i v e d from

the Fredholm a l t e r n a t i v e .

When t h e body i s i n i t i a l l y a t e a s e ,

these conditions collapse into the l i n e a r conditions of Signorini,and generally determine the global r o t a t i o n .

Introduction.

In

1971, d u r i n g

Stability i n Elasticity",

a Symposium on " E x i s t e n c e and

W.

w h i c h he s k e t c h e d a method o f

N o l L delivered a lecture i n formulating t h e boundary-value

problem o f f i n i t e e l a s t o s t a t i c s . l e c t u r e remained u n p u b l i s h e d ,

Though t h e t e x t of t h i s

l a t e r P r o f e s s o r N o 1 1 was k i n d

enough t o p r o v i d e m e w i t h a copy of h i s m a n u s c r i p t .

The

q u e s t i o n s o f e l a s t i c i t y I i n t e n d t o d i s c u s s h e r e a r e most

*

Presently at Istituto di Scienza delle Costruzioni, Facolth di Ingegneria, Universita di Pisa, Italy.

226

POD10 GUIDUGLI

P.

conveniently dealt with within N o l l ’ s framework:

therefore,

I will adopt that framework, in a somewhat modified f o r m , to serve my purpose.

I identify a n elastic body

03

with the following

collection o f objects: (i)

B

A regular, bounded region

3B

has boundary

B

E ;

of Euclidean space

with outer unit n o r m a l n.

Whenever

convenient, I write vvmeas’v for the pair of the usual volume and surface measures on

B:

meas = (vol, surf). (ii)

b

A class

B,

o f deformations of

i.e., of inver-

B

tible and locally orientation-preserving mappings of open subsets of tries between

E ;

B

@

is assumed to contain all the isorne-

and open subsets of

b.

Given any

y E

g,

E ,

s o that, in

v x E B,

i(x) = x

particular, the identity mapping: to

onto

below

the deformation gradient

F(x)

def = VYI,

(A-1) x;

is the basic measure of deformation at

which are invertible second-order tensors. u(x) = y(x)

x

is the displacement vector at H(x)

has values

F o r each

x E B, (A-2)

x

in the deformation

y,

def = vulx

is the displacement gradient; F(x) =

-

F

clearly, for each

1 + H(x),

and

(-4-3) x

E B (A-4)

where 1 = Vi.

(A-5)

E L A S T I C B O D I E S I N A SIGNORINI-TYPE E N V I R O N M E N T

(iii) each

A n elastic response function

x E R

227

such that, f o r

5,

y E l ~ ,

and f o r each

( A- 6 ) x

i s the measure of the stress at the deformation

y;

when

R

has undergone

in particular,

is the ground stress at

F o r convenience, I think of S(x)

x.

as the first Piola-Kirchhoff stress tensor at

X.

The

prescription (6) satisfies the axioms of material frameindifference and balance of angular momentum if one assumes that T

5(F) = F & ( F F ) with

fJ

( A- 8 )

I

a mapping of the space of symmetric p o s i t i v e - d e f i n i t e

tensors into t h e space of symmetric t e n s o r s . A system of loads for

S

is any pair

4, = ( b , s )

such that:

b,

the body force, is a volume-integrable vector

field over

B;

s , the ___ surface _ traction, _ _ i s~ a surface-

integrable vector field over fB Ld(meas)

def =

aB;

1

Ld(meas) = 0, where

IB

( n b d(vol)

+

s d(surf).

The collection of

all such systems of loads is the load space, denoted b y Once a n elastic b o d y

fi

and a load space

6:

2.

are

given, I introduce the elastic response operator

.e:

l$

+

x,

Z(Y> =

(&),:(u)),

where

(A-9)

P.

228

POD10 GUIDUGLI

( A - 10) i(y)

i s t h e s y s t e m of

location

A

1.oads which hol-ds

i n t h e deformed

Y(B). F u r t h e r , i n t h e manner of N o l l , I i n t r o d u c e t h e

environmental r e a c t i o n o p e r a t o r

i(y)

63

i s t h e system o f

Z:

Q

+ C,

X(y) = (i;(y) , ~ ( y ) ) ;

l o a d s t h e e n v i r o n m e n t would e x e r t on

i f i t o c c u p i e d t h e deformed l o c a t i o n

y(B).

A typical

d i f f i c u l t y i n f i n i t e e l a s t i c i t y i s t o model p h y s i c a l l y m e a n i n g f u l l o a d i n g d e v i c e s by means of s e n s i b l e a s s i g n e m e n t s o f environmental r e a c t i o n o p e r a t o r s ( v i d . e . g . Batra (1972); Ericksen d e s c r i b e s a s y s t e m of

(1967);

Sewell

( 1 9 7 3 ) ) . The s i m p l e s t c a s e i s when dead l o a d s :

U s i n g t h e above m a c h i n e r y ,

t h e t r a c t i o n boundary-

v a l u e problem o f e l a s t o s t a t i c s i s f o r m u l a t e d a s f o l l o w s : A

Given a n e l a s t i c body

63,

with e l a s t i c response o p e r a t o r

f,,

which i s p l a c e d i n t o a n e n v i r o n m e n t d e s c r i b e d by t h e e n v i r o n -

-

mental r e a c t i o n o p e r a t o r

f,,

t(Y) I call (12).

(y,x(y))

find

y

E &

such t h a t :

= Z(Y).

(A-12)

a n e q u i l i b r i u m p a i r when

y

For an e q u i l i b r i u m p a i r , an easy consequence of

solves

(8)

and t h e d i v e r g e n c e theorem i s t h a t :

\

(y(x)-q) A t ( y ) l ,

d(me-1

= 0

(A-13)

B (here o f

q

i s any f i x e d p o i n t of

t h e m u t u a l c o n s i s t e n c y of

e);

( 1 3 ) i s an e x p r e s s i o n

a n e q u i l i b r i u m p a i r and i s

ELASTIC BODIES I N A SIGNORINI-TYPE E N V I R O N M E N T

i n t e r p r e t e d a s t h e requirement t h a t t h e system o f has vani shi ng t o t a l torque i n

229

loads

-

L

y(B).

S i g n o r i n i - t y p e environment.

The f o r m a l s e t t i n g d e v i s e d by Nol.1

( 1 9 7 1 ) a l l o w s one

t o d e a l i n a n a t u r a l way w i t h t h e f o l l o w i n g q u e s t i o n : When

63

i s a t r e s t i n t o a g i v e n e n v i r o n m e n t , what happens i f

t h e e n v i r o n m e n t i s changed? P e r s e , s u c h a q u e s t i o n l e a d s one t o c o n s i d e r p e r t u r b a t i o n techniques. non-trivial

I n t h i s s e c t i o n I o f f e r an answer f o r a classical c a s e f i r s t s t u d i e d by A .

Signorini i n the thirties;

more d e t a i l s w i l l b e found i n C a p r i z & Podio G u i d u g l i

(1977).

(i,% ( i ) )be a n e q u i l i b r i u m p a i r for a n e l a s t i c

Let

n

63

body

c h a r a c t e r i z e d by t h e e l a s t i c r e s p o n s e o p e r a t o r

.t(i) = l o ( i ) .

t,: (B-1)

A Signo r i n i - t y p e e n v i r o n m e n t i s d e s c r i b e d by a r a y -____ in

2 ,

with o r i g i n a t G - %

E (y)

zo,

o f s y s t e m s of dead l o a d s :

Eo(i) +

=

c

l,(i)

++

Y E 14.

(B-2)

One t h e n s e e k s a r e g u l a r F - f a m i l y of d e f o r m a t i o n s , i.e.,

a path i n

I$ E-Y

originating a t

yo = i

(B-3)

E

and s u c h t h a t i n a n e i g h b o r h o o d of

e q u i l i b r i u m p a i r for e a c h

t(Y& = tE(YE)' I n the s p i r i t of

0:

(B-4)

t h e s t a n d a r d methods o f a n a l y t i c n

p e r t u r b a t i o n , I assume t h a t

L

i s analytic a t

yo:

P.

230

with

v(n)i

Iyo

POD10 GUIDUGLI

a n-linear

c omple te ly symmetric transformati on.

Further, I seek a solution of type

(3) of equation (4) in the

form o f a formal power ser es in

:

E

-

-

Thus, I replace equation (4) by its Oth-order

“approximation” (l), plus a n infinite hierarchy of linear equations f o r the successive approximations solution

un

of

yE:

-

L u1 = C1,

with

L

def =

oil

and

t,

= t,(yo),

YO

L un =

the

xn,

n=2,3 ,..., with

-

cn

def n- 1 = -Qn(Yo;Cukll ).

(B-9)

If the system (9) i s solvable, the system of (1) and

(9) may be regarded as a reasonable model of equation (4)+ , o r rather of (A-12). -k

See next page.

The next Section will b e devoted to

231

ELASTIC BODIES I N A S I G N O R I N I - T Y P E E N V I R O N M E N T

s o l v a b i l i t y f o r t h e s y s t e m (9).

e s t a b l i s h c e r t a i n c o n d i t i o n s of

Fredholm c o n d i t i o n s .

L

Let

b e r e g a r d e d a s a l i n e a r t r a n s f o r m a t i o n from

a Banach s p a c e

X

i t s transpose.

I n t h e c a s e of

i n t o a Banach s p a c e

L

e l a s t o s t a t i c s both ker L

that

2[O]

t h e t r a c t i o n problem of are not injective.

L*

t h e n o n - l i n e a r problem ( A - 1 2 ) ; ker L

t h e s t r u c t u r e of

classification of

i s of

s o l u t i o n of n t h datum in

n-1

t h e nth

ln

thus, a

impnrtance f o r a

equations

(B-9)

when

Accord-

k e r L*$

[ O ] ,

have been s o l v e d , t h e

equation does e x i s t i f

and o n l y i f

the

b e l o n g s t o t h e o r t h o g o n a l complement o f k e r L*

Y*:

(tn,v*) Remark Fichera of

The f a c t

t h e d i v e r s e t y p e s of b i f u r c a t i o n s .

i n g t o a g e n e r a l lemma of a l t e r n a t i v e , a f t e r the first

be

t h e n i n d i c a t e s t h e e x i s t e n c e o f a bifurcation

i n t h e s o l u t i o n of knowledge of

a?d

L*

and l e t

Y,

-

= 0

I n h i s treatment

v*

of

k e r L* + +

.

(c-1)

a b s t r a c t l i n e a r transformations,

(1954) s t a t e s c l e a r l y the primitive algebraic nature

t h e lemma of a l t e r n a t i v e ,

‘Notice

E

that

and a d j u s t s i t s o a s

t o apply i n

( 9 ) 1 coincides formally with t h e equation o f t h e

t h e o r y o f i n f i n i t e s i m a l d e f o r m a t i o n s s u p e r i m p o s e d on l a r g e , but o f course the p o s i t i o n o f ( 9 ) a s t h e l S t - o r d e r approxima t i o n o f a n o n - l i n e a r problem w i t k i n a g i v e n p e r t u r b a t i o n scheme i s n o t a t a l l same.

++ I d e n o t e by

Y*

Y,

the dual of

p a i r i n g between e l e m e n t s of

Y

and by and

Y*.

(

,

)

the

P. PODIO GUIDUGLI

232

in various concrete contexts.

The role of the lemma in the

theory of branching o f solutions of non-linear equations is made clear in the book by Vainberg & Trenogin ( 1 9 7 4 ) through the notion of I1normally solvablef1operators.

L

of normally solvable operators

that

ker L

and

ker L*

A relevant class

is Fredholm's (those such

have finite, and equal, dimension);

it should be possible to find reasonable hypotheses under h

which the elastic response operator

L

approximated by a Fredholm operator

I,.

turns out to be

I call the solvability conditions(1) Fredholm I will provide below a n interpretation, i n terms

conditions.

of elasticity theory, of these conditions, o f their use to remove indeterminacies , and of the cases of "incompatibility", i.e.,

the cases when they fail to be fulfilled for some

n.

The Fredholm conditions will now be given a less general, but more explicit form. Let 0 < dim ker L = dim ker L"

and let ker L

r

{v,]

and

lcer L*.

and

(.:If

(c-2)

+m

be, respectively, a basis of

Then, if

equation ( B - 9 ) ,

(n-l)th

= r <

cn-

is any solution of the

the general solution of the same

equation has the representation

where

i (yn-l]y

numbers, (B-9)

is a list o f , i n principle, arbitrary real

However, the Fredholm condition for the nth equation

restricts the choice of the numbers

(U~];-~ and

n- 1

are given.

i yn,l

once

yo

)

233

ELASTIC BODIES IN A SIGNORINI-TYPE ENVIRONMENT

I introduce the following notation

(c-4) I then write (1) in the case

r

equations of second degree in the r

C,< +

i

Y1:

r

bki

c

unknowns

algebraic

r

as a system o f

n=2

Y; +

Gkij Y i Y p =

1

i= 1

'

0

k=l,2,...,r;

(C-5)

i,j = 1

i n the case

n

2

3

i unknowns Yn-1: r + C 'k,n-l i= 1

as a linear algebraic system in the

r

bki+ z C

Cikij Y j~ )i Y ~ =- 0~

... , r ,

k=1,2,

j=1

r

(C-6)

where the coefficients are defined in the following way:

st

Thus, the 1 response operator

i

-order approximation of the elastic

determines the possibility of solving

i

system (B-9), whereas the 2nd-order approximation of i determines the possibility of finding the y n for any

n.

A simple and yet non-trivial instance is when all the coefficients numbers

Yn-1

aki

vahish.

Then, if

det

are uniquely specified f o r each

as a consequence, series ( B - 6 )

bki n

0, the 2

2,

and,

is completely determined, the

indetermination in the solution of each equation (B-9) notwithstanding. from

(4)-(7)

Another example of the information deducible is obtained when

r=l.

In the instance, systems

P. POD10 GUIDUGLI

234

(5)

and (6) reduce to

c

+XY1

+

a(Y,)2 =

0,

bn-l P

+(J:+2aY1)Yn-1

(c-8)

= 0

and conditions of solvability, as well as incompatibility, for (8) can be read off at once.

Body initially at ease

In the case originally studied by Signorini (vid. e.g, Capriz & Podio Guidugli at ease in

B,

(1974)), (i) the body

@

is

i.e., the ground system of loads is null:

and the ground stress vanishes identically over

(ii)

@

B:

is hyperelastic, i.e., a strain-energy function

u = o(F TF)

exists such that

I put

with

L

a self-adjoint transformation whose kernel, as can

be seen at once, contains isometries

+,

+

3,

+. Further, I put

See next page.

the set of all inPinitesima1

ELASTIC BODIES IN A SIGNORINI-TYPE ENVIRONMENT

235

where

(v(2)sl-c, vVii ) c Ovji

1,r- vVii ) c v V j ~+

= d2)Q

+ (vvi)scvv.l J + (vvj)srvvil + sr (VVj)

T

,

vvil

(D-7

1

so that, i n particular,

Assume now, following Signorini, hypotheses sufficient to guarantee that

ker L = 3.

S

symmetry properties of

and

Then it follows from the that all the coefficients

Q

vanish; thus, although the (n-1) th equation (B-9) %i j determines u ~ to - within ~ an infinitesimal isometry, the situation occurs which I mentioned at the end of the previous section, and the Fredholm conditions of order

n

can be used,

in general, to specify the infinitesimal rotation of Moreover, when

ker L = 3 ,

ordern-1.

Capriz & Podio Guidugli ( 1 9 7 7 )

prove the following characterization of the Fredholm cofitions: The following three statements are equivalent: (i)

xn,

The systems of loads

n=1,2,...,

satisfy the

Fredholm conditionsfor a l l infinitesimal r o t a t i o n s . (ii) The systems of loads

Ln -

have vanishing total torque

in B: (x-q)

A

znlx d

(meas) = 0 ,

(iii) The system of loads

ic

n=1,2,...

.

(D-9)

has vanishing total torque in

Ye ( B ) : (D-10) X

'Here

sym is the symmetrizer o f second-order tensors.

An infinitesimal isometry is a displacement vector field which has the form w(x)

= w(q)

+ w(x-q),

with W a skew-symmetric second-order tensor, independent of X.

236

P.

POD10 GUIDUGLI

Aknow-1 e d gme n t Williams for h e l p i n g m e w i t h

I t h a n k P r o f e s s o r W.0.

t h e E n g l i s h w o r d i n g a n d p r o v i d i n g some u s e f u l comments.

References B a t r a , R.C.

(1972).

Capriz, G.,

and P o d i o G u i d u g l i , P.

Mech. C a p r i z , G.

F i c h e r a , G.

57,

Anal.

(1973).

(1974).

Anal.

(1954).

(1971).

4 8 , 163.

Arch.

Rational

1.

(1977). I n p r e p a r a t i o n .

I n N o n l i n e a r E l a s t i c i t y , R.W.

(Academic P r e s s , New York-London),

( L i b r e r i a V. N o l l , T$.

R a t i o n a l Mech.

and P o d i o G u i d u g l i , P .

E r i c k s e n , J.L. Ed.

Arch.

Dickey

161.

Lezioni s u l l e trasformazioni

lineari.

V e s c h i , Roma).

What i s a “ w e l l - p o s e d “ p r o b l e m i n f i n i t e

elasticity?

Italian

-

American Symposium on

“ E x i s t e n c e and S t a b i l i t y i n E l a s t i c i t y 1 ’ , U d i n e , I t a l y , J u n e 18-23. Sewell, M.J.

(1967).

V a i n b e r g , M.M.,

Arch.

R a t i o n a l Mech. A n a l .

a n d T r e n o g i n , V.A.

(1974).

T h e o r y of Branch-

i n g o f S o l u t i o n s of N o n - l i n e a r E q u a t i o n s . (Noordhoff, Leyden).

3 , 327.

G.M. de La Penha. L.A. Medeiros (eds.) Contenporary Developments i n Continuum Mechanics and Partial Differential Equations @North-Holland Publishing Conpany (1978)

ON THE NONLINEAR THEORY OF ELASTICITY

MORTON E. GURTIN Department of Mathematics Carnegie-Mellon University Pittsburgh, PA 15213

1. Introduction.

In these lecture notes we give a brief introduction to the nonlinear theory of elasticity.

We present a concise

derivation of the basic equations followed by a detailed discussion of the underlying boundary-value problems.

Our

discussion demonstrates why this theory is far more difficult than most nonlinear theories of mathematical physics.

It is

hoped that these notes will convince analysts that nonlinear elasticity is a fertile field i n which to work. The most important general references to the subject are :

[11

Truesdell, C. and W. Noll,

The non-linear field theories

Handbuch der Physik. III/3.

of mechanics.

Berlin:

Springer-Verlag (1965). [ 2 ] Wane;, C.-C. and C. Truesdell,

Elasticity.

[3]

Leyden: Noordhoff

Antman, S.S.,

Introduction to Rational

(1973).

Ordinary differential equations of non-linear

elasticity. Arch. Rational Mech. Anal.

[4] Ball, J . M . ,

a,307-393 (1976).

Convexity conditions and existence theorems

in non-linear elasticity.

337-403 (1977)

Arch. Rational Mech. Anal. 63,

MORTON E .

238

GURTIN

I have a l s o p r o f i t e d f r o m :

151

Antman, S . S . ,

Fundamental m a t h e m a t i c a l problems i n t h e

theory o f nonlinear e l a s t i c i t y . P a r t i a l D i f f e r e n t i a l Equations, Press

Numerical S o l u t i o n of 111.

New York:

Academic

(1976).

I n t h e t e x t when I r e f e r t o a t h e o r e m , c o u n t e r e x a m p l e , e t c . by name b u t o m i t t h e e x a c t r e f e r e n c e , t h e l a t t e r can b e found in [

2.

11

or [ 2 3 .

Notation.

W e u s e l'ower c a s e Greek l e t t e r s f o r s c a l a r s , l o w e r case Latin l e t t e r s f o r vectors

(elements o f

~ 3 1 ,u p p e r c a s e

Latin l e t t e r s f o r tensors ( l i n e a r transformations o f R3).

R'

into

Also:

C a I. b i '

a.b =

A * B = C A . .B IJ

ij'

AT

i s t h e transpose o f

A-T

= (AT)-17

A,

I = the identity tensor, a 8 b,

t h e t-. e n so r product o f

-

and

b,

is the tensor

a . b ., 1 3

w i t h components a A b = a 4 b

a

b 8 a,

Lin = t h e space o f a l l t e n s o r s , Lin'

= { F E L i n : d e t F > 01

Sym = [ F

t L i n : F = FT1

,

Orth+ = t h e space o f a l l proper orthogonal t e n s o r s . F i n a l l y , f o r a smooth ( i . e . , on a r e g i o n i n

R3,

div T

class

C')

tensor f i e l d

xcT(x)

i s t h e v e c t o r f i e l d w i t h canponents

239

ON THE NONLINEAR TIBORY O F E L A S T I C I T Y

aT..

(div T). = C

.

3 ax. J

3. ___Kinematics. 63

A body

p E 8

Points

is a compact, regular region in

are called material points. __ _____

R3.

A deformation of -.

~

n

fl

is a smooth one-to-one map

A motion is a

t E iR.

deformation for each x = r(p t) = rt(p)

at

= r

WJ

4

r: 9xIR + IR3

map

C2

r: 63

with

with

det

vr >

0.

rt = r(-,t) a

We use the following terminology:

is the place-occupied by

p

at __ time t;

t B) i

5 = {(x t):

x E R,,

t E W]

t)

is the trajectory;

is the veloc-iiy; is the spatial description of

p=ril (x) the velocitv: F(p,t) = vr(p,t)

is the deformation gradient.

We also use the notation summarized i n the following table:

63xR

domain gradient w.r.t. argument

first

divergence w.r.t. argument

first

derivative w.r.t. argumen t

second

V@ Div @

5 grad div

A

A

A‘

I t is also convenient to define the material time derivative

2 40

MORTON E .

12

GURTIN

A:

of a s p a t i a l f i e l d

Then

= v’

4.

+

(grad v)v.

Mass. A m a s s distribution f o r

t h e Bore1 s u b s e t s o f

8 )

B

i s a measure

m

(on

o f t h e form

i

po

with

po

a smooth, s t r i c t l y - p o s i t i v e

i s called the reference density;

C16 = n g ( p )

where

s c a l a r f i e l d on

R.

clearly,

i s the b a l l w i t h radius

6

centered at p,

n

V

and p

i s t h e Lebesgue volume measure on

i n a motion

where

r

x = r(p,t)

R’.

The d e n s i t y

i s t h e s p a t i a l f i e l d d e f i n e d by

R6

and

i s as a b o v e .

O f course,

conservation o f m a s s i s a t a c i t p a r t of t h i s d e f i n i t i o n . Proposition

-

(a)

p(x,t) det F(p,t) = po(p),

(b)

p’

+

x = rt(p);

d i v ( p v ) = 0.

5 . Stress. A s y s t e m of of:

f o r c e s for

8

i n a motion

r

consists

241

ON THE NONLINEAR THEORY OF ELASTICITY

(i) XI-

surface forces

s(n,x,t)

(ii)

x 3

s: {unit vectors)

with

-+ lR3

smooth;

9ody forces -

b: 3 + EL3

with

x-b(x,t)

continuous.

These fields are assumed to be consistent with balance of momentum which asserts that -___

f o r every part

with

nx

P

T,

63

and time

t.

the outward unit normal to

-

Cauchy's Theorem field

of

Here

apt

P t = rt(P) and

at

x.

There exists a symmetric spatial tensor

called the Cauchy stress, such that s(n,x,t) = ~ ( x , t ) n

and div T

+

b = p;

6. Constitutive _- Equation.

(equation of motion).

Change i n Observer.

Finite elasticity is based on the constitutive equation A

T = T(F) giving the stress at a material point gradient at

p

is known.

Here

p

when the deformation

MORTON E . GURTIN

242

is a smooth function; for convenience we have assumed that the material is homogeneous so that the response function . independent of the material point (r,T)

A pair

with

r

,.

T

is

p.

T = ?(vr)

a motion and

is

called an admissible process. A change in o b s e r v e r is a s m o o t h m a p

+ Q(t)x,t)

(x,t)l-(a(t) with

a(t) E R 3

and

Q(t) E Orth+

(R

4

-+

t E W.

for each

Axiom (invariance under a change in observer)

(r*,T*)

-

Given any

(r,T) and any change in frame (C), the

admissible process process

(c)

R4)

defined by r*(P,t) = a(t) + Q(t)r(P,t),

= Q(t)T(x,t)Q(t)T,

T*(x*,t)

x* = r*(p,t),

x = r(p,t)

is also admissible. Proposition

-

For every

F E Lin’

and

Q E Orth’,

Q+(F)Q~ = +(QF). Remark

-

(1)

It is not difficult to show that (I) holds if and

T = $(F)

only if the constitutive relation

T = F?(C)F T,

c

is of the form

= FTF.

This type of ”reduced constitutive relation”, while popular in the literature, seems to be of little use in studies of existence and uniqueness.

ON THE N O N L I m A H TIEORY O F E L A S T I C I T Y

7 . The P a r t i a l D i f f e r e n t i a l E q u a t i o n .

The c o n s t i t u t i v e r e l a t i o n and t h e e q u a t i o n o f motion combine t o g i v e t h e e q u a t i o n

+

d i v ?(Vr)

b = p;,

where

.

p det F = p

It i s important t o n o t e t h a t

div,

i s with r e s p e c t t o t h e p l a c e

x

t h e s p a t i a l divergence,

at.

in

Generally,

problems i n v o l v i n g s o l i d s 1 t h e deformed r e g i o n

Et

in i s not

known i n a d v a n c e , and for t h a t r e a s o n i t i s u s u a l l y more convenient

t o use t h e m a t e r i a l p o i n t

variable.

O f course,

one c a n c o n v e r t

p

a s independent div

t o an o p e r a t o r

i n v o l v i n g only d i f f e r e n t i a t i o n with r e s p e c t t o

p,

b u t then

t h e u n d e r l y i n g e q u a t i o n w i l l no l o n g e r b e i n d i v e r g e n c e form. F o r t h e above r e a s o n we i n t r o d u c e t h e Piola-Kirchhoff

stress S = ( d e t F)TFmT (considered a s a material f i e l d ) .

where

ap,

m

and

n

Then g i v e n any p a r t

a r e t h e outward u n i t normals t o

respectively, s o that

Sn

apt

63,

and

r e p r e s e n t s the surface f o r c e

measured p e r u n i t a r e a i n t h e u n d e f o r m e d

c o n f i g u r a t i o n . This

o b s e r v a t i o n , b a l a n c e of l i n e a r momentum, and t h e symmetry o f T

y i e l d the following A s opposed t o f l u f d s , where @ i s g e n e r a l l y known a p r i o r i , and where t h e s p a t i a l d e s c r i p t i o n i s u s u a l l y t h e m o s t appropriate.

244

MORTON E.

GURTIN

Proposition Div S

+

bo =

T

SFT = FS

where

bo = (det F ) b .

Remark

-

..

par,

,

Note that the above equations involve only material

fields and only material differentiation. I n view of the definition o f

relation for

S

T = +(F)

S,

the constitutive

is equivalent to a constitutive equation

of the f o r m S

= :(F)

with

5: and (I) and the relation

.S:

restrictions on

L i n + + Lin; SFT = FST

yield the following

Henceforth we neglect body forces and write

.

S;

S

for -

the underlying partial differential equation then takes

the form Div S(vr) = p 0 ? . Remark -

-

This is a partial differential equation i n divergence

form (recall that material field

Div

is with respect to

(p,t)t-r(p,t).

p)

for the

Note that, since

po

is the

reference density, we do not need balance o f mass; given a

245

ON THE NONLINEAR T H E O R Y OF ELASTICITY

r,

solution

p

if the density

computed using the expression

in

is needed it can be

r

p = po/det Vr.

The following table compares the salient features of the equation of elasticity with those o f

the Navier-Stokes

equations. ~~-

-

~

-

~

Navi er-S t okes

elasticity

~

Div S(Vr) = p

equations

-

v’+(grad v ) v = yAv grad T T , div v = 0

-

0

~-

restrictions on field

v a vector field, can live in a linear space

r one-to-one,

det vr > 0

__

S

nonlinearity

(grad v)v -

.

material properties _

__._

function

_

~

po

-

.

constitutive restrictions -

S,

__

?

- -~

W > O

-

8. The Elasticity Tensor. The derivative

A(F) (of

at

S

with respect to

F E Lin+;

to each tensor

A(F) H

= DS(F)

F) is called the elasticity tensor

is a linear transformation that assigns a tens or

A(F)H. When the reference is natural, that is when

s(1)

= 0,

(R) yield the following important restriction on

A(1):

246

MORTON E. GURTIN

Proposition

-

If the reference is natural, then A(I)W = 0, A(1)

W = 0

W.

for every skew tensor Remark .__

T

The above relations show that (when the reference is

natural)

A(I)

cannot -_ .

be positive definite;

in fact,

H.A(I)H = E*A(I)E, where E = z1 ( H + H T) is the symmetric part of

H.

It does, however, make sense to

talk about the positive definiteness of the restriction

The main open question of finite elasticity is: what physically acceptable restrictions should be placed on the response function

S

to insure successful mathematical

analysis of meaningful problems.

1

One possible restriction is the strong ellipticity condition: (a 4 b)*A(F)(a 1

8 b)

>

0

Generally, studies in partial differential equations center on investigating a given set of equations with a priori knowledge of the type of nonlinearity, etc. Note that here the thrust is different: one tries to deduce restrictions on S by investigating their consequences. Even negative results such as uniqueness everywhere (as we will see, we cannot expect to have unqualified uniqueness) are important, because they imply that the underlying restrictiomare not meaningful.

ON THE N O N L I N E A R THEORY O F E L A S T I C I T Y

f o r a l l nonzero t e n s o r p r o d u c t s

a 0 b.

2 47

T h i s assumption

renders the underlying p a r t i a l d i f f e r e n t i a l equations t o t a l l y h y p e r b o l i c and hence a p p r o p r i a t e f o r wave p r o p a g a t i o n s t u d i e s .

9. The Boundary-Value Problems o f E l a s t o s t a t i c s

.

W e now l i m i t our a t t e n t i o n t o t h e s t a t i c t h e o r y and consider the d i f f e r e n t i a l equation Div S ( V r )

= 0

supplemented by boundary c o n d i t i o n s of t h e form S(vr)n r where

a8 =

IJ

w i t h /J.

p r e s c r i b e d on d

t h e outward u n i t normal t o The f i e l d

p ,

p r e s c r i b e d on and

,

d i s j o i n t , and where

f

n

is

an.

S(Vr)n

i s c a l l e d the surface t r a c t i o n .

The s i m p l e s t example of a t r a c t i o n boundary c o n d i t i o n i s

dead

l o a d i n g i n which s(vr)n = s with

p k so(p)

a f u n c t i o n of

F o r many problems of

on

4

the m a t e r i a l p o i n t only. i n t e r e s t the prescribed

t r a c t i o n i s a f u n c t i o n of t h e d e f o r m a t i o n p r e s s u r e h a d i n g i n which each p l a c e

x

r.

An example i s

on t h e deformed

surface

r ( @ ) i s a c t e d on by a p r e s s u r e

terms o f

t h e Cauchy s t r e s s ) t h e t r a c t i o n boundary c o n d i t i o n

lT(x).

Thus ( i n

has t h e form

T(x)m = -n(x)m where

m

for

x E r(p),

i s t h e outward u n i t normal t o

ar(6).

This c o n d i m ,

248

MORTON E. GURTIN

in turn, is equivalent to (cf. the definition of the PiolaKirchhoff stress) S(Vr)n

= -(det Vr)Tf(r)Vr-Tn

(PI

on 0 .

It is not difficult t o verify that (de t Vr ) vr-Tn depends only on the tangential gradient

vTr

of

r

on p ;

thus the boundary condition (P) can be written more compactly as follows: S(vr)n

= so(r,v7r)

on

0 .

We now give some counterexamples which demonstrate, quite vividly, that uniqueness general is not to be expect-_ _ _ i n- -~

fi.

We assume i n examples ( A b ) ,

(Ba), and (Ca) that the

reference is natural. A.

The traction problem (a)

(D

= 38).

F o r the traction problem with dead loading a

translation of a solution yields another solution. (b)

(Armanni) Consider a thin hemispherical shell with

zero surface tractions.

Then

r = identity is a solution.

But there should be a second solution consisting o f the everted shell. (c)

(Ericksen)

Consider a rod subject to equal and

opposite tractions on its ends.

This type of loading should result in two types of solutions as shown below.

ON T H E NONLINEAR TIIEORY OF ELASTICITY

249

-I (d)

Also,

i n problem (c) w e would expect “buckled

solutions” of the form

provided the loads are sufficiently large.

B.

The displacement problem (a)

(John)

(jb =

as).

Consider a spherical shell with boundary

an.

condition

r(p) = p

identity.

But there are other deformations which leave the

on

Orie solution, o f course, is the

boundary unmoved, but deform the interior.

Indeed, consider

the deformation caused by a rotation of the inner boundary by a n integral multiple of

ZTI

about an axis through the center

of the sphere.

C.

The genuine mixed problem (a)

( 0

f 0,

p

f 0)

Consider a finite cylindrical rod with sides traction

free and ends rigidly fixed.

One solution is the identity.

Another corresponds to the deformation caused by a rotation ( i n its plane) of one of the ends b y a n integral multiple of ~ T T

about an axis through its center. (b)

We would also expect a situation similar to ( A d ) for

a rod which is loaded at one end, but which has the other end

MORTON E . GURTIN

2 50

fixed. Another difficulty intrinsic to finite elasticity A s noted before, to be meaning-

concerns the solution space. ful a solution

r: @ -+ R3

must b e

(a)

one-to-one, and have

(b)

det Vr > 0 . Condition (b) is severe and makes the theory quite

difficult.

Indeed, the collection of fields satisfying (b)

is not convex;

8

as a matter of fact, f o r

a torus this

collection can have an infinite number of connected components, none of which is convex (cf. Antman [ 3 3 ) .

Further, it is

usually not possible to extend the domain of to tensors

F

det F = 0,

with

becomes infinite as

since

S(F)

S

continuously

generally

det F + 0 .

Condition (a) is even more severe, since it is global.

O f course, one can drop this restriction provided

one is willing to accept solutions of the form:

R

An interesting question i n global analysis is: (b)

+

what

3

(a) 7

F o r the displacement problem an answer is furnished by the

fo 11owing

ON THE N O N L I N E A R T m O R Y OF ELASTICITY

-

Theorem (Meisters, Olech)'

r: 63

Let

4

R3

be smooth, and

suppos e that

det vr > 0, and

(i)

r

(ii) Then

r

I as

is one-to-one.

is one-to-one.

10. Variational Characterization o f the Problem.

The material is hyperelastic if there exists a

u: B +

stored energy function

such that

R

I n this case the mixed problem can be characterized by the principle o f minimum potential energy, at least f o r dead loadingc.

Thus consider the boundary conditions S(vr)n

= s

on P ,

r = r

On

0

with

so

E Ll(n ),

ro E

Ll(p);

7

and let

Def = the collection o f all deformations o f Kin = (r E Def: rIp = ro]

8,

.

Then the mixed problem is formally equivalent to the variational principle: 'Meisters, G . H . and C. a classical theorem on 6 3- 80 ( 1 9 6 3 ) . Here 363. separating set o f ~ 3

minimize 3 Olech, Locally one-to-one mappings and 30, Schlicht function, Duke Math. J. is assumed to b e an irreducible .

20r more generally when the loading is conservative.

3Ball [h]

has established the existence o f minimizers f o r a particular class o f stored energy functions.

MORTON E .

252

m(r) =

i

u(vr)

-

GURTIN

10

s *r

over

Two necessary c o n d i t i o n s f o r a deformation

4

Kin.

n

0

r E Kin

t o render

a l o c a l minimum a r e :

(i) (ii)

J

vu.A(Vr)Vu z

for all

0

B ( a 8 b).A(Vr)(a 8 b ) 2 0

u

E

Var,

f o r a l l tensor products

a 8 b. Here

and l o c a l means w i t h r e s p e c t t o t h e L,

norm of

t o p o l o g y g e n e r a t e d by t h e

t h e deformation g r a d i e n t .

C o n d i t i o n ( i i )i s

u s u a l l y r e f e r r e d t o as t h e Legendre-Hadamard c o n d i t i o n ; a theorem o f Hadamard t e l l s u s t h a t

11.

( i ) implies (ii).

U -n i q u e n e s s .

To a v o i d c o m p l i c a t i o n s w e assume t h a t

30.

( r e l a t i v e l y ) open s u b s e t o f t h e ( p u r e ) t r a c t i o n problem.

This,

pf

$I

is a

of course, rules out

We a l s o r e s t r i c t o u r a t t e n t i o n

t o dead l o a d i n g .

I t i s n o t d i f f i c u l t t o show t h a t a of

C2

solution

t h e mixed problem s a t i s f i e s t h e i d e n t i t y

Our r e s u l t s r e g a r d i n g u n i q u e n e s s p e r t a i n t o weak s o l u t i o n s ; t h a t i s , deformations A deformation

r E Kin r

E

Kin

that satisfy (W). is internally stable i f

r

ON THE N O N L I N E A R TIEORY OF ELASTICITY

253

B

a __ set

Pl c Kin

is internally stable i f every

f E

K

has this

property. The next result',

which is a simple consequence of

the above definition and the mean-value theorem, shows that stability implies uniqueness.

-

Theorem

X

C

Uniqueness holds i n any convex, internally stable

Kin.

Corollary fi)

(it)

-

Assume that the reference is natural and

A(I)ISym

P

= a63

elliptic.

for some

is positive definite,

or

(displacement problem) and

A(1)

is strongly

Then uniqueness holds i n

6 7 0.

The above theorem can be extended to situations

involving non-dead loads, but the extension i s nontrivial.

In particular, for loading of the form show that if

X c Kin

s0(r,vTr)

one can

is convex and uniformly internally

stable in the sense that vu-A(vr)Vu

2

xllull * H

for all

u E Var

and

r E

X,

(0)

and if the first two derivatives of s

0

are sufficiently si'iall

on K , t h e n uniqueness holds i n K.

'Gurtin, M.E. and S.J. Spector, On uniqueness i n finite elasticity. Arch. Rational Mech. Anal. Forthcoming. 3

'Spector, S.J., On uniqueness in finite elasticity with general loading. Forthcoming.

G.M. de La Penha. L.A. Medeiros (eds.) Contenporary Devel opmnts i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations @North-Holland P u b l i s h i n g Company (1978)

CONSTITUTIVE EQUATIONS AND FREE SURFACES

DANIEL D. JOSEPH Department of Aerospace Engineering and Mechanics University of Minnesota, Minneapolis

Abs t ra c t The general theory of perturbations o f rigid body motions of simple fluids with applications to free surface problems is discussed.

The

general theory is utilized to explain phenomena exhibited i n the movie "Novel Weissenberg Effectst1by G . S .

Beavers and D.D. Joseph.

Introduction. The ideas behind the simultaneous perturbation of the domain and the constitutive equation may be explained without the extra, and largcly incidental, mathematical complications which follow from analysis of the nasty equations which govern rheologically complicated materials.

In a

simpler setting, we shall first consider a model problem in which we show that the distortion of a free surface due to motion may be expressed in terms of unknown rheological constants.

In the model problem we perturb the "state of

rest'?. A rich theory of perturbations of the 'Irest state" of real viscoelastic fluids following along lines laid out in this lecture has already been given by Joseph and Beavers

CONSTITUTIVE E Q U A T I O N S AND FREE SURFACES

(1977).

Many of

255

t h e s i m p l i f y i n g f e a t u r e s o f t h e t h e o r y of

P e r t u r b a t i o n s of t h e r e s t s t a t e o f a v i s c o e l a s t i c f l u i d a r e

a l s o p r e s e n t i n t h e y e t r i c h e r t h e o r y of p e r t u r b a t i o n s o f s t a t e s o f r i g i d motion of a v i s c o e l a s t i c f l u i d .

In fact,

s t a t e s o f r i g i d motions o f a v i s c o e l a s t i c f l u i d a r e g e n e r a l i z ed r e s t s t a t e s i n t h e s e n s e t h a t t h e Cauchy s t r a i n the extra s t r e s s

5

b o t h v a n i s h on r i g i d m o t i o n s .

G(s)

and

Moreover,

i n t h e t h e o r y g i v e n h e r e , and by J o s e p h ( 1 9 7 7 ) , t h e same m a t e r i a l f u n c t i o n s s u f f i c e t o d e s c r i b e t h e m o t i o n s which perturb a l l s t a t e s of steady r i g i d rotation including the rest s t a t e i n which t h e r e i s no r o t a t i o n .

I t i s t h e r e f o r e p o s s i b l e t o s e p a r a t e t h e problem o f f r e e s u r f a c e s on v i s c o e l a s t i c f l u i d s which a r e c l o s e t o steady r i g i d motions i n t o t w o p a r t s .

I n the f i r s t part

( $ 1 )we

c o n s i d e r t h e model problem i n which t h e i d e a s i n v o l v e d i n t h e s i m u l t a n e o u s p e r t u r b a t i o n o f t h e domain and c o n s t i t u t i v e e q u a t i o n a r e exposed i n a s i m p l e c o n t e x t .

I n t h e second p a r t ,

( $ 2 - 7 ) we p e r t u r b s t a t e s of s t e a d y r i g i d r o t a t i o n o f r e a l v i s c o e l a s t i c f l u i d s , b u t l e a v e t h e domain f i x e d . domain i s n o t f i x e d t h e c a n o n i c a l forms of

When t h e

the s t r e s s ( $ 6 )

and t h e e q u a t i o n s o f motion ( $ 7 ) a r e unchanged and t h e e q u a t i o n s which h o l d on t h e f r e e boundary may b e e a s i l y d e r i v e d by t h e methods u s e d i n t h e model problem ( $ 7 ) .

1. A model problem f o r domain p e r t u r b a t i o n s of

the r e s t state.

We s u p p o s e t h a t t h e r e i s a n a n a l y t i c f u n c t i o n

5(@) @ = 0

which i s n o t known e x c e p t t h a t i n a neighborhood o f

i t has a Taylor s e r i e s

256

DANIEL D.

1

a ( @ )= a = 5

where

(0)

a@

2

b = 5

and

1

+

a0

JOSEPH

bo3

00@

...

+

,

(1.1)

We may t h i n k t h a t

(0).

5 ( @ ) i s representative of the nonlinear p a r t of the stress Our i d e a i s t o f i n d t h e T a y l o r

i n some f i c t i t i o u s m a t e r i a l .

c o e f € i c i e n t s f o r (1.1) by m e a s u r i n g t h e f r e e s u r f a c e i n d u c e d by a dynamical p r o c e s s

v 20 + 5 ( @ )

F(E) =

= 0

in

b6,

(1.2)

subject t o the condition that

b6

where

i s a bounded r e g i o n o f s p a c e d e p e n d i n g on a

parameter

6

parameter

C .

and

g(x,E:)

a r e g i v e n d a t a d e p e n d i n g on a

I t i s i n s t r u c t i v e t o c a r r y o u t our a n a l y s i s i n e a s y stages.

F i r s t we p e r t u r b

leaving

6 ,

6

f i x e d and assume

that @(IE,E

1

1

=

@a(d

(1.4)

gn(d = 0,

(1.6)

c fG l 0

where,

in

6' 00 +

a n d , on

'

No0)

= 0,

ab6; Gn

where

gn(x)

we c o u l d s o l v e etc.

Y

i s nth

=

on(&)

-

d e r i v a t i v e of

( 1 . 5 ) and ( 1 . 6 ) for

g(&,s)

Go

at

c =

0.

we c o u l d f i n d

as t h e s o l u t i o n of l i n e a r b o u n d a r y v a l u e p r o b l e m s .

If

5,,@,, It

CONSTITUTIVE EQIJATIONS AND FREE

would b e h a r d t o s o l v e ( l . 5 ) l ,

i t i s nonlinear. we d o n ' t know n e i g h b o r h o o d of

3(0,),

even i f we knew

But we c a n n o t s o l v e ( 1 . 5 ) 1

3(o0)

2 57

SURFACES

bccausc.

a t all b e c a u s e

e x c e p t a s a power s e r i e s i n a s m a l l

zero.

The " r e s t s t a t e " i s now d e f i n e d as t h e s t a t e f o r

go(xlab

which

6

= 0.

assumption t h a t

(1.5)1

@,(x)(~,, =

Then

0.

W e couple t h i s d e f i n i t i o n with t h e has @

6

0

110

(x) -

E

solutions

b6

in

0

rijo

#

0

when @ o ( x _ ) - O

and, r e p l a c i n g

we g e t

etc.

These p r o b l e m s a r e l i n e a r and e a s y t o s o l v e even rihen

a

i s unknown ( i n f a c t ,

, @,

a r e independent o f

o2

= @21

+

a @ 2 2 , where

021

and

S o we c a n u s e t h e s o l u t i o n

a).

which p e r t u r b s t h e r e s t s t a t e t o f i n d

a

a n d , w i t h more work,

we c a n g e t e x p r e s s i o n s which i n v o l v e t h e o t h e r T a y l o r coefficients of

(1.1) a s w e l l .

Now suppose t h a t suppose f u r t h e r t h a t

of

convenience i s ,

symmetry. in

b6

6

i s f i x e d and

6

varies.

And

b o i s s o m e c o n v e n i e n t domain whose point

say, that

Ir0

h a s a h i g h degree o f

W e a r e g o i n g t o t r y t o s o l v e t h e dynamic problem

a s a s e r i e s whose c o e f f i c i e n t s c a n b e d e t e r m i n e d from

b o u n d a r y v a l u e problems posed on t h e symmetric domain F i r s t we map

Ir6

into

bo

t o one mapping, which i s a n a l y t i c i n p o i n t s on

ab6

into

all o :

b0.

w i t h a n i n v e r t i b l e one

6

and which t a k e s

D A N I E L D.

258

x_ = ~ ( ,0) x

(identity),

-1 = x_ (5,s)

(inverse),

-0

xo alr Let Ir6

H(x,6)

JOSEPH

alr o .

(1.8)

be any function defined i n the family of domains

and introduce the notation:

and

(1.9) etc.

It follows from the equations that

when

x

E b6,

x

-0

E lro,

and that (1.10)

W e may easily establish by mathematical induction, using the

the connection formulas, that

F(n)(so) = For example,

+'

F( 0 ) ( 5 , ) = 0

xC1] -VF(0)(~o)

in

= F(I)(x0) = 0.

0

b0

in

bo*

and

r (x,)

F 11

(1.11) = F(

' ) (&,)

+

It is a bit more complicat-

259

C O N S T I T U T I V E E Q U A T I O N S AND FREE SURFACES

ed a t t h e boundary. on

aho

Since

G(6) = 0

L mJ ( 5 , )

and t a n g e n t i a l d e r i v a t i v e s o f

G

C ml

must

It f o l l o w s

alro

t h a t on

n-v =

bo

on

v a n i s h b u t normal d e r i v a t i v e s need n o t v a n i s h .

n_

where

= 0

&his, G

on

i s t h e outward normal t o

[: 11 .n_ h g t vn = 5

and

a/an. W e may s e e k t h e s o l u t i o n o f

as a s e r i e s d e f i n e d i n

(1.2)

and

(1.3) i n

b5

b0

(1.13)

(1.14) etc.,

and on

The problems can't

ab,

(1.14),

( 1 . 1 5 ) a r e l i k e ( 1 . 5 ) and ( 1 . 6 ) .

s o l v e them b e c a u s e

W e

5 ( @ ) i s g i v e n only as a Taylor

s e r i e s (1.1) w i t h unknown c o e f f i c i e n t s and a n unknown c i r c l e of

convergence. The " r e s t s t a t e " f o r t h e domain p e r t u r b a t i o n ,

t h e " r e s t s t a t e " f o r t h e p e r t u r b a t i o n of may b e d e f i n e d b y t h e c o n d i t i o n condition implies t h a t Q,(O) P

0

in

b,

and

0 '0'

g[: 0 3

= @( o ) =

t h e boundary d a t a

(x,)] o

like

=

0.

This

0

on

aho,

hence,

2 60

D A N I E L D.

JOSEPH

(1.16)

The linear problems (1.16), (1.17) and higher order

etc.

problems are solvable and not too hard to actually solve, even when

a

is unknown.

In our rheological problems the boundary data

(e

in

our first example) pe’rturb the boundary (6 in our second example).

S o we may put

E

= 6

and construct a simple

example of a domain perturbation of the rest state with a free surface.

(c)

By a free surface we understand that there is a one

parameter family of domains

bc

which are unknown

Supposing now that o u r dynamical process (1.1) and (1.2 in

Uc

we might expect solutions i n each and every

responding to some possibly small

e

Ire

hold co r -

interval of the origin.

But no, this will not be possible because in addition to (1.2) we pose an additional boundary condition, which is analogous to, but much simpler than, the condition that the jump i n the normal component of stress is balanced by surface tension times mean curvature.

Because we have this extra condition

we can’t solve (1.1) and (1.2) i n every condition can be satisfied only when

kc

-

Ire ’

the extra

is properly chosen.

A s an example of the foregoing consider the two

dimensional problem specified in polar coordinates figure 1 where the boundary data correspond to a rest state ical process

@(r,B,c)

g(8,O)

g(8,c)

in

are given and

= g(O’(8)

and the function

(r,e)

= 0.

f(e,c)

The dynamwhich gives

261

C O N S T I T U T I V E E Q U A T I O N S AND FREE S U R F A C E S

r = 1 -+ f ( e , c )

t h e boundary

of

be

a r e unknown.

5(@);

t h e r e a d e r t h a t o u r aim i s t o show how t o f i n d

is,

W e remind that

t h e T a y l o r c o e f f i c i e n t s i n (1.1) b y ( f i c t i t i o u s ) e x p e r i -

m e n t a l measurements o f

r = 1

t h e (made u p ) b o u n d a r y

+

f(8,g).

(1.18)

Fig.

1.

Model of a f r e e s u r f a c e problem

F i r s t we s o l v e ( 1 . 1 8 ) when

v2@

z ( @ )=

+

0

in

bo.

bo

with

@ ( r y e y o=)

has

I# E 0

that

the r e f e r e n c e c o n f i g u r a t i o n

r

= 1.

where

in

Then,

since

W e seek t h e s o l u t i o n o€

f

[ 01 (8,)

= 0[

01 ( r o , e o )

r e l a t e d by a s h i f t i n g map

e = 0

= 0

Z(0)

bo

0

= 0,

and we f i n d t h a t on

r = 1 + f(8,o)

f(8,o) = 0

i s the unit circle

( 1 . 1 8 ) i n powers o f

and

9

and

bo

6

are

SO

2 62

D A N I E L D.

JOSEPH

h a v i n g a l l t h e p r o p e r t i e s r e q u i r e d of map t h e d e f o r m a t i o n o f

bo

Note t h a t f o r any f u n c t i o n f(n'

(8)

c

f E n 3( 8 ) .

is

(1.8).

F o r the shifting

a l o n g r a y s and

f(e)

of

8 = Bo

a l o n e , we have

Using t h e c o n n e c t i o n f o r m u l a s ( 1 . 9 )

we find

that

0[ 11

(ropeo) =

@

( 1) (ro,eo)

and

b o , r o = 1 we h a v e , from ( 1 . 1 8 ) 3 t h a t

On t h e boundary of

- f c 11 ( e o )

3m(0)@(1'(l,eo) that

f

c 11 ( 8 , )

o(')(ro,e0)

Fig.

= 0. and

2.

= 0.

Since

3@(0) = 0 ,

we f i n d

The boundary v a l u e problems s a t i s f i e d by

@( 2 )

(ropeo)

a r e g i v e n i n f i g u r e s 2 and 3.

The problem s a t i s f i e d by

0

( 1)

(ro,eo)

263

CONSTITUTIVE E Q U A T I O N S AND FREE SURFACES

c 21 ( e o )

f

3 that

We f i n d f r o m f i g u r e

These p r o b l e m s a r e e a s y t o s o l v e .

= a @( 1) 2( 1 9 6 o ) 9

where

@( 1)

(ro,eo)

f'igure

2.

I t f o l l o w s t h a t t h e f i r s t approximation t o the

b,

shape o f

i s t h e s o l u t i o n o f t h e problem shown i n

i s g i v e n by

2 r = I +

f(e,e)

= 1

+

a@(')

The n e x t a p p r o x i m a t i o n depends on

b

(i,e)s2+

qE3).

a s w e l l as

t h e r e f o r e deduce t h e v a l u e s o f d e r i v a t i v e s

of

by m o n i t o r i n g t h e c h a n g e s i n t h e s h a p e o f

a.

3(@)

We may at

@ = O

is

when

n ear t o zero.

2. C o n s t i t u t i v e e x p r e s s i o n s which p e r t u r b t h e r e s t s t a t e of -_

a simple f l u i d . ~

I n our r h e o l o g i c a l s t u d i e s t h e r o l e o f t h e nonlinear function

5 ( @ ) i s assumed by t h e n o n l i n e a r f u n c t i o n a l -

g i v i n g t h e c o n s t i t u t i v e l y determined p a r t of O S an i n c o m p r e s s i b l e f l u i d .

domain of

s = t

-

T

-3

the s t r e s s

The argument f u n c t i o n s i n t h e

a r e symmetric t e n s o r v a l u e d f u n c t i o n s of

f o r fixed

t

aiid

r,

called histories,

which a r e d e f i n e d o n t h e r e l a t i v e d e f o r m a t i o n t e n s o r

= V,X~(X,T)

of

T

where

the p a r t i c l e

&

Xt(x,T)

= {(&,T)

F (T)

-t

=

i s the position vector

which i s p r e s e n t l y a t

&.

Eq.

(2.1),

which assumes t h a t t h e s t r e s s d e p e n d s o n l y o n t h e f i r s t spatial

2 64

D A N I E L D.

JOSEPH

g r a d i e n t of t h e d e f o r m a t i o n , i s s t i l l t o o g e n e r a l t o be u s e d

t o s o l v e t h e problems which l e a d t o u n d e r s t a n d i n g how v i s c o e l a s t i c f l u i d s respond

t o applied forces.

I t i s w e l l known t h a t t h e e x t r a s t r e s s v a n i s h e s on n(0) = 0.

the zero h i s t o r y expansions o f

~(G(S))

I t i s t h e n n a t u r a l t o seek f u n c t i o n a l e x p a n s i o n s on

i n terms of

the r e s t h i s t o r y

gCo;G(s)l

3[G(s)l =

,G_(s)l

+ a,Co;G_(s)

3;2r:O;G(s),G;(s),G(s)l

+

+

9

. . a

where

i s an

n

linear form.

W e s h a l l assume ( G r e e n and R i v l i n

( 1 9 5 7 ) , Coleman-No11 ( 1 9 6 1 ) ~ P i p k i n ( 1 9 6 4 ) t h a t t h e s e n - l i n e a r forms

may b e r e p r e s e n t e d b y i t e r a t e d i n t e g r a l s o f t h e f o r m

a

'i j k k

. ..mv'

Gkk(sl)

...

1' 2 '

'

YSrl

1

G m v ( s n ) d s 1d s 2

. ..

d sn *

It i s v e r y e a s y t o f i n d t h e i s o t r o p i c forms o f t h e s e i n t e g r a l s f r o m invariance theory f o r a single tensor

94.7 i n Joseph ( 1 9 7 6 ) ) ,

a(")(C+(s))

G ( s ) ( s e e Exercise

If we i n t r o d u c e t h e n o t a t i o n

f o r t h e p a r t i a l sum o f

(2.2)

after

n

terms i t

(2.4)

CONSTITUTIVE E Q U A T I O N S AfJD FREE SURFACES

m

m

(n

So we may e x p e c t

i n which

G_(s)

stresses

a(").

l i n e a r forms i n G _ ( s i ) ) d sl...dsn

t h a t t h e s t r e s s i n any motion c l o s e t o one

= 0

can he e x p r e s s e d i n t e r m s of

But t h e s t r e s s e s

3(n)

t h e FrLchet

a r e n o t i n t h e form which

I c a l l canonical f o r perturbation o f the zero h i s t o r y .

The

c a n o n i c a l forms a r e t h e forms which t h e s t r e s s and e q u a t i o n s o f m o t i o n t a k e when d a t a g i v i n g r i s e t o s t e a d y r o t a t i o n a r e perturbed.

W - e d o n o t a l l o w k i n e m a t i c s which a r e i n c o n s i s t e n t ~~

~

F o r example,

w i t_ h __ t h e equ a t i o n s o f m.otion. _ -

though i t i s well-

known and e a s y t o d e m o n s t r a t e t h a t o n a l l r i g i d body m o t i o n s

G_(s) = 0,

only t h e s t e a d y r i g i d motions a r e compatible with

t h e equations.

R e l a t i v e t o a c o o r d i n a t e system t r a n s l a t i n g

w i t h t h e t r a n s l a t i o n a l v e l o c i t y o f t h e r i g i d body we may assume t h a t body a l s o r o t a t e s r i g i d l y w i t h v e l o c i t y Then

= 0,

3[0] = 0

A

5.

and t h e e q u a t i o n s o f motion

0

a r e s o l v a b l e i f and o n l y i f

a

A

x

is a gradient;

that is,

0

if

= 0.

S o we a r e r e s t r i c t e d t o p e r t u r b a t i o n s of r i g i d

r o t a t i o n s with constant

g.

T o f i n d t h e c a n o n i c a l f o r m s , t h e f o r m s which t h e s t r e s s and t h e e q u a t i o n s o f m o t i o n t a k e when d a t a g i v i n g r i s e

t o steady r i g i d rotation are perturbed, i t w i l l suffice t o i m a g i n e t h a t for ~ E a l j ( t ) , where by f l u i d ,

lj(t)

i s t h e r e g i o n occupied

t h e p r e s c r i b e d boundary v e l - o c i t y

266

D A N I E L D.

U(,li,t,€) =

61

A

X_

+

JOSEPH

Ff(x_,t),

VtER,

is a steady rigid rotation plus an arbitrary part proportional to

Now we suppose that the solutions o f all the govern-

11.

ing equations depend on

6

through the prescribed data and

that they may be differentiated a certain number of times at 6

= 0.

In the best case we would have analytic solutions and

convergent power series in

E .

In less good conditions we

suppose that some low order partial sums are asymptotic to true solutions.

In either event we must identify the boundary

value problems which govern the derivatives o f the solution at

E

= 0

and we call the stress and equations f o r these

derivatives canonical; canonical in the sense that the derivatives are independent of

E.

This natural method of

doing perturbations requires that we consider only those forms of the stress which are compatible with the solutions o f the equations, s o after all is done we get a good theory with which we can actually compute solutions to problems.

For

example, we have already noted the only rigid body rotations compatible with the equations of motion are steady. The canonical forms o f the stress and the equations o f motion are easiest to understand by actually deriving them.

We shall find that at each stage of the perturbation we shall need to ~ o l v efour linear partial differential equations f o r three components of velocity and a reaction pressure, as in the Navier-Stokes equations.

The strain history comes in as

an after-thought after the velocities are computed.

CONSTITUTIVE E Q U A T I O N S AMD FREE SURFACES

2 67

3. Kinematics f o r p e r t u r b a t i o n s .

Since

we h a v e , a s s u m i n g t h e p a r t i c l e

label

1

5 = Ijt(X,t,C i s independent o f

( n)

ax_t a T Moreover, u s i n g

n = 0,1,2,

that f o r

E ,

=

(g,T)

(3.4)

E(n)(~,~).

( 3 . 1 ) , (3.4) and ( 3 . 2 ) ,

xt(

0)

...,

(litt)

-p t( x _ , t )

=

(3.5)

we f i n d t h a t

x,

= 0

and

i-@ ( X , t ) The f u n c t i o n

,.,

E(&,T)

=

p ( x , t ) .

i s an a u x i l i a r y f u n c t i o n used t o

f a c i l i t a t e o u r c o m p u t a t i o n of p a r t i c l e p a t h s . To s i m p l i f y n o t a t i o n s , we d e f i n e

X(n) and

(TI

sin)

(_x,T),

n=1,2,.

..

2 68

D A N I E L D.

JOSEPH

Then, using (3.3) and the chain rule, we find that

where

The functions

z ( ~ ) ( T )

may be computed by integrating (3.10)

subject to the conditions (3.6) and (3.7). We turn next to the computation of derivatives of the strain tensor.

From the definition of the relative

deformation gradient given i n $2, we find, u s i n g (3.1) and

( 3 . 9 ) that

where

Then

where

and

2 69

CONSTITUTIVE EQUATIONS AND FREE SURFACES

4. Functional derivatives

the stress and the equations

of

governing the of ~ special _ ~ perturbation ~ _. __ _ motions _ ~with

-_

arbitrary motions.

Suppose that

G(s,E)

is the series given by ( 3 . 1 2 ) .

This series may be assumed to induce a functional expansion of the stress in powers of

Apart from a factorial,

6

(see Joseph, 1976; p.197):

3n is a functional derivative,

typically a Fr6chet derivative, evaluated on the history

g")(s)

of the special solution,

The linear arguments of

these derivatives, those following the vertical bar, are to be determined sequentially by solving the perturbation equations of motion which have yet to be specified.

The

functional derivatives given i n ( 4.1) are still too generally specified to be useful in the solution of problems. However, the first Frgchet derivative

z1[G(O)

assumed to b e in integral form when 2

Lh(O,m)

1 *]

may be

G _ ( s , c ) lies in a

Hilbert space whose scalar product is defined by a n

integral with a weight

h(s),

h(s)

-+

0

as

s

-+

-.

Such a

representation may be justified by appeal to the representation theorem of F. Riesz. Identifying independent powers of

6

i n the

270

D A N I E L D.

expansion o f

JOSEPH

t h e e q u a t i o n s o f m o t i o n , we may i d e n t i f y a n The z e r o t h

ordered sequence of p e r t u r b a t i o n problems. problem i s d e f i n e d by ( 5 . 1 ) , we f i n d t h a t i n

A t f i r s t order,

t > 0

b(t),

+

( 5 . 2 ) and ( 5 . 3 ) .

order

5 [ G ( O) -1 -

v and (7

-

) =

i'

The h i s t o r y of t h e v e l o c i t y

- (l) U

(5 ( 7 ) ,'r

Since

)

_F

i s prescribed i n

l~

(T

),

i s the g r a d i e n t of

T

S

(4.6)

0.

, (VI.2),

-

with

(4.2),

(4.3)

and

( 4 . 4 ) may b e viewed s e v e n l i n e a r e q u a t i o n s

i n t h e s e v e n unknown f u n c t i o n s

X(-t" ( ? , ' r ) ,

C(l)(z,t)

and

d l ) ( 5 ,t ) . A s i m i l a r l i n e a r problem f o r

(n 2

2)

a r i s e s at higher orders.

, -(n)

,

U(n) -

and p ( n)

I f t h e s e problems a r e

s o l v a b l e , t h e y a r e s e q u e n t i a l l y s o l v a b l e and t h e m o t i o n and

,

27 1

CONSTITUTIVE E Q U A T I O T J S A N D FREE SURFACES

s t r a i n h i s t o r y may b e g e n e r a t e d a s power s e r i e s . p e r t u r b a t i o n problems, integral,

1 61

S,[:(i(O)

with

These linear

r e p r e s e n t e d by a n

a r e not t o o g e n e r a l f o r mathematical s t u d i e s o f

e x i s t e n c e and u n i q u e n e s s .

5 . K i n e m a t-_ ics of a r b i t r a r y .m- o tio n s p e r t u r bi n g s- t e a d y r i g i d ~~

r o t a t i o n s of a simple f l u i d . ~~

N o w I am g o i n g t o d e r i v e a n a l g o r i t h m f o r computing

I want

m o t i o n s which p e r t u r b s t e a d y r i g i d r o t a t i o n s . solutions of

t h e b a s i c e q u a t i o n s for which

E‘O’ ( S

R i g i d body m o t i o n s have

)

I

but,

0

G ( O ) ( s ) z 0. i n general, 0

motions w i l l n o t s a t i s f y the e q u a t i o n s because conservative (see ( 2 . 6 ) ) .

I, therefore,

-m

<

< t

T

&=

5

0

i s not and p u t

= c-2 A S(l-1

U -( O ) ( 5 ( T ) , T )

for

set

A

such

a t a l l points i n

l j

(7

).

(5.1) Then

Z [ G-( O ’

(s)]

0

and p(O)(r,t) a t each The p a t h

x_

E

b(t)

xio’ -

+

2- 1~

xi2

r\

= const

and a t e a c h and e v e r y i n s t a n t

(X,T

)

z

S(T)

for

‘r

S

t

<

( t )= 5.

(5.2) t >

-m.

i s o b t a i n e d by

integrat i n e

5

-’T

= 61 A

C(T), -

Without l o s i n g g e n e r a l i t y , normal b a s i s

el,

g 2 , E~

such t h a t

Ri s a constant vector.

we c h o o s e a f i x e d o r t h o -

=

“7n

(5.3)

Then t h e p a r t i c l e p a t h i s g i v e n by

JOSEPH

DANIEL D.

g(ns)

where

is the unique orthogonal tensor rotating the

orthonormal basis is,

$,(ns),

e = Q(ns).&(Rs). -i

$,(ns),

Qij(ns)

ei

=

=

-

c o s 62s

sin 0s

ns

c o s 62s

sin

It follows from (5.4) that

3")

E

0.

z2, e3;

that

Qij(Qs)gj

0

and

g19

~ l ~ ~ - ( R s ) . ~that j ; is,

[Qij(ns)]

Hence,

into

Relative to the fixed basis,

Q(ns) where

g3

0

(5.5) 1

27 3

CONSTITUTIVE EQUATIONS AND FREE SURFACES

It follows that

The most natural measure of deformation for rotating simple fluids is the time derivative o f the Cauchy strain

We shall need the following expansion formula for

where the

3(n)

L(n'

=

are defined in terms

v[c(n)(x,7.)

Equations (5.8) show that & L n

and on

4 X,(Z,T)

-

&("'

for

A

Q(ns),

X( n ) ~ , n

depends o n

2

(')

and

1.

~ " ) ( $ ( T ) , T )

(5.12) for

k < n. In fact,

(5.13) and for

re2

To prove the expansion formula (5.11)*, that (5.10) reduces to (5.11)1. consequence o f the identity

we must first s h o w

This reduction is a direct

27 4

D A N I E L D.

''"''"'

JOSEPH

( 5 * 1 3 ) and ( 5 . 1 4 ) , we expand

F ( s , c ) = Q _ ( n s )+

c E'E"'

and E x , using ( 5 . 9 ) and (5.8), and collect the 1 coefficients o f independent powers o f E i n the induced expansion o f ( 5 . 1 1 ) . Some further transformations of the tensors are used in the analysis.

,(n)

(s)

These transformations are motivat-

ed by the fact that the perturbation problems to be derived lead to the sequential determination of the velocity coefficients

,(n)

( g ( )~,T

)

whose natural arguments are the

Components of the rotating vector now that w h e n b(t),

T = t,

we find easily that

Noting next that

we find that

and

where

5 = x

and

xio)

( 5 , ~ =)

div

g(n)(,,t)

S(T).

= 0

Noting in

CONSTITUTIVE EQUATIONS AND FREE SURFACES

275

and

Using ( 5 . 8 ) , we find that

and

where

and

L.0.t.

= lower order terms.

6. Canonical forms for the stress.

My constitutive hypothesis is that the FrGchet derivatives of ed by integrals.

Z[G_(s)] -

o n the zero history can be represent-

I also assume that kernels in these

integrals vanish at a rate sufficient to justify integrating by parts; for example,

DANIEL D.

27 6

JOSEPH

and

Explicit expressions for Beavers (1977). duce

B(s) -

g3

and

&

are given by Joseph 8s

This integration by parts allows us to intro-

= -dC_(s)/ds

as the fundamental measure of defor-

mation and leads ultimately to a theory in which perturbation velocities are sequentially computed from four equations governing three components of velocity and the pressure,as in an incompressible, Navier-Stokes fluid,

stress

?[G(s)]

Assuming that the

admits a FGechet expansion i n integrals with

good kernels, we get

T o obtain the canonical forms of the stress for the

theory o f rotating fluids, we identify independent coefficients

277

CONSTITUTIVE EQUATIONS AND FREE SURFACES

i n t h e s e r i e s expansion of

(5.12)

g(s,E).

of

5 [ 3 ( s , c ) ] i n d u c e d by t h e e x p a n s i o n

This l e a d s us t o

The c a n o n i c a l f o r m s o f t h e s t r e s s f o r p e r t u r b a t i o n s of

s t e a d y r i g i d r o t a t i o n a r e g i v e n t h r o u g h o r d e r two by m

Z [ ~ ( S , E ) ]=

E

G_(s)

G(s) Q T ( n s ) * [-* ( * ) ( s )

where

S -' " ( s )

+

sT(ns).A(l)(s)*y_T(,s)ds

+ B(s)]

.Q(ns)ds

i s d e f i n e d by (5.18) a n d

B(s)

+

by

The h i g h e r o r d e r s t r e s s e s a r e n o t h a r d t o d e r i v e . i n t h e form

(5.22). They a r e

278

D A N I E L D.

+

7. C a n o n i c a l f o r m s

JOSEPH

lower-order

of

terms.

t h e e q u a t i o n s of m o t i o n .

A f t e r e x p a n d i n g i n powers of

When P"'

n = 0,

y ( O ) ( ~ y ; t )=

+ 1/2 p l c A s 1 2

(6.7)

QA?,

3")

i s constant.

=

E ,

we f i n d t h a t

a!(')/at

= 0,

and

N o w we s h a l l d e m o n s t r a t e

that

[

m

div %(n) =

+ To e s t a b l i s h E(5,T)

=

g( s ) Q T ( n s )* V52 U- ( n )

lower-order

(7.2),

hl(S,T),

(C- ( t - s ) , t - s ) d s

terms.

(7.2)

w e f i r s t show t h a t f o r any t e n s o r

we have

T a k i n g components i n t h e f i x e d C a r t e s i a n b a s i s , we f i n d t h a t

where we h a v e u s e d

Q , ~=

we may a l s o v e r i f y t h a t

a c k / a x j'

Since

auj") ( 5 , T

)/ati

=

0,

CONSTITUTIVE E Q U A T I O N S AND FREE S U R F A C E S

Cornbilling

( 7 . 3 ) and ( 7 . 4 ) w i t h ( 6 . 7 ) , we p r o v e ( 7 . 2 ) .

(7.1)

Equations components

el,

1

,

g2,

e

TO

-3'

identically i n b(t)

U(n'(x,t)

( 7 . 1 ) and

b(t)

t

when

r e l a t i v e t o the f i x e d

e x p r e s s ( 7 . 2 ) i n terms of

S

5 E ab(t)

for

fern

The i n i t i a l - h i s t o r y y("(x,t)

and

t 2 0,

and

Q

U -")

n = 2,

When

are also we may n = 1,

(7.6)

( 7 . 6 ) determine

The s o l u t i o n i s i n d e p e n d e n t of

5 , ~ )a t f i r s t o r d e r .

p a t h may b e computed a s a n a f t e r t h o u g h t once known:

i s prescrib-

( 5- ( t - s ) , t - s ) d s = O .

problem a s s o c i a t e d w i t h

3 ( 1) (

!(n),

A

i n sequence.

I . ( S ) Q T ( W *V!

p ( 1) ( 5 , t ) .

the p a r t i c l e path

(x,t)

y(n'

(~(n).o)&o)=

w r i t e t h e p e r t u r b a t i o n problems

-

and

hold

Boundary c o n d i t i o n s , s a y

0.

Noting t h a t

div v ( n ) ( x , t ) = 0

t > 0

for

= g(I1)(T,t)

prescribed.

When

the

we n o t e t h a t

Equations

ed i n

may b e e x p r e s s e d i n t e r m s o f

g(n) - ci ui( n )

~ 1( n ) of

c a r t e s t a n basis U(n)

27 9

The p a r t i c l e

!(')

( 5 ,t )

is

280

D A N I E L D.

JOSEPH

G( s ) Q T ( n s )adiv

B(s)ds

5 -

[

+ div

where

4

is given by (6.19) and

terms on the right of are known.

(7.8)

Y(s1,s2)L ( l ) ( ~ l ) * J ( l(s2)dslds ) 2

B(s)

by ( 6 . 2 2 ) .

dl) -

(7.8) are known when

The

and

Hence, we may solve the initial-history problem

associated with (7.8) for

y(2)(x(,t)

and

p(2)(x,t).

Then

we can compute the path at second order:

It follows that at each order we may compute three velocity components

c(n'(s,t)

and a pressure

~ ( ~ ) ( x , t ) from a n

inhomogeneous, linear, initial-history problem associated with (7.1) and

div

c(n) =

0.

&( n, appears

T h e particle path

(.L)

as an auxiliary quantity which may be computed when .L < n

is known.

,

In other words, at each stage o f the

sequence, we solve four equations i n four unknowns. The mathematical problem defined by this perturbation sequence may b e stated as follows: b(t) on

for ab(t)

t > 0, h(x,t) for

div a(x,t) = 0

in

t > 0, find and

b(t) g(5,t)

~ ( 5 , t )= &(lf,t)

for

and in

Given

f(s,t)

t

and

S

0

@(&,t)

b(t)

in

q(5,t)

such that

for

t < 0,

28 1

CONSTITUTIVE E Q U A T I O N S AND F R E E SURFACES

g(r;,t)

.a(x,t) =

ba p { s

+

(c

+

?)-VS

A

-[

t > 0

x E ab(t),

for

+

51

T

and

V@

2

G(s)Q (Qs)*V? ? ( < - ( t - s ) , t - s ) d s = f ( x , t ) .

I t i s known ( S l e m r o d , 1 9 7 6 ) t h a t u n d e r v e r y mild c o n d i t i o n s G(s),

on t h e s h e a r - r e l a x a t i o n modulus

R

= 0

n f

The s t a h i l i t y r e s u l t p r o v e d by

has a unique s o l u t i o n .

Joseph (1977) s u g g e s t s t h a t i f a 0,

t h e problem w i t h

s o l u t i o n o f t h i s p r o b l e m e x i s t s when

t h e n i t i s unique. Finally, I note that i n l i m i t

+

0

the theory o f

r o t a t i n g s i m p l e f l u i d s c o l l a p s e s i n t o my p r e v i o u s t h e o r y o f p e r t u r b a t i o n s of

the s t a t e of r e s t

The c a n o n i c a l forms o f

(Joseph,

1976).

t h e s t r e s s and t h e e q u a t i o n s

of m o t i o n t a k e a p a r t i c u l a r l y s i m p l e f o r m i n a r o t a t i o n a l l y s y m m e t r i c c o o r d i n a t e s y s t e m d e f i n e d by particles

a t zeroth

order.

t h e c i r c u l a r p a t h s of

These e q u a t i o n s a r e d e r i v e d and

t h e i r p r o p e r t i e s a r e d i s c u s s e d and some r h e o l o g i c a l problems a r e s o l v e d by J o s e p h

(1977).

I n the derivation of

t h e c a n o n i c a l f o r m s of

e q u a t i o n s o f m o t i o n we assumed t h a t

5 =

Xt(&,t,~)

of

t h e domain

problem,

labels

u s e d i n t h e backward i n t e g r a t i o n o f p a t h l i n e s

a r e independent

xt(,,T,e)

the particle

the

b,(t)

of

If

E.

d e p e n d s on

t h e boundary

a b C( t )

as i n a f r e e s u r f a c e

C ,

we p r o c e e d a s i n t h e model problem e x c e p t

that

the

mapping f u n c t i o n w i l l depend p a r a m e t r i c a l l y o n t h e t i m e ;

i s , we map

b,(t)

into

bo

where

t i m e w i t h a n i n v e r t i b l e mapping

bo

i s independent

that

of

5 = ~ ( ~ o , tp o, s ~ s e s)s i n g

D A N I E L D.

282

the properties and i f

that

E abo,

x

-0

xo then

JOSEPH

z ~ ( & ~ , t , O ) i s i n d e p e n d e n t of t i m e , 5

E abc(t).

A f t e r c a r r y i n g out

a n a l y s i s l i k e t h a t g i v e n i n $ 2 , we c a n d e r i v e e x a c t l y t h e same c a n o n i c a l e q u a t i o n s w i t h

b0.

a r e posed o n t h e domain s e t on

b0

x

-0

replacing

5.

A l l equations

The f r e e s u r f a c e e q u a t i o n s a r e

a n d , s i n c e t h e r e a l f r e e s u r f a c e depends on t i m e ,

t h e s e e q u a t i o n s c a n c o n t a i n d e r i v a t i v e s of t h e boundary v a l u e s of t h e mapping f u n c t i o n . The main a p p l i c a t i o n s o f

the theory j u s t described

t o f r e e s u r f a c e problems have s o f a r been c o n f i n e d t o

1977

p e r t u r b a t i o n s o f t h e r e s t s t a t e ( s e e J o s e p h 6% B e a v e r s ,

The m o s t s u c c e s s f u l

for a r e v i e w of t h e s e a p p l i c a t i o n s ) .

a p p l i c a t i o n s o f a r h a s been t o rod climbing ( t h e Weissenberg effect).

O n t h e b a s i s of

t h e s e c o n d o r d e r t h e o r y a l o n e we

h a v e b e e n a b l e t o e x p l a i n and e v e n t o p r e d i c t many of n o v e l e f f e c t s which a p p e a r i n t h e movie by G . S . myself.

B e a v e r s and

These e f f e c t s a r e p e c u l i a r t o non-Newtonian

l i k e STP.

They i n c l u d e c l i m b i n g on r o t a t i n g r o d s ,

bifurcation

( t h e b r e a t h i n g i n s t a b i l i t y ) of

symmetric c l i m b ,

the c r i t i c a l radius,

t e m p e r a t u r e and achesion remove g r a v i t y on e a r t h

b u c k l i n g of

fluids

a Hopf

the steady axi-

t h e b i g e f f e c t of

a normal s t r e s s a m p l i f i e r

,

the

f u i d towers,

(how t o t h e mean

c l i m b on a n o s c i l l a t i n g r o d and t h e symmetry b r e a k i n g b i f u r c a t i o n of a x i s y m m e t r i c t m e - p e r i o d i c

li t y )

f l o w ( t h e flower instabi-

. T h i s w o r k was s u p p o r t e d by t h e U . S .

O f f i c e and u n d e r N S F G r a n t

19047.

A r m y Research

283

C O N S T I T U T I V E E Q U A T I O N S AND F R E E S U R F A C E S

References Coleman, R., and Noll. W. ( 1 9 6 1 ) . Viscoelasticity. Green, A.E.,

Foundations of Linear

33, 239. Rev. Modern Phys. -_

and Rivlin, R.S. ( 1 9 5 7 / 5 8 ) .

The Mechanics of

Non-linear Materials with Memory, Part I . Rational Mech. Anal. Joseph, D.D. ( 1 9 7 6 ) .

Arch.

1,1.

Stability of Fluid Motions 11.

(Springer: New Y o r k , Heidelberg, Berlin). Joseph, D.D.

(1977).

Rotating Simple Fluids.

Arch.

Rational Mech. Anal. __ 0 6 , 311-344. Joseph, D.D., and Beavers, G.S.

(1977).

in Rheological Fluid Mechanics.

Free Surface Problems Rheol. Acta. L6-,

169- 189. Pipkin,

A.C.

(1964).

Small Finite Deformations of Visco-

elastic Solids, Slemrod, M. ( 1 9 7 6 ) .

Rev. Mod. Phys. 36, - 1034.

A Hereditary Partial Differential

Equation with Application in the Theory o f Simple Fluids,

Arch. Rational Mech. Anal.

52-,

303-322.

G.M. de La Penha, L.A. Medeiros (eds.) Contemporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations ONorth-Holland P u b l i s h i n g Conpany (1978)

ON SOME Q U E S T I O N S I N BOUNDARY VALUE PROBLEMS O F

MATHEMATICAL PHYSICS

J.L.

LIONS

College de France

Introduction. I n t h e s e Notes we g i v e a n i n t r o d u c t i o n t o some q u e s t i o n s which a r i s e i n Boundary Value Problems o f Mathematical Physics. I n t h i s f i e l d , which i s s t i l l r a p i d l y e x p a n d i n g , we have chosen f o u r t o p i c s : ( i ) some problems

o f hydrodynamics of i n c o m p r e s s i b l e non

homogeneous f l u i d s ; t h e e x p o s i t i o n made i n C h a p t e r I f o l l o w s t h e work of Antonzev and K a j i k o v

( c f . Bibliography

o f Chapter

1);

(ii)

some non l i n e a r h y p e r b o l i c e q u a t i o n s ( c o n n e c t e d w i t h

n o n l i n e a r v i b r a t i o n s ) ; we f o l l o w t h e work o f Pohozaev i n d i c a t e d i n t h e B i b l i o g r a p h y of ( i i i ) i n C h a p t e r I11 we s t u d y a

C h a p t e r 11;

l i n e a r e q u a t i o n a r i s i n g in

t h e k i n e t i c t h e o r y o f g a s e s and which c o n t a i n s some non standard aspects; (iv)

i n C h a p t e r I V we g i v e a n i n t r o d u c t i o n t o t h e method

o f homogenization f o r composite m a t e r i a l s .

BOUNDARY VALUE PROBLEMS O F MATHEMATICAL PIIY SICS

285

Chapter I On the Navier-Stokes equations -~ Introduc ti on. We consider in this Chapter the equations o f flows of incompressible fluids which are non homogeneous

-

in the sense

of not having a constant density.

[d,

The classical Navier-Stokes equations (cf. J. Leray

O.A. Ladyzenskaya 1 1 1 ) are a particular case of the equations studied here. Except in some details of presentation, we follow the notes of Antonzev and Kajikov [

11 ,

Kajikov [ 11,

Other problems studied in Antonzev and Kajikov [11 by the same techniques

as

those given below

-

-

include in

particular the existence of strong solutions in the two dimensional case. Similar problems can be considered for non newtonian fluids, i n particular for Bingham's fluids; we shall return to that elsewhere.

( F o r the case when

Binghamws fluids, cf. G. Duvaut and J . L .

p = constant in

Lions [l]).

The plan is as follows: 1.

2.

Position of the problem. 1.1.

Classical formulation.

1.2.

Weak formulation.

Statement of existence theorem. 2.1.

The hypothesis.

2.2.

Existence theorem.

2.3

Plan of the proof of existence.

286

J.L.

3.

4.

5.

6.

7.

LIONS

Galerkin's approximation

-

Standard a priori estimates.

3.1.

Spaces

3.2.

Galerkinls approximation.

3.3.

Standard a priori estimates.

' m a

Time estimates.

4.1.

An identity.

14.2.

Time difference quotients.

A compactness result.

5.1.

Interpolation estimates.

5.2.

A compactness result.

Passing to the limit i n Galerkinls approximation.

6.1.

Use of the a priori estimates.

6.2,

Proof

of

(1.20).

Problems.

1. Position of the problem. 1.1.

Classical formulation. Let

r

n

be a bounded open set o f

R

3,

with boundary

(not necessarily smooth). Q =

In the cylinder system of equations f o r

nX] O,T[ ,

u = { ui'

velocity and the density) and for (1.1) (1.2)

(1.3)

aU

p ( f i

+ (U.V)U)

-

T <

1s i

p

YAu = p f

(u.v)p

,

we consider the

33, p

(the

(the pressure):

div u = 0,

Q + at

S

m

= 0

-

VP,

287

BOUNDARY VALUE PROBLEMS OF MATHEMATICAL PHYSICS

with the initial conditions (l.l+)

(1.5) and the boundary conditions

(1.6)

o

u =

on

C =

G,

r

lo,~[.

x

p.

Remark 1.1: There are no boundary conditions on

0

Equivalent formulation.

If we multiply (1.3) by

u

and add up the correspond-

ing result to equality (1.1) we obtain

a

+

-(pu)

at

Since

p(u.v)u

div u = 0,

+

u(u.v)p

-

pAu =

(1.3) is equivalent to p + d i v ( p u ) = 0.

0

at 1.2.

p f - vp

Weak formulation. We introduce some classical functional spaces.

We

define

(1.9)

b = {v

I

v E 8(n))3,

8(n) = Cz(n)

support in

div

= functions

n

I real valued);

v Cm

=

01,

in

2-t

with compact

(all functions i n this chapter are

2 88

(1.10)

J.L. L I O N S

H

1

(0)= u s u a l Sobolev s p a c e o f o r d e r 1

= {v

I

% !

~ ~ ( n ) ,

v E

E L

2

(n) +

iJ,

p r o v i d e d w i t h i t s u s u a l H i l b e r t norm,

(1.11)

We n e x t d e f i n e V = c l o s u r e of

(1.12)

b

into

(H1(n))3,

O n e has

(1.14)

V = {v

I

v E

W e a r e looking f o r -

(1.15)

u

(H;(n))?,

u

and ___

2

E

01.

such t h a t

P E Lm(Q),

L (O,T;V),

and s a t i s f y i n g ( 1 . 7 ) ( 1 . 8 )

p

div v =

and t h e i n i t i a l c o n d i t i o n s

(1.4)

(1.5). But t h i s f o r m u l a t i o n h a s t o b e made more p r e c i s e , p a r t i c u l a r in o r d e r t o g i v e a p r e c i s e m e a n i n g t o

in

(1.6)(1.5).

One i n t r o d u c e s

(1.16)

+

=

{v I

q

E L~(o,T;v),

2

E L ( o , T ; v / 1, tp(T) = 0,

where

(1.17)

V c H

C

V’

We m u l t i p l y

= d u a l of V when H i s i d e n t i f i e d t o i t s duaL

( 1 . 7 ) by

v

a n d we i n t e g r a t e o v e r

Q.

~

(1) This c o n d i t i o n can be modified,

a s we s h a l l s e e b e l o w .

We

289

BOUNDARY VALUE PROBLEMS G F MATTIEMATICAL J'IIYSTCS

set (1.18)

(1.19) We o b t a i n

By t h e S o b o l e v ' s i m b e d d i n g t h e o r e m

Remark 1.2: _______

v

(1.21) so t h a t (1.20)

c (L6(n))3

1 and t h e second i n t e g r a l i n u.Pui E L (0,T; L3(n)) 3 acpi Lm(O,T; L 3 I 2 ( 0 ) ) . 0 makes s e n s e if ax. .J ~

We noh

s e t t h e d e f i n i t i o n o f weak s o l u t i o n s :

{u,p]

will b e a weak s o l u t i o n o f the p r o b l e m i f o n e h a s ( 1 . 1 5 ) , ~. ~~~

(1.20)

ic

cp

E @,

(1.8) and ( 1 . 5 ) . ~

We r e m a r k t h a t i f

1

(1.22)

div(pu)

H-'(n) s o that

o n e has

if

and

(1.5)

Remark has

makes s e n s e .

1.3:

If

u

E

takes place,

then

2 L ( 0 , T ; H-'(62)),

= dual space of

1 Ho(n),

(1.8) then

$$

(1.23)

(1.15)

E L2 ( 0 , T ; H - l ( n ) )

c.

i s a weak s o l u t i o n o f

(1.20), t h e n o n e

(1.7).

Remark 1 . 4 :

We s h a l l f i n d ( c f .

(5.1)

and Theorem 2 . 1 )

below)

290

J.L.

a solution

which s a t i s f i e s

u

uipu

a

(1.24)

Ft

(in particular)

4 E L (0,T;

ui so that

LIONS

L3(n))

2

E L (0,T; (L3I2(n))'),

E L2(0,T;V-')

(pu)

Then one c a n d e f i n e

for

and 0

(1.7) implies:

l a r g e enough (1)

( P U ) ~ , a~ n d f r o m ( 1 . 2 0 ) w e h a v e

(1.25)

2.

Statement of

2.1.

e x i s t e n c e theorem.

T _ -h_ e_ h y p o t h e s i s . We a s s u m e :

(2.1)

f

2

E L (o,T;H),

(2.2)

E H

0 < a c p

Po E L a m ,

(2.3) Remark 2 . 2

(Open p r o b l e m ) :

(x)

5

p.

I t would h e v e r y i n t e r e s t i n g t o

k n o w w h a t h a p p e n s when c o n s i d e r i n g i n i t i a l f u n c t i o n

po

which can have z e r o s . ( 2 ) 0

2.2.

Existence-Pheorem.

~.

Theorem 2 . 1 - We a s-s u m e t h a t ( 2 . 1 ) ( 2 . 2 ) ( 2 . 3 ) --__ u p Tush t&a& exists-functions

and w h i c h s a t i s f y ( 1 . 1 5 ) ( 1 . 2 0 ) ( 1 . 8 )

-~

(1) V-'

trxe.

There

E L2 ( o , T ; v ) n L ~ ( o , T ; H ) , P t L ~ ( Q )

(2.4)

V'

hold

= closure of

I, i n

V" = d u a l space of we c a n t a k e 0 =

(1.5).

(~'(n))3; ( w h e n H i s i d e n t i f i e d t o i t s duai);

3/2.

( 2 ) O n e c a n p r o v e g l o b a l e x i s t e n c e f o r t h e c a s e when ( x p"_ ( x ) S 6 f o r a m o d i f i e d m o d e l w h i c h i s a n ' ! a p p r o x i m a t i o n b y penalty!' of

the condition

" d i v u=O1' ( c f . J.L.

L i o n s [2]),

and

using the idea o f compensated compactness; c f . F. Murat [ 11, L. T a r t a r [l] a n d a n e x a m p l e o f a p p l i c a t i o n t o n o n linear equations in J.L. Lions [ 3 ] . Added in Proof: An existence theorem €or 0 5 p 5 $ has been found by J. Simon, C.R.A.S., Paris, 1978.

BOUNDARY VALUE PROBLEMS O F MATHEMATICAL P I I Y S I C S

Remark 2 . 2 :

291

We obtain the supplementary result (with respect

to the definition o f weak solutions) that

u E Lm(O,T;H);

therefore ui E L ' ( o , T ; L ~ ( ~ ) )n L ~ ( o , T ; L ~ ( ~ ) )

(2.5) and consequently

Theref ore (2.7)

and in (1.16) it suffices to assume that

Remark 2.3: ~ .

.

If we have

p0(x)

(2.9)

= p o = constant

then

p = po

(2.10)

satisfies (1.3)(1.5)

and the problem reduces to the classical Navier-Stokes situation.

0

Remark 2 A : Uniqueness -~~is ~an open question. in the particular case (2.9) (l). 2.3.

It i s still open

0

Plan- of the- proof of existence.

The plan of the P r o o f o f Theorem 2.1 is as follows: (i) (ii) (iii) (iv)

construction of a "Galerkin's approximation"; a priori estimates; a compactness result; passing to the limit in the Galerkin's approximation.

(1) I n the t w o dimensional case, existence and uniqueness of a classical solution (assuming of course all datas smooth enough) has been proved by A . V . Kajikov [l], and also by

Ladyzenskaya and Solonnikov, personal communication, September 1 9 7 7 .

Added i n P r o o f : D e t a i l s

t h e Leningrad Seminar, 1975.

of Proof a r e given i n

J.L.

292

LIONS

This is the usual plan followed for the standard Navier-Stokes equations (cf. for instance J . L . technical differences appear in (ii)(iii), p.

due to the presence of

3. Galerkinls approximation. 3.1.

Spaces

Lions 113);

mainly i n (ii),

Standard a priori estimates.

Vm.

We consider a family of "internal approximationsq' Vm

c V;

we assume that

(3.1)

Vm

(3.2)

I

V-

I

is a subspace of v E V,

V

o f dimension

vm E Vm

there exists a sequence

such that

vm -+ v

in

V

m +

as

m,

m.

We also assume that

(3.3)

a l l components of functions v i n Vm belong to

Since u

E Vm

V

is dense in

given

u

E H,

we can find

such that u

(3.4) 3.2.

H,

C1(n).

om

+ u

in

H

as

m + m .

Semi-Galerkin's approximation. .-

To start with, we u s e the equations (1.1)(1.2)(1.3) that we "approximate" as follows:

we l o o k f3r

urn,p,

such

that

(3.5

(l)This is not a Galerkinls type approximation; this is why we call this approximation a Semi-Galerkin's approximation. (*)This is all right if the functions of V, are supposed to is also smooth. If p o E Lm(h2) one can

b e smooth and if p o

293

BOUNDARY VALUE PROBLEMS O F MATHEMATICAL I ’ H Y S I C S

(3.8)

IIm(O)

= u

(3.9)

P,(O)

=

om’

Po(+

We c a n e x p r e s s t h e s o l u t i o n o f t o be known) m the particles

u

as follows;

(3.7)(3.9)

(assuming

w e c o n s i d e r t h e tr a j-e cto. r.i.es of

I

(3.10) Y(0) = x; if

denotes

y:(t)

or

the s o l u t i o n

(3.11)

(3.10),

then

m = P,(Y,(t)).

p,(x,t)

A l l t h i s makes s e n s e s i n c e we assumed ( 3 . 3 ) .

Using

(3.11) i n ( 3 . 6 ) g i v e s a s y s t e m o f n o n l i n e a r d i f f e r e n t i a l e q u a t i o n s h a v i n g l_ o_c_a l e x i s t e n c e . The s t a n d a r d a p r i o r i e s t i m a t e s w h i c h f o l l o w p r o v e t h e global-existence

3.3.

um y P r n .

of the solution

S t a n d a r d a p-rio-ri e s t i m a t e s . We c a n r e p l a c e i n U

(3.6)

2

+

(3.12)

C

j

‘n

fn

v

pmujm

( 3 . 7 ) by

Adding ( 3 . 1 2 ) ( 3 . 1 3 ) approximate p smooth,

a

S

0

S

L e t us a l s o r e m a r k of

p

,

that

”

( 3 d x

+

a ( u m , u m )=

= b,f9um).

urn 7 and

we obtain (since

(3.5) ...(3.9) equations

q

<

div u co,

m

= 0)

with

i s NOT a

(because of

and a f i x e d p o i n t argument.

P om

t o the l i m i t . u s u a l system

(3.7)); the

e x i s t e n c e o f a s o l u t i o n i s p r o v e n by t h e e s t i m a t e s o f Section 9.3

n;

we i n t e g r a t e o v e r

a n d one c a n p a s s

ordinary d i f f e r e n t i a l

2

j

by p o r n i n t h e L q ( n i norm,

porn

i t comes

*

In’

a ax

2 We m u l t i p l y

u

by

294

J.L.

i

U

2

in j ax-;

U

a

+

Pm( ;-)dx

n

i:

J

LIONS

(P

2

mujm -2)dx

+ a(um,um) =

"

( p f , u ) i.e. m ni

(3.14) By vir

(3.15) It fol

'n

-

'0 0 'R Using this identity and (3.15) we obtain global (in t)

existence and

(3.17)

um

is bounded i n

L2(0,T;V) n Lm(O,T;H).

4 . Time -~~ estimates We now proceed with the m a i n estimates, established in Antonzev and Ka jikov [ 11. ( 1 )

4.1. An identity. We multiply over

R,

(3.7) by

um.v,

and we add to (3.6).

v E Vm,

we integrate

We obtain:

We have'-'

(l)Estimates o f the type ( 4 . 1 7 ) b e l o w have been obtained independently by H. Brezis [l] for Variational Inequalities connected with Navier-Stokes System. the (2)The C I S denote various constants independent of m ; norms in H and in V are respectively denoted by I 1 and by

I1 I / .

BOUNDARY VALUE PROBLEMS O F M A T I I E M A T I C A L P H Y S I C S

We i n t e g r a t e interval

(4.5)

(4.1)

t,

in

t + 6 r: T.

(t,t+b),

with a fixEc

i n the

We o b t a i n

( P m ( t + 6 ) u m ( t + 6) - p , ( t ) u , ( t ) We t a k e now

v,

v = u,(t+b)

,v) =

-

u,(t)

(

([liE in

Fm(s)ds, v).

(4.5).

L e t us set:

Then:

/t+6

(4.7) Since

p,

2

a > 0,

we h a v e

(4.8) L e t u s now t r a n s f o r m

Y , .

0

I t f o l l o w s f r o m ( 3 . 7 ) (which i s e q u i v a l e n t t o

that

(4.9)

P&+fj)

-

PJt)

=

- c ax. a ( 't

so that

295

J.L.

296

LIONS

(4.10)

Theref o r e

hence ( s i n c e

V t ( L4( n ) ) 3 ) :

and s i n c e we have

(3.17) i t follows t h a t

W e i n t e g r a t e ( 4 . 1 3 ) on

(3.17);

(0,T-6)

and we u s e a g a i n

we o b t a i n

(4.14)

lYmldt

5

c 6 1/2

W e now e s t i m a t e t h e s e c o n d term of of

( 4 . 7 ) , t h a t we d e n o t e

by

Z

m'

.

@

t h e r i g h t hand s i d e

W e have, u s i n g ( 4 . 4 ) :

BOUNDARY VALUE PROBLEMS O F MATIIEMATICAL I'HYSICS

297

,T-6

Theref ore

(4.15)

(with the convention that

um = 0

on

(-6,O)).

But as above ,s

s o that

(4.15) gives T-6

fI T-6 lZmldt S

(4.16)

1

C 6

'0

Using

[k(s)

+

.

Il~~(s)ll~1dsS 61 C/2

0

ci

(4.8), (4.14), ( 4 . 1 6 ) , we have proven f

1

(4.17)

T-b lum(t+6)

-

um(t)I2

dt

S

C 6 1/2

.

0

'0

5.

A compactness result.

5.1.

Interpolation estimates. Let us observe first that

(5.1)

1

urn

remains i n

a

bounded set of LP(0,T;(Lq(n))3),

298

J.L.

Indeed, i f

LIONS

we d e n o t e ( c f .

Lions-Magenes

[ 1 ] ) by

Ve

t h e i n t e r p o l a t i o n s p a c e ( o b t a i n e d by t h e complex method) between

V

and f o r

v E V:

H,

and

(0 <

w e have

< 1)

(5.3) We u s e ( 5 . 3 ) w i t h follows

v = urn;

(3.17), i t

s i n c e w e have

that

b

so that

T

It um( t )IIP

dt s

c

if

Bp = 2 ,

hence

( Lq ( ) (62 ) ) 3

(5.1) follows.

5.2.

A c ompac t n e s s r e s u l t

W e a r e g o i n g t o prove t h e Lemma

-

5.1

8

Let -

be a s e t of f u n c t i o n s which s a t i s f y

*

(5.4)

v

(5.5)

there e x i s t s a constant

R

v

iTm6

2 L (0,T;V)

i s bounded i n

(v(t+6)

-

v(t)I2 dt

c 5

n

Lm(O,T;H),

such t h a t c

W 6.

'0

Then

(5.6)

fl

i s a r e l a t i v e l y compact s u b s e t o f

-p E [ 2 , m [ , when

qE[2,6[

LP(0,T;(Lq(62))3)

and s a t i s f y (compare t o

(5.1))

BOUNDARY VALUE PROBLEMS O F MATHEMATICAL

P r o -o f. : -

1) W e h a v e t o p r o v e t h a t

weakly and i n

Lm(O,T;H)

if

v

m

-v

0

in

,IIYSICS

299

2

L (0,T;V)

weak s t a r and i t s a t i s f i e s

(5.5)

then

2 ) The p r o p e r t y

( 5 . 8 ) w i l l be p r o v e n i f we v e r i f y t h a t

(5.9)

v

+

m

o

L

in

2

(o,T;E)

I n d e e d we u s e ( 5 . 3 ) w i t h being;

0

strongly.

a r b i t r a r y f o r t h e time

we h a v e :

1 > 1

we c h o o s e

such t h a t

(5.11)

p(1-e)x

= 2

( w e s h a l l v e r i f y below that a l l t h e s e c h o i c e s a r e c o m p a t i b l e ) .

c >

Then, g i v e n

1

0,

there exists

c(c)

such t h a t

1

x+h’= Theref o r e

We f i x

E

arbitrarily s m a l l ,

and

(5.8) follows i f

300

J.L.

LIONS

s o t h a t e v e r y t h i n g i s reduced t o p r o v i n g ( 5 . 9 )

3 ) w e c o n s i d e r t h e components o f t o the following:

(1)

a

The problem i s r e d u c e d

v.

we a r e g i v e n a s e t

e

of

( s c a l a r ) functions

such t h a t

v E:

(5.13)

e *

(5.14) Then

2 1 2 v i s bounded i n L ( O , T ; H o ( R ) ) n L m ( O , T ; L ( n ) ) ,

r6i, e

(5.15)

) v ( x , t + 6 ) - v ( x , t ) 1 2 dxdt s C 61 / 2

i s r e l a t i v e l y compact i n

L ~ ( Q ) .

By t h e c l a s s i c a l c h a r a c t e r i z a t i o n of M . compact s u b s e t s o f ( i ) for e v e r y

R3d0,T[

E

t h e r e e x i s t s a compact s u b s e t

0,

) v I 2 dxdt L E

e >

0,

there exists

? = extension

of

v

(Y

Th

>

v E

q

K

of

e;

w v

such t h a t

e,

f ( x , t ) = f(xl-hl,

x2-h2,

by 0 o u t s i d e x3-h3,

i n p a r t i c u l a r a number Lp(Q),

we have

I

p > 2

(0,T)

and where

t-hq).

P r o o f o f ( i ) : S i n c e we h a v e s e e n i n ( 5 . 1 )

in

R i e s z of

we h a v e t o v e r i f y t h e f o l l o w i n g :

such t h a t

( i i ) l f o r every

where

L2(Q),

.

such t h a t

t h a t there exists v E

e

v

bounded

2 1--

( v 1 2 dxdt

S

c(meas C K )

CK

(')We

8

choose

by

7+ F =p

O n e v e r i f i e s t h a t by v i r t u e o f

that

ph'

2

2.

and n e x t 1 by x p ( l - 8 )

( 5 . 7 ) one h a s

X > 1

= 2. and

301

BOUNDARY VALUE PROBLEMS OF MATHEMATICAL I’IIY S I C S

hence (i) follows by choosing meas CK Proof o f (ii):

small enough.

I t will be enough to prove (ii) f o r

We observe that in all what we have done

T

arbitrary. Therefore we can assume

(5.14) for T

finite arbitrary

and by smooth truncation, we can assume that

1

v

with compact support in

hence (ii) follows for the first choice o f

t, h

in (5.16).

It only remains to consider space translation.

But i t

is a standard result that

(

L3

_ 1 --

s

Iv(x-h,t)-v(x,t) 1

2 dx)’j2

5

Clh

1

2 - C ’

hence the desired result follows.

0

6. Passing t o the limit in the Galerkin’s approximation. 6.1. Use of the a priori estimates. By virtue of the a priori estimates and by Lemma 5.1, we can extract a subsequence, still denoted by

u

m

and

p,

such that

(6.1)

2

um + u in L ( 0 , T ; V )

weakly and in L m ( O , T ; H )

weak star,

J.L.

302

u

m

-t

u

p

and

pm

-t

in

Lp(O,T;

(Lq(n))3)

s t r o n g l y , where

satisfy (5.7),

cl

p

LIONS

weak s t a r .

Lm(Q)

in

W e a l s o k.now t h a t

aspss:.

(6.4)

I t follows from ( 6 . 2 ) and ( 6 . 3 ) u. pm Jm

(6.5)

-t

u.p J

in

that

LP(O,T;Lq(n))

weakly;

we u s e ( 3 . 7 ) i n t h e f o r m :

hence i t f o l l o w s t h a t

-a p+m -

at

(6.7)

a~ at

in

2 L (o,T;H-’(~)) ( i n p a r t i c u l a r ) .

The e q u a t i o n ( 6 . 6 ) g i v e s i n t h e l i m i t

a aaP+,r t J

(u.p) = 0 J

and i t f o l l o w s f r o m ( 6 . 3 ) a n d ( 6 . 7 ) pm(x,O)

-t

p(x,0)

in

H-l(n) P(X,O)

weakly and t h e r e f o r e

=

P0(X).

It r e m a i n s o n l y t o s h o w t h a t ( 1 . 2 0 ) smooth enough and s u c h t h a t

6.2.

Proof of

h o l d s t r u e f o r every

e p ( x , T ) = 0.

(1.20):

We r e w r i t e ( 4 . 0 )

that, in particular,

i n t h e form:

cp

BOUNDARY VALUE PROBLEMS OF MATHEMATICAL P H Y S I C S

We t a k e

mo

f-ixed and

'

cp = smooth f u n c t i o n f r o m

v

Then w e c a n r e p l a c e i n ( 6 . 8 )

By v i r t u e

u

[O,T]

.$

Vmo

V(T) = 0.

'

i n t e g r a t i n g over

90 3

(O,T),

of

we o b t a i n ,

by

for

L'(Q)

in

m 2 m

r > 1

(6.2) there exists

u 4 u u i m jm i j

cp(t)

and a f t e r

0'

such t h a t

strongly

so that p u. u 4 pu.u. m im j m 1 J

in

v~ep

Using t h e f a c t t h a t

of

t h e form

I1Vm -t V "

(1.20) i s s a t i s f i e d for every

weakly

(6.10).

and we c a n p a s s t o t h e l i m i t i n W e obtain (1.20)

Lr(Q)

(cf.

cp

E

(6.9). (3.2)) i t follows t h a t and t h e p r o o f

is

completed.

7.

Problems.

I t would b e v e r y i n t e r e s t i n g t o s t u d y t h e above problems when t h e v i s c o s i t y

p 4 0,

at least i n

Rn

or on

J.L.

304

LIONS

a v a r i e t y w i t h o u t boutidary. The l i m i t c a s e problem

( w i t h boundary c o n d i t i o n

l o c a l l y i n t i m e by

p a r a l l e l t o t h e boundary) has been solved

J.E.

v

MARSDEN [ l ] .

I t would a l s o be i n t e r e s t i n g t o c o n s i d e r non lianogeneous l i q u i d c r i s t a l s and e x t e n d t h e w o r k of J . P .

o S- Chap txc--1

Eib 1i o gpapliy Antonzev a n d A.V.

S.N.

K a j i k o v [l],

M a t h e m a t i c a l s t u d y of'

S l o w s o f non homogeneous f l u i d s

Lectures a t t h e University H.

DIAS [ 11.

-

-

-

Novosibirsk

1973 ( R u s s i a n ) .

B r e z i s [I], V a r i a t i o n a l I n e q u a l i t i e s c o n n e c t e d w i t h N a v i e r S t o k e s and p e r s o n a l cornmiinication.

J.P.

D i a s [ l ] , Sur l e s G q u a t i o n s c o u p l 6 e s b i d i m e n s i o n n e l l e s d l u n c r i s t a l l i q u i d e n6matique in c o mp re s s ib le .

C.R.A.S. G.

Paris,

Duvaut and J . L . Physics.

1977.

L i o n s [ 11

Paris,

,

Dunod,

Inequalities i n Mechanics._a@g

1972 ( i n F r e n c h ) ; Springer,l974.

A.V.

K a j i k o v [ l l , Doklady Akad.

A.V.

Kajikov [ 2 ]

(Sem. O.A.

,

Nauk,

2 1 6 ( 1 9 7 4 ) , p.1008-1010.

I n S e m i n a r on N u m e r i c a l Methods i n M e c h a n i c s Y a n e n k o ) , p a r t 11, N o v i s i b i r s l c ,

N.N.

L a d y z e n s k a y a [l],

The m a t h e m a t i c a l t h e o r y - o f

incompressible f l u i d s .

p.

65-76.

viscous

Gordon B r e a c h , N e w Y o r k ,

1963.

J . L e r a y [ l ] , E t u d e d e d i v e r s e s b q u a t i o n s i n t g g r a l e s non l i n b a i r e s e t d e q u e l q u e s p r o b l h m e s que p o s e l ' h y d r o dynamique. p.

J.L.

J . Math.

Pures e t AppliquLes, X I 1 ( 1 9 3 3 ) ,

1-82.

L i o n s [l]

,

Q u e l q u e s m e t h o d e s d e r j s o l u t i o n -des

aux l i m i t e s non 1 i n Q a i ~ e P~a. r i s ,

Villars,

1969.

probl5mes

Dunod-Gauthier

ROIJNDARY VALUE PROHLEbIS O F MATIIEMATICAL 1 ’ I I Y S I C S

305

L i o n s r21, On s o m e p r o b l e m s c o n n e c t e d w i t h N a v i e r - S t o k e s

J.L.

equations. Lions [

J.L.

71,

Conference Madison, October,

1977.

O n ~ o m enon l i n e a r s y s t e m o f e v o l u t i o n - e q u a t i m

Workshop on n o n l i n e a r p r o b l e m s o f M e c h a n i c s .

1977.

March,

J.L.

Austin,

Lions and E .

Magenes [ 13, Non homogeneous b o u n d a r y v a l u e

problems- ( I ) . P a r i s , Dunod,

1968 ( i n F r e n c h ) ; S p r i n g e r

1971. J.E.

Marsden [l],

W e l l p o s e d n e s s o f t h e e q u a t i o n s o f a non

homogeneous p e r f e c t f l u i d . Equations, F.

in Partial D i f f .

( 3 ) , ( 1 9 7 6 ) , p. 215-270.

M u r a t [11, C o m p a c i t g p a r c o m p e n s a t i o n . Pisa,

L.

1

Comm.

Ann.

Sc.

Norm.

1977.

T a r t a r [l],

Cours Peccot,

CoLlSge d e F r a n c e ,

1977.

Sup.

LIONS

J.L.

306

Chapter I1 ___Some non linear hyperbolic equations ~. .-

Introduction. We present here some results o f Pohozaev [l], ed with non linear vibrations.

connect-

Somewhat similar problems are

considered by R.W. Dickey [ 11. The technique o f Pohozaev, loc. cit., to obtain a priori estimates is very interesting; it applies only f o r initial conditions which belong to some special classes of functions (which can be characterized using the "iterate" theorem o f J.L. Lions-Magenes [ 13 )

.

Hyperbolic problems with unilateral constraints ( w h e n _______.__ the functions .~

M(X)

= constant in what follows) have been

considered by Amerio-Prouse [l] and M. Schatzman [l] ;

in the

present situation, unilateral constraints seem to lead to a very challenging problem...

.

The plan is as follows: 1.

Position of the problem. 1.1.

First example.

1.2.

Second example.

1.3.

A general problem.

1.4. Orientation. 2.

Standard a priori estimates.

3.

The class

3.1.

(P) and the main theorem.

Definition o f (P).

BOUNDARY VALUE

PROBLEMS OF MATHEMATICAL

3.2.

Statement of t h e theorem.

3.3.

Orientation.

iwysIcs

307

4. Galerkin's approximation and a priori estimates. 4.1.

Galerkin's approximation.

4.2.

Notations.

4.3.

A

priori estimates.

1. Position of the problem.

(1.1)

X -+ M(X)

M(X) 2

is continuous on

and satisfies

0

mo > 0 .

[We implicitly

assume in (1.2) that

lvu(x,t)12

Remark 1.1:

5

dx <

m

a.e.

in

t

1.

T h e hypothesis

is not useful in the setting o f the problem, but it will play an essential role in the solution given below.

J.L.

308 The case when Remark 1.2:

M(1)

LIONS

has zeros in an open problem

0

O n e could more generally consider

'n b u t this case seems to be essentially open'').

Remark 1.3:

M(h)

Of course if

classical wave equation. Remark 1.4:

= mo,

r?

(1.2-)(1-3)(1*4) is the

[7

F o r the solution given below, we shall assume

that

M E C1 (120).

1.2.

Second example. The second example we want to mention here is due to

R.W. Dickey [l].

It is a particular case o f (1.2) with

respect to the dimension

(=1) but i n an unbounded __ . __ __. domain:

(1.7)

L-2

=

lo,d ,

with

(1.9) This equation describes the motion o f a semi-infinite string (cf. Narasimka [ 11 )

1.3.

.

A general problem.

We consider now a n I1abstract1'situation which contains the above situations as particular case. ( 2 ) (l)Cf. Problem 11.10, Chap. 2, of the book J . L . LIONS [ l ] quoted in Chap. 1. (*)Added in Proof: Other problems o f a somewhat similar nature have been studied by L . A . MEDEIROS, J.M.A.A., to appear.

309

BOUNDARY VALUE PROBLEMS O F M A T H E M A T I C A L P ’ I I Y S I C S

Let

V

H

and

V c H,

(1.10)

1) 1)

w e d e n o t e by and by

be t w o H i l b e r t s p a c e s o n

(

,

)

dense i n

V

I I)

(resp.

We identify

H

H;

the n o r m i n

the scalar product

in

R (1)

V

(resp.

H),

H.

t o i t s d u a l ; then the dual

V’

o f

V

c a n be i d e n t i f i e d i n s n c h a w a y t h a t V C H C V’.

(1.11)

(1.12)

E x a m p l e 1.1

-

A E 6:(V;V’),

A*

( ~ v , v )= a(v)

2

If

(1.15)

(1)

-

alJvl12

1 = H,(n),

+

v

c

V.

w e f i n d the p r o b l e m

C

(1.2)(1.3)(1.4). E x a m p l e 1.2

v

A = -A,

= A,

If

A = -A+I,

a 2u

+

at

M(

( ’n

1 V = H (‘2)

u 2d x +

[

we find

IVU 2 d x ) ( - A u + u )

R

W e c o u l d as w e l l c o n s i d e r t h e c o m p l e x c a s e .

=

f

i n Q,

310

J.L. L I O N S

Example 1.3

-

A =

If

alu_ +

(1.18)

A

M(

2

1

2

2 V = Ho(n),

,

H = L

2 ( 5 ~ dx) ) ~5 u = f

u = 0,

(1.19)

-- 0 aU hV

Q,

C,

on

0

and the initial conditionsas before. Problem: Let us consider, for

-+’ *U

in

we find

n

at2

(1.20)

(n),

M(e

i

u2 dx

+

lvul

(compare to (1.15)): 2

dx)(-Au

+ CU)

= f

n

R

at2

E > 0

with conditions (1.16)(1.17) unchanged. What can be said o f the solution

u

E

(which will b e proven

to exist under conditions of Section 3) when

1.4.

E + 0 ?

Orientation. We shall show, according to Pohozaev, loc. cit., that

if - u

0’

u

1

and -

f

problem (1.13)(1.14)

are taken i n a (very) special class, admits a unique (strong) solution.

3) we give the

Before defining this class (Section standard a priori estimates

-

and we indicate why they are

(apparently) not sufficient to conclude.

2.

Standard a priori estimates. Let us multiply (1.13) by

(2.1)

_I2 -dt d

Iu’(t)I2

If we introduce:

+

u‘ =

aU at

M(a(u))a(u,u’)

-

We obtain = (f,u‘).

3 11

BOUNDARY VALUE PROBLEMS C P MATHEMATICAL 1 ’ I i Y S I C S

x i ( X )

(2.2)

=

M(CI

) C u 1 9

then (2.1) can be written

If we assume that

it follows from (2.3) that

L

(2.5)

Since

(2.6)

fi(X)

Iu’(t)I2 O S

Of

-+

+

m

as

liu(t)l12

t S T .

X

-+

S

c[

i t f o l l o w s that

a ,

lf(s)I2

ds + lu1

2

+

II uoll2I,

0

course one would use (2.6)

approximation (cf. Section

on a Galerkin

4 below),

s

But these a priori

estimatesare not sufficient for passing t o the limit, i n particular in the term M(a(um(t))) if

u

m

9

= Galerkin’s approximations.

In order to obtain further a priori estimates, a natural idea is to multiply ( 1 . 1 3 ) by

-

A* u‘

for suitable

r

but this leads to estimates which necessarily ( ? ) involve

the whole sequence of

( A p u‘

I

for all integers

p

(or

something close to that). This fact, and technical estimates given in Section

4

312

J.L.

LIONS

justify the introduction of the class o f functions presented in the next section.

3.

The class (6) and the main theorem.

3.1.

Definition of.)'6( We set

I

D(Am) = ( v

(3-1)

We shall say that

v E D(Ak)

[uo,ul,f) E

k 2 A f E L (0,T;H)

Remark 3.1:

V

k

E

N,

T h e condition (3.3) is introduced in order to

useful information from the technical estimates

given in Section

(3.4)

(P) if

uoyulE D(Am),

(3.2)

obtain

w k E IN).

4.

A = second order elliptic operator i n

si

with analytic coefficients i n

(3.5)

hl

R,

has an analytic boundary

r

then (3.3) is equivalent (according to the theorem on "elliptic iterates" of Lions-Magenes [ 11 ) to the property:

(3.6)

uo,ul are real analytic in

-n,

BOUNDARY VALUE PROBLEMS OF MATHEMATICAL PHYSICS

13.7)

t

is continuous in

f

’313

with values i n real analytic

func t i ons

and

u

0’

u

f

1’

satisfy a n infinite number o f boundary

conditions (corresponding to the belonging to Remark 3.3: . ~

If we assume that

-

m > 1

is elliptic of order

u o , u1

(3.3) i s very difficult to interpret.

n

should be restriction to

special classes.

Functions

of entire functions of

117

Statement o f the theorem. - ..

Theorem 3.1

-

(3.8)

the injection mapping

(3.9)

M

(3.10)

{Uo,Ul,f3 E ( P )

We assume that

V + H

satisfies (1.1) and is (i.e.

is compact,

C1 ( X > O ) ,

(3.2)(3.3)

Then there exists a unique function

and -

Zm,

even with analytic coefficients (or even with constant

coefficients),

3.2.

A

n

D(Am)).

u

in theorem;

which satisfies

In fact we shall prove more than what is stated

-

if we denote by

[co,cl,?], f

defined in

(P(s))

(s,T),

analogous conditions to (3.2)(3.3)

(3.13)

u

satisfies to (1.13)(1.14).

Remark 3.4:

(s ,T),

hold true).

the class of functions

and satisfying the with

(0,T) replaced by

then Eu(t),u’(t),f(t)?

€

(P(t)).

n

314

J.L.

3.5: The h y p o t h e s i s ( 3 . 8 ) i s n o t s a t i s f i e d i n t h e c a s e

Remark of

LIONS

t h e example o f

Section 1.2.

I n t h e p a p e r q u o t e d of R.W.

t

the local existence i n

Dickey,

t h i s a u t h o r proves f = 0,

of a s o l u t i o n , w i t h

w i t h a f i n i t e number o f c o n d i t i o n s on

u

u

and

1'

and

0

3.6 (Open p r o b l e m ) : I s t h e problem n o n w e l l s e t i f one

Remark

f = 0,

takes, say

3.7

Remark

and

uo E V ,

u1 E V ?

M

(Open p r o b l e m ) : What h a p p e n s when

i s only

assumed t o be c o n t i n u o u s ( a n d s a t i s f y i n g (l,l))?

3.3.

Orientation. The u n i q u e n e s s

i s standard.

s o l u t i o n s and i f we s e t :

(3.14)

w"

If

a(u(t)) = a,

+ M ( a ) A w = (M(g)-M(a))AG,

u

and

G.

a(G.(t)) = w(0)

= 0,

are t w o

g,

w = u-G:

w'(0)

It f o l o w s t h a t (by t a k i n g t h e s c a l a r product w i t h 2w'

): d dt

hence

w'

+

M(a)

d

2~ a ( w )

= 2((M(g)-M(a))Ac,w')

= 0.

315

BOUNDARY VALUE PROBLEMS O F M A T I I E M A T I C A L P I I Y S I C S

’0

w =

hence

n

0.

We give now the proof of the existence o f a solution.

4. Galerkin‘s approximation and a priori estimates. 4.1.

Galerkin‘s approximation. urn

We define

V m c V,

Vm

as the solution of

finite dimensional.

The estimates of Section 2 show the global (in time) existence of a solution of (4*1)(4.2)(4.3). We choose:

(4.4)

Vm = space generated by the first

of

A.

This choice allows to replace in (4.2) for any (l)It

eigenfunctions

m

v

by

k

A v

k.

suffices in this proof that

u E L~(o,T;D(A)), and this can even be weakened.

(2) We take f o r

uom and

Fourier series of u o functions of A (see

u

lm and

the u1

(4.4)).

u

(and

U’

m

G)

satisfies:

E L~(o,T;v),

first terms in the

i n the basis o f eigen-

J.L.

LIONS

I n what follows we shall derive a priori estimates for ._____u

(without index

m)

satisfying ___ the equation; by taking some

care with respect to the choice o f initial conditions, this __ does not restrict the generality.

4.2.

Notations. - ~_____ We suppose therefore that (u”,v)

(4.5)

+

(Au,v) = ( f , v ) .

M(a(u(t)))

We set

(4.6) We already know that

(4.7)

Ilu(t)ll

+ Iu’(t)l

c

i.e.

(4.7

bis)

B O W

+

vows

c-

(4.8)

4.3.

A priori estimates. We take

(4.9)

v = 2A2k u’ M(a(u(t)))

in

(4.5).

=

(t)

d

8,

After dividing by

we obtain 1 d

Y, + or

2

=i-r

(Akf, Aku‘),

BOUNDARY VALUE PROBLEMS O F MATHEMATICAL P I I Y S I C S

2 (Akf, = -

d d t hk

( I + , 10)

Since

’k ~

!J

S

Ak u’

11

)

+

!J‘ ’k -

I J P

3 17

.

i t f o l l o w s from ( 4 . 1 0 ) t h a t

hk,

(4.11)

L e t u s now e s t i m a t e

.

!J

We have

11’ = 2 M ’ ( a ( u ) ) a ( u , u ’ ) .

But

= ( A 2ku , u ’ )

lAku’I2

gives 1/2 Yks

Yo

’

1/2 2k

s o that

Therefore (4.12)

We u s e the fact that for

(4.14)

k=2J,

gives

(4.13) i n (4.11) p

2

mo;

jzl,...,

dhk

(for

k=2J)

and we u s e a g a i n

w e obtain: t h e r e e x i s t s a c o n s t a n t K such t h a t

s

(F1 + K h 1/4k ) h k + I A k f I 2 . k

0

I t f o l l o w s from ( 4 . 1 4 )

that

J.L.

LIONS

lAkf

dt]

318

Ke

+

:[

e

The hypothesis on the datas are exactly those such that

Tk

(4.16)

2

7

>

0.

Therefore we have: lAku’ (t) I

(4.17)

for

+ 1Ak+ll2 u(t)

1

s

constant

t E [o,Tol

so that one can proceed one step further in

13

This completes the proof of Pohozaev’s theorem. Remark in

4.1:

(4.4) but

The compactness o f the injection n o t in a n essential manner.

V + H

is used

One could use

spectral subspaces of the spectral decomposition o f f o r an application o f

etc.

(To,2?o),

A

(cf.

this Remark to a non linear hyperbolic

problem, J.L. Lions and W.A.

Strauss [l], p.62).

What is more important is t o being able to pass to the limit in

M(a(um)).

Since we know that, in particular

719

BOUNDARY VALUE 1’KOBLEMS O F MATIIEMATICAL P H Y S I C S

u

2

i s bounded in L ( O , T ; D ( A ) )

m

a

---is

bounded i n

at

then

u

m

i s r e l a t i v e l y compact i n

true i n the situation o f

l i m i t i n t h e term

L2(0,T;H), L2(0,T;V)

Section 1.2)

( t h i s i s not

and one c a n p a s s t o t h e

t!

M(a(u,,)).

Bibliography ~

~~~

o f C h a p t e~ r I1 _ _

L. Amerio, G. P r o u s e [l], S t u d y o f t h e m o t i o n o f a s t r i n g v i b r a t i n g a g a i n s t an o b s t a c l e .

R.W.

Rend.

d i Mat.(2),8,1975.

Dickey [l], The i n i t i a l v a l u e problem f o r a non l i n e a r semi i n f i n i t e s t r i n g , U n i v e r s i t y o f Texas, Workshop, March

J.L.

Austin,

1977.

Magenes 111, Eon homogeneous boundary v a l u e

L i o n s and E .

~~~

~

p r o b l e m s and a p p l i c a t i o___ n s- , V o l . 3 .~

,

Dunod, 1970 (in French),

Springer, 1972. J.L.

S t r a u s s [l], Some non l i n e a r e v o l u t i o n

L i o n s and W.A. equations.

S.M.F. 9 3 ( 1 9 6 5 ) , P . 43-96.

Bull.

R. Narasimka [l], Non l i n e a r v i b r a t i o n o f a n e l a s t i c s t r i n g . J. S.I.

Sound Vib.

Pohozaev [l], On a c l a s s o f equations.

M.

8 ( 1 9 6 8 ) , 134-146.

Schatzman [l]

,

Mat.

quasi l i n e a r hyperbolic

USSR Sbornik.

Thesis.

Paris,

25 ( 1 9 7 5 ) , 1 , p.145-158.

19’78.

J.L.

LIONS

C h a p t e r I11 A l i n ea r problem a r i s -i n-g i n k i n e t i c t h e o r y of g a s e s

Introduction. W e b r i e f l y s t u d y i n t h i s c h a p t e r a l i n e a r problem which a r i s e s i n k i n e t i c t h e o r y o f g a s e s ( c f . Kaper [l]).

The

equation i s

where

u

and for

i s s u b j e c t t o boundary c o n d t i o n s f o r x < 0,

t=T.

I f we w r i t e

we h a v e a s i t u a t i o n where c o e r c i v e and where

8

G.

x > 0,

( * ) i n t h e form

i s unbounded,

20

but not

has a k e r n e l n o t reduced t o

0.

A s y s t e m a t i c s t u d y o f s u c h s i t u a t i o n s i s made i n

Beals

11. We g i v e h e r e a d i r e c t t r e a t m e n t o f

(*).

The p l a n i s as f o l l o w s :

1.

S e t t i n g of

t h e problem.

1.1.

Introduction,

1.2.

Functional spaces.

1.3.

S e t t i n g o f t h e problem and main r e s u l t .

t=O

BOUNDARY VALUE PROBLEMS OF MATHEMATICAL P I I Y SICS

2.

3.

321

Proof o f uniqueness. 2.1.

A Lemma.

2.2.

Uniqueness.

Proof

of existence.

3.1.

Reduction of the problem.

3.2.

Elliptic regularization.

3.3.

A priori estimates

3.4.

A priori estimates (11).

3.5.

Proof of existence.

Appendix:

(I).

A trace result.

1. Setting of the problem. ~-

1.1.

Introduction. _ _ I

In the domain

8 = (x,t

I

-m

< x <

m,

0 < t < T]

we

consider the 1.1 Problem _~_.___

-

Find a function

u

defined i n

8,

with real

values, such that +m

x

aU - + at

u -

-X2 u(x,t)dx = 0,

with the boundary conditions

Of course the integral i n (1.1) should make sense! Therefore

u(.,t)

is subject to growth conditions;

it will

be quite natural in this context (and also for physical reasons!) to impose

J.L.

322

LIONS

2

u ( x , ~ d) x~ <

(1.4)

a.e.

m

T h i s w i l l imply i n t u r n c o n d i t i o n s on

in

t.

go

and on

gl'

I n o r d e r t o make a l l t h e s e c o n d i t i o n s p r e c i s e , i t i s necessary t o introduce functional spaces.

1.2.

Functional spaces. We d e f i n e

J-m

For

u,v E H,

we s e t

tV,Vl

=

Cvl

for t h i s s t r u c t u r e ,

1/2.

i s a Hilbert space.

H

We d e f i n e n e x t

~r = cv

(1.7)

I

v

E

2 L (o,T;H),

x

2

E L (o,T;H)I;

provided w i t h t h e s c a l a r product

Ir

171

becomes a H i l b e r t s p a c e .

T r a c e s..of .

elements o f

b.

~

L e t u s i n t r o d u c e for

(1.9)

H(xo,m)

= [v

I

x

> 0:

v

m e a s u r a b l e on

lx:

e-x2 v ( x ) ~ dx <

m}

(x0,m),

323

BOUNDARY VALUE PROHLEMS O F M A T I E M A T I C A L I ’ H Y S I C S

equipped w i t h i t s n a t u r a l H i l b e r t n o r m .

v E

Then i f x > x

t E ]O,T[,

0’

Ir

we can c o n s i d e r i t s r e s t r i c t i o n t o and

(1.10) 2 v E L (0,T; H(x0,m)))

v

where e q u a l t o ) a c o n t i n u o u s f u n c t i o n

t

so that

(since

[O,T] + H ( x o , m ) .

Since

is

x

+

( a l m o s t every-

v(t)

from

a r b i t r a r y and s i n c e one

>O

c a n as w e l l c o n s i d e r t h e i n t e r v a l

is

]-m,-x

[

,

we h a v e

uniquely defined

The t r a c e s p a c e s ( s p a n n e d by

v

spans

II)( I )

Indeed i f

and

v(T)

when

H.

vo E H ,

the function

v(x,t) = v

(1.12)

to

belongs

1.3.

c ontain -

v(0)

II

S e t t i n g of

and

I?

v ( x , O ) = v0.

t h e problem and main r e s u l t .

With t h e n o t a t i o n s o f S e c t i o n 1 . 2 ,

t h e e q u a t i o n (1.1)

becomes : aU

x -+ u

at

(1.13) Therefore i f

u

E L

2

-

[U,ll

(0,T;H)

= 0.

i s s o l u t i o n of

(1.13),

one h a s

(1)

They a r e c h a r a c t e r i z e d i n t h e Appendix o f

t h i s chapter.

J.L.

324

u

and therefore

LIONS

E b,

We now set Problem 1.1 i n the precise form which follows : Problem 1.2 satisfying ~ _

-

We look for

u

E Ir

(defined i n ~ _

(1.13) and ..such _ ~. -__- that

(1.14)

u ( 0 ) = go

for

x > 0,

(1.15)

U(T) = gl

for

x < 0,

with .__ go

(1.7)) _

and -

gl

x > o

restriction to

elements i n the trace space of

(resp. x < 0 )

of

b.

We shall solve in what follows Problem 1.2 under a somewhat stronger hypothesis on Theorem 1.1

-

and

gl:

We assume that m

(1.16)

go

-x

0

2

2

-X

e

gl(x

2

) dx <

m.

T h e n Problem 1.2 admits a unique solution.

Remark 1.1: __

It will follow from the proof of Theorem 1.1 that

~

if

N(go,g,)

denotes the square root of the expression i n

(1.16), then

where here and i n what follow the

CIS

denote various constants.

n

BOUNDARY VALUE PROBLEMS O F M A T H E M A T I C A L P H Y S I C S

2.

325

P r o o f o f uniqueness.

A Lemma.

2.1.

Let us define

(2.1)

I

b 0 = [v

~ r ,V(O)

v

for

x > 0,

V(T) =

o

01.

x <

for

o

=

Then

Proof:

Since the regions

x > 0

x < 0

and

play s y m m e t r i c

roles, i t w i l l be s u f f i c i e n t t o prove t h a t

But J = l i m Jm,

(2.4)

W e have, s i n c e

m +

v(x,O) = 0

m ,

for

x > 0:

il/m hence ( 2 . 3 )

2.2.

follows.

Uniqueness.

Let

u

be solution o f P r o b l e m 1 . 2 w i t h

L e t us prove t h a t

of

0

(1.13) w i t h

u,

u = 0.

g o = gl

Taking the s c a l a r product i n

w e obtain

= 0. H

J.L.

325

Q'

aat, U uldt

[x

(2.5)

LIONS

+

iT ,

[x

that

u

(2.6)

-

[ u , 1 ] I 2 d t = 0.

i'

Since according t o ( 2. 2) f o l l o w s from ( 2 . 5 )

-

[u

'0

[u,1] =

a u uldt at,

o

2

0,

it

a.e.

The r e f o r e a U

x--=

at

(2.7) and

= Lu,1]

[u]

to

u(x,O) = 0

>

3.

Proof

3.1.

for

of

u ( x , t ) = cp(t)

so that

t h e n ( 2 . 7 ) reduces x

9 dt -

0

independent o f

u(x,t) = k

so that

one h a s

0,

0

k = 0

and s i n c e

u = 0.

and

0

existence,

Reduction o f

t h e problem,

Let us s e t w(x,t) = go(x) for x

(3.1)

B y v i r t u e of

(1.16)

>

g,(x)

0,

w(x) E H

w E L2(0,T;H) Therefore

w E

Ir

f o r x < 0 = w(x).

so that and

x

aw at =

0.

and i f we s e t

0 = u-w)

(3.2)

Problem 1 . 2 i s r e d u c e d t o t h e e q u i v a l e n t : Problem

(3.3) where

3.1

-

Find

0 E Ir0

(defined i n (2.1))

a@ + 0 at

X-

[@,1] = F,

x;

such t h a t

327

BOUNDARY VALUE PROBLEMS O F MATHEMATICAL PHYSICS

(3.4)

F = -W

F(x,t) = F(x)

(in fact

3.2.

+

[w,11

E H

and

E L2 (o,T;H) [F,1]

= 0).

@

Elliptic regularization. For

E

> 0, one considers the equation'')

b

subject to the boundary conditions a6(0) = 0

(3.6) (3.7)

= 0

-? ( lBOE)

at

for

for

x > 0, $I

c

%

a x < 0, -(T)

(T) = 0 for = 0

at

for

In what follows we solve (3.5)(3.6)(3.7) then

E

4

0.

x < 0

x > 0.

and we let

@

Variational form o f (3.5)(3.6)(3.7). On

(3.8)

Iro

we define the continuous bilinear form

E

[X

aU at,

av

x -1dt

at

[u

-

+

[X

aU , E

vldt

t

[u,l] ,vldt.

Using (2.2) we have

Let us notice that

Ir0

is ___ closed i n

Ir

for

\lvIl,,

and let us verify that (l)It is called the "elliptic regularization" of (3.3) although the operator appearing i n (3.5) is not an elliptic operator!

328

J.L.

(3.10)

2 ac ( v , v ) 2 ceIIvllb

W e define

[

LIONS

c

T

(3.11)

111 111

IIIVII!

=

i s a n o r m on

av 2 d t

Ir0

+

(same p r o o f

*

[v-[v,11l2 d t ;

than i n Section 2 ) .

I f we check t h a t

k 0 i s complete f o r

(3.12)

t h e n ( 3 . 1 0 ) follows from ( 3 . 9 ) ,

Proof o f ( 3 . 1 2 ) :

(3.13)

Let

- [Vn,11 av

(3.14)

x-

at

L e t u s d e n o t e by i n d e p e n d e n t of

x

(E

M

n

E

+ g

+ h

in

i s equivalent

I(( 111.

Then

L~(o,T;H)

2

L (0,T;H).

in

t h e s e t o f f u n c t i o n s o f L2 ( 0 , T ; H )

L2(0,T)).

2 L (o,T;H) = E

(3.15)

11 /I

b e a Cauchy s e q u e n c e for

vn

wn = v I 1

s i n c e then

(11 111

Then 2

e

L~(o,T;H)

where 2 L ~ ( o , T ; H =)

(3.16) If

(V

I

[v(t),lI =

P = o r t h o g o n a l p r o j e c t i o n on

o

a.e.1.

2 Lo(O,T;H),

(3.13)

means Pvn + g Therefore,

+

en

+

On t h e o t h e r hand ( 3 . 1 4 ) av (3.18)

at

en = e n ( t ) E E

there exists vn

(3.1-7)

h/x

in

2 Lo(O,T;H).

in

k

in

such t h a t

2 L (0,T;H).

implies: 2 L (O,T;H(x0,m))

(x,

>

0).

BOUNDARY VALUE PROBLEMS OF MATHEMATICAL PHYSICS

Since

(3.19)

vn(xyO) = 0 vn

x > x

for

329

(3.18) implies:

0'

T

+ . x(

h(x,s)ds

in

2

L (O,T;H(x

ym)).

,O Comparing (3.17) and (7.19) it follows that (3.20)

2

+ e in L

en

+ k-e = v in n so that (3.12) is proven. c1 But then

Application.

v

L2(OyT;H)

There exists a unique function

that ac(gE ,v) =

(3.21)

3.3.

( 0 , ~ ) .

c

One verifies that

@e

A priori estimates

(I).

We take

v E

GC

at

E Ir0

Ira.

Since

CF,~]

= 0,

can write (3.22)

=

ac(QcYQ

i'

[F,

@ e - [ @ c ,111dt-

Using (3.9) it follows that

(3.23)

@ C - [ $ $ is c ybounded l] in

(3.24)

dG

3.4.

at

x a @ e is bounded in

A priori estimates (11). We write ( 3 . 5 ) as follows: x -a=@, at

Qc

9

such

(3.5)(7.6)(3.7).

satisfies

in (3.21).

v = @

+

[F,vldt

h = x av -,

and

L2(0,T;H) 2

L (0,T;H).

0

we

330

LIONS

J.L.

(3.27)

~ , ( x , o )= o

for

x < 0,

(3.28)

$ g ( ~ , T )= 0

for

x > 0.

We are going to show that these conditions imply

Ji,

(3.29)

= x $t

is bounded in

2

L (0,T;H).

We observe that G

(3.30)

E

2

is bounded in L (0,T;H) (by ( 3 . 2 3 ) ) .

It follows from ( 3 . 2 6 ) ( 3 . 2 7 )

that

If we set 1

0

for

t > 0,

for

t < 0

I

then (3.31) is equivalent to

where we think of and for

t > T.

Gs

as extended by, say,

0

for

t < 0

Therefore,

T dt)1/2.

We shall have a similar estimate for

x > 0

(integrab 2

ing backward i n time),

s o that after multiplying by

integrating, we obtain ( 3 . 2 9 ) .

c1

e-x

and

331

BOUNDARY VALUE PROBLEMS OF MATIIEMATICAL PIIYSICS

Application.

It follows from (3.23)(3.29)

(3.33)

3.5.

is bounded i n

Qc

and (3.12) that

2

n

L (0,T;H).

Proof - o f existence. According t o (3.33) and (3.29) we can extract a sub-

oC,

sequence, still denoted b y

!be

such that

Q

-$

in

bo

weakly i.e. QF1 + Q

xat and

@

E bo.

fact i n

Then

L2(0,T)

-t

x

in

L2(0,T;H)

a@

in

weakly,

L2(0,T;H)

weakly

at

+

[QE,l]

[@,l]

in

L2(0,T;H) weakly (in

weakly) and one c a n pass to the limit i n

(3.5) to obtain (3.3) so that the existence of

I$ is proven

and the proof is completed.

Appendix:

A trace result.

We prove in this Appendix:

(All

f o r every

v E Ir

(cf. definition in

uniquely define its trace v(0)

v(0);

R

v

when

spans the space 2 of function

measurable on

(1.7))

g

one c a n

spans

Lf,

which are

and such that

In other words, (A2)

1 I

(')Of

2 % = L (W; mdx), 2

m(x)

= 1x1

for

1x1 s 1,

e

-X

for

1x1 > 1. (1)

course "1" does not play any essential role here and can be replaced by any finite xo > 0.

332

J.L.

LIONS

Proof:

where

as

H(1,m)

i s defined as i n ( 1 . 9 ) ,

H(-m,-l)

with

I f we d e n o t e by restriction of

x < -1),

v

i n s t e a d of

t o the s t r i p

1x1

resp.

< 1

v,)

the

( r e s p . x 7 1, r e s p .

we c a n c o n s i d e r t h e mapping

v

+

which i s a n i s o m e t r y from spaces

'I;

H(1,m).

( r e s p . v:,

v1

and l e t u s d e f i n e

b l , b;,

b

+

tv1'V2,v;I onto

bl

X b;

x L';

(where t h e

a r e equipped w i t h t h e i r n a t u r a l n o r m s ) .

T h e r e f o r e ( A l ) w i l l b e p r o v e d i f we v e r i f y t h e f o l l o w ing results:

v1 + v1(0)

maps c o n t i n u o u s l y

2 L (-1,l;

of f u n c t i o n s

(A51

v2

+

v2(0)

L2(lym); e

onto t h e space

b;

onto t h e space

Ixldx);

maps c o n t i n u o u s l y -X

b,

2 dx).

(Of c o u r s e we h a v e t h e n a n a n a l o g o u s r e s u l t f o r

2 ) Proof of ( A h ) :

Let us s e t , f o r

v1 = v E bl:

b;)

333

BOUNDARY VALUE PROBLEMS OF MATHEMATICAL PIIYSICS

Since we are now on a bounded interval, we have

(A71

w E L~(o,T;L'(-I,~)),

and according to ( A 6 )

A

Let us denote by operator i n

L2(-1,1)

the unbounded self adjoint

of multiplication by

X

A

be the domain of

.

Let

provided with its graph norm.

D(A)

Then ( A 7 )

and (A8) are equivalent to

and by hypothesis

aw

(A10)

at E

2

L ( 0 , T ; L2(-l,l)).

Therefore (cf. J.L. L i o n s [l]) i.e.

the space of functions

Then

v(0)

3 ) Proof of

(since

-

spans

D(A'/'),

such that

spans

L2 (-1,l; Ixldx).

(A5):

If v2 = v E b 2 ,

+

is bounded on

X

h(x)

w(0)

we have in particular

(l,+m))

av v, - €

at

2

L ( O P T ;H ( 1 , m ) )

so that v ( 0 ) E H(1,m).

Reciprocally i f

+ b2

belongs to H(l,m)

2

g E H(l,m)

and

= L (1,m; e

the function

v(x,O) = g(x),

-x2

dx).

0

so that

v(0)

spans

334

J.L. LIONS

Bibliography of Chapter I11

R.

Beals [ 11, An abstract treatment of a class of “forwardbackward” differential and integro-differential problems. To appear.

H.G. Kaper

111,

Full-range boundary value problems in the

kinetic theory of gases transport theory.

-

Proc. Fourth Conf. on

Blacksburg. Va. 1975.

J.L. Lions El], Espaces interm6diaires entre espaces hilbertiens et applications.

Bull. Math. R . P . R .

Bucarest, 2 , (1958), p. 419-432.

Chapter I V Introduction to the theory of homogenization

Introduction, I n this chapter we consider only the very introduction to the theory of homogenization i n connection with composite materials. This chapter can be thoughtof as a very preliminary introduction to the book o f A . Bensoussan, J.L. G.

Papanicolaou [ 2 ] .

Lions and

We consider here only the simplest

situation for stationary problems.

Other problems are studied

BOUNDARY VALUE PKOBLEMS OF MATHEMATICAL PIIYSICS

335

i n the b o o k quoted: evolution problems,

n o n linear problems,

and, in particular, problems o f wave propagation i n composite materials, where also other methods than those indicated here are studied. For the Bibliography, we refer to the Bibliography of the b o o k .

Let us only mention here that the method indicated

i n Section 3 i s due to L. Tartar [l]

.

The plan is as f o l l o w s : 1.

2.

3.

Elliptic homogenization.

The problem.

1.1.

Notations and hypothesis.

1.2.

Problem.

1.3.

Homogenization.

Method o f asymptotic expansions. Homogenization formulas. 2.1.

Multiple scales.

2.2.

Asymptotic expansion.

Weak convergence,

3.1.

Statement o f the result.

3.2.

A priori estimates.

3.3.

Adjoint functions.

Bibliography.

335

1.

J.L.

LIONS

E l l i p t i c homogenization.

The problem.

N o t a t i o n s and h y p o t h e s i s .

1.1.

n We c o n s i d e r i n

Rn

t h e cube

Y =

lO,yg[ ;

we

j=1 ai j

consider functions

(1.1)

I

ai j E aij

n

Let

such t h a t

L ~ ( I R ~ ) , i s Y-periodic

,

i.e.

be any open s e t i n

0

admits t h e p e r i o d y . i n y J j'

assumed t o be bounded

Rn,

t o f i x ideas.

n

In

we c o n s i d e r t h e f a m i l y of

operators

As

given

bY

(1.3) where

0

>

0.

We want t o s t u d y t h e a s y m p t o t i c b e h a v i o u r o f operator

A'

e -+ 0.

when

the

L e t u s s t a t e t h e problem more

precisely.

1.2.

Problem.

B y v i r t u e of space o f

Hi(n))

(1.1)(1.2), g i v e n

f

in

H-'(n)

t h e r e e x i s t s a unique f u n c t i o n

u

(dual

c

that

(1.4)

uE E Hl,(%

(1) I n t h i s c h a p t e r w e u s e t h e summation c o n v e n t i o n .

such

337

BOUNDARY VALUE PROBLEMS OF MATHEMATICAL I'IIYSICS

E

(1.5)

A u

G

= f

in

n.

We want to study the behaviourLc

1.3.

u

E

-

as

E

-b

0.

Homogenization.

What we intend to prove in what follows is that, weak sense, ___ .

u

F

converges to

u,

6~

the solution o f a boundary

value problem for a (unique) elliptic operator

G

with

constant coefficients:

The operator A'

G

is called the homogenized operator of

.

Remark 1.1:

We shall give below explicit formulas for comput-

ing (or a1 least allowing to make numerical computations) the coefficients of

G.

Formulas are not straightforward, as one

can imagine after realizing that already in the one dimensional case the homogenized operator of d

is given by

(1.8)

Remark 1.2:

From a physical view point, one can say that the

coefficients of material.

0

G

are the effective coefficients of the

J.L.

338

2.

LIONS

Method of asymptotic expansions

2.1.

-

Homogenization formulas.

Multiple scales.

Let us consider x E 0,

y E Rn

x

and

as independent variable,

y

and let us consider functions such that @ ( x , y ) , measurable in

Y-periodic in The operator

a ax j

x, y,

y. X

applied to

@(x,r)

gives

Therefore

where

A

(2.3)

1

= - -

a

a yi

(2.4) (2.5)

The idea of expanding functions in series of functions of

x

and

y

-

and to replace

y

by

x/k

at the end

-

a variant of the technique of multiple scales in singular

perturbations.

2.2.

Asymptotic expansion. We try to find

fo r m

u

6 '

solution of

(1.4)(1.5),

in the

is

339

BOUNDARY VALUE PROBLEMS O F MATHEMATICAL PHYSICS

u

y

P o

+ c2

u2

n,

x E

+

...

u. J

Y-periodic,

x/c).

r e p l a c e d by

We u s e

u1

uj(x,y),

uj

(with

+ c

= u

c

( 2 . 2 ) and we i d e n t i f y t h e powers o f

i n equation

(1.5).

C

-2

,

0

-1

,

We f i n d : Aluo

= 0,

Alul

+

AZuo = 0 ,

A1u2

+

AZul

+ A 3u 0 =

f.

0

We remark t h a t : (2.10)

the equation

A1@

solution i f f

I,

is satisfied,

8 Y-periodic,

= F,

F(y)dy = 0;

if

admits a

t h i s condition

i s d e f i n e d up t o an a r b i t r a r y

@

0

additive constant.

I t f o l l o w s f r o m ( 2 . 1 0 ) ( 2 . 7 ) and f r o m t h e f a c t t h a t i s Y-periodic

in

y,

that

uo(x,y) = u(x)

(2.11) Then ( 2 . 8 )

uo

i n d e p e n d e n t of

y.

0

becomes

(2.12) ul(x,y)

i s Y-periodic.

The r i g h t hand s i d e o f of f u n c t i o n s o f

x

(2.12) i s a t e n s o r i a l product

by f u n c t i o n o f

y.

i m p o r t a n t from a p r a c t i c a l view p o i n t ' ' ) . i n t r o d u c t i o n of

xj(y)

This f a c t i s very

It leads t o the

such t h a t

(')We r e f e r t o L a b o r i a R e p o r t s , numerical a p p l i c a t i o n s .

i n p a r t i c u l a r by B o u r g a t ,

for

J.L.

340

- ayi i3

=

AIXj

(2.13)

The c o n d i t i o n o f

LIONS

xj

aij(y),

(2.10)

i s Y-periodic.

is satisfied,

s o t h a t (2.13)

admits a s o l u t i o n , defined up t o an a d d i t i v e c o n s t r u c t .

X j

f i x i d e a s we c h o o s e

such t h a t

f

(2.14)

” T

X j ( y ) d y = 0.

’Y 2.12)

Then t h e g e n e r a l s o l u t i o n o f

(2.15)

A1u2

(2.16) a Y-periodic iff

+ G+).

x)

We now u s e ( 2 . 1 1 ) ( 2 . 1 5 )

&

= f

s o l u t i o n of

‘(

F dy = 0

-

-

A2u1

(2.16) IYI

i n (2

is

0

9):

A u = F;

3

exists,

according t o (2.10),

= measure o f

Y),

i.e.

/

(2.18)

Gu = f,

(2.19)

GU =

(2.20)

qij

-‘ij

=

axiaX.J (aij

+ 4,

-

u9

axj

a . -) ik a Y k

dY9

(2.21) T h e s e a r e t h e f o r m u l a s g i v i n g t h e homogenized o p e r a t o r .

0 Remark 2 . 1 : directly,

T h i s c o m p u t a t i o n i s formal..

I t can be j u s t i f i e d

b u t w e g i v e i n S e c t i o n 3 below a b e t t e r s o l u t i o n . 0

BOUNDARY VALUE PROBLEMS O F MATHEMATICAL P H Y S I C S

Remark 2.2:

341

One c a n v e r i f y t h a t t h e above f o r m u l a s g i v e

@

(1.8).

Remark 2 . 3 :

The o p e r a t o r

W(Y) = s p a c e o f f u n c t i o n s on o p p o s i t e s i d e s o f

G

Let us d e f i n e :

is e l l i p t i c .

which t a k e e q u a l v a l u e s

J, E H1(Y)

Y;

We o b s e r v e t h a t

(2.13) i s equivalent t o

and t h a t

(2.23)

I Y I qij

I n d e e d by v i r t u e o f

x i- y i ) . = xi ,

= a,(xj-yj,

(2.2)

with

Theref o r e

IYI and i f

qij

5.5. = 1

5 1. 5 J. = a , ( e , e ) ,

qij

J

then

0

a,(e,e)

e

= S,(X

= 0;

i

-yi>,

therefore

8 = 5

( a c o n s t a n t ) and c o n s e q u e n t l y

:ixi - => o But

5, +

siyi

is

r e s u l t follows.

3.

3.1.

Y

periodic i f f

+ FiYi'

5,

= 0

V

i

and t h e

4

Weak c o n v e r g e n c e . Statement o f

the result.

W e c o n s i d e r more g e n e r a l boundary v a l u e problems t h a n

342

J.L.

(1.4)(1.5).

V

W e introduce Ho(h2) 1

(3.1)

E V

and f o r

LI,V

(3.2)

aE(u,v)=

LIONS

E

such t h a t V 6 H1(n),

we s e t

'h2 B y v i r t u e of

(3.3)

ay(v,v)

al(vll

2

6:

E V

2

I1 /I

,

f E V'

Therefore, given u

we have (1)

(1.1)(1.2)

,

,

= norm i n

V.

t h e r e e x i s t s a unique

such t h a t

(3.4)

ag(ue,v) = ( f , v )

a~ v

V.

W e d e f i n e next

(3.5)

G(u,v)

where t h e

=

t h e r e e x i s t s a unique

(3.6)

E

u

G(u,v)

Theorem 3.1

-

When

6:

3.2.

V =

0,

-b

u

(3.7)

a r e g i v e n by ( 2 . 2 0 ) ( 2 . 2 1 ) .

qo

and

qijfs

E

Then

such t h a t

( f , ~ )

V v

E

V.

one has

+ u

in

V

weakly.

A p r i o r __ i estimates,

By virtue of

(3.3),

one has

(3q8)

If we s e t

(1)

a

0

2

a > 0

can be r e l a x e d i f , i n p a r t i c u l a r ,

1 V = H,(n).

343

BOUNDARY VALUE PRODLENS O F M A T I E M A T I C A L P I I Y S I C S

(3.9) we have

(3.10) W e can w r i t e t h e e q u a t i o n

(3.11) and i n v a r i a t i o n a l ( a n d more p r e c i s e ) f o r m

(3.12) We c a n e x t r a c t a s u b s e q u e n c e , s t i l l d e n o t e d by u E ,

<:,

such t h a t

(3.13)

Then

u

-b

u

in

L2(62)

s t r o n g l y , or a t l e a s t l o c a l l y

€

strongly,

L2(h)

in

so that weakly,

X

a.(-)u = E

c

h(ao)u,

-b

and ( 3 . 1 2 )

-) av

a xi

+

h(ao) =

1 Jy)

a.

dy,

gives i n the l i m i t

(h(ao)u,v) =

( f , ~ )

+

v E V.

n s t o show t h a t

(3.15)

3.3.

Adjoint

functions.

We d o n o t r e s t r i c t t h e g e n e r a l i t y by a s s u m i n g t h a t a

= 0.

Let

P(y)

b e a n homogeneous p o l y n o m i a l o f l S td e g r e e

344

J.L.

be t h e s o l u t i o n ( d e f i n e d up t o a n a d d i t i v e

w

and l e t

LIONS

c o n s t a n t ) of

A*w = 0 , 1 w-P

i s Y-periodic.

If we s e t

=

=

AZP,

A;,*

w

We d e f i n e

-X*,

w-P

xI

= Y-periodic.

by

E

(3.19)

w (x) = E

= p

E W (X 4

6

-

EX*(,-,. X

We h a v e

(P)*w

(3.20)

E

=

0.

w

W e now u s e t h e a d j o i n t f u n c t i o n s let

Cp

be i n

multiply (3.20)

~Q(hl); by

we t a k e

cpu

-

E '

as follows;

E

v = cpw

i n (3.12)

E

and w e

a f t e r s u b s t r a c t i o n we f i n d

But

in

L2(hl)

weakly

s o t h a t t h e l e f t hand s i d e o f ( 3 . 2 1 ) c o n v e r g e s t o I

We a l s o o b s e r v e t h a t (since

a

(EX*(;)) j

X

4

0

in

cpwE + cpP L2(n)

in

H:(n)

weakly),

weakly SO

that t h e

BOUNDARY VALUE PROBLEMS O F MATIIEMATICAL I ' H Y S I C S

r i g h hand s i d e o f

(3.21)

dx =

(3.23)

to

converges

(5i,

a

~ ( C p p ) )

(using (3.14))

1

LL

comparing w i t h ( 3 . 1 2 ) :

hence,

Consequently

i.e.

I f wo t a k e

P = y. Then

w = y

W e find

-

i

X,

9

i Az(X*

-

yi)

= 0,

(3.15) with qij = b ( a i j

-

and ( 3 . 2 4 ) g i v e s

Xi q). a

akj

The d e s i r e d r e s u l t w i l l b e p r o v e n i f we v e r i f y t h a t

345

J.L.

346

LIONS

Bibliography o f Chapter IV

I. Babuska [l], Tech. report, Univ. o f Maryland, 1974. N.S.

Bakbalov [l], Doklady Akad. Nauk, 218

(1974),

p. 1046-1048. A. Bensoussan, J . L .

Lions and G . Papanicolaou [l],

C.R.A.S.

281 (1975), P-89-94,317-322. A.

Bensoussan, J.L. Lions and G. Papanicolaou [2], Asymptotic methods __in Periodic Structures.

North-Holland, 1978.

E. de Giorgi and S. Spagnolo [l], Sulla convergenza... Boll. U . M . I .

8 (1973), p. 391-411.

L. Tartar El], French Japan Colloquium.

1976.

Tokyo, September

G.M. de La Penha, L.A. Medeiros (eds.) Contemporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations @North-Holland Publishing Company (1978)

MEMORY EFFECTS IN ONE-DIMENSIONAL PROBLEMS OF CONTINUUM MECHANICS

R.C. MACCAMY' Department o f Mathematics Carnegie-Mellon University Pittsburgh, Pennsylvania 15213

1. Introduction.

This paper concerns equations o f the form, ut(x,t) = OX(X,t)

+ f(x,t)

0 < x < 1,

t > 0,

(H)

utt(x,t) = ax(x,t)

+ f(x,t)

0 < x < 1,

t > 0.

(El

The designations (H) and (E) are used since we think of these equations as representing one-dimensional heat flow and elasticity respectively. require

u(0,t)

In both cases

u(1,t)

e

f

is given and we

0.

Equations (E) and (H) represent balance laws.

One

obtains models by making constitutive assumptions which connect

a

u.

and

There are three choices we want to consider: a(x,t) = g(u,(x,t))

(NM)

( N o memory);

(SM)

(Short memory);

( LM)

(Long memory) ;

a(x,t) = g(ux(x,t))

o(x,t) = cpbx(x,t))

+

d

+ h(ux(x,t))uxt

a(r)$(ux(x,t-'))dT.

The special forms are chosen on the basis o f two

*

~

This work was supported by the National Science Foundation.

348

R.C. MACCAMY

considerations:

(i) some relation to experiment (ii) some

chance of analysis of the resulting models. is so limiting for nonlinear models.

It is (ii) that

The form (LM) seems to

be the simplest nonlinear model consistent with the fading memory theory of continuum mechanics (see Section 2).

(NM)

and (SM) are obtained as approximations to (LM) as follows. Write,

t in (LM).

W (u,(x,t))

(UX(X,t-T))

-

TVI'

(Ux(xtt))uXt(x,t)

Then (NM) is obtained by keeping only the first

terms and (SM) by keeping both.

This is the specialization

to o u r case of the approximation theorem of Coleman and N o 1 1

C41. There are four properties of models which we want to consider.

These are:

(E)

Existence of solutions, (U).Uniqueness of solutions,

(A)

Asymptotic stability, (F) Finite propagation speed.

t +

(A) means that if

f

4

0

(f+r)

( F ) means that if

f

=_

0

and the initial data have compact

support there will be a such that

u(-,t)

T > 0

as

m

then u

4

0 (u-bu).

(depending o n initial data)

has compact support in

(0,l)

for

t .< T.

The properties of existence and uniqueness are further subdivided according to what one is willing to accept as a solution.

One can ask for classical solutions,that is

o n e s i n which all derivatives are continuous.

Then there are

two different kinds o f generalized solutions.

The first are

those obtained by Galerkin methods as in parabolic problems (see [ 1 6 ] ) .

The second are "shock" solutions obtained by

finite difference methods as in nonlinear hyperbolic problems

MEMORY EFFECTS I N O N E - D I M E N S I O N A L

(see

171 ) .

W e us’e ( C ) ,

and

(G)

349

PROBLEMS

( S ) t o d e n o t e t h e s e t y p e s of

solutions. We summarize t h e r e s u l t s known t o u s , w i t h a few references,

i n the following t a b l e . Model

Equation

E

Solution

U

F

A

References

H

Yes Yes Yes

No

[ 141

H

Yes Yes Yes

No

[: 161

H

Yes Yes Yes

No

H

Yes*Yes*Yes*Yes*

11 ,161 , [ I 8 1 C ~ I , r l 9 1 , [211 :

[

E

Yes Yes

E

No

E

Yes*

E

Yes Yes Yes

No

E

Yes Yes Yes

No

E

Yes*Yes*Yes*Yes*

(*):

N o Yes

[: 151 ?

Yes*Yes*

C 71 ,[ 81 ,C 131 [ l o ] ,[I71 C 161 191 ,L: 201

f o r small data. If a l l t h e c o n s t i t u t i v e a s s u m p t i o n s a r e l i n e a r ,

the

e n t r i e s i n t h e t a b l e can b e v e r i f i e d by e l e m e n t a r y methods, i f one pays a t t e n t i o n t o s i g n s .

a l l the entries

(except

O n t h e o t h e r hand,

go

(5)

It i s t o be understood t h a t

=

0)

c o v e r some n o n l i n e a r i t y .

t h e a s s e r t i o n i s o n l y t h a t t h e e n t r i e s hold

__

u n d e r ._. some ( u s u a l l y v e r y s p e c i a l ) h y p o t h e s e s . We b e l i e v e t h e t a b l e r e v e a l s t h e i n c r e a s e d f l e x i b i l i t y i n t h e models p r o v i d e d by t h e i n t r o d u c t i o n o f g e n e r a l i z e d s o l u t i o n s and o f memory. questions.

It a l s o s u g g e s t s some open

The q u e s t i o n mark u n d e r

o n e example.

E,

f o r

NM,

The c a s e o f l a r g e d a t a i s a n o t h e r .

c a n p r o b a b l y b e removed u n d e r E , NM, n o t f o r t h e LM c a s e s . (H,LM(vsO),S)

U

and

S

is

The a s t e r i s k

b u t almost c e r t a i n l y

Note t h a t t h e c a s e s

(E,LM,S)

S

(H,LM(ep&O),C),

a r e a l l missing.

For t h e s e

350

R.C. MACCAMY

interesting cases, we know

of no results, although a beginn-

ing for the last is provided in

121.

In this paper we present a detailed outline of one case,

(H,LM(q=O),C)

and comment briefly on

(E,LM,C). These

cases seem interesting physically and they also illustrate two o f the important tools at one‘s disposal, energy integrals and Riemann invariants.

We remark that results for classical

solutions in more than one dimension are almost nonexistent because of the absence of the second tool.

2. Fadinn Memory.

The theory of fading memory ( [ 2 ] ,[ 31 ,[41 ) can be For

described as follows.

where

k

p

2

1

is essentially positive.

cesses i n which one state variable

y(t)

according to the history of tional over

H

g

= $(pt)

3

and

with

p

t

8

Hp(k)

by,

Then one considers pro-

y

where

is related to another pt(T) = p ( t - 7 ) defines

is to be a continuous nonlinear func-

E H.

q(%)

Suppose that

define

and

j, ( 5 )

are continuous func-

grows algebraically of order

tions on

R

and that

Then, if

a

is suitably restricted, the formula,

s(u)

= Cp(P(0)) +

[

a(T ) $ (!J ( T ) )dT

defines a continuous n o n l i n e a r f u n c t i o n on

Hp(k).

tion (LM) amounts to requiring that, for each

p-1.

(2.1)

Thus assumpx,

MEMORY EFFECTS I N O h % - D I M E N S I O N A L

PROBLEMS

351

t a(x,t) = a(u,(x,-)). f o r either (E) o r (H)

Appropriate i n i t i a l d a t a ,

u n d e r t h e a s s u m p t i o n (LM) would be a knowledge o f

t < 0.

We assume h e r e ,

for

t S 0.

where

,

u(x,t) for

f o r s i m p l i c i t y only, t h a t u ( x , t )

Then problems

(H)

and

0

( E ) become,

t La[ vl ( t ) =

a ( t-7 ) V ( T )dT

.

3 . Energy I n t e g r a l s . S e c t i o n s t h r e e and f o u r a r e c o n c e r n e d w i t h t h e problem ( H * ) , essence o f

a >

0.

utt where

that

(H*)

H,

is

LM(cp=O),C.

by c o n s i d e r i n g t h e s p e c i a l c a s e

Differentiation yields,

+

-

a ut

$(uX), = f t

u (x) = f ( x , O ) . 1

equation.

One can u n d e r s t a n d

utt

a(t)

,

i n this case, u ( x , O ) = 0 , ut(x,O)=ul(xh ( 3 . 1 )

T h i s i s a "damped" n o n - l i n e a r wave

Nishida [ 2 1 ]

f o r the equation

+ af,

the

-at = e

+

c o n s i d e r e d t h e i n i t i a l v a l u e problem aut

-

~

(

u= ~ 0

)on ~

(-m,m)

x

(0,m).

H e showed t h a t t h i s problem h a s a u n i q u e c l a s s i c a l s o l u t i o n , f o r s u f f i c i e n t l y small data. Our p o i n t h e r e i s t h a t a s Nishida's

(11*)

i s e s s e n t i a l l y t h e same

problem f o r a l a r g e c l a s s o f

we o b s e r v e t h a t i f

we d i f f e r e n t i a t e

a's.

To s e e t h i s

the equation i n (H*)

we

$ ( u ~ )i n~ t h e f o r m ,

can s o l v e f o r

1

d

(3.2)

352

R.C.

where

i s a new f u n c t i o n , d e t e r m i n e d by

k

F(x,t) = Equation

1 a( o> f t ( x , t )

a,

and

+ k(O)f(x,t) + Lk[f(x,*)] ( t ) .

(3.3)

( 3 . 2 ) c a n be r e w r i t t e n a g a i n a s ,

+

utt where

MACCAMY

a =

a(O)k(O)

aut

-

a(o)w ( u x )X

(3.4)

= R #

and,

R ( x , ~ ) = a ( o ) { ~ ( x , t +) i c ( o ) u ( x , t )

We c a n now d e s c r i b e t h e t w o p a r t s of (1) O b t a i n a p r i o r i bounds for

T,

independent of the r e c t a n g l e ( 2 ) Show t h a t t i o n s of

IIR(

*

, .)I1

our strategy:

L1(RT),

f o r any s o l u t i o n of

which a r e

(H*).

(RT i s

(O,l)x(O,T)).

t h e a p r i o r i bound i n (1) y i e l d s g l o b a l s o l u -

(3.4).

The s e c o n d t a s k i n v o l v e s N i s h i d a ‘ s i d e a s and i s outlined i n the next section. (1).

Suppose t h e f o l l o w i n g c o n d i t i o n s a r e s a t i s f i e d : a(0) > 0,

Then

Here we s k e t c h t h e i d e a s f o r

a = a(O)k(O) >

k(0)

0,

c,:

(3.6)

E Ll(O,m).

and one c a n v e r i f y t h a t t h e r e q u i r e d

0

bound i n (1) w i l l h o l d if independently of

>

f,ft

E L1(R,)

and

I)ll\\L1(RT) g

M,

T.

Suppose we h a v e a n e s t i m a t e o f t h e f o r m ,

-

v ( t ) z-t d Lk[v](T)d7

2

Y

Then i f we m u l t i p l y ( H ” )

Local t h e o r y a p p l i e d t o

(t)dt,

by

ut

(3.4)

Y >

0 for a l l

and i n t e g r a t e o v e r

s h o w s that

T.

RT

(3.7)

we

(F) i s satisfied.

MEMORY EFFECTS I N ONE-DIMENSIONAL PRODLEMS

9 15

153

ll~llL~(R~) 2 + 2 11 IL~(*,O)IIL~(~,~)' 2

If we assume

ut(x,O) = f(x,O).

where we have used

(3.8)

it follows from (3 3) and (3.5) that

f,ft E L ~ ( R , )

If we assume, in add tion, that

F E L2 R,).

$'(<)

2

E

> o

(1.9)

then (3.8) yields,

where

C

T.

i s independent of

from the next to last and

T h e last inequality follows

u(0,T) = u(1,T) = 0.

We obtain a second energy inequality by multiplying (H*)

by

u

and integrating over

little algebra and the use of

RT.

(3.6),

This yields, with a

(3.9) and ( 3 . 1 0 ) , the

n e w estimates,

The constants

C

and

C1

arbitrarily small by making

in (3.10) and (3.11) can be made f

small.

The two sets of estimates (3.10) and (3.11) c a n be extended by multiplying successively by integrating with an induction process

.

tmut

and

tmu

and

The result is that

if (3.6) i s strengthened to tm;,

tmi; E L1(O,m),

(3.qm.

354

R.C. MACCAMY

Tm/lU (

t h e n one f i n d s

*

,T ) )I L# 2 ( o , 1 )

.

I: C

For

m=2

t h i s shows

< C where C + 0 a s f + 0 . The l e n g t h y L1(RT) d e t a i l s o f t h e above c o m p u t a t i o n a r e c a r r i e d o u t i n [ 1 9 ] . that

IIUII

(In the calculation f o r assumption

I$(;)]

w i s e bounds

for

1

'r

KlFl.

S

u

m

one makes t h e p r o v i s i o n a l

S i n c e i n t h e end we o b t a i n p o i n t -

t h i s i s only a t e c h n i c a l d e v i c e . )

X

The above s k e t c h shows t h a t t h e c o m p l e t i o n o f (1) i s r e d u c e d t o e s t a b l i s h i n g

( 3 . 6 ) m and ( 3 . 7 ) .

I n order

t h a t t h e theorems have meaning t h e s e e s t i m a t e s m u s t , c o u r s e , b e i n f e r r e d from c o n d i t i o n s on

a.

task

of

T h i s c a n be done

by means of what a r e c a l l e d f r e q u e n c y domain methods i n t h e t h e o r y of

feed-back

control (see

151).

The f r e q u e n c y domain method i n v o l v e s t h e L a p l a c e transform.

A

Let

b e t h e t r a n s f o r m of

a

checked t h a t t h e t r a n s f o r m of

in (3.2)

k

Then i t c a n b e

a. is,

I n o r d e r t o proceed we need some a d d i t i o n a l t e c h n i c a l assumpt i o n s on

a,

namely,

a E C(')[O,m), This implies t h a t the estimate,

A

a (s)

a ( k ) E L1(O,co) exists in

A

a (s) = a ( 0 ) s

-1

+

##

Re s 2

A(0)s

-2

+

0

(3.13)

2.

k S

and s a t i s f i e s

o ( s - ~ )a s

s

4

m.

(3.12) y i e l d s then, A

k (s) = -A(O)a(O)

-2

( 3 . 1 4 ) , i n t u r n , suggests t h a t

s

-1

+

o(s-l)

as

s

+

m

k ( 0 ) = -&(0)a(O)-2

(3.14) thus the

7 T h i s e s t a b l i s h e s (A) once we have g l o b a l e x i s t e n c e .

## I t

can b e shown t h a t aEL ( 0 , m ) a t l e a s t i s t o b e e x p e c t e d i n t h e c o n t e x t of h e a t f t o w p r o b l e m s , s e e [: 191.

A ( 0 ) < 0.

(3.6) i s s a t i s f i e d i f

of

f i r s t part

The above argument i s v e r y f o r m a l . e s s e n t i a l assumption s t i l l needed.

in

-m

< n <

a

A

t h e s e c t i o n ) , If

(3.14) i s defined i n

kA(s)

-

-1 kms

recover

,

cannot v a n i s h i n

(s)

Re s

km = a A ( O ) - ' ,

Re s

so

0

kA

Note t h a t

0.

s = 0.

near

2

O n e can now

by u s i n g t h e i n v e r s i o n f o r m u l a ,

k

+m

k ( t ) = km

f

s

0,

2.

(3.15)

m.

comment o n t h i s c o n d i t i o n a t t h e end o f

(3.15) holds then

T h e r e i s one

This i s ,

Re a A ( i n ) > 0 (We

355

PROBLEMS

Ml3MORY EFFECTS I N O N E - D I M E N S I O N A L

+

i.

(2ri)-'

e i q t ( kA ( i q )

-

km(in)-')dn.

(3.16)

The t e c h n i c a l d e t a i l s of t h e above a p p e a r i n [ 1 9 ] .

It i s shown t h a t i f of

t j a ( k ) E Ll(O,m)

t h e form

(3.2)

satisfies

s a t i s f i e s s o m e a d d i t i o n a l assumptions

a

(3.6).

and

( 3 . 7 ) a s we i n d i c a t e now. by

t > 0,

= k(t)

v,(t)

k,

then

The c o n d i t i o n Let

vT(t)

(3.15) also yields

b e d e f i n e d on

t < 0.

for

=O

(-=,a)

Then by P a r s e V a l ' s

r e l a t i o n and t h e c o n v o l u t i o n theorem t h e l e f t s i d e of is,

(3.7)

+=

N

where

A

kA(s) A

-nk

(in)

-

4

A

l

I ~ G (2~ d)nI ,

~m k " ( i q )

> 0

(iq) 7 0

-n I m k A ( i q ) 2 Y

for

q f 0.

>

as

0

s = 0

near as

(3.17)

m

-a(0)a(O)-2

kms-l

= a (O)-l

n

i s t h e F o u r i e r transform o f

vT

implies -qk

(3.16),

a s d e f i n e d by

n

4

0.

for all

q

vT.

Now

(3.14) s h o w s that 4

yields

m

and t h e e s t i m a t e A

-n I m k

Hence t h e r e i s a

n

and

(3.15)

( i n ) -+ km = y >

0

such t h a t

(3.7) follows f r o m (3.17)

and a n o t h e r a p p l i c a t i o n o f P a r s e v a l f s r e l a t i o n . W e comment o n c o n d i t i o n

(3.15).

We have, f o r n > 0 ,

R.C.

356

c

A

Re a ( i q ) =

a(.t)cos q t d t =

‘a(t) >

This w i l l be p o s i t i v e i f

=

Irn

a(t)dt > 0

MACCAMY

a ( t ) > 0.

if

con2itions are s a t i s f i e d if f o r example,

a ( t ) = e-at.

f o r example,

a ( t ) = e-atcos

t h e c l a s s of

a’s

-

fern

n

A Re a (0) =

Also

0.

i ( t ) s i n q t dt

Thus a l l t h e r e q u i r e d

(-l)k a ( k ) ( t ) 2 0.

k = 0,1,2;

This condition i s not necessary;

B t

s a t i s f i e s (3.15).

s a t i s f y i n g (3.15)

Note t h a t

i s closed under p o s i t i v e

l i n e a r combinations,

4.

Riemann I n v a r i a n t s . ~We c o n s i d e r h e r e p a r t

( 2 ) of

t h e program o u t l i n e d

i n S e c t i o n ( 2 ) t h a t i s , we t r e a t ( 3 . 4 ) .

These i d e a s w e r e

u s e d by N i s h i d a [ 211 b u t were o r i g i n a l l y i n t r o d u c e d b y Lax

h5].

The f i r s t o b s e r v a t i o n i s t h a t t h e b o u n d a r y p r o b l e m for ( 3 . 4 ) c a n b e c o n v e r t e d t o a n i n i t i a l v a l u e problem b y p e r i o d i c e x t e n s i o n of t h e d a t a ( s e e [ 2 0 ] ) .

We r e w r i t e ( 3 . 4 )

as a

system,

Vt =

x’

w

t

+

aw

-

v =

= R,

x’

w = ‘

t’

(4.1)

W e i n t r o d u c e the Riemann i n v a r i a n t s ,

r

= w 5 r(v),

r(v)=

c

C o n d i t i o n ( 3 . 9 ) i m p l i e s t h a t t h e map t o one from

RxR

onto

RxR.

Jm

ds.

(v,w)

Equations

-b

(r,s)

( 4 . 1 ) become,

(4.2) i s one

MEMORY EFFECTS I N O N E - D I M E N S I O N A L

3 57

PROBLEMS

The next step is to introduce the characteristic curves, x = x 1 (t,e) =

We let

D

p

h

D !J

and

h

6’

+

x = x (t,)’) = y 2

dT,

+

denote differentiation along these

curves, that is,

a

a D,,==+X-,

a

a

a

(4.5)

a

Dp = -a t f ’ ” Y ’

The classical local theory is obta ned by solving

(4.4)

(4.6)

and

on a time interval

[O,T].

If, in this local

theory, one can obtain a priori bounds for the derivatives of r

and

then one can extend to a global theory.

s

It is

here that the ideas of L a x and Nishida enter as we describe now. We differentiate ( 4 . 3 ) ,

x

with respect to

and

obtain, rxt

+ hrxx

= -hrrx

Next we have from D s = st

A

=

2

-

hssxrx

-

a(rx+sx)+ R ~ .

(4.7)

(4.5) and (4.6),

+ Isx

+

= st

usx

+ (h+)sX

= D s

CI

+

2XsX

-a(r+s) + 2 h s x + R,

or

D s

We note that

X

and

x

=p

a

2h

s

+

Q (r+s) 2X

-

are functions o f

R 2T ’

(r-s)

(4.8) o n l y with

358

X <

R.C. MACCAMY

0.

1 h = - log(-X(r-s)) 2

If we set

then

(4.6) yields, (4.9)

(4.8) and (4.9) into (4.7) and rearrangement

Substitution of yields, finally,

h

Dx(e r

X

+

)

e

h

h

r x ( a + h r rx} = Dx g + Rxe ,

(4.10)

(4.10) is the basic result used by Lax and

Equation

Nishida (in the case

R

existence result f o r

E,NM,C.

g = 0

and

(4.2) yields

xz +

Then (4.10) becomes,

a = 0, R =

(with of

D

0)

the initial data.

assumption).

0).

Then

First we indicate Lax's nonF o r that problem the above

a = 0 and

theory applies with case

E

R

E

if

0

h r = -$"/4$',

e-harz2 = 0.

r

Xr

2

6 > 0

and

In this

0.

h z = e r

Set

X

and

$''(E)

Suppose then that e

=

Integration of

yields bounds for

-h

f

s

< 0

.

(4.3)

in terms (Lax's

becomes infinite

z

in finite time. The existence result o f Nishida and its extension a r e , in principle, just as simple,

z = e

With

-h

r

X

equation

(4.10) has the form, D z

x

+

az

+

e-hhrz

2

=

x.

(4.11)

One can consider this as an ordinary differential equation for

z

along the characteristic.

Since

perturbation theory argument s h o w s that

a > 0, a standard

(4.11) will have a

global solution, for sufficiently small initial values if one can keep

x

and

emhXr

accomplished by using

under control.

This latter can be

(4.3) and our a priori estimates for u.

MEMORY EFFECTS I N O N E - D I M E N S I O N A L

A n a n a l o g o u s argument i s u s e d f o r

159

PROBLEMS

The d e t a i l s a r e a g a i n

S .

q u i t e t e d i o u s and c a n b e found i n [ 2 0 1 .

5.

Remarks

on Case

E,LM,C.

(E")

leads t o

E,LM,C

The problem

i n S e c t i o n 2.

T h i s c a s e h a s p r o v e d somewhat more d i f f i c u l t . h i n t of

One g a i n s some a ( t ) = -e

t h i s by c o n s i d e r i n g a g a i n a s p e c i a l c a s e

D i f f e r e n t i a t i o n of

(E*)

-

=

uttt+ autt

yields ~(u.),

-0 t

.

then, + acp(ux)x +

af

+

(5.1)

f t *

T h i s r a t h e r u n p l e a s a n t l o o k i n g e q u a t i o n w a s a n a l y z e d by Greenberg

[9].

Remarkably, he was a b l e t o e s t a b l i s h e x i s t e n c e

and u n i q u e n e s s , f o r s m a l l d a t a , w i t h h i s o n l y e s s e n t i a l assump

I' ( q ) ,

0 <

tion being

0

< cp' ( { ) .

T h e r e i s a n o t h e r s p e c i a l c a s e which h a s been t r e a t T h i s i s t h e one i n which

ed s u c c e s s f u l l y .

t h e same f o r m , t h a t i s h ( t ) = !-l +

i,"

a(r)dP

cp(5) = then,

p$ (

5).

cp

and

$

have

I f we s e t

i n t h i s c a s e , i n t e g r a t i o n of

(E*) y i e l d s ,

Ut

) + F.

= Lb(WbX)

(5.2)

X

Equation

( 5 . 2 ) looks e x a c t l y l i k e (H*).

however a c r u c i a l d i f f e r e n c e . a E

2 we u s e d t h e f a c t t h a t

I n a l l t h e a n a l y s i s of Ll(O,m).

L1( O , m )

,

i n fact

,

one h a s

b(t )

t h e e f f e c t of d e s t r o y i n g t h e p r o p e r t y t o the analysis i n Section 2.

Section

I t c a n be shown ( s e e

[ 2 0 ] ) t h a t i n t h e c o n t e x t of e l a s t i c i t y problems b be i n

There i s

4

bm

>

0.

w i l l not T h i s has

( 3 . 7 ) which was c e n t r a l

Indeed suppose

b ( t ) = bm

+

b

1

R.C. MACCAMY

3 60 E Ll(O,m).

where

bl

b;(s)

is continuous in

-1

bA = b m s

Then one has Re

(3.11) one sees then that

-nk

A

If one forms

0.

s 2

+

(in)

-*

0

as

n

+

A

where

bl(s)

kA

as in

0.

The difficulty just raised is overcome i n [20] but with considerable effort, k = 0,1,2

and (3.9) holds then (E")

solution for small data. and (F). on

a

The result is that if (-l)kb(k)(t)>O

Moreover, one has properties ( A )

It is indicated in [ 2 0 ] that the above conditions

are actually suggested by stress relaxation experiments cp

The case in which a

has a unique classical

is not a multiple of

is a general function still remains open.

W

and

This would offer

a model with considerably more flexibility.

References

[l] Barbu, V. (1976).

Nonlinear Semigroups and Differential

Equations in Banach Spaces, Noordhoff. [2] Coleman, B . D . ,

(1965).

G u r t i n , M.E.,

Herrera, I., and Truesdell, C.

Wave Propagation in Dissipative Materials,

Springer-Verlag.

[3l Coleman, B.D. and Mizel,

V,J.

(1966).

Norms and semi-

norms in the theory o f fading memory, Arch. Rat. Mech. and Anal. 23, 87-123.

[4]

Coleman, B.D.

and N o l l ,

W. (1960).

An approximation

theorem for fimctionals with applications i n conkinuum mechanics, Arch. R a t . Mech. and Anal.

[ 5 1 Corduneaunu, C.

(1973).

6,

355-370.

Integral equations and Stability

o f Feedback Systems, Acad. Press.

MEMORY EP'FECTS I N ONE-DIMENSIONAL PROBLEMS

Crandell, M.G.,

London, S . - 0 . ,

and Nohel, J.A.

361

(1976).

An abstract nonlinear Volterra integrodifferential equation, MRC Tech. S u m m . Glimm, J. (1965).

Report 1684.

Solutions i n the large for nonlinear

hyperbolic systems of equations, Comm. Pure and Appl. Math. 18, - 697-715. Glimm, J., and Lax, P.D. (1970).

Decay o f solutions o f

systems o f nonlinear hyperbolic conservation laws, Amer. Math. SOC. Mem. 101. Greenberg, J.

A-priori estimates f o r flows in dissipa-

tive materials, to appear in Journ. of Diff. Eqn. Greenberg, J., MacCamy, R.C.,

and Mizel, V.J.

(1968).

O n the existence, uniqueness and stability o f the equation

~'(ux)uxx + huxtx = poutt,

and Mech. 17,

707-728.

Greenberg, J. and MacCamy, R . C .

(1970).

nential stability o f solutions o f putt,

Journ. o f Math.

O n the expo-

E(ux)uxx

+ Xuxtx =

Journ. Math. Anal. and Appl. 3 1 , 406-417.

Gurtin, M.E.,

and Pipkin, A.C.

(1968).

A general theory

of heat conductionwith finite wave speeds, Arch. Rat. Mech. and Anal. 3 1 , 113-126. Johnson, J.L, and Srnoller, J.A.. (1969).

Global solutions

f o r an extended class o f hyperbolic systems of

conservation laws, Journ. Math. Mech. 16, 201-210. Ladyzenskaja, O.A., Solonnikov, V.A., and Ural'ceva, N.N. (1968). Linear and Quasilinear Equations o f Parabolic Type, Amer. Math. SOC.

36 2

R.C.

Lax,

(1964),

P.D.

Development of

tions of non-linear equations,

[ 1 6 ] Lions, J.L.

MACCAMY

Journ.

(1969).

s i n g u l a r i t i e s of s o l u -

hyperbolic p a r t i a l d i f f e r e n t i a l

Math.

Phys.

5 , 611-613. -

Q u e l q u e s m6thodes d e r 6 s o l u t i o n d e s

p r o b l & m e s aux l i m i t e s non l i n & a i r e s , G a u t h i e r - V i l l a r s .

[17] MacCamy, R . C .

(1970).

Existence,

u n i q u e n e s s and s t a b i l -

i t y of s o l u t i o n s o f t h e e q u a t i o n u

a

tt

20, -

= ax ( ~ ( u , )

+

h(ux)uxt)..

I n d . Univ.

Math.

Journ.

231-238.

[I81 MacCamy, R . C .

(1976).

S t a b i l i t y t h e o r e m s f o r a c l a s s of

f u n c t i o n a l d i f f e r e n t i a l e q u a t i o n s , S I A M Journ. Math.

Appl.

3 0 , 557-576. -

[ l 9 ] MacCamy, R . C .

(1977).

An i n t e g r o - d i f f e r e n t i a l

with a p p l i c a t i o n i n heat flows,

Quart.

equation

of A p p l .

Math.

35, 1-20. [ 2 0 ] MacCamy, R . C .

(1977).

A model f o r o n e - d i m e n s i o n a l non-

l i n e a r v i s ' c o e l a s t i c i t y , Quart.

of Appl.

Math.

35, -

21-33 [21]

N i s h i d a , T.

G l o b a l smooth s o l u t i o n s f o r t h e s e c o n d

order quasilinear equation with f i r s t order dissipation (preprint).

G.M. de La Penha. L.A. Medeiros (eds.) Contemporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations W o r t h - H o l l a n d Pub1i s h i n g Company (1978)

A GENERAL FRAMEWORK FOR PROBLEMS I N T H D STATICS O F F I N I T E ELASTICITY

WALTER N O L L Department o f Mathematics Carnegie-Mellon U n i v e r s i t y Pittsburgh Pa 15213

-

1. I_n t r_ o d u_ c t i o_n .

~

I t i s a c l i c h 6 t h a t llmathematical p h y s i c s ” i s t h e s t u d y of that

” b o u n d a r y - v a l u e ” and

i n itial-value

such problems a r e ltwell-posedll i f

uniqueness,

and s t a b i l i t y o f

solutions.

Th is i s a narrow

t h e zero-load

t h a t govern t h e

i t i s t o t a l l y i n a d e q u a t e when

these equations a r e non-linear, d i s c u s s i o n of

a s w i l l b e e v i d e n t from t h e

problem i n S e c t i o n

5.

I n S e c t i o n s 2 and 3 I w i l l g i v e a b r i e f of

t h e concepts of

a c o n t i n u o u s body and o f

a c t i n g on s u c h a b o d y .

I n Section

meant b y a n e l a s t i c b o d y , s u c h a body.

of

t h e body t h e l o a d - s y s t e m

Section

4, a f t e r

description

a load system d e f i n i n g what i s

I w i l l d e s c r i b e the load o p e r a t o r

of

t h i s placement.

and

one c a n p r o v e existence,

v i e w e v e n when t h e d i f f e r e n t i a l e q u a t i o n s given s i t u a t i o n are l i n e a r ;

problems

This o p e r a t o r a s s o c i a t e s with each placement n e c e s s a r y t o k e e p t h e body i n

A f t e r d i s c u s s i n g zero-load

problems i n

5 , I p r e s e n t , i n S e c t i o n 6 , a g e n e r a l framework f o r

problems i n e l a s t o s t a t i c s .

The new f e a t u r e o f

t h i s framework

i s t h a t one p r e s c r i b e s , n o t b o u n d a r y d a t a , b u t a n environmental r e a c t i o n o p e r a t o r w h i c h s p e c i f i e s how t h e e n v i r o n m e n t would

3 54

WALTER NOLL

a c t on t h e body f o r e v e r y c o n c e i v a b l e p l a c e m e n t . then i s t o f i n d the placements,

i f any,

The problem

f o r which t h e l o a d

o p e r a t o r and t h e e n v i r o n m e n t a l r e a c t i o n o p e ' r a t o r h a v e t h e I n a n o t h e r t y p e of problem one p r e s c r i b e s a one-

same v a l u e .

p a r a m e t e r f a m i l y of s u c h e n v i r o n m e n t a l r e a c t i o n o p e r a t o r s , which s p e c i f i e s a c o n t r o l l e d g r a d u a l change of t h e environment.

7

I n Section

I discuss the modifications necessary t o

accommodate e x t e r n a l c o n s t r a i n t s and i n S e c t i o n 8 I t o u c h on the subject of s t a b i l i t y . Notation:

If

,

:=

a c o l o n i s s e t b e f o r e an e q u a l i t y s i g n ,

t h e n t h e l e f t s i d e o f t h e e q u a t i o n i s c o n s i d e r e d t o b e defined by t h e r i g h s i d e . denoted by A

Dom f

i s a s u b s e t of

The domain o f a mapping

:= D D,

a n d i t s codomain by t h e n t h e image o f

denoted by

f > ( A ) := ( f (x)

d e n o t e d by

Rng f

:= $(Dom

denote i t s i n v e r s e by t o p o l o g i c a l space

f+.

R,

numbers i s d e n o t e d by by l~

if' and

1

x E A]. f).

Cod f

:= C. is

The r a n g e of

f

is

i s i n v e r t i b l e , we

Clo(A).

A

t h e s e t of a l l n o n - n e g a t i v e

a r e l i n e a r spaces, then

Lin(L,$)

L

into

t h e s e t o f a l l l i n e a r isomorphisms from abbreviate

L i n ( b ) := L i n ( L , b )

and

lr

Lis(L)

Syrn2(lr

2

,W).

l y isomorphic t o a subspace := Lin(b,tR).

We w r i t e

The s p a c e Sym(lr , b * )

IJJ and onto

P

X

.

If

Lis(b,W)

Llr.

We

:= L i s ( b , L ) .

2

Sym2(lr , R ) of

reals

denotes t h e

The s p a c e o f a l l symmetric b i l i n e a r mappings f r o m

i s d e n o t e d by

of a

The s e t o f r e a l

and t h e s e t o f a l l s t r i c t l y p o s i t i v e r e a l s by

u1

If

f

A

The c l o s u r e of a s u b s e t

s p a c e of a l l l i n e a r mappings f r o m

L*

is

C

under

f

If

i s denoted by

D

f: D

Lc

i n t o lb

i s natural-

Lin(l, ,b * ) ,

where

355

PROBLEMS I N THE STATICS OF FINITE ELASTICITY

If set

a,

f: fi fi

4

5

is a mapping of class

from some open sub-

C2

of a finite-dimensional affine space

then

vf: r9

4

and

Lin(b,lb)

8

V(')f:

the first and second gradient of

If

f.

3

E

into another,

Sym2(b 2 ,111) denote

A C Dom f, yoECod f,

we read '2 x E A , as "Find the elements

2.

x

in

f(x) A,

= yo

if any, such that

f(x)=

yot'.

Placements _________of a continuous body.

In this section we review some basic notions concerning continuous bodies.

More details can b e found in [N1].

Before defining the mathematical structures that idealize the concept of a continuous body, one must fix a class D

D

of "displacements".

F o r definiteness, we here take

to b e the class of all invertible restrictions to open

sets of C2-diffeomorphisms between Euclidean spaces. The structure o f a continuous body on a set

03

of

"material points" is determined by the prescription of a class of placements, which are bijections from F o r our

sets of Euclidean spaces.

B

onto open sub-

purposes here, it is

sufficient to confine attention to placements of fixed (3-dimensional) Euclidean space

C!?.

B

in a

This space

E

is

a mathematical idealization not only of the background against which one wishes to observe the body

f3,

but also of the

environment which may act onand react to the body translation space

l~

of

C!?

a,

The

is a (3-dimensional) inner

366

WALTER NOLL

product space, whose elements are called -spatial vectors. _-___ ___

63

Let a body €

the environment

be given.

form a set

P

The placements of

8

in

with the following

properties:

u E

Each

Rn

set

P

E ,

of

n.

n,Y E

then

uoYt E

E D,

R u = Dom h ,

f,

onto an open sub-

63

called the region occupied by --____

the placement

If

63

is a bijection from

> 0

det v(n0Y')

and

ID

in -

(valuewise).

u E P,

If

det(vX) > 0

(valuewise), then

Rng X holz.

C

€,

and

E P.

F o r simplicity we make the following assumption:

F o r some (and hence every)

B

in

class

C1

by

E

u E P,

the region

is bounded and has a boundary

aRn

occupied that is of

. It is useful to imbed the body

closure

Ru

8.

8

into an -_.-___ abstract

This closure, uniquely determined by

ha

to

within a n isomorphism, is characterized by the property that every placement

E: i

4

Clo(Rx)

The closure space, and write

-

8

6!

u E P

has a unique invertible extension

which makes the diagram shown commutative.

has the natural topology of a compact Hausdarff is the closure of

Q3

in that topology.

a63 := fi\E and c a l l i t the boundary of the body

We Q3.

3 67

PROBLEMS I N THE STATICS OF FINITE ELASTICITY

5

Let

be some finite-dimensional affine space with

L!J.

translation space

the mapping sense.

If

C1(R,a)

placement

-

Vuf

-b

3

is of class

x

u E

and

P,

is the mapping

the gradient of

Cr(63,5).

vUf: 8

J

f

Lin(lr,L!J)

in the defined by

~

:=

Similar definitions describe other

(V(fox+))ou.

differential operators in a given placement. f E C2(S,3),

v(2)f U

then

:=

We denote by mappings

-+ 3

f: R

Co(E,3)

f

in

F o r example, if

63

(o(2)(fout))~x:

denotes the second gradient of

g.

-+

Sym2(L

~~

extension of

is uniquely determined by

f

to

E co(m,3)

u E P,

all)

8.

Cr(G,3)

f

to

8.

,L!J)

the set of all continuous

which have a continuous extension

Of course,

2

x.

the placement

f.

We denote b y

C1(i,3)

-

f

to

We often

omit the superimposed bar and use the same symbol

all

is of

in the ordinary

Cr

The set of all such mappings is denoted b y

E

f

mu

foNe:

5

J

i f f o r some (and hence all) N E P,

( r = 0,1, o r 2)

Cr

class

f: 63

We say that a mapping

f

for the

the set of

with the property that f o r some (and hence

the gradient

Vuf

has a continuous extension C2((,3).

A similar definition applies to ( r = 0 , 1 , o r 2)

The set

has the natural structure of an

infinite-dimensional affine space whose translation space is the linear space

Cr(E,b).

If

3

i s a Euclidean space,

there are natural magnitudes, denoted by

b , b, Lin(L,b),

Syrn2(lr

2

,LO),

etc.

can define, f o r each placement C2(R,Ur)

1 1,

on the spaces

Using these magnitudes one N.

E P

a norm

11 11%

on

by

These norms are all equivalent and give

C2(0?.,b)

the structure

368

WALTER NOLL

of a complete normable (but not normed) topological linear space.

Of course,

acquires the structure o f a

C2(i,$)

complete topological affine space. The set

P

03

of all placements of

in

E

can be

identified, by extending the codomains of the placements to

e,

with a subset of the affine space

C2(B

,e ) .

Using an P

extension theorem of Whitney one can prove that ly open in u E C2(i,lr) N+U: of

@

is such that

+ E ,

P

Specifically, if

C2(G,€).

IIuIln

K

€ P

is actual-

and if

is small enough, then

defined by value-wise addition, becomes a member

after its codomain is restricted to its range, P

I t is not hard to see that subset of

is a connected open -_ -

The proof of this fact depends not only

C2(i,E),

on the assumed positivity o f the determinant in ( P 2 ) , but also very strongly on the assumption that the displacements have extensions that are diffeomorphisms from __ all of itself.

E

onto

If we had assumed merely that displacements are E ,

diffeomorphisms between open subsets of

we would not

have excluded, for example, a mapping from a simple torus in E

onto a knotted torus in The body

63

e.

has the natural structure of a

C

(3-dimensional) manifold of class

Zx

at ‘ X E 63

2

.

The tangent space

serves as a mathematical model for the

infinitesimal body-element surrounding the material point With every placement associated a placement at -

X.

w. E P

o f the whole body

W ( X ) E Lis(Tx,b)

Given any mapping

there is a unique gradient

f: &

4

Vf(X)

3

8

X.

is

of the body-element _-

of class

E Lir1(3~,h)

C’

and

X E hd,

such that

3 69

PROBLEMS IN THE STATICS OF FINITE ELASTICITY

vf(X) holds for all placements at

VK(X)

X

=

(vwf(X))(vx(X))

k

E P.

In particular, the gradient

n

of the placement

is the placement of the

X.

body-element at

3. Load-systems. ______ Let

63

be a continuous body as described in the

previous section. k ,

values i n

We consider external actions on

as described in [ N

1,

Sect. 8 , 9 .

1

bd

Bor(z)

volume-measure -

the set

%,(ff)

,

Bor(E)

Vx

-b

k

is the volume of

and

k

is the

Ax(ff)

K

.

that is absolutely

n E P

Bor(i)

into A

It

will be called a

l~

that is abso-

with a continuous

f o r some (and hence all)

called a surface-load.

Bor(E)

with a continuous density

lutely continuous with respect to

tx: 363

on

o f the boundary a R

into

for some (and hence all)

A measure f r o m

A,

e

n bRn

n,(6))

continuous with respect to

body-load. -

one can introduce a

V,(p)

in the Euclidean space

A measure from

density

0,

of

E Bor(ii)

surface area of the piece

-+

x

and a surface-measure

VK

F o r every ' 6

as follows:

6

on

of all Bore1 subsets of the closure

Given any placement

w.

Li,

R.

of

b :

with

Such external

actions determine vector-measures, with values in the collection

a,

A measure from

Bor(E)

x E

P

into

will be

Ir

that

is the sum of a body-load and a surface-load will be called a load-system o n -

63

will be denoted by

and the collections of all load-systems L.

Thus, if

Z, E L

and

x E

P,

there

370

are

WALTER NOLL

bR E C o ( i , b )

(3.1)

tX E C o ( a b d , l r )

and

i

&(P) =

bWdV

+

tWdAll

pnaba

P

63 E B o r ( i ) .

for all

to two placements

L E L

The densities o f

n

and

nx(Y)

where the value

such that

of

Y

are related by

n

n '* at3

unit normal to the surface

lr

-+

at

f:

-+

111

Y E aR

at the point

ed towards the exterior of the region

If

corresponding

'n

is the

K(Y),

direct-

*

is a continuous function with values

in some finite-dimensional vector space

111

and i f

4, E L ,

one can f o r m integrals

i

(3.4)

fad& =

'k

E

p

for all

Lin(b ,111)

i

(f@bw)dVll

+

Bor(i).

(f@tu.)dAW

a n,!

63

The values

o f such integrals belong to

and we have

The resultant force of a given load system is defined to be the value load

A 21 99

( 4 , ) E Lin(lr)

placement

w E

P

&(i)

of a system

and the point

(3*6)

AN,,(&)

:=

1-

4,

of

& E L

q E fl!

(u.4)

e.

at

& E C

The astatic ___

relative to the

is defined to be @

d.e,

B

where

n-q:

i+

Ir

The skew part of

is defined by value-wise point-difference. A

n,,

(k)

is called the resultant moment of

PROBLEMS I N THE STATICS O F F I N I T E ELASTICITY

the load system relative to

L

The set

n

and

q.

of all load systems has the natural

structure of a n infinite-dimensional linear space.

n

define, for each

37 1

E P,

11

a norm

on

L

One can

by

(3.7) where

b

X

and

are the densities which make (3.1) valid.

tn

These norms are all equivalent and give

L

the structure of

a complete normable (but not normed) topological linear space. The set Lo := (4, E L

I

t(E) = 01

of all load systems with zero resultant force is a closed subspace of L . Remark: The absolute continuity requirements o n the loadsystems may b e too strong f o r some purposes, and it may be necessary to replace

L

by a suitable space of vector

measures of more general type.

4. Elastic

response.

By an elastic body we mean a continuous body

fl

endowed with additional structure by the prescription of a family

(gX I x

E

a)

of intrinsic-stress functions

An explanation o f this definition is given i n [ N 2 ] , I t is useful to consider, for each placement Piola-Kirchhoff-stress functions

n E

Sect. P,

the

14.

,372

WALTER NOLL

d e f i n e d by

for all

n E

E

F

Lis(lr).

We assume t h a t f o r some (and h e n c e a l l )

t h e mapping

P,

Lin(b)

c

i s of class

6

from

(X,F)-+n,x(F)

x Lis(b)

into

.

1

n E

We s a y t h a t t h e p l a c e m e n t

i s homogeneous

P

A

R

f o r t h e e l a s t i c body We s a y t h a t

da

placements.

If

aon

n

n

:=

We s a y t h a t

H

then

a

Rn

? n ( L ) = 0.

If

n

i s a n a t u r a l place-

Rn

onto a n o t h e r s u b s e t

i s a g a i n a n a t u r a l placement.

i n a g i v e n placement

c e r t a i n load system

p

ft

E

P,

%"(U) E L 0 a

b e c a l l e d t h e e l-a s t i c - l o a d

be g i v e n .

I n o r d e r t o keep

one must a p p l y t o

z:

The mapping

o p e r a t o r of

8.

P

f i r s t define the stress operator

W e then d e f i n e t h e body-force

-

TU

: P

operator

+

a

+ Lo w i l l

n

and

C1(z,

Lin(b))

by

: P

Co(i,b)

by

. H.

.rr

En(il)

8

To d e s c r i b e t h i s

o p e r a t o r , we c h o s e a n a r b i t r a r y r e f e r e n c e p l a c e m e n t

(4.3)

i s homo-

Tn ,x' E P i s a n a t u r a l p l a c e m e n t i f L f i.;

i s a congruence f r o m

uon

x

If

is

e,

onto a n o t h e r s u b s e t o f

L e t a n e l a s t i c body

63

a

A

T

homogeneous and i f

t?,

i t a d m i t s homogeneous

i s a g a i n a homogeneous p l a c e m e n t .

ment and i f

X.

i s a homogeneous p l a c e m e n t and i f

g e n e o u s , we w r i t e

of

d o e s n o t depend on

w. ,X

i s homogeneous i f

an a f f i n e b i j e c t i o n f r o m then

T

if

:= - d i v n ( T , ( M ) )

J

37 3

PROBLEMS I N THE STATICS O F FINITE ELASTICITY

w

and t h e s u r f a c e - t r a c t i o n

t U :P

operator

Co(aba,b)

4

by

(4.4) n : a63

where

i s t h e e x t e r i o r u n i t normal f'unction

b

4

k

described i n t h e previous of

63

s i d e of

6'

operator

R,

Also, p

placement

and t h a t

63

.e"(p)(E) = 0 P

a n d maps

q E

and a n y p o i n t

i s zero.

for all

t!

if

%

#

E P

so that

as indicated

relative t o the

i s symmetric, i . e . ,

.e"(u)

t h e load system

However,

!J

Lo,

into

%,q(i(b))

the a s t a t i c load

r e s u l t a n t moment o f q

the right

( 4 . 5 ) d o e s n o t d e p e n d o n t h e c h o i c e of t h e r e f e r e n c e

d e p e n d s o n l y on

above.

O f course, i t turns out that

E Bor(6).

placement

and

The e l a s t i c - l o a d

i s t h e n g i v e n by

for all

L

section.

p,

relative to

then

A

N.19

the LI

(.e"(p)) n e e d

not be symmetric.

It i s e a s i l y seen t h a t i f 8,

ment of

then

z(p)

= 0.

-L

The l o a d o p e r a t o r If

p E P

s u b s e t of

where

and i f

t!,

va E

a

then

Orth(b)

n If

has the following property:

i s a congruence from

aop E P

i s the gradient value

f o r each

Rp

onto another

and

6:

The l o a d o p e r a t o r e n t i a b l e because,

i s a n a t u r a l place-

ll

k

P

4

Lo

E P,

a.

i s Frgchet-differ-

the body-force

and t h e s u r f a c e t r a c t i o n o p e r a t o r i s a homogeneous p l a c e m e n t ,

of

-t N

operator

are differentiable.

then the FrGchet-differ-

WALTER N O L L

37 4

DG : P

ent a l s

P

DTN

4

2

Lin(C ( 6 , b ) , C o ( a , b ) )

Lin(C'(fi,b),

4

bN

C o ( a @ , b ) ) of

and and

tx

a r e given

bY

(4.5)

2

where

:=

K

v'?

: Lis(b)

elasticity-tensor ment of

K .

4

is k n o w n a s t h e

Lin(Lin(b))

It

f u n c t i o n o f t h e body r e l a t i v e t o t h e p l a c e -

The F r G c h e t - d i f f e r e n t i a l

P + L i n ( C * ( f i , b ) ,Lo)

Di:

t h e l o a d o p e r a t o r i s g i v e n by

(4.9) ((DC(P))u)(P)

\

=

(Dcn(p)u)dVN+

P for all

63 E B o r ( 6 ) .

5. Z e r o - l o a d p r o b l e m s . Let

B

b e a n e l a s t i c body w i t h l o a d o p e r a t o r

a s described i n the previous section.

G i v e n any

-

.C,

L o E Lo'

one c a n c o n s i d e r t h e p r o b l e m

Such a p r o b l e m i s c a l l e d a d e a d - l o a d i n g

problem.

If

do

#

0

s u c h problems a r e p h y s i c a l l y a r t i f i c i a l b e c a u s e i t i s d i f f i c u l t , i n the r e a l world, a load-system ment

on t h e body t h a t d o e s n o t v a r y w i t h t h e p l a c e -

. If

(5.2)

t o c r e a t e an environment t h a t e x e r t s

do = 0,

then

?cI E P ,

( 5 . 1 ) becomes t h e z e r o - l o a d problan X(P) = 0 ,

375

PROBLEMS I N THE S T A T I C S OF FINITE E L A S T I C I T Y

which has the following vivid physical interpretation: x,

the body, in some given placement

and then "let it go"

s o as to end u p completely free of external forces.

placements

from

RCI

(5.2).

of

What

can it assume?

il

If

Take

1-I

is a solution o f (5.2)

onto a subset of

E ,

then

and

aoU

W e say that two solutions o f

a

a congruence

is again a solution

(5.2) that are relat-

ed by such are congruence are equivalent solutions. We assume n o w that

03

has natural placements.

It

is evident that these are solutions to the zero-load problem. Physical experience tells u s , however, that they need not be the only solutions.

F o r example, if the region

Rx

occupied

by a body made of rubber has the shape o f a hemispherical shell

i n a natural placement

to (5.2), not equivalent to

CI

x, x,

one can expect a solution that corresponds to an

"eversion" of the shell (see Fig. 1).

Figure 1

If the shell is thin enough, one can experimentally produce 1) a third solution

M I ,

corresponding to eversion of only the

middle o f the shell, as shown i n Figure 1.

Actually, if the

eversion or partial eversion is produced by pushing down with a concentrated force while holding the rim alone; the circle, one can produce additional solutions to (5.2).

They correspod

1)

I am indebted to R. Fosdick for showing this to me.

37 6

WALTER NOLL

t o t h e o n s e t o f ”popping” i n t o t h e e v e r t e d o r p a r t i a l l y e v e r t e d p l a c e m e n t and a r e p h y s i c a l l y u n s t a b l e . A more c o m p l i c a t e d s i t u a t i o n c a n b e e x p e c t e d when

the region has

Ex

o c c u p i e d by

63

x

i n the n a t u r a l placement

t h e s h a p e of a “ r u b b e r m a t ” w i t h many h e m i s p h e r i c a l i n -

dentations ( s e e Figure 2 ) .

Figure 2

(5.2),

One e x p e c t s t h a t t h e r e a r e many i n e q u i v a l e n t s o l u t i o n s t o e a c h c o r r e s p o n d i n g t o “ e v e r s i o n ” of a p r e s c r i b e d s u b s e t of t h e s e t of

indentations.

one e x p e c t s a t l e a s t

Zn

Thus, i f

there

solutions.

are n indentations,

I n addition,

t h e r e may

be p h y s i c a l l y u n s t a b l e s o l u t i o n s . I n t h e examples a b o v e ,

the region

o c c u p i e d by

t h e body i n i t s n a t u r a l p l a c e m e n t s f a i l s t o b e convex. o f f e r the following conjecture:

If

I

t h e s t r e s s f u n c t i o n of a n

e l a s t i c body s a t i s f i e s c o n d i t i o n s a p p r o p r i a t e f o r m a t e r i a l s s u c h as r u b b e r and i f

t h e body o c c u p i e s a convex r e g i o n i n

i t s n a t u r a l placements,

t h e n t h e s e n a t u r a l placements a r e t h e

only s o l u t i o n s t o t h e zero-load

problem.

If a n e l a s t i c body h a s no n a t u r a l p l a c e m e n t ,

there

a r e c i r c u m s t a n c e s when one c a n e x p e c t t h a t t h e z e r o - l o a d problem h a s no s o l u t i o n s a t a l l . bodies

8,

and

B2

shown i n F i g u r e 3.

Consider,

w i t h n a t u r a l placements

f o r example,

nl

and

n2

two as

177

PROBLEMS I N TllE STATICS C F F I N I T E ELASTICITY

Figure n

Assume t h a t

T

A

= T

“I1

62

.

One c a n t h e n

together as indicated i n Figure which i s

BI

‘lgliie”

n

X

8

in

and

fi2

3 t o o b t a i n a s i n g l e body

l o c a l l y homogeneous i n t h e s e n s e t h a t

material point n

3

fl

f o r every

u

t h e r e i s a placement

such t h a t

n

= T = T i s indepeiideilt o f Y €or a l l Y i n some “I1 N2 n e i g h b o r h o o d of X. However, B h a s no n a t u r a l p l a c e m e n t s T

N,Y

because,

6al

before glueing

t h e i r upper p a r t s o u t of expect t h a t

a part

t h e way.

t h e zero-load

t h i s case, because,

if

and

B2

together,

one m u s t bend

It i s unreasonable t o

problem h a s any s o l u t i o n a t a l l i n

one

” l e t s go‘! of

63,

one would e x p e c t

t h e b o u n d a r y t o t o u c h a n o t h e r p a r t and h e n c e e x e r t

of

a non-zero

force.

6 . Environmental r e a c t i o n s . I n g e n e r a l , i f a n e l a s t i c body

operator

,.!.

fi

with load-

i s p l a c e d i n t o a n e n v i r o n m e n t r e p r e s e n t e d by t h e

Euclidean space

C

,

t h e e n v i r o n m e n t w i l l e x e r t f o r c e s on

and t h e s e f o r c e s w i l l depend on t h e p l a c e m e n t

of

8,

B.

M a t h e m a t i c a l l y , t h e s e f o r c e s a r e s p e c i f i e d by t h e p r e s c r i p t i o n

WALTER NOLL

37 8

of an environmental-reaction operator

z: t(P)

whose value exerts on

63

P

.-)

L,

gives the load-system the environment

when

i3

is held i n the placement

kl.

The

problem

then has the following physical interpretation: x,

body, i n some given placement

f ,

described by

Take the

put it in the environment

a n d then "let i t g o " .

What placements, if

any, can it assume? Given any reference placement

one can describe

K ,

i n terms of a n environmental body-force operator ~

gx:

P -+ C 0 ( G , b )

TK:

operator

(6.2)

and an environmental surface-traction

P -+ c o ( a B , l r )

t(P)(63) =

(

I

for all

$2

E

Ror(ii

.

so that "(kl)dVR

I n view of

+

(

Tpl(!J)dA1l

'mas

P

(4.5), the problem (6.1) is

then equivalent to

(6.3)

?!J E P

In most physical situations, t h e body-force operator is local in the sense that it is determined by a prescribed function

(6.4) Using

gK E C 0 ( i x e , b )

G K ( p ) ( x ) := g , ( x , P ( x ) )

in such a w a y that

for all

x

(4.3) and (6.k), we see that the condition

of the problem (6.3) becomes Y

(6.5)

E R.

div

K

(Tx(ll)) + g K 0 ( l B , P ) = 0.

gR(v)

=

LUG)

37 9

PROBLEMS I N THE STATICS OF FINITE E L A S T I C I T Y

If

t h e r e f e r e n c e placement

( 6 . 5 ) c a n be

i s homogeneous,

N

w r i t t e n i n t h e more e x p l i c i t form

If

we i n t r o d u c e t h e d i s p l a c e m e n t

r e f e r e n c e placement

4-

r : = clan

f r o m the V

t o t h e “unknownlt p l a c e m e n t

M

then

( 6 . 6 ) g i v e s r i s e t o t h e c l a s s i c a l d i f__. f e r e n t i a l e q _______uation of ~ion-lineare l a s t i c i t y :

\\here

kN := g N o ( K

4-

,I,)

E C0(Clo(RK)xe,b).

If t h e o n l y e n v i r o n m e n t a l body f o r c e i s g r a v i t a t i o n a l o r e l e c t r o s t a t i c , then the f u n c t i o n

(6.3)

g,(X,x)

where pl.t

:= p , , ( X ) ~ c p ( x )

E C o ( E ,E)

gu

has the f o r m

x

for a l l

E 6 ,

x E

e,

d e s c r i b e s t h e d e n s i t y of g r a v i t a t i o n a l

mass or e l e c t r i c c h a r g e , and where

Q

E C1(e

i s the

,W)

g r a v i t a t i o n a l or e l e c t r i c p o t e n t i a l . I n some p h y s i c a l s i t u a t i o n , surface-traction

t

operator

by a p r e s c r i b e d f u n c t i o n

h

M

t h e environmental

i s also local,

i.e.

E Co(a63xexLis(Ir),b)

determined i n such a

way t h a t

(6.9) If

z,,(V)(Y)

t h i s i s the case,

t h e problem ( 6 . 3 ) if

:= hn(Y,u(Y),vxU(Y))

f o r all

then t h e c o n di t i o n

T x ( p ) = z,(rJ.)

becomes a b o u n d a r y c o n d i t i o n .

i s homogeneous,

Y E 263.

of

F o r example,

(6.9) takes the e x p l i c i t f o r m

A r e a l i s t i c s p e c i a l c a s e of

(6.10) corresponds t o a

380

WALTER NOLL

hH

hydrostatic environment, i n which

has the special form

(6.11) Y E 363,

for all

y E E ,

F E Lis(Ir),

and

p: E -+ P

where

X

gives the pressure exerted by the environment as a function of

the possible places of the boundary points of

63.

In many physical situations, the environmental surface traction operator

-tu

is not local and hence does

not give rise to boundary conditions i n the conventional sense. An example is the balloon problem: 363

of

6

has two connected components

W,

may think o f

v>(S2)

region in

E

S2

of

d,.

and

One

W,

Given any placement

as the the

p,

is the boundary of a unique compact

whose volume we denote b y

on the exterior surface

v(P).

T h e pressure

s assumed to depend only on the

81

position of the points of ment.

8,

as the exterior surface and of

interior surface of a balloon. image

Assume that the boundary

81

as in a hydrostatic environ-

If we think of the interior of the balloon as filled

with a given amount of compressible g a s , the pressure on the interior surface will depend o n l y on the volume

v(1-I).

Thus,

the environmental surface-traction operator has the form

where

+

p: E

functions.

PX

and

TT:

Elx

+ Px

Note that the value of

depends on the global nature of values of

p

near

Y.

1-I

are given continuous zu(!J)

at a point

Y E S2

and not merely on the

PROBLEMS I N T I E STATICS O F F I N I T E ELASTICITY

I n e x p e r i m e n t a l s i t u a t i o n s one o f t e n d e a l s , n o t w i t h a f i x e d e n v i r o n m e n t , b u t w i t h a n environmeiit c o n t r o l l e d gradual changes.

Mathematically,

such a changing

e n v i r o n m e n t i s s p e c i f i e d by t h e p r e s c r i p t i o n of m e t e r f a m i l y of erivironmcrital mapping o f

of

a onc-para-

reaction operators,

i.e.

by a

the type

z: where

subject to

d E Px.

[O,dl

The v a l u e

+ Map(P,L)

zs

of

1

at

may b e t h o u g h t

s

a s d e s c r i b i n g the environmental r e a c t i o n a t time

u

i s assumed t h a t t h e i n i t i a l p l a c e m e n t i s g i v e n and t h a t

N

of

t h e e l a s t i c 1,ody

lo,

i s compatible with

It

s.

so t h a t

c o ( ~ =) if.). The p r o b l e m

then has

the following physical

interpretation:

u

t h e body i s i n i t i a l l y i n t l i c p l a c e m e n t d e s c r i b e d by

si-L

- .

do.

i n t h e environment

Change t h e e n v i r o n m e n t a s s p e c i f i e d b y

How d o e s t h e p l a c e m e n t cllange w i t h t h e p a r a m e t e r If

L

-

Assume t h a t

t h e mapping

(s,y)+- ls(y)

is FrGchet-differentiable,

and i f

from

=

Le(Ps)

w i t h respect to

s

(DX(?t))u = z o ( u )

(6.14)

at

+

s=O

r o n m e n t a l body l o a d s a r e l o c a l , second-order

linear differential

example,

w.

if

i s homogeneous,

of

=

xs(Us)

t o obtain

i s used. then

I-I

a given s o l u t i o n

(DLo(U))u,

where t h e n o t a t i o n

into

[O,d]xt-'

( 6 . 1 3 ) i s d i f f e r e n t i a b l e , one c a n d i f f e r e n t i a t e

s?

:=

11

If

Po the envi-

(6.14) g i v e s r i s e t o a

equation f o r

u:

t h i s equation has

8

4

IJ.

the f o r m

For

g

WALTER N O L L

382

n

(6.15)

div

X

(An(I,)vtlu)

a E Co(i,b),

f o r suitable

+ Bu

= a

B E Co(G,Lin(b)).

The e q u a t i o n

(6.15) i s , i n essence, the d i f f e r e n t i a l equation of c l a s s i c a l infinitesimal elasticity. The problem ( 6 . 1 3 )

i s also a useful basis f o r

p e r t u r b a t i o n a n a l y s e s i n f i n i t e e l a s t i c i t y , such as t h o s e i n i t i a t e d by S i g n o r i n i ( c f . [ T N ] , C a p r i z and P o d i o [ C P ]

6 . 3 ) and d e v e l o p e d by

Sect.

.

7. External c o n s t r a i n t s . A c o n s t r a i n t i s a l i m i t a t i o n on t h e placements t h a t

a r e c o n s i d e r e d p o s s i b l e i n a g i v e n body. a r e l i m i t a t i o n s on t h e v a l u e s o f

I n t e r n a l constraints

t h e g r a d i e n t s of t h e p l a c e -

ments and e x t e r n a l c o n s t r a i n t s a r e l i m i t a t i o n s on t h e v a l u e s of

W e c o n s i d e r t w o examples.

t h e placements themselves.

( a ) Boundary c o n d i t i o n o f p l a c e .

Such a c o n d i t i o n i s d e t e r -

8

mined by t h e p r e s c r i p t i o n of a p i e c e of

C

1

t h e body

.

and a n i n j e c t i v e mapping

t h e boundary

7 :

8

4

E!

of

363

class

The p l a c e m e n t s t h a t s a t i s f y t h e c o n d i t i o n form t h e s e t

Physically, part

R

of

8

surface

of

t h e c o n d i t i o n e x p r e s s e s t h e assumption t h a t t h e t h e boundary o f

Rng 1~

I n t h e space

( b ) Confinement c o n d i t i o n .

63

i s " g l u e d 1 ft o t h e r i g i d

E .

Such a c o n d i t i o n i s d e t e r m i n e d by

t h e p r e s c r i p t i o n o f a n open s u b s e t

C

of

l?.

The p l a c e m e n t s

PROBLEMS I N THE STATICS O F FINITE E L A S T I C I T Y

that

s a t i s f y t h e c o n d i t i o n form t h e s e t

I

: = (I-I E P

P

(7.2)

C

Physically,

Ru

= Rng

t h e c o n d i t i o n e x p r e s s e s t h e assumption t h a t

C

body i s c o n f i n e d t o t h e p o r t i o n example, i f

C

t h e p r e s c r i p t i o n of

a suitable subset

E .

If

b e t a k e n a s t h e domain of

X(i-1)

63,

z(u)

of

P,

must

t h e environmental r e a c t i o n operator

PC

but

4

L

leaves out t h e forces exerted

= e"(p)

z(u)

b u t one must add t o

a p p r o p r i a t e t o t h e a c t i o n of Mathematically,

P

is present, the

r a t h e r t h a n a l l of

by t h e c o n s t r a i n i n g o b j e c t s .

longer require that I-I

of t h e s e t

no l o n g e r g i v e s t h e e n t i r e load system t h e

environment e x e r t s on

bd

Pc

such a c o n s t r a i n t

e:

on

a rigid

011

a n e x t e r n a l c o n s t r a i n t i s s p e c i f i e d by

"admissible" placements,

whose v a l u e

For

would b e a h a l f - s p a c e .

I n general,

a l l placements i n

the

t h e environment.

of

t o consider a rubber b a l l

one w a n t s

plane surface,

s e t of

CI c C ] .

%:

f o r t h e unknown p l a c e m e n t s

an i n d e t e r m i n a t e f o r c e system the constraining objects.

t h i s corresponds

projection operators

T h e r e f o r e , one c a n no

L + L

t o t h e c h o i c e of for a l l

suitable

Ll E P c .

The problem

- I@))

= 0.

( 6 . 1 ) must t h e n b e r e p l a c e d by

(7.3)

7 I-I E P c , F o r example, i f

pc

i s d e f i n e d by

t h e c o n s t r a i n t i s a boundary c o n d i t i o n of d o e s n o t depend on

(7.4)

R(C)(P)

i-I

(7.11, i . e . i f

place,

R : = R,,

and i s g i v e n by

:= C ( p \ S )

for all

6 E Bor(ba).

384

WALTER N O L L

In this case, (7.1) is equivalent to

where

R~ : = { r

E

L

I

r(P) = o

P

if

is the set of all load systems on ed i n the glued part 8

I€

Pc

63

E nor(E)

arid

P ~ I S = q,}

whose support is contain-

of the boundary of'

8.

is defined by (7.2), i.e. if' the constraint

is a confinement condition, then the choice o f the projection

R I-r

operators

depends o n additional physical assumptions.

F o r example, if one assumes that there is n o friction at the

points o f contact between the body and the confining region @,

%(G)

then

G

is obtained r'rom

by annihilating the

normal component of the surface-load of 38

points of

whose place in

G

at all those

is at the boundary of

p

@.

8. Stability. -

Let

8

be a continuous body and

e,

placements i n the environment A one-parameter family

s d.

E [O,dl, Let

L

p : [O,d] + P

(8.1)

o f such placements 8

be the space of all load systems o n

If

4 : [O,d]

family of such load systems L

as described in Sect. 2.

can be interpreted as a motion o f

described i n Sect. 3.

done by

the set of its

P

Cs, p

during the motion d W

:=

(

'0

s

(

1-

ps,

of duration B

as

is a one-parameter

4

L

E

[O,d]

,

then the w o r k

is defined by

;s*d.Ls)ds.

63

To make this meaningful, we assume that

IJ

is of class

C1

385

PROBLEMS I N THE STATICS O F F I N I T E ELASTICITY

and t h a t

G

i s of

class

Co.

El

Assume now t h a t

-

-

load operator

4 ,

is prescribed,

and t h a t

(6.1), of

i.e.,

Class

cs

that

i s a n e l a s t i c body w i t h e l a s t i c -

t h a t an environmental-reaction

e"(,)

with

I-lo

n E P

i s a s o l u t i o n t o t h e problem

= z ( x ) . = 'h

G

operator

u:

F o r any motion

t E [O,dl

and a n y

[O,d]

+ P

w e can

consider (8.2)

and

f

/t

(8.3)

The f i r s t i s t h e w o r k d o n e by t h e e l a s t i c l o a d s y s t e m i n t h e time i n t e r v a l f r o m

to

0

t

a n d t h e s e c o n d t h e work d o n e b y

t h e e n v i r o n m e n t a l l o a d s y s t e m i n t h e same t i m e i n t e r v a l . The f o l l o w i n g d e f i n i t i o n o f s t a b i l i t y i s a r e a s o n able generalization of Coleman a n d N o 1 1 ( c f .

Definition: -

t h e o n e s g i v e n by Hadamard, Duhem, f o o t n o t e 3 on p .

The s o l u t i o n

(8.4)

w.

? n E P

class

such -

c1

s t ______ arting at

3 2 8 of [ T N ] ) :

t o t h e problem

=

l ( K )

i s s a i d t o be l o c a l l y s t a b l e i f of

and

T(%) V:

f o r e v e r y motion

P o = It,

there is a

d'

E

[O,d]

+P

1 O,d]

that

(a.5 )

wt(p) 5 i t ( l i )

If

for a l l

t h e r e i s a stored-energy

body,

t h e n t h e work

ment

Po = n

it(p)

t E [O,d']. f u n c t i o n for t h e e l a s t i c

d e p e n d s o n l y on t h e i n i t i a l p l a c e -

and t h e f i n a l d i s p l a c e m e n t

Ut.

Specifically,

WALTER N O L L

386

we h a v e

where

-

P

E:

4

W

g i v e s t h e t o t a l e n e r g y s t o r e d i n t h e body a s

a f u n c t i o n of t h e placement. work done b y

It; may h a p p e n , a l s o , t h a t t h e

t h e e n v i r o n m e n t a l l o a d s y s t e m i s d e r i v a b l e from

-

a p o t e n t i a l energy

P

E:

R,

4

so that

t h i s i s the case,

t h e environmental r e a c t i o n o p e r a t o r i s

called conservative.

The s p e c i a l c a s e s c o n s i d e r e d i n S e c t .

If

a re c o n s e r v a t i v e .

I f t h e r e i s a s t o r e d e n e r g y and i f t h e

environmental r e a c t i o n i s c o n s e r v a t i v e , t h e problem ( 8 . 4 ) N

of

x

in

P

is locally stable i f

y E

N,

Acknowledgment.

then the s o l u t i o n t o t h e r e is a neiehborhoal

such t h a t

E(u) for a l l

6

i.e.

- E(7t)

if

FE(V)

- - E-

- E(y)

h a s a l o c a l minimum a t

E

x.

The r e s e a r c h l e a d i n g t o t h i s p a p e r was

s u p p o r t e d by G r a n t MCS75-08257

f r o m the N a t i o n a l S c i e n c e

Foundation. References

[ 13 N o l l , W.,

“ L e c t u r e s o n t h e f o u n d a t i o n s o f continuum

mechanics and t h e r m o d y n a m i c s ” , A r c h i v e f o r R a t i o n a l

8, 1-12 Mechanics and A n a l y s i s 3 [ Z ]

Noll,

W.,

“ A new m a t h e m a t i c a l

(1970).

t h e o r y o f simple m a t e r i a l s “ ,

A r c h i v e f o r R a t i o n a l Mechanics and A n a l y s i s

1-30 ( 1 9 1 2 ) .

5,

3 87

PROBLEMS I N THE STATICS OF F I N I T E E L A S T I C I T Y

[ 3 ] Truesdell, C. of

and W.

Noll,

The N o n - l i n e a r F i e l d T h e o r i e s

Mechanics, Encyclopedia of

602 pages.

Springer-Verlag,

Physics, V o l .

111/3,

Berlin-Heidelberg-New

Y o r k 1965.

[4]

Capriz, G.

and P .

Podio G u i d u g l i ,

"On S i g n o r i r i i ' s

P e r t u r b a t i o n Method i n F i n i t e E l a s t i c i t y " , A r c h i v e f o r R a t i o n a l Mechanics and A n a l y s i s

57, -

1-30

(1974).

de La Penha. L.A. kdeiros (eds.) Contemporary Developments in Continuum Mechan i cs and Partial Differential Equations @North-Holland Publishing Company (1978) G.M.

ELLIPTIC METRICS ON LORENTZ MANIFOLDS

PEDRO NOWOSAD Instituto de Matemgtica Pura e Aplicada (IMPA) Rio d e Janeiro

-

Brasil

When studying the wave equation i n a normally hyperbolic manifold signature

Vn

(i.e. a pseudo-Riemannian manifold with

...,-l),

(l ,-l,

f o r short a Lorentz manifold) one

is naturally led to associate elliptic metrics (i.e. definite metrics) in order to define the energy F o r instance, given the global chart

E

IR4

of the solutions. o f Minkowski space,

the wave operator is the standard d'alembertian

av2

a2 aZ

Clearly

C?

- -a2- -

a 2 - -a 2 --

-

at2 a x2 and the energy associated with a solution u is

depends

011

the chart, and i n particular on the

decomposition of the space into its space-sections and time-axis

(x,y,z)

t.

In the context of physical problems it is customary to assume the decomposition like

[,I.

V

n

= R x C,

where

C

is space-

A s a consequence this essentially fixes the

reference system and hence the associated energy

e.

The purpose of this Note is to analyze the different energies (i.e.

elliptic metrics) that one can naturally

associate with the me-tric Lorentz manifold

Vn

g

o f a given time-oriented

without special assumptions.

ELLIPTIC METRICS ON LOF33NTZ MANIFOLDS

389

Although elementary this is an important question which does n o t seem to have been conclusively analyzed.

In

[l] Avez considered this question with regard to the problem

of completeness. The vector fields considered are assumed to be real v

A vector

and continuous.

in the tangent space is called g(v,v) > 0,

time-like, isotropic, or space-like, according as

g(v,v) < 0, respectively; in the first two

g(v,v) = 0, o r

Similarly for vector fields.

cases it is also called causal. A vector field 1. Definition

vector field

T

T ,

such that

lg(7

,T)

I

is called unitary.

1

I

Given a-~ unitary time-like time-oriented . -- -- - -

~

its associated elliptic metric

is

'g

defined bv

Clearly (1) arises from the bilinear form

x

Choosing a local chart at any coordinate axis tangent to = diag(1,-1,

at

x

so

gT

is

...,-1)

diag(l,O,

at

x,

...,0)

is indeed positive.

that in any local chart

at x and

T

in

with

Vn, g. . = 1J

then the matrix of and hence

g(v,.r)g(v,T')

T

g . . = diag(1,l 1J

One also shows , using

ldet g .

define the same volume element.

,I

1J

with one

T

= det g. .; 1J

so

g

g

...,l), 7 ,'T )

=

and

g'

1,

390

PEDRO NOWOSAD

2. E q~u i v a.-l.e nt e l l i p t i c metrics. ..

T'c; t h e s e t o f u n i t a r y t i m e - l i k e v e c t o r

Denote by

i n t h e f u t u r e , say.

f i e l d s , time-oriented Then

T , T ' ~

%

are called equivalent i f they define

uniformly e quivalent e l l i p t i c m e t r i c s , M > 0

i.e.

if

such that

(3)

Y

Clearly write

f o r t h e e q u i v a l e n c e class of

[?]

Tx

i s t h e tangent space a t One c-hecks e a s i l y t h a t

distance

Lemma -

TIC

For

T ,TIE

e

projective

X.

d

i s indeed a p r o j e c t i v e

1.1

i f and o n l y i f

I

As

g

?

,

2 2

g

s u p --- ( v , v > vfo (v,v)

-

v

homogeneity,

-

I'

) =

s u p cosh Vn

-1

g(?,T1).

are definite metrics, for fixed

i s a c h i e v e d a t some

? f

0.

x

Due t o t h e

i s a c r i t i c a l p o i n t o f t h e q u a d r a t i c form

I

XgT

d ( g T ,gT ) < m .

:

d(gT,gT

gT(v,v)

on

I

(5)

X =

...

O,O1,

. Observe t h a t

Proof:

we

Therefore on t h i s

s e t i s n a t u r a l l y d e f i n e d t h e extended-valued

where

t ;

T.

i s a cone u n d e r p o i n t w i s e o p e r a t i o n s .

Vn,

v E E.

( 3 ) i s an e q u i v a l e n c e r e l a t i o n i n

The s e t o f p o s i t i v e d e f i n i t e m e t r i c s

3.

there exists

(v,v),

hence n e c e s s a r i l y

391

E L L I P T I C M E T R I C S O N LORENTZ M A N I F O L D S

Putting

v =

T

in

( 6 ) and u s i n g

I

g(T,?)=

and

(2) o n e g e t s

The r o o t s a r e

which a r e r e a l b e c a u s e t h e r e v e r s e d Schwarz i n e q u a l i t y v a l i d

for c a u s a l v e c t o r s

(p.

111 [ 21

)

The t w o r o o t s a r e r e c i p r o c a l , and a s

g ( ~ , 7 ’ )> 0 ,

T h e r e f o r e t h e r e s u l t f o l l o w s from d ( g T, g T ’ ) =

4.

Corollary

i s bounded on

-

T ,T‘€

‘n *

1

sup h i

X

2

the

( 1 0 ) and from

.

C a r e e q u i v a l e n t i f and o____ nly i f _.____

g ( 7 ,T

’)

PEDRO NOWOSAD

392

If at each point we take a local chart such that at g i j = diag(1,-l,..,-l),

this point

then

will be

7’

T ,

i j S: gijv v = 1

represented by points o n the unit pseudo-sphere

and their scalar product g ( T , T ‘ ) is simply cosh of the arc distance on S between these points. By ( 5 ) d ( g T , g T ’ ) is the sup

of this arc distance. Clearly if Vn is compact all

equivalent.

Vn

Conversely if

non-equivalent T, _ I’ - _ _ ~ ~ _ _

i -_ n

Indeed take for vector field on

Vn.

S.

are

is not _compact there exist _

.

“G

any

r

unitary

time- like

Choose at each point a local chart in

the above manner and so that fixed point o n

Z

in

T ,T’

is always represented by a

T

Choose a continuous vector field

that its representative on asymptote light cone.

S

such

T ’

approaches indefinitely the

By the above

T‘

@

[TI.

This is the general situation as shows the following

5. Proposition

-

If -

,T’€

an isotropic vector field sequence o f points i n 7

g

between

T ’

Vn,

and u‘,

G

a’

and __

then there exists

r ‘ @[ T I ,

with

gT ( 0 ‘

)u‘ )

I

1

and a

such that the angle in the metric ______ tends to zero along-.that sequence.

Proof: Define two isotropic vector fields

a,

0‘

oriented in

the future and such that

and

~ ( x ) and

~ ’ ( x ) lie i n the subspace spanned by

and

~ ’ ( x ) for each

x E Vn.

Since

T(x)

time-like this is always possible at each run along the trajectories of

r

and X.

T ’ ( x )

Letting

T(x) are

x

we do get vector fields.

393

E L L I P T I C METRICS O N LORENTZ M A N I F O L D S

Therefore by construction there are real functions

Vn

such that

+ $a‘,

= aa

T /

) = 2apg(a , a / ) .

g(7‘ , 7 ’

Similarly from (11) g(7

As

7,

7’

) =

17’

4 09

0 ’

).

are unitary we get from the two last

express ions (13)

and

(14) (15) In particular are positive as

T‘@

and

T /

have the same sign and hence are equally time-oriented.

(13) we get

From

Hence as

7

a, B

[TI

the corollary.

,

either

sup @ =

inf $ = 0 , by

Without loss o f generality assume

F r o m ( 2 ) , ( 1 2 ) , (ls), g‘(U’

and/or

+m

,T‘

) =

sup

(16) and (14) we get

2g(U‘ ,T)g(T’

,T

)

-

g ( U ’ ,T’

I

)

B

=

+m.

PEDRO NOWOSAD

394

1 = (=

+

@)2

-

1 =

B

2

1

+Z’

Therefore

as claimed,

6. Comments _ _

-

i) The

elements

of

can be thought

[T]

of as physically equivalent time-reference systems as they define equivalent energies.

According to the above results,

an observer whose time axes are in the class his own positive metric

g

T

,

sees, in

[ T I

the angle between

T’$

and

[ T I

the light cone approach zero along some sequence, i.e. he sees some observer in the class

[T’l

eventually moving as

close to the speed of light as pleased, and conversely.

To

pass from one to the other reference system an infinite amount of energy is required. Clearly non-equivalence is a n asymptotic property.

In other words the reference systems of the above kind are gauged at infinity, where local finite perturbations have no effect.

Therefore the present results give

also a

classification of (physical) reference systems on

ii) Instead o f unitary vector fields have assumed the weaker condition

1

g(7 , T )

5

2:

T

M,

‘n’

we could

M consM.

We obtain the same results, as this case can be reduced to the previous one by observing that the equivalence relation

[ ]

T

/

J

z

)

is preserved.

E t;

and that

ELLIPTIC METRICS O N LORENTZ MANIFOLDS

F i n a l comment. ___-____-

395

The above a n a l y s i s w a s c a r r i e d o u t a s a

preliminary s t e p t o a tentative generalization o f radiation conditions, i . e .

boundary c o n d i t i o n s a t i n f i n i t y , for g e n e r a l

Lorentz spaces. The r e s u l t s o b t a i n e d were i n ( p a r t i a l ) c o n t r a d i c t i o n

t o t h e d e f i n i t i o n of t h e r e v i e w e r o f [l]

7

the class

i n [l].

and shown i n t h e p r e s e n t a n a l y s i s ,

s e c o n d c o n d i t i o n i n D Q f i n i t i o n 2 [l] free. not ment

S o i n non-compact

Vn

,

is i n d e e d n o t c o o r d i n a t e

7.

(and t h e replacement o f t h e r i g h t - h a n d

f o r m u l a i n Lemma 2 [ 11 by

X

the

t h e r e a r e i n f i n i t e l y many and

j u s t one ( p r o p e r l y d e f i n e d ) c l a s s

With t h i s ammends i d e of t h e

g i v e n by ( 1 0 ) ) t h e r e s t o f

a n a l y s i s i n [ 11 g o e s t h r o u g h . complements

A s q u e s t i o n e d by

the

The p r e s e n t Note t h e r e f o r e also

C 11. References

-.

[l]

Avez, A .

D g f i n i t i o n des

i n d g f i n i e s , C.R.

(MR 1 6 , [Z]

Avez, A .

Acad.

Sci.,

240,

1955,

485-487

856).

E s s a i s de g6om6trie riemannienne hyperbolique

-

globale Ann.

v a r i g t 6 s compl&tes & m6triques

applicationsa l a r e l a t i v i t 6 ggnQrale,

I n s t . F o u r i e r 1 3 , 2 ( 1 9 6 3 ) 105-190.

[ 3 ] Trautman, A.

Boundary C o n d i t i o n s a t I n f i n i t y f o r

Physical Theories, Bull. math.,

astr.

403-406.

e t phys.

-

Ac.

Vol.

Pol. S c i . VI,

nP

se'rie s c i .

6 , 1958,

G.M. de La Penha. L.A. Medeiros (eds.) Contemporary Developments in Continuum Mechanics and Partia\ Differential Equations @North-Holland Publishing Company (1978)

BOUNDARY VALUE PROBLEMS FOR EQUATIONS O F M I X E D TYPE

STANLEY O S H E R Mathematics

-

UCLA

Los Angeles, C a l i f .

90024

U. S.A.

We a r e c o n c e r n e d w i t h t h e q u e s t i o n :

which boundary

v a l u e p r o b l e m s a r e w e l l posed f o r t h e d i f f e r e n t i a l e q u a t i o n o f mixed t y p e :

K(y)hxx+u + a u + b u + c u = F YY X Y

(1) where

~ ( y =) s g n y

and

a, by c

+1

if

y > O

-1

if

y < O

=

a r e smooth f u n c t i o n s of

(x,y).

We s h a l l a l s o d e v e l o p a n e f f e c t i v e n u m e r i c a l algorithm (obtained j o i n t l y

w i t h A r t h u r Deacon) b a s e d

on a

r e d u c t i o n o f t h e boundary v a l u e problem i n t o a n e l l i p t i c problem w i t h u n u s u a l non p s e u d o - l o c a l

boundary c o n d i t i o n s f o r

which t h e G a l e r k i n p r o c e d u r e w o r k s w e l l .

T h i s problem i s

s o l v e d b o t h a n a l y t i c a l l y , u s i n g and m o d i f y i n g r e s u l t s

of

K o n d r a t i e n on e l l i p t i c e q u a t i o n s i n c o n i c a l r e g i o n s , and n u m e r i c a l l y u s i n g and m o d i f y i n g s t a n d a r d L a p l a c e i n v e r t e r s . We t h e n u s e t h e i n v e r t e d Cauchy d a t a on t h e p a r a b o l i c l i n e t o o b t a i n t h d s o l u t i o n i n t h e h y p e r b o l i c region. A s a very simple i l l u s t r a t i o n w e c o n s i d e r t h e follow-

i n g problem

3 97

E Q U A T I O N S O F M I X E D TYPE

Yt

in

Qo

we s o l v e

in

fll

we s o l v e

on

C0’

on

C1

which i s s p a c e l i k e , we p r e s c r i b e

on

rl

which i s a c h a r a c t e r i s t i c ,

hxx

+

u

-

xx

uY Y = F o U

YY = F 1

which i s a smooth c u r v e , we p r e s c r i b e

u =

Cp

u = ep 1

nothing i s given.

The aim i s t o r e d u c e all t h e h y p e r b o l i c i n f o r m a t i o n t o the l i n e

y = 0.

T h i s i s done as f o l l o w s :

u = f(x-y)

I n the hyperbolic region write f

and

g

+ g(x+y),

unknown ( h e r e we assume f o r s i m p l i c i t y that F r O ) 1

thus

u

- u

X

(2)

hx

+

u

Y

= 2f’(x)

y=o

Y y=0

= 2g‘ ( x )

Again,

f o r s i m p l i c i t y , we t a k e t h e c u r v e

line,

y = -cx

0 < c < 1

we t h e n h a v e

= cp,(x)

= f(x(l+c))

u(x,-cx)

Cl

+

g(x(1-c))

or

(3)

(-1

cp‘ 1 l + X c 1-c ( K = -) 1+c

= f’ (x)

+

Kg‘

(X

t o be a s t r a i g h t

(1-4) l+C

S T A N L E Y OSHER

398

Using (2) arid ( 3 ) , we have

where

Kx

x

ep,

is always a known function. Thus we have the following n o n pseudo-local boundary

Y

value problem

A

LII = F

B

A F o r the problem analyzed above,

general boundary conditions on

C1,

! ,

= k

X

for

is easily cornpuked.

We now build a parametrix f o r this problem and obtain coerciveness in weighted Sobolev spaces as in Kondratien.

Moreover, we can write an asymptotic expansion u

near

r

A

and

and pj

B

-c

, x j Pj(V)

R e X . + m J

on the boundary with simple formulas for the

moreover it turns out that if

li;le

is

sufficiently small, the hypotheses o f the Lax-Milgran theorem are obeyed f o r this elliptic problem on

HI,

thus a Galerkin

procedure can be shown to converge. We also use as elements i n this procedure the functions having the appropriate singular behavior at A and B. The resulting numerical procedure works very well.

399

E Q U A T I O N S O F M I X E D TYPE

Bibliography

and O s h e r , S.

[l] D e a c o n , A.

121 K o n d r a t i e n , V.A.

-

-

-

T o appear.

B o u n d a r y v a l u e p r o b l e m s for e l l i p t i c

e q u a t i o n s i n d o m a i n s w i t h c o n i c a l o r a n g u l a r points, T r a n s . M o s c o w Math.

SOC.,

T r d y , Vol.

R u s s i a n 2 0 9 - 2 4 2 , T r a n s l a t e d A.M.S.

[3]

O s h e r , S.

-

16 ( 1 9 6 7 )

(1968).

B o u n d a r y v a l u e p r o b l e m s f o r e q u a t i o n s of

m i x e d t y p e I. Cornm. P . D . F .

T h e L a u r e n t i e n - B i t s a d z e model.

2, ( 1 9 7 7 ) , '199-547.

G.M. de La Penha, L.A.

Medeiros (eds.) Contemporary Deve 1opmen t s i n Continuum Mechan i cs and P a r t i a l D i f f e r e n t i a l Equations @North-Holland Publishing Company (1978)

NONLINEAR E I G E N V A L U E PROBLEM

A F R E E BOUNDARY,

PUEL

J.P.

4

D6partement de MathGmatiques Universitg Paris V I pl. Jussieu, 7 5 2 3 0 Paris Cedex 0 5

The problem we are going to study has been introduced by R .

Temam [1] and is related to the shape, at equilibrium,

of a confined plasma in a torus. Let

n

Rn,

be a bounded open set in

and

r

boundary that we will assume regular f o r simplicity. problem consists in finding an open subset denote the boundary o f

a function

UJ),

and a real positive number (i)

JJ

(ii)

h

on

~ 1 ,

t

u

1

ft)

of

be its The

0

defined on

(Y will R,

such that:

0; is the first eigenvalue o f the Dirichlet problem

and

u1

is the corresponding positive eigenfunction,

i.e.

(iii)

u1

conditions on (iv)

i

-Au,

= Xu,

u1 2 0

in

in

ul/Y

= 0;

is harmonic in

n-w,

LI

u)

with compatibility

Y.

The value of

(a priori unknown) and

u1

-

on

r

[2 r

prescribed positive constant pointing normal at a point o f

I

has to be constant dl?

(?

has to be equal to a denotes the outward

r).

It turns out very easily that this problem can be

A FREE n O t J N D A R Y ,

NONL1L:JEAR E I G E N V A L U E PROULEM

formulated a s f ollow s:

we

u1

look f o r

-au,

= xu,

A

and

1401

such t h a t

+, (unknown) ,

= C s t

I f we d e f i n e

I

= {x

*I

x E 0,

> 03,

UJX)

t h e n t h e c l a s s i c a l r e g u l a r i t y r e s u l t s f o r problem (1) show that

i s a n open s e t .

W

Notice t h a t

t h i s problem

(1) i s n o t

q u i t e e q u i v a l e n t t o t h e o r i g i n a l problem b e c a u s e we a r e r i o t sure that

C

0.

S o t h e problem can be t a k e n a s a n o n l i n e a r e i g e n v a l u e problem,

R.

or a s a f r e e boundary problem. (1) c o n s i s t s i n

Temarn's method f o r s o l v i n g problem

looking f o r c r i t i c a l points of

I

M = {v

on t h e m a n i f o l d

v

E

the functional

H

1

(n) ,

I+

= Cst,

v dx =

62

I n f a c t , he m i n i m i z e s

T

M

over

and s h o w s t h a t

i s a c h i e v e d a t a p o i n t which s a t i s f i e s ( 1 ) H. problem.

B e r e s t y c k i and H.

Brezis

[Sl

t h e minimum

( c f [2]).

have a l s o s t u d i e d t h i s

They c o n s i d e r t h e convex s e t

K = {p

I

p

E

L2((n),

p

2

0

a.e.

on

0,

f

p dx = I ) .

'R

They d e f i n e t h e f u n c t i o n a l

where

I -3. x

H =

( - A ) -1

( w i t h D i r i c h l e t boundary c o n d i t i o n s ) .

402

PUEL

J.P.

They s h o w t h a t

i s bounded from b e ow on

B

p o E K.

i t s minimum a t a p o i n t

= Hpo

u

Then

K

and a c h e v e s

+- B ( P 0 ) I

0

is a

s o l u t i o n o f our problem ( 1 ) . We s h a l l p r e s e n t a n o t h e r method f o r s o l v i n g t h i s problem and a l s o a s i m p l e p r o o f

1.

values o f

( T h i s uniqueness

collaboration with A.

of u n i q u e n e s s f o r c e r t a i n r e s u l t h a s b e e n proved i n

Damlamian.)

More p r e c i s e l y we s h a l l

prove t h e f o l l o w i n g r e s u l t .

-

Theorem 1

p <

W3”(n)

Moreover i f we d e n o t e by

Xi%

t h e D i r i c h l e t problem on

R,

X <

( i )F o r

n.

on

X1/n, w =

(Then

for all

p

n E

for all

u

1

such t h a t [O,l[

).

the ordered eigenvalues o f we h a v e :

every s o l u t i o n o f

(1) i s non n e g a t i v e

every solution of

(1) d e f i n e s a n open

n!)

X > X,/n,

( i i )F o r

there exists a solution

us E C”‘(5)

(Then

+m.

E

u1

o f problem (1) w i t h 1S

k ,

For a l l positive

set

w = {x

I

x E 0,

UJX>

7

03,

and a boundary

y = a w = {x with

y

C

x E

A,

U,(X)

= 03

,

R.

( i i i )F o r

Remarlds: -

I

0

c X

5

X,/R,

problem (1) h a s a u n i q u e s o l u t i o n .

1) The r e g u l a r i t y d f

the solution

u1

i s shown by

a p p l y i n g many t i m e s t h e r e g u l a r i t y r e s u l t s f o r t h e D i r i c h l e t problem. 2 ) The r e g u l a r i t y of t h e boundary D.

K i n d e r l e h r e r , L.

y

N i r e n b e r g and J .

h a s b e e n s t u d i e d by Spruck

([ 41 ,[ 57 )

.

A FREE BOUNDARY,

N O N L I N E A R EIGENVALUE PROBLEM

3 ) In t h e g e n e r a l c a s e , t h e u n i q u e n e s s r e s u l t seems t o be D.

optimal.

Schaeffer

161

has

u n i q u e n e s s f o r a bone-shaped

The p r o o f the results

g i v e n a n example of non

n

1 >

and

o f Theorem 1 i s a n immediate c o n s e q u e n c e o f

t h a t a r e s t a t e d below f o r a n e q u i v a l e n t problem.

An e q u i v a l e n t problem:

of

Let

C

finding

u

be a f i x e d c o n s t a n t and c o n s i d e r t h e problem

1

and

2

such t h a t

*

N o t i c e t h a t we h a v e d r o p p e d a c o n d i t i o n o f problem ( l ) ,b u t

i s fixed.

C

that here

i s non n e g a t i v e , e v e r y s o l u t i o n o f

As

u2 2 C

u

and e i t h e r of

r.

C

in

n,

aU2 < 0 av

either

-

If

X

p

constant

Proof:

If

-

such t h a t u2

PU,

C,

i s not i d e n t i c a l t o

a u 2 d r = p > 0.

there exists a positive

Let

C,

B =I il

we h a v e and

au2 -< 0 .

u1 =

av

Bu,.

i s a s o l u t i o n o f problem ( 1 ) . We s h a l l now s t u d y t h e non c o n s t a n t s o l u t i o n s o f

problem ( 2 ) , t a k i n g

and

u2

i s s o l u t i o n o f problem ( 1 ) .

'r ul

a t each p o i n t

i s p o s i t i v e , f o r every s o l u t i o n

( 2 ) which i s n o t i d e n t i c a l t o

Then

R,

in

Then we have

Proposition 1 of

=

2 -

( 2 ) satisfies

C

as parameters.

We b e g i n w i t h t h e s i m p l e c a s e where

0 < A < h,/n.

Then

404

J.P.

-

Proposition 2 problem

E

I?

For

(2) possesses

problem

c

u (x) > 2

[O,X,/n[,

If

( 1 ) ) . If

C > 0,

f o r every

x

C 4

(and s o i t won't

C

C E R,

and f o r a l l

a unique s o l u t i o n .

solution is identical t o of

PUEL

the solution

this

0,

give a solution

u2

satisfies

n.

i n

T h i s r e s u l t c a n b e proved by t h e f a c t t h a t t h e operator

-A

-

X I

i s coercive. N e x t we g i v e , w i t h o u t p r o o f ,

two s i m p l e r e s u l t s which

c a n b e shown w i t h t h e a i d o f t h e F r e d h o l m a l t e r n a t i v e .

-

Proposition 3

C < 0,

(i) if u2

For

(iii) i f

= h,/n,

we have:

problem

( 2 ) h a s the u n i q u e s o l u t i o n

C > 0,

problem

( 2 ) has n o s o l u t i o n

c

t h e o n l y non t r i v i a l s o l u t i o n o f

c

E

(ii) if

X

= 0,

(2)

(up t o a m u l t i p l i c a t i v e p o s i t i v e constant) i s t h e positive eigenfunction

Proposition

4 -

no s o l u t i o n ;

For

associated w i t h

X > X,/R,

C = 0,

if

cpl/R

C > 0,

if

p r o b l e m ( 2 ) has

t h e o n l y s o l u t i o n o f problem

(2) is

u 2 = 0. N o w w e come t o t h e e s s e n t i a l r e s u l t

Theorem 2

-

solution

u2

>

If

t h i s work.

C < 0 , p r o b l e m ( 2 ) has a

and i f

which i s non i d e n t i c a l t o

( 2 ) possesses a t Proof:

X,/n

of

C.

(Thus p r o b l e m

l e a s t two d i s t i n c t s o l u t i o n s . )

L e t us c o n s i d e r the f u n c t i o n a l , d e f i n e d on

J(v) =

] g r a d v I 2 dx

-

A

\

I(v+C)

+ 2

I

Hi(n)

by

dx.

62

J

i s of class

C1

on

Hk(n),

and the c r i t i c a l p o i n t s of

J

A F R E E GOUNDARY,

a r e s o l u t i o n s of

kO5

N O N L I N E A R E I G E N V A L U E PROBLEM

t h e problem

T h i s problem i s e q u i v a l e n t t o ( Z ) , u

We a r e g o i n g t o show t h a t and t h a t t h e r e e x i s t s a

-- u3

~

+

c.

i s a r e a l l o c a l minimum of

0 vo

modulo t h e t r a n s f o r m a t i o n

#

0

which s a t i s f i e s

J,

J ( v o ) r; 0.

Then we s h a l l be a b l e t o a p p l y a t h e o r e m o f A m b r o s e t t i Rabinowitz

[ 7 ] showing t h e e x i s t e n c e of a non t r i v i a l c r i t i c a l

point.

Lemma 1

-

y

There e x i s t p o s i t i v e c o n s t a n t s

t h a t f o r all

v € Ht(h?)

with

IlvlI

r;

6,

and

6

such

we have

H O P )

I t s u f f i c e s t o show t h a t f o r

If we s e t

l i m P-++m

(

w

6

s m a l l enough, and f o r all

i t s u f f i c e s now t o show t h a t

= * 9

H O P )

1 (w-p)+(

2

dx = 0 ,

uniformly i n

w

on t h e u n i t

I,

sphere of

HA(n).

F r o m t h e S o b o l e v imbedding t h e o r e m ,

we know t h a t t h e r e e x i s t s

406

J.P. PUEL and

q 7 2

because i n T h e n if

Cq 7 0

such that

Ap(w),

Iw-pl

H;(n)

< lw]

lim p++m

-

n

and in

and

Ap(w),

1

belongs to the unit sphere o f Ho(n), 2 s-2 l(w-p)+l d x < C 2 [meas A (w)] q

w

9

just

N o w we

cLq(n),

1w1 2

we have

.

0

have to show that

= 0

[meas A (w)] P

1

uniformly on the unit sphere of H , ( n ) .

and Lemma 1 is proved. Lemma 2

-

J(vo)

0.

5

There exists a

Proof: Let

0.

q 1 = Cp/n l

associated with

A 2

l/n,

1

v0 E H o ( n ) ,

v0

#

0,

such that

be the positive eigenfunction and consider

J(UV1). 2

A FREE BOUNDARY,

x

As

>

large

a

we s e e t h a t f o r

1,

we have

aO

N O N L I N E A R E I G E N V A L U E PROBLEM

J(acpl)

5;

g r e a t e r them a s u f f i c i e n t l y

v o = aocpl.

W e s h a l l take

0.

W e a r e now r e a d y t o a p p l y t h e r e s u l t of A r n b r o s e t t i and Rabinowitz.

x

Let

= {k

1

1 k E C([0,11 ; Ho(n)),

k(O)

(K

i s t h e s e t o f continuous paths i n

to

vo).

Hi(n),

k(l) =

vO1 *

g o i n g from

0

t h e r e e x i s t s a non z e r o c r t i c a l p o i n t of

J

Then i f d = Inf

is a critica

T h i s means

that

J(k(t))

SUP

kEX d

= 0,

t€[O,l]

v a l u p o-f

J,

and i t f i n i s h e s t h e p r o o f

with

> 0

d

of Theorem 2 .

I n o r d e r t o complete t h e proof

o f Theorem 1, we have

t o show t h e f o l l o w i n g u n i q u e n e s s r e s u l t . Theorem 3 ( i n c o l l a b o r a t i o n w i t h A . If Proof:

of

h E

[0,x,/n],

For a fixed

1,

problem if

u1

Damlamian).

(1) h a s a u n i q u e s o l u t i o n .

and

G1

a r e two s o l u t i o n s

(l), t h e i r boundary v a l u e s have t h e same s i g n .

just

Then we

have t o s h o w u n i q u e n e s s f o r non c o n s t a n t s o l u t i o n s of

problem

(2). Let

(2).

u2

and

t w o d i f f e r e n t s o l u t i o n s of problem

We t h e n have

i Define

G2

- A ( u , - ~ ~ =) h ( u 2+

-

+.u,)

in

n

408

0 C h(x) S

W e have

12,

in

1

(-A(u2-G2) u2-u2 = 0

and

and

= X*h*(u2-c2) i n

1 ^ p z 0

G 2(x)

u (x) = 2

if

If

PUEL

J.P.

r.

on

f 0, denote by

p

R

pi(p)

the (ordered)

eigenvalues of the problem

-A v = p i ( p ) * p . v v/P pi(p)

W e know that

h

4

0, as

0

S

h

pi(h)

If

u2-C2 $

Now

h

=

X,/R

X = wl(h).

1,

and

p

2

(h) 7

A,/n

i=1,2,...

for

So, i f

A =

=

u2

A.

4

.

Then if

pi(h).

If

and then we must have

4

p.

1, we have

i s one of the

we necessary have

k > Al/n,

= 0.

pi(i)

0, A

R

depends monotonically on

5

2

in

A,/R, U;

X < k,/n,

we have and

G2 $ G?.

So again we have

1 = Cll(h). NOW

pl(h),

in

R.

(u2-G2)

i s an eigenfunction associated with

and then it has a constant sign, for example

2 j2.

I€ w e set

w = {x

(j = Ex we have

u2

jc

w.

But, as

Dirichlet problem on

w

I 1

x

E 62,

u2(x)

> 03

n,

G2(X)

> 03

x E

1

is the first eigenvalue

and o n

. w,

of the

^

we have

w = w,

and

409

A FREE BOUNDARY, NONLINEAR EIGENVALUE PROBLEM

then

u

A

2 = u2’

This finishes the proof of the uniqueness result. An interesting open problem is to find if for some ”good” geometrical shapes of

0

we can have uniqueness f o r all

(This result is true in 1-dimension if

fl

k.

is an interval).

Another problem consists in showing howthe set u depends on k. Here again i n 1-dimension we can show that ly decreasing in

w

is monotonic&-

1.

Remark: All these results have been announced in [ 8 ] and are _ detailed in [9]. References [l] R. Temam, A nonlinear eigenvalue problem: the shape at

equilibrium of a confined plasma, Arch. f o r Rat. Mech. and Anal., V o l .

6 0 ( 1 9 7 5 ) p 51-73.

[ 2 ] R. Temam, Remarks on a free boundary value problem

arising in plasma physics, Cornmunicatiomin P.D.E.

(1977).

[31 H. Berestycki

-

€I.

Brezis, Sur certains p r o b l h e s de

fronti&re libre, Note CRAS

-

Paris, 2 8 3 , Serie A ,

1976, p . 1091.

141 D. Kinderlehrer

-

J. Spruck,

151 D. Kinderlehrer

-

L. Nirenberg

to appear.

-

J. Spruck,

to appear.

[ 6 ] D.G. Schaeffer, N o n uniqueness i n the equilibrium shape of a confined plasma.

[ 7 ] A. Ambrosetti

-

Communications in P.D.E.

P.H. Rabinowitz, Dual variational methods

in critical point theory, Journal of Functional Analysis,

(1977).

14, 1 9 7 3 , p. 349-381.

410

[81

J.P.

J.P.

Puel,

PUEL

S u r un p r o b l & m e d e v a l e u r p r o p r e non l i n 6 a i r e

e t d e f r o n t i G r e l i h r e , Note CRAS p.

[91

J.P.

-

Paris, 284, 1 9 7 7 ,

861. P u e l , U n p r o b l 5 m e d e v a l e u r s p r o p r e s non l i n 6 a i r e s

e t de f r o n t i & ? - e l i b r e , P u b l i c a t i o n s d u L a b o r a t o i r e d ' b n a l y s e Numg'rique

d e l ' U n i v e r s i t 6 Paris V I ,

no

76016

G.M. de La Penha, L.A. Medeiros (eds.) Contemporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations QNorth-Holland P u b l i s h i n g Company (1978)

TIIli: CONCE FITS OF THERMO I)Y N AM I C S

.TAMES

SERRIN

School o f M a t h e m a t i c s University Minrieapoliy

,

of Miniiesota Miniiesota, 5 5 ' + 5 5

ABSTRACT E l e m e n t a r y thermodynamics h a s been much n e g l e c t e d a s an o b j e c t

of

i n q u i r y b o t h by m a t h e m a t i c i a n s

The b a s i c c o n c e p t 5 of h e a t and h o t n e s s c a n , a precise

hobever, be giben

s t r u c t u r e consis t e n t with the foundations o f c l a s s ~ a l

continuum m e c h a n i c s , n o t i o n of

and e n g i n e e r s .

and i t

i s p o s s i b l e t o develop a r i g o r o u s

a b s o l u t e t e m p e r a t u r e arid a r i g o r o u s d i s c u s s i o n o f

the Clausius inequality.

I n t h i s v i e b entropy i s a d e r i v e d

c o n c e p t , and t h e e x i s t e n c e o f

i n t e r n a l e n c r g y and e n t r o p y must

h e e s t a b l i s h e d f o r any g i v e n c l a s s o f m a t e r i a l s . To c o m p l e t e t h e f u n d a m e n t a l n o t i o n s o f one m u s t i n a d d i t i o n c o n s i d e r t h e problem of s t a t e s , arid t h e q u e s t i o n o f

thermodynamics

s t a b i l i t y of rest

i n t e r n a l l y constrained materials.

To i l l u s t r a t e t l i e problems i n v o l v e d i n t h e foundations o f

the

thermodynamics we g i v e examples o f

a

m a t e r i a l o b e y i n g t h e C l a u s i u s i n e q u a l i t y which d o e s r i o t have a u n i q u e e n t r o p y f u n c t i o n , and o f

an u n s t a b l e m a t e r i a l

sati5fh

i n g t h e Clausius-Duhem i n e q u a l 1 t y which i s c o o l e d b? t h e a d d i t i o n of h e a t . of

We a l s o examine t h e thermodynamic s t r u c t i r e

an i n t e r n a l l y c o n s t r a i n e d m a t e r i a l b h o s e d e n s i t y i s a

JAMES SERRIN

412

function of temperature, and finally show how a heat-conducting compressible Navier-Stokes fluid can be introduced as a thermodynamic entity without using either the notion of internal energy or o f entropy.

1. Introduction. -

It is now one hundred and fifty years since Sadi Carnot laid the foundations of theoretical thermodynamics, one hundred and twenty five years since Thomson and Clausius established the concepts of absolute temperature and entropy, and Joule overturned caloric theory, and one hundred years since Gibbs' great theory of equilibrium appeared.

It would,

therefore, be reasonable to believe that, by now, this traditional subject should be a mature and completed chapter of science, with developed and honed ideas, refined and delicate i n its axiomatic structure,

But, surprisingly, even

paradoxically, in disregard of scientific decorum thermodynamics still remains without a clear and definite structure. Of course, we cannot help but be every day aware of its manifold utility i n chemistry, physcis and mechanics; we can find the axiomatization of entropy admirably carried out by Callen at the classical level and by Truesdell in a generality suitable for continuum mechanics; and with Truesdell's work on Carnot engines we have gained new understanding of this classical device in a clear setting.

More-

over, Coleman and Owen in their theory of actions have initiated a strong attack on the entropy problem for complex

THE CONCEPTS OF THERMODYNAMICS

413

materials, and Coleman and others have carefully refined Gibbsf theory of equilibrium. Nevertheless, even with these successes, thermodynmics remains a patchwork of disjoint results, only partially complete and without acknowledged unity of pattern.

At

the most elementary level there is neither a mathematically acceptable nor physically clear treatment of the ideas o f Clausius and Thompson.

T o gain a most vivid impression of

this remarkable state of affairs, it is only necessary to contrast the answers one will receive from a good scientist if he is asked, on the one hand, to state the classical laws governing the motion of bodies and o n the other to formulate the Zeroth and Second Laws of thermodynamics!

A d ,

just at

one remove from the most elementary ideas of thermodynamics, neither have the classical notions been integrated with continuum ideas nor have the concepts of Gibbsian equilibrium been given more than ad hoc formulation. Generations have learned to tolerate, nay venerate, this ignorance*.

I t is noted and then disclaimed in count-

less books, occasionally with honest dismay, more often as a club to stifle inquiry. How often do we hear that an attempt to clarify the foundations of thermodynamics will cramp our physical intuition?

H o w often that we need not press deeper

since after all we all know how to -apply the subject anyway? _ _ Contrary to tradition I will interpose here the view that we now possess sufficient mathematical and mechamal -

* Some

books leave the real impression that thermodynamics springs god-like from pure introspection.

414

JAMES SERRIN

m a t u r i t y t o p l a c e thermodynamics

on a sound f o o t i n g b o t h a s

It

a n e l e m e n t a r y s c i e n c e and a s a handmaiden t o m e c h a n i c s . i s my f u r t h e r b e l i e f

t h a t t h i s need n o t be a s t e r i l e u n d e r -

taking, but t h a t i t w i l l i n t o the

2.

p r o v i d e f u n d a m e n t a l i d e a s and b r i n g

open a number o f u n s o l v e d

problems.

M a t e r i a l behavior.

Naturally,

i t i s n e c e s s a r y t h a t we s o u l d g r a n t a t

the outset c e r t a i n basic concepts, exactly a s i s the case f o r other theoretical sciences.

F o r thermodynamics t h e s e b a s i c Heat i s t o be c o n s i d e r e d

c o n c e p t s a r e h e a t and h o t n e s s . ~

i n t u i t i v e l y a s a form o f e n e r g y f l u x , work a s a n e n e r g y s o u r c e .

Hotness,

prime a t t r i b u t e o f m a t e r i a l s , of

interconvertible w i t h

on t h e o t h e r h a n d ,

is a

determining the e f f e c t i v e s t a t e

a material along ~ i t h o t h e r v a r i a b l e s o f mechanical o r

c h e m i c a l o r i g i n s u c h as

density

and c h e m i c a l c o n c e n t r a t i o n .

W e s e t t o one s i d e , once and f o r a l l , heat

o r hotness must

p r i n i i tive ideas.

t h e view t h a t

either

be d e r i v e d f r o m o t h e r p r e s u m a b l e more

*

The b e h a v i o r o f a m a t e r i a l , m a t e r i a l s , i n response

o r o f a system o f

t o a s y s t e m o f f o r c e s and t o t h e

*

To c o n s i d e r c l a s s i c a l thermodynamics as a n o u t g r o w t h of s t a t i s t i c a l mechanics i s s i m i l a r t o l o o k i n g on c l a s s i c a l mechanics a s t h e l i m i t of quantum or r e l a t i v i s t i c mechanics and need n o t c o n c e r n u s h e r e . The s t a t u s of s t a t i s t i c a l m e c h a n i c s , m o r e o v e r , h a s b e e n a r g u e d a t l e n g t h i n o t h e r places, p a r t i c u l a r l y i n t h e monographs o f Rridgman, Buchdahl and T r u e s d e l l . C e r t a i n l y i t i s v e r y much open t o q u e s t i o n w h e t h e r t h e s t a t i s t i c s of a g g r e g a t e s o f r e l a t i v e l y s i m p l e e n t i t i e s , d i f f i c u l t even a s t h a t i s , can s e r v e as a b a s i s f o r t h e complex s t r u c t u r e o f thermodynamics a s a whole.

This is not t o say, o f course, t h a t molecular i d e a s a r e r e j e c t e d - but only t h a t t h e y need n o t e x p l i c i t l y e n t e r t h e p o s t u l a t i o n a l f o u n d a t i o n s of thermodynamics any more t h a n t h e y e n t e r t h e f o u n d a t i o n s o f c l a s s i c a l mechanics.

application o f

external

SOII~CPS

of' h e a t

i s d e t e r m i n e d by f o u r

t actors :

( i ) the f a m i l y o f p r o c e s s e s k i n e m a t i c a l l y a d m i s s i b l e t o the m a t e r i a l ,

(ii)

(iii) (iv)

t h e c o n s t i t u t i v e s t r u c t u r e of

the m a t e r i a l ,

tlie l a w s o f m e c h a i i i c s , the l a w s

of

tFiermociyriarnics.

W e s h a l l be i n t e r e s t e d i n tlie f i r s t t h r e e of they

impinge o n t h e f o u r t h . Before

observe f i r s t

t a k i n g tliis up i n d e t a i l i t i s i m p o r t a n t

r a t h e r t e l l us a b o u t t h e

special cyclic processes.

berves

A s we s h a l l

t o r e s t r i c t the form of

materials.

to

t h a t t h e laws o f thermodynamics d o n o t a p p e a r

a s balance s t a t e m e n t s but o f

t h e s e only a s

see,

this

behavior 111

tlie c o n s t i t u t i v e r e s p o n s e o f

T h a t t h e r e i s c o n t r o v e r s y - o v e r t h e laws of

dynanucs i s a n a t u r a l r e s u l t pattern f o r the subject.

or

turn

tlie l a c k o f

therino-

an a c c e p t e d

T h i s i n t u r n may be a t

least partly

e x p l a i n e d by a n u n f o r t u n a i r d e s i r e t o c i r c u m v e n t a b s t r a c t i o n ,

a t e n d e n c y which m a n i f e s t s i t s e l f m o s t s i r o n g l y i n t h e proliferation

of s p e c i a l k i y p o t h e s e s , a t b e s t a d e q u a t e t o r

r a t h e r s i m p l e c l a s s e s of m a t e r i a l s ,

t o a p p l y beyond t h e i r n a r r o I v l i m i t s .

d i f f i c u l t or i m p o s 5 i b l e Even t h e c e l e b r a t e d

m a t h e ma t i c a 1 a t t a c k o n t h e rmod yriaini c s whi c h C a r a t h 6 od o rq initiated in

1909

s u f f e r s from

just t h i s defect.

I t seems t o me a d e s i d e r a t u m t h e r e f o r e t o acknoivl e d g e o p e n l y and d i s t i n c t l y t h a t t h e laws of must n e c e s s a r i l y b e a r a n e l e m e n t of

t o cover t h e broad range of m a t e r i a l

thermodynamics

abstractness i f behavior

they a r e

which we

416

JAMES SERRIN

presently desire for continuum mechanics.

3. The laws

of

Let

L

universe

thermodynamics. B

be a material body chosen from a given

of thermodynamic materials.

there is a preassigned set able to

8.

each process

of admissible processes avail-

While o n the one hand it is presumed that only

63 E P

processes

P

are allowable to

P E P

8 ,

we suppose also that

is dynamically possible for some

appropriate (unique) set o f forces and inputs o f heat. P

let

be a n admissible process of

I = [a,b]

there is a time duration and

I -+ R

Q(t):

8

Corresponding to

8 .

Now

Corresponding to

6

and two functions W ( t )

which represent, respectively, the cumulat-

ive w o r k done by the process against exterior forces, from

t = a

time 8

to time

t,

and thecumulative heat addition to -~ -

from the _ exterior, _ _ _~ during ~the same period. The function

W(t)

associated with a process will

in practice be determined by mechanical considerations (see Section 4).

That there exists a function

Q(t)

representing

the cumulative addition of heat to a material body during any admissible process can be considered a preliminary axiom of thermodynamics, on a par with the Zeroth Law.

The latter

can be stated as follows. Zeroth Law

-

There exists a topological line ..

m

which serves

~

as a coordinate manifold o f material behaviour. __ The points and

h

L

of

m

are called hotness levels,

is called the universal hotness manifold.

Note that

h

t h e manifold

i s t h e same f o r a l l m a t e r i a l s .

[ The word " e q u i l i b r i i n n " d o e s n o t a p p e a r i n t l i i s formulation o f

t h e Z e r o t h Law, s i n c e a t tlie f o u n d a t i o n a l le-cel

e q u i l i b r i u m has a t b e s t only a \ague

oper&ixre m e a n i n s ,

l y t o a x i o m a t i a e and u n n e c e s s a r y t o b o o t . we may o b s e r v e t h e the Zeroth Law:

If

~ a m eh o t n e s s l e v e l ,

imgairi-

A t t h e same t i m e ,

t r a d i t i o n a l c o i i c l u s i o r l t o be g a i n e d C r o m the material bodies and i f

rieTs l e v e l i s t h e same a s

C

H,

A

and

H

have t h e

i s a t h i r d m a t e r i a l whose h o t -

tlien

A

arid

R,

011

C

(trivially)

h a v e t h e same h o t n e s s l e v e l . ] A l o c a l coordinate of

m

il

T :

-+

a n open i n t e r v a l

i s called a l .o c a l e m p i r i c a l t e m p e r a t u r e s c a l e . ~

~~

I n e v e r y d a y t e r m s , e m p i r i c a l t e m p e r a t u r e c a n he measured by a n a r b i t r a r i l y c a l i b r a t e d t h e r m o m e t e r b a s e d a n y one o f a number of a i r , helium,

\\ay) just

etc.

substances:

mercury,

on

a l c o h o l , wine,

The Z e r o t h L a w e x p r e s s e s ( i n a n a b s t r a c t together w i t h the f a c t t h a t hot-

this possibility,

ness i s a c o n s t i t u t i v e determinant of m a t e r i a l behavior. First Law -.

-

Every c y c l i c

work i t d o e s :

interval

proce s s a b s o r b s a s much h p a t- a s t h e

thus i n a c y c l i c p r o c e s s occupying a time

I = [a,b]

we have

Q ( h ) = W(b).

I t i s n a t u r a l l y c r u c i a l i n t h e i n t e r p r e t a t i o n of t h e F i r s t Law t o b e a b l e t o r e c o g n i z e and i d e n t i f y p r o c e s s e s which a r e c y c l i c . dynamics i s b r o a d h o w e v e r ,

those

The i n t e n t i o n a l s c o p e o f and v a r i e d ;

emphatically c l e a r that no s i n g l e ,

tliermo-

i t i s therefore

generally applicable

p r e s c r i p t i o n c a n be g i v e n by which one c a n c h a r a c t e r i z e t h e s e processes,

Accordingly,

t h e i d e n t i f i c a t i o n of

cyclic

418

JAMES SERRIN

processes must i n point of fact be considered part of the axiomatic structure of the subject, a particular item i n the constitutive makeup o f each individual body.

This point of

view is nevertheless not to be construed as a counsel of benign neglect:

for any class o f materials under special

study the corresponding family of cyclic processes should and indeed must be defined with care and concern.

In the present circumstances, where our chief interest resides in continuum mechanics, one may obtain a rigorous, not to say strict and exacting, definition of cyclic processes by invoking the notion of periodicity.

Specifical-

ly, a motion of a continuous medium is said to be periodic if it is defined for all time, and if the physical path and the hotness level o f each particle involved i n the motion is a -periodic function of time (with the same period). _____ is then ___ cyclic if and only if it consists of a

A process

period

of a full periodic process.

It may, of course, occur for a particular material that it can effect no non-trivial cyclic processes whatever, in which case the First Law cannot, o n t h e

veryface of it,

be of particular use i n determining the constitutive behavior o f the material.

This is not to say that one cannot imagine

generalizations of the First Law, as well as of the Second Law,

strong enough to apply i n these circumstances.

We shall

not, however, consider such matters here. To state

the Second Law in a n effective way we need

a refinement of the notion of cumulative heat addition, taking into account not just the total heat supply but in addition

T H E C O N C E P T S O F THRRMODYNA?.IICS

t h e hotness let

U

t h e l e t t e r U denote i n t e r v a l s

(t)

6

t o t h o s e p a r t s of

8

and f o r any s u c h

U

whose h o t n e s s

i s a b s o r b -~ tive i f

Second Law

-

the exterior)

level i s i n

Note i n

U.

we now s a y t h a t a p r o c e s s

With t h i s d e f i n i t i o n ,

-~. -___

we l e t

‘+,,(t)= Q ( t ) .

particular that

QU(b)

Thus

of t h e hotness manifold.

denote t h e cumulative h e a t a d d i t i o n ( f r o m

function

8 .

l e v e l a t which t h e h e a t i s a c c e p t e d by

F o r any a d m i s s i b l e p r o c e s s Q

4 19

Q(b) > 0

U c h,

and i f , f o r e a c h

6

the

is non-negative.

A b s~. o r b t i v e p r o c e s- s_ e s

cannot be c y c l i c .

I t i s h e r e t h a t t h e n o t i o n of a u n i v e r s a l h o t n e s s m a n i f o l d g a i n s s i g n i f i c a n c e , and h e r e a l s o t h a t t h e axionfs o f thermodynamics d i s t i n g u i s h t h e q u a l i t y of h e a t i t s e l f i n terms of

t h e t e m p e r a t u r e a t which i t i s a b s o r b e d .

The p h y s i c a l

n o t i o n s which o r i g i n a l l y m o t i v a t e d t h e v a r i o u s n i n e t e e n t h century statements of

t h e second law a r e f a i r l y o b s c u r e today.

S u f f i c e i t t o say t h a t ,

i n t h e form s t a t e d h e r e , we h o l d t h e

view t h a t i n a p r o c e s s i n v o l v i n g a t o t a l

a d d i t i o n of h e a t

(throughout a l l temperature r a n g e s ) a m a t e r i a l w i l l never return t o i t s original state. p r e c i s e terms, must e m i t h e a t

I n popular,

though n o t q u i t e

e v e r y c y c l i c p r o c e s s which a c c o m p l i s h e s work a t some s t a g e o f

i t s operation

I t s h o u l d be emphasized t h a t , w h i l e a positive

. Q(b)

must be

f u n c t i o n i n an a b s o r b t i v e p r o c e s s , t h i s i n i t s e l f

i s not a s u f f i c i e n t condition i f

the various parts of

8

a t different hotness levels.

*

S t r i c t l y speaking, non-decreasing.

though t h e p h r a s e i s n o x i o u s ,

monotone

are

JAMES SERRIN

The thermodynamic structure outlined above can easily be generalized to systems, the associated processes then involving the simultaneous motion and interaction of one o r more material bodies.

P

process

functions

involving Wi(t)

and

F o r example, to an admissible

material bodies Qi(t),

S1,.

i=l,. ..,m,

. . ,8 m

w c associate

representing

respectively the cumulative work done and heat absorbed by each

Si

relative to. the exterior of the system, and functions _ ~ ~ i = l,..

Qi,(t),

.,m,

representing the cumulative heat

absorbed at hotness levels

P E U.

The total work done and

heat absorbed from the exterior o f the system is then defined to be =

Q(t) and s o forth.

C Qi(t),

W(t) =

c

Wi(t),

The laws of thermodynamics continue t o apply

in unchanged form.

B.

Perfect gases.

In the absence of a particular universe of materials only very limited conclusions can be drawn from the laws of thermodynamics.

Therefore it is in practice necessary to

introduce materials with specific sorts o f constitutive behavior.

I n the context o f continuum mechanics, which will

be our main concern here, the function

where

n

= n(t)

W(t)

has the form

is the instantaneous volume occupied by the

material, p = p ( x , t )

its density,

+ v

its velocity vector,

42 1

THE C O N C E P T S OF T E R M O D Y N A M I C S

-b

and

t

t h e s t r e s s v e c t o r a c t i n g a t t h e boundary

By C a u c h y ' s page

laws

of

motion t h i s may b e w r i t t e n

138)

-

= K(a)

K(t)

I

i s t h e k i n e t i c e n e r g y of

K(t)

tensor,

(see

151 ,

t

W(t)

where

[b

R.

of

D

arid

t h e r a t e of

8 ,

T:D

T

dx d t

the s t r e s s

deformation t e n s o r .

Q ( t ) s i m i l a r l y can b e e x p r e s s e d i n

The f u n c t i o n

4

where

r

i s t h e r a t e of h e a t s u p p l y p e r u n i t mass and

the heat f l u x vector.

A n analogous formula f o r

QU(t)

h

is

can

be w r i t t e n by r e s t r i c t i n g t h e s p a t i a l i n t e g r a t i o n s t o t h e a p p r o p r i a t e s u b s e t s where t h e h o t n e s s l e v e l l i e s i n mechanics a l o n e i t i s n o t

possible t o express

d i r e c t function of the process,

U.

From

Q ( t ) as a

a s was t h e c a s e f o r

W(t).

I n g e n e r a l f u r t h e r c o n s t i t u t i v e r e l a t i o n s a r e r e q u i r e d both

i'or

Q(t)

and

+ h.

A s a p a r t i c u l a r example o f c o n s i d e r a non-viscous viscosity

perf'ect g a s .

t h e s t r e s s t e n s o r has

i s t h e p r e s s u r e and

I

the simplest s o r t ,

we

I n t h e absence of

t h e form

T = -PI,

the i d e n t i t y matrix.

where

To d e t e r m i n e

p

p

we a d o p t B o y l e ' s law

P = p the funclion

0 : h + R+

t h e h o t n e s s manifold

W(t)

h.

0

(P),

being a coordinate f o r points

P

on

Now u s i n g t h e p r e c e d i n g formula f o r

t o g e t h e r w i t h t h e e q u a t i o n of

c o n t i n u i t y , we g e t

J A M E S SERRIN

422

-

W(t) = K(a)

[[

-

K(t)

0;

d x dt

where the superposed dot denotes the material time derivative. (It is assumed that the processes considered are smooth enough so that the integrals which appear here and later are well-

defined. ) T o complete the definition o f a perfect gas we must

In

add a constitutive formula f o r the cumulative heat. analogy with the expression for

[

W(t),

we put

t

Q(t) where

A

and

B

=

+ B6)dx d t ,

(Ab

are functions of

0

and

p

still to be

determined, and we require in addition that Q(t) = W(t)

+

K(t)

-

for each isothermal admissible process. formula for work, the integrand when

‘p =

0;

A;

indeed the extra term

+ Bb

Bb

K(a)

(In contrast to the does not vanish is essential because

the addition of heat to a material will, in general, change its temperature.

Special cases of the formula for

of course been traditional,

B

A

Q(t) have

being called latent heat and

heat capacity.)

-+

Finally we put h =

0 ,

this being the condition

generally appropriate for a non-viscous material. B y virtue of the condition on isothermal processes it ri-s evident that we m u s t have

To determine

B

we shall make use of the First Law.

42 3

THE C O N C E P T S O F TIIERMODYNAMICS

I n particular,

H

l e t u s s a y t h a t a p r o c e s s i s of

8

f o r a m a t e r i a l body

type __

i f

( i ) the h o t n e s s l e v e l a t each i n s t a n t o f time i s uniform

( i i ) t h e mapping

P = P(t),

that i s

over t h e body,

P(t):

I

+ h

and

i s c o n t i n u o u s and p i e c e w i s e

monotone. Consider a process of

p = p(t).

such t h a t a l s o W(t)

f o r a p e r f e c t gas 8 ,

H

A simple r e d u c t i o n then y i e l d s

-

= K(a)

type

-

K(t)

Lt

P dV

t

Q ( t )=

V

R

b e i n g t h e volume of

m

naturally

(clearly

V = m/p,

by t h e f o r m u l a

Q ( b ) = W(b)

Consequently t h e r e l a t i o n closed paths i n the

0,

m

where

V

and

p

are related

i s t h e mass of

8;

i s c o n s t a n t f o r a f i x e d body of m a t e r i a l ) .

t h e F i r s t Law we h a v e

o n l y on

+ p dV),

( B V do

(0,p)

f

By

f o r cyclic processes.

(B/p)dO = 0

plane.

must h o l d for a l l

Hence

B/p

c a n depend

that i s

B = pc(0). ( I n p h y s i c a l terms t h i s r e s u l t of

s t a t e s that the specific heat

a p e r f e c t g a s d e p e n d s o n l y on t h e h o t n e s s l e v e l . ) From t h e p o i n t of v i e w of continuum m e c h a n i c s ,

i s w o r t h w h i l e t o show how t h e s t a n d a r d e n e r g y b a l a n c e law f o l l o w s from t h e above c o n s i d e r a t i o n s .

L e t us d e f i n e t h e

s t a t e functions

E =

p e dx,

e =/c(@)d@.

it

424

J A M E S SERRIN

Then a s i m p l e c a l c u l a t i o n * s h o w s d

- (B+K) = dt as required. e n e r g y of

The f u n c t i o n

E

that

b(t)

-

G(t),

t h u s serves a s the i n t e r n a l

t h e e n t i r e body, w h i l e

e

i s the s p e c i f i c i n t e r n a l

energy.

then

O n t h e other hand from t h e f o r n l u l a f o r

a t t h e beginning

Q(t)

+ of

t h e s e c t i o n we h a \ e ,

since

h = 0,

Q ( t )= where

r

p r dx

i s t h e r a t e of h e a t s u p p l y p e r u n i t mass.

can be supposed a r b i t r a r y ,

i t follows t h a t

Or, =

r,

which i s t h e s t a n d a r d d i f f e r e n t i a l f o r m o f law.

Because i-2

t h e energy balance

F r o m t h i s e q u a t i o n i t i s clear i n a d d i t i o n t h a t

t h i s e q u a l i t y shows immediately t h a t n o c y c l i c p r o c e s s can be x

Namely

( f r o m t h e T r a n s p o r t Theorem)

i t b e i n g assumed f o r t h i s f o r m u l a t h a t continuously d i f f e r e n t i a b l e .

Q,

W

and

K

are

THE C O N C E P T S 01“ TERI\IOD1’NAMICS

1+2j

a b s o r b t i v e , S i n c e t h e integral o f t h e r i g h t s i d e is p o s i t i v e for such Processes.

H e n c e t h e S e c o n d Law h o l d s f o r n o n -

v i s c o u s p e r f c c t gases.

s

function

n ,I

is the entropy o f the body,

the -.__ specific entropy. -

We emphasize that there has been no

,

i 111 c ~ ? - n , t l energy) o r e’lcii of absolute

-

1)l-i

ori 11-c of^ ~

r i l t i’o})?

tenlperature in this d e r i v a t i o n .

What is required rather is a

constitutive formula for the cumulative heat supply traditional notion in classical thermodynamics. L

V

and

Q(t),

a

In Section 9

shall treat a more complicated example using the same

~

techniques.

In classical thermodynamics one seldom has nearly as much detailed constitutive information about the behavior of materials as i n the considerations above.

Because of this,

the formulas f o r heat and w o r k are usually stated only for processes which are spatially homogeneous at each instant of tinie.

F o r the purposes of many thermodynamical calculations

this proves to be sufficient, though obviously i t is not enough f o r continuum mecliaiiics

.

5. The Clausius inequality -~ and absolute temperature. Given a universe of material bodies, one may, if the constitutive response of at least some of the materials is relatively simple, and i n particular if the universe of materials includes a non-viscous perfect g a s , prove the

42 6

JAMES SERRIN

following key theorem: There exists a global coordinate system ~-

8:

h

4

R+

on manifold with the property that ~ the hotness _ ___

for any cyclic process whatsoever of type

H.

Moreover

8 is

unique up to a constant positive multiple. Any one of the multiplicatively related coordinate systems

8

can obviously be used as a canonical empirical

temperature scale on to normalize

8

h.

In practice it is customarily agreed

by the convention

I ~ ( L ~ -) where

Ls

and

Li

e(Li)l

= 100

( o r 180)

are two conventionally fixed points i n h ,

say the boiling point and freezing point of pure water at standard atmospheric pressure.

The resulting temperature

scale is called Cbsolute temperature.”

We shall use it hence-

forth i n the paper.

For w a n t of space we shall omit the proof of ( * ) .

*The

actual determination of absolute temperature clearly requires some further theoretical analysis (see for example, Epstein, Section 2 9 ) as well as accurate laboratory techrrique. This need not, of course, concern us here. Note also the absolute values i n the normalization condition: it is not or 9 ( L ~ ) will be the a priori evident which of 8 (Li) larger.

427

THE CONCEPTS OF TERMODYNAMICS

We n o t e h o w e v e r t h a t t h e r e a r e s e v e r a l s t a n d a r d d e m o n s t r a t i o n s a v a i l a b l e u n d e r s t r o n g e r a s s u m p t i o n s o r i n more r e s t r i c t i v e contexts."

6. Internal energy and entropy. We have seen the concepts of internal energy and entropy arise in the example o f a perfect gas i n Section

4.

One of the principal discoveries of early thermodynamics (in essence due to Clausius) is that the same thing occurs for a large class of simple materials.

Rather than give a rigorous

definition of these materials, which i n any case would not be particularly useful here, w e simply note that their prime characteristic is the existence of a finite set of configuration variables which determine their state in spatially homogeneous processes. The internal energy state space of

8

E

o f such a body

8

maps the

into the reals, and has the property that A(E+K)

= AQ

for all admissible processes of

8

- aw whose initial and final

configurations are spatially homogeneous ( s o that

AE

is

*

F o r the validity of ( * )

it is in fact enough i f the hotness level at each instant is uniform o v e r just those parts of the body which are either absorbing o r emitting heat. A more general result can be conjectured. Since the s . s function Q,(t) generates a Bore1 measure p t on b , the integral

is defined f o r each t E I.. One may then presume that J(b) < 0 for all cyclic processes. (Note added, May 1978: T h i s h a s i n fact r e c e n t l y been proved by t h e a u t h o r t o g e t h e r w i t h R. Hummel and hl. Ricou.)

428

J A M E S S E R R IN

well-defined). Similarly the entropy

maps t h e s t a t e s p a c e o f 8

S

i n t o t h e r e a l s and h a s t h e p r o p e r t y t h a t

‘I t h e i n e q u a l i t y h o l d i n g f o r a l l a d m i s s i b l e p r o c e s s o f type H whose i n i t i a l and f i n a l c o n f i g u r a t i o n s a r e s p a t i a l l y homogeneous. The r e s t r i c t i o n t o s p a t i a l l y homogeneous c o n f i g u r a t i o n s and t o p r o c e s s e s o f

end

type H i s of c o u r s e a

t h e p r e o c c u p a t i o n o f c l a s s i c a l thermodynamics

r e f l e c t i o n of

with t hese concepts.

The s i t u a t i o n i s c l a r i f i e d i f we consider

t h e example o f a p e r f e c t g a s d i s c u s s e d i n S e c t i o n 14.

For

s p a t i a l l y homogeneous c o n f i g u r a t i o n s , t h e i n t e r n a l e n e r g y and e n t r o p y

d e f i n e d t h e r e reduce t o

S

(We have r e p l a c e d

0

0

by

i s proportional t o

i n the standard texts. BdS = d E This formula,

E

s i n c e i t i s easy t o prove t h a t

Re,

0 ) .

I t i s t h e s e f o r m u l a s which a p p e a r

One may a l s o c h e c k e a s i l y t h a t

+

pdV

(Gibbs r e l a t i o n ) .

o r v a r i a n t s o f i t , occurs a s a u n i f y i n g t h r e a d

t h r o u g h o u t c l a s s i c a l thermodynamics. If we go beyond t h e m a t e r i a l s t r e a t e d i n thermo-

dynamics t e x t s t h e e x i s t e n c e of a n e n t r o p y ( a n d e v e n o f i n t e r n a l e n e r g y ) i s n o l o n g e r a n o b v i o u s or d i r e c t m a t t e r .

423

TIIE C O N C E P T S O F THERMODYNAMICS

F r o m t h e w o r k of Coleman and Owen [ 9 ] , that i f

hobever,

it follows

a s u i t a b l y l a r g e f a m i l y of p r o c e s s e s i s a v a i l a b l e then

a ( u n i q u e c o n t i n u o u s ) i n t e r n a l e n e r g y i n g e n e r a l e x i s t s , and that

there i s a (not necessarily continuous)entropy function

r e l a t i v e t o processes o f

type H .

A s we h a v e a l r e a d y s e e n f o r t h e c a s e o f a non-viscous p e r f e c t g a s , and a s w i l l b e s h o w n l a t e r f o r a N a v i e r - S t o k e s r l u i d , g i v e n a r e l a t i v e l y simple c o n s t i t u t i v e s t r u c t u r e i t i s p o s s i b l e t o o b t a i n n o t only a d i f f e r e n t i a b l e i n t e r n a l energy f u n c t i o n b u t a l s o a unique d i f f e r e n t i a b l e e n t r o p y , b o t h t h e s e q u a n t i t i e s moreover b e i n g tlefiried i n t e r m s o f i n t e g r a l s of m a s s specific functions.

To u n d e r s c o r e t h e n a t u r e of

tlie e n t r o p y problem f o r

g e n e r a l m a t e r i a l s , we e m p h a s i i e t h a t even when a n e n t r o p y f u n c t i o n e x i s t s , i t need n o t be u n i q u e .

A s i m p l e example o f

t h i s phenomenon is i n c l u d e d i n t h e a p p e n d i x .

7 . S t a b i l i t y of m a t e r i a l s . S t a n d a r d t e x t b o o k s h a v e c o n f u s e d t h e f o u n d a t i o n s of thermodynamics n o t o n l y by i n a d e q u a t e l y f o r m u l a t i n g t h e i r i d e a s , b u t a l s o by o m i t t i n g c o n s t i t u t i v e a s s u m p t i o n s ,

by

t a c i t l y u s i n g f r i c t i o n l e s s m a t e r i a l s under t h e t h i n l y disguised p r e t e n c e o f q u a s i - s t a t i c patently incorrect logic.

processes,

Nor d o e s t h e s i t u a t i o n improve

when e q u i l i b r i u m i s d i s c u s s e d . the impression that

and by r e s o r t i n g t o

I t i s i n f a c t easy t o g a i n

e i t h e r ( a ) thermodynamics c a n b e d e r i v e d

f r o m e q u i l i b r i u m c o n c e p t s a l o n e , or ( b ) e q u i l i b r i u m i s a c o n s e q u e n c e o f t h e F i r s t and Second Laws a l o n e . N e i t h e r i s

430

J A M E S SERRIN

the case. To see this, consider the following material, which was recently introduced by Green and Naghdi for rather different purposcs,

dt

where

a

and

c

(a <

are constants

0,

*

c > 0).

~t is

immediately clear that the First Law is sat sfied for all

e

cyclic processes (we assume o r course that

return

to its

original values at the end of the cycle) and that there is an internal energy function, namely E = i

p

e = c0

edx,

+

aB.

To verify the Second Law it is enough to show that there

q(8,b)

exists a function

such that

ci + a; s 8

d dt

q(e,i).

This inequality, however, is easily verified when c log 8 + a 6 / 8 .

(The function

q

q =

is of course nothing more

than the specific entropy.) Now consider the response of this material in a simple heat conduction problem p 6 = X A ~+ p r ,

r = heat supply rate,

p = constant

subject to the initial (rest) condition )t

The material may be supposed incompressible if one wishes to account for the omission of the pressure integral in the formula for W(t).

431

THE CONCEPTS O F THERMODYNAMICS

and t o i n s u l a t e d w a l l s .

r = r ( t ) , we s e e

Assuming t h a t

t h a t a l l s o l u t i o n s w i l l h a v e t h e form

6 =

e(t),

Consider i n p a r t i c u l a r a uniform r a t e of h e a t withdrawal

r = -c

Assuming

a <

d u r i n g t h e time

with i n i t i a l conditions

Since

-c/a

t > &

> 0

a f t e r wich

r=O.

we t h e n have t h e problem

0,

i- €

s o l u t i o n for

t < &,

0 5

0 s t s g

6 = 0o,

9 =

0

at

t = 0.

The

i s e a s i l y found t o b e

i t is clear that

Therefore s u b t r a c t i o n of s m a l l e s t amount of p o s i t i~ v e i n finity. -

t h e s m a l l e s t amount of h e a t f o r t h e

time d r i v e s t h e t e m p e r a t u r e u l t i m a t e l y t o O f course,

t h i s v i o l a t e s not

only our

s e n s e o f t h e r e l a t i o n between h e a t i n g and t e m p e r a t u r e , b u t a l s o shows t h a t t h e m a t e r i a l i s u n s t a b l e . t h e F i r s t and Second L a w s a l o n e do

We c o n c l u d e t h a t not guarantee s t a b i l i t y .

a

7

0

Moreover by c o n s i d e r i n g t h e c a s e

i t i s e v i d e n t t h a t s t a b i l i t y does n o t g u a r a n t e e t h e

F i r s t and Second L a w s ( i . e .

a s t a b l e m a t e r i a l could d r i v e a

p e r p e t u a l m o t i o n machine of

t h e second k i n d ) .

a

7

0

one f i n d s t h a t

I n d e e d when

432

JAMES SERRIN

f o r every n o n - t r i v i a l

differentiable

c y c l i c process

of c l a s s

s o t h a t t h e Second Law p a t e n t l y f a i l s ; on t h e o t h e r h a n d ,

H,

s t a b i l i t y i s guaranteed s i n c e t h e h e a t conduction equation c a n be t r a n s f o r m e d into t h e wave e q u a t i o n by w r i t i n g 9 = e- c / a t

$

,

t h e exponent being n e g a t i v e .

I n v i e w o f t h i s e x a m p l e , a n d i n d e e d as h a s b e e n evident since t h e pioneering work o f Gibbs,

it is

necessary to augment the classical laws o f thermodynamics by conditions guaranteeing t h e stability of materials. F o r the universe o f materials introduced by Gibss and s t i l l s t u d i e d i n modern t e x t b o o k s ,

one may ( a l b e i t o n

i n t u i t i v e g r o u n d s ) p h r a s e t h e s e c o n d i t i o n s i n t e r m s of

restrictions

o n t h e geometry of v a r i o u s s t a t e f u n c t i o n s .

I t would seem b e t t e r , however, t o i n t r o d u c e t h e s t a b i l i t y of m a t e r i a l s i n a more f u n d a m e n t a l way of course u s e i t t o deri-ce ~t h e G i b b s i a n r u l e s of

equilibrium.

S i n c e a g e n e r a l t r e a t m e n t of of

the question,

s t a b i l i t y may be o u t

I w i l l consider here only certainelementary

p o i n t s , a t an a d m i t t e d l y i n f o r m a l l e v e l o f d i s c u s s i o n .

_ e~f i n_i~ _A s t a t i o n a r y c o n f i g u r a t i o n D t i o_ n -

50

of a thermo-

dynamical s y s t e m w i l l b e s a i d t o b e i n i__ s o l a t i o n equilibrium u n d e r a s e t of

constraints i f

t h e e f f e c t i v e thermodynamic

s t a t e changes a s l i t t l e a s we p l e a s e f o r .any. p r o c e s s

63 E P

which

(i)

starts at

aO

,

and

(ii) s a t i s f i e s the constraints, and f o r which t h e c u m u l a t i v e h e a t s work

W(t)

a r e s u i t a b l y small.

QU(t)

and t h e c u m u l a t i v e

433

THE CONCEPTS O F THERMODYNAMICS

T o examine the consequences of isolation equilibrium,

suppose that an internal energy and a n entropy exist.

Then

for processes starting from a stationary configuration we have

-

AE = -K(t) + Q(t) Now let K(t)

2

E > 0

0

-

(AE = E(t)

W(t),

be an arbitrary (small) constant.

E(a)). Since

we can guarantee the relation

by making the cumulative heats and the cumulative work suitably small. inequality*

Similarly, by virtue of the Clausius

we can make

as

-E

2.

by choosing the cumulative heats even smaller (if necessary). Let

S = So

and

E = Eo

at

S

strong local maximum of

63

Suppose that

is a

So

with respect to the side condition

E 5 E 0 - i n the sense that if

a process

uo.

S

2

So

-

E

and

E

S

E0

+

on ~

satisfying (i) and (ii) above, then the

effective thermodynamic state change can be made as small as we please by making

E

small.

Then clearly the configuration

is i n isolation equilibrium with respect to the given constraints. Conversely if

So

is not a strong local maximum in

the sense described then the configuration cannot be i n isolation equilibrium.

When this conclusion is specialized

to the classical circumstances discussed i n thermodynamics, where the constraints are those of fixed volume and mass,

* Such

a n inequality is not proved i n general, as we have already noted, but it certainly holds for large classes of processes.

434

JAMES SERRIN

this yields Gibbs' necessary condition f o r isolation equilibrium, ~

~~

namely that

must be a maximum for fixed

S

V

E,

M.

and

A parallel discussion obviously applies to the

situation when

Eo

the side condition

is a strong local minimum with respect to

S

2

S

0'

As is well known, quilibrium considerations play a r o l e of considerable importance not only i n general. dynamical

problems but also in the determination of the constitutive behavior o f materials.

To take a case in point, i n the

example which opened this section it was shown that a homo-

a <

geneous rest state o f the material was unstable when

a

Accordingly one may simply reject negative valies o f this material since they lead to anomalous behavior.*

0.

for At the

same time it does not seem to be a simple matter to give a general formulation to such ideas, since the reasoning cannot be based simply on the notion that homogeneous rest states are stable, because this is i n fact not always true.

These

remarks motivate the following rule, which is both somewhat special i n character and proposed only for tentative consideration,

Let __ level

be a material bod-y.

8

~

Then f o r any hotness ~-

~

and any volume

Po

ation of

8

~~

Vo

there exists a rest configur._ - ~

~

which is____ in isolation equilibrium under the -~

constraint of fixed volume and mass, has a uniform hotness __ -.-__ ~~

level -

Po

*Likewise,

and occupies the volume __

~

~

vO

~

a

positive values of a cannot occur because of the second law, leading to the conclusion that the coefficient a necessarily is zero.

THE CONCEPTS O F T T I C R I I O D Y Z I A ~ I I C S

When a p p l i e d t o

435

tlie s t a n d a r d m a t e r i a l b o d i e s

of

c l a s s j c a l thermodynamics t h i s i m p l i e s t h e well-known i n e q u a l i t i e s r e l a t i n g the s p e c i f i c heats of m a t e r i a l s , a s a number o f

additional

conclusions.

d i s c u s s e d a t t h e beginning v f

for o t h e r w i s e , temperature

as shown,

Also,

f o r the material

the section s e find that

a

= 0,

i n p u t o f heat causes a l a r g e

a small

change. thermodynamics i s

Besides i s o l a t i o n e q u i l i b r i u m , a l s o concerned with equilihriirm of

sys tems which a r e immersed

i n an e n v i r o n m e n t a l b a t h a t c o i l s t a n t t e m p e r a t u r e tliis

a s well

Be.

In

c a s e t h e p r e v i o u s d e f i n i t i o n i s t o be r n o d i f i e d b y a d d i n g

a t h i r d r e s t r i c t i o n on t h e p r o c e s s e s , namely t h a t

( T h a t i s , t h e body c a n a b s o r b h e a t from t h e e n v i r o n m e n t o n l y if

i t s a b s o l u t e temperature i s

if

i t s temperature i s

on t h e m a g n i t u d e o f

ee.)

2

At

and c a n e m i t h e a t o n l y

t h e same t i m e n o r e s t r i c t i o n

i s imposed:

Q,

p a r a m e t e r i s t h e s m a l l n e s s of If

< Be,

W ( t )

the s t a b i l i z i n g alone.

we a g a i n s u p p o s e t h e r e e x i s t s an i n t e r n a l

energy

and e n t r o p y t h e n u n d e r t h e p r e s e n t c o n d i t i - o n s we h a v e

3 E = -K

From t h e c o n d i t i o n elimination of

+

Q

IW(t)l

-

W,

C

E

-

eens s

LS

2

Q/ee.

there results, a f t e r

4, LI:

I n general unless

8

E

9e

Q.

there a r e processes s a t i s f y i n g

t h i s c o n d i t i o n which d o r i o t i n v o l v e c o n f i g u r a t i o n changes

436

JAMES SERRIN

tending to zero as 0 = Be

E

becomes small.

Hence we conclude that

at equilibrium.* Isolation and heat bath equilibrium are of course

not identical concepts.

I t can be proved, however, that a

system in heat bath equilibrium is necessarily in isolation equilibrium, and conversely if certain natural hypotheses hold, that a system i n isolation equilibrium at a uniform temperature €Io

is i n heat bath equilibrium for the same environmental

temperature. In reference

[14] the reader will find a very

complete discussion of stability when the environmental temperature is held fixed.

8. Internal constraints. ~__ ._ __ The final element o f thermodynamic structure which

I should like to consider is the matter of internal constraints.

*A s

a check on these formulas, o n e may note i n the case of spatially homogeneous configurations of a perfect gas that

when B 0 is the temperature of the original configuration. we have Thus for processes with V E V

exist processes It follows that whenever 8 # B e there will obeying AE eeAS S 0 bu? for which 8 is __ not arbitrarily We conclude, as noted, that e 0 necessarily equals near B 0 .

-

Be

at equilibrium.

THE CONCEPTS OF TIIENMODYNAMICS

4 37

In general, when the class of kinematically admissible processes is restricted we can expect dynamic forces to arise to maintain the constraints in the face of external forces which would otherwise deform the material.

Alternately,

without the new dynamic forces one would expect to find smooth external force fields for which no corresponding admissible processes would be dynamically possible.* A s a n example, consider a non-viscous fluid body

under the constraint of a temperature dependent density p = p(B),

a relation commonly encountered i n liquids.

TR

introduce a reaction stress tensor

5

and make the key

as sump ti on w

T = T + T R ,.

where

T =

-G(e)I.

Then from the formulas i n the first

paragraph of Section W(t)

4 we find

= K(a)

-

ft c

-

K(t)

!a

In{PP p

+

T

R:

D)dx dt.

Correspondingly, the cumulative heat supply can be supposed to have the form

1’, [ t

Q(t) = Where

B

and

C

depend on

can be absorbed into

B’e

ing differentiation with

e.)

A;

ic

{Bi

8

+

C: ?R}

and since

TRw

d x dt, (The expected term

= p’i

,

primes denot-

There is a third possibility: to consider the constraints to arise in the limit as some parameter of constitutive behavior becomes singular ( s a y as the compressibility approaches zero). We do not consider this here.

J A M E S SERRIN

438

I t i s e a s y t o s e e from t h e F i r s t L a w t h a t where

is a s c a l a r r e a c t i o n .

p

c l e a r l y b e c y c l e s for which

TR = -PI,

I n d e e d o t h e r b i s e t h e r e would

Q(b)

f W(h),

since

c o n s i d e r e d t o be a n i n d e p e n d e n t dynamic v a r i a b l e .

TR

is

It f o l l o N s

that

-

( h e r e we h a v e a b s o r b e d and c o n s i d e r

C

p

into

as c a n o b v i o u s l y be d o n e ,

p,

now t o b e a s c a l a r ) .

F o r s p a t i a l l y homogeneous p r o c e s s e s , we f i n d i n particular

W(t)

= K(a)

Q(t)

=

f BV

1 (BV

By t h e F i r s t Law

so also i s

where of

Y(8)

(BV/e)dfI

-

+

dfI

K(t)

+

pV’)de

p V‘

dB

CV dp.

+

(GV/e)dp.

i s a r b i t r a r y and

\

+

T

C V dp

i s e x a c t , and by ( * )

S o l v i n g , we f i n d e a s i l y that

= T(8)

i s the reciprocal

p = ~ ( 8 ) - Thus i n t u r n t h e r e e x i s t s a s p e c i f i c i n t e r n a l

energy

e

and

a

s p e c i f i c entropy

The dynamic v a r i a b l e

p

7 ,

havine; t h e forms

t h u s c o n t r i b u t e s n o t only t o t h e

s t r e s s t e n s o r , b u t a l s o adds a l i n e a r term t o t h e i n t e r n a l

e n e r g y , t h e e n t r o p y and t h e s p e c i f i c h e a t a t c o n s t a n t pressure.

Late a l T o t h a t t h e Gibbs r e l a t i o n eclr, = de c o n t i n u e s t o hold e v e n though r u r t l i e r d i s c u s s i o n of

+

p

pdr

i s a dynamic v a r i a b l e .

t h i s example, i n c l u d i n E a d e r i p a t i o n o f

t h e energy b a l a n c e eqiiati

oil

aiid v e r i f i c a t i o n

L a u h o l d s f o r a l l adniissAb1P c y c l i c p r o c e s s ,

i s standard.

we s p e c i a l j / e t h e p r e c e d i n g c o m p i i t a t i o n t o t h e

If

case bhere

t h a t t h e Second

p = c o n s t a n t , namely t o a n i n c o m p r e s s i b l e m a t e r i a l , B/p

then t he specj f i c h e a t

becomes s i m p l y

y(Q),

while

/

I t i s a l s o o f i n t e r e s t t o o b s e r s e t h a t t h e dynamic f'orce

c o n t r i b u t e s to t h e work e x p r e s s i o n case

p = p(0)

,

T:

D

p

i n tlie c o m p r e s s i b l e

but not i n the incompressible case.

The i d e a

or

a s t r e s s r e a c t i o n seems t o be d u e t o

G r e e n , Naghdi and T r a p p .

A general theory within the context

o € t h e Clausius-Duhem i n c y r i a l i t y was p r e s e n t e d by S e r r i n a t a conference o f

t h e S o c i e t y l'or

N a t u r a l Philosophy i n P i s a i n

1974.

9 . C o m p r e s s i b l e N a v i e-__ r-Stokes

fluids. .~

S i n c e t h e p r e v i o i i s examples h a v e b e e n r e l a t i v e l y simple,

i t w i l l b e u s e f u l i n c o n c l u s i o n t o c o n s i d e r a more

general situation.

We s h a l l s h o w how a h e a t - c o n d u c t i n g

compressible Navier-Stokes

r l u i d can be i n t r o d u c e d a s a

thermodynamic e n t i t y w i t h o u t u s i n g e i t h e r t h e n o t i o n o f

440

JAMES SERRIN

o f i n t e r n a l energy o r o f e n t r o p y , * Without l o s s of g e n e r a l i t y , we may b e g i n w i t h t h e s t a n d a r d formula f o r t h e s t r e s s t e n s o r ( s e e r e f e r e n c e

P.

[151,

233) T = (-p p , 1,

where

Section

4 it

+ X

+

&D,

and

0.

div v ) I

a r e f u n c t i o n s of

9

Then from

i s clear that W(t)

= K(a)

-

ftf

K(t)

+

I

P

T r a c e D dx d t

The f i r s t i n t e g r a n d i s t h e m o s t g e n e r a l l i n e a r f u n c t i o n of and

D

which v a n i s h e s when

i n t e g r a n d ( b a r r i n g a term of

D = 0,

w h i l e t h e second

t h e form

6

Trace D)

s i m i l a r l y t h e most g e n e r a l q u a d r a t i c e x p r e s s i o n i n which v a n i s h e s when

D = 0,

is

6

and

D

t h e assumption of m a t e r i a l

o j e c t i v i t y b e i n g used throughout.

I n a n a l o g y we m a y t h e n

require

*

I n [9] v i s c o u s f l u i d e l e m e n t s were s t u d i e d w i t h t h e p u r p o s e of e x h i b i t i n g a n i n t e r n a l e n e r g y and e n t r o p y ( s e e p a r t i c u l a r l y S e c t i o n 1 3 ) . A l t h o u g h t h e r e i s some r e l a t i o n between t h a t a n a l y s i s and t h e p r e s e n t w o r k , t h e r e a r e a l s o n o t a b l e d i f f e r ences, I n particular the s t a r t i n g point here i s the basic Clausius i n e q u a l i t y ( * ) , while t h e r e t h e corresponding formula i s ( 9 . 7 ) w i t h s 2 0 on c y c l e s . These r e l a t i o n s are q u i t e d i s t i n c t i n content since (*) r e f e r s t o s p e c i a l p r o c e s s e s of an e n t i r e m a t e r i a l body w h i l e ( 9 . 7 ) r e f e r s t o a f l u i d e l e m e n t . To some e x t e n t t h e c o n c l u s i o n s r e a c h e d a r e a l s o d i f f e r e n t , s i n c e t h e i r r e l a t i o n ( 1 3 . 3 3 ) f o r example c a n n o t b e o b t a i n e d s o d i r e c t l y ( o r even a t a l l ) u n d e r t h e p r e s e n t hypotheses. The r e s u l t s o f t h i s s e c t i o n were r e p o r t e d f i r s t a t t h e N i n t h B i e n n i a l C o n f e r e n c e on F l u i d M e c h a n i c s , h e l d a t B i a l o w i e z c a , P o l a n d , i n September 1975.

44 1

THE CONCEPTS OF THERMODYNAMICS

[[ t

Q(t)

=

where

A,

{A

Trace D

{a(Trace

B, a , B , y , 6

D)

+

B8)dx dt

2

+

@D:D

+

are functions of

remain t o be determined,

'2 )dx dt

+ 68

y i Trace D

p

and

which

We shall see that these coefficients

cannot b e arbitrary i f the laws of thermodynamics are to hold, and that i n fact

Q(t)

must reduce precisely to the usual

formula. * The key is to consider cyclic process of type

H

in which the velocity vector has the form

where

al, a2, a3

are functions o f

t

ing that the density is uniform at time easily that

p = p(t)

alone.

t = a,

Then, assumwe find

and that

Trace D = (j/p D:D

2 = a: + a 2

= al + a2 + a3, 2 +

a3'

From the First Law it can be shown that BTde

+ (A-p)dT

(. = l / p )

is an exact differential, as well as that

*The

same conclusion can b e reached under even weaker initial hypotheses o n the constitutive form of Q(t), namely that

where 2 is an arbitrary continuously differen$iable function of its arguments, which vanishes when D and 8 are zero. Naturally there is no reason to assume that 2 vanishes when D = 0 but 9 # 0.

442

JAMES S E R R IN

Similarly using

( * ) we f i n d t h a t

+

Ed8

e

i s exact,

A -dd7

e

and

LL

Consequently t h e r e e x i s t f u n c t i o n s

P

d e p e n d i n g o n l y on

+

de = BTd0

p ,

and

+

dn = B d 8

e

q,

-Ad T .

e

a r e a r b i t r a r y (except f o r

i t i s e a s i l y shown t h a t

periodicity)

and

such t h a t

(A-p)dT,

al, a 2 , a 3

Moreover, s i n c e

e

3h +

and

1.I 5 0

Zp.20.

W(t)

A f t e r s u b t r a c t i n g the o r i g i n a l e x p r e s s i o n f o r

from

Q ( t ) , an easy c a l c u l a t i o n then y i e l d s t h e r e l a t i o n

Gdt where

Jn

E =

p e dx.

b(t) -

=

(E+K)

+(t)

We h a v e t h u s o b t a i n e d t h e s t a n d a r d

e n e r g y b a l a n c e law of

continuum m e c h a n i c s .

Note ( i n summary)

that T = (-p

where

l-l

2

0,

31 +

2

+

+

0

div v)I and where 8dq = d e

r e l a t e d by t h e Gibbs r e l a t i o n

+ 211

D

e, p,

+

and

n

are

pdr.

I f we add t h e f o r m u l a

t ( t )= where

r

[

p r dx

+

h - n ds

%o"

-4

i s t h e h e a t s u p p l y r a t e p e r u n i t m a s s and

heat flux vector, dS dt

-

h

i s the

a standard calculation then y i e l d s E d x

-

fa

+

- t +

G

e

d

s +

h'grad

e2

8

) dx

443

T H E CONCEPTS O F TIIERMODYNAMICS

S =

where

p q dx

and

I = X ( T r a c e D ) 2 + 2p D:D. I n a heat

p r o c e s s t h e f i r s t t w o i n t e g r a l s on t h e

absorptive

r i g h t hand s i d e would be n o n - n e g a t i v e

t.

a s f u n c t i o n s of that

I

Thus if

0.

2

The c o n d i t i o n

+

h.grad

8

2

and n o t i d e n t i c a l l y z e m

p 2

0,

31 +

2

shows

0

t h e n no c y c l i c p r o c e s s

0

c o u l d b e h e a t a b s o r p t i v e , v e r i f y i n g t h e Second Law ( s i n c e grad 8 = 0 ever

+

h

i n a process o f type

i t i s c l e a r t h a t , what-

H

may b e , no c y c l i c p r o c e s s o f t y p e €3 can b e h e a t

absorbtive). Except i n s p e c i a l c o n s t i t u t i v e c a s e s i t i s n a t u r a l l y

+

hsgrad 8

not p o s s i b l e t o prove t h a t

c o n s i d e r t h e s i m p l e Newton-Fourier

+

h =

where

K

= u ( p ,E,).

-x

grad

>

0.

On t h e o t h e r h a n d ,

law

8,

We s h a l l show t h a t n e c e s s a r i l y

(For s i m p l i c i t y we g i v e t h e p r o o f o n l y when

K

=

n

constant.)

C o n s i d e r i n p a r t i c u l a r a p r o c e s s i n which E,

while say

+ xl.

0

E

T h i s p r o c e s s i s c l e a r l y c y c l i c , and a

c a l c u l a t i o n from t h e e n e r g y e q u a t i o n shows t h a t t h e n Supposing f u r t h e r t h a t the coordinate axes,

+ +

h-n f

a r e constant.

0,

n

r

E

0.

i s a parallelopiped oriented along

one s e e s t h a t o n l y t w o o p p o s i t e s i d e s o f

and on t h e s e i n f a c t b o t h

0

and

+ +

h-n

Thus h e a t i s a b s o r b e d by t h e m a t e r i a l o n l y a t

two d i s t i n c t t e m p e r a t u r e s

l y holds:

+v

i s a s p a t i a l l y l i n e a r function, constant i n time,

8 = const

ahl have

0.

2

and t h e C l a u s i u s i n e q u a l i t y c e r t a i n -

444

JAMES SERRIN

t, &in. 4 - b

-

ds dt S 0 .

This easily reduces to

U(R,-O,)

higher o f the two temperatures.

s

0,

where

Hence

U 2

e2

is the

0.

In contrast with the standard treatment of the energy balance law for a compressible Navier-Stokes fluid, in which it is olympically assumed that internal energy and specific entropy exist, we have here required only a general functional form for the cumulative heat, together with the two laws of thermodynamics.

10. The Clausius-Duhem inequality. ~ _ _ _ . _ _ _ _ _

Rational thermodynamics, as a subject developed within the context of continuum mechanics, has taken the energy balance axiom d - (E+K) dt

..

= Q-w

as the fundamental expression o f the First Law, where

6

=

p rdx

+

h.n ds,

t - v ds,

and has similarly expressed the Second Law by the ClausiusDuhem inequality

I think it fair to grant that this change o f emphasis, turning from a set of procedural laws to a pair of elegant but non-intuitive analytic relations, has made the student, if not his mentor, somewhat uncomfortable.

The rationale for

445

THE CONCEPTS OF THERMODYNAMICS

this shift of emphasis has of course not been lacking and is at least twofold:

(i) the application of the laws of thermo-

dynamics in their classical form has not seemed practical (partly, of course, because the laws themselves were not clearly understood in general contexts), and (ii) the energy balance law and the Clausius-Duhem inequality provide an elegant generalization of the classical formulas (see Section

6 ) , consistent throughout with that structure and with the relatively few situations which had earlier been treated on an ad hoc basis.

Even more, as Coleman and No11 showed in a

brilliant paper more than a decade ago, this fondation of thermodynamics implied _ _ a number of important constitutive conclusions.

They found, f o r example, that when

E, S, and

are appropriately simple in form then a Gibbs relation can be derived, whereas previously it had been the custom simply to lay down this relation i n analogy with classical theory. Similarly, f o r a Navier-Stokes fluid the non-negativity of and of

3x

+

2U

p

could be proved, and o f course, more imprtant,

ly, a series o f analogous restrictions were, by the same means, discovered for more complex materials. Recently Fosdick and Serrin, generalizing earlier workof Truesdell, showed that any material body obeying the energy balance law and the Clausius-Duhem inequality also obeys the standardly phrased versions of the Second Law.

In

the present context, of couse, this is essentially obvious both for the First and Second Laws:

the energy balance law

immediately implies that

Q(b) = W(b)

for a cyclic process,

since for this case

= K(a)

E(b)

K(b)

and

= E(a),

while

446

JAMES SERRIN

f o r an absorbtive process the Clausius-Duhem

inequality yields

the conclusion

dS> dt s o that i n turn

0, but not

S(b) > S(a)

In light of

I

0,

and the process cannot be cyclic.

these remarks and the earlier conclusims

of the paper, how might we categorize o r classify the ClausiusDuhem inequality today?

To begin with, the first of the two

reasons adduced earlier for adopting the Clausius-Duhem inequality as an axiom loses a good deal of its force

-

we

can in fact carry through a considerable analysis of materials directly on the basis of the First Law and the Clausius inequality ( * ) .

Since this analysis moreover provides explicit

constitutive information, as we have seen i n Section

4 , 7, 8,

and 9 , it may n o t be unreasonable to expect that more complex materials may be treated similarly.*

Moreover as an axiom

the Clausius-Duhem inequality invokes both the notion of entropy and absolute temperature, neither of which can be considered physically intuitive o r mathematically obvious. At the same time, the elegance and the analytical simplicity o f the energy balance law and the Clausius-Duhem inequality cannot be gainsaid.

And, as we have just noted

above, any material which is axiomatically required to satisfy these conditions will in turn satisfy the First and Second Laws.

I would argue accordingly that the energy balance law

and the Clausius-Duhem inequality first of all play a subsidiary role to the First and Second Laws, that in the

*We

have noted earlier the attack on this problem provided by Coleman and Owen.

$47

THE C O N C E P T S O F THERMODYNAMICS

tyeatment of simpler materials they are an unnecessary (or tautological) adjunct to classical ideas, even though they certainly emerge as proved formulas which relate the desired quantities of internal energy and entropy with a constructive and canonical absolute temperature scale.

Nevertheless,

beyond these cases, they serve as a preeminent device f o r the construction of thermodynamically admissible materials, materials which, f o r lack o f better terms, may be said to b e

If is apparent that, whatever else

o f Clausius-Duhem type.

might develop, such materials will continue to be studied f o r their own sake because of their relative analytic and structural simplicity.

APPENDIX

We present here a simple example of a material body

5

satisfying the laws of thermodynamics, but not having a

unique entropy function.*

It will b e sufficient to refer only

to spatially homogeneous processes. W(t) =

Q(t)

where

c

=

lat

P d

We put

v

(cd8

+ pdV),

is a positive constant, and

p = f(e,V,c)

is

defined by

I[ a

a

p = 7, R8 where

*O f

R =

if

V 2 0

if

+ < 0,

course, we mean that 8 has two entropies which d o not differ simply by a constant,

448

JAMES SERRIN

a

the constants and

and

p

being positive.

Note t h a t

Q(t)

a r e L i p s c h i t z c o n t i n u o u s , though n o t in g e n e r a l

W(t)

piecewise s m o o t h , f o r c o n t i n u o u s l y d i f f e r e n t i a b l e p r o c e s s e s

e

=

e(t),

v

=

v(t).

Put

sl(e,v)

= c log

e + u

v,

log

,v)

s,(e

= c log

e +

log

v.

Then f o r a n y p r o c e s s o f t h e t y p e n o t e d , we h a v e

where

I’

i s the subset o f

I

then that AS 1 =

(y+

(a-s)

I

\

< 0.

I t follows

d;.

I’

It i s c l e a r l y necessary t h a t on c y c l i c p r o c e s s e s :

b

o v e r which

< a

@

in o r d e r f o r ( * ) t o h o l d

a t t h e same t i m e ,

a b s o r b t i v e p r o c e s s c a n be c y c l i c .

p

if

Thus

I;

a

then no

s___ a t i____ s f i e s ( ___ as is

e v i d e n t ) b-~ o t h t h e- F i r s t and Second. _ Laws o f thermodynamics i f _ ~

and o n l y i f

B

s a. $ <

Suppose i n p a r t i c u l a r t h a t AS,

a.

Then

2

I and

S1

s e r v e s a s an e n t r o p y f u n c t i o n f o r

8 .

On t h e o t h e r

hand i t i s c l e a r t h a t

where

I” = I\I’

i s the subset o f

‘I

I

where

?

2

0.

Hence

449

THE CONCEPTS OE TERMODYNAMICS

and

S2

also serves as an entropy.

(It is not hard to show,

i n fact, that any function

s

is an entropy function f o r between

a

+ y

= c log 8 8 ,

l.0g y

where

v is any constant

e.)

and

Acknowledgement.

This research was partly supported by the

National Science Foundation.

References [l] Bridgman, P.W.

The Nature o f Thermodynamics, Harvard

Univ. Press, Cambridge, 1941. [Z]

Buchdahl, H.A.

The concepts of classical thermodynamics.

Cambridge Univ. Press, 1966.

[3] Callen, H.B.

Thermodynamics, Wiley, New Y o r k ,

[ 4 ] CarathGodory, C.

1960.

Untersuchunger ttber die Grundlagen der

Thermodynamik.

Math. Ann.

67 ( 1 9 0 9 ) , 755-386.

Cf. a l s o Sitz. Preuss. hkad. Wiss. 1925.

[ 5 ] Carnot, S.

Reflections on the Motive Power o € Fire

(Translation o f the 1824 edition by H.R. Turston), Dover,

[6] Clausius, R. W.R,

New Y o r k ,

1960.

Mechanical Theory of Heat (Translation by

Browne), London, 1879.

See especially pages

76-79* [ 7 ] Coleman, B.D.

On the stability of equilibrium states o f

general fluids.

( 1 9 7 0 1 , 1-32.

Arch. Rational Mech. Anal.

36

JAMES SERRIN

450

I81

Coleman, B.D. and Noll, W.

The thermodynamics of elastic

materials with heat conduction and viscosity.

Arch.

Rational Mech. Anal. 1 3 (1963), 167-178.

C 91

Coleman, B . D .

and Owen, D.R.

f o r thermodynamics.

A

mathematical foundation

Arch. Rational Mech. Anal.

54 -

(1974), 1-104.

r 103

Epstein, P.

A Textbook o f Thermodynamics.

Wiley, New

York, 1937.

r 111

Fosdick, R.L. & Serrin, J.

Global properties o f

continuum thermodynamical processes.

Arch. Rational

59 (1976), 97-109. Mech. Anal. -

r 121

Gibbs, J . W .

O n the equilibrium o f heterogeneous

substances.

108-248, 343-524.

c 131

Green, A . E . ,

3 -

Trans. Conn. Acad. Sci. Also

(1875-1878),

in Collected Works, V o l .

Naghdi, P.M., and Trapp, J . A .

1.

Thermodynarrdcs

of a continuum with internal constraints.

Int. J.

Eng. Sci. 8 (1970), 891-908.

[: 141 Gurtiri, M.E.

Thermodynamics and stability.

Rational Mech. Anal.

r 153

Serrin, J.

Arch.

59 (1975), 63-96.

Mathematical Principles of Classical Fluid

Mechanics.

Handbuch der Physik, v o l .

8/1, Springer-

Verlag, Berlin, 1957.

L: 161

Thomson, W. (Lord Kelvin) Papers.

Mathematical and Physical

Cambridge University Press, 1882.

See

especially pages 100-106 and pages 178-181.

[ 171 Truesdell, C.

Rational Thermodynamics.

Springer-Verlag

Berlin, 1972.

[: 181 Truesdell, C . engine.

349-371.

The efficiency of a homogeneous heat

J. Math. and Phys. Sciences

(1973),

THE C O N C E P T S OF T E R M O D Y N A M I C S

1193 T r u e s d e l l , C .

451

T h e concepts and logic of classical

thermodynamics, developed u p o n the foundation laid

b y S. C a r n o t a n d F . [20]

Truesdell, C.

R e e c h . S p r i n g e r - V e r l a g , N e w York, 1977

Irreversible heat engines and the second

law o f thermodynamics.

Letters i n Heat and Mass

Transfer 3 ( 1 9 7 6 ) , 267-290.

G.M. de La Penha, L.A. Medeiros (eds.) Contemporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations @North-Holland Publishing Conpany (1978)

THE NONLINEAR SCHR~DINGER EQUATION

WALTER A. STRAUSS Mathematics Department Brown University

A survey is presented o f some physical applications and recent mathematical results on the Schr6dinger equation with a power nonlinearity.

1. Trapping and focusing of laser beams.

Consider the electromagnetic wave equation 2

i 3a c

(gi)E)

-

-4

CIE = 0.

at 4

Assume a linearly polarized wave (E vector

-4

e)

parallel to a fixed unit

which is monochromatic with frequency

which propagates along the z-axis.

ul

and

Thus

and (1) reduces to

(2)

2ik

3 Z aU

Au

+

2 (k

-

gw2/c2)u

= 0.

The high intensity of a laser beam can produce significant local changes in the density of the medium and hence i n the dielectric constant

g .

This work was supported by the National Science Foundation under Grant MCS75-08827.

453

THE N O N L I N E A R SCIIRdDINGER E Q U A T I O N

C h i a o , Garmire and Townes [ 2 ] l i n e a r dependence

+ c 2 1312

= €,

6

assume t h e s i m p l e non-

They show how t h e r e s u l t -

.

i n g n o n l i n e a r t e r m may g i v e r i s e t o a n e l e c t r o m a g n e t i c beam which p r o d u c e s i t s own waveguide and p r o p a g a t e s w i t h o u t spreading.

T h i s phenomenon i s c a l l e d " s e l f - t r a p p i n g " .

corresponds t o a s o l u t i o n of b2u

-2-av

(3)

a2u

Kelley

p a r t of called

( k2

2

-

IuI2 u = 0.

C

can produce a b u i l d - u p i n t h e i n t e n s i t y o f

t h e beam a s a f u n c t i o n o f

Z.

W

~

~

~

/

(2)

b l o w s up a t a c e r t a i n v a l u e of

IuI2

If t h e " p a r a x i a l t t approximation k =

T h i s phenomenon i s

I t c o r r e s p o n d s t o a s o l u t i o n of

'+self-focusing".

and we c h o o s e

z:

2

-

w c oW ~ ) u e 2 2 c

i n which t h e i n t e n s i t y z.

( 2 ) which i s i n d e p e n d e n t of

[8] and T a l a n o v [ 1 6 ] show how a n o n l i n e a r

c

dependehce o f

+

It

luzzl

<< \ k u Z l

i s used

eCq u, a t i o n ( 2 ) r e d u c e s t o

(4) Under a p p r o p r i a t e a s s u m p t i o n s a b o u t t h e n a t u r e of t h e a b s o r b i n g medium,

o r bent instead of blooming". Aitken

[71

t h e beam c a n b e s p r e a d o u t , d i s t o r t e d These phenomena a r e c a l l e d t l t h e r m a l

focused.

G e b h a r d t and Smith [ 31 and H a y e s , U l r i c h and t a k e i n t o account t h e e f f e c t s o f a s t e a d y t r a n s So

v e r s e wind a l o n g t h e y - a x i s . intensity

(5)

IuI

2

upstream.

-

-

€

co

d e p e n d s on t h e beam

I n t h e s i m p l e s t c a s e they have Y

c0

= const

T a n i u t i and Washimi

[ I 7 1 have found a s i m i l a r

e q u a t i o n t o d e s c r i b e t h e long-time b e h a v i o r of

one-dimensional

4 54

WALTER A .

STRAUSS

hydromagnetic waves i n a plasma:

-

ax where

a, b, c

are constants.

(a1u12

+

b)u =

o

Such an equation also occurs

in the Ginzburg-Landau theory of superconductivity [ ? I ] .

It

can also b e regarded as a simple non-relativistic quantum field equation.

2. Existence and blow-up.

We consider the problem

(7) where

@

p > 1

and

(0 E 8

is a nice function is real.

if not stated otherwise),

T h e two standard conservation laws

(7) by

are obtained by multiplying parts, and by multiplying by

Gt

ii

and taking imaginary

and taking real parts. They

are IuI E =

and

d x = constant

p+l x lulP+']dx

\(y 1 louIz +

= constant.

They would lead us to believe that solutions ought to exist globally and be stable i f

7

0, but that if

X < 0

instability may be possible. Theorem 1 val at

~x E

u = 0

-

There exists a unique solution in some time intelr

I R ~ , It of

integer it is

< tl].

the function

ern

It i s as smooth as the singularity /U(~-'U

allows; i f

p

is an odd

455

THE NONLINEAR SCIIRdDINGER EQUATION

This local existence result follows easily by the standard Picard method. Vela [

51

proof may be found in Ginibre and

A

.

Theorem 2

-

Let

X < 0

and

p

2

1 +

4/n.

Let

I$

satisfy

the condition

(7) can exist for all

Then no smooth solution o f

t.

This result is a variation o f one of Glassey C61, 0< T <

who shows under certain conditions that there exists

<

a

such that

n = 2,

[ lvu(t) I

2

dx + =

t

as

7

T.

p = 3,

In case

(4) and Theorem 2 provides a

the equation reduces to

precise initial condition for the self-focusing of an electromagnetic beam.

The opposite situation is described i n the

next theorem. Theorem 3 n > 2

-

If

assume

1 < 0 p < 1

p < 1

assume

+ 4/(n-2).

a unique solution o f

(7)

continuous function of regularity statement

If

f o r all

t

t

+ 4/n. @

If

E Hs(IRn),

h > 0

and

there exists

which is a bounded

with values in

H1(Rn).

of Theorem 1 is valid for all

The t.

In particular, equation (6) has global smooth solutions.

Theorem 3 has been proved by Ginibre and Velo [51

except for the regularity.

Particular cases have also been

proved by Baillon, Cazenave and Figueira [11, by Glassey [61 and by Lin and Strauss [ 9 ] .

A l l four proofs were done

independently. In the case

p=3,

n=2,

Baillon et al. also prove

the global existence and regularity provided

11 I

1 @ I2dx

< 2.

456

WALTER A .

STRAUSS

T h e i r r e s u l t d o e s n o t c o n t r a d i c t Theorem 2 b e c a u s e t h e inequality

\\@lL'

dx S

,f lv@l2

1]@I2

dx

dx

p < 1

+

4/n

Theorem

4 -

p

2

1

+ 4/n

>

3 , we c a l l

I n v i e w o f Theorems 2 and

1 < 0,

implies

and

0

< 0,

t h e s t a b l e c a s e s and we c a l l

the unstable case. Let

1

7

0

and

p < m.

If

@

E H1(!Rn) ,

there

e x i s t s a s o l u t i o n which i s a bounded, weakly c o n t i n u o u s f u n c -

t

tion of

with values i n

H1(Rn).

T h i s theorem i s p r o v e d below.

We d o n o t know

whether t h i s weak s o l u t i o n i s u n i q u e or smooth. B a i l l o n e t a l . [l] h a v e a l s o proved t h e f o l l o w i n g r e s u l t for equation

(5).

The i n t e g r a l i n t h e n o n l i n e a r term

p r e v e n t s blow-up, Theorem 5

-

F o r any

1 , t h e r e e x i s t s a u n i q u e smooth s o l u t i o n

_.

of

with

u(x,y,O) = @ ( x , y ) .

Proof

of Theorem 2 :

= Xlul

P+1 /(p+l).

r = 1x1 ) d

Write

F ( u ) = Xlul

u

M u l t i p l y e q u a t i o n ( 7 ) by

G(u) =

and

2r; r

+

n;

where

and t a k e r e a l p a r t s :

I m Jru,;

dx = -2 (lvuI2

dx

[(Zn+4)G(u)-n~F(u)]dx 2 Now multiply

(7)

by

2-

r u

+ -4E

n

I

[2G(u)-;F(u)ldx

since

p > 1

and t a k e imaginary p a r t s :

+

4/n.

THE N O N L I N E A R SC1lRe)DINGER EQUATION

457

Hence

the l a s t i n t e g r a l i s p o s i t i v e .

T h i s i s nonsense because

Proof p

o f Theorem

< 1 +lr/(n-Z),

3: We m e r e l y s k e t c h t h e i n g r e d i e n t s .

H1 c LP+'.

S o b o l e v ' s Theorem s t a t e s t h a t

( i i ) The e v o l u t i o n o p e r a t o r

Uo(t):

@I

( i )I f

-+ ~ ( t f) o r t h e f r e e

( l i n e a r ) Schrddinger equation s a t i s f i e s t h e e s t i m a t e

2 S

for

q S

+

l/q

m,

6 = n/2

= 1,

l/q'

o b t a i n e d by i n t e r p o l a t i o n f r o m t h e c a s e s Y

= p+l,

t = 0.

E

t-'

so that

a r e non-negative

X < 0,

q=2

dx 5:

C

+

n

2a = ( 2 - n ) p + 2

a >

we h a v e

and

q=m.

i s integrable a t

1 > 0 , both

If

and h e n c e bounded f o r a l l

uniqueness,

H1

u

let

\,I2

([

and

1

B

and i n

v

( 1VuI2 dx]'

= n(p-l)/h.

If

p

< 1 + 4/13,

LP+'.

( i v ) To p r o v e t h e

be t w o s o l u t i o n s .

(t-sP

II l u l p - l u -

IvIp-lvlI

By ( i i )

( p + l )(/s p )ds

0

ft s c

I

(t-s)

-b

~ ~ u - ~ l ~ ~ + ~ ( s ) d s

'0

implies

u

t.

Hence t h e c o n s e r v a t i o n l a w s

ft

l l u ~ t ~ - v ( t ~ l l p + cl

dxIa

and

$ < 1.

and

0

p r o v i d e a bound i n

since

If

Sobolev's inequality s t a t e s t h a t

\lulp+l where

6 < 1

as c a n b e

n/q,

( i i i ) We u s e t h e c o n s e r v a t i o n l a w s .

terms i n If

0 <

then

-

and u = v

v

a r e bounded i n because

6 < 1.

LP+',

by ( i i i ) . T h i s

( v ) The e x i s t e n c e and

458

WALTER A .

STRAUSS

r e g u l a r i t y a r e p r o v e d by combining ( i ) - ( i i iw ) i t h Theorem 1. Proof

4: L e t

of Theorem

approximate such t h a t g N ( s ) -+

g ( s ) = 1sP+ 1/ ( p + l

s 2 0.

by a s e q u e n c e of smooth f u n c t i ons

g(s)

gN(s) = c o n s t s 2

g(s)

for

for a l l

s.

= g~([u])u/lul. L e t

for Let

u,(x,t)

s

l a r g e and

0 5

F(U) = Xlu P- 1 u

We

gNb) g N (s )

5

4.1

FN(u)=

and

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

the

problem

-

iu

(8)

N t

riuN

It e x i s t s because

+

F ~ ( u = ~ 0 ) ,

FN(u)

= ~ ( x ) .

u,(x,o)

u

i s linear for large

e a s y v a r i a t i o n o f Theorem 3 i s a p p l i c a b l e .

s o t h a t an

Then we h a v e t h e

o b v i o u s a p r i o r i bounds

By c o m p a c t n e s s ,

t h e r e i s a subsequence ( s t i l l c a l l e d

which c o n v e r g e s

t o some

where.

Hence

L

locally i n

,

weakly i n

l u N l ) + g( I u I )

g,(

1

u

and

u

FN(uN) + F N ( u )

by Theorem 1.1 o f S t r a u s s [ 1 2 ] .

pass t o t h e l i m i t i n each term i n ( 8 ) . of

and a l m o s t e v e r y -

H1

a.e.

UN)

We may

The o t h e r p r o p e r t i e s

f o l l o w e a s i l y a s i n C121.

3. S o l i t o n s , I n t h i s s e c t i o n w e assume ion i n t o equation

( 7 ) shows t h a t ,

parameter f a m i l y o f

if

1 <

Direct substitut-

0.

n=l,

t h e r e i s a four-

ttsolitary-wavett solutions

459

THE N O N L I N E A R SCEIRbDINGEK E Q U A T I O N

u ( x , ~ )= f ( x - c t ) e x p ( i g ( x - b t ) )

I n any d i m e n s i o n

n,

u = @ ( x ) exp(iult)

s t a n d i n g wave s o l u t i o n s

( 7 ) s a t i s f y a n e l l i p t i c e q u a t i o n which i s t h e same as ( 3 )

of

except f o r c o n s t a n t s .

V a r i a t i o n a l methods

p < 1

show t h e e x i s t e n c e of s u c h s o l u t i o n s p r o v i d e d p <

n=l

if

m

2.

or

t r a p p e d l a s e r beams

T h i s proves

(the case

[l51)

(see Strauss

+

4/(n-2);

the existence of s e l f -

n = 2,

p =

3).

These s o l u t i o n s o b v i o u s l y do n o t d e c a y t o z e r o a s

t +

m.

O n t h e o t h e r hand,

if

t h e i n i t i a l datum

s m a l l enough ( i n some Sobolev norm) and

p 2

u(x,O)

3,

n

s o l u t i o n c a n b e shown t o d e c a y t o z e r o u n i f o r m l y

[l3]

or [ l 4 ] ) .

2

is

3,

the

( s e e Strauss

I n t h e u n s t a b l e c a s e w e t h u s have t h e

s i t u a t i o n t h a t c e r t a i n s o l u t i o n s go t o z e r o ,

others maintain

t h e i r a m p l i t u d e s and s t i l l o t h e r s blow up i n a f i n i t e t i m e . The s t a b l e c a s e

p =

n = 1,

3,

1 < 0

has been

s t u d i e d i n g r e a t d e t a i l s i n c e t h e i m p o r t a n t work of Zakharov and S h a b a t [ I S ] . properties.

We now o u t l i n e some o f i t s s t r i k i n g

The e q u a t i o n i s

(9) The s o l i t a r y waves m e n t i o n e d above a r e c a l l e d s o l i t o n s i n this case. f

u = feig,

They c a n b e w r i t t e n a s

= s e c h [ 211 (x-ll )

+ 8$ tl

and

I t i s e x p o n e n t i a l l y s m a l l as

g = -25x

1x1

-b

m

-

where

4(5 2-q 2 ) t +

for each

t.

V.

Its

460

WALTER A. STRAUSS

*Z.&'q

amplitude oscillates between

while traveling along

45.

the x-axis at speed

Zakharov and Shabat found that the solitons interact almost independently in a manner highly analogous t o that of The general solution of ( 9 )

the Korteweg de Vries equation.

breaks up into a finite number of solitons having a set of four parameters

(s,q,b,V),

component whose amplitude decays like

Ill,.

.. , uN ,

each

plus a spreading

It

as

I tI

+

m .

Certain solutions, the pure combinations o r a finite number of solitons

ul,

...,uN, can be

Suppose the speeds

They are the I1multisolitons". are distinct.

t

As

-b

written explicitly.

..

4C1,. ,4%N

the multisoliton breaks up into

m,

(single) solitons arranged s o that the fastest soliton is in front and the slowest is at the reararrangement is reversed,

As

t +

I n the passage from

each soliton is completely unchanged In shape.

Ej,

a phase shift:

The changes

:l.t

q j , CI-

j'

- b yJ

J

and

v: V+

j

J

goes into

-

V-

j

-m,

t =

the to

-m

+m,

There is only

tj, n j ,

-b

b j , v:.

are given by a simple

formula which shows that the shifts are the same as i f the solitons collided only pairwise,

Thus triple collisions have

n o effect!

On the other hand, if some of the speeds coincide, the solitons do not separate.

u

multisoliton

5,

= I2 = 0

depends on

formed by a pair of solitons with parameters

and

t

F o r instance, consider the

ql

#

q2.

The formula shows that

only through the expression

lu(x,t)l

461

THE N O N L I N E A R S C I I R d D I N G E R E Q U A T I O N

and s o i t i s p e r i o d i c i n t i m e .

These s o l u t i o n s a r e c a l l e d

" b r e a t h e r s If. A couple o f important f e a t u r e s o f

u n d e r l i e t h e s e s t r i k i n g phenomena.

One i s t h e e x i s t e n c e o f

an i n f i n i t e number o f c o n s e r v a t i o n laws. association of

u,

The o t h e r i s t h e

( 9 ) w i t h a f a m i l y o f l i n e a r o p e r a t o r s , namely,

Lt

Here

(9)

equation

[

= i

the eigenvalues o f and t h e o p e r a t o r s

[ 11.

i

( 9 ) , p l a y s t h e role o f a time-

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

dependent p e r t u r b a t i o n .

-

A f t e r a minor change o f v a r i a b l e s , t u r n o u t t o b e i n d e p e n d e n t of

Lt Lt

t o be u n i t a r i l y e q u i v a l e n t .

time

A general

r e f e r e n c e on s o l i t o n s i s [11].

4.

Scattering.

W e t u r n now t o t h e a s y m p t o t i c b e h a v i o r o f

k >

s o l u t i o n s i n the case

0.

A s with the case

s o l u t i o n decays t o z e r o u n i f o r m l y i f s m a l l enough i n some norm and

k <

the the

0,

the i n i t i a l data are

p 1 3

(n 2

3).

For

X > 0,

t h i s i s the general situation. Theorem

6

-

It1

4

m.

as

( b ) If

(a) If

n = 3

and

p

2

1

8/3 <

+

4/n,

'then

p

< 5,

then

dx = O(t-')

max

IuI

= O(lt1

- 3/'

)

X

as

It1

m.

Part

( a ) comes from t h e llpseudo-conformalll conserv-

a t i o n law o f G i n i b r e and Velo

L-51

which may b e w r i t t e n a s

4 62

WALTER A. STRAUSS

I t i s a c o m b i n a t i o n o f t h e i d e n t i t i e s i n t h e p r o o f of Theorem 2.

Note t h a t t h i s i s a n e x a c t c o n s e r v a t i o n law i n t h e c a s e

p =

3,

other

n = 2.

They have a l s o p r o v e d s t r o n g e r d e c a y r a t e s i n

Lq-norms

for

q

< 2 + 4/(n-2)

and f o r more g e n e r a l n o n l i n e a r t e r m s . of L i n and S t r a u s s

p < 1

and Part

+

4/(n-2),

(h) is a result

C91, who u s e e x p l i c i t e s t i m a t e s o f t h e m

Green's

f u n c t i o n t o j a c k a n L P + l - e s t i m a t e up t o an L - e s t i m a t e .

U s i n g ( a ) , s i m i l a r r e s u l t s can h e d e r i v e d i n o t h e r d i m e n s i o n s . By a f r e e s o l u t i o n i s meant a s o l u t i o n o f t h e l i n e a r Schr6dinger equation

ivt

a s k s whether a s o l u t i o n o f free solution

as

t +

-a.

u

+

as

t +

= Av,

I n s c a t t e r i n g t h e o r y , one

( 7 ) looks a s y m p t o t i c a l l y l i k e a +m

u

and l i k e a f r e e s o l u t i o n

U s u a l l y one u s e s t h e e n e r g y norm

(L2

or

-

HI)

t o express the l i m i t . Theorem

7

-

(a) If

1

+ 2/n < p , u(t)

f r e e s o l u t i o n s such that

(b) If

1

+

4/n < p < 1

pair of f r e e solutions

+ u*

-

u,(t)

4/(n-2),

+ 0

weakly i n

uk of H1(fRn).

t h e r e e x i s t s a unique

such t h a t

where t h e norm i s t h e LP+l-norm, Uo(t)

there exists a p a i r

8 = n/2

-

n/(P+l),

and

i s the f r e e evolution operator.

( c ) If

n =

3 and

i n the

H1

norm.

8/3 < p < 5 ,

we h a v e

lIu(t)-uk(t)ll

+ 0

463

THE NONLINEAR SCHRdDINGER E Q U A T I O N

( a ) f o l l o w s f r o m the ideas o f I.E.

Part

Part

Matsumura [lo].

[5].

and Velo

Part

( b ) i s one o f

Remark:

vu

-

[91.

p < 1

2

+ ; ,

t h e r e do n o t

exist

u

+ 0.

u,(t)\l

The c o n s e r v a t i o n l a w a l s o i m p l i e s a r e g u l a r i z i n g

property: then

IIu(t)

t h e r e s u l t s of Ginibre

( c ) i s a r e s u l t o f L i n and S t r a u s s

I t c a n b e shown, t h a t , i f such t h a t

S e g a l and

if

p 2

1

is locally

+ 4/n, k > L

2 -

for t

0

and

$,

x@

are

L2,

> 0.

References

[l]

J.B.

B a i l l o n , T.

Cazenave and M.

F i g u e i r a , Equation de

SchrGdinger non l i n z a i r e , C.R.Acad.Sci. 284 (19771, 869-872. [2) R . Y .

Chiao, E.

G r m i r e and C . H .

o p t i c a l beams,

[ 3 ] F.G.

G e b h a r d t and D.C.

Phys.

Rev.

Tomes, Lett.

Self-trapping of

1 3 (1964), 479-482.

Smith, S e l f - i n d u c e d thermal d i s t o r -

t i o n i n t h e n e a r f i e l d f o r a l a s e r beam i n a moving medium IEEE, J .

Quantum E l e c t r o n i c s QS-7

(1971),

63-73.

[43

P.G.

deGennes, S u p e r c o n d u c t i v i t y of M e t a l s and A l l o y s , B e n j a m i n , New York,

1966, ch. 6.

[ 5 ] J . G i n i b r e and G , V e l o , On a c l a s s o f n o n l i n e a r Schrddinger e q u a t i o n s , I , I1 and 111, t o a p p e a r .

[ 6 ] R.T.

G l a s s e y , On t h e blowing-up

o f s o l u t i o n s t o t h e Cauchy

problem f o r n o n l i n e a r S c h r d d i n g e r e q u a t i o n s , Math.

Phys. 1 8 ( 1 9 7 7 ) ,

1794-7.

J.

464

C 71

WALTER A .

J.N. Hayes, P . B .

STRAUSS

Ulrich and A.H. Aitken, Effects o f the

atmosphere on the propagation o f 10.6 laser beams, Applied Optics 11 (1972), 257-260.

c 81

P.L.

Kelley, Self-focusing of optical beams, Phys. Rev. Lett. 15 (1965), 1005-1012.

[ 91

J.E. Lin and W.A.

Strauss, Decay and scattering o f

solutions o f a nonlinear Schr8dinger equation,

J. Funct. Anal., to appear.

r 101

A.

Matsumura, On the asymptotic behavior o f solutions o f semi-linear wave equations, Publ. Res. Inst. Math. Sci. Kyoto 12 (1976), 169-189.

I: 111

A.C.

Scott, F . Y . F .

Chu and D.W. McLaughlin, The soliton:

a new concept i n applied science, Proc. IEEE 61

(1973)9 1443-1483. [12] W.A.

Stranss, On weak solutions o f semi-linear hyperbolic equations, Anais Acad. Brasil. CiGncias 4 2 (1970) , 645-651.

[l3] W.A.

Strauss, Dispersion o f low-energy waves for two conservative equations, Arch. Rat. Mech. Anal. (1974)

55

86-92.

[14] W.A. Strauss, Nonlinear scattering theory, i n Scattering Theory i n Math. Physics, Reidel Publ. C o . ,

Dordrecht,

1974, 53-78. [l5] W.A.

Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 ( 1 9 7 7 ) ,

149-162.

THE N O N L I N E A R S C H R d D I N G E R E Q U A T I O N

[16] V . I .

465

Talanov, Self-focusing o f wave beams in nonlinear media, JETP Lett. 2 (1965), 138-141.

T. Taniuti and H . Washimi, Self-trapping and instability of hydromagnetic waves along the magnetic field in a cold plasma, Phys. R e v I Lett. 21 (1968), 209-212. [IS]

V.E. Zakharov and A.B.

Shabat, Exact theory of t w o -

dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media, JETP 34 (1972)

62-69.

G.M. de La Penha, L.A. Medeiros (eds.) Contemporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations ONorth-Holland P u b l i s h i n g Company (1978)

D E R I V A T I V E DEPENDENT INFINITESIMAL DEFORMATIONS OF DIFFERENTIABLE FUNCTIONS

GEORGE SVETLICHNY P A U L OTTERSON

P o n t i f i c i a Universidade C a t 6 l i c a R i o de J a n e i r o , R J

-

Brasil

W e p r e s e n t h e r e a r e s u l t a l r e a d y announced p r e v i o u s -

l y [ 1,2]

though now i n a more g e n e r a l form.

The demonstration

b e i n g e x t r e m e l y t e c h n i c a l and t e d i o u s s h a l l b e p u b l i s h e d s e p a r a t e l y , and we l i m i t o u r s e l v e s h e r e t o t h e s t a t e m e n t and i n t e r p r e t a t i o n of

t h e main theorem.

a m u l t i i n d e x and d e n o t e by (0,

...,0

1 , O q...,O)

RnxlRA

j e t bundle TT:

jARn

+ Rn

I

(J,a)

( J , a ) E A),

with generic point

,is a

Cm

Let

jkn

(x,ya),

E

a

I n what

d e n o t e by

T S f : /Rn

and

rAf Let

b e any s e t of

+

X

j A f : IRn

jqRn

t h e g r a p h of

the

Let

A.

follows

f : Rn + LRm

we

map d e f i n e d on some open s e t

s u c h a s above we d e f i n e

(J,a) E A;

A

...,m } .

and g e n e r a l l y f a i l t o m e n t i o n t h i s open s e t .

U c lRn, f

f

be

E {I,

We d e n o t e by

ahead we d e n o t e maps b y e x p r e s s i o n s u c h a s

an

J

where

be the canonical p r o j e c t i o n .

s h a l l mean t h a t

...,a n )

i s i n the i - t h place.

t h e form

[ la\

A \ = sup

(al,

=

the m u l t i index

1

where

be a s e t o f p a i r s of We s e t

i

a

Let

by

-b

A

R

Given

by x t - - ( a a f J ( x ) ) ,

A x++(x,j f(x)).

W e

A j f.

b e any t o p o l o g i c a l s p a c e ,

S C X

f u n c t i o n s d e f i n e d on n e i g h b o r h o o d s

of

and S

3 with

4 67

DEFORMATIONS O F DIFFERENTIADLE F U N C T I O N S

Y.

values i n a s e t

t h e r e i s a neighborhood

if

of

f

g

and

c

f u n c t i o n of

A

f

t h e germ of

i n t h e c l a s s of

Let

X

i n t e g r a l curve o f

t r a n sf or m

mtAm

= C@,(A)I (m, t)

@

each

Am E C

It\

T.

5

the characteristic

form

with

j%n

(rAf)

II,

-+

II, c WxR,

manifold

Cm

+-

t

i s an

@ (x,t )

i s an

at

II,

and

@

m

at

by

$

be any s e t of germs of

t

X

of

=

@

if for

for all

v e c t o r f i e l d on a n open s e t

C o n s i d e r t h e s e t o f germs

03.

Cm

m

@,

GtAm E'

such t h a t

0

Cm

be a

IAl <

>

A

subsets o f

i s @ - p r e s e r v i n g , or p r e s e r v e s T

for

we c a n d e f i n e t h e

A C W

A

Gt

We w r i t e

functions

f : IRn

+ IRm

o f the

such t h a t

TAf(X)

rA f

C W.

If

2

p r e s e r v e s t h i s c l a s s we s a y

2

i s graph

preserving.

T h i s n o t i o n a p p e a r s i n t h e s t u d y of a t i o n t h e o r y of d i f f e r e n t i a l e q u a t i o n s . e q u a t i o n we mean a map f:

En + Rm

T s f c F'l(0).

M

W C M.

where

W

and

t = 0.

x

there i s a

Let n o w in

E

Let

X

we s a y t h a t

W,

Cp:

o f t h e germ of

.

If

d e f i n e d o n a maximal open i n t e r v a l o f

(m,t)

Given

S.

d e f i n e d o n a n open s e t

x 101 c

and p a s s i n g t h r o u g h

C(*,t).

W

W

the

a l l characteristic functions.

be t h e maximal f l o w map

Cp

open s e t s u c h t h a t

[R

X

vector f i e l d

Cm

fS

at

f

C o n s i d e r now a n a r b i t r a r y f i n i t e d i m e n s i o n a l and a

i f and o n l y

and d e n o t e b y

c a l l e d t h e germ o f

AS

g

N

c o n t a i n e d i n t h e domains

S

f l V = glV

f

we d e n o t e by

X

we s e t

of

V

such t h a t

equivalence c l a s s o f A

f,g E 5

For

satisfies I f now

F:

F = 0

C

+

j%ln i f

Rk

t h e transform-

By a d i f f e r e n t i a l

and we s a y t h a t

ForAf = 0 ,

that is i f

i s t h e f l o w of a v e c t o r f i e l d

468

GEORGE SVETLICHNY and

jAWn

d e f i n e d on a n open s e t i n

FoC t .

d i f f e r e n t i a l equation A

new equationifI+,r

A function

C ~ fT

f o r some f u n c t i o n

X

1t 1

for

we g e t a

satisfies this

f

a

i s not necessarily

C t does

that is,

ft;

n o t n e c e s s a r i l y r e l a t e two s o l u t i o n s o f I f however

t

then f o r each

f c ~ " ( 0 ) but

rAft

o f t h e form

P A U L OTTERSON

the t w o equations.

i s g r a p h p r e s e r v i n g , t h e n for any we have t h a t A

c ,(rAf)

s u f f i c i e n t l y small

= (I-f t ) @ tTAf (X)

T A f (X)

and t h u s t h e f l o w c a n be u s e d t o t r a n s f o r m l o c a l s o l u t i o n s o f o n e e q u a t i o n t o t h a t of

the

ker.

FoIt t F ,

c a s e i s t h a t of

A much s t u d i e d p a r t i c u l a r

i n o t h e r words t h a t of

symmetries

of a g i v e n e q u a t i o n . Let now

A'

= A U {(J,n+i)

{ (J,a) E

A].

X

t h e f l o w of a g r a p h p r e s e r v i n g v e c t o r f i e l d

on

t h e n we c a n d e f i n e a g r a p h p r e s e r v i n g v e c t o r f i e l d j

A'

R

n

as follows.

P E

If

r A ' f ( x ) = P.

such t h a t

=

We s e t

t h i s d e f i n e s t h e f l o w of

A' T f

t

X

@

f

is

j%n

x'

on

function

Cm

f

defined

(nstTAf(x))

and

once i t i s s h o w q t h e d e f i n i t i o n

E'

depend on t h e c h o i c e o f t h e f u n c t i o n

the l i f t i n g of ing.

find a

Consider t h e f u n c t i o n

i n the p a r a g r a p h a b o v e .

doesn't

JA'(Rn

If

f.

W e call

2'

and one s h o w s t h a t i t t o o i s g r a p h p r e s e r v -

C o n t i n u i n g i n t h i s manner one d e f i n e s a r b i t r a r i l y h i g h

liftings

~

(

i~n

1j. A

( N ) ~ ~ and s i n c e e v e r y t h i n g depends

o n l y on a f i n i t e number o f v a r i a b l e s one e v e n t u a l l y a r r i v e s a t a vector field

3'")

Define t h e f u n c t i o n s

in

€

( JYU

j

)

by t h e c o n d i t i o n A(m)

A(m)

j

ft(x) = ( a Q f J ( x )

+ t(c(JYu)OT

f)(x)

+

B(t),

469

DEFORMATIONS OF DIFFERENTIABLE FUNCTIONS

Di

and the differential operator

by

5

We find that the following equations relate the with the

n

and

G :

E

(J,U)

-

-

'(J,a)

'(J,a+i)

*i

and that graph preservation is equivalent to either one of the following two conditions:

or = Di '(J,a)

'(J,a+i) E

We note that the

(J,O

)

- c&

Y (J,a+&) i'&'

5,

and the

determine

Z(m)

completely. We need one last concept before stating the main theorem.

X

Let

manifolds

XXY

the equation

k = x(x),

b e a vector field in a product of two and suppose

(G,?)

= x(x,y)

X(x,u) = (X1(x),X2(x,y))

can be solved by first solving

substituting the solution

In other words

= x,(x(t),y(t)).

x(t)

a central function of the equation x2

is a vector field i n

Theorem

-

Let

be a

set

W C j%".

...,m] Y~,.. . ,yr [l,

Suppose Cm

A

Y

then

x(t)

into

;(t)

=

can be thought of = x,(x,y).

controlled by

is finite with ((J,O)

x

I

a

We say that

1'

...,rn}

J=l,

C

A.

graph preserving vector field on an open Then there is a partition

Pl,.

.. ,63

r

of

and a sequence of graph preserving vector fields such that

47 0

GEORGE S V E T L I C H N Y and PAUL OTTERSON

is a vector field i n j

(J , U )

controlled by a n appropriate lifting of

Yi-1.

(1) Each

x

(2)

yi

I J E P ~ Ju, 1S1lRn

is a restriction of an appropriate lifting of Furthermore each of the functions

si

the components

of

x

by

vector field

Y1 =

c

ya ' 5i

qo

m = 1 we see that

Interpretingthis result for are independent of

YJ,O)

' r .

determines

la\ > 1, Hence i n

ax a +

qo

ay a

-b

c

qi

and

j

lalsllRn

a

is graph

ti the

2

preserving and

X

is a restriction of a lifting o f it.

Furthermore the function

~o(X1'.",Xn;

Y; Y1'".'Yn)

determines the flow since we now find

where we note that if

'

a 'iGi

a -c0

is independent of

y

then

is an infinitesimal contact transform-

+ ' " j q

ation with

c

being the Hamiltonian.

We conclude there-

fore that an infinitesimal deformation of a differentiable function that depends only on a finite number of derivatives involves essentially only derivatives of order no higher than the first and the deformation generalizes slightly the notion of a n infinitesimal contact transformation. The case now

m

m > 1

Hamiltonians

is somewhat analogous, there being

-'( J ,

0)

'

J = l,.. .,m,

but again the

DEFORMATIONS OF DIFFERENTIABLE FUNCTIONS

47 1

appearance o f higher order derivatives comes aboiit only through lifting,the difference being n o w that such lifting c a n be used as controls for defining the f l o w in another part

of the jet space.

References [l]

‘IRemarks on Symmetry G r o u p s o f Partial Differential Equations!’ Atas d o

10

Semingrio B r a s .

d e Anglise.

[ 21 “Infinitesimal Deforina tions of Differential Equations that depend o n the Derivatives of Solutions” Talk at the 49 Semingrio Brasileiro de Anglise.

G.M. de La Penha, L.A. Medeiros (eds.) Contenporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations Worth-Holland Publishing Company (1978)

NON LINEAR CONSTITUTIVE RELATIONS AND HOMOGENIZATION

L. TARTAR University of Paris

-

Sud,

France

1. Introduction. A few years ago there were two classes of methods to

handle non linear partial differential equations arising i n Continuum Mechanics o r Physics: the first one used a compactness argument and started with the w o r k of Leray, the second one a monotonicity or convexity argument initiated by Minty and Zarantonello and developped by BrGzis, Browder, Lions and others (cf. Lions Ell).

Although Leray's work was motivated

by Mechanics (cf. Leray [l])

the origins of the other method

were numerous but its applications to Mechanics were soon recognized and investigated (cf. Duvaut-Lions [l]).

It was

known, but not really emphasized, that these methods were inadequate for most

practical problems.

After a few years of darkness light came with the work of Ball [l]; adding ideas from homogenization gave rise to a new concept: compensated compactness (cf. Murat El], [l])

Tartar

and then t o the principles of the method presented here

(cf. Tartar [ a ] ) . Weak convergence plays an essential role and will be related to microscopic and macroscopic properties.

This

method seems perfectly adequate f o r non linear partial differential equations coming f r o m Mechanics and Physics (it was

NON LINEAR CONSTITUTIVE RELATIONS A N D HOMOGENIZATION

developed

473

with this purpose) but the approach is philosophic-

ally quite different from the classical one (except for Statistical Mechanics and Quantum Mechanics perhaps). Although the method could be used for stationary

or

evolutionary equations, it cannot yet be used for nonlinear hyperbolic systems: indeed one has first to know what a solution is, the concept o f entropy (which is not completely understood) being a n important restriction added to the equations.

We will avoid this problem here and accept all

solutions (in the sense of distributions) of the equations that we will consider. We will work on partial differential equations and not o n boundary value problems, but the method can o f course be

used to treat non linear boundary conditions.

2. Microscopic-macroscopic properties and weak convergence.

Partial differential equations are used as a mathematical model to describe the values o f physical quantities and

their adequacy is tested through prediction

o f the output of an experience knowing

the input.

Although there is no mathematical difficulty to work with functions there is a physical impossibility t o know everywhere its values (except for analytic cases); measurements o r identifications of some type will be used giving a finite number of parameters and from them an approximation of the function will be derived.

There is no a priori reason

that this approximate function satisfy any equation.

Of

course a large number of measurements may give a better know-

474

L.

TARTAR

ledge of the function but one is usually forced to work above some length (or time) scale in order to avoid perturbing the phenomenon under study (cf. Heisenberg principle of incertitude in Quantum Mechanics). The information one can obtain by measurements is called macroscopic, but the equations are usually valid for a physical quantity which is microscopic.

There is an implicit

belief that, if the phenomenon has been well analysed, all the relevant quantities appear in the equations and the macroscopic quantities will satisfy the same equations as the microscopic ones.

[It is my belief that some classical

equations have not been derived with sufficient care]. T h e numerical analysis approach is similar: to compute the solution of a n equation an approximation, depending o n a finite number of parameters, will be derived;

by increasing

the number of parameters it is hoped that this approximation converges(in

a weak o r strong sense) to the solution.

If the

macroscopic quantities were satisfying a different equation than the microscopic ones the limit of numerical approximations will usually satisfy the macroscopic equations (this fact will in this case depend upon the numerical method used).

Weak convergence seems to be the appropriate mathmatical tool to handle the above situation.

As,

o n bounded sets of

functions, the weak topology can usually be defined by a metric we will interpret the above analysis by saying that if two functions are nearby for the weak topology they are almost undiscernable through measurements.

NON LINEAR C O N S T I T U T I V E R E L A T I O N S A N D H O M O G E N I Z A T I O N

jt7

5

A s a n example c o n s i d e r a measurement o f t h e e l e c t r i c field

u

E(x)

i n a conductor.

is the electrostatic potential.

x0

w i l l c o n s i s t i n measuring

example

u(xo)

and

u(xo+h),

Suppose t h a t

u1

u(x)

if

0.

E

then

x

C

ul(:)

u(x)

-

h

0’

for

E(xo) the

and t h a t t h e

where 1.

u

is

Within t h e

w i l l be i d e n t i f i e d with which i s a l m o s t

duo du1 x - dx (F); dx compared

to

-

uo(x)

-

the difference

du1 x - z(z)

- b u t i t s mean dx

0

i t is quite f a r f r o m i n a weak t o p o l o g y

X

i n any s t r o n g

0

(if

cp

T ( ~ ) c o n v e r g e s weakly t o t h e mean v a l u e o f to

near

T h i s d i f f e r e n c e i s a l m o s t i m p o s s i b l e t o measure

topology b u t n e a r

goes

is

i s t o o small;

6

= uo(x) +

11

important v a r i a t i o n s

value i s

a t two p o i n t s n e a r

and t h u s d e r i v e f o r

uo ( x o ) - u o ( x o + h )

with

But t h e e x a c t has

E

i s periodic with period

a c c u r a c y o f measurement

E(x)

where

A measurement of

i s s m a l l compared t o

C

exact p o t e n t i a l i s

and

(x)

u(x,)-u(x,+h) h

approximation

smooth and

u

- du d x

E(x) =

We h a v e

i s periodic

cp

when

G

0).

I t seems t h a t a l l measurements a r e done t h r o u g h a v e r a g e s of p h y s i c a l q u a n t i t i e s ;

from t h e s e measurements

i d e n t i f i c a t i o n o f p h y s i c a l parameters

can b e o b t a i n e d :

other

n o t i o n s of c o n v e r g e n c e a r e r e l a t e d t o t h i s f a c t as i n homogenization

(cf. Tartar

11 )

.

The r e a d e r w i l l h a v e remarked t h a t

t h e above

c o n s i d e r a t i o n s a r e more p h i l o s o p h i c a l t h a n m a t h e m a t i c a l : u s i n g a compactness r e s u l t weak c o n v e r g e n c e i n some s p a c e may i m p l y s t r o n g c o n v e r g e n c e i n an o t h e r ;

as t h e above a n a l y s i s

d e p e n d s on t h e s p a c e u s e d i t h a s no i n t r i n s i c v a l u e arid e a c h

47 6

L.

TARTAR

m a t h e m a t i c i a n w i l l want t o c h o o s e i t s p r e f e r r e d s p a c e .

But

e q u a t i o n s w i t h d i s c o n t i n u o u s c o e f f i c i e n t s and h o m o g e n i z a t i o n show t h a t t h e r e a r e w e l l d e f i n e d s p a c e s a s s o c i a t e d t o a g i v e n p a r t i a l d i f f e r e n t i a l e q u a t i o n coming f r o m Mechanics or Physics;

o u r a n a l y s i s r e l i e s on t h i s f a c t .

3 . A p p l i c a t i o n s t-.o. __. n o.____ nlinear e l a s t i c i t y . The p a r t i a l d i f f e r e n t i a l e q u a t i o n s g o v e r n i n g t h e motion of a n e l a s t i c body a r e ( c f . G u r t i n [ 11 )

(1) (2)

bo(x,t) = (det F ( x , t ) ) b ( x , t , r ( x , t ) )

(3)

Fij

--

i n general

a ri

ax. J

(4)

S

(5)

det F 7 0

= :(F)

A

The f u n c t i o n a l

S

(6)

satisfies

;(F)

(7)

F~ = F

G(QF) = Q$(F)

for a l l for a l l

F

F

and

Q E Orth'

Of c o u r s e t h e r e a r e u s u a l l y some b o u n d a r y c o n d i t i o n s . Remark 1: I f

one adds t h e s t r o n g e l l i p t i c i t y c o n d i t i o n on t h e

e l a s t i c i t y tensor

A(F)

= DS(F):

( a @ b ) * A ( F ) ( a0 b ) > 0

for

a 0 b

f 0,

t h e above s y s t e m becomes h y p e r b o l i c a n d , by a n a l o g y w i t h other

more or l e s s u n d e r s t o o d s i t u a t i o n s , o n l y p a r t i c u l a r s o l u t i o n s

NON L I N E A R C O N S T I T U T I V E R E L A T I O N S AND I I O M O G E N I Z A T I O N

477

of this system are believed to be physical: some kind of

inequalities, called entropy conditions, are added;

at the

moment only a few examples are understood and the above system seems out of reach. Quite naturally one is

led to consider the stationary

equations, where functions only depend on stable

x.

stationary solutions will be observed;

Of course only but stability

involves the complete system so we prefer to forget about this point. A more curious point is that, when dealing with

stationary solutions, nobody

thinks of restricting the class

of solutions with entropy conditions as if they were automatically satisfied f o r stationnary discontinuities;

to be

sure o f this point one should know what these entropy inequalities are and this is not the case, but as for some simple hyperbolic systems the analog is false we have to be careful.

In order to avoid this question we add the follow-

ing Postulate: all discontinuous solutions (F or S, not of the stationary

r)

system of elasticity are accepted.

F r o m this and the philosophical approach of paragraph A

2 we will derive a necessary condition on the function

S;

it is an interesting fact that this condition implies that stationary discontinuities (along a smooth surface) satisfy the entropy inequalities whatever they are (because they cannot rule out discontinuities in the linear case).

Of

course this does not prove (8).

If we do not want to accept (8) we may as well work

47 8

L.

TARTAR

d i r e c t l y w i t h t h e c o m p l e t e e v o l u t i o n problem.

Presumably we

have t o do s o f o r e l a s t i c f l u i d s .

Our p h i l o s o p h i c a l a p p r o a c h t e l l s u s t h a t a weak l i m i t

(1) t o ( 5 ) i s a l s o a s o l u t i o n .

of stationnary solutions of

T o e x p r e s s t h a t we f i r s t h a v e t o p r e c i s e what weak t o p o l o g y

we u s e . I n l i n e a r e l a s t i c t h e o r y , u s i n g v a r i a t i o n a l methods and d i s c o n t i n u o u s c o e f f i c i e n t s , we know t h a t a n a t u r a l s p a c e is

~ ‘ ( n ) , sij

F.. E 1J

E L2(61);

by u s i n g m o r e s o p h i s t i c a t e d

r e s u l t s one c a n f i n d t h a t t h e s o l u t i o n s a t i s f i e s a n e s t i m a t e

E LP(n)

Fij,Sij

f o r some

(2 s

p

p 5:

a t the best

+a);

(when

c o e f f i c i e n t s a r e d i s c o n t i n u o u s ) we may e x p e c t a l l f u n c t i o n s

F . ., I J

S

t o b e bounded.

,

i J

We a r e l e d t o s a y t h a t ( a s s u m i n g ( 8 ) ) i f

i s a s e q u e n c e of weak

*

s o l u t i o n s of then

(r,F,S)

to

(rn,Fn,Sn)

( I ) t o ( 5 ) converging i n

~ ~ ( 6 2 )

is a l s o a solution.

(r,F,S)

A

T h i s i s a n i m p l i c i t h y p o t h e s i s on t h e f u n c t i o n motivates

the

Definition 1

,.

-

which

S

i s an a d m i s s i b l e c o n s t i t u t i v e r e l a t i o n i f

S

i t s a t i s f i e s the preceding conditions. Let us n o t e f i r s t t h a t t h e only r e a l d i f f i c u l t y i s t o know i f

F

and

theorem o f B a l l [l], det F 2 0;

(5),

s t r o n g l y and

det Fn-det

F

L”

in

I n d e e d , by a

Ig(G)l +

+

det F

det G

As

r

n

4

r

0

w i l l imply

s t r o n g l y we have

bz(x,t)-bo(x,t)

in

L”

*

weak

a n a t u r a l growth c o n d i t i o n

s t a y s bounded and c a r e of

S = g(F).

a r e r e l a t e d by

S

>

+m

0

giving when

G

taking

b(x,t,rn)+b(x,t,r)

weak

*

taking care

479

N O N L I N E A R CONSTITUTIVE RELATIONS AND IIOMOGENIZATION

of

(2).

(1) a n d

('4)

remains;

Thus o n l y a

( 3 ) b e i n g l i n e a r p r e s e n t no d i f f i c u l t y .

Then

result

as f o r

of Murat-Tartar

(6)

note than

,

( c f . M u r a t [ 13

SnFnT--SFT

T a r t a r [ 11

by

).

I t i s n o t h a r d t o d e r i v e a n e c e s s a r y c o n d i t i o n for admissibility Theorem 1

-

(cf.

T a r t a r [ 21 )

n

If

is admissible then i f

S

d e t Fi

> 0;

F2-F

F1,

F2

satisfy

5

- 1 8

1 -

T h e n we h a v e

(10)

5((1-e) F 1 + 8 F 2 ) = (1-8) g ( F 1 )

Remark 2 : from

As

F1 t o

+

( 9 ) i s the Rankine-Hugoniot F

2

€I;(F,)

for 8

E

[O,ll.

condition f o r a

5

a c r o s s a n h y p e r s u r f a c e of n o r m a l

jump

this

n

s h o w s t h a t for a d m i s s i b l e

s

d i s c o n t i n u i t i e s o c c u r o n l y on

A

lines where wise

(a.e)

i s a f f i n e a n d t h u s c a n b e o b t a i n e d as p o i n t -

S

l i m i t o f smooth s o l u t i o n s :

a r e believed contradictory

entropy inequalities

t o hold f o r these solutions s o ( 8 ) i s not ( b u t may n o t b e p h y s i c a l ) .

I S the s t r o n g e l l i p t i c i t y condition holds o n l y o c c u r s for

F1

= F2;

then

presumably s o l u t i o n s of

e q u a t i o n s may b e s m o o t h i n t h i s c a s e

(F

and

S

(9)

the

HBlder

continuous), F o r h y p e r e l a s t i c m a t e r i a l s whose s t o r e d e n e r g y f u n c t i o n a l s a t i s f i e s t h e Legendre-Hadamard necessary

condition, the

c o n d i t i o n o f Theorem 1 i s s a t i s f i e d .

I t i s n o t known i f sufficient o r not.

this necessary condition i s

Some s u f f i c i e n t c o n d i t i o n s f o r actnisiibility

480

L. TARTAR

can be obtained but the main problem remains that it is hard to check on particular examples if they apply.

-

Example 1

F

for all

matrix

A sufficient condition for admissibility is that

(satisfying

MF

det F > 0)

such that

5

(We assume o f course that if

G

there exists an invertible

is bounded and

is continuous and I;(G)I

det G + 0).

thesis corresponds to the case

MF

+

+co

The monotonicity hypoI

I.

T o obtain a wider class of admissible conditions we will use the following important notion which is adapted to equations (1)(3). Definition -_

2

-

A functional

(rn,Fn,Sn) converging in

sequence

and such that then

is admissible if for any

cp(F,S)

Div Sn

cp(Fn,Sn) If

L"

is bounded in

converging weakly to depends only on

to the quasiconvexity o f

ep

F

(and

L" J,

++

weak

to

(r,F,S)

Fn = O r n )

implies

~

z q(F,S).

this notion is equivalent

(cf. Ball [l]).

The exact

structure of these admissible functions is not known but simple examples, generalizing Ball's polyconvex functions,

All quadratic admissible functionals are

can be obtained.

known (cf. Tartar 111 , [ 2 ] ) : if

F = X 8

Example 2

-

5

and

S5

they must satisfy

cp(F,S)

2

0

= 0.

A sufficient condition for admissibility is that

there is a family

(cp,)

of admissible functionals such

a€A

that (12)

s

= :(G)

is equivalent to cp ( G , s ) c Q

o

for all

a E A.

481

N O N L I N E A R C O N S T I T U T I V E R E L A T I O N S AND H O M O G E N I Z A T I O N

Then Example 1 is only a particular case where the functions VU

take the form

Example 3 (PU)u:A

-

.

A sufficient condition is that there is a family

of admissible functionals satisfying cpa(F-G,

(13)

T

c p ( G , S ) = MF(:(F)-S(G))'(F-G)

< 0 f o r all F,G and all

;(F)-:(G))

u E

A

and the maximality condition cpu(F-G,

(14)

;(F)-S)

s

0 for all

F and a implies S = : ( G ) .

This is also a particular case of Example 2 but it will he more convenient to handle homogenization. Remark 3: It is not known if hyperelastic material having a polyconvex stored energy functional (cf. B a l l [l]) have an admissible constitutive relation.

4. Homogenization. Homogeneous materials are very

often heterogeneous at

a microscopic level (we stay o f course far above the molecular level).

If the different components are small enough compared

to the experience scale the material will behave like a homogeneous material. as

(Of course this is the same approach

in paragraph 2). T o avoid technicalities we will work with a material

having a periodic structure of size

6

and assume that there

are no exterior forces; we have functions satisfying

(15)

Div Sc = 0

( r E , F E, s e )

L. TARTAR

482

a Tie

F6. . =

ax. J

IJ

Sc

,(:

=

Fe)

> 0

det F' G

,;(: If when (r,F,S)

QF ) = US(%, F')

g o e s to

E

we expect that

for

Q F Orth'.

(re ,FG , S ' )

0

(r,F,S)

converge weakly to

will satisfy the equations (Of c o u r s e

corresponding to some homogeneous material. does not go to

0

but if

is small enough

E

almost undiscernable f r o m

e

(rE ,FE ,S')

is

(r,F,S)).

-

We will obtain the homogenized constitutive relation S by saying that

(21)

1

-

S = S(F)

(re ,FE, S @ )

if there exists a sequence

m

satisfying (15)(16)(17) converging i n L

-S

It is a belief that such an

*

weak

exists: a priori

-S

to (r,F,S)

may be

multivalued. Properties (18)(20) f o r

5;

corresponding properties of automatic by its definition. obtain information on

5:

-S

follow easily from

admissibility of

5

is

The only kind of question is to

i f for every

x,

...

S(x,F)

is of

the type of example 1, 2 , 3 ( o r in any other interesting class) what can be said on same family

To.

-S.

F o r the class of Example 3 , if the

is used f o r all

inequality ( 1 3 ) is true for

x,

then the same

4 . ,

S;

on the contrary Example 1

does not seem to b e a good setting for homogenization.

NON LINEAR CONSTITUTIVE RELATIONS AND HOMOGENIZATION

483

5. Comments. Non linear partial differential equations o f Continuum Mechanics or Physics should be stable under some kind of weak convergence, the natural spaces being pointed out by the case o f discontinuous coefficients (heterogeneous materials) and

homogenizati on. The main open problem is related to the so c a l l e d entropy inequalities and consists in asking which are the physical discontinuous solutions of the equations.

If one accepts all weak solutions o f the equations, which we have done here, w e are led to a simple necessary condition and some implicit sufficient conditions which are difficult to check.

An important problem is to derive a

necessary and sufficient condition or at least to give a simple way to check a sufficient condition. The same analysis can be done with boundary conditions and of course writing all this f o r manifolds with or without boundary is left as a good exercise for specialists in translations. Existence theorems are now reduced to construct approximations with a good a priori estimate, the admissibility property dealing with the passage to limit. The philosophical approach used (which is more or less known in Statistical and Quantum Mechanics) is i n opposition

with the classical use of strong topology, implicit function theorem and other local results (in n o r m topology).

484

L. TARTAR

Bibliography

Ball,

-

J.M. [l]

Convexity conditions and existence theorems

in nonlinear elasticity,

63, Duvaut, G.

337-403

-

Arch. Rational Mech. Anal.

-

(1977)

Lions, J.L.

[l]

-

Inequalities in Mechanics and

in Physics, Paris, Dunod 1972 (in French); Springer

1974.

-

Gurtin, M.E. [l]

O n the n o n linear theory of elasticity

in this volume. Leray, J. [l]

-

Etude de diverses Qquations int6grales n o n

lin6aires et de quelques probl5rnes que pose l'hydrodynamique, J. Math. Pures et AppliquGes

a, 1-82

(1933). Lions, J . L .

-

[l]

Quelques d t h o d e s de r6solutions des

probl6rnes aux limites n o n lin6aires.

Paris, Dunod-

Gauthier Villars, 1969. Murat, F. [ 13

-

Cornpacite par compensation, t o appear in

Annali d i Pisa. Tartar, L . [ll

-

HomogenGisation dans les Gquations aux

d6rivGes partielles.

[a]

Cours Peccot

1977, to appear.

Weak convergence in non linear partial differential equations in "Existence theory in nonlinear

elasticity", Austin

1977.

G.M. de La Penha, L . A . Medeiros (eds.) Contemporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations @North-Holland P u b l i s h i n g Company (1978)

QUALITATIVE PROPERTIES OF NAVIER-STOKES EQUATIONS

R.

TEMAM

DGpartement d e Math6matiques U n i v e r s i t 6 d e P a r i s Sud

91405 - Orsay, France

The p u r p o s e o f t h i s p r o p e r t i e s o f t h e s e t of

l e c t u r e i s t o p r e s e n t some new

s t e a d y - s t a t e s o l u t i o n s t o t h e Navier-

S t o k e s e q u a t i o n s of a v i s c o u s i n c o m p r e s s i b l e f l u i d .

I t i s known t h a t f o r s m a l l Reynolds numbers, i f a steady e x c i t a t i o n i s applied t o the f l u i d then there i s a u n i q u e s t a b l e s t e a d y s t a t e which a c t u a l l y a p p e a r s f o r

(t +

If

m).

Joseph

171 )

Benjamin

A s f a r as we know, v e r y l i t t l e

h a s b e e n p r o v e d c o n c e r n i n g t h e s e t of the equations.

( c f . B.T.

t h a t new s t e a d y s t a t e s a p p e a r some being

s t a b l e and some b e i n g u n s t a b l e .

of

large

t h e Reynoldsnumber i n c r e a s e s , t h e n i t i s

c o n j e c t u r e d and e x p e r i m e n t a l l y well-known

[ 1,21, D.

t

a l l s t a t i o n a r y solutions

I n j o i n t works w i t h C .

Foias (cf

[ 4 1 [ 5][ 61)

t h e a u t h o r has a t t e m p t e d t o f i n d some q u a l i t a t i v e i n f o r m a t i o n o n t h i s s e t , and we a r e g o i n g t o summarize t h e main r e s u l t s of

C51[61. S e c t i o n 1 c o n t a i n s t h e d e s c r i p t i o n of N a v i e r - S t o k e s

e q u a t i o n s and t h e i r f u n c t i o n a l s e t t i n g . the description of the r e s u l t s . 1. S t e a d y - s t a t e N a v i e r - S t o k e s 2.

P r o p e r t i e s of 2.1

S(f,v,v)

General p r o p e r t i e s

Section 2 c o n t a i n s

The p l a n i s t h e f o l l o w i n g : Equations.

R.

486

TEMAM

2.2

Generic p r o p e r t i e s

2.3

Generic b i f u r c a t i o n .

.

-

1. S __t e a d y S t a t e Navie r- S t olce s equa t i o n s

R

Let

4 = 2 o r 3.

be t h e domain f i l l e d by t h e f l u i d ,

W e assume t h a t

62

62

C

RL,

i s bounded w i t h a smooth

F.

boundary

,...,u , ( x ) ) ,

u ( x ) = (u,(x)

Let

velocity of the p a r t i c l e pressure a t

x

(x

and

p(x)

x

of f l u i d a t point

E 0);

u

then

and

p

be the

and t h e

s a t i s f y the

equations

where of

f

r

r e p r e s e n t s volumic f o r c e s ,

i s t h e g i v e n velocity -1

CQ

which i s assumed t o be m a t e r i a l i z e d and s o l i d , V = Re

i s t h e i n v e r s e of a Reynolds number. g i v e n , t h e problem i s t o f i n d

u

f , cp,

For

and

p

v

(and

n)

s a t i s f y i n g (1.1)-

(1.3)I n t h e f u n c t i o n a l s & t t i n g of

lL2(n) = L 2 ( n ) &

t o i n t r o d u c e t h e space

2

orthogonal decomposition o f

[8]

or R.

Temam

the equation, i t i s usual

IL.

(n)

and t o c o n s i d e r t h e

( c f O.A.

Ladyzhenskaya

1141): 2

L (62) G = [vp

I

= H @ G,

2 P E L (R),

3P ax. E

L

2

(n),

1 s

i s L]

1

H =

with

v

{U

E

2

IL

(62)

I

V U = 0,

u.vlr. =

t h e u n i t outward normal v e c t o r on

r.

01, We d e n o t e by

487

QUALITATIVE PROPERTIES O F N A V I E R - STOKES EQUATIONS

P

the If

u

l y regular,

and

p

u

then

(1.4)

t o modifying satisfies

H.

clearly satisfies

Pf = f ,

+ ( u - v ) ~ )=

f

in

0,

which i s a l w a y s p o s s i b l e and amounts

181) t h a t i f

Conversely i t i s c l a s s i c a l ( c f

p.

(1.4)

t o g e t h e r with ( 1 . 2 ) p

exists a scalar function

(1.3).

onto

s a t i s f y (1.1)-(1.3) and a r e s u f f i c i e n t -

P(-WAU

assuming t h a t

u

a2((n)

projection i n

and (1.3),

such t h a t

u, p

Therefore the equations ( 1 . 2 ) - ( 1 . 4 )

then t h e r e

s a t i s f y (1.1)for

u

are

e q u i v a l e n t t o t h e o r i g i n a l problem.

N o w we w r i t e function

cp

+

u =

@

0.

inside

CP

where

i s some e x t e n s i o n

It i s convenient t o d e f i n e

as t h e s o l u t i o n o f t h e nonhomogeneous S t o k e s problem

I n t h i s case

1

-A@

vCP

+

VTI

= 0

= 0

R,

in

H = ep

n,

in

r.

on

satisfies

VU

=

o

R,

in

W e i n t r o d u c e t h e l i n e a r unbounded o p e r a t o r

A

in

H,

whose domain i s D(A)

= ~

~ ( n0 ~((61) )

n

H,

and AV = -PAv,

here

Hm(h2)

v

i s t h e S o b o l e v s p a c e of

E

D(A); order

. m,

H”(n) =[H”’(n)l”

485

R.

H:(n)

and

TEMAM

u E H1(n)

i s t h e s p a c e of

the theory of

J.L.

Sobolev spaces cf

[141)

I t i s known ( c f

that

A

Lions-E.

H,

(n

F

(for

Magenes 1111).

i s a self-adjoint

p o s i t i v e and i n v e r t i b l e o p e r a t o r i n

i s compact

v a n i s h i n g on

strictly

and t h a t

A-'

E e(H)

bounded).

B

W e a l s o introduce the operators

= P[

B(u,v)

such t h a t

(~*V)vl,

B ( u ) = B(u,u).

4, = 2 o r 3 ,

For

)

that

B(',

D(A)

x D(A

we i n f e r from t h e S o b o l e v imbedding theorems

H2(n) x k12(n)

maps

I

(1,6)-(1.8)

G

To f i n d

(1.9) VAu

+

and i n p a r t i c u l a r

H.

into

The e q u a t i o r i s p r o b 1em

H,

into

D(A)

in

are now equivalent t o the

such t h a t

B(G+Acp) = f.

T h i s i s t h e f u n c t i o n a l f o r m of

t h e s t e a d y - s t a t e Navier-

S t o k e s e q u a t i o n s which we had i n v i e w and on which i s b a s e d the f o l l o w i n g s t u d y .

6 E D(A)

2.

satisfying

Properties of

W e d e n o t e by

S(f,cp,v).

B 3 / 2 ( 1 - ) = [H3'*(r)]'.

cp

the s e t of

(1.9).

W e assume t h a t

(1.2),

S(f,cp,V)

f

i s given i n

H,

cp

i s given i n

By t h e S t o k e s f o r m u l a , and b e c a u s e o f

must v e r i f y

W e w i l l impose a s l i g h t l y s t r o n g e r c o n d i t i o n o n

cp:

Q U A L I T A T I V E : P R O P E R T I E S O F NAVIER-STOKES

q3.v

(2.1)

489

EQUATIONS

1 s i s N,

dr = 0 ,

‘i

rl,.. . , r N

where

N=l

and

cp

s e t of

r

i f

a r e t h e c o n n e c t e d components of

i s connected).

We d e n o t e by

~ ” ~ ( sr a )t i s f y i n g

in

F

(P

if13’2(r)

F,

=

the

(2.1).

General p r o p e r t i e s .

2.1

F o r every the s e t

H

given i n

f

i~s-nonempty.

S(f,Cp,V)

and

Lions

1101

equations.

[S],

Ladyzhenskaya

w i t h s t r o n g e r a s s u m p t i o n s on f,cp E H x k3’ * (T ),

t h e weaker a s s u m p t i o n

S(f,cp,v)

The s e t

k3j2(r),

T h i s i s a n e x i s t e n c e theorem

f o r t h e s t e a d y s t a t e Navier-Stokes r e s u l t a p p e a r s i n O.A.

given i n

J.

f

This existence Leray [ 9 ] , J . L .

and/or

cp;

for

c f . C.Foias-R.T.[6].

i s reduceed t o a s i n g l e p o i n t i f

J!

i s s u f f i c i e n t l y l a r g e , more p r e c i s e l y i f

where t h e f u n c t i o n

Uo:

R+xR+-R+

i s increasing with respect

t o e a c h of i t s t w o a r g u m e n t s . Now l e t

consisting of adjoint i n H

w.,

J

j 2

1,

b e t h e orthomormal b a s i s i n

the eigenvectors of

H).

Let

Pm

(A”

A-1

H

i s compact s e l f -

denote the orthogonal projector i n

o n t o t h e s p a c e s p a n n e d by

wl,

...,wm.

We have t h e follow-

ing:

(2.3)

For

m

sufficiently large,

one t o one mapping on

rn

S ( f ,Cp , W

a r e a l compact @ - a n a l y t i c s e t .

2

)

m*(f,rp,V), and

Pm

pm S(f,cP,V)

is a is

R.

490 T h i s means t h a t

TEMAM

( a n d i n some s e n s e

Pm S ( f , c p , V )

S ( f , c p , ~ ) i t s e l f ) i s a f i n i t e u n i o n of p o i n t s ,

regular

a n a l y t i c c u r v e s , r e g u l a r a n a l y t i c manifolds of h i g h e r

[ 31 ) .

d i m e n s i o n s ( c f Bruhat-Whitney

(2.4)

Either

S(f,Cp,V)

points o r

A s a c o r o l l a r y we g e t :

i s t h e u n i o n o f a f i n i t e number o f

~ ( f , c p , v ) c o n t a i n s a t l e a s t an a n a l y t i c

curve.

2.2

Generic P r o p e r t i e s . We a r e now g o i n g t o d e s c r i b e g e n e r i c p r o p e r t i e s of A r e s u l t which i s t y p i c a l of

S(f,cp,V). ed i n

[4][5]

Theorem 1

-

F o r every

V

>

is finite,

and

0

8

1

C

cp E

H,

&3/2(r) f i x e d ,

and f o r e v e r y

c a r d S(f,cp,V)

The p r i n c i p l e o f t h e p r o o f t h e non l i n e a r mapping

N;

from

is a regular value of

f

of

N~

N~

such t h a t

= f).

Whence t h e

i s compact,

v a l u e s of

N1

The f a c t t h a t

we c o n s i d e r

H:

the Fr6chet derivative

?its

i s o l a t e d and t h e r e i s a f i n i t e number of

Nyl(f)

into

i s r e g u l a r a t every preimage p o i n t

N1(;)

E Q1,

@.

i s as follows: D(A)

N1,

f

there

i s odd, and c a r d S(f,cp,V)

i s c o n s t a n t on e v e r y c o n n e c t e d component of

When

establish-

i s the following

e x i s t s an open d e n s e s e t S(f,cp,V)

the r e s u l t s

in

N;'(f)

such

the s e t

(i.e.

Q1

;Is,

every are since

of r e g u l a r

i s d e n s e i s t h e l e s s t r i v i a l r e s u l t and follows

from t h e i n f i n i t e d i m e n s i o n a l v e r s i o n o f S a r d ' s theorem due

t o Smale [13].

ii

The o t h e r p r o p e r t i e s a r e c o n s e q u e n c e s o f t h e

QUALITATIVE

PROPERTIES

OF

NAVIER-STOKES

491

EQUATIONS

o f N1.

i m p l i c i t f u n c t i o n t h e o r e m and some s p e c i f i c p r o p e r t i e s F i n a l l y t h e o d d n e s s of

c a r d S(f,ep,V)

f o l l o w s from a

t o p o l o g i c a l d e g r e e argument. A s i m i l a r r e s u l t when

f , Cp,

a r e simultaneously

V ,

allowed t o var y i s t h i s one:

-

Theorem 2

x R+,

T h e r e e x i s t s a d e n s e open s e t f,cp,V E Q 2 ,

and f o r e v e r y

and odd. ____

Furthermore card ___________

c o n n e c t ed component o f

82

(r) x

i n Hxk3'2 -

c a r d S(f,cp,V)

is finite

i s c o n s t a n t on e v e r y

S(f,cp,V)

82'

Same p r o o f as Theorem 1. We may n o w t h i n k of a r e s u l t

s y m m e t r i c a l t o Theorem 1

i n the sense of a generic r e s u l t w i t h respect t o and

are fixed.

V

Theorem

a (-a )L

3

n

-

(a >

C

k2'a(r)

+

tp

E Q3,

??& (9

> 0

f

in

f

f i x e d , f o r every f i x e d

8

t h e r e e x i s t s a d e n s e open s e t

3

C

i s f i n i t e and odd,

card S ( f , c p , V )

c a r d S(f,cp,V)

i s c o n s t a n t on every connected

3.

The p r o o f author ( c f [ 1 2 ] ) Sard-Smale's

V

such t h a t

(l),

component o f

0),

when

We h a v e :

For every

H

cp

of

t h i s r e s u l t due t o J . C .

S a u t and t h e

involves d i f f e r e n t technics,

I n particular

theorem i s r e p l a c e d by a t r a n s v e r s a l i t y t h e o r e m A s a t o o l f o r t h i s p r o o f we a l s o

d u e t o Abraham and Quinns.

need t h e f o l l o w i n g uniqueness

t h e o r e m f o r a Cauchy problem

a s s o c i a t e d t o Stokes e q u a t i o n s : For

(1)

b

given i n

H

n

H1'm(a)t,

if

*

Cs(r)

i s t h e s e t of f u n c t i o n s

satisfy (2.1).

cp

in

v

@*(T')'

and

q

satisfy

which

R.

492

+

-Av

I v = 0

then Remark

1

-

and

2.3

-av = 0

and

+

Vq = 0

hl

in

r

on

av

i s a constant.

The f a c t t h a t

c o n j e c t u r e d by B.T.

(v*V)b

hl

in

0

v = O q

+

(b.V)v

vv =

I

TEMAM

i s g e n e r a l l y f i n i t e was

S(f,rp,V)

Benjamin.

Generic B i f u r c a t i o n . W e now d e s c r i b e a r e s u l t o f g e n e r i c b i f u r c a t i o n f o r

the equation ( 1 . 9 ) .

S i m i l a r r e s u l t s a r e proved i n

C6l f o r

t h e c l a s s i c a l T a y l o r and B h a r d p r o b l e m s . Theorem 4 exists

-

Q4(rp)

f E G4(ep),

a dense

h3'*(T')

cp E

W e assume t h a t

s u b s e t of

G

i s fixed.

There

and f o r e v e r y

H

t h e manifold

s

u

=

S(f,cp,V)

V 70

h a s t h e f o l l o w i n g form: ( i ) I t i s c o n s t i t u t e d of i s o l a t e d p o i n t s and i s o l a t e d

a n a l y t i-c. _ ma n_ i f o_ l d_ s _ which l_ i e _above s_ o l a t e d v a l u e s of _____ _ ___ __ - _iThe number of s u c h v a l u e s o f

v

2

V.

& s f i ? i . t e on e v e r y semi a x i s

V

vo > 0.

( i i ) It; i s c o n s t i t u t e d of one ( o r more) a n a l y t i c manifold(s)

of d i m e n s i o n 1, whose p r o j e c t i o n on t h e __ i-n_ t e_ r v_ al

]O,+m[.

V

a x i s i s t h e whole

The s e t o f s i n g u l a r p o i n t s o f t h i s manifold

i s f i n i t e i n every r e g i o n

V

A s a C o r o l l a r y we g e t

2

V

>

0.

QUALITATIVE PROPERTIES OF NAVIER-STOKES EQUATIONS

(2.6)

493

Generically, the set of (primary and secundary) hi-

v

furcating values of only accumulate at Remark 2 .___

-

V

for (1.9) is countable and can = 0.

As far as we know, this is the first information

available concerning all the primary and secundary bifurcating points of a nonlinear equation, Remark 3

-

The methods used for the proof of the above results

are quite general and probably apply to the equations of nonlinear elasticity.

References [l] T.B.

Benjamin, Applications of Leray-Schauder degree

theory to problems of hydrodynamic stability, Math. Proc. Camb. Phil. SOC. 79, 1976, p.373-392. [2]

T.B. Benjamin, Bifurcation Phenomena i n Steady flows of a viscous fluid, Part 1 Theory, Part 2 Experiments, Reports no 83, 84, University of Essex, Fluid Mechanics Research Institute, May

-

[ 3 ] F. Bruhat

1977.

H. Whitney, Quelques proprietes fondamentales

des ensembles analytiques Rgels, Comm. Math. Helv. 3 3 ,

1959, p. 132-160.

[4] C. Foias

-

R. Temam, On the stationary statistical

solutions o f the Navier-Stokes equations and Turbulence, Publication Math6matique d’orsay, no 120-75-28, Universit6 de Paris

-

Sud, Orsay, 1975.

494

R. TEMAM

[ 5 ] C. Foias

-

R. Temam, Structure o f the set of

stationary

solutions o f the Navier-Stokes equations, C o m m .

Pure

Appl. Math., XXX, 1 9 7 7 , p. 149-164.

[ 6 ] C. Foias

-

R. Temam, Remarques sur les kquations de

Navier-Stokes stationnaires et les ph&norn&nes succesifs de bifuraction, Annali di S c .

Norm. Sup. di Pisa, vol.

d6di6 & J. Leray, & para?tre.

[ 7 ] D. Joseph, Stability of fluid motions, vol. I and 11, Springer-Verlag, New-York-Heidelberg, 1976. [81 O.A. Ladyzhenskaya, The mathematical _ _ theory ~ o f viscous incompressible flow, Gordon and Breach, New-York, 1969.

191 J. Leray, Etude de diverses Qquations int6grales lin6aires et <e

quelques probl&mes que pose l'hydro-

dynamique, J. Math. Pures Appl. XIII, 193 [lo] J.L.

non

,

p.

1-82.

Lions, Quelques m6thodes de rbsolution des probldmes

aux limites n o n linGaires,Gauthier-Villars-Dunod, Paris 1969. [ll] J.L.

-

Lions

E. Magenes, Non-homogeneous boundary value

problems and applications, Springer-Verlag, Heidelberg/New-York, 1 9 7 3 . [12]

J.C. Saut

-

R. Temam, Propri6tQs de l'ensemble des

solutions stationnaires ou p6riodiques des Qquations de _ Navier-Stokes: _

g6nQricit6 par rapport aux donn6es __

- -

aux limites, Comptes Rendus Ac. S c . Paris,

[I31

S.

1977.

Smale, An infinite dimensional version o f Sardts Theorem, Amer. J. Math., 87, 1965, p. 861-866.

1141 R. Temam, Navier-Stokes equations, Theory and Numerical Analysis, North-Holland-Elsevier, Amsterdam, New-York,

1977

rn

G.M. de La Penha, L.A. Medeiros (eds.) Contemporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations @North-Holland P u b l i s h i n g Company (1978)

SOME CHALLENGES OFFERED TO ANALYSIS BY RATIONAL THERMOMECHANICS

C.

TRUESDELL

D e p a r t m e n t o f Mechanics The J o h n s H o p k i n s U n i v e r s i t y B a l t i m o r e , Maryland

T h r e e L e c t u r e s for t h e I n t e r n a t i o n a l Symposium on Continuum Mechanics a n d P a r t i a l D i f f e r e n t i a l E q u a t i o n s

a t the I n s t i t u t o d e Matemstica U n i v e r s i d a d e F e d e r a l d o X i 0 d e Janeiro

August 1-5,

1977

P r o l o g ue

497

L e c t u r e 1.

Comments on E l a s t i c i t y

501

Lecture 2 .

Comments on Thermodynamics

541

Lecture 3 .

Comments o n t h e K i n e t i c Theory of Gases

569

Epilogue

598

This Page Intentionally Left Blank

497

A N A L Y S I S BY R A T I O N A L TEERMOblECIIANICS

Prologue 1 cannot summarize recent progress in rational continuum

mechanics, because there is too much of it for me to study, and much of the best of it is too analytical €or me to master. There is very difficult and very important work by men who are equally at home in modern rational mechanics and in modern pure analysis: ANTIUIAN, DAFERMOS, and SERRIN. more names deserve to be on that list already.

Perhaps

Certainly I

wish it were a longer list, and the best possible outcome of this meeting would be to make it grow. There are men who have solved important analytical problems in the traditional fields of fluid mechanics and elasticity: FICHERA, FINN, FRAENKEL, GILBARG, HARTMAN, PLkRRIS, SERRIN, TOUPIN, and several others.

I call particular atten-

tion to SERRIN's work on hydrodynamic stability, which gave new life to a theory of central importance that before then was simply dying, throttled by a vested profession. The new foundation of therinomechanics, which has greatly enlarged the conceptual content of "classical" mechanics and thermodynamics, is primarily due to NOLL.

Most of the best

work done today is expressed at least partly in terms of his concepts and approach.

Among the many others who have helped

and are still helping to complete or enlarge this foundation, not always in just the same spirit, are COLEMAN, DAY, ERICKSEN, FOSDICK, GURTIN, OWEN, WANG, and WILLIAMS.

Their work is by no

498

C.

TRUESDELL

means limited to axiomatics.

An outstanding example of

basic research on a Eairly informal basis is provided by ERICKSEN's deep investigations into the nexus of material symmetries and the symmetries that define simple classes of solutions. The greatest activity, perhaps in the end too great, has been in formulation of new theories and exploration of elementary examples.

Here the list would be longest, but I

shall shorten it to two: RIVLIN's discovery of pilot examples - examples of theories and examples of special solutions - in 1948-1958, and ERICKSEN's patient, persistent, and successful development of theories of liquid crystals. Finally there is the solution of major analytical problems in branches of thermomechanics that scarcely existed twenty years ago.

The list would be fairly long,

but I will mention today only two examples, examples both of which seem to me particularly important for this meeting. Both concern work done by men whose schoolings did not provide them with much mathematics but who saw that they could not solve the problems they faced unless they first mastered a good deal of analysis.

It was the background in nature as

represented by the concepts and methods of modern continuum thermomechanics that enabled them to set their problems in precise terms and guided them to the aspects of pure analysis needed to solve them. I refer first to COLEMAN'S thermodynamics of materials with fading memory, which has replaced

ANA LY S IS BY RATIONAL T€€ERMOPIECI IAN IC S

499

a century of vague claims by a concrete mathematical structure that serves as a model of what thermodynamics can and should do.

Second, I refer to JOSEPH’S development of a

method of domain perturbation for determining the free surfaces of simple fluids.

His results enable us to predict

a wide class of motions of non-linear fluids with a degree of explicitness and accuracy that ten years ago was undreamt

of.

Our field is not a popular one, but it is beautiful in

itself and central to man‘s understanding of the world around him.

When I read the publicized and publicly rewarded

achievements of the most celebrated protagonists of the natural sciences in our day, I do not see in them anything superior to the work by COLEMAN and JOSEPH that I have just described.

As I said at the outset, I cannot summarize even a small part of the researches I have just now listed.

For-

tunately nearly half of the persons I named are present at this meeting, and they are all eloquent speakers.

Rather

than make a futile attempt to speak for them, I will explain a few things - very, very simple things - that I hope will interest the analysts, and especially the powerful and renowned group from Paris. I shall speak about unsolved problems, problems the

solution of which will call in equal measure upon rational thermomechanics and analysis. Many of us have been here before.

We know ALBERT0

500

C. TRUESDELL

COIMBRA and G U I L H E R M E DE LA PENHA; we know what they did for Brazil, for young Brazilian scientists and engineers, and for science and engineering in South America and North America.

This symposium, we know, is one of the fruits of

their tireless labor, their devotion, their wisdom.

What-

ever may be found worthwhile in the modest lectures I shall present, I dedicate to ALBERT0 COIMBRA and G U I L H E R M E DE LA PENHA.

A N A L Y S I S BY R A T I O N A L T H E R M O M E C I I A N I C S

LECTURE 1.

501

COMMENTS ON ELASTICITY

Every mathematician knows what a minimal surface is. Sometimes he calls it ''a soap film"; that does not necessarily mean he is nuch interested in soap (at least, if appearances and odors may be trusted), or that he finds blowing real bubbles fun; rather, it indicates that one of the tools in a mathematician's mind is a sensual basis for the abstract. "Rubber" topology is another example.

There the mathematician

pictures topological transformations by stretching a thin sheet of rubber in its own plane.

I have never heard of a

mathematician who actually used a sheet of rubber to try or incorporate his pictures.

If he did, he might find them not

only tedious but even difficult to effect.

It takes a lot

of force to stretch a rubber sheet non-uniformly, and even all ten fingers can provide only ten spots where local friction can enable the necessary tangential forces to be applied. Both the soap films and the rubber sheets are put to conceptual use by the mathematician in ways that do not require him to think about forces.

Without forces, however, no soap

film would deform its original, arbitrary shape so as to assume the shape the mathematician takes to represent a minimal surface.

Without forces the fantastic topological

transformations which the mathematician in his imagination sees the rubber sheets assume could not be produced or maintained.

I suggest that forces, too, belong in the

502

C.

TRUFSDELL

mathematician's tool box. Let us begin from an example that might seem not to involve our knowing much about forces.

I show you two pieces

of rubber tubing (Figure 1). They look pretty much alike. In fact, twelve years ago they were contiguous parts of one common hose.

If we lay them together now, end to end, we

find that they do not quite match.

One has plane ends,

while the ends of the other flare outward.

If you look at

them, something else may make you doubt that they once lay end to end: one piece is smooth and shiny on the outside; the other, rough, with helical scoring.

If you look down

into the tubes, you will recognize the smooth surface inside the one whose outside is rough and whose ends are plane as being the same as the smooth surface now outside the one whose ends are flanged, and inside the smooth piece you will feel the same rough surface and helical scoring as you see on the outside of the other piece.

The two pieces were in-

deed once contiguous, both having been parts of a hose made

as uniformly as the manufacturer could at small cost effect, out of raw material likewise made uniform within common tolerances.

The piece with flaring ends has been turned

inside out.

My friend JAMES BELL caused a machinist to

evert it.

To do so without tearing the tube required skill

as well as much patience, many hours of work, and much force, but let us join the rubber topologist in neglecting all that.

We see that the everted piece is a little longer

A N A L Y S I S BY R A T I O k A T , TIWRR?IOblCCIlANLCS

F i g u r e 1.

Tube a t ease and t u b e e v e r t e d .

503

504

C.

than the other.

TRUESDELL

Knowing that solid rubber is nearly in-

compressible, we expect that a specimen made longer will also shrink in diameter.

A calipers would enable us

verify that such is the case here.

to

With the naked eye we

can see that the wall Of the everted piece is a little thinner than it was originally.

If we consider the part of the tube

that lies a distance from the ends greater than one-fifth of the diameter, we can say that the everted piece, like the undeformed one, is very nearly a right-circular cylinder. We can idealize what we have seen by saying that an infinitely long, elastic, right-circular cylinder can be turned inside out so as to form another right-circular cylinder, havinq different radii. Has the severe treatment suffered by the rubber changed its properties?

Imitating the famous experiments of DANIEL

BERNOULLI on the vibrations of metal bars, I hold first one

and then the other piece lightly by the tips of two fingers and subject them to the simple tests by which we assure ourselves that a garden hose is in good condition.

They vibrate

transversely at the same frequencies; also longitudinally. Grasping first one and then the other by an end, we may bend them severely and see that they spring back in just the same way.

They respond in the same way also to severe stretch.

Both pieces exhibit all the properties we think of as lielasticii. I have no doubt that if we should slit both pieces along a generating plane, the original piece would

505

ANALYSIS BY R A T I O N A L T € E R M O M E C I I A N I C S

remain nearly cylindrical, while the everted one would snap around into a nearly cylindrical form with the helical scoring outside, just as it was before it was everted.

I will not

try the experiment and so spoil my toy, for the machinist who effected the

eversion has retired, and even experimen-

tists now have trouble getting money to pay for their research.

No doubt both pieces would show effects of aging: possibly also the everted piece would take some time to recover the natural shape it has been .forced for twelve years to try to forget, but that is another matter.

I am pretty sure that

anybody given the two slit pieces would convince himself easily that they still consist of the same material: surely that would have been the case at a time just after eversion was completed.

I demonstrate to you with soft rubber bands

the effects which I expect the tubes would show.

There is

no question that the piece first everted and then cut is indistinguishable from its fellow that is cut without first having been everted. At the time the machinist everted the hose for me I asked for a tube big enough that I could turn it inside out all by myself on the lecture platform. had to be of much weaker stuff.

Necessarily it

I was given a tube of foam

rubber two feet long, one foot in outside diameter, with a wall one inch thick.

The tubes of solid rubber were an inch

in outer diamter and about 1/2 of an inch thick, so the proportions of the tubes were the same, but because my arms are

C.

506

TRUESDELL

short I could not reach through a tube twelve feet long and so preserve full similarity.

(You will note that I continue to

use the natural measures established by my ancestors.

Much

as I admire France and French mathematicians, especially those whom I am honored to be permitted to address today, I cannot go so far as to endorse Napoleonic,decimal uniformity.) It was easy to evert the foam rubber tube.

To my astonish-

ment, what came out was nothing like a cylinder!

The two

ends remained roughly circular, but the mantle was depressed into several big lobes, convex inward.

The everted tube

resembled in shape a tin can that has been crumpled by great inward pressure.

When I everted the tube once more, it

sprang smartly back into its original form.

I repeated the

demonstration many times, always with just the same outcomes. No change of the material resulted; it remained perfectly elastic.

Of course, foam rubber is easily compressed, but

as the body showed little change of volume, in this case I doubt that the difference lay there, especially since the only compressive load on the everted body was the negligibly small pressure of the circumambient air. When I forced the everted tube into roughly cylindrical form by propping struts across its cavity, the moment I removed these struts the surface snapped back into lobes.

All

tests indicated that for the large cylinder of foam rubber, after eversion the lobed shape was stable, just as for the small cylinder of solid rubber the cylindrical shape was

A N A L Y S I S BY R A T I O N A L T H E R M O M E C I I A N I C S

stable.

507

(Unfortunately I have to say "was" in the case of

the big cylinder, for after a few years it began to stink as well as to tear, and I could no longer live with it. Recently I tried to have a new tube prepared, but I was told that foam rubber is no longer manufactured.

With some other

substance, it would not be the same experiment.

Here we see

a puzzle for the philosophers of physics, for a real experiment has become a Gedankenexperiment through no fault of its own.

Perhaps a sociologist of science should be called in

to consult.) Ten years ago I showed my rubber tubes to an assembly of physicists, some youthful and some mature, gathered together from all over the world to discuss non-linear effects. one showed any interest.

Not

Perhaps they were uncomfortable

with a phenomenon the naked eye could see, a specimen the human hand could touch without mortal danger.

I could help

them little, for I could not suggest what pseudo-Hermitian operator of unspecified domain and co-domain would give the appropriate wave functions.

Thus the physicists could not

see how to begin to code the problem for a computer, of capacity greater than any yet constructed.

Small wonder they

saw no avenue of progress. I hope more from analysts. Everybody known which they are.

Differential equations? The theory of elasticity was

fairly well formulated nearly a century ago, and by now there are several concise and clear expositions.

Like all theories

508

C. TRUESDELL

of mechanics, it rests upon the concept of force.

In

continuum mechanics we visualize fields of stress tensors inside bodies.

They represent the forces exerted upon a

body by all its neighboring parts: contact forces. body

8

be subject to a transplacement

-K

placement

5,-

from a reference

:

Present place

x

=

x,(x,t)

-

,

x E ~ ( f l ,)

being a bounded, open, connected set in

- (#)

K

assume

5,

-

Let a

.

E G2[5(73)x7t'1

7e3

.

We

Let the material of the body

be elastic, and let the body be homogeneous. mines the stress and does

te7e

Then

"$

deter-

in just the same way at every

SO

body-point:

The function choices of

&

fi is the response of the material.

Different

allow us to model different degrees and kinds

of elasticity as shown, for example, by solid rubber and by sponge rubber, but all choices must be such that the restriction of

$

to positive tensor arguments U

for all invertible tensor arguments

-

--

F = RU

,

-

R

orthogonal

,

-

U

-

F

positive

determines

as follows:

,

If

then

This requirement is equivalent to the condition that the

$'

A N A L Y S I S BY R A T I O N A L T H E R M O M E C I I A N I C S

--

Piola stress be Euclid-invariant:

&(QF)

v

F -

orthogonal

9

I

invertible

;

509

-- -

= Q &(F)

it is a consequence

also of more general requirements, such as the principle of material frame-indifference. Often we feel we can justly regard a specimen as being free of stress, not only upon its boundary, which we can see, but within its interior, which we cannot see or treat directly. Such a placement we call a placement at ease.

Some industrial

processes are designed just so as to relieve such stresses as there may be.

If a body in a placement at ease is cut into

pieces, those pieces will not spontaneously move.

The rubber

band as it comes from the package passes this test.

The

everted band does not, for after it is split once along a generator, it snaps back into its shape before eversion.

If

a body has a placement at ease, we can choose that as the reference placement.

Then

Appealing to Cauchy's First Law of Motion, we obtain the differential equation of elasticity:

We have supposed that body forces such as gravity are absent. The constant placement.

p

K

is the mass-density of

73

in the reference

If we denote the Cartesian components of

xK.

by

510

C.

xm , we

TRUESDELL

can write the differential equation as a scalar

system:

,

which axe functions of the nine

-

,

The coefficients Aka,' components

of

Xq,r

VZK

are determined as follows from

the response

They are called the elasticities of the elastic materials. For static problems such a s eversion, the differential system reduces to

Boundary conditions? For the problem of eversion, nothing could be simpler: The traction upon the boundary is null.

If

nK

-

denotes the unit outer normal field to

our condition is

the argument

TKnK - -=

VzK

-

,

a K ( d ) *

0 : in components,

~

being understood as an interior limit at

5

.

Thus the solution must be such as to make the normal at a point o the boundary of the reference shape lie in the nullspace of the Piola stress there. The problem of eversion has the trivial solution

A N A L Y S I S BY R A T I O N A L T H E R M O M E C H A N I C S

5,

-

- , for then t K -( V 1 ( $- = R - -$(U)- , it is clear that a

(X) = X

j(F) =

=

-

_o

511

Because

rotation of the whole

body carries one solution into another, and I shall refer to the whole class so obtained as being just one solution.

If

eversion is possible, our problem has at least one solution beyond the trivial one.

For the cylindrical tube, nature

suggests to us that the problem has at least two other solutions.

One of them retains the symmetry of

5 (fl)

:

Each

parallel cylindrical section is mapped into a parallel cylindrical section.

Thus we are led to expect that a cylin-

drical tube has at least two everted forms, one of them a cylinder, the other not; one of them stable, the other unstable. Which of the two is the stable one, depends upon the response

4 that defines the particular elastic material.

The differential equation and boundary condition do not reveal their secrets easily to the analyst. given shape at ease

-IC (13)

Certainly if the

is a solid sphere, nobody expects

to find any solution but the trivial one.

As you see from

this bisected baby's ball, a thin hemispherical shell easily everts into a nearly hemispherical shape with edges flanged outward, reversing the curvature of the boundary surface (Figure 2).

Next I show you also a wrinkled, auricular,

everted shape.

Here is a third everted shape, like a hat

with brim turned back.

A l l of these everted shapes are

stable: they bounce smartly when cast against a hard surface, and if left alone they persist indefinitely. AS Pastor

Figure 2 .

Above: Half of b a b y ' s b a l l a t e a s e and

i n mastoid, o t o i d , and p i l o i d e v e r t e d shapes.

Below: Half of

t e n n i s b a l l a t ease a n d . i n mastoid and o t o i d shapes.

A N A L Y S I S BY R A T I O N A L T I I E R M O M E C I I A N I C S

513

VAN DYKE wrote in Little Rivers (1895), "To be able to call the plants by name makes them a hundredfold more sweet and intimate.

Naming things is one of the oldest and simplest

of the human pastimes."

Indulging my humanity in this pastime,

I follow the path, well worn by the hands and knees of experimentists, that leads from the workbench (Figure 3) to the Greek lexicon, which empowers me to name these three everted forms mastoid, otoid, and piloid, respectivelyL.

I have

p a o T o E i b f i s (ARISTOTLE), ~ T o E l t i q s (RUFUS MEDICUS) , . r r i ~ o E i 6 i s (CLEANTHUS STOICUS and HELIODORUS EROTICUS). The photograph reproduced here is not the one projected at the symposium. During the discussions there after my lecture Mr. FOSDICK succeeded in effecting a symmetrical everted form, which is shown at the left front of the bench in Figure 3. A s it has a rather narrow range of stability, we sought a Latin rather than Greek name for it; then and there we hit upon pateric. The transitoriness of this shape is suggested by the fact that the paterae it resembles were of Hellenistic rather than Roman use. (The Roman empire fell.) I found the pateric shape much harder to effect in my Baltimore workshop than it was below the equator. The difficulty may lie in the variations of the earth's magnetic field, reversed Coriolis forces, astrological factors such as lack of guidance from the Southern Cross, or perhaps the Odyssean syndrome associated with return from a long absence passed in voluptuous climes.

-

never been able to reach the otoid or piloid shape without first going through the mastoid.

A hard blow usually makes

the otoid or piloid shape snap back to the mastoid, not the

C. TRUESDELL

514

Figure 3 .

Experimental s e t - u p and results.

515

A N A L Y S I S R Y R A T I O N A L THERMOMECIIANICS

hemispherical.

With half a tennis ball (Figure 2) the

mastoid shape is easy to get; the otoid is fairly difficult; and I have never been able to get the piloid at all.

The

etymology of "piloid" suggests that compressed felt might be a good material for experiments on large elastic deformation. Now suppose that

-

K(*)

is a large sheet with two

hollow hemispherical bosses far apart from each other Figure 4 ) .

Figure 4 .

Bosses.

The big boss represents a vice-president; the little boss, a dean.

Surely we can evert the vice-president and the dean

separately, they being both empty and distant from one another. Thus we should get four solutions (Figure 5).

Figure 5 .

For two identical

Major Reorganization (Everted Bosses).

C. TRUESDELL,

516

little bosses (both deans), we can rotate one of the everted solutions with another, so only three are distinct (Figure 6);

Figure 6.

Minor Reorganization (Everted Deans)

otherwise all four are distinct.

.

In both cases we expect

each stable everted form to give rise to at least one corresponding less stable or unstable form as well, so there should be at least nine solutions for distinct bosses and at least six solutions for identical bosses.

Two big

bosses might interfere with each other, leaving the outcome beyond conjecture.

Going on in this way, we can get to an

arbitrarily large number o€ vice-presidents and deans, any subset of which may be inverted - and, as anybody knows who has watched a university grow ordinary material.

-

all from the same,

Without wishing to conjecture on the basis

of so special an example the full range of possibilities, we can state as follows what we have securely seen:

Let

Lin

9

517

A N A L Y S I S BY R A T I O N A L THERMOMECHANICS

and

Invlin

@

d e n o t e t h e s p a c e s of t e n s o r s and of i n v e r t i b l e

t e n s o r s , r e s p e c t i v e l y , over a three-dimensional v e c t o r space

fl .

a be an open c o n n e c t e d s u b s e t of %: - u Lin fl . Suppose a l s o t h a t

Let

and l e t

Invlin

,

-+

and t h a t

t h e n t h e d i f f e r e n t i a l system o v e r t h e bounded and connected open s e t

Ic

($1

cfl

w i t h nil1 1 boundary d a t a :

may have a r b i t r a r i l y many s o l u t i o n s i n a d d i t i o n t o t h e t r i v i a l one. region

P r e c i s e l y how many t h e r e a r e depends upon t h e g i v e n

~ ( sand ) the

expect t h a t i f

given function

8.

As I s a i d , I

~ ( 7 3 ) i s a s o l i d s p h e r e and i f

& repre-

s e n t s w e l l any material i n n a t u r e , t h e number of n o n - t r i v i a l s o l u t i o n s w i l l be

0

.

W e note t h a t only the r e l a t i v e

518

C. TRUESDELL

magnitudes of the elastic stresses, not their absolute magnitudes, enter this problem: into itself if we multiply constant.

& by

The problem is transformed an arbitrary non-null

It is a problem that need involve only arbitrarily

small forces but may involve arbitrarily large relative rotations and strains2

.

The informed reader may note here a pitfall in the "physical" reasoning commonly used in the "derivation" of the linearized theory. Sufficiently small strains and rotations justify linearization of the constitutive relation TK = f i ( V ; , ) in a neighborhood of V z X , = 1 For small st?ains androtations, small results. The linearized constitutive might seem relation is invertible, so small values of IT,I to imply small strains and rotations. The exigtence of everted shapes shows that such is not always the case. Although many "engineering" treatments presume that one or another form of the constitutive relation is invertible, there is no justification in nature for such an assumption.

- .

For a very special

8

an everted solution for an entire

spherical shell was found many decades ago.

(To evert a closed

shell by experiment, you can borrow a proctoscope from an anal surgeon and furnish it with a pair of internal pincers.) About thirty years ago we learned that for incompressible isotropic elastic materials several problems could be solved in some generality.

For these materials the constitutive

equation reduces to a form simple enough to exhibit in detail in terms of the Cauchy stress:

A N A L Y S I S BY R A T I O N A L THERWOMECHANICS

(Cauchy) stress

J, ,

J-,

=

-

T

=

-

+

pl + J B

1-

5 19

2

trB ... and

given functions of

,

B-’

-1-

trB-’ v

The important difference here is that the pressure field

.

p

is not determined by the deformation; we may adjust it so as to help satisfy not only boundary conditions equations of motion.

Now

p = p,

-

= const.

,

but also the and the equations

of motion become

-

pa =

-

grad p

+

div(>

B

The independent variables are now

+

2

B-l) -1-

1-

-

x

and

t

, and a- is

the spatial acceleration field:

This kind of full spatial description is familiar from classical hydrodynamics. shape

x,(s,t)

-

The spatial domain is now the unknown actual instead of the known reference shape

-K ( $ )

While simple particular solutions were fairly easy to discover and study, the analyst may find this system in general even more difficult to master than the one for unconstrained materials. problem is

The differential equation for static

.

520

C. TRUESDELL

curl d i v

if

( J1B - +

The corresponding boundary condition for eversion can be stated similarly in terms of the unit normal

ax,(B)

boundary

Y

n

to the

of the everted shape:

%

is a proper vectox of

n Y

The corresponding field of proper numbers on

ax,(fl)

-

is

also the field of equilibrating hydrostatic pressure there. Any smooth hydrostatic field in one on

a -x , ( e )

-

x,(B)

that reduces to that

equilibrates also the interior of

~ ~ ( . 8 ) -

Within the theory of incompressible isotropic materials the general problem of eversion, first for an infinitely long cylinder and second for an entire sphere, was reduced about twenty-five years ago to the inversion of certain known functions.

Explicit and illuminating results have been

extracted therefrom only in the special case when the two coefficients

; I ,and

J-l

axe constants.

There the

problem of eversion rests at this time. Even if we could effect the functional inversions for the cylinder, the shape of the everted tube which we have seen would not be fully predicted thereby.

The mathematical

condition imposed on the infinite cylinder is that while the curved surfaces are free, the plane cross-sections merely suffer no resultant normal force.

That is much weaker than

A N A L Y S I S BY R A T I O N A L T H E R M O M E C H A N I C S

521

the correct condition for the cylinder of finite length, which requires the traction vector to vanish at each point on the free cross-sections. From the analysis we can conclude that if the everted shape is again a right-circular cylinder, this correct condition cannot be satisfied. speaks to us directly.

Here nature

We look back at the everted rubber

tube (Figure 11, which of course satisfies the correct condition, and we see again that the free ends flare outwards and are not plane. not exactly so.

The everted shape is nearly a cylinder but The difference is limited to regions distant

from the two ends not more than about 1/26 of their distance from the middle.

This very limited zone of influence suggests

that something like the vague statement called St. Venant's Principle in linear elasticity must remain true even for severely deformed bodies.

1 shall return to that topic at

the end of this lecture. Some work done recently offers limited prospects of progress.

HADAMARD's method of domain perturbation, conceived

in linear contexts, has been extended to non-linear problems by Mr. JOSEPH.

He has achieved notable successes with it in

determining flows of Rivlin-Ericksen fluids with free surfaces. Perhaps his kind of analysis could be applied here to transform the known solution for the infinite cylinder into the solution with flared ends subject to null traction.

While SIGNO2INI's method

of perturbation in finite elasticity, developed some forty years ago, began from a placement at ease, CAPRI2 have shown how

&

POD10 G U I D U G L I

522

C.

TRUESDELL

it can be modified to allow the possibility of bifurcation, and BHARATHA

&

LEVINSON in work not yet completed have obviated

a long-standing block and can begin from any known solution. With these refined methods one might approach this problem: By using the standard tests of infinitesimal elasticity: extension, inflation, torsion,

e., can we

tell whether or

not a tube has been everted? But, after all, is the problem ready for pure analysis? Let us look back at the differential equations and boundary conditions. They are defined by the response

&

of the

elastic material, and I have left that function arbitrary to within matters of smoothness, so long as its argument be a pure stretch.

In doing so I have neglected something we all

learned in school.

Namely, if we limit attention to infini-

tesimal strains and rotations from a shape at ease, the solution of the traction boundary-value problem is unique to within an arbitrary infinitesimal rotation; in fact it is both well set and stable.

And why is it well set and stable?

The constitutive relation, expressed in Cartesian coordinates, is

Is c

kmP 9

no means!

,

the tensor of elasticities, really arbitrary?

By

Several different lines of reasoning - appeal

to common sense, demand for well-set boundary problems, invocation of pseudo-thermodynamics, requirement that waves may propagate in all directions

-

all these, more or less,

A N A L Y S I S BY R A T I O N A L T J B R M O M E C H A N I C S

induce us to assume that

-

C

523

taken as a tensor over the

6-dimensional vector space of symmetric tensors be positive:

In particular, we know that if this condition is not satisfied, the standard mixed boundary-value problem will not have a solution f o r certain perfectly reasonable cases, that in others more than one solution will exist, and that not all such solutions as exist will be stable.

While traditionally both

these latter possibilities have seemed anomalous in a theory of infinitesimal strain and rotation from a shape at ease, some students, pointing to the elastica for support, now regard the linear theory’s total denial of bifurcation as a defect in that theory. Over twenty years ago the problem of finding a reasonable adscititious inequality for finite elasticity was proposed to the specialists, and much work has been done upon it.

Many

interesting inequalities have been proposed tentatively and their consequences explored and interrelated.

Although one

principal authority regards the whole program as futile, most others have favored one or both of two lines of attack upon it.

One line exploits experience in mechanics and the intuition

gained thereby; the other brings to bear the concepts of analysis: 1.

Mechanics:

To make one or another inequality

plausible, then explore its consequences.

524

C. TRUESDELL

2.

Analysis:

By trial to determine an inequality

that delivers the right kind of existence, the right degree of uniqueness dependent upon shape and material, the right allocation of stability and instability. Either approach might work.

Most of this lecture has been

spent on showing that the second would not be easy even for an analyst of supreme skill, for what are "the right" existence, uniqueness, and stability? Perhaps the first thing an analyst might try would be to assume that

#

is a monotone transformation:

This is impossible for any body having a shape at ease, for then, as we have seen,

-

&(Q) = 0

scalar product vanishes if we let orthogonal tensors.

-

orthogonal Q

F1

and

F2

, so

the

be distinct

The first existence theorem proved rested

on the assumption that

&

was such as to make every placement

infinitesimally stable, so unqualified uniqueness for the static boundary-value problems followed outcome.

- an equally unsatisfactory

The second avenue may lead to a final solution some

day, but the end is not yet in sight. Much work has explored the first avenue. result may turn out to give the final answer.

The earliest HADAMARD in

his magnificent researches about eighty years ago - researches that DUHEM first stimulated and then subsequently augmented

-

525

A N A L Y S I S BY R A T I O N A L THERMOMECHANICS

inferred that a classical kind of stability implied the local condition

d DUHEM found a gap in HADAMARD's proof; many years later correct proofs were provided, but in the interim GRAVES, who was working in the calculus of Variations, had obtained a necessary and sufficient condition which MORREY later showed to include and generalize HADAMARD's.

HADAMARD himself showed that a

necessary and sufficient condition for all formal wave speeds to be real and non-zero was the slightly stronger inequality

This

condition is now called strong ellipticity.

Recently

WHEELER has shown that it suffices for unqualified uniqueness of regular solutions to the initial-value problem for boundary conditions of place.

HUGHES, KATO,

&

MARSDEN have shown as a

special case of a far more general theorem that the condition of uniform ellipticity suffices for the initial-value problem

for a body filling all space to be well-posed in a sufficiently short interval of time.

These results are plausible.

They leave

open the possibility that some problems may have solutions that do not remain regular, or no regular solutions for more than a very short time. Mr. ANTMAN regards as natural from the physical interpretation

C.

526

TRUESDELL

the following requirements upon the constitutive relations

of elasticity : 1.

They must preserve some order in the mapping from strain to stress.

2.

Their domain must be a subset of the orientationpreserving transplacements: det F > 0

, so

those for which

we can visualize them as the out-

comes of smooth, progressive changes of a reference shape. 3.

They must satisfy suitable growth conditions for extreme changes; for example, that in order to reduce the volume of a specimen to nil an arbitrarily large stress should be needed.

These properties must be compatible with each other and with the essential condition of invariance:

p(€tt)

- .

= R --..$(U)

Mr. ANTMAN himself has obtained remarkably complete and natural theories of elastic rods and shells by assuming that the parent 3-dimensional material is strongly elliptic.

His solutions

are classical solutions, and they do not exclude the most striking phenomena these theories are designed to model: buckling, necking, and -shear instability.

On the one hand,

they provide the greatest advance in the theory of rods since EULER's analysis of the elastica; on the other, they rest upon

reduction of the basic problem to the solution of ordinary differential equations and hence do not clearly guide us toward solving 3-dimensional problems.

A N A L Y S I S BY R A T I O N A L T€€ERMOMECIIANICS

527

Long before these analytical researches but long after HADAMARD's work, the avenue through arguments based on ideas

of mechanics that seem natural to many persons had led to results of a different kind.

COLEMAN & NOLL appealed to ideas

very close to what GIBBS had taken as axiomatic in his theory of the equilibrium of fluids.

They proposed that an elastic

potential function exhibit a restricted kind of convexity. We may express the part of their work that concerns elasticity in more general form as follows:

\J That is,

$

positive

- 3 U--F

U

domain

shall be monotone in arguments that differ by a pure

stretch other than the identity.

This adscititious inequality has

been more extensively studied than any other.

It leads to a very

restricted type of uniqueness, which does not exclude buckling or necking or shear instability; it can be interpreted as a neat work theorem; it leads to many conclusions that seem plausible in the

statics of isotropic bodies; finally, it

implies as a special case the universally accepted inequality of the theory of infinitesimal strain from a shape at ease, something that the condition of strong ellipticity is notoriously

too weak to do.

The adherents of strong ellipticity

could remedy this defect by simply adjoining as a further

528

C.

TRUESDELL

requirement the classical inequalities for infinitesimal strains.

I should not find this way out pretty.

NEWTON said, is simple.

Nature, as

While an engineer must make do with

what he has, a natural philosopher has no such social or economic obligation and ought not be satisfied with patchwork or any other kind of ugliness. COLEMAN

&

NOLL's condition, too, has drawbacks.

Mr.

COLEMAN himself showed that if applied to elastic fluids it would exclude critical points and indeed an open set of perfectly reasonable states of a Van der Waals fluid. in general, their inequality is too severe.

Thus,

Later writers

were quick to seize this fact, sometimes through rediscovery, and to propose substitutes.

Most critics of COLEMAN

&

NOLL's

work have failed to examine with equal severity their own efforts to improve upon it.

For example, some have proposed

substitutes that exclude all fluids, not just those capable of changes of phase.

Of course the condition of strong ellipti-

city has this same failing

- failing if such it is. Some

students regard elastic fluids and elastic solids as being so different as to require altogether separate treatment.

For

my part, still now as twenty-five years ago, I should like to see a corpus of existence theory valid for solids and for fluids alike, subject to one and the same physically reasonable adscititious inequality.

As an example of partial success I

point to the unified theory of the propagation, growth, and decay of waves in elastic materials, a theory in which the

A N A L Y S I S BY R A T I O N A L T H E R M O M E C I I A N I C S

529

previously known results for elastic fluids find their special but easy places. Both lines of attack upon adscititious inequalities have been brought to bear in a major memoir by BALL, who is the first analyst to come to grips with the real 3-dimensional boundary-value problems of elastostatics. several conditions upon polyconvexity.

8. ..,

BALL

lays down

The most important is one he calls

He proves that polyconvexity is a special kind

of quasiconvexity in MORREY's sense, and he relates it to the conditions of ellipticity and strong ellipticity. he takes account of the celebrated requirement

Of course

- --

&(RU)

-

-

= R&(U)

He obtains a general existence theorem that does not imply uniqueness.

He shows how this theorem applies in detail to

various static boundary-value problems, both for compressible and for incompressible materials. solutions.

His solutions are generalized

The problem of regularization remains.

It is too soon to estimate the scope and success of BALL'S monumental work.

He gives examples to show that his condition

of polyconvexity is satisfied by some interesting special materials; he shows how to construct materials that not only are strongly elliptic but also are orientation-preserving and have satisfactory growth conditions for large strains.

On

the other hand, his criterion of polyconvexity, like strong ellipticity, cannot be satisfied by any elastic fluid. than that, it is not effective.

6

Worse

Indeed, a particular response

either does or does not satisfy it, but if we are given an

C. TRUESDELL

530

4

, the

burden is left with us to be clever enough to devise

a proof one way or the other. clear these questions.

Perhaps further study will

Certainly BALL'S work is unique among

studies of elasticity by analysts in its mastery of concepts and methods of modern rational mechanics and in its grasp of what the problems of non-linear elasticity really are.

I mentioned static conditions that seem natural.

One

of these is that shear stress shall be an increasing function of shear strain.

For a solid elastic material this seems

plausible enough (Figure 7):

f

-

Figure 7. From a molecular standpoint, on the other hand, this conclusion is not plausible at all. angle 45',transforms (Figure 8) ;

A shear of amount 1

, that

is,

of

an infinite cubic lattice into itself

53 1

A N A L Y S I S BY R A T I O N A L THERMOMECHANICS

0

0

0

0

0

0

0

0

0

* 0

0

0

0

0

0

0

0

*,I

a

0

0

I

0

k

K < 1 : reversed stress

A

0

ease, unstable

0

0

1

I . I .

0

Figure 8 .

/

:

0

0

0

1

K =

0

0

0

K infinitesimal

0

12. <

0

0

0

ease, stable

0

0

0

P

o

0

0

0

K = 1 : ease, stable

Shear of a Cubic Crystal.

For shears of amounts between

1

and

1 the shear stress would

have to reverse in order to maintain equilibrium.

A continuum

model for this effect would yield an S-shaped stress strain curve, not a monotone function; considerations of stability and snapthrough would have to be brought to bear in determining a solution. Neither COLEMAN

&

NOLL's inequality nor the condition of strong

ellipticity is compatible with this effect.

ERICKSEN has shown

how to define peer groups for continuous materials whose behavior is of this kind.

According to NOLL's classification, which is

now used by most theorists, these materials are neither solids nor fluids. We have begun to discuss boundary conditions more general

C. TRUESDELL

532

than t h a t f o r eversion.

A

r e a l model f o r s t a t i c a p p l i e d

t r a c t i o n i s n o t e a s y t o f o r m u l a t e , because as soon a s w e a p p l y t r a c t i o n t o t h e boundary of a n e l a s t i c body, t h a t boundary w i l l g i v e way and go somewhere e l s e , b u t where t h a t somewhere

e l s e i s w e do n o t know u n t i l a f t e r w e have s o l v e d t h e problem, which w e c a n s c a r c e l y d o b e f o r e w e f o r m u l a t e it.

A way o u t

was

found l o n g ago: P r e s c r i b e t h e P i o l a t r a c t i o n on t h e boundary of t h e r e f e r e n c e shape

T h i s makes s e n s e .

- (6) K

:

I t i s a p r e s c r i p t i o n of how t o l o a d t h e

s u r f a c e as i t deforms:

The d i r e c t i o n of t h e t r a c t i o n i s f i x e d ,

b u t i t s magnitude a d j u s t s so a s t o keep c o n s t a n t t h e f o r c e p e r u n i t a r e a on t h e r e f e r e n c e s h a p e .

ERICKSEN p o i n t e d o u t t h a t

t h i s v e r y n e a t s t a t e m e n t b r i n g s w i t h it a n o t h e r f a i l u r e of u n i q u e n e s s , as F i g u r e 9 shows:

533

A N A L Y S I S BY R A T I O N A L THERMOMECIIANICS

Solution 2. Compression. area increased, force decreased Rotation: 180°

Solution 1: Extension. area reduced, force increased Rotation: N u l l Figure 9.

The corresponding homogeneous transplacements are well known and for incompressible materials can be exhibited explicitly.

As

we

have formulated the problem, the same Piola stress on the boundary gives rise to two different shapes, with different Cauchy stresses.

If, on the other hand, we formulate the problem in terms of Cauchy stress, we can expect there to be exactly one solution that is affine.

We know that the affine solutions deform parallel

planes into parallel planes.

For a general load, or for other

solutions, we have no corresponding knowledge, so assignment of

534

C.

TRUESDELL

the Cauchy traction fails to make sense in general.

As DUHEM

noted long ago, it does make sense to assign a hydrostatic pressure, since then the direction of the traction vector is determined trivially by the loaded boundary. It seems to me that most of the remaining fundamental problems of elasticity are problems of pure analysis.

The

older types of specialist in elasticity, often with interests that grow from engineering, will contribute little here. Engineers and "applied mathematicians" have owned elasticity for a century; they have done what they could, and we must be grateful for that.

As far as analysis goes, on the other hand,

the main problems are not yet unequivocally specified.

Analysts

often feel uneasy when asked to consider analytical problems that are not yet distinctly set.

They should not.

The history

of mechanics should teach them that the greatest work in it was done by mathematicians who were just as secure in the concepts of mechanics as in the methods of analysis of their time. Mes amis, don't you think rubber elasticity your trouble? Let me not leave you with the idea that only non-linear elasticity is interesting.

Imagine with me an engineer of a

strange k i n d : a man who has to design shafts for engines yet is curious about the formulae he finds in his handbook and wonders why they are true or if they are really true.

He

has a cross-section and a material he is content to regard as isotropic and linearly elastic in infinitesimal strain, say a

535

A N A L Y S I S BY R A T I O N A L TIBRMOMECIIANICS

steel shaft of triangular section. a torsional couple of torque the twist

T

He knows that if he applies

to the ends of this shaft,

L

that results will satisfy

L = PAT

I

u

= shear modulus of steel

and his handbook will give him the factor A

,

for a triangle.

Our engineer has used his table frequently, and he has confidence in it, but this time he wonders how the torsional rigidity of a triangle was determined.

He checks the references.

In LOVE'S Treatise on the Mathematical Theory of Elasticity he finds an elegant chapter on torsion. him exactly how to calculate A tion.

In a few pages it tells

from an assumed, special solu-

At the end of the chapter he finds a disturbing addition

that shows him, by explicit example, that there are in fact infinitely many other solutions to his problem!

For none of

them is a twist or a torsional rigidity mentioned.

In the

example, which refers only to a circular shaft and only to a particular kind of loads, he is shown that the "local perturbations" die off exponentially with the distance from the ends. He is also referred to an earlier article, a half page of words, which tells him about the "principle of the elastic equivalence of statically equipollent systems of loads", usually called "St. Venant's principle".

He has slipped into the great

maelstrom of classical linear elasticity.

His head spins.

Our engineer is unusually perspicacious as well as unusually diligent.

He keeps on swimming until he reaches

536

C.

TRUESDELL

Part 2 of Volume VIa of the Encyclopedia of Physics. he can set some of his doubts at rest.

There

He reads FICHERA's

famous article on existence theory, and he no longer suspects that some of those competing distributions of torsional loads may not lead to solutions at all.

FICHERA's theorems assure

him that solutions with different boundary data will be different solutions.

He reads GURTIN's article and finds in

it all the general theorems of elasticity there are, as well as a masterful summary of the status of the rational aspects of "St. Venant's principle" today.

A beautiful theorem of

T O U P I N shows him that in a shaft the differences of energy

due to equipollent torsional loads will decrease exponentially with the distance from the ends of the bar and estimates the rate of decay.

Other theorems give him pointwise estimates

of the differences of respective stresses and strains. Our engineer is still unhappy. the energy to me? at all the points?

He says, "What good is

And what can I do with stresses and strains I have a shaft.

I twist it.

I don't

know how the grips apply the torque, and I don't wish to know.

That is superfluous information, superfluous because

in practice we cannot determine it. way.

But the bar twists any-

By experiment from COULOMB'S day to mine I know that

the relative rotation of the ends is proportional to the torque.

I call the factor 'torsional rigidity'.

How much

is it affected for a shaft of given length by variation in the class of equipollent torsional loads?"

A N A L Y S I S BY R A T I O N A L THERMOMECIIANICS

I think my imaginary engineer is right.

537

He has recog-

nized the fact that twist itself is a concept that refers to ST. VENANT's solution only.

The other solutions do not have

a twist, because the cross-sections do not simply rotate while uniformly warping. I think what my engineer wishes in order to justify his

confidence in applying ST. VENANT's formulae to his designs is, first of all, a definition of twist that refers to all solutions of the torsion problem. would do:

Let

Perhaps the following

be the volume of the shaft; then ue = azimuthal displacement

1 T S -

p zpJ - araz dv

r

=

radial co-ordinate

[z

=

axial co-ordinate

,

.

In ST. VENANT's solutions the integrand is assumed to be constant: for the others, it is not.

For the general problem

of torsion we may define the torsional modulus as follows: A S -L !JT

A

'

is a functional of the fields of tractions applied to the

two ends of the given bar.

As infinitesimal elasticity is a

linear theory, abundantly supplied with theorems of existence, uniqueness, and qualitative behavior, it is a purely analytical problem to study this functional and to assess and delimit the deviation of its values from the particular one found by ST. VENANT.

Perhaps a correct interpretation of

,

538

C

.

TRUESDELL

my imagined engineer's confidence is that this deviation is in some sense small. Be that as it may, I think that studies of "St. Venant's principle" have strayed too far from the physical circumstances that gave rise to it.

I believe that a suf-

ficiently clear and precise treatment of the torsion problem is necessary before we can even formulate a "St. Venant's principle" correctly for more general boundary-value problems.

Acknowledgements

Theory.

I thank Messrs. ANTMAN and ERICKSEN for their

helpful criticisms of the first draught: I am grateful also for various comments made on the occasion of the lecture which have enabled me to improve the text as printed here. Experiment.

I thank Mr. BELL for the original tubes,

which he caused to be prepared some twelve years ago, and for directing that photographs be made for this lecture: the wife of a Vice-president of The Johns Hopkins University for donating a supply of used tennis balls: and my own wife for selecting and bisecting the baby's balls.

ANALYSIS BY RATIONAL THERMOMECHANICS

539

References for further reading (not necessarily the sources of the lecture):

S.

ANTMAN, "Ordinary differential equations of non-linear elasticity, I

&

and Analysis

61 (1976), 307-393. _-

11," Archive for Rational Mechanics

J. M. BALL, "Convexity conditions and existence theorems in

non-linear elasticity," Archive €or Rational Mechanics

_-

and Analysis 63 (1976/7), 337-403 (1977). P. J. CHEN, "Growth and decay of waves in solids," Encyclopedia

of Physics VIa/3 (ed. C. TRUESDELL), pp. 303-402, Berlin,

etc., Springer-Verlag, 1973. J. L. ERICKSEN, "Special topics in elastostatics," Advances

-_

in Applied Mechanics 17, in press. G. FICHE-

,

"Existence theorems in elasticity" and "Boundary

value problems of elasticity with unilateral constraints," Encyclopedia of Physics VIa/2 (ed. C. TRUESDELL), pp. 347-424, Berlin, etc., Springer-Verlag, 1972.

M. E. GURTIN, "The linear theory of elasticity," Encyclopedia of Physics VIa/2 (ed. C. TRUESDELL), pp. 1-295, Berlin,

etc.,

Springer-Verlag, 1972.

T. J. R. HUGHES, T. KATO,

&

G. MARSDEN, "Well-posed quasi-

linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity,"

--

Archive for Rational Mechanics and Analysis 63 (1976/7), 273-294 (1977).

C. TRUESDELL

540

R. J. KNOPS

&

E. W. WILKES, "Theory of elastic stability,"

Encyclopedia of Physics VIa/3 (ed. C. TRUESDELL), pp. 125-302, Berlin, R.

A.

e., Springer-Verlag, 1973.

TOUPIN, "Saint-Venant's principle," Archive for Rational

--

Mechanics and Analysis 18 (1965), 83-96. C.-C.

WANG

&

C. TRUESDELL, Introduction to Rational

Elasticity, Leyden, Noordhoff, 1973. L. WHEELER, "A uniqueness theorem for the displacement in finite elastodynamics," Archive for Rational Mechanics

-_

and Analysis 63 (1976/7), 183-189 (1977).

ANALYSIS BY RATIONAL THERMOMECHANICS

LECTURE 2.

541

COMMENTS ON THERMODYNAMICS

Mathematicians rarely refer to thermodynamics except for an occasional joke.

Indeed, in some recent works we find

definition and use of various quantities labelled "entropy", but it is difficult to relate these entropies to what is called by that name in books on thermodynamics; certainly they are not shown to have the formal properties of entropy; in many cases they are just some non-decreasing maps, which the classical entropy usually is not.

The reason mathematicians

rarely refer to old-fashioned thermodynamics is that they cannot understand it.

The reason they cannot understand it

is that it is not understandable.

It can be memorized and

repeated; repetition of the truths of a faith bring comfort to the faithful but do not enlighten the heathen.

When it

comes to thermodynamics, mathematicians usually remain heathen. They are rarely converted by hearing rituals intoned. Everyone knows that rational thermodynamics seems to be a science different from classical thermodynamics - different in notations, in fundamental concepts, in interpretation, in results, in style, and in mathematical rigor.

Of course

the contrast in style and rigor is essential; the difference in notations is, I think, desirable.

There is no reason, on

the other hand, that the fundamental concepts, interpretations, and results should seem as different as they do.

The sticky

muddle in which classical thermodynamics has come down to US

542

C. TRUESDELL

results from an accident of history; thermodynamics, unlike mechanics, optics, electromagnetism, and the kinetic theory of gases, was discovered mainly by scientists who were uncomfortable with mathematics and deficient in the faculty of logical criticism. &

In a booklet now in press

I present classical thermodynamics

Mr. BHARATHA

- the thermodynamics

of "reversible" processes in homogeneous fluid bodies the basis of a set of axioms.

- on

The style and level of rigor

is similar, we hope, to HILBERT's in his presentation of Euclidean geometry.

Moreover, our axioms are, with one

exception, exactly the axioms used in effect by CARNOT.

The

one exception, of course, is CARNOT's assumption that heat is conserved in cyclic processes.

That we discard, and in its

place we assume that at least one ideal gas has constant specific heats.

Furthermore, while CARNOT's development of

his ideas is notoriously obscure, ours is simple, explicit, and direct. I think mathematicians should know that this axiomatic development is available and have an idea of how it goes.

Of

course I am not going to present the details to you, but I shall sketch the bones in such a way that any mature mathematician could clothe them with flesh.

Only because it is

thermodynamics I discuss, do I feel I must ask you to believe, or verify for yourselves, that rigorous proofs are possible. In fact they are easy.

The mathematical style is that of a

conscientious first course in mechanics using infinitesimal

543

ANALYSIS BY RATIONAL THERMOMECIIANICS

calculus. Primitive concepts:

E+

v

volume

0

temperature

E

ti^

%+

,&

constitutive domain (open, connected, non-empty) c x?+ x g +

p

pressure, a Cauchy-integrable function from [t,,t,~ into X'

Q

heating, ditto

A process is a piecewise smooth map from

x+

;

x

[t,,t,l

for interpretation we denote it by

into

(V,0,

.

A

dot denotes the derivative with respect to time, which €or V

and

exists almost always. The work done by a process

8

and the net gain of heat in a process are defined thus: L z

If and

2 '

and

Q < 0

,

Ipcdt

C E

I

7

3- denote

the subsets of

the heat absorbed

/Qdt 3

3

on which

Q > 0

and the heat emitted

C+

C-

are given by C+

I Qdt

J+

I

C-

-

1 Qdt 3-

I

544

so

C. TRUESDELL

c

c+

=

-

c-

I pass over the definitions of reverse, simple, path, cycle, isobar, isotherm, etc., as being standard. Axiom 1 (Thermal equation of state)

. 3 m E C' [.&I >

and

E av <0 Axiom 2 (Calorimetry: latent and specific heats). 3 AV,KV

Q

c'[4?13

r?

whenever

%'O

and 6

exist (that is, almost always); and

.

These axioms delimit a constitutive class, the elements of which are the following tetrads: :

constitutive domain

:

pressure function

A"

:

latent heat with respect to volume

%

:

specific heat at constant volume

7i8

Each choice of these four quantities defines a particular fluid body.

For example, for an ideal gas

A N A L Y S I S BY R A T I O N A L THERMOMECIIANICS

545

&+x R +

& =I

fl= R0/V R

=

const. > 0

To complete specification of the body we need to prescribe

%

.

we may call

A

and

A,

and

It is customary to do so indirectly.

If

the latent heat with respect to pressure

P

the specific heat at constant pressure, because

K P

.

Q = A ; + K B

P

P

Nowadays we think it is reasonable to allow the possibility that both A,

= (K

P

-

K

P %)/R

and

.

€or an ideal gas be constants.

K,

Then

On the basis of Axioms 1 and 2 alone

this possibilty is only one of many.

We shall come back to

it. Axioms 1 and 2 enable us to define an adiabatic process; Q = 0

.

The path of adiabatic processes are called adiabats.

They satisfy the differential equation

Once a body is specified, so are its adiabats. point is one at which

A,

9

0

.

An ordinary

In a neighborhood of an

C. TRUESDELL

546

ordinary point, the adiabats and isotherms decussate.

There

are two possible local patterns (Figure 1):

Figure 1.

Adiabats and isotherms near an ordinary point.

We may define a Carnot cycle as follows: a cycle such that and Q >

The temperatures of the cycle.

e

o

and

=

8-

e+

= const.

are the operating temperatures

An ordinary Carnot cycle is a simple Carnot

cycle in a region of

where

Av

90 .

Thus ordinary

Carnot cycles are of two forms only (Figure 2 ) :

Figure 2.

C+ > 0

Ordinary Carnot cycles.

547

A N A L Y S I S B Y R A T I O N A L THERMOMECHANICS

Ordinary Carnot cycles exist and are dense in a small enough neighborhood of any ordinary point. 8-

,

Va

,

Vb

The four numbers '8

specify an ordinary Carnot cycle

If

,

uniquely.

Thus everything associated with an ordinary Carnot cycle is determined by those same quantities.

For example, f o r a

given body,

AS

Av

is of one sign in the neighborhood we are discussing,

Moreover, we may solve this relation for of

Va

and

C+(

g) .

Vb

as a function

Thus

If we take some one ordinary Carnot cycle with operating temperatures

B

and

A

,

within it we may construct and call a

Carnot w e b the dense system of ordinary Carnot cycles indicated by Figure 3 , which is drawn for the case in which A v > O

:

548

TRUESDELL

C.

v

B -

=

+,(el

e+given, fixed ordinary

e-A-

Fiyure 3 .

If we fix the adiabats two

C'

$1

- functions h4~@2

Carnot web.

, on

and

4

and

g9192

[A,Bl

exist

such that

Anybody can prove this statement by simply exhibiting the functions, which are defined in terms of %r' KV

.

, AK

,

and

The former equality implies obviously that

All this entirely elementary matter follows from Axioms 1 and 2 alone.

calorimetry.

It is part of the mathematical theory of It lay within the grasp of LAPLACE, CARNOT,

A N A L Y S I S BY R A T I O N A L THERMOMECIIANICS

549

CLAPEYRON, KELVIN, RANKINE, CLAUSIUS, and the lesser figures. None of them recognized it.

Had they mastered the Doctrine

of Latent and Specific Heats and put it into the simple, explicit form I have just presented, the history of thermodynamics might have been different. Of the conclusions with which Axioms 1 and 2 leave us, I remark now the two expressions for the work done by a Carnot

cycle

If

in a Carnot web for a given body.

From them we

conclude that that work depends upon the operating temperatures €3'

and

€3-

and upon the two adiabats of the cycle, or,

instead of them, upon the heat absorbed

C+(r)

and one adiabat

of the cycle, and that the work done can be positive or negative.

CARNOT had the insight to see that some of these

possibilities are excessive.

He imposed a restricting axiom

which marks the beginning of thermodynamics. his axiom, but we divide it into two parts.

We adopt precisely The first of these

is Axiom 3 (Part I of CARNOT's General Axiom). body, let

by any ordinary Carnot cycle.

For a given

Then

That is, the work done is always positive, and in calculating it we may disregard the two adiabats except insofar as they determine the heat absorbed.

On the basis of this axiom, in

combination with what the Doctrine of Latent and Specific Heats

550

C. TRUESDELL

tells us, we can demonstrate and unfold the entire formal structure of a thermodynamics that includes both the "caloric theory" and CLAUSIUS' theory as mutually exclusive special cases. First we apply CARNOT's axiom to an ordinary Carnot cycle subdivided into two smaller ones (Figure 4 )

:

e

I

V

Figure 4 .

Cycles used to prove that G is linear in its third argument.

Thereby we can easily show that

G

is additive in its last

argument

We can show also that G

By a slight adjustment of a proof due to CAUCHY, we conclude that

G

is linear in

C+($)

:

ANALYSIS BY RATIONAL THERMOMECHANICS

This statement, though not its proof, is due to REECH.

551

Now

considering the cycles in a Carnot web, we have

The left-hand side is independent of the choice of the adiabats and [A,B]

$2

.

Therefore, so is the right-hand side.

there are

g

C'-functions

and

h

Hence on

such that f o r

all

cycles in the web

Moreover, by Carnot's General Axiom, We easily prove that obvious:

Kh

g

and

h

will do, and then

g

is monotone-increasing.

are unique to within the g

+

const.

will do.

The

same argument proves also that

We easily convert the foregoing results to local form:

These are the basic local constitutive restrictions of thermo-

552

C. TRUESDELL

dynamics.

To derive the first of them, we note that

= C+(&)

- c-(&

I

and then we let the cycle shrink down to a point. The functions g The interval

IA,B]

a single interval.

and

h

are defined only for a web.

may be small; two disjoint webs may share The analysis thus delivers, so far, infi-

nitely many different webs and hence infinitely many different pairs o f functions g,h

for one and the same body.

We escape

from the webs by considering the following diagram:

e

AB-

V Figure 4 .

Escape from the web.

ANALYSIS BY RATIONAL THERMOMECIIANICS

The points therm.

and

P

Q

553

are ordinary points on the same iso-

A fresh appeal to Carnot's axiom, followed by a

limiting argument, shows that

-a -e - a% a'

at

-a-e - a% = 'a

P

Av

*V

at

Q

A crucial definition is suggested by this lemma. of

A point

is normal (could I but learn modern English, hopefully

I would call these points "generic") if

Each of its neighborhoods contains an ordinary point of kJ-

1.

.

The isotherm through it contains an ordinary point of ,A-

2.

.

The set of all normal points of of O&

.

G&

is the normal set

a-

This set need be neither open nor connected.

Since

is open, the set of all temperatures corresponding to

ordinary points is an open set

8 c %+ .

Thus

&

is

the union of countable collection of disjoint open intervals

d k.

An argument based on the smoothness assumed at the

outset for d

,

AV

, and

KV

enables us to prove the following

global constitutive restrictions: ............................... h

C

C' [ 41

,

g E C'

[a] , h

> 0

,

There are functions g

monotone increasing on

554

TRUESDELL

C.

gk to within the obvious constants, such that at every point of -8,

each 9

,

both functions being unique on

aa e(nv) ii -

&(%)

,

= 0

Conversely, if we assume Axioms 1 and 2, these constitutive restrictions imply Axiom 3 .

Thus CARNOT's ideas are expressed

once and for all as differential relations connecting h

, Av , and %

with the constitutive functions 4

sequently, if we can determine

g

and

h

g

.

and

Con-

from other consider-

ations, the local restrictions become a set of partial differential equations to be satisfied by 57

,

, and %

Av

.

These

interconnections between calorimetry and the motive power of heat are the essence of classical thermodynamics. A simply connected open subset of thermodynamic part

ath of

,&.

an is called a

In such a part we may

regard the local constitutive restrictions as conditions of integrability and hence demonstrate the existence of potential functions.

----

Thus we obtain the following Theorem:

There are functions Hh

€

C2 [&,I

and

Eg,h

such that A , = h - aHh

av

K V = h -aHh

ae

I

e c2 [&h]

ANALYSIS BY RATIONAL THERMOMECHANICS

Hh

is the pro-entropy;

555

is the internal pro-energy.

E

g,h

The theorem expresses generalized "First and Second Laws" of Thermodynamics.

There is a converse to this "energy-

entropy formulation" also.

If Axioms 1 and 2 are assumed,

and if the energy-entropy relations hold in some open subset of

,& ,

then Axiom 3 holds in that subset.

Thus in

Gththe energy-entropy formulation is equivalent to Part I Carnot's General Axiom. Before imposing axioms such as to specify

g

and

h

it is worthwhile to estimate the efficiency of a general cycle

by applying the results already obtained.

Using

an approach suggested by a more difficult problem solved by Messrs. FOSDICK

&

for any constant

SERRIN, we note a simple formula that holds a

and any cyclic process in Qth

simple cyclic process in ,Qn L = aC+

- I

zf+

a (

-

or any

:

$Qdt

- I %(-Q)dt J-

This is true because the local constitutive restrictions

C. TRUESDELL

556

tell us that

for these processes.

By suitable choices of

off obtain upper and lower bounds for

L

a

we straight

in terms of the

minimum and maximum temperatures of the process:

The upper bound is achieved if and only if

e = em i n

on

3-

I

If, further,

g

-

9(emin) h

+

the condition on 7

is increasing on reduces to

9 = 8

max

[emin, emax I ,

and thence

we conclude that if C+ , emin , and emax are given, Carnot cycles achieve the maximum efficiency, namely,

I

55 7

A N A L Y S I S BY R A T I O N A L THERMOMECHANICS

and they alone achieve it. We are now ready to specify the essential functions g

and

.

h

, so

C+ = C-

We note that for CARNOT's caloric theory h = const.

, while

for CLAUSIUS' thermodynamics

it is easy to see that h = KO, g and

J

+ const.,

= JKO

K > O

I

J > 0

I

is the mechanical equivalent of a unit of heat.

How

do we arrive at CLAUSIUS' theory and exclude the caloric theory? Apart from a naked assumption that these functions g

and

h

are the right ones, there axe over twenty pairs

of further axioms that will do the job.

For the first member I

preferto follow CARNOT and assume that his function G a universal function, the same for all bodies.

More precisely,

Axiom 4 (Part I1 of CARNOT's General Axiom). G GU

is

The function

for a given body is the restriction of a universal function to arguments compatible with ordinary Carnot cycles for

that body. z > o

GU (x,y,z)

is defined on the set x > y > 0

,

. It follows at once that

are the restrictions to €unctions gu

and

h,

&

g

and

h

for a given body

f o r that body of universal

, defined

on

fe+

.

Axiom 4 tells us that if we can find a single body for which Q h

=

x' x+ x

and €or which also we can determine

C. TRUESDELL

558

g

and

h

, we

shall thus have

gu

and

hU

-

Just as,

centuries ago, experiments on real gases suggested the concept of the ideal gas, experiments on real gases about 125 years ago suggested that an ideal gas with constant specific heats would be a good model to consider.

Accordingly I prefer to

lay down Axiom 5.

8,, = 74'

There is one body of ideal gas for which

z'

and botk: K and KV are constants. P A simple direct calculation shows that for an ideal gas x

with constant specific heats

h

= KB

and

g = JKB

+ const.

All of the structure of the classical thermodynamics of homogeneous simple fluid bodies then follows by specialization

of the results already obtained. One alternative should be mentioned here: Axiom 5.

There is one body such that

&,

=

and for which the work done by all Carnot cycles

%+ x

z'

satisfies

the relation L (?

That is, the classical efficiency formula for Carnot cycles if adjoined to CARNOT's own axioms delivers all of CLAUSIUS' thermodynamics, including both the "First Law" and the "Second Law".

This variant is interesting because DAY, who

for several years has been studying ways to define entropy in terms of work, heat, and temperature, in collaboration

559

A N A L Y S I S BY R A T I O N A L T H E R M O M E C I I A N I C S

with ZILHAVP has shown that for a rather general constitutive class something slightly similax holds.

DAY ti SILHAV?

assume the equivalent of the "First Law"; their processes, essentially reversible, are mappings of

into Banach

spaces; they assume that the classical efficiency is maximal for given extremes of temperature; and they assume that Carnot cycles do achieve that efficiency. Usually in rational thermodynamics we take the energyentropy formulation as a basis for generalization. General "First and Second Laws" are these axioms: For any process

L

The sign

A

means increment:

We still consider homogeneous processes, so all quantities are functions of

t

alone.

The process is said to be

reversible if equality holds in the Second Law. it is irreversible.

Otherwise,

We can prove rather easily that

C. TRUESDELL

560

and that equality holds if and only if the body undergoes a reversible Carnot process, which need not be cyclic. Messrs. FOSDICK

Ei

to deformable continua.

SERRIN have shown how to extend these results For them the heating

in terms of a heating flux field

r

a heating supply field

-

h

is expressed

Q

on the boundary

in the interior,

8

ad and

:

and the Second Law is most commonly taken to be expressed by the Clausius-Duhem inequality:

FOSDICK

&

SERRIN consider a body immersed in a heat bath,

the temperature of which at the time

t

is

r(t)

.

They

impose MAXWELL'S heat transfer inequalities:

That is, heat flows into those parts of a body where it is colder than its environment, out of the body at those points where it is hotter.

FOSDICK

&

SERRIN define a reversible

Carnot process and prove that for the deformable body just the same estimate of efficiency holds as for the homogeneous process, except that

T

replaces

0

:

ANALYSIS B Y RATIONAL THERMOMECHANICS

561

F O S D I C K & SERRIN on the basis of this result are able to

state clearly and demonstrate simply a number of the "Second (1) A

Laws" pronounced by physicists in the last century: cycle cannot absorb heat unless it emits heat, that emits no heat cannot do positive work,

(2)

A cycle

( 3 ) A cycle

that absorbs and emits heat at only a finite set of temperatures, at all but one of which the heat emitted equals the ( 4 ) If a cycle

heat absorbed, cannot do positive work,

absorbs heat only at temperatures not greater than those at which it emits heat, it cannot do positive work.

The cycles

involved may correspond to "reversible" or "irreversible" processes. All these matters are elementary.

I have gone over

them today because they were not cleared up until a decade or more after the major advances in rational thermodynamics had been made.

Those advances were based on the interpretation

of the Clausius-Duhem inequality proposed by COLEMAN in 1963.

&

NOLL

They regarded it as an identical restriction on

constitutive relations. They required the constitutive relation itself to exclude automatically any process that does not obey the Second Law.

I do not need to tell the

participants in this meeting that COLEMAN

&

NOLL's approach,

when applied to assumptions that satisfy the rule of

C. TRUESDELL

562

e q u i p r e s e n c e , d e l i v e r s t h e goods: s p e c i f i c and e x p l i c i t res t r i c t i o n s t h a t g r e a t l y s i m p l i f y any c o n s t i t u t i v e c l a s s l a i d down f o r s t u d y .

I t h a s been used a g a i n and a g a i n : i t i s now

r e g a r d e d as a s t a n d a r d t o o l . Thermodynamics i s become a branch of m a t h e m a t i c a l s c i e n c e ; even more, thermodynamics and mechanics are now f u s e d i n t o a u n i f i e d d i s c i p l i n e t h a t might w e l l b e c a l l e d r a t i o n a l thermomechanics.

The a n a l y s t s h o u l d be a b l e t o

c a l l upon t h e c o n c e p t s and methods of t h a t s c i e n c e , i n much t h e same way as h e u s e s hydrodynamics and t h e l i n e a r t h e o r i e s of e l a s t i c i t y and of t h e c o n d u c t i o n of h e a t .

In t h i s lecture I

have a t t e m p t e d t o s k e t c h for t h e d i s t i n g u i s h e d a n a l y s t s p r e s e n t a c l e a r and d e c e n t way t o see t h e c l a s s i c a l p a r t of the subject.

I hope t h a t no-one w i l l i n f e r t h e r e f r o m t h a t

thermomechanics i s a dead and d r i e d f i e l d .

By no means.

I

c l o s e t h i s l e c t u r e by a l i s t of some of t h e avenues of c u r r e n t research i n it. 1.

Some p e r s o n s d o u b t t h a t t h e Clausius-Duhem i n e q u a l i -

t y b e a p r o p e r s t a t e m e n t of what t h e y r e g a r d as "The Second Law".

O f t h o s e , some r e g a r d t h e i r Second Law as a f f e c t i n g

o n l y t h e e n d p o i n t s of a p r o c e s s b e g i n n i n g from and e n d i n g i n what t h e y choose t o d e s c r i b e as " e q u i l i b r i u m " .

Others regard

"The Second Law" as r e f e r r i n g o n l y t o p a i r s of b o d i e s , n o t t o a l l bodies.

One member of t h i s p a i r t h e y c a l l "environ-

ment", and t h e y are l o t h t o s p e c i f y any o f i t s p r o p e r t i e s . I n s o f a r as t h e s e p e o p l e can r e d u c e t h e i r i d e a s t o forms p r e -

ANALYSIS BY RATIONAL TERMOMECHANICS

563

c i s e enough t o b e comprehensible t o o t h e r s , t h e y d e s e r v e respect.

Usually they l i m i t t h e i r c o n s t i t u t i v e r e l a t i o n s ,

o f t e n t a c i t l y , t o a narrow c l a s s , so i t i s no wonder t h e y c a n g e t some r e s u l t s o u t o f a "Second Law" w e a k e r t h a n t h e Clausius-Duhem i n e q u a l i t y .

I t i s of c o u r s e p o s s i b l e t h a t

some g e n u i n e g e n e r a l i z a t i o n may u l t i m a t e l y emerge from t h i s group o f s t u d i e s . M i x t u r e s have always been i m p o r t a n t i n thermodynamics. MULLER h a s shown t h a t f o r them t h e f l u x of e n t r o p y c a n n o t

g e n e r a l l y be t a k e n as t h e f l u x o f h e a t d i v i d e d by t h e t e m p e r a t u r e , and some of t h e r e s e a r c h e s o f G U R T I N , NOLL, and WILLIAMS l e a v e u s d o u b t i n g t h a t even t h e c o n c e p t of p a r t i a l

stress c a n w i t h s t a n d s c r u t i n y .

Thermomechanics w i l l n o t be a

s e t t l e d s c i e n c e u n t i l a s a t i s f a c t o r y , b a s i c s t r u c t u r e t o desc r i b e m i x t u r e s h a s been c r e a t e d .

WILMA~SKIhas presented

a n a t t e m p t a t a v e r y g e n e r a l , a b s t r a c t , and formal thermodynamics, s u f f i c i e n t t o c o v e r a l l s i t u a t i o n s t h e t e r m u s u a l l y suggests. 2.

H i s work d e s e r v e s t o be s t u d i e d and a p p l i e d .

Some p e r s o n s l a y g r e a t w e i g h t on d e f i n i n g tempera-

t u r e i n terms of a l l e g e d l y s i m p l e r c o n c e p t s such as " e q u i l i brium", w i t h o u t u s i n g c o n s t i t u t i v e p r o p e r t i e s of p a r t i c u l a r

materials. possible.

I doubt t h a t such a program of d e f i n i t i o n b e I t h i n k t h e c o n c e p t of h o t n e s s s h o u l d j o i n t h e

c o n c e p t s of motion, m a s s , sensual experience.

and f o r c e a s a p r i m i t i v e based on

W e ought t o r e c a l l t h a t f o r c e i t s e l f

h a s o n l y r e c e n t l y been r e c o g n i z e d f o r m a l l y , though i n l a y i n g

C. TRUESDELL

564

down s p e c i f i c axioms f o r it M r . NOLL w a s f o l l o w i n g t r a d i t i o n . H e made e x p l i c i t and p r e c i s e some i d e a s used h a l f - c o n s c i o u s l y by NEWTON, EULER, and many p i o n e e r s , much a s BOREL and LEBESGUE

had i s o l a t e d t h e e s s e n c e of m a s s and c h a r g e from t h e shadows of p h y s i c s and metaphysics. S E R R I N ' s r e c e n t work.

In t h i s sense I i n t e r p r e t M r .

I t seems t o me t h a t he s t a r t s from a

c o n c e p t o f h o t n e s s and shows how t o c o n s t r u c t a scale of t e m p e r a t u r e by use of a p a r t i c u l a r c l a s s of materials, which w e might c a l l thermometers.

These a r e d e f i n e d , o f c o u r s e , by

special constitutive relations. 3.

Some p e r s o n s wish t o d e f i n e e n t r o p y o r b o t h energy

and e n t r o p y i n t e r m s o f h e a t and work.

How t h i s can be done

w i t h p e r f e c t r i g o r i n t h e c l a s s i c a l c o n t e x t of r e v e r s i b l e p r o c e s s e s i n homogeneous b o d i e s , I have o u t l i n e d a t t h e beginning of t h i s l e c t u r e .

For b o d i e s s u s c e p t i b l e of i r r e v e r -

s i b l e p r o c e s s e s it i s n o t so e a s y . s u g g e s t e d some y e a r s ago by M r .

An extreme p o s s i b i l i t y ,

ERICKSEN, i s t h a t a l l "Second

L a w s " are d i s g u i s e d s t a t e m e n t s of p u r e l y mechanical s t a b i l i t y . A noteworthy s t e p i n t h i s d i r e c t i o n may b e found i n a monumen-

t a l w o r k by Messrs. COLEMAN

& OWEN.

They p r o v i d e a g e n e r a l

t h e o r y of a c t i o n s on dynamical s y s t e m s , i n t e r m s of which t h e y d e f i n e a " C l a u s i u s p r o p e r t y " and a " c o n s e r v a t i o n p r o p e r t y " and t h e y i n t e r p r e t t h e l a w s o f thermodynamics as assumptions t h a t t h e r e a r e a c t i o n s h a v i n g t h e s e p r o p e r t i e s a t some s t a t e . E x i s t e n c e of an e n e r g y f u n c t i o n and an e n t r o p y f u n c t i o n f o l l o w . COLEMAN & OWEN p r o v e t h a t many r e s u l t s known t o f o l l o w from t h e

,

ANALYSIS B Y RATIONAL THERMOMECHANICS

565

o l d e r , more r e s t r i c t i v e thermodynamics based on t h e C l a u s i u s Duhem i n e q u a l i t y are consequences of t h e i r new and weaker assumptions.

I r e g a r d t h i s work,

i n t r i c a t e and d i f f i c u l t

as i t i s t o u n d e r s t a n d i n d e t a i l , as t h e arrow p o i n t i n g t o t h e f u t u r e of s e r i o u s r e s e a r c h i n thermodynamics. 4.

S t a b i l i t y b r i n g s u s back t o t h e realm of a n a l y s i s

i n t h e o r d i n a r y sense.

There are i n f i n i t e l y many c o n c e p t s

o f s t a b i l i t y , some so broad as t o l e t u s c o n c l u d e any s y s t e m o r any s t a t e o f a s y s t e m t o be s t a b l e o r u n s t a b l e by d e f i n i tion.

I t i s a long-standing

problem t o r e l a t e s t a t i c c o n c e p t s

of s t a b i l i t y t o dynamic ones.

The a d v a n t a g e of a s t a t i c

d e f i n i t i o n i s t h a t it does n o t r e q u i r e t h e u s e r t o know t h e e q u a t i o n s of motion.

The p r a g m a t i c s u c c e s s of s t a t i c con-

cepts shows t h a t many d i f f e r e n t e q u a t i o n s of motion may l e a d t o a common s u f f i c i e n t c o n d i t i o n f o r s t a b i l i t y .

I t becomes

a m a t h e m a t i c a l problem t o p r o v e t h a t s u c h i s t h e case. &

KNOPS

WILKES have p r o v i d e d us w i t h a f i n e e x p l a n a t i o n of t h e s e

problems and an e x p o s i t i o n of r e s u l t s , many o f them t h e i r own, up t o 1973.

A long-forgotten

r e s e a r c h by DUHEM was t h e f i r s t

t o a p p l y L I A P U N O V ' s c o n c e p t s t o t h e thermodynamics o f deformable b o d i e s and t o i n t r o d u c e , on t h e b a s i s of t h e r m o s t a t i c c o n c e p t s and a c e r t a i n c o n s t i t u t i v e c l a s s , a " b a l l i s t i c energy" which s e r v e s a s a Liapunov f u n c t i o n . by M r .

S i m i l a r i d e a s w e r e p u t forward

ERICKSEN and have been v e r y i n f l u e n t i a l i n the r e c e n t

and i n t e n s i v e s t u d y of t h i s f i e l d .

An e l e g a n t , abstract general

t h e o r y which p r o v i d e s b o t h a summary and an e x t e n s i o n of what

566

C.

TRUESDELL

w a s known b e f o r e h a s been p r e s e n t e d by M r . G U R T I N .

T h i s work

a l s o c a l l e d upon a modern c o n c e p t of a dynamical system as a mathematical o b j e c t . I n d e s c r i b i n g t h i s r e c e n t work I a m m a t h e m a t i c a l l y i n above my d e p t h .

I f you have q u e s t i o n s , p l e a s e a s k them o f

t h e p r i n c i p a l e x p e r t s , most of whom are p r e s e n t .

I hope I

have shown you t h a t c l a s s i c a l thermodynamics i s e v e r y b i t

as n a t u r a l and c l e a r and r i g o r o u s as t h e o l d a n a l y t i c a l dynamics.

I hope t h a t my t e l e g r a m a t t h e e n d , f o r more t h a n

t h a t i t i s n o t , makes p l a i n t h a t t h e problems of modern thermo mechanics are i n l a r g e p a r t problems of a n a l y s i s .

They a r e

n o t problems of t h e k i n d t h a t c a n b e handed t o an e x p e r t on e x i s t e n c e theorems o r n u m e r i c a l c a l c u l a t i o n who w i s h e s t o f o r g e t t h e i r physical contexts.

Like most g r e a t problems of

n a t u r a l s c i e n c e i n t h e p a s t , t h e y must b e f o r m u l a t e d p r o p e r l y by him who i s t o s o l v e them.

They are problems f o r a

ggombtre: a man who s e e k s b e a u t y and o r d e r everywhere, i n n a t u r e and t h o u g h t a l i k e .

567

A N A L Y S I S BY R A T I O N A L THERMOMECHANICS

References for further reading (not necessarily the sources of the lecture).

B. D. COLEMAN

&

D. R. OWEN, "A mathematical foundation for thermo-

dynamics," Archive for Rational Mechanics and Analysis 54 - 5

(1974), 1-104.

W. A. DAY

&

M. sILHAV9, "Efficiency and the existence of entropy

in classical thermodynamics,'' Archive for Rational Mechanics and Analysis, in press. M. E. GURTIN, "Thermodynamics and stability," Archive for Rational Mechanics and Analysis 59 (1975), 63-96. *?

R. J. KNOPS

&

E. W. WILKES, "Theory of Elastic Stability," in

Handbuch der Physik VIa/3 (ed. C. TRUESDELL), Berlin

e. , Springer-Verlag, 1973.

I. MULLER, "A thermodynamic theory of mixtures of fluids," Archive for Rational Mechanics and Analysis 28 (1968), *1-39. C. TRUESDELL

& S.

BHARATHA, The Concepts and Logic of Classical

Thermodynamics as a Theory of Heat Engines, Rigorously Developed upon the Foundation Laid by S. Carnot and F. Reech , New York , Springer-Verlag, to appear in 1977. C. TRUESDELL, "Improved estimates for the efficiencies of irreversible heat engines," Annali di Matematica Pura ed Applicata (IV) 108 (1976), 305-323. *** C. TRUESDELL, "Irreversible heat engines and the second law of

-

thermodynamics," Letters in Heat and Mass Transfer 3

C. TRUESDELL

568

(1976)I 267-289.

W. 0. WILLIAMS, "On the theory of mixtures," Archive for Rational Mechanics and Analysis 51 -- (1973), 239-260. K. WILMAgSKI, Podstawy Termodynamiki Fenomenologicznej, Warsaw,

Pafistwowe Wydawnictwo Naukowe , 1974. Postscript: I have been allowed to see a manuscript by JOHA" WALTER in which he quotes several astounding pronouncements in the literature of thermodynamics, pronouncements about Carnot cycles which I believe the foregoing outline shows to be totally absurd : 1. BORN : "NO ordinary mathematics 2 . BOURBAKI : "Anomalous'I 3 . JAUCH: "Dusty pieces of Victoriana" (1824!) , "Not purely analytic , not mathematically rigorous" 4. LANDE: Itsuperfluoustrimmings" 5. VAN DER WAALS: "Not strictly logical", "What has the useful work of heat engines to do with the dependence of caloric quantities upon thermal ones ! I' 6. ZEMANSKI: "All-powerful but mysterious" Only two statements from WALTER'S list survive the analysis outlined in this lecture: 7. CARATHfiODORY : "Can be conveniently and convincingly dealt with

only in the case of a system with two degrees of freedom" & HARTKE: Near an isobaric maximum of the density Carnot cycles degenerate "catastrophically".

8. THOMSEN

A major motive for the axiomatic treatment Mr. BHARATHA & I have constructed is to prove mathematically that despite the truly "catastrophic" inability of Carnot cycles to include a segment of a neutral curve, nevertheless their properties suffice to prove thermodynamic relations at points on such a curve. The real catastrophe, I maintain, i s the failure of thermodynamicists to master elementary calculus sufficiently to do more with it than apply ritual algebra.

A N A L Y S I S B Y RATIONAL THERMObmCHANICS

LECTURE 3 .

569

COMMENTS ON THE KINETIC THEORY OF GASES

The physicists tell us that matter really is composed of little bits they call molecules.

They are less successful in

explaining just what properties they attribute to those little bits.

Indeed, about that among themselves they squabble, while

presenting a reinforced concrete front to laymen and dispensers of funds.

Be that as it may, mathematicians have sought for a

long time to prove that a sufficiently large number of discrete little bodies would be indistinguishable, as far as human senses can discern, from the usually continuous hunks of material that matter seems to us to be. The most sucessful result of this endeavor is MAXWELL'S kinetic theory of gases. theory to mathematicians.

It is hard to lecture on the kinetic Mathematicians distrust a lecture

entirely in words; they like to see a lot of formulae on the board.

In the kinetic theory that is all too easy, for equa-

tions which fill half a page or more are the rule, but there we encounter two worse problems: first, to write out a dozen formulae of the kinetic theory would take the whole hour and leave no time to explain what they are good for; second, the calculations used to derive them are €or the most purely formal, and if there is anything mathematicians distrust more than pure words, it is purely formal calculation.

The kinetic theory is

both the Scylla and the Charybdis of the lecturer, for whilst a display of calculation cripples him before he can penetrate

C. TRUESDELL

570

to his proper subject, as in other parts of mathematics here too it is the formulae that present the matter most clearly as well as succinctly, and without them he spins in dizzy incomprehensibility, as vague as physics or philosophy. Since my lecture is doomed to failure, I sail ahead with confidence on the only course I know how to steer: toward concepts, made clear as well as precise by mathematical definitions and axioms, concepts thereupon ennobled as well as enriched by proof or at least projection of proof of theorems. The kinetic theory is designed to represent a moderately rarefied monatomic gas.

MAXWELL, employing a skillful mixture

of mechanical and stochastic ideas, made it do just that.

His

theory is formally a continuum theory, which concerns massdensity, gross velocity, pressure, internal energy (or temperature), stress, and flux of heating.

These are not introduced

a priori but defined in terms of a molecular density function F :

$?x

Ex

[7e+ [

.

This function is interpreted as

a density of probability at the time the place

x ..,

t

and the molecular velocity

phenomenological fields are defined. number-density

in a neighborhood of v ..,

.

In terms of it,

First,

n(t,z) : J F(t,x,y)dy

v

-

,

or, in a briefer notation, number-density

n :IF

mass-density

p

I

EWln

QYb being the mass of one molecule, which we regard as an assigned

571

A N A L Y S I S BY R A T I O N A L THERMOMECIIANICS

positive constant. function

g

-

The average or expectation

g

of a

is the field defined thus:

Then gross velocity

u -

v -

-

random velocity

c

-

_

v -a.

5

-

The further fields of main interest are defined in terms of u - 1 7 2 -

energetic

E = - c

pressure tensor

M

mean pressure

p

pc@c - 1 -trM

energy-flux vector

s

1z p-zc2c

-

3

:

I

I

-

I

-

These fields are to be interpreted just like their counterparts in the classical mechanics of fluids or in modern continuum thermomechanics. An immediate consequence of the definitions is

Thus the concepts of the kinetic theory make the kinetic gas an ideal gas with constant specific heats, the ratio of which is 5/3. The molecular density

F

is governed by an equation of

evolution:

+ To render this equation definite, we must specify

I

and

tlt,

a: .

K i s the retrogressor corresponding to a given field of body

.

572

TRUESDELL

C.

- .

force b

It simply evaluates F

at the argument obtained

by going back along the molecular trajectory to the place and velocity the molecule now at t

-

to

, had

would have had at the time

5.y

it moved subject to the field

alone.

is linear, multiplicative, and monotone.

collisions operator


intermolecular forces.

*

is

b is of a simple kind; in ...

easy to calculate explicitly when general,

b

represents the change of

F

The

due to

I will discuss only the results of

using the classical collisions operator invented by the singular genius of MAXWELL. pair of vectors

y,y*

An encounter is a transformation of a into another pair

y',y:

in such a way

as to leave the total momentum and total kinetic energy unchanged :

-

v'

+ v; -

=

- + v* -

v

Traditionally the pair

-

v'

, and

vI2

~4

+

v'*2 = v2

+

v :

.

is calculated from a

by solving a problem of analytical dynamics

given pair

in which these two pairs appear as asymptotic velocities at the times

-m

and

+m

,

respectively.

We can extend this

idea by considering an encounter operator

9%Pu

,

?F4

: p

x

yxd

+

being a non-empty disk in a Euclidean

plane which we shall specify presently:

The plane in which

lies contains the place

E

and is

A N A L Y S I S B Y R A T I O N A L THERMOMECHANICS

normal to

v* - v

;

pd

x

is a disk centered at

.

is an arbitrary place in

573

;

d

and

We require the operator

a

not only to conserve momentum and energy but also to be idempotent, symmetric in

? and

?* , and orthogonally invariant.

MAXWELL'S collisions operator is specified in terms of these definitions:

The adjoined

*

and

; indicate v, - , y ' , and

and

replaced as an argument by

The ordinary field variables we can regard tions of

t

and

-

x

-

that

v

,

-

v;

is to be

respectively.

are silent here, so

as a mapping of functions of

- .

-

v

onto func-

v

A mathematician when he defines an operator usually specifies its domain.

It is difficult to do so for

except indirectly: all those non-negative over

F

@

, integrable

fl , €or which the indicated integrals exist.

Recently

MUNCASTER, in work not yet published, has succeeded in proving that certain particular function spaces are subsets of the domain of

when

is of one of the kinds commonly considered

The operator (13 has wonderful properties, properties that give MAXWELL'S kinetic theory its celebrated difficulty and its fascinating power to survive the many objections that have been raised against it.

The most important of these

properties flow from BOLTZMANN's Monotonicity Theorem:

,

5 74

C. TRUESDELL

equality holds if and only if

F = ae-bc2

.

An easy corollary

states that ~ = a- bec 2

CF=O The theorem suggests that

d!

represents the effects of colli-

sions as being in some sense irreversible unless null. corollary shows us that the operator

The

gives a special role,

associated with a concept of equilibrium a s a steady, unaltering condition] to the Maxwellian density

ae-bc2

The foregoing results ought not be seen as consequences of the kinetic theory.

Rather, they are built into it as part

of its apparatus. When we put MAXWELL'S specific operator

d:

into the

equation of evolution, it becomes an integral-functional equation. to

-

b

If we differentiate it along a trajectory corresponding

, we

obtain @F

in which

@

=

C F

I

denotes the derivative along the trajectory.

This

statement is an ordinary integro-differential equation; we get one such equation for each trajectory that corresponds to If the solution F

-

b . varies sufficiently smoothly from trajectory

to trajectory] we can calculate the directional derivative by the gradient rule:

A N A L Y S I S BY R A T I O N A L THERMOME C I I A N I C S

5 75

The resulting partial integro-differential equation, namely 8 F = d3F

I

is the Maxwell-Boltzmann Equation. classical solutions.

Its solutions are called

Solutions of the mother and grandmother

equations from which we have obtained it are among the several kinds of generalized solutions that arise in the kinetic theory. Smoothness seems to be of the essence.

The most inter-

esting results that the kinetic theory delivers presume that solutions are not merely classical but also have moments of orders 0, 1, 2 , 3 , and even 4 , moments which are themselves smooth fields on a subset of

f xe .

For example, there is

MAXWELL’S Consistency Theorem: The fields cj

obtained from any smooth solution

F

p

,2 ,

E

,M ,

and

satisfy the field

equations of continuum mechanics: p =

pdivu = 0

PU + -

pE

+

+

grad ( $ E )

2 -pEdivu 3 .”

I

+

-

divP

pb = 0

P-E+ divq

- _

-

= 0

I

I

here

f and

-

P

tensor :

S

a,f

+

-

u-grad f

is the pressure deviator, while

I

-

E

is the distortion

576

C. TRUESDELL

E :-gradu 1 2 -

+ 21 (gradu)T *

1 (divu)l

3

-..,

It follows that the kinetic gas is a continuum. difference, however, from continuum mechanics.

There is a big There we move

into special problems by defining ideal materials.

A particular

material is characterized by a particular set of constitutive axioms.

To solve a particular problem, we substitute constitu-

tive equations into the general field equations and so obtain "equations of motion" for a body consisting of that material. Here, on the contrary, there is no further room for axioms.

The

field equations do not express principles to be used in obtaining equations of motion. smooth solution.

They are satisfied automatically by every

The kinetic theory does not need equations of

motion and does not provide them except in very special and unusual cases.

The kinetic constitutive quantities are neither

more nor less than IyM

molecular mass

I

J'

cross-section

I

encounter operator

.

Once these three quantities have been prescribed, all the rest must follow by mathematics alone.

Therein lies one of the

charms of the kinetic theory for the analyst.

It might seem

to be like the old theories of hydrodynamics, elasticity, and heat conduction: nothing but analysis disguised.

I do not

577

ANALYSIS BY RATIONAL THERMOMECHANICS

think that is quite so, although I am the first to admit that soon it may become so.

Now, I think, conceptual problems re-

main, problems which must be solved indeed in the end by analysis, but to pose which some experience in rational mechanics might be helpful. We begin to see the difference when we look back at the

, which

cross-section normal to

--

v

y*

.

is a disk in the plane through

x

This section is that part of the plane

that can correspond to interactions of the molecules.

If the

molecules are spheres that actually collide like billiard balls, the radius of the disk is the diameter of the ball. More

generally, if the intermolecular forces are of finite

range, the radius of the disk is finite.

Unfortunately the

only example of such forces with a cut-off that seems interesting for comparisons with nature is the ideal sphere.

All the molecular models which lend themselves to physical interpretation correspond to forces of infinite range, and for them the radius of the cross-section is infinite; that is,

d

is the entire plane.

Without any exception, every

analyst who has touched the kinetic theory has restricted himself to the case in which

,d is finite.

has done so is plain from a glance at

If

is finite, and if

then C F

,

F

The reason he

:

has moments of orders 0 and 1

,

if it exists, is the difference of two absolutely

C. TRUESDELL

5 78

convergent integrals.

For such a (I:

the equation of evolu-

tion lends itself to several neat iterative schemes of solution.

Existence theorems, while far from trivial, can be

proved with tools presently available.

On the other hand,

if the intermolecular forces are of infinite range, the integrals defining

cf

are only conditionally convergent.

Most

of the calculations that have been made for such models are merely formal.

After over L O O years of study, theorists are

only a little closer to justifying them than were MAXWELL and BOLTZIUXN.

The difficulty, as we shall see a little later,

lies much deeper than the proof of existence theorems for the initial-value problem. The difference between the classes of molecular models to which they devote their efforts is by no means the greatest of those that divide the analysts and the mechanicists when it comes to the kinetic theory.

The analyst looks at

the equation of evolution, F(t,x,v) = -

5

t-t

oF(t,x,v) - - +

t

/&-. ((i:F) (t,fS,y)ds

t o

and he says, aha! An initial-value problem:

I w i l l specify a

function space to which the initial value and the solution must belong, and then I will prove that the solution exists and is unique ! I do not belittle this problem.

problem.

First of all, it is a

Second, it is a difficult problem.

Third, the physi-

579

A N A L Y S I S B Y R A T I O N A L THERMOMECHANICS

cists are simply stupid when they say it does not need to be solved, because their "physical intuition" tells them it can be solved.

This is one more example of religion-science,

which is of no concern to this group. existence and un'iqueness

A s I said, theorems of

have been proved for a kinetic gas

consisting of molecules with a cut-off.

These refer to two

different cases. Case 1. dependent of p =

Spatially homogeneous solutions:

- .

x

, u.-

const.

F

is in-

In this case the gas is grossly at rest:

= const.

,

E =

const.

From the standpoint

of continuum mechanics nothing at all is happening, and no prob-

lem could be more trivial.

Within the framework of the

kinetic theory, on the contrary, something more than mere existence and uniqueness can be proved.

Kinetic equilibrium

corresponds to a Maxwellian molecular density:

F

=

-bC2

ae

For a spatially homogeneous problem there is one and only one Maxwellian solution, and it has constant parameters and

-

u

;

equivalently, p

,

,

and

E

.

a

,

b

The question

is, does every other solutionawith the same constant values of those parameters approach the Maxwellian one?

This is the

most restricted, the narrowest interpretation of the trend to equilibrium. ARKERYD.

The best results so far published are those of

They establish the narrow trend to equilibrium in

the sense of weak convergence for a very extensive class of spatially homogeneous solutions in a gas of molecules with a cut-off.

A necessary preliminary is a theorem of existence

,

580

C i TRUESDELL

and uniqueness.

Modern methods of analysis usually deliver

only solutions generalized in one sense or another.

In this

regard ARKERYD has treated the Maxwell-Boltzmann equation as an ordinary differential equation for a function of time taking on values in the set of integrable functions. tions, also, are generalized.

Hence his solu-

In work not yet published MUN-

CASTER has shown that ARKERYD's technique may be modified so as to deliver classical solutions for essentially the same group of problems. Here I interpellate the summary by two remarks.

First,

generalized solutions do not presently provide more than a way-station in the kinetic theory. Theorem is but

Maxwell's Consistency

one of the major theorems that refer only to

smooth solutions.

Either regularization is of the essence,

or the basic theory has to be refounded in such a way as to make generalized solutions sufficient to yield concrete results.

Second, the one existence theorem that has been proved

for a gas of molecules whose mutual forces are of infinite range refers to spatially homogeneous solutions.

It is

special in that the molecules are of the kind called Maxwel-

lian: They repel each other with forces inversely proportional to the fifth power of the distance. It is restrictive in that it refers not to the Maxwell-Boltzmann equation itself but to the corresponding infinite differential system satisfied by the moments of a solution and so concerns only solutions the moments of whlch of all orders.exist and are differentiable.

ANALYSIS BY RATIONAL THERMOMECHANICS

In return for these limitations explicit results.

581

the theorem delivers fully

The moments are exhibited as functions of

time, determined by their initial values.

The trend to equi-

librium is almost trivially established and shown to be exponential; and it is proved that among all the infinitely many moments the slowest to relax to its equilibrium value is the flux of heat. elementary.

Unfortunately the proof of this theorem is

Perhaps that is why it is rarely mentioned in

the literature. The kinetic theory represents gross dissipation.

A

theory of dissipation should refer not merely to the molecular density but to the gross flow.

Grossly, if we stir up

a body of gas in a stationary vessel, we expect that body to come to rest in due time.

It is this kind of trend to equi-

librium we should be able to establish in the kinetic theory.

So far as I know, there is no trace of proof of anything like that at the present. Coming back to molecules with a cut-off, I turn to existence theorems for solutions that may vary with position. These are the only solutions that are of any interest for calculating gas flows.

Of course, guided by what we know

about the classical theory of elastic fluids, we expect to establish existence only for a finite interval of time.

Two

theorems, both for a gas that fills all of space, are known. GRAD considered solutions whose initial values are close to uniform Maxwellian density.

He proved that such solutions

C. TRUESDELL

582

exist, are classical colutions, and remain close to the uni-

form Maxwellian density.

The more nearly Maxwellian they are,

the longer is their interval of existence.

This is a pretty

theorem, but little about gas dynamics can be learned from flows that are nearly a state of rest.

GLIKSON proved a

theorem that allows a broad class of initial values, but it does not yield classical solutions. Be all this as it may, even if we could solve the initialvalue problem in full generality we should not know much about the kinetic theory. heat flux

q ..

ple we know Pof

F

that we seek. and

-

It is the pressure deviator

q

True, if we know

,

F

and the in princi-

and a lot more, but the mere existence

,.

does not tell us how to relate

condition of the gas.

P

P -.. and

s

to the gross

A physicist's book on the kinetic theory

pays no attention to existence and regularity but has a lot to say about

-

P

and

cj

.

For example, the author somehow stum-

bles along toward the Stokes-Kirchhoff laws:

The coefficients p

and

m

are the shear viscosity and the

Maxwell number of the kinetic gas.

The physicist calculates

these "transport coefficients" for many molecular models, and he compares his values with experimental data.

Except for

the Maxwellian density, his whole book does not exhibit a single F

satisfying the Maxwell-Boltzmann equation; neither

does it calculate a single gas flow.

Looking at an analyst's

ANALYSIS BY RATIONAL THERMOMECHANICS

583

paper on the kinetic theory and a standard book on that theory, you would be hard put to it to discern any connection beyond the titles. Very little skill in the kinetic theory suffices to show that a typical solution of the Maxwell-Boltzmann equation does not satisfy the Stokes-Kirchhoff laws or any other relations that determine p

,y ,

and

E

.

P -.,

and

in terms of the fields

cj

Bluntly, a typical solution does not seem to

have a viscosity or a Maxwell number.

Of course I really do

not mean to refer only to the classical formulae of the StokesKirchhoff theory of ideal gases; I regard far more general relations of the same kind, relations which resemble the constitutive equations of modern thermomechanics. Such, too, MAXWELL and later physicists found by methods not only purely formal but even mysterious.

Such relations correspond, if they hold

at all, only to a special class of solutions. they important?

Then why are

Because they have some asymptotic status in

regard to all solutions.

There we see, dimly, the central

-------

problems of the kinetic theory: l - - - - _ _ - - - - l _ _ _ - _ - _ _ _ I _ _ _ _ _ _ _ _

1.

To specify a class of solutions for which

hence also

-

P

and

-

q

in a neighborhood of

are determined by

x

t,x I

2.

, or

at the time t

in a time-space neighborhood of

p

.

,2 , by p

F

,

E

,

and

and b -

- , and

u

E

To prove the existence of these special solutions,

which we may call grossly determined and momentally determined, respectively.

To make these concepts formal, we let

-p

denote

C. TRUESDELL

584

the fields p

,

u

and

E

Then a solution F

collectively.

is grossly determined if there is a map

such that

, momentally determined if there is a map

such that

-

The notation x+- denotes a restriction to some neighborhood

x

-

;

about

t

of

the notation

.

the map

t+.

G

The map

,

restriction to some interval

is the gross determiner of

is the momental determiner of

F

.

F

The term

"momental" refers to the first five scalar moments of

-p

which we have denoted collectively by

;

;

,

F

we shall call

-p

the principal moment. 3.

To calculate the corresponding gross or momental

determiners its

-

P

and

9f!

and

'9-/

F

If

F

is grossly determined,

are the values of maps on a domain of fields

p and b-" f o r fixed t

If

. :

is momentally determined, its

-

P

and

-

q

are the

values of maps on a domain of time-dependent fields

-p

:

A N A L Y S I S BY R A T I O N A L T H E R M O M E C H A N I C S

585

Grossly determined solution are the values of maps of the fields of principal moment and body force at the present time

only; momentally determined solutions are the values of maps of the principal moment alone, over a time-space neighborhood. We know that the two classes just defined are not empty, for a Maxwellian molecular density is both grossly and momentally

-

determined, and for it both pairs of maps 3 their arguments into

0 -

.

and

9 -

map all

The gross determiners resemble

constitutive functionals of continuum mechanics, but only after

-

b

has been specified; the momental determiners resemble con-

stitutive functionals more closely in that they require no commitment to a particular

b -.

,

but they are of a more complica-

ted kind, involving time rates.

The gross and momental rela-

tions have sometimes been confused with constitutive relations of continuum mechanics, but that they are not: The arguments of a true constitutive function are independent, but MAXWELL'S Consistency Theorem imposes differential relations among the

-

fields p

,P ,

and q associated with any solution of the I

Maxwell-Boltzmann equation. After exploring these intricacies, which seem to have escaped the physicists but should be easy for mathematicians to understand, we return to the central problems of the kinetic theory and state another one: _-.,----

4.

To establish an asymptotic status for these special

solutions, either (i) as

t + tm when un is fixed,

tm

being the

586

C. TRUESDELL

time (possibly

m)

at which the special solution ceases to

exist, or (ii) as m + 0

when

t

is fixed.

We can put the matter roughly in another way. molecular density

F

contains too much information.

not need all that; we should prefer not to know it.

The We do It is

all very well to say that a molecular density is determined by its initial value and then it in turn determines all, but how would we ever decide what initial value to prescribe? What we seek is information that is largely independent of those initial conditions, information that belongs in common to a large class of solutions, the difference between the members of which we should not easily be able to discern in any application to flows of real gases.

We encounter here a group

of problems similar to those that surround "St. Venant's principle" in linear elasticity.

There, as I mentioned in my first

lecture, we wish to eliminate the details concerning the way the grips that twist a rod are applied to the ends.

We are

convinced that these details could not be determined in practice and usually do not matter.

The difference is that the

general theory of boundary-value problems in linear elasticity stands at hand, on call whenever we wish it.

In the kinetic

theory there is no such corpus of analytical theorems.

The

kinetic theory is essentially non-linear, and we have only a few examples and formal calculations on which to base conjectures and intuition.

The last thing we should wish to handle

ANALYSIS B Y RATIONAL THERMOMECHANICS

587

in detail would be a solution of the initial-value problem; certainly, however, we should be able to use with great profit the fact that solutions of it exist and are unique.

A fairly

general proof to this effect would provide the main lemma of the kinetic theory.

For the rest, freely admitting the central

role of the molecular density

F

,

we wish to use its proper-

ties to extract from it essential characteristics of gas flows. While this attitude is clear in the work of MAXWELL and BOLTZMANN, it was first avowed by HILBERT in a masterful paper that even today stands as a lighthouse, though the strong, steady light it has beamed seems to have been black to most eyes.

It was HILBERT who first sought to characterize and

calculate a special class of solutions.

With brilliant in-

sight he proposed a scheme for obtaining these solutions.

His

paper is brief; his explanations are laconic; the claims he put forward for his solutions are not true in general: his work, with one exception, is purely formal, and the exception, the one part where he appeals to his theory of linear integral equations, is marred by a major gap.

Nevertheless, his is the

first systematic attack on the central problems of the kinetic theory.

Whatever his reason for introducing his formal process,

he showed that if there were any solutions of the type he sought they would be determined by the initial values of their own fields

p

, y

, and

E

.

By "determined" I refer here

to the solutions of initial-value problems.

Solutions of the

Maxwell-Boltzmann equation of this kind may be called gas-dynamic,

588

C. TRUESDELL

because the initial-value problem they suggest for and

p

, 5 ,

is of the type we know in classical gas dynamics.

E

HILBERT avowed that he sought gas-dynamic solutions only; only these does his formal process deliver. In unpublished work based in part on an earlier study by GRAD, MUNCASTER has proved that if such solutions exist, their fields

P .”

and

-

q

are grossly determined.

A solution

of the fourth fundamental problem might show that grossly determined solutions are gas-dynamic and that every solution is a member of a family of solutions that are asymptotic to a gas-dynamic solution.

The meaning and value of gross deter-

miners would then be clear.

It may be something of this kind

that the physicists have in mind when they tell us that a kinetic gas has a viscosity and a Maxwell number. Another scheme for approximating special solutions was conceived by ENSKOG.

His work is difficult for mathematicians

to follow, though physicists find it convincing. Looking at it today, we may grasp it in principle by saying that ENSKOG sought to calculate directly, without recourse to any initialvalue problem, the most important quantities the existence of which HILBERT’s work implies, namely,

and

9

.

EI\TSKOG suc-

ceeded in formulating a few steps of an iterative process toward that end. 1-1

and

He himself obtained general expressions for

this way.

CHAP-IAN, using a heuristic approach, had

previously obtained essentially the same expressions for these two quantities.

ANALYSIS BY RATIONAL THERMOMECHANICS

589

HILBERT's work is clear at the general stage, while ENSKOG's is not.

At every stage past the first in HILBERT's

process, we must solve an initial-value problem for a horrid set of partial differential equations.

S o far as

nobody has ever done so for even one stage.

I know,

It would be hard

to start, for even the type to which these equations belong has never been determined; perhaps it varies from one problem to another.

ENSKOG's method, devised to set aside HILBERT's

initial-value problem, has been carried through the first non-trivial step explicitly for a broad class of molecular models, through the second step at the cost of some dubious approximations, and partly through one more.

Beyond the

original work of MAXWELL and BOLTZMANN, ENSKOG's approach to gross determiners is the only one so far that has led to results that appeal to those who perform or analyse experiments.

Both HILBERT and ENSKOG obtained their results through

the intermediary of approximate molecular densities

.

F

The

results we desire, as I see them, do not refer directly to but only to and its flux.

-

s

,

-

P

and

cj

,

F

and also, of course, to the entropy

These latter are proportional to fields

h

and

respectively, defined as follows:

In work still unpublished MUNCASTER has divided the HILBERT-ENSKOG problem into two parts.

Neither HILBERT nor

ENSKOG stated precisely what his process was designed to

590

C. TRUESDELL

approximate.

We may see now that both processes referred only

to grossly determined solutions.

But do grossly determined

solutions exist? MUNCASTER has derived from the MaxwellBoltzmann equation a functional equation that any gross determiner must satisfy.

The celebrated elimination of time deriva-

tives, which has always been the most thorny point in ENSKOG's process of approximation, is effected once and f o r all at the outset, with no reference to approximation. Here is a problem for analysis: to prove that MUNCASTER's equation has a solution, perhaps a unique solution, that is, to prove that one or more gross determiners exist. We may summarize much of the history of the kinetic theory by saying that the applicable solutions are the solutions of the Maxwell-Boltzmann equation that correspond to the solutions of MUNCASTER's equation.

A solution of MUNCASTER's equation is

by itself not a solution of the Maxwell-Boltzmann equation. It becomes such when the fields

p

, y

,

and

E

,

the

independent variables upon which it depends, are made to satisfy the field equations expressing the balance of mass, momentum, and energy: A solution of MUNCASTER's equation is to be used just like a constitutive relation of continuum mechanics.

That

is no surprise, for the results at the first few stages of ENSKOG's process of approximation were always used in that way.

The next step is to calculate the gross determiner. As usual, an existence theorem would help, but none such being known, we can set up a formal process and hope later to prove

A N A L Y S I S BY R A T I O N A L THERMOMECHANICS

a mathematical status for it. &

591

Here MUNCASTER recalled COLEMAN

NOLL's retardation theorem in continuum mechanics.

of stretching the time axis, he held x -

a neighborhood of

.

t

Instead

fixed and stretched

The method is systematic and explicit.

The student may calculate as many terms as he has patience for.

The molecular density need never be determined in full,

as only the terms in its expansion that give rise to terms in the expansion of

-

P

CJ

h

-

s

and

are of interest.

The

results of the first two non-trivial stages confirm and extend those for the gross determiners that were obtained earlier by ENSKOG's process and complete them by adjoining counterparts for s-

and

:

Here to

h

s

q( I

is CHAPMAN

&

COWLING'S approximate gross determiner

and the coefficients 78,

I

denoted by the same symbols in CHAPMAN

O2 &

,

and

B,,

are those

COWLING'S book.

MUN-

CASTER'S formulae for the coefficients are explicit and valid

for all spherically symmetric molecules. The next mathematical problem is to specify and then demonstrate the sense in which MUNCASTER's general series converges.

592

TRUESDELL

C.

Independent of that is another, more important analytical problem arising from the existence of gross determiners: What is the precise status of grossly determined SOlutiOnS? Here we have a lot of examples, but all of them are very special.

The first of these is MAXWELL'S relaxation theorem,

which has been generalized, as I said already, to an estimate of the rates of decay of all the moments of a Maxwellian gas MAXWELL'S theorem is

grossly at rest.

q(t)

= q

This theorem states that

P

and

for any

cj

olution te d to

I

0 both I

as

t+==

as m+O The molecular mass

, ,

T

fixed, and

t

fixed.

enters here because for a Maxwellian

gas Cp=

1.3103...

g = intermolecular force constant. Now

,O

is the solution that corresponds to the particular

initial values

-

P(0) = 0

I

,

q(0) = 0 I

.

This solution is the

ANALYSIS BY R A T I O N A L THERMOMECHANICS

593

only one that the methods of HILBERT, ENSKOG, and MUNCASTER deliver when applied to this problem.

Thus, for this problem,

those methods pick out a special solution, which we may call the principal solution.

The principal solution provides also

a common asymptotic form for all solutions that correspond to the same gross problem. Several other special exact solutions have been discovered in the past twenty years.

They deliver the

-

P

-

and

q

that

correspond to the solution of the initial-value problem for steady shearing, dilatation, cylindrical expansion, and simple extension.

For each of these problems a principal solution is

easily calculated and shown to provide, if parameters measuring departure from equilibrium be small enough, a common asymptotic

form for all the other solutions as long as it and they exist. For simple shearing the principal solution conforms with COLEMAN &

NOLL's theory of viscometry of simple fluids.

In dilatations

and extensions that make the gas denser the solutions exist only for a finite interval of time, the length of which depends on the initial conditions, and the approach to the principal solution is not exponential but like a power of Only one convergence theorem is known.

t

.

It involves a

different iterative process of approximation, which Mr. IKENBERRY &

I introduced as an extension of MAXWELL'S ideas.

It rests on

successive differentiation and makes no use of integral equations. The formal method, which we called Maxwellian iteration, seeks to approximate momenta1 determiners.

It is more easily explained

C. TRUESDELL

594

by an example which is somewhat similar to the differential equations that lead to MAXWELL'S relaxation theorem:

Y

dt

T =

I

const. > 0

Of course the general solution is

y(t)

=

y(O)e

-t/r

The principal solution results if

y(0)

=

0

.

Then

y = 0

.

The Maxwellian iterative scheme is

- Y

(n+l>

~

d

(n) L

dt

Hence

We note at once a major difference between this scheme and those of HILBERT, ENSKOG, and MUNCASTER.

Those force the

zeroth iterate to correspond to a Maxwellian density.

Max-

wellian iteration allows us to use any initial iterate y ( 0 ) we like.

We need not take equilibrium as our starting point.

Glancing back at our formula, we see that for a large class of

Y

(0)

, including all polynomials, Maxwellian iteration

converges to the principal solution. choice of

y(O)

Moreover, with any

it provides ever better asymptotic approxi-

A N A L Y S I S BY R A T I O N A L THERMOMECHANICS

mations to the principal solution as

T +

0

.

595

If it seems

surprising to approximate by repeated differentiation, notice that the principal solution is a surprising solution, the only solution that when T

at

T

.

T

= 0

.

t

is fixed is a continuous function of

Every Maxwellian iterate is a polynomial in

After all, differentiation is a good way to get rid of

initial conditions, and in the kinetic theory it is just the initial conditions that we do not wish to have to know if we can help it. Coming back to convergence, let me say only that a variant of Maxwellian iteration when applied to simple shearing of a Maxwellian gas does converge if the amount of shearing is small enough, and that it diverges otherwise; it converges, moreover, to the gross determiner of that problem.

At present this theorem,

established in 1956, is the only convergence theorem ever published for any of the processes designed to approximate gross or momenta1 determiners in the kinetic theory. Maxwellian iteration has one other virtue.

Everyone knows

that according to FOURIER'S linear theory of heat conduction disturbances of temperature travel at infinite speed.

Everyone

knows also that according to the molecular view of matter, this cannot be s o , for heat must be transmitted through mutual actions of molecules a finite distance apart.

The results of ENSKOG's

process share this defect of FOURIER'S theory, €or they make the temperature satisfy a parabolic differential equation when the fields of velocity and density are regarded as known.

The

596

C.

TRUESDELL

second stage of Maxwellian iteration, on the contrary, makes the temperature satisfy a differential equation of second order in time and space jointly, and according to that equation disturbances of temperature generally propagate at finite speeds.

ANALYSIS BY RATIONAL THERMOMECHANICS

597

References for further reading

S . CHAPMAN & T.G. COWLING, The Mathematical Theory of Non-

Uniform Gases, Cambridge, Cambridge University Press, 1939, and later editions.

H. GRAD, "Principles of the kinetic theory of gases," in

_-

Encyclopedia of Physics 12 (ed. S . FLUGGE), Berlin

etc. ,

Springer-Verlag, 19 58.

C. TRUESDELL, Mathematical Aspects of the Kinetic Theory of Gases (Notas de Matemstica Fisica, Vol. III), Institute de Matemdtica UFRJ, Rio de Janeiro, 1973. C. TRUESDELL

&.

R. G. MUNCASTER, Fundamentals of Maxwell's

Kinetic Theory of a Simple Gas, treated as a Branch of Rational Mechanics, now being polished for publication by Academic Press in its series "Pure and Applied Mathematics".

Acknowledgement.

I am obliged to Mr. MUNCASTER for criticism

of the text as delivered in Rio de Janeiro.

C. TRUESDELL

598

Epi l og ue The many f i n e l e c t u r e s w e have h e a r d a t t h i s meeting have convinced you, I hope, i f you were n o t a l r e a d y convinced, t h a t r a t i o n a l thermomechanics o f f e r s something of i n t e r e s t

-

t h a t many of i t s q u e s t i o n s now l e a d t o deep and d i f f i c u l t problems of p u r e a n a l y s i s , t o f o r m u l a t e which o u r modern c o n c e p t s and methods may b e u s e f u l o r even e s s e n t i a l . o f u s i s too busy w i t h h i s own work t o do much e l s e .

Each Few of u s

c a n t u r n a s i d e t o s t u d y r e g u l a r l y and t h o r o u g h l y a n o t h e r d i s c i p l i n e , even a c l o s e n e i g h b o r t o o u r own.

Young p e o p l e ,

p e o p l e who a r e j u s t s t a r t i n g t h e i r s e r i o u s s t u d y , have more

t i m e and p e r h a p s a l s o more energy.

I a s k t h e eminent a n a l y s t s

h e r e p r e s e n t t o encourage some of t h e i r s t u d e n t s t o s t u d y modern r a t i o n a l thermomechanics s e r i o u s l y

- as t h e y might

s t u d y c o m b i n a t o r i a l topology o r l a t t i c e t h e o r y , n o t necess a r i l y t o s p e c i a l i z e i n t h o s e f i e l d s , b u t because of i n t r i n s i c i n t e r e s t and p o t e n t i a l l a t e r use.

A f t e r a l l , none of us c a n

a p p l y knowledge h e does n o t have. But more t h a n p u t a t i v e a p p l i c a t i o n i s a t i s s u e .

a m a t t e r of method.

It is

The mathematics through which p r o g r e s s

i n mechanics w a s made i n t h e p a s t grew i n two ways: 1. A t o o l i s a v a i l a b l e .

T h a t i s l i k e a n equipped machine shop s e e k i n g

solve. orders

W e look f o r problems i t can

.

2. A problem a r i s e s n a t u r a l l y .

matics t h a t w i l l s o l v e i t .

W e seek t o create mathe-

Then we have t o d e s i g n

599

A N A L Y S I S BY R A T I O N A L THERMOMECHANICS

and c o n s t r u c t t h e machines a n d o r g a n i z e t h e shop. Both a v e n u e s h a v e b e e n f r u i t f u l .

F o r p r e c e d e n t w e may l o o k

t o o u r masters o f e a r l i e r t i m e s .

EULER, who was t h e g r e a t e s t

o f a l l m e c h a n i c i s t s as w e l l as t h e g r e a t e s t o f a l l m a t h e m a t i c i a n s , p r o v i d e s u s w i t h m a j o r examples o f b o t h .

T o s o l v e t h e problem

of t h e v i b r a t i n g s t r i n g , h e had t o c o n c e i v e a d e f i n i t i o n of f u n c t i o n f a r more g e n e r a l t h a n any t h e r e t o f o r e p r o p o s e d : a mapping of a s u b s e t of t h e r e a l numbers i n t o t h e r e a l

numbers.

T h i s c o n c e p t o n c e s p e c i f i e d , h e had t o r e c o g n i z e

t h a t t h e s o l u t i o n of a p a r t i a l d i f f e r e n t i a l e q u a t i o n i n t h e i n t e r i o r o f a r e g i o n c o u l d n o t be e x p e c t e d t o s a t i s f y t h a t e q u a t i o n a t o r beyond b o u n d a r i e s .

Once t h e s e i d e a s , a b s o l u t e -

l y new f o r h i s d a y , were c l e a r t o him ( n o t w i t h s t a n d i n g t h e f a c t t h a t nobody e l s e t h e n g r a s p e d t h e m ) , h e q u i c k l y s e t t l e d a l l t h e main p r o b l e m s a s s o c i a t e d w i t h t h e o n e - d i m e n s i o n a l l i n e a r wave e q u a t i o n , u s i n g h i s s o l u t i o n i n terms o f a r b i t r a r y functions.

Emboldened t h e r e b y , h e s o u g h t o t h e r problems o f

m e c h a n i c s t h a t c o u l d b e s o l v e d by t h e same method. many.

H e found

When, t o d a y , w e l o o k a t h i s work on t h e p r o p a g a t i o n of

sound i n c o n i c a l o r h y p e r b o l o i d a l h o r n s , a n d i n f i n i t e l y many o t h e r f o r m s b e s i d e s , w e are a s t o n i s h e d a t what h e c o u l d do. N e v e r t h e l e s s , h i s method w a s i n h e r e n t l y i n c a p a b l e o f t r e a t i n g a t u b e o f g e n e r a l c r o s s - s e c t i o n o r a s t r i n g o f a r b i t r a r y massdensity.

Deeper a n a l y s i s was n e c e s s a r y .

I n t i m e i t came:

t h e Sturm-Liouville theory, expansions i n proper functions, and RIEMANN's t h e o r y o f h y p e r b o l i c p a r t i a l d i f f e r e n t i a l

600

C. TRUESDELL

equations i n t w o var iables.

T h i s work, a t a l e v e l d e e p e r t h a n

EULER's e x t e n s i o n of h i s i n i t i a l triumph, d i d n o t s e e k problems t o which a known method would a p p l y .

Rather, it faced t h e

problems l e f t unsolved by EULER and found more powerful, e n t i r e l y new methods. I do n o t b e l i t t l e t h e former and easier approach, b u t I

c o n s i d e r it i n f e r i o r t o t h e l a t t e r .

I n t h e end i t i s

the

problem t h a t must b e s o l v e d . HOW,

t h e n , do w e r e c o g n i z e a problem?

O f course, a

s o c i a l n e c e s s i t y o r , what t o d a y seems t o be t h e same t h i n g , a sum of money a t hand, can make u s t u r n a s i d e from t h e

s e r i o u s business of our s c i e n c e , b u t only f o r a l i t t l e while. The s t r u c t u r e of a t h e o r y d i c t a t e s i t s problems away, b u t one a f t e r t h e o t h e r .

- not r i g h t

Here a g a i n t h e o l d mechanics,

t h e grand mechanics of t h e e i g h t e e n t h c e n t u r y , i s o u r g u i d e . The h i s t o r y of c l a s s i c a l mechanics i s a h i s t o r y of s p e c i a l problems, b u t n o t j u s t any problems, o r problems s e l e c t e d because s u b s e r v i e n t t o some known a n a l y t i c a l method, o r problems e p h e m e r a l l y r e g a r d e d as h e l p f u l t o mankind o r t o t h e s t a t e and hence worth s p e n d i n g money o n , b u t problems germane t o t h e d o c t r i n e .

I n t h e e i g h t e e n t h c e n t u r y a key

problem s o l v e d l e d always t o a new key problem posed. H e r e t a s t e i s of t h e e s s e n c e .

Taste i s n o t l e a r n e d

from sermons b u t from e x p e r i e n c e - c o s t l y p u r c h a s e , s t u d y , e n l i g h t e n m e n t and d i s a p p o i n t m e n t , loss, new p u r c h a s e w i t h a s h a r p e r e y e , and so on, j u s t l i k e a c q u i r i n g works of a r t .

A N A L Y S I S BY R A T I O N A L THERMOMECHANICS

Good analysis takes time as well as talent.

601

Neither

talent nor time should be wasted on small matters.

Let me

take the v. K&m&

An analyst

theory of plates as an example.

may regard that theory as handed down by some higher power (a Hungarian wizard, say) and study it as a matter of pure analysis. To do so for the v. K&m&

theory is particularly tempting

because nobody can make sense out of the "derivations". If ever there was a case for the inspirations of a mad genius who like the Pythian priestess discloses the truth in riddles that can be interpreted only after the predicted events have occurred, it is provided by the v. K&m&

theory.

I mention

it for two reasons: Analysts seem to love it, and it makes no sense to critical students of mechanics. That is not to say that the theory is bad.

Rather,

whether it is good or bad is assessable only after its predictions shall have been compared with a rational theory. Being unable to explain just why the v. K&m&

theory has al-

ways made me feel a little nauseated as well as very slow and stupid, I asked an expert, Mr. ANTMAN, what was wrong with it.

I can do no better than paraphrase what he told me:

It relies

upon 1) "approximate geometry", the validity of which is assessable only in terms of some other theory. 2) assumptions about the way the stress varies over

a cross-section, assumptions that could be justified only in terms of some other theory.

C. TRUESDELL

602

3) commitment to some specific linear constitutive

relation - linear, that is, in some special measure of strain, while such approximate linearity should be the outcome, not the basis, of a theory.

4) neglect of some components of strain - again, something that should be proved mathematically from an overriding, self-consistent theory.

5) an apparent confusion of the referential and spatial descriptions - a confusion that is easily justified for classical linearised elasticity but here is carried over unquestioned, in contrast with all recent studies of the elasticity of finite deformations. These objections do not prove that anything is wrong with

v. j ? & ~ N ' s strange theory.

They merely suggest that it would

be difficult to prove that there is anything right about it. The basic fallacy, Mr. ANTMAN remarks, lies in the following argument:

"The physical system being modelled has

small solutions: therefore, terms that are very small when the solution is small may be discarded from the equations." The fallacy, says Mr. ANTMAN, is that "the discarded terms may be the very ones that would keep the solutions small, and in their absence the emasculated equations may fail to describe what they were intended to.

I'

Moderncontinuum mechanics is a branch of mathematics. It restsupon accurate geometry, precise concepts of stress and force and power, and systematic, principled development of

A N A L Y S I S BY R A T I O N A L THERMOMECHANICS

constitutive relations.

603

The worst danger of the v. K&m&

theory lies in its magical quality.

Those who devote their

efforts to it are tempted to doubt that mechanics is a mathematical science.

It is easy for them to forget that particu-

lar theories of mechanics come inexorably from clear, mathematical assumptions expressed in terms of concepts just as precise as those every mathematician today associates with geometry. Theories that require intensive pseudophysical indoctrination are not mathematical theories.

They are religions -

religions that are not made truer by their claim to be "science" because of their mere godlessness and amorality.

Mathematics

is the province of the mind - trained, indeed, wary, and armed, but still naked, like LEONIDAS and his Spartans at Thermopylae.

Acknowledgement.

The research reported here is a part of a

program generously supported by the U . S . Foundation since 1960.

National Science

G.M. de La Penha, L.A. Medeiros (eds.) Contemporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations @North-Holland Publishing Company (1978)

SOME SIMPLIFIED EQUATIONS FROM THE THEORY OF MIXTURES

w. 0.

WILLIAMS

Department of Mathemat ics Carnegie-Mellon University Pittsburgh, Pa 15213 Introduction. The theory of mixtures is based on a n idea which is both obvious

-

at least after the fact

-

and appealing.

One

models a mixture as a number, let us agree to say as usual that this number is two, a number of continuous bodies superposed in space.

The form of the balance equations for each

of the bodies is then taken to b e essentially the same as for

In the balance of forces, with which we

an isolated body.

will be concerned, one adds in the end only one volume-dense interaction force, although of course each of the usual terms must be interpreted differently than in the standard case. Thus, all of the many mixture theories presented since Truesdell's systematization of the theory [l] have individual momentum equations of the form

Here

TA, TB

body forces,

are the partial stresses, pA,

pB

accelerations, and constituents

A

e bA, ki

the mixture densities,

kAB

and

B.

the external

g A , gB

the

the net interaction force of the Balance of mass, ignoring the

SOME SIMPLIFIED EQUATIONS FROM

THE THEORY OF MIXTURES

605

possibility of exchange of mass, is even more familiar:

Here

v v -A’ -B

p i

+

div(p A-A v ) = 0,

pb

+

div(p

v ) = 0.

B-B

are the velocities of the constituents.

The significant

differences in the manifold mixture

theories arise at the stage of formulating constitutive equations.

Because the interactions of the constituents are

not accessible to measurement or observation, this formulatim cannot be guided directly by experience. procedure

The only rational

is t o adopt the rule o f equipresence and then to

use the second law of thermodynamics to reduce the complexity of the equations.

Problems arise: there is not yet universal

agreement as to exactly what form the second law should take, SO

that various sets of reduced constitutive equations are

derived from the same initial equations, and, more unfortunately, even these reduced forms are of such generality andinvolve so many inaccessible quantities that they are difficult even

of qualitative analysis.

In general it is only after

significant linearization that it is possible to obtain sets of equations which admit elementary solutions, let alone their

being experimentally deterministic.

However this process of

obtaining constitutive equations is preferable to the only other alternative process, which consists of arguments leading from microscopic models of the mixture to continuous models. The transition here involves averaging operations which are, except perhaps in the kinetic theory of gases, not made precise and probably cannot be made precise.

Moreover I

maintain that since even the interpretation of the partial

606

W. 0. W I L L I A M S

stresses i n terms of elementary forces is not well understood, such a

procedure is essentially random. M y point, then, is that the theory of constitutive

equations f o r mixtures is not well understood.

I t follows

that the justification of a mixture theory cannot arise from observing the final form of the constitutive relations (indeed, the best argument often advanced for support of a particular set of such relations seems to b e that they reduce when linearized to Darcy’s law).

The appropriate course f o r

the mixture theorists i s to seek, in the most rational possible way, simple constitutive equations for which one may obtain solutions t o the balance equations and then to justify the theory through comparison of these solutions to known phenomena.

It is pleasing to see increasing numbers of

examples of this in the literature.

In this paper I present two sets o f constitutive equations basedon the formulation of the thermodynamics of mixtures by R.

Sampaio and myself

121.

T h e peculiar virtue

of these models is that they are quite simple; they involve only coefficients which should be experimentally accessible. But the observation about them which I think is most significant is that although they are linear constitutive equations, the equations of motion i n both cases turn out to be non-linear, unavoidably so, and thus it is at least very plausible that any more complicated, and hence more realistic, constitutive model should lead to similar non-linearities. Before presenting the equations I introduce some further terminology which allows more insight into the nature

SOME SIMPLIFIED E Q U A T I O N S FROM THE TIIEORY O F MIXTURES

607

I n t h e t h e o r y which I u s e , b a s e d on a

of t h e e q u a t i o n s .

s y s t e m a t i c d e r i v a t i o n f r o m g l o b a l laws o f b a l a n c e ,

the

momentum e q u a k i o n s have t h e form

represents

Here t h e t e n s o r

SA

constituent

SAB t h e

A

ween by

B

A,

and on

B If

A.

( s u r f a c e ) f o r c e of i n t e r a c t i o n b e t -

bAB

and

8

the

8

by

A.

n,

1s

then

But if

o f t h e volume of

fraction

(volume-dense) f o r c e e x e r t e d

i s a s u r f a c e i n t h e s p a c e o c c u p i e d by

t h e b o d i e s , w i t h normal v e c t o r transmitted across

t h e f o r c e t r a n s m i t t e d through

A

S n A-

i s the f o r c e

occupies only a

t h e r e g i o n then

SA/cpA

b e t h e " t r u e s t r e s s " c a r r i e d by t h e b u l k m a t e r i a l o f Similarly i f

-p A

were t h e b u l k d e n s i t y of

A

then

would A.

pA =

cpApA.

F i n a l l y we may d e f i n e

TA = SA

1 - S

2

1

TB =

2AB =

+ +

1

2

2

AB'

'ABP

(+AB-ijBA),

t o o b t a i n (1).

A Mixture o f Incompressible Viscous F l u i d s .

I n a l l m i x t u r e t h e o r i e s f o r t h i s s i t u a t i o n one f i n d s equations o f t h e f o r m

608

W.O.

WILLIAMS

p r o v i d e d t h e p a r t i a l s t r e s s e s a r e assumed Here

I’

-e

removal o f

t o b e symmetric.

d e n o t e s t h e o p e r a t i o n s o f s y m m e t r i z a t i o n and the trace.

The d i f f e r e n c e b e t w e e n t h e o r i e s involves

only t h e p r e c i s e f o r m given t o

p A , pB.

I t i s common, i n

s e e k i n g s o l u t i o n s t o t h e c o r r e s p o n d i n g momentum e q u a t i o n s , t o suppose t h e v i s c o s i t i e s

p2

been observed i n experiments

t o be constant.

I n f a c t , i t has

o v e r a p e r i o d o f more t h a n 50

years t h a t the v i s c o s i t i e s a r e not constant, but vary i n a c o m p l i c a t e d way w i t h c o m p o s i t i o n . With o u r v e r s i o n o f

t h e s e c o n d law of

thermodynamics

one f i n d s [ 2 , 3 ] SA = - p A l

+

+ 2 p A vv -A’

Experimental r e s u l t s , as w e l l a s h e u r i s t i c r e a s o n i n g , s u p p o r t t h e f o l l o w i n g f o r m u l a s , o r i g i n a l l y d e r i v e d by c r u d e k i n e t i c t h e o r y arguments [ 31 :

where and

‘IA

QAB’

and

nB,

the v i s c o s i t i e s of

t h e unmixed

liquids,

t h e m u t u a l v i s c o s i t y , m a y r e a s o n a b l y be t a k e n a s

constant.

Incompressibility arguments relate pA’ total, indeterminant pressure p. l i n e a r i z e d e q u a t i o n s for have t h e form

bAB

P B ~P A B t o t h e

Using these relations and

the equations for the mixture

SOME SIMPLIFIED EQUATIONS FROM THE THEORY OF MIXTURES

with similar equations for constituent B.

609

Clearly, if

variations in the composition are to be considered the

If variations i n composition are

equations are non-linear.

not expected to occur then even the use o f mixture theory is a n over complication in the solution of the problem.

Flow of an Incompressible Viscous Fluid through a Rigid Porous Medium. We deal here with an even simpler situation.

If the

solid constituent is rigid then we need deal only with the equations of motion of the fluid. emerge from our theory

The linear equations which

[2,41 are T = -pl + ;sI %Vx,

2

=

-ax + pvcp.

The equations of motion are

Guided by the results for mixtures o f fluids we take

where

X,

presumed constant, is a v.scosity augmentation

factorwhichcan be determined from experiments on flow of fluid o v e r a porous s l a b [ 41

.

To proceed further, consider first a situation in

610

W.O.

WILLIAMS

which the acceleration and the viscous effects may be ignored, and i n which the uniformly porous medium is saturated with the fluid, so that

Vcp

vanishes.

+ ax

-vp

This is Darcy’s law.

Then

0.

=

Accounting for the usual interpretations

one finds

a = -cp &rl K

where

K

’

is the specific permeability, usually taken as

constant. Similarly, i f the pressure is assumed uniform and the acceleration and viscosity of negligible effect we have

-ax

+

BOY =

0

which with ( 2 ) , yields Fick’s law of diffusion.

Hence

B = where

D

is the diffusivity, usually supposed constant o r

proportional to some power o f

cp.

Thus we have for ( 2 )

Again the equations are non-linear. Boundary Conditions. Finally I may point out that formulation of practical boundary conditions is very difficult i n mixture theory.

Let us look at the porous medium flow.

If the

boundary is a surface at which the porous medium terminates

SOME SIMPLIFIED EQUATIONS FROM TIE TILEORY OF' MIXTURES

then t h e r e a r e s e v e r a l conceivable s i t u a t i o n s . be f r e e t o f l o w o u t i n t o a r e s e r v o i r , o r

611

The f l u i d may

i n t o f r e e s p a c e , or

t o r e t r e a t away from t h e b o u n d a r y , o r i t may e x t e n d p r e c i s e l y t o t h e b o u n d a r y w i t h t h e f l u i d r e s t r a i n e d by c a p i l l a r y a t t r a c t i o n or by a n impermeable l a y e r . C o n s i d e r i n g j u s t t h e r e s e r v o i r s i t u a t i o n we h a v e s e v e r a l amusing r e s u l t s

cpv_-n,

that

with

normal f l o w of

C o n t i n u i t y o f t h e f l u i d shows

t h e n o r m a l v e c t o r , must e q u a l t h e

the f l u i d i n the reservoir.

A rough c a l c u l a t -

t h a t perhaps i t i s even r e a s o n a b l e t o r e q u i r e

ation indicates

cpv_

2

[4].

e q u a l t h e v e l o c i t y o f t h e r e s e r v o i r f l u i d a t t h e boundary

(which i s n o t n e c e s s a r i l y normal t o t h e boundary). ing of

s h e a r i n g f o r c e s shows t h a t

normal d e r i v a t i v e of f o l l o w s , even i f

Xcpvy;

must b e e q u a l t o t h e

the r e s e r v o i r v e l o c i t y .

X = 1,

that

A match-

As a result it

t h e r e must b e a t r a n s f e r o f

s h e a r s t r e s s from t h e s o l i d t o t h e e n t r a p p e d f l u i d , concent r a t e d on t h e b o u n d a r y .

Thus i t i s e s s e n t i a l t o a l l o w a

s i n g u l a r i t y i n t h e s t r e s s a t t h e boundary

(cf.

[51).

Equating

normal f o r c e s r e v e a l s t h a t t h e r e s e r v o i r p r e s s u r e must e q u a l 2

cpp

+

cp(l-cp')p

V 7 ~ _ . g-

u n l e s s s u r f a c e e f f e c t s such a s s u r f a c e

compaction a r e s i g n i f i c a n t . The f r e e b o u n d a r y c o n d i t i o n s , i n c l u d i n g t h e s u r f a c e e f f e c t o f c a p i l l a r y f o r c e s , i s even more c o m p l i c a t e d t o model: t o my knowledge t h e s u r f a c e c a p i l l a r y t e r m s have b e e n c o n s i d e r e d i n no p o r o u s media t h e o r y .

F i n a l Comments.

I must acknowledge t h a t

f l o w model i s e n t i r e l y o v e r s i m p l i f i e d .

t h e p o r o u s medium The m o s t i m p o r t a n t

d e f e c t i s t h a t c a p i l l a r y f o r c e s a r e o n l y c r u d e l y modeled by

612

the

W. 0. W I L L I A M S

pvcp

term.

A linear model can never be expected to

predict such important effects as the hysteresis of imbibitiondrainage or the self-limiting of diffusion o f finite volumes of fluid.

For this reason the equations must suffer further

complication in order to be brought into a respectable form.

References [l] Truesdell, C.,

Sulle basi della termomechanica, Atti

Accad. Naz. Lincei Re. Ser 8, [2]

Sampaio, R . & W.O.

22,

3 3 , 1957.

Williams, Thermodynamics of a diffusing

mixture, J. de Mgcanique, t o appear.

[ 3 ] Sampaio, R. & W.0.

Williams, O n the viscosities of liquid

mixtures, ZAMP, 28, 607-619, 1977.

[4]

Williams,

w.o.,

Constitutive equations for flow of an

incompressible viscous fluid through a porous medium,

C51 Williams,

0.

w.o.,

Appl. N a t h . ,

t o appear.

A note on surfaces of separation in

mixtures, ZAMP,

3, 516-568, 1975.

AUTHOR INDEX

Antman, S.S., 1 hvila, G.S.S., 25

Lions, J . L . , Maccamy, R . C . ,

Baillon, J.B., 37 Benjamin, T.B., 45 Brezis, H., 74, 81

347

Noll, W., 363 Nowosad, P., 388

Cardoso, F., 90 0 Carrol, M., 102 Chadam, J.M., 37 Coleman, B.D., 135 Costa, D.G., 25

Osher, S., 396 Otterson, P., 466 Pironneau, P., 171 Podio Guidugli, P., 225 Puel, J.P. 400

Do, C . , 112 Fosdick, R . L . ,

284

143

Serrin, J., 411 Strauss, W.A., 452 Svetlichny, G., 466

Glowinski, R . , 171 Gurtin, M . E . , 237 Hounie, J., 90

Tartar, L . , 472 Temam, R., 485 Truesdell, C., 495

Joseph, D.D., 254

Williams, W.O., 604

613

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