A Nonlinear Theory Of Generalized Functions

Lecture Notes in Mathematics Edited by A. Dold, B. Eckmann and E Takens

1421 Hebe A. Biagioni

A Nonlinear Theory of Generalized Functions

Springer-Verlag Berlin Heidelberg NewYork London ParisTokyo Hong Kong

Author

Hebe de Azevedo Biagioni Departamento de Matem~.tica Universidade Estadual de Campinas Caixa Postal 6065 13081 Campinas, S P - Brasil

Mathematics Subject Classification (1980): Primary: 46F10 Secondary: 35D05, 35L60, 35L67, 35K55, 65M05, 65M10, 73D05, 73J06, 7 6 L 0 5 ISI3N 3-540-52408-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-52408-8 Springer-Verlag N e w Y o r k Berlin Heidelberg

This work is subiect to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1990 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper

PREFACE

This 1987

and

tica

of

has

book

is

the

reproduced

in

1988

the

Universidade

benefitted

different. more

OF

from

The

recent

of

SECOND

second in

EDITION

edition

the

Estadual

a few

set

THE

preprint de

a text series

Campinas.

improvements

references

of

has

but

Notas

This

it

however

written

1986,

de M a t e m a -

second

is not been

in

edition

substantially

enriched

by

many

papers.

INTRODUCTION

In on

the

that

the

space

there

been

of

tory

results,

addition

of

problem

underlying years

all

natural

form

constructed.

involving The

ago,

relative Recently

ly w e l l - a d a p t e d

to such aim

the

and At

even

to

the

been

in c l a s s i c a l cases of

to

(of

the

alge-

time

"illegal

leads

numerical

it

multi-

theories

to

satisfac-

codes

used

that

they

a solution

algebra

course

one

impossibility

recognized of

the

in should

in f o r m

of

theory.

a differential

solution

to

physical

this

find

having

that

to m a t h e m a t i c i a n s

try

Schwartz'

using

prov-

A containing

and

differentiation

consisting

it was

line)

multiplication. had

result

distributions",

algebra

to

properties to

real

a celebrated

and

mathematical

Seven having

(on

In m a n y

as

of

a differential

suggests

so

published

multiplication

physicists

them

situation

the

a suitable

that

Mechanics.

some

This

reconsider

the exist

distributions",

Continuum

industry.

L.Schwartz

relative

of

recognized

as

weakened

of not

properties

such

ing

1954

distributions

operations

plications

and

does

~' of

classical

had

year

"impossibility

ing

braic

the

that

problems

this of

~ of

containing them

is

result), theory

physics

in has

was

and

~', a been

perfect-

engineer-

multiplications. of

this

book

is

to p r o v i d e

a simple

introduction

iV

to this

nonlinear

J.F.Colombeau. physics, tions,

pure of

Now

passing

both

from

the

mathematics:

main

physics:

this

agreement

large

one

sical

the

of

corresponding

it p r o v i d e s

the

suggests to n e w

a faithful

to equa-

viewpoints.

generalization

encompassing

a synthesis

by

mathematics

differential

These

more

all

of

its

of m o s t

existing

when

they

been

to the

of

physics

involve

"ambiguous

precise

differential

have

no

solutions

checked

equations:

solutions

case

these

instance

solutions

of

formulas,

the in

new

for

in the

setting

by

the

of e q u a t i o n s solutions

piecewise

expected

in this

results

are c o h e r e n t w i t h

In the

that

for

results"

formulations

existence-uniqueness

exist.

functions,

give

facts.

which

new

equations and

(and u n a m b i g u o u s )

of p a r t i a l general

equations

often

classical

presents

experimental

solutions

solutions

it has

fact

and

can o b t a i n

classes

pure

introduced

the n u m e r i c a l

C ® functions,

in w h i c h

leading

of d i s t r i b u t i o n s .

from

of p a r t i a l

and

of d i s t r i b u t i o n s "

allows

theoretical theory

of

functions

distributions.

cases

thus with

extends

theory

theory

Further

in some

theory

the

theoretical

the

of

"multiplications

equations,

theory

theory

properties.

multiplications

this

of g e n e r a l i z e d

through

the c l a s s i c a l

ics

theory

clasof p h y ~

are

in

C" f u n c t i o n s

physicists

and

,

engi-

neers.

numerical which

solutions:

permit

(numerical

form,

theory

this

is m a d e

one

systems

in n o n c o n s e r v a t i v e

obtains

in this

classical

system

the

very

of f l u i d

of

important

form

new numerical

systems

for

treatment

more

to t r a n s f o r m

way

provides

solutions

of c o l l i s i o n s

of m a s t e r i n g which

allows

theory

to c o m p u t e

simulations

possibility tive

one

instance).

of

systems

by

the

conservative (in c e r t a i n

efficient

numerical

dynamics).

used

methods

in i n d u s t r y

It g i v e s

the

in n o n c o n s e r v a -

fact

that

this

systems

into

equivalent

circumstances schemes

even

one for

the

This now

were

only

maticians,

it m a y

ration

an

notions

over

very

of

preprint

form.

perbolic

systems,

conservative fore

not

be

useful

for

its

reading

we

should

theory

basic

points

We

give

this

facilitate

and

simplest most

some

and

will

them:

useful

it

them

to

of

is

the

integ-

recent

we

give

still

systems

in hy-

in

schemes.

a wide

and

to

semilinear

numerical

articles

The

reduced theory

appl !

presentation.

of

equations, new

be

of

of

its

understood;

its

until mathe-

engineers.

the

and

formulas

reading

and

using

of

that for

from

En . S i n c e

widely

parabolic

survey

the

theory

essentially

space not

results intended

recent

a sketch

new

are

calculus

is

more

nonlinear

form,

believe

yet

the

physicists

euclidean

this

It

dissociate

differential

are

is

do

of

the

and

form.

we

n-dimensional

applications

concepts

article

for

original,

account

The

also

basic

in

prerequisites

classical

and

presents

but, since

cations, needed

text

published

books

non-

There-

audience on

and

this

sub-

ject.

Chapter on

an

arbitrary

we

sketch

C=

or

for

the

convex

acquainted

with

tion

be

obtained, We

also

in

1.6

texts

in on

ordinate tion the

a natural

a way this

products, dimension

space.

In

well §1.8

of

has

of

to

the

who

is

the

remainder

not

this

dimensional

gener~lized space

of

familiar

the

this

natural

simplifications

an

elementary

definition

which

is

significantly

We

obtain

free

En).

as

concepts

we

In

define

of

§1.7 of a

the we

set

desired

subspaces

of

generalized

strong

subspace

and of

natural

weak ~(~),

sec-

can

be

viewpoint. the

first in

the

and

§§

other

properties

to

introduce

concept trouble

this

studied from

and

readers

concept

of

different the

restrictions

independence of

all

For

1.1

a study

the

believe

is

§

~(~)

any

mathematical

then

subject.

from

with

book. we

functions

. In

spaces

without

original

a purely

~n

concept convex

section

spaces,

how

from

this

locally

drop

explain

these

euclidean

obtained

over

way,

to

the

may

successive

and

invariance,

• (Q), as

~

reader

infinite

useful

expose

construction to

the spaces

understanding

might

introduction

functions

Fortunately

locally

an

subset

J.F.Colombeau

holomorphic

~' ( ~ ) of

how

I is open

(co-

composi-

numbers

of

topologies

convergence ~s (~), w h i c h

in

on this

al-

1.2

of

VI

though

simpler

applications space of

~' (~)of

~s (~) m a y

theless cal

than

the

fying

proofs;

the

slightly ized

We

more

the

technicalities

in

of

previous

definitions

various

Chapter

authors

in

of

sarily

open

tions

these

on

been

published

Whitney sion

which

C®

tions.

The

classical particular

we

cases:

when

proof

due

nonlinear

in n e w and

make

formulas

engineers.

Hooke's

law

form.

a

generalclassical additional

2 and

theory

3.

In

unifies

distributions

is

theory

the the

proposed

generalizing 2 is

recall

the

the

by

and

without for

proof

an

extension

theorem

(§2.4).

reduced

to

X is

a closed

half

to

(by

neces

this

the

We

C®

func-

natural

has

not

yet

of

the

Whitney

exten-

generalized

analysis

of

prove

a single

space

defi-

not

concept

Colombeau's

from

the

any

which

classical

state

that

to

classical

devoted

holds

obtained

is

easily

applications

We

result

applications

partial and In

new

numerical

elasticity

that

are

exposed

differential

in n o n c o n s e r v a t i v e

it e v i d e n t

in w h i c h

in C h a p t e r s

when is

modi-

of

in d e t a i l

point

following

func

a proof

(Borel's a very

to R . T . S e e l e y ) .

Several cern

X

by

(§1.10)

straightforward

extends

Chapter

some

on X and

Whitney

and

thus

in

sketch

~(~)

Neverphysi-

it g i v e s

just

automatically

of

elements

some

since ~(~

prefer

to d i s t r i b u t i o n

sense.

in b o o k

in

Colombeau's

functions

~n

is u s e d

proof

the

simple

X of

A similar

two

theorem)

relative

functions

theorem.

in

results how

book

might of

for

the

the

cases.

generalized

X in W h i t n e y ' s

extension,

of

expose

special

subset

this

of

of

distribution.

adequate

are

consists

most

~s (~) : s e v e r a l

worked

One

for

inclusion

given

of m u l t i p l i c a t i o n s

A novelty nition

in

have

way.

proofs

I we

any more

equations

to pay

some

Appendix

in

~

definition

algebraic

price

on

~s (~)

in a r o u t i n e

of

natural

and

could

sophisticated

solutions

solutions;

in

we

setting

is no

well

clearer

work

anyway

proofs

There

equally

is p e r h a p s

applications.

, is a s u f f i c i e n t 3.

distributions

represent

this

shorter

~(~)

in C h a p t e r

the

and

schemes

of

some

of

see

Appendix

partial

3.

of

interest

elastoplasticity

form,

systems

in C h a p t e r

equations;

They

them

con-

consist

to p h y s i c i s t s

engineers

state

I. N u m e r i c a l

differential

tests

equations

VII

thus by

obtained

have

classical

nonconservative not

make

ized

Roux

§§3.1

for

to

for

how

agreement

Appendix

2.

In

§3.3

of h y d r o d y n a m i c s

which

which

gives

to n e w

These

new

formulas

experimental of

makes that of

very is

papers

dimensions.

In

we

bolic

present

such

We

an with

we

only

more

specialized

ably

self-contained

In

applications

described

a few

and

for

We

in

schemes

a general with

the we

reader

two

data

distributions. solutions

and

agree

with

the

classical and

we

fact

interest only

to

sketch

special-

three

space result

distributions.

for It

in

the

which

existence-uniqueness

result

results

book

the

in

Cauchy

have

simpler

with

obvious

hence

this

refer

do not

the

the

3.

simulations

microseconds,

(besides

and

in A p p e n d i x

only

course

analysis,

formulation

last

expensive)

show

formulation

in a g r e e m e n t

Of

we

numerical

classical

show

discontinuous

numerical

existence-uniqueness

obtain

give

the

difficult

data

of

we

§3.2

with

for

systems

Cauchy

In

Le

compute

are

extremely

present

equations

solutions

exist.

we

form.

be

given

3.4

can

used

dimension.

results

hyperbolic

equation

general

more §3.5

semilinear

~§3.6

the

for

and

we

is

(nonconservative)

to

schemes

simulations. space

can

are

They

phenomena

in one

§§3.1 how

or

Colombeau,

theory

problems

schemes,

numerical

by

their In

does

general-

problems

observations a new

numerical

experimentation numerical

the

obtain

experimentation

schemes

4.

Cauchy

equivalent

the

these

for

these

solve

such

a few ized

we is

and

and

in

"meaningless"

recently of

are

solutions

to

these

done

elastoplasticity

with

observations.

collisions;

a sketch

may

in

rise

and

2,3

mathematically

systems

Colombeau's

in n o n c o n s e r v a t i v e

we

data,

rise

in w h i c h

been

Appendices

systems

systems

these

discontinuous gives

has

and

represented

Since

distributions.

a setting

co-authors, and

of

this

of

This

elasticity

formulas

Cauchy see

provide

3.4

of

these

concept

multiplications

their

in m o d e l s jump

the

successfully.

and

solutions,

functions.

mathematically:

functions

treated

in

form,

sense

"ambiguous"

shock-wave

discontinuous

a nonlinear is k n o w n

the

that

classical

solutions refer

the

In

parain

sense;

when

reader

they to

papers.

short

we

hope

that

panorama

accessible

to

of

this

book

this

new

a wider

gives theory

audience),

an

easy

(thereby

and

reason-

making

its

V;ll

Section 2.

The Toda Stems

Section

3.

The Oda Stems

Section

4.

Tentative

Chapter

6:

Section

The Chicago i.

(~,

9 ~ N ~ 19) ............ 99

(~,

20 ~ N ~ 31) ............ 104

D i f f e r e n t i a l s ....................

Stems

(~,

113

32 ~ N ~ 45)

introduction ...............................

139

S e c t i o n 2.

Computation

of ~ ,

32 ~ N ~ 38 ............. 139

S e c t i o n 3.

Computation

of ~ ,

39 ~ N ~ 45 ............. 149

Section

Tentative

Chapter

7:

4.

D i f f e r e n t i a l s ....................

The New Stems

(~,

162

46 s N ~ 64)

Section

i.

Introduction ...............................

Section

2.

Computation

of ~ ,

46 ~ N ~ SO ............. 212

Section

3.

Computation

of ~ ,

51 ~ N ~ 55 ............. 220

Section

4.

Computation

of S

56 ~ N ~ 60 ............. 230

Section

S.

Computation

of ~ ,

S e c t i o n 6. Chapter

8:

Section

Tentative

The Elements I.

S e c t i o n 2.

N'

212

61 ~ N ~ 64 ............. 242

Differentials .................... 253

of Arf Invariant

One

introduction ...............................

284

The Existence

285

of 8 ........................ 4

S e c t i o n 3.

The Existence

of 8 ........................

289

S

Appendix

I:

The Stable Stems ...............................

Appendix

2:

Multiplicative

Appendix

3:

Toda Brackets ..................................

303

Appendix

4:

Leaders

308

Appendix

5:

The

Computer

Appendix

S:

The

Adams

Appendix

7:

Representing

Bibliography

294

Relations ....................... 297

........................................ Programs Spectral

.......................... Sequence

....................

Maps ..............................

................................................

312 317 327 328

IX

duced it,

me

for

of his

to his several

papers.

Donohue, genberger text.

research

I a m also

J.E.Gale, and

already

discussions

and

very much

M.Langlais,

B.Perrot

in

for

1982

when

for h a v i n g indebted

A.Y.Le

their

help

Roux, and

he was

developing

sent me m a n u s c r i p t s to J . A r a g o n a , A.Noussair,

J.T.

M.Obergug-

for c o r r e c t i o n s

in the

INTRODUCTION CHAPTER

I

I

-

GENERALIZED

The

original

FUNCTIONS

AN

OPEN

SUBSET

OF

E~ ........

§

I

§

I 2

An

§

I 3

Local

§

1 4

Nonlinear

§

I 5

Pointvalues

and

§

I 6

Association

processes

§

I 7

Topologies

§

I 8

The

§

I 9

Heaviside

§

1.10

Generalized solutions of algebraic differential equations and classical solutions ................................ 55

APPENDIX-A

CHAPTER

elementary

on

subspace

§2.2

Essential of W h i t n e y

§2.3

Generalized

§2.4

Whitney's

§2.5

Borel's

§

2.6

~

and

~s (~)

products

GENERALIZED

Generalized

................................

6

of

compositions functions

. . . . . . . . 21 ..........

27

theory .....................

30

.................................. ~(~)

34

.............................

40

................................... functions

of

FUNCTIONS

functions

and

generalized

integration

generalized

§2.1

I

restrictions

properties

on

.................................

definition

properties,

survey

2 -

definition

ON

on

........................

distributions

ON the

AN

45

..................

ARBITRARY

closure

49

of

SUBSET an

open

OF

En

65

.69

s e t .... 6 9

facts concerning C ~ functions in t h e s e n s e ............................................. functions extension

theorem

for

on

an

theorem

arbitrary for

generalized

subset

generalized functions

of

70

E n . . . . 72

functions..74 ..............

75

Extension of a generalized function defined on a halfspace ..................................................

80

Xll CHAPTER

3

GENERALIZED

-

EQUATION~

SOLUTIONS

OF

NONLINEAR

PARTIAL

DIFFERENTIAL

............................................

83

§ 3.1

Explicit computations for shock wave solutions of s y s t e m s in n o n c o n s e r v a t i v e form ................................. 83

§ 3,2

Discontinuous solutions of the C a u c h y p r o b l e m for a system in n o n c o n s e r v a t i v e form ................................. 89

§ 3.3

A new

§ 3.4

Jump formulas ticity . . . . . . .

formulation

of

for .

.

.

.

.

the

shock .

.

.

.

.

.

equations waves

.

.

.

.

.

.

.

in .

.

.

.

of

Hydrodynamics

Elasticity .

.

.

.

.

.

.

.

.

.

.

and .

.

.

.

.

.

.

.... 106

Elastoplas. 114

.

§ 3.5

Existence-uniqueness for semilinear hyperbolic systems irregular Cauchy data ..................................

with 121

§ 3.6

Existence-uniqueness for a nonlinear parabolic equation irregular Cauchy data ..................................

with 133

APPENDIX

I

-

Systems used collisions . .

by

engineers

for

numerical

simulations

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

APPENDIX

2

Numerical

tests

in

a system

APPENDIX

3 - Numerical

tests

in

fluid

APPENDIX

4

Numerical

tests

in m o d e l s

APPENDIX

5 - Semilinear hyperbolic systems with irregular coefficients and systems of e q u a t i o n s in A c o u s t i c s . . 1 9 0

-

-

BIBLIOGRAPHIC REFERENCES ALPHABETICAL

in n o n c o n s e r v a t i v e

of 142

dynamics of

f o r m . 148

...................

elastoplasticity

157

. . . . . . . 175

NOTES ..............................................

....................................................... INDEX ...............................................

198 201 212

CHAPTER

GENERALIZED

1.1

-

THE

ORIGINAL

J.F. of of

Colombeau,

definition

differential We

spaces

and

trying

to

successive which

find

ideas

requires

only

a general until

a very

this

chapter

with

Colombeau's reasoning. are not familiar with

fi d e n o t e s a n o p e n valued functions f i r s t i d e a w a s to

~(~).

He

their

product

these

arrived

elementary

ideas

so

at

a

knowledge

that

This paragraph the theories of

the

reader

m a y be d r o p p e d locally convex

thought

that,

might

be

if

s u b s e t of ~n a n d ~(fl) t h e s p a c e of on ~ with compact support, u s e C ® or h o l o m o r p h i c functions on

T l and

the

T2

were

function

the

usual

and

~ E ~(~)

E ~

~ denotes ; but

multiDlication we

have,

the

this of in

distributions

on

~,

T on

the

map

E ~(fi) ~-~ < T I , ~ > . < T 2 , ~ >

test

multiplication

he

distributions.

If C® complex Colombeau's

where

E~

OF

calculus.

begin

might follow by those who

ON AN O P E N S U B S E T

DEFINITION.

distributions,had

simple

FUNCTIONS

1

value

definition C"

of

E C

the

would

functions

,

distribution not

since

even for

fl,

generalize f2

E C'(fl)

general,

(1)

functions

Without abandoning the o n ~ ( ~ ) a n d in o r d e r to

(i),

considered

he

the

idea

of

i d e a of u s i n g identify the

taking

a

quotient.

C ® or two

holomorphic members of

We

recall

t h a t ~ ( Q ) is

contained

and

dense

in @,(fl)(the

space

of all distributions o n fi w i t h compact support $' (fl)isthetopological dual of C'(fl~, which is a strong dual of a Fr6chet-Schwartz space. C®(g' (fl)) and C®(~(~)) denote respectively the spaces of all complex valued Co restriction

functions map

on

$" ( ~ ) a n d

C®

is injective, consider that

see

fl,

fm

($, (fl)) - ~

Colombeau

C" If

on @(fl),

(,¢" (~))

E C®(Q),

the

T E $'(fl)~-*

C®

see

0.6.9

c®

(@(fi))

following

[1].

Then

(@(~))

[i,

c

Colombeau

and

1.1.6].

So w e

map E {E



coincides in t h e s e t 0

[ (DR) (~ for The @H(@(fl)).

It

all

x (

set is

K and

of

that the product in @ M ( ~ ( ~ ) ) . As some C ' ( @ ' (~)) a n d t h e

in

is N E ~

~>0

such

that,

for

each

satisfying

, x ) ] ~< c g - ~

00

R

E and

Finally

the

Many

e~ ~ - ~

C®(@"

the (~))

initial

requirement

=

.

Ker

A

:

quotient

~lgebra

containing

(0))/Ker

~

O 0 s m a l l e n o u g h a n d x E 2, w e r e r e q u i r e d by the definition. inductive limit with

the

Then he of s p a c e s

following

replaced C®(~(Q)) by an algebraic C ® ( U ) , w h e r e U is a n o p e n s e t of ~ ( ~ )

property

:

I f o r e v e r y KCCO t h e r e is N E ~ s u c h t h e r e is n > 0 w i t h W ~ , ~ E U f o r a l l

(PI)

that for all ~ E ~N x £ K and 0O

Lebesgue' change absolute

s by

and

the

a

value

basis.

If

Vn

measure

in

which

integral If

sense,

we

the

has the

on

take

positive of

that

the

define

composed

a

Haar

( c | foj(xl,...,X )JR"

V. where

us

the

factor

of

determinant an cube

inner

basis

hand in

side

V~,

the

product

matrix

there by

is

Lebesgue

f by

an

is

taken d×

equal of

a

is

I__ f ( x ) #V n

proportionality,

of

determined

foj of

n) d x l . . . d x n

second

another

map

measure

privileged

orthonormal

to

change basis

in

will the of Haar has

measure basis

i.

This

since

a

determinant

measure

is

matrix

of

with

consider

the

absolute

Haar

change

of

of

value

the

chosen

orthonormal

equal

to

1

dx I

...

orthonormal basis

has

Then

we

will

measure

f(x)

f

dx

|

E~

with

independent

foj(xl,...,x,)

dx.

J~

c=l. 1.2.2

-

If

En

integrable

Remark.

=

~,

function

constant

of

given

proportionality -

1.2.3

were

privileged by

its

equal

Haar Lebesgue

to

measure

of

integral

a

Lebesgue

(with

the

1) .

Remarks.

Those who Colombeau's books

functions

the

is

have may

defined

already notice

on

~

known that,

(or

open

the theory through one of until now, generalized

subsets

of

~")

with

a

fixed

orthonormal b a s i s i.e. t h e c l a s s i c a l basis. The definition_~given in t h i s b o o k is i n d e p e n d e n t of t h i s c h o i c e s i n c e the set ~ ( E , ) d e f i n e d b e l o w is i n d e p e n d e n t of an o r t h o n o r m a ] b a s i s in E n. Furthermore compositions of g e n e r a l i z e d functions defined in s p a c e s of d i f f e r e n t functions to s u b s p a c e s Hore

dimensions,and restrictions of g e n e r a l i z e d are allowed with this new definition.

precisely

1.2.4

-

Set

~0 (~)

:

Definition.

=

{~i

E @(~)

such

+® O-neighborhood

and

if

and

I J 0

that

~i

is

even,constant

in a

i ~i (A) d k

= -- } 2

,

q=l,2, . . .,

~q (R)

=

( ~ i E ~ o (~)

such

that

~

/t o

~

(Tt)d ~ = O

if

l~j~q,

l~m~q}

1.2.5

Proposition

The with

any

Proof

-

with

~q(~)

constant

We numbers,

set

compact

is

value

want

to

there

-

at

prove

exists

E

if

or

~1

can t a k e ~lE~q(~)

real

0 ~

C(~)

by 8 ~ ),

functions

such

(if

is

there

t h e r e is, ~E]0,~0[.

then, ~ 0 E ] 0 , 1 [ Finally set

such

(6')

= [

Xi(~> (A)~(A-x)

dA.

f(A)

we

= diam(supp

set X ~ m 0 ) . F o r a n y K C C ~ identical to 1 on K for R~(~,x)

map

C !~(~), define

~

now, if

an

Ks]

denoted

: ~0 (En) the

have

no

~) . that such

that

X~

X,E~(~) x then

is

E n

Notice

that,

for

all

~ E ~ 0 (Eu)

and

KCC~

there

is

~>0

such

that

r (7)

Rt ( ~ ,x)

= I

f (~)

~

(A-x)

d~,

E n

for all xEK and 00

there

and

(13)

~>0

is N

E ~ such

that,

for

all

satisfying

[R i (~, ,x) ] ~< c8 -N

for

all

x

have

from

(12)

(14)

E K,

0 0 a n d w>0 s a t i s f y i n g

i lajf(y)l ~< c" (l+lyl)~ lR~(~,x)l ~< c" ~'~ l (R~-Rj) ( ~ , x )

I 0 and lZ(~)

We

of

define

an

I ~< c~ -N

ideal

of

for

@m

all

generalized

~0(~)

complex

into

E ~ such that W>0 satisfying

C.

We

if

0 0 s a t i s f y i n g IZ(~+)I

~< e ~

algebra

The

of

(q)-N

for

generalized

all

00

,~>0.

elements

of

C

to

as

each

constant ~

E ~

the

admits x 6

all

~ and

~ E ~0(E~)

Z 6 @m is a r e p r e s e n t a t i v e of Z) is o b v i o u s that the topology on

~(D)

induces

II.Ilx,k

: f E C'(~)

x E

1.7.1.

set

G 6 QK,k,~

on

is

C®(~)

its

on

C®(~)

coincides

~-*

sup

(sup

lal~k Remark

1,

topology.

on

since

we

associating

which

for

own

topology

space topology, seminorms

all

(that

= Z(I~),

on ~

The

proposition

if

consider

as its representative, where t h e n ~ is c o n t a i n e d in ~ ( ~ ) . It • (~)

of

: G 6 Q~,k,~}

functions

R(~,x)

to

KCC~, k E ~ a n d G 6 ~ ( ~ )

If

+~

-

generalized

is

obvious

own

Fr~chet

with

its

using

the

dense

in

[D~f(x)~)

xEK

that

the

map

G ~-~ G ( x )

is

linear

and

continuity ~(Q) . Remark

since

of

4

it

-This has

Rosinger

[4]

"stability

As

above

Since

map,

topology

on

nonabsorbing p.39-47

paradoxes"

differential

structure.

continuous.

the

we

~

is

have

~(~)

is

not

that

not

dense C®(Q)

a vector

zero-neighborhoods.It that for

this solutions

is

a of

in ~, is

not

space is

necessity

topology shown to

nonlinear

in avoid

partial

equations.

in

1.7.1,

the

topology

on

~(~)

arises

from

a

uniform

44

Definition by

the

4

In ~(~)

-

UK,k,.

: 0 s u c h


0 and W>O such

that I(D~R)(~,x)I for

all

x E K and

(c) (f E C ® ( ~ ) ~ function functions

on

00, a there corresponds

family of f u n c t i o n s in @(~) as a generalized function in ~ ( ~ ) ,

[ (~,x)

To compensate this classes in ~ (~) a r e

1.8.4

-

The

The

~-~ [ J

let

~

be

~

the

following

~[~]

= {Z

ideal

6 @~.,s

denotes

the

A(@~)

n>0

is

constructed

~s(~)

:

let

naturally

obvious

~ C

C @M£~]

that

E~

the

the

same

denote

N 6 ~, 00 ;

that

for

n~1~ural

each

JZ(~)I

q 6 ~,

there

~< c~ q for

all

are

c>0

00 s u c h t h a t IZ(~)I ~< c~ -N for

and

then

~ (x+~)~(~)d~]

"~s.

subalgebra

subalgebra

~

If A : ~ o ~

~

(b) ,

fact, it is e a s y to c h e c k t h a t all t h e s e associated with each other (see 1 .8.8) .

simplifications as d o n e @ ~ , s the f o l l o w i n g set @~,~

f(z+~)~(~

in 1 . 2 . 1 7 c l a s s of

is

(M is

: 4~, ~/$~

of ~.

open,

complex

= M-I (A(J~)) defined by

KCC~,

x 6

generalized

~ and numbers

G 6 ~s(~), G(x)

and

then

it

G(~)

[

is

dA

K

(see

§1.8)

tively,by

are

in

fact

in ~ ,

~ ** K ( ~ z / = , x )

representative

of

G

(see

and

with ~ ~

representatives

II%(~/~,A)dA,

1 . 2 . 1 1 (c)) .

if

given,

respec-

K6@~,s[~ ]

is a

49

1.8.5

-

The in

macroscopic

aspect

of

a

generalized

function

~(~).

Let T be a d i s t r i b u t i o n on ~. We say that a generalized function G E ~(~) has the distribution T as its m~crosco~c mspect if t h e r e is a r e p r e s e n t a t i v e R E @M,,[~] of G s u c h t h a t for all ~ E @(~) we have

]im f

The now,

unique d u e to

difference t h e l a c k of

R(~,x)

~(x)

dx



=

with the concept of a s s o c i a t i o n is t h a t , a sanonical inclusion ~" (~) C ~ s ( ~ ) , we

c a n n o t s a y t h a t T is a n e l e m e n t of ~ ( ~ ) (see e x e r c i s e 1.6.9) . Considering the i n c l u s i o n ~(~) C ~(~) then G has T as macroscopic aspect if a n d o n l y if G ~ T (T b e i n g a w e l l d e f i n e d element of ~ ( ~ ) ) .

1.8.6

Fn'if

QI

-

Restrictions

of

With the notations of 1 . 3 . 5 , if is a n o p e n s u b s e t of F s u c h t h a t

GE~9 s (~),

then

a

representative

K~(~,x)

for ~>0 1.2.11)

s-generalized

and

1.9

-

As inclusion

xE~1,

of

where

HEAVISIDE

we have C(O) (or

of

G]

GENERALIZED

noticed in Cf(O)) into

~

is

a

to

subspaces.

is a n o p e n s u b s e t of ~l=II1 (~f]jl ( F ) ) / g and

E ~s ( ~ )

= R(~i/n,(x,0

R E @ M , s[~]

functions

....

is

given

by

0))

representative

of

G

(see

FUNCTIONS.

1.8.3, ~s(~).

there Since

is no c a n o n i c a l a generalized

function in ~(~) may be c o n s i d e r e d as a " l i m i t " of a f a m i l y { R ~ } e > o of C ® f u n c t i o n s o n ~, w e p r e s e n t b e l o w s o m e e x a m p l e s of functions R~ which, at the limit ~0, give back the classical Heaviside function Y E C t ( ~ ) d e f i n e d b y Y ( x ) = 0 if x < 0 a n d Y ( x ) =

I

if

x>0

:

50 2

/

I/ If we consider the c l a s s in ~ ( ~ ) of a p o w e r n > l { R e } o ( e < I d r a w n , for i n s t a n c e , at the left hand then, representing both families { R e} a n d { R ~ } below (R~ b e i n g r e p r e s e n t e d by the d o t t e d l i n e ) ,

/

/

/', i

of t h e side, on the

family above, figure

R~

Rt

/ #,

>

we see that {Re} and {R~} represent different generalized functions (lock, for instance, at the difference (R~-R~)(O) which does not necessarily t e n d to z e r o w h e n ~*0) in ~ s ( ~ ) , each of t h e m b e i n g c a l l e d Heaviside generalized function. Let us give, then, a definition of w h a t we w i l l m e a n by a Heaviside generalized function : 1.9.1

-

Definition.

A generalized function H g e ~ e P a Z i z e d ~ u a c t ~ o n (*) if it has

in ~ s ( ~ ) is c a l l e d a H e a ~ s ~ d e a representative R E @.,.[~]

(*) T h i s d e f i n i t i o n is n o t r e a l l y f i x e d ; e x a m p l e s of r a t h e r w i l d Heaviside generalized functions w i t h p e a k s at the o r i g i n a r e g i v e n in A p p e n d i x 2 of C h a p t e r 3; t h e y c a n be e l i m i n a t e d in a p h y s i c a l context,for instance by i m p o s i n g a condition stronger t h a n (iii) , for i n s t a n c e , s u p ] R ( ~ , x ) I0 xEi

51

satisfying

:

there

is

A(~)>0, (i)

R(~,x)

(ii)

3

H~K. have

If H Indeed,

f where

of

6

1 for

all

~J(x)

*(X)

classical Heaviside

-

are

dx

and

~>0

xA(~)

0

as

~

0.

dx

--~

0

as

~

representatives

Definition --*

1.9.1.

f°

0 Heaviside

~(x)

-*

of

Note

also

dx=

function

generalized

is

as the

0

,

H

and

K,

that ~0

,

macroscopic

function.

Remark.

n~l,

deriving

both

terms

H "÷I

H

¢>0

IK(~'x) ldx "~

A(~) for

all

~*0, for

is an e l e m e n t such that

such

8 of

that

all

~>0

I I R 6 (~,x) ]dx 0

-

f u n c t i o n (*} on ~n (~,x) ~--* Rs (~,x)

is A ( ~ ) > 0 ,

Ra(~,x)

a>O

1

Definition,

A Dirac generalized with a representative (a)

let

n

if n¢l.

1.9.5

• ,(~)

2

to

)

The

Let ~ £ be g i v e n .

generalized

function

~(x2-a2).

~ , ( ~ ) be a D i r a c g e n e r a l i z e d function Let R~ be a r e p r e s e n t a t i v e of $ . S e t R~ (£,x)--

and

R s (~,A) dA b

for some b0

IR(6,x)-a

K

such ~

g,

i

the

set

account in

G

is

classical

representative

compact

R

polynomial

if

n be

a

IP(R(~,x))I

continuity

S~(Q)

of

term

~(~)

has

can

technical

N~(Q)

a nonzero

is

30>0

One

get

and

be

one

introduction,

give

al,...,a

given

to

construct to

equations

and

[~]

we as

the

concept

Then

of

more

place

For

case

N~.

Proposition.

be

if

6 @IM , S

that,

are

in in

have,

-

P

G 6 ~(EI),

~

e

#i[~]

R(~,x)

setting

~i(~) of

1.10.5

Proo~

the

we

all

on

~s(~).

simplification of

@i[~]

simplified fixed

informal

with

with depend

reproduced.

so

an

solutions

First

for

representatives

proofs

that

x

proofs

the

character

not

similarly

the

is

in

the

that

the

do

representatives C®

pattern

: but

@~I[~],

~

Define

Define

only

our

~(~)

like

c and

0<e

complement

{ (x,~+n)

contain (this

is

f~

~ ( f x S)

distribution

fE~

multiplied

of

case

SE~,

every

on

that there distributions

product

of

for

fxml

of

=

; for

the

(T)} T

can

was one

be

given S and

T

LIMIT

net

provided

0

all

g. be

x ~(--)

constructed

carefully used from

in a

distinguished this single

book ~.

, and

from

the

which

67

If S,T are distributions then S*~ ~ and T*~ ~ are C" functions, a n d so t h e p r o d u c t s (S*~)T, S ( T * ~ e ) , (S*~e) (T*~ ~) e x i s t in the classical sense. There are four natural definitions of t h e product of S a n d T :

(1)

= lira

ES]T

(S*~)T

e-0

(2)

SET]

: lim

S (T*~ ~)

e~0

(~)

[S][T]:

(4)

[ST]

~im (S*~ ~) ( T * * ~) ~

provided {~} for

the limits exist (3)) . D e f i n i t i o n s

definition

(3)

:

PROPOSITION

1

(4)

by

PROPOSITION may

be

2

:

replaced

strict delta nets (~} (and been given by Hirata-0gata,

Mikusinski The

Definitions

:

is m o r e

( S * ~ ' ) ( T * 9 ~)

in @' ,for a l l (i) , (2) h a v e

Antosik-Mikusinski-Sikorski.

definition

0

lim

and

following

(1),

(2)

definition results

and

(3)

are

(c)

on

(4)

have

been

by

proved:

equivalent,

and

general.

The

positivity

condition

the

functions

~

by

(e') sup,~ l~'(x)Idx < and

the

products

(2),(3),(4)

thus

obtained

obtained By

with

restricting

are

the

class

course more general definitions, ~nd references there. 4

-

COMPARISON

WITK

THE

3 -

Let

S,T

(4)

then

the

product

associated

distribution.

E@'.

of see

PRODUCT

PROPOSITION exists,

respectively

equivalent

to

(1),

(c).

IN

If of

delta

nets

used,

one

gets

Colombeau-0berguggenberger

of Eli

@(~n)

the S and

product T

in

of

S and

~(~n)

T

admits

given

by

EST]

as

68

Therefore, products (~n)

defined

senses

of

(1),(2),(3)

the

-

If

3 are

the

the with

natural), Several

1

section

product

consistent is

4

sections of

Thus

this

proposition

section

product

and

2,

and

proposition with

the

2,all

the

product

in

(since

it

from

in ~ ( ~ n ) . A s each

other be

other

uses

S

and it

T

exists

in

in

exists

any

the

the

of

senses

3.

products

to

of then

sections

and

appear,

particular products

more

Colombeau-0berguggenberger

the

i and

a consequence

exist

also

these the

product

that

are

non

process).

are

association

the

non commutative in ~ ( ~ n ) is m o r e

consistent

products

of

association

Eli,

2 are

all

through

cases

the product in ~ ( ~ n ) (for i n s t a n c e whose consistence with the product shown

1

consistent

.

PROPOSITION

with

from in

(as

in ~(~n) .

consistent

with

products) delicate to See

the

or be

survey

CHAPTER

GENERALIZED

From now onwards In t h i s c h a p t e r we is ~ n y s u b s e t , n o t

~s(~). C Ea with

2.1

~ ( ~ ) ,see

-

ON AN A R B I T R A R Y

SUBSET

E.

OF

we s h ~ l l w o r k o n l y w i t h t h e s u b a l g e b r a e x t e n d the d e f i n i t i o n of ~s(fi) w h e n necessarily o p e n . W e m i g h t ~s w e l l w o r k

Biagioni-Colombe&u

GENERALIZED

ralized set.

FUNCTIONS

[1,2].

ON T H E

CLOSURE

OF

AN O P E N

SET

First, let us e x t e n d the d e f i n i t i o n of C o l o m b e a u ' s g e n e functions to the c ~ s e X C E ~ is the closure of an open

2.1.1

an

FUNCTIONS

2

-

Definition

If ~ is ~ n o p e n s u b s e t of orthonorm~l b a s i s of E n, we set

$~C~]: 0 }

its ;

derivatives

@M,,[~]={REgs[~]

for

all

7>0

such

a E ~ a,

there

are

such NE~,

that

e>0

and

I(D~R)(e,x)lO

such

lal4m that,

and for

xoEX

there

every

are

pair

of

V,

a

points

have 1

(1)

If~(x)-

2

--

I~l<m-l~l We

denote

2.2.2

-

by

Ix-x'l

~ f=+B(x')[~A

"-'~'÷~

B!

C~(X)

the

set

of

all

these

families

(f,)

n

Remark

(f~)

If

(x-x')

~ E

C;(X),

the

f~

functions

are

continuous

aE~

in

X

and,

It

follows

set

~,

each

that, or

closure, have,

at

if

the

then,

interior is

if

X

X

contains

functions by

point

(i)

open,

f~

that

x

or

an

are

of

X,

X

is

if

open

set

(f=)

_

have

the

is

of

a

an

contained

determined gives

u

f~(x)-Dsf0(x).

closure

and

uniquely

f0EC®(X).This

we

by

open

in

its

fo.

We

map

E C~(X)~---. foe C®(X)

n aEl~

2.2.3

-

Let Then

there

Whitney's

be

X

a

extension

closed

F6C®(E.)

is

theorem

subset

such

of

and

En

let

.EC; (X).

(f~)

that D~F(x)=f~(x)

for

all

Hestenes

xEX

and

The

proof

of

this

theorem

can

be

found

in

Whitney

[1]

or

[i] . 2.2.4

-

The consider

aE~".

X

(]an,bn[).>

map as

Remark

from

the

2 of

2.2.2

closure

disjoint

is in

not ~

open

of

onto the

in

general

For

union

of

countable n-1

intervals

a

defined

by

an-

1 bn=

a n + I - -n 3

as

n*~.

Let

example, family and n

Then f be

the

an0

such

0<e<w

;

a moderate

x0EX,

there

all

xEvnx ii)

and

for

each

~)0,

V,

when a

~0.

This

neighborhood

that [f~ (~,x) I~

for

growth are

the

family

c~ -~

{x*f~(g,x)}

~ satisfies ~EN

the usual requirements defining a Whitney C ® function o n X; t h a t is, f o r all mEM, ~ E ~ n, with [~[~m, and xoEX there is a neighborhood V of xo s u c h t h a t , f o r e a c h ~ > 0 t h e r e is c ( E ) > O satisfying

1 (2)

IB[~m-I~l

for

each

pair iii)

by

xoEX that

x,x'EVAX

~ f~, j (e,x,)

iO and C>O independent of i,n and h such that I

o ~ u~

< A

I° 1

A

(20)

(stability

I

n

(2:)

n

zlu,.1 i

-

Z

- ~i

~i+:

for

B (stability ~

the

for

for

the total in time in constants A>O,

L®-norm)

the

B

total

variation

in space)

i

I

~lu~

~+1

i

(22)

+I

i Proof.

Let,

for

n

n

each

R?

~< C

u i

-

i and

:

(stability time

~< C

for the total variation in in Tonnelli-Cesari's sense).

n,

+ k u':

~?

(23)

From

(13)

and

(14)

I

we

n

have

n

:

from

(15)

n

+ r

(l-r

w i)

~

:

(l-r

w,)

and

Q~

(16)

R~ ÷1 (25)

+

= Ri

(24)

and,

n

Pi

n

n

r w i RL~

n Qni - 1

wi

:

= 0t

(l-r

k)

+ r

k

Ol÷ 1

92

We

will

prove

that,

(i)

for

all

n,

Iorl,~x< u~,

(ii)

for

have

all

i

u i

+

~i~)

~

kg

(k u ~

-

~)

~

kM

(k

max

we

I

(iii)

max I

~ I~+,

(iv)

- R~I ~ z

i

z IQ~'+,

(v)

- Q~I < z

i

for

some

positive The

from

(17)

the to

proof

;

(ii)

bounded (v)

is

and

done (iii)

variation

hold

for

o

~

by

induction

from

(18)

assumption

some

(26)

K.

constant

n. RT

From 4

kM

~

M.

on

(i),

on

;

(u o,

(ii)

and

n.

(iv)

n=o (v)

~

(iii) Q~

~

0

we

we

have

from

~o) . A s s u m e ,

and

- kM

For

and

now, have,

which

,

from

(19)

~

(24)

O ~0~

(2S)

and

o ~ from

~÷i

,

(26)

~

kM

and

and

(27)

:

- kM

~

~

~

o,

for

all

i.

o,

for

all

(28),

(29) Since,

w~

,

(28)

From

o

implies

(27) Then,

u i

(23)

=

R ~ +I

~< k M

and

- kM

~< Q T + I

~

i.

,

(R~÷i

+ Q~+i)/2

and

k

u "+~ i

=

(R~ +~

(i) and

that for

hence o ~

:

(23)

Q'~+ ~ ) / 2 ,

all

(i) i,

93

assertions

(i) ,

and

(ii)

(iii)

for

(n+l)

may

be

respectively

written

as

[R? +~ max

+ m?+'[

R~ . 1

~

~ R? +,

_ Q,:+~,

kM

and max which

(_Q~+I)

follow

from

Let

us

g

kM

(29)

prove

, (iv)

and

t

(v)

:

from

-

Dt

(19)

and

~i+t

(l-rk)+Z

rk

i

similarly,

(+o.) Now,

(+l>

Pitt

-

have

(24)

xlp:+,

Dt

i

, i

mlm,+"+:m,+'l" i from

we

i

~

and,

(25)

and

-

(28)

+:I

ml+,+," - +,I i :

= m l R ? + , - r w , + , (+R ~ +R+ - R ": > - R- : + rRw : "

i

"

,

,

,-,)I

i

.< ~IR~ + ,-

R '~

(l-rw~ + ,)

+~rw~

!

:

I R," - R " , - , I

i

~+IR:+~-R:I

"< x

i

by

induction

hypothesis.

Similarly

we

have

that

i

which

completes

From stable

for

the

the

induction

(23) ,

(i) ,

L'-norm.

and 1

i

i 2

(ii)

Further, 1

proof+

and from

(iii) (iv)

we

get

and

that

(v),

we

the get

scheme

is

94

t h a t is, t h e the s t a b i l i t y sense)

,

we

scheme is s t a b l e for for total variation

have,

from

(23)

, (25)

total variation in space. For in t i m e (in T o n n e l l i - C e s a r i ' s

, (24)

, (31)

, (27)

n

n+l

n

, (iv)

, (v)

and

(19)

:

1

+lu: +'

"

i

t

2k

I+IQ,

-Q,I>

1 n

m-

An

analogous

Remark,

If

estimate

hm

*

holds

c and

for

um=u ~

, gm=~ ~ m

[]

n Z I ( ~ + I -(~il '

, we

have

from

(20)

that

(u m)

m

a n d (Gm) a r e b o u n d e d sequences in L® ( ~ x ] o , + ¢ [ ) . Using the following result on w e a k - * topology : "if E is a s e p a r a b l e Banach s p a c e t h e n t h e u n i t b a l l in E' is m e t r i z ~ b l e and compact for the ~(E' , E ) - t o p o l o g y " we obtain t h a t the a b o v e s e q u e n c e s in L ® = ( L I ) ' admit subsequences converging respectively to u and ~ for the ~(L®,L1)-topology, t h a t is I for

all

3.2.2.

Um f f ~s f

u

and

fELl ( R x ] o , + m [ )

A generalized

f

0m f f '* Is g

as

s*~

,

.

solution.

We will explain here how the functions u a n d q m a y be considered as s o l u t i o n s of the s y s t e m (2) . F u r t h e r we construct here two generalized functions U, Z E ~ ( [ o , + ~ [ x ~ ) solutions of the system (2), w h i c h h a v e u a n d ~ as r e s p e c t i v e macroscopic aspects in the s e n s e of 1 . 8 . 5 .

95

T h e o r e m . The U

and t:o

system

EI ] t:o

(2)

have

has

a

U,

Z~

u o and

~o

solution

respectively

([o,+m[X as

~)

such

macroscopic

that

aspects.

f Let D~(~)be such that n~o, supp ~ [ - i ; i ] rand I ~(k)dA=lLet R~,R~ be real f u n c t i o n s d e f i n e d on ] O , l ] x ~ x [ O , + m [ by ~

Prao'f.

R, (£,x,t)

= Iu~ (x-£3~,

t - £ 3 r ~ ) p ( ~ ) ~ (z)d~ dz

g~ (g,x,t)

= Ige (x-Sa~,

t - £ 3 r z ) ~ (~)~ (z)d~ d~

I

(32)

if g=h m for some m and R~ (~,x,t)=Ru R, (~,x,t)=R~ (hm,x,t)) if h ~ + : ~ s < h ~ . By the for the L'-norm, in 3.2.1, we have that Let U and Z be their r e s p e c t i v e c l a s s e s in We must P~=

prove

that,

for each ~ E @

if a a {-- K~ (£,x,t)+R~ (~,x,t)-- Ru (g,x,t) at 8x

(hm,x,t) (respectively, s t a b i l i t y of the scheme Ru, R a E @ ~ [ ~ x E o , + m [ ] . N~ (~x[o,~[) .

(~x]o,+m[) , the i

8

quantities

Ra (£,x,t)}~ (x,t)dxdt

~ o 8x

and (g

(33)

Q,

a

a

=I I ( JJ

tend to o as £ - 0 .

at

3

Ro (~,x,t)+Ru ( ~ , x , t ) - R~ (~,x,t)-k 2 - R~ (~,x,t)} ax 8x (x, t) dx dt

We may

consider

only

the case

~E{h m : mE~}.

96

functions

The u~

following a n d {~c :

figure

2~

3

(

)

explains

the

values

2~ 3

of

the

step

2~ 3

: n+l :

!:.i

..........

)

':

:

:

:

;

:

:

::

U n i-I

~-I

x=(i-}~}~

n+l

'~i

:Ui

...........

l

i

l

:

:

:

n U i

o'~

x=(i-X)~

i!:! n Ui+

:

:

:

:

1

:

: o',+ I

n

x=(i+~)

Let gg

(34)

Q2

=

H

8 R.(a,x,t)~

JJ

Since constant

supp

except

It- (nT~>

~c[-l,l],

on

the

ra

Io

consider, integral,

is

first,

independent

for

each

of

i,n

fixed

and

tE[(n-N)

~.

Now,

for

r ~ + r ~ 3,

J~,t,n,

(n+N) r ~ - r ~

let

us

3] , t h e

98

K~,~,~

(~8)

On the R~ ( e , x , t ) only respectively. and

R~.

-

f x = (i-M) a + e 3 I R~ ( ~ , x , t ) x: (i-~) £ - ~ ~

8 -8X

R, ( ~ , x , t )

dx.

rectangle R~,. the functions x --~ R u ( ~ , x , t ) a n d x --~ do a C" j u n c t i o n between u ~ _ I a n d u~, g ~ - 1 and ~, Also, from (32) t h e s a m e f u n c t i o n o was used in R u

Therefore

there

is

a

C"

function

H,

of

the

form

1

such

that

I R.(£

'x J t)=u~_1+(u~-u9

l- I )H,

(x-(i-~)~)

(32') R e (8,X,t)-

if

(x,t)ERi,

(32') , i n t o

a.

Replacing

(38) , w e

--(yn ,.I+

the

n (0~-i

(~-1) vp then

~ (v,p)=

The

system

(50)

becomes

¥-i V t + U V x - V U x = O (Sl)

U t + U U x + V P x = 0 Pt

3.3.2.

+ U

Study

of

the

Let,

as

usual,

Px

Shocks

+ ¥ P U×

of

~

(50).

u (x,t) = A u . H (x-ct) +u, (52)

p(x,t)=Ap.N(x-ct)+pt v (x, t) = A v .M (x-ct) +v t

0.

where

(v,p)]u x ~ 0

association

in

110

where

H,N

velocity

and of

We Lemma.

The

M

are

the

Heaviside

will

prove

that,

equations

(,53)

generalized

functions

and

c

is

the

shock.

-Ao

(44)

in

fact,

and

(45)

H=N=M. give

that

ut+cAn=(Ao+o,)Au

and D t

(54)

Ap=(Au)

~o~ ( i + - - ) . AO

It

suffices

(we

use

also

to

put

(53)

(52) to

(53)

Condition

O (x,t)=Ao.L(x-ct)+ot

into

(44)

and

($4)) . []

gives Au

O r

C

and

obtain

+ U r

--

,

where

AO

=

or-O t •

Ao Then O r Au (55)

where

c-u,:

Av=vr-v

t .

Au

- -

Replacing

and

V~

(54)

(55)

in

,

(50a)

V~

M'-H'M-~

Av Condition

H'=O Av

gives

:

Ap

O e

-

(Au) ~

Au

-

(52)

(H+ ~ )

(56)

-v,

:

O,

D r

D~

(i+~

AO

Ao

we

have

:

(48)

111

or, 1 (57)

- -

-- ~

Ap Replacing

(52) , (55)

(58) If

D r

and

set

(~=v~/Av,

(56')

(56)

P r

in

we

= (Av.H+ve)H'

and

(58)

-

Pe

P~

(50b)

(H+~)M' -H'M-~H'

(58')

-VrX-AV

.

have

.

become,

=

Vt

Or

respectively

:

0

(M+~)N' :(H+~)H'

use

the

(59)

explicit

formula

y (x) = -

~

the

solution

I

of

exp (

a(~)

-~

for

-

D~,

(57)

(Av.i~+v~)N'

we

We

=

the

d~)

-

dA

-- e x p ~

differential

-

J

-co a ( ~ )

-

d~

-co a (4)

equation

(x)y' +b(x)y+c (x)=O y (-~) = 0

where

b(-~)=

function Note in

of

for

the

order

to

c(-~)=0,

that reader

consideration,

We

solve

mathematician show

mathematician differential

to

(56')

and

thus

to

obtain

14 a s

a

been

chosen

H.

formula

reader.This

classical has

presentation

automatisms to

(59)gives

equation.

have

b

-H'

c

-c~H'

a

H+~

a

H+o~

check the

set

of

that,in of

all

has

calculation the solutions

work.The case of

under this

112

Note that we always have ~0 (which corresponds to vt>vr or ve0 small enough, H+~/0. The computations run as follows : A

b(~)

I

H(A)

. d,~ =-.~nlH(X)+~l+'nl=l=-'enl ~o (A)

I

i exp(f

a(k)

b (~) --

a(~)

-~

d~) dl---

1+-

I ;

Fx

ell' (A)

j-o~

H(A)+(~

dh

H(A)

ll+

-m

H' (A)

ff

'-I

i dA-- (*)

_--~fX -CO

H(A)

I+

H(A) i+'~

H I +I if I+-- >0 where H -I if i+-- >

i

we

have

k ~ (62)

H'

K' Ao"

o e

H- (I+ --) A~ Any generalized function G on ~ h a s a p r i m i t i v e P i.e. t h a t P' =G, f u r t h e r m o r e if G E ~ (~) we h a v e that P E g S (~) , a n d such primitives d i f f e r o n l y by a c o n s t a n t generalized function Colombeau [3] , § 2 , 3 f o r d e t a i l s ) . T h e n we h a v e f r o m (62) t h a t

such two (see

116

k2 (63)

K:

--

~ 8n

(i+--

A~ where

~n

denotes

the

-H)+C.

A~

logarithm.

By d e f i n i t i o n of a H e a v i s i d e generalized function, t h e r e is A(~)>O such that a representative R z of K (resp. R~ of H) s a t i s f i e s RK(~,X) =0 (resp. R H ( ~ , x ) : O ) for all ~ > O a n d x < - A ( ~ ) . T h e n

C:

[l1

--

~n

-

-

I+-AO

and

(63)

becomes

k 2

(64)

K=

-A~

H ~n

ID~ i+

Am The all

representatives RH ~>0 and x>A(g) . Then

and

R K satisfy

also

D t

k 2

Am

A~

D t

=

i+ Am i .e.

1

R H (~,x)=R~ (~,x)=l

for

117

I (6S)

~n

-Aa

Ao i+ - -

k 2

Setting

(66)

~

:

i+

-Ap

we

have

by

elimin&tion

of

in

v-u~ -k ~

1

Ao"

c~

(67)

(ii .a)

A

From

(66)

we

and

(ii .c) ,

.

have AD

(68) ~, Replacing

(67)

and

(68)

in

(69)

A

(65)

i-~ we

1 = -- +

get 1

~n(l-~) Remark.

value

When A=~

for

a ~

0 we

- -

have

small

A ~

(which

~

and

is

the

thus case

formula in

(69)

perfect

gives

the

elasticity).

De

T h u s the d i f f e r e n c e between (9') a n d (9'') is r a t h e r i n s i g n i f i c a n t in the p r e s e n t case. The numerical s c h e m e s u s e d in C a u r e t [l] h a v e b e e n b u i l t so as to correspond to s o l u t i o n s with A=~. Their comparison with experiments, via more classical schemes, is v e r y p o s i t i v e , s e e C a u r e t [i, chapter

5] .

Exercise. no

solution

Write of

(9')

the

with

form

three

(I0) .

equalities

and

prove

that

there

is

118

Final

conclusion.

The system (9) (with three associations) has an infinite number of different kinds of shocks, i.e. ambiguous Rankine-Hugoniot jump conditions ((ll~,b,c) depending on a r e a l v a l u e A). The conditions. tions.

more precise systems The system with three

(9') (9") equations

have has

nonambiguous no shock

Exercise. Study

the

jump

conditions

of

the

systems

1 I

Ut

+ U U x -~. - -

g×

Po

(2') gt

+ U gx

= k2

Ux

and 1

I

U t + U U x z -Do

(~×

(2 ")

Gt as

we

3.4.2.

have

done

for

the

+ U g×

system

(9')

A S Y S T E M OF FOUR E Q U A T I O N S SPACE D I M E N S I O N .

~ k 2 U~

.

MODELLING

ELASTICITY

I N ONE

L e t us c o m e b a c k to the s y s t e m (1), w r i t i n g it w i t h equality and association, as it w a s d o n e w i t h the s y s t e m of Hydrodyn&mics in § 5 . 3 :

shock solu-

119

t +

(~u) ~ = 0

(~u) t +

(pu ~) ~ +

(p-S) ~ = 0

(pc) t+ (seu) ~+[ (p-S) u] ×=0

(I')

St + U

S~

- A~2 ux'~

0

p ~ ~(P,I)

From

§3.3,

give

if

that, are

two

constitutive

represented

ambiguous

jump

additional on p h y s i c a l

by

of

equations

form

the

studied

Heaviside

written

conditions.The

idealized

fourth

with

first

up

to

two

association,this

ambiguity

equations

now,then

function. Since

e o u a t i o n , or b y Colombeau [13].

model

line

I St (70)

+

and

being

This

can

be

some

additional

u,v

and

there

are

system

has

resolved

by

an

assumption

possible

~ u ~ 0 ( a c c o r d i n g to

The

constant that,

is m o d e l l e d becomes

by

the

show

that,in

studies the

~

The in

given

by

< So way this

value

S O is

(S) :0

two §3.3

the

states

the

the

v,

function.

roughly

reported there give

phenomena

to

in an

the

Fung i d e a of

considered

plasticity is

point.

elastic

here

This

JSJ

reaches

S o the

elastic

and

plastic

are

u and

and p are

in

means

(therefore

when

(hydrodynamics) case

ISJ~S 0 .

corresponds

material

(l'))and

function and S-p

if

i)

called

JSJ<So

plastic

generalized

(however

Appendix

system

Heaviside generalized elastic case,v, u, Heaviside

equation

S o is

when

the

plastic.

Thus

the

otherwise,i.e.~2

functions

(i')

is

by

relationship of steel . Figures 9.9b and 9.9c

different,see

physically

elastoplasticity

if JSI

u s~ S:±So

constitutive

load-elongation [l,§9.7,fig.9.9a] quite

of

replaced

obtained)

are

the same

in

ELASTOPLASTIC MODELS

the

other

equalities

the

constitutive ground, see

An with

P shocks

for

p-S

3.4.3,

1 = -- , t h e

v

3.4.2

obtained

it

material

juxtaposed. (elasticity)

with

the

H ( a n d S = ± S o is c o n s t a n t ) , a n d , i n are also obtained with the

same the same

120

one

has

It has been first a shock

observed by wave (called

physicists t h a t in elastic precursor)

certain cases in w h i c h the m

•

material is in the e l a s t i c state (and w h i c h r a i s e s the v a l u e of ISI to So), f o l l o w e d by a l e s s r a p i d s h o c k w a v e in t h e plastic state.. For stronger s h o c k s the v e l o c i t y of the p l a s t i c shock can increase so as to r e a c h the v e l o c i t y of the e l a s t i c shock,and so the two shocks are put together in a unique elastoplastic shock . See appendix

4.

In t h e of A p p e n d i x 4 represent v, below, since S

elastoplastic shocks presented in F i g u r e s 9,10,ii,12 , the H e a v i s i d e generalized functions H, (used to u a n d p) a n d K, ( w h i c h r e p r e s e n t s S) h a v e the a s p e c t c e a s e s to v a r y a f t e r the e l a s t i c - p l a s t i c transition:

K

elastic stage

plastic t

stage

transition

This figure shows that, when the plastic s t a g e is n o t negligible (in f a c t , in very strong collisions, it is u s u a l l y the most important s t a g e ) the H e a v i s i d e generalized functions H and K are very different. T h i s f a c t s h o w s t h a t it is indispensable, in physical applications, to u s e s e v e r a l H e a v i s i d e functions and not o n l y one, l i k e in d i s t r i b u t i o n theory,or in m o s t a t t e m p t s to d e f i n e multiplication of distributions. When the elastic stage is negligible

we

have H K'

A detailed studies [ I] , a n d

theoretical

can

be

found

Appendix

4.

study in

can

be

Colombeau-Le

~

0.

found

in

Colombeau

Roux-Noussair-Perrot

[13].

Numerical

[I],

Noussair

121

3.5

-

EXISTENCE WITH

of

-

After engineering

gives

UNIQUENESS

IRREGULAR

the and

general

-

us

-

consider

the

Aj(t,x)

=

at

the

SYSTEMS

results.

Uniqueness

result.

system

fl ( t , x , u l , . . . , u ~ ) ,

l~i~N

ax

is

hyperbolic

closed

interval

Ai are real characteristic the

HYPERBOLIC

3u~ -

which

SEMILINEAR

- uniqueness

Existence

3u~ (71)

FOR DATA.

r e s u l t s e x p o s e d t i l l n o w on p a r t i c u l a r systems p h y s i c s , we s h o w t h a t C o l o m b e a u ' s theory ~iso

existence

3.5.1

Let

CAUCHY

ordinary

in a r e g i o n [a,b]

functions c u r v e s L: differential

of

E of

the

on E. through

the

x-&xis,

(t,x)-plane

that

is,

containing

the

functions

We a s s u m e t h a t A ~ A ~ . . . ~ A N. e a c h p o i n t of E a r e defined

The by

equations

dx = - A,(t,x)

, l~i~N

dt L e t ~ be the o p e n s u b s e t of ~2 b o u n d e d by the e x t e r n a l characteristic c u r v e s at a a n d b a n d by the x - a x i s , let I = ] a , b [ and

let

X

= ~ U

({0}×I)

Each function infinitely differentiable as

all

assumed compact

its

partial

to be uniformly subsets of ~

q E ~ ~+2 a n d E, a c o m p a c t t h e r e is C > O s u c h t h a t sup ID~fi(t,x,ul, (t,x)EK (u I .... , u N ) E ~ N

(see

the

figure

below).

ft : ~ N ' 2 * ~ is a n d ,as w e l l

L,I.//

derivatives,is bounded i.e for subset

of

Then f.i(t,x,G1,...,Gs) may functions A i are supposed to

/

'

~2

.... U N ) I ~ C

be defined be in C ® ( X ) .

,¢ /

on all

kN

\ # .t,,_

\

X'

¢

if

G i

E

@~(~2).

The

122

Theorem(71) has

(72)

I f ¢~ unique

Ui I t= 0

E !~,(I) is real valued, l~0,

(t,x)

= f' ( t ' x ' R i ( g ' t ' x ) . . . . . R s ( 8 , t , x ) )

f~(t,x,R~(e,t,x),...,Rs(e,t,x))

E X,

~nd

(R~-Rt)(e,O,x) for all curves we

(Ri-Ri)

~>0 have

and

($,t,x)

+

x

+ Ni(e,t,x)

6

= ~i (£2,xi

I.

= n~(tZ,x)

Integrating

(0,t,x))

along

the

characteristic

+

[fi(~,xi(z,t,x),R~(~,~,xi(Z,t,x)),.,.,RN(e,~,xi(z,t,x))) 0

fiO:,xi(~,t,x),

+ N i (¢,z,x

i (z,t,x))]

1 (~,z,xj

dz

;

(~,t,x))

.....

RN(¢,~;,xf(~,t,x)))

127 thus,

by

(85)

+

the m e a n

value

(R~-R:)(e,t,x)

ft

0 s a t i s f y i n g i

and

n~

E Z~[ I],

for

each

q E ~,

there

are

In i ($~,x i (O,t,x)) ] ~< (c/2)e q

(86)

I

for

all

(87)

side

(t,x)

] (Ri-Ri)

of

,Z,xi

E ~(K)

(¢,t,x)

(z,t,x))dzl

and

00.

L i

Then,

given

= xi({0}×~(K))ccI

implies

z i

same

way.

such

E QK,o,~,

that

i=l,...,N

.

i

The

proof

for

(a)

These

results

result,

under

fi,

is

Remarks.

proved

(b)

(for ruled

coefficients. extended and

to

the 5.

(c)

In

apply.

the

Then

results

-

are

We

with

one

linear

hyperbolic

reason

the

U t

Ill

the

see

case

x

ranges

above,

as

well

as

the

much

apply

in to

which

in

the

case

FOR

CAUCHY

interested 0

in

~n

more

several

NONLINEAR

have

Q

the =

nonlinear

~×]0,T[

been

nonlinear

n>l,

the

terms

(b)

sophisticated is

done

existencespace

f6]

method

(a) , in

of does

method Lafon-

uniqueness

dimensions.

PARABOLIC

EQUATION

DATA.

in

Ca

piecewise

0berguggenberger

remarks

abstract

of

A

in

This

the

the

Ill,

Ca

immiscible

results

or/and

Lafon

with

func-

with

several

systems

the 5.

medium

of

above

coefficients

UNIQUENESS

AU@U 3 =

made

existenceon

Appendix

heterogeneous

operators.

IRREGULAR

are

a

[i] . A n

assumptions [4];see

a medium

can

extended

EXISTENCEWITH

(97)

in

functions,

used

Biagioni

instance

pseudo-differential

0berguggenberger

3.6

case

from

restrictive

0berguggenberger

this

generalized

characteristics of

by

For

Appendix

not

in

follow less

Acoustics

is

fluids)

are

the

-

characteristics

fi

follows

3.5.4

uniqueness tions

k>0

p~rabolic

problem

134

(98) and

u(x,t) the

(99)

= 0

initial u(x,O)

with

~

the

in

condition

= $(x)

Dirac

regular

mass

~

with

no

solution

in

6

L~oc(Q)

satisfying

It

the

8~x]0,T[

is

~,

at

the

boundary,

proved

following

in

the

in

Q,

T

~

an

of

open

finite

Br6zis-Friedman

sense

sense

origin,

0 £

or

[i]

distributions

(97),

in

the

bounded

subset

that

this

problem

in Q

i.e.

there

of

~' (Q),

and

sense

of

not.

is

has no

u

(99) , in

:

f ess

(lO0)

(ess set

lim of

lim

|

t-~O

)

means

that

Lebesgue

concept

of

u(x,t)

this

measure

functions,

limit

0

;

of

the

to

if the

one

functions is

is

this

(~

~

~

the

The

when

slight

t ranges

outside

weakening

of

a

the

by

gives

~(0). two

the

The

limits

for

frame

may

f

in

is

is

that be

we

having

o n of

~o(x)

loss

initial than

(i00)

initial

a

loss

generalized

this

the

the

sense

r

cannot

sequence verifying

so

without

written

u~ ( x , t )

our

n>O,

weaker to

be

a

(98)

Colombeau's

that

l i m [ u~ (x,t) t*O J key

all of

explained

condition

lim | ) t-~O ~-~0

£~0

of

a

lim

lira

order

is

(I00)

condition

which

the

$ by

(97), in

[,×T]×~

corresponding

then

initial

u,~ of

difference

justified solution

i n ~" (~))

the

taken a

= ~n(X)

In

phenomenon

(ess)

while

in

condition.

condition.

naturally

C'(Q)

is

approximates

solutions

0 uniformly

initial

initial

= ~(0)

this

Un(X,O) converge

dx

limit).

Furthermore regular

~o(x)

of

the

condition : if

condition

u~

E 8~

as

dx

= 0

amounts

to

do

~(x) dx

are

permuted.

in

a

case

in which

the

135

Let with

us

initial Let

C®(Q)

when

of

us

shall

restriction

to

find ~

is

function

restriction

of

a

~s(8~x[O,T]). what

this

compact

8QX[O,T]

is a

following

(see

§2.1

function

defined on

local

Q

0

the and

defi-

to

(the

1.3.2) .

local

maps

and

8QX[0,T]

uniqueness

in

whose

in ~s,c(Q) see

by

and

solution

for

is

support,

function

also

unique

8~x[O,T]

generalized

is

the

to

given

generalized

establishes

to ~(fl).

in Ns (Q)

a

(98)

6.

has

with on

condition

than

problem

restriction

a

with

general

belongs

a solution

There

(97)

more

condition

means

generalized

t=0

that

whose

"c"

equation

in

initial

~(Q))

subscript

the

recall

the

We nition

retake

condition

A the

is

in

result.

That

is

solution

of

:

- Theorem,

3.6.1 Given

E

u o

Ss,c(fl)

there

is

E ~s(Q)

u

the

problem

i a) (101)

b)

u[ ~ = u o

e)

U[a~x[O,T

where

equalities

where

fl

Qx]O,T[,

is T

open

finite proof

For the

each

initial

solution

taken

or

not.

is

based

u E C®(Q)

- First, lu(x,t)l

on

sense

of

of

with

the the

k=O,l,2,..,

~"

generalized C®

solution

of

following

functions,

boundary,

(101)

lemmas

the

there

u o of

the

is

a

polynomial

problem

belongs

satisfies k

c

(lO2)

the

subset Then

condition

liull

Proof

in

bounded

is

Q

=

unique.

:

- Lemma.

3.6.2

the

I : 0

are

an

The

if

- Au + u S = 0

u t

~

maximum

~< Iluoll

c

principle

. L

2~+i

PkClluoll

(Q)

, (n)

(x,t)

)' (n)

states E Q

.

that

such

that,

to ~ ( ~ ) ,

Pk

then

136

By u t and

differentiation

u x

,

14i~n,

are

of

the

equation

solutions

(101.a)

we

have

that

of

i zt so

that

u t and

u x

,

Az

+

l~i~n,

3u2z

:

attain

0

in

their

Q,

extremum

over

~

x



. L

the

+

(Q)

solutions

I

Then

~gO

attains

:

±[u(x,t)

-

Aco +

used

:

±(Auo-u2uo)

tO(x~O)

=

0

~(x,t)

:

-at

in

Q

its

; since

maximum

gives,

in eGO

in

in

the

from

(104)

llntli

.

X

also

-

-

( 0

at

~

in Q

in

~< 0

ut

]0,T[ 8~

.

x

]0,T[

, its

follows

that

Then for

definition

of

(x,O)

~< i i a u o i ]

to

(Q)

all

x

E

fl ,

~,

~< ff

(see

estimates

of

Ilull

the

. L

f~.

in

u

+

IluoiiS®

(Q)

on

8~

Ladyzenskaya

L

×

]0,T[,

[1,

(Q)

we

construct,

chapter

VI,

analogously,

3])

which

kind

~ (Q)

. L

estimate

functions

(los)

)]

(103) ,

L

order

3~

t=0.

z Thus,

uo(x

in

Cdt ( x , O ) which

-

of

~t

In

(Q)

L

functions ¢d(x,t)

are

Iluol[So

cEUuoll

® L

+ (Q)

tluoli S . L

+ (0)

llAuol]

. L

] (~)

give

137

From

(102) ,

each

~ E

(104) ,

]0,1[

(105

, the

Ilull

~

we

rewrite

co(u)

~

c

If

f =

equation

- u 3,

relative for

to

cr E

lOl.a)

together

the

of

c

(~)

(i01)

satisfies

for

heat

as

Au

with

=

in

f

Q

conditions

equation

give

(lOl.b),

(see

(lOl.c) , estimates

Ladyzenskaya

[1]

) :

]0,1[ ,

c

(~)

By

iteration

P~

with

we

c

have

that

coefficients

Ilull

have

the

(Q)

c

for

all

independent

of

:

~,+¢,~+¢~

~

result

of

(~)

integer u 0

~

c

there

such

that

~

;

P~(/JD*Uoll

(5)

c

We

u

Iluol]

~)

u t with

solution

:

is

a

(~)

polynomial

li1~2~).

(~)

c

lemma. []

3.6.3

Let

i (106)

where Pk

k c

The

Lemma.

E

~

and

v t - Av

+

E

C" (Q)

a0v

=

f

=

g

in

v(x,t)

= h

in

in

coefficients

~Pk(tlaoll (~)

proof

v

v(x,O)

ao(x,t)~O

with

Ilvll

T

-

in

in

a

solution

of

Q

8~x]O,T[

Then

for

each

independent

~+i c

is

Q.

be

).[llflr (~)

Ladyzenskaya

of

k=O,l,..., a 0,

2~+~ c

[l,

f,

+llgll (Q)

chapter

there

g,

h,

such

~k+~ c

IV,

is

+llhll (~)

5].

a

polynomial

that

2~+1 c

] (~ox[ O,T])

138

Proof

of

theorem

3.6,1

Given

-

u o 6 ~s,o(~),

let

R,

E @M,~[~]

be

o

a representative

of

u o . For

each

6>0,

Ru

(8,.)

6 ~ ( ~ ) . By

the

0

classical results there (101.a) and satisfying

v~[~

= Ru

6 C" (Q)

solution

of

equation

(8,.)

V~I~xl0,T[

u

v~

0

(I0~)

Let

is

: 30,+~[

x Q ~

:

0

~ be

defined

u(8,x,t)

by

: v~(x,t)

We h a v e , s i n c e ve E C ® ( Q ) , for e a c h 6, t h a t u E $ C Q ] . The p r o o f that u E $M,,[Q] follows directly from lemma 3.6.2 and the fact that E @M,s[~] .

Ru

0

6 ~9 (Q)

In be

o r d e r to p r o v e u n i q u e n e s s of the s o l u t i o n , let u I, two s o l u t i o n s of (101) . T h e n u = u l - u 2 s a t i s f i e s

(1o8)

i uU I Q- Au0 + (u~+u~u~+u~)u = o

u~

=

UI~Qx[O,T , = o

If RI, R~ E @~, sCQ] a r e there are f 6 Ss[Q],gExs[O] each 6>0,

i

(109)

respective representatives and h 6 Ss[8~×[0,T]] such

[Rt-AR+(R~+RIR2+R~)R] R(8 n/n-l,.,0) R(sn/~+~,.)

(8,.)

= g ( 8 , .) = h(8,.)

:

f(8,.)

in in

3~×]0,T[,

in

of u I , u 2 , that, for

139

where

R

g and

h we

= R]-R~. have

3.6.4

-

Let representatives property :

f

fl

using

lemma

3.6.3

and

the

Uo, 1, uo, 2 E ~ 5 , o ( ~ ) be Ro, i E ~1, ~ [ ~ ] , i=1,2,

are

c>0

and

~>0

such

~ c for

JRo,~(~,x)[dx

such with

all

~,

00

and

suuset

operator n>0

such

K of

of

~2

order

and one

D in

a

partial

(x,t)

there

that 1

(5)

ID~(~,x,t)J~

sup

i

o logm

(x,t)EK

E

if These ~s ( ~ )

- But

0

estimates

they

are

not

< ~