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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