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

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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

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These

more

all

of

its

of m o s t

existing

when

they

been

to the

of

physics

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precise

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have

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solutions

checked

equations:

solutions

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instance

solutions

of

formulas,

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new

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setting

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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

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results"

formulations

existence-uniqueness

exist.

functions,

give

facts.

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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

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of

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equations,

theory

theory

properties.

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this

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

through

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ics

theory

clasof p h y ~

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C" f u n c t i o n s

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,

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neers.

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classical

system

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systems

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solutions

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of m a s t e r i n g which

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to c o m p u t e

simulations

possibility tive

one

instance).

of

systems

by

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conservative (in c e r t a i n

efficient

numerical

dynamics).

used

methods

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maticians,

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systems,

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theory

basic

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n-dimensional

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cations, needed

text

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audience on

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sketch

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or

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acquainted

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tion

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1.6

texts

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ordinate tion the

a natural

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products, dimension

space.

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well §1.8

of

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of

to

the

who

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remainder

not

this

dimensional

gener~lized space

of

familiar

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this

natural

simplifications

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definition

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significantly

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obtain

free

En).

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concepts

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In

define

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strong

subspace

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natural

weak ~(~),

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viewpoint. the

first in

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and

§§

other

properties

to

introduce

concept trouble

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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

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spaces

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original

a purely

~n

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section

spaces,

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locally

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euclidean

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• (Q), as

~

reader

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understanding

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introduction

functions

Fortunately

locally

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subset

J.F.Colombeau

holomorphic

~' ( ~ ) of

how

I is open

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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

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the

fying

proofs;

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slightly ized

We

more

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technicalities

in

of

previous

definitions

various

Chapter

authors

in

of

sarily

open

tions

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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

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to

(by

neces

this

the

We

C®

func-

natural

has

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Whitney

exten-

generalized

analysis

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prove

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space

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not

concept

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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

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Chapter

some

on X and

Whitney

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sketch

~(~)

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just

automatically

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elements

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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 <5~ : x E ~) ( ~ is t h e D i r a c m e a s u r e at with the classical product fl.f2. This fact led Colombeau consider in C ® (~' (~)) t h e e q u i v a l e n c e relation r : R1 r If

A denotes

R~ ~=~ R 1 ( 5 ~)

the

= R~($~)

may

for

all

x

x) to

E fl

map C" (~" (~)) ----* C" (fl) R e---. (x ~ R ( ~ x ) )

then R E Ker Therefore

the

algebras

A ~

R r

C®(@ • (O))/Ker

0

A and

C" (~)

= L(@" (fl),~)

are

isomorphic (if E is a l o c a l l y c o n v e x s p a c e , L(E,~) denotes the s p a c e of a l l c o n t i n u o u s linear maps from E into ~ ; one proves classically that L(~" ( ~ ) , ~ ) = C®(Q) : reflexivity of C ® ( ~ ) )

which

A new concept of derivatives generalized the derivation in t h e

and which i n C" (fl) .

C'(@(f~))

corresponded,

via

the

map

A,

in C ® ( ~ ( ~ ) ) was defined sense of distributions to

the

usual

derivatives

For these reasons, Colombeau was seeking an ideal whose intersection w i t h C ® ( @ " (~)) w o u l d be K e r A.

of

the

The

following

given

f

REEer

x i ~(x)

(KCC~,

A,

dx

for

result

gave

qE~,

= 0

~E~q

if

short)

(9.)

a convenient := <~

i E ~

there IR(~.~)I

for all xEE defined by

and

for for

all given

the

It was this definition of a n

e>0

<

such and

and

~>0

that

K a such

of

[~(x)

compact

dx

Ker

A:

: i

subset

of

that

ce q+~

0<8<~.

Here

~,~

denotes

~,×(A)

= 8 -)'

~(~-~(A-x))

the

A E ~. T h a t is, R ( C ® ( @ ' (~)) s a t i s f i e s ~ a n d K if, a n d o n l y if, R E K e r A.

an element of grow very fast consider t h a t is,

E ~(~)

i~IiJ~q}

are

characterization

element

of ~q

inequality

(2)

characterization of K e r A t h a t w o u l d p r o d u c e i d e a l of C® ( @ ( ~ ) ) . H o w e v e r , when multiplying

~er A by an element w i t h 8-~, w h e n g ~

of O.

C®(~(~)), the product might Therefore he decided to

only elements of C ® ( @ ( ~ ) ) with a moderate elements R w i t h the p r o p e r t y :

growth

in

8-i

alk,

for

every

ECC~

and

D =

(0<~]k[ <~)

k

partial

ax~ ~ . . .Ox~ "

derivation E ~N

operator,

there

are

c>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 - ~

0<~<~.

these

obvious

there and

elements

of

that

DR

E @~l(~(~))

C®(@(fl))

of

two

C ® (@(~]))

is

if

@~(@(~))

elements

important examples distributions o n ft.

there

R of

E

denoted

@M(~(~))

are

the

is

by and

still

elements

of

@~(~(~))

Now, with for

every

such (N)

an the

n>0

ideal to following KCCO

that,

for

and

be c o n s i d e r e d property : D derivation

each

~ E ~,

all

This

x E K and

ideal

•

X

there

there

of

all

is N

are

c>0

R

E and

Finally

the

Many

e~ ~ - ~

C®(@"

the (~))

initial

requirement

=

.

Ker

A

:

quotient

~lgebra

containing

(0))/Ker

~

O<~
satisfies N

an

q)N,

set

satisfying

for

is

the

operator

with

I(DR)(~,~)I

C®(~)=C'(@'

is

A,

results

C®(~)

and

containing

were

proved

as

a

@' (~), by

subalgebra, since

using

this

since

~N@' (~)={0}. definition

but,

working with representatives of elements of ~(~), Colombeau realized t h a t t h e y d i d n o t n e e d to be d e f i n e d on the whole ~(~), since only their values on functions of t h e k i n d ~ , x , with ~ E ~q (q l a r g e e n o u g h ) , e > 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 0
I

This elements their

were

set

appear

{~,x}, and

: an then

sometimes

was

defined

being

Again, to

space

denoted

denoted

in

the by

working

same

on

applications,

of

@~(~))

it

to

be

this

@(~(a)) as

and those

its of

moderate C®(~(~)),

@~(~(~)).

element had

by

manner

extension

was

extended led

to

to

another obtained an

open

complications

difficulty first subset in

on U

the of

proofs.

used set ~(~)

where

Therefore might be

U

satisfying

the a

property

~(U)

:
sets subset

(PI),

C®(U) were of @(fl),

: U *

In We

dropped. algebraic

C such

that,

if

~ ~--'~ R ( ~ , ~ )

~

~ E ~N

and

inductive

when

a

constructed,

is

limit not

was

the

is

denoted

simplest

representative

of

the

U w~s

chosen

set

operator

are

fixed

C~}.

by

definition

yet.

~ generalized often

U

((~x~) (y)=~(y-x)).

in W later.

It w a s

function

x~o,

E is new

noticed

on

where

~ xx

wms is

the

~

Then,

why not

s e t U as t h e c o m m o n d o m a i n of a l l r e p r e s e n t a t i v e s functions on ~ ? It is p r o v e d in Colombeau the consideration of functions with domain ~×~0 gquivalent tothe morecomplicated u s e of t h e a b o v e Besides the simplific&tion modification o n t h e s e t s ~ q . In

~nd The

@~(~(0)).

xE~

tr~nsl~tion

~>0

set {~,~ : x ~ ~} is c o n t a i n e d in a n o p e n s u b s e t of fl, t h e n t h e f u n c -

this definition differentiability sh~ll come b~ck to this point

This that,

x

@~(U), open,

where

such that the U, w h e r e ~ is tion

repl~ced by sets not necessarily

sketched order to

to

t~ke

this

of g e n e r a l i z e d [2], § 7 . ? , that is indeed inductive limit

above, t h e r e is ~ i s o define composition

products we shall

and restrictions of g e n e r & l i z e d functions to s u b s p a c e s define different sets ~q, which further,like the

concepts

sketched

which and

does

of

not

the

one

bound The

Rosinger ~q

&

has

a concept

change

to

general bounds from ~ into

technic&l

sets

give

on

of

of

generalized

b~sis,

a

change

of

the

functions of

origin

norm.

Also (N) m o r e function

above,

depend

[4] would

is

needed

elementary is be

consider

for

the

replaced

one

by

where to +~

certain

definition

exactly

instead

c~ ~ ( q ) - N , ~+ tending

the

as

properties, given

given

in

sets

~q

bound

C~ q-N

in

¥ is an increasing ~t i n f i n i t y . This more

in

§ 1.2 In

see

1.2.16

Colombe~u below

below.

[3]

in w h i c h

the definition

and our given

in C o l o m b e a u [3] a n d R o s i n g e r [4], t h e r e s t r i c t i o n of a g e n e r a l i z e d function on ~ to a subspace of dimension l e s s t h a n n is n o t d e f i n e d ~s a g e n e r a l i z e d function on the subspace, since the sets ~q

depend

too

much

on

the

space dimension.

This

is

quite

n~tural

since the concept on the behaviour

of of

restriction to the generalized

to the subspace. In t h i s in w h i c h the restriction any generalized function

a subspace

depends naturally function"infinitely close"

book we adopt anoriginal to any linear subspace on ~n is a g e n e r a l i z e d

V. T h u s w e g e t n i c e r properties at t h e the sets ~q whose definition is s l i g h t l y one

1.2

of

-

the

AN

sets

-

Let deduced

E~

DEFINITION.

an

an

n

dimensional

inner

real

vector

<.[->-

product

Let

space

with

~(mE~)

be

norm the

vecior • (En)

space of t h e m - h o m o g e n e o u s complex polynomials in E~. be the algebra (for t h e u s u a l operations of f u n c t i o n s )

t h e C" denote

functions by dx a

by

an

how

from Haar

orthonormal a

Haar

obtained

E n into measure

basis

measure

: take

isomorphism

with the

Notations.

be

from

price of working more technical than

~q

ELEMENTARY

1.2.1

presentation V of ~n of function on

a

on

on

basis

between

~ on

real

~"

and

V,

measure

vector

el,...,e

~

in

given

j (X1,...,Xn)

real Let of

with compact support. Let us E~ s u c h t h a t t h e c u b e d e t e r m i n e d

E n has

a

I'IE

=

1

(let

space

V..

us

V.

of

j

: ~*

Let

recall

here

dimension --* V,

n

be

is the

by

X = ~

xie i •

i=l

Let

f

:

V.

integrable

--* ~

be

on

~

and

f

f ( x ) dx

such

let

c>O

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<~i<...<~

valued.

p are

C® ( O , + ~ ) , c o n s t a n t

such

in

a

given

rational

O-neighborhood,

that

÷®

(3)

Further,one

origin,

that,

~

support,

nonvoid. the

1 dA:

f

~ (,~)

o

-2

+m

(4)

f

(ii i t~(i) d / = O

i:1,2 .....

if

p

0

constant functions

We in

start with an a O-neighborhood

~l, . . .,~p+16~(0,+~) = ~0

satisfies

(3)

and

arbitrary function and with compact and

+ xi~i

(4) , i . e .

the

+

xl,...,xp+16~ ••. +

~i's

Xl

xi's

such ~p+l must

satisfy

+m

1

+m

o

o

2

o

f. ~

+~

i ~i (,~)di+ . . . + X p + I

A

+~

A

I ~ p ÷ ] (A) dA---

i

A

igyo (i) dA

f. +m

A o

f. 0

+m

P ~i ( A ) d i + . . . + x p +

:

+I

0

f. o

f.

which is We seek

that

+m

0

xl

xp+l

and

+m

(5)

~06C® (O,+m) support.

P *p+ i (A)dA=-

A

0

P~o ( A ) d A

If

(Sn)n:1,z,..

is a 1

1 supp

~n

C

[- - - ' +

--],

n

will

regularizing f~n(A)

sequence,

dA

=I

, ~n~O

that on

~

is , for

, ~nE~(R), all

n,

we

so

that

n

choose

~j (l)

= ~n

(l- Aj) o

where Aj>O , j=l, . . . , p + l , and (5) h a s a s o l u t i o n (x I ..... x p + 1 )

n06~ will Since

be

determined

+~

lim

(A) ~ ~n (A-Aj)

dA

=(ij~ i

0

and

f in to

order to prove that

+®

1

~n(A-Aj)

dA

=

1

if

n > --

0

Aj

find n0E~ such the determinant

1

that

(5

has

a

1

solution,

it

suffices

1

a

(Ai)

I

(A2)

i

(hp+i)

1

P

(A2)

p

(Ap,i)

p

a

(Ai)

is

different We

from may

O.

write

O/ i

=

--

, where

O
,

and

¥

are

¥ i/w

natural becomes

numbers.

Then

setting

/~j

z

Aj)

,

the

determinant

10

1

gl

1

I

1

gz

p+ i

P

/Z 2 P

P

p+ 1

Setting

~i

=

~i

1

for

@>0

,the

determinant

1

1

p~ 1

p

~

is

_--

~:~ %

+.

.

. +B

1

p

1

~

1

~

choose

~ /

P

der

Monde

p~

B

p+l)~

p

a Van

1

p+l)~

2~

which

becomes

determinant,

if

we

p

p

i.

Then

we

take Aj

1.2.6

The ~n~n

is

-

= ~'J

[]

Remark

n-dimensional

defined

=(~j)'

by

spherical

coordinates

mapping,

T n

:

11

and

maps

it

86[0,2~]} of

for

T n

some

of

z

D

COS

=

D

sin

ffl

cos

q2

X 3

z

D

sin

ffl

sin

~2

Xn_ I =

D

xn =

sin

Q

onto

is

Xl X 2

=

the

the

The

-

sin

.....

ball

that

we

is

the

If

~3

~n-2 ~n-2

~n-~,

unit

e)6~n B~

cos

sin

@

:

D6[0,1],

~i6[0,~]

Notice

that

the

sphere

S n-1

of

F,

and

x~

+

. . .

+ x~

have =

~2

Remark

constant

0

(n-1)-dimensional at

the

origin

1.2.8

-

Definition

q=0,1,2

..... , we

in

o

area

of

the

radius

~n

set 2n

~ q (E n) =

( W 6 ~ ( E n)

such

that On

for

some

Notice

~16~q

that

,

Jacobian

form

o

centered

cos

~i.,.sin

~i,..sin

((~,~1 closed

function

1.2.7

D

G I

~

(~)). is

C®

since

~i

is

constant

in

a

O-neighborhood.

l,

12

1.2.9

Proposition

If ~E~q(En) , q=0,1,..,

~(x)dx=l

then f

and,

if q~n,

E n

PE~(iEn),l~

i ~q,

f

P(x)

~(x)dx

= O°

E

Proof

~

Let

;

~E~ o (En) . Then

2n

E

Qn n

III

+~ n

2n E

Qn

o

o

n ~n . . . . .

o

0

n

dpd~ l ..d~n_2d8

÷o

= 2n I

on-~

~i (On) dO

0 ~m

= 2 f

~i (~)d~=l 0

If P6~(~En)

with

i

odd then

it

is clear

r

that I

P(x)~(x) E

if ~6~q (En),since Let

~ is an even

jE~ be such

I

P(X)

E

that

function,

2~<2j~
~o(x) dx = -

2nl ~n

n

n

E n

if ~E~q (Ea) , q~n,

P(x)

~i (Ixln)dx

dx=O

13

2n f + ~ : -D n - l + 2 j ~ i (pn) ~ 0

(where

c n independs

~n definition

Notation

If

and ~

all

~ E ~ ( E n ) , we d e n o t e 1 x (x) : - - ~ ( - )

by

~

the

function

8

xEE n

1.2.11 Given

Proposition m,n=l,2,..,

there

I~ verifying

0 D

8n for

I +~ g2j /n ~l (a) dg= 0

of ~ q ( ~ )

1.2.10

~>0

cn

on ~i)

= _~Cn _

by

d~

the

following

is

a

: ~o (Era) ~ ~o (En)

properties

(a)

I m (~q(E~))=

(b)

I~

=

:

~q(En),

(E

I~

q:1,2,3

I~

)

(~)=(I~o)

o

;

m

m/~,

for

all

(~) (d)

....

id 0

(e)

bijection

I~

=

I~

for

~ 6 ~ o ( E ~) all

all

p=l,2,...

and 8>0;

14

Proof -

By

definition

of

2q(En)

: 2 o (~)

In defined

is

a

bijection

-~ 2 o ( E ~ )

by

(I~l)

(x)

2n --

=

~(Ixl

n)

R

for

, there

all

~i

E 2 0 (~)

and

l

x E E n . If

z~

m,n=l,2

: 2 o(E~)

....

define

-. 2 o ( E ~ )

by Iron =

It notice

or

if

that

~E~0

is

obvious

if

~iE2o

that (~)

and

In

0

(Ira)-i

I nm s a t i s f i e s E>O we have

(a)

(b)

and

(d)

Let

us

(E~) , [(im

) - i (~)]

~

__

~

(i m

l

¢

i zm)

Then i m (~.) = (inO =[inO

(Ira) - I) (~)

~_in[(im

(Is)

ra/n

-1 (~)]

~-

) - i (~)] (Inm~)

m/n

_-in ([(im)

- I (~)]

m)

15

1.2.12

If orthonormal

Notation.

-

~ is basis

an open s u b s e t of of E,, f E C®(fi) a n d

E n, ~ = { b l , . . . , b ~} is a n a = (al,...,a ~) E ~n, let

1

D~f

=

o...o

(f)

where 8

f(x+tbi)-f(x) f(x)

= lim t~O

8x i and 8

~

[8__~i]

8 o...o 8x i

-

If ~ = < c l , . . . , c 8f by -8y~

(x)

of

we

ei,

the have

1.2.13

If ~ (k~)

~

~}

is

the

and

(~i

orthonormal

derivative

following

times)

8x~

another

directional

-

of

8

i =

of

relation

basis f at

between

x

of

E n,

denoting

in

the

direction

D~

and

D~

:

Lemma.

~

real

are

as

above

numbers

and

such

a 6 ~n

there

is

a

family

that

IBI=i~I

J~l=ial

The

proof

is

1.2.14

set

of

classical

(matrix

of

change

of

basis)

-Notations.

(a) If Q is a n all functions

open

R

subset

: ~o(E~)

of

×

E a,

Q---*

we

denote

by

@[Q]

the

16

which a r e of c l a s s ~0(E~). Notice that subalgebra

¥ from

(with

C ® in t h e v a r i a b l e x E Q, f o r e a c h f i x e d ~ E ~ [ Q ] is a n algebra of which C®(~) is a

the

inclusion

(b) W e d e n o t e i n t o R+ s u c h

~

fEC®(0)

by F the that lim

~-¢ [ ( ~ , x )

set

of

=

+m

~(q)

all

~-~ f ( x ) ] E @ [ ~ ] )

increasing

.

functions

(c) If • denotes an orthonormal b a s i s of E . , w e d e n o t e by : @~,s[~] t h e s e t of a l l R 6 @ [ ~ ] s u c h t h a t f o r all compact subset K of ~ (KCC~, f o r s h o r t ) ~ n d a l l q E M" t h e r e is N ~

such

th&t

for

each

~ E ~N(E.),

there ]

iD~R(~,x)

for such

all that

that,

if

are

c>0

and

~>0

satisfying

~< c~ "N

x E K a n d 0 < ~ < n ; a n d b y ~ s [ ~ ] t h e s e t of a l l R E @ [ ~ ] f o r a l l K C C ~ a n d q E ~" t h e r e a r e N E M a n d ¥ E F s u c h ~ E ~q(E,),

for all x E g and equivalently state

with

q)N,

there

ID~

R(~,x)I

are

and

n>O

satisfying

c~ 7(q)-~

~

O<~
c>O

(by

modifying

N and

¥ )

we

may

It is clear that ~M,s[~] is a s u b a l g e b r a of $ [ Q ] a n d that ~s[~] is a n i d e a l of @ M , s [ O ] . It is a l s o s t r a i g h t f o r w a r d to verify, from lemma 1.2.13, th&t @.,s[~] a n d ~ s [ ~ ] do n o t d e p e n d on the chosen orthonorm&l b a s i s ~. T h u s w e o m i t t h e s u b s c r i p t in @ M , s [ O ]

and

1.2.15

is

if all

an

it

Definition.

-

If ~ is element

A has

an of

o p e n s u b s e t of E n, the quotient space

a generaZized

generalized function o n 0 is s a i d to a representative RE$~[Q] which takes

~E~o(En) 1.2.16

modified

ds[~].

real -

valued

and

x E

function

on

be reel valued real values for

~,

Remark.

Property (N) in the by the introduction of

definition ~ E F,

of in

d (see §l.1) w a s order to obtain

17

uniqueness of a s y m p t o t i c e x p a n s i o n s a n d a l s o u n i q u e n e s s of the h o l o m o r p h i c c o n t i n u a t i o n for h o l o m o r p h i c generalized functions (see Colombeau [2, § 12.8] and [3, A p p e n d i x 5], r e s p e c t i v e l y ) . This i n t r o d u c t i o n of ~ is a l s o n e e d e d in § i . i 0 for s o l u t i o n s of algebraic equations.

1.2.17

-

Inclusion

Let C(~) f u n c t i o n s on Q.

of

denote

a)

We

the

Rf (~,x)

= [ J

f(l)

into

~(~).

space

consider,

If fEC(En) , let R~ (6)

C(Q)

of all

first,

the

case

: ~0 (En)XEn ~ $ be

~(l-x)dl

continuous

, ~E~0 (E~),

complex

2=E n.

defined

by

xEE n

E n

given

Then, clearly R ~ 6 @ [ E n ] . Let ~6~ n a n d ~ an o r t h o n o r m a l b a s i s

(D~Rf) (~,x)

: I

us v e r i f y t h a t of E~, we h a v e

f(k)

R~6@~[E~] that

D ~ ( ~ x ~ ) (A) d~

E n

where

~x

denotes,

('cx~o) (y)=~(y-x)

.

as

usual,

the

operator

translation

Then

(D~R,) ( ~ , , x ) = ~ ) - i ~ a ~

f (A) zx ( D ~ ) ~ (A) dA E n

1

A-x dA

E

~n n

=~y,

• rf

f (x+~#) ( D ~ )

(g) dg

E n

If K is a c o m p a c t

for

all

x6K and

subset

0<~
of E n t h e r e

is a c o n s t a n t

c>0

such

that

18

We

have,

then,

a

f~C(E~) given

by

(6).

linear

map

~ m, ~ ~,[E~]

Since R~ (~e,x)

= ~

f(x+~g)

~(g)dg

~

f (x)

as

~0

,

E n

then,if

ReE,#[E~]

,

we

get

f=O.

f6C(E~) b) for a n y o p e n first a map

set

In

Therefore

~

[class

order

~ (whose

to

associating Let, and

X, (x):l

to

each

~ E ~ 0 (En)

of

define

boundary

i

we

is

6 ~ ( E n)

the

inclusion

injective

number

i (~)

(Xe}e> 0 be a f a m i l y of i xE~, Ixl~ -- a n d d ( x , 8 ~ ) > ~

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 0<£<~. It is important to n o t e t h a t the arbitrariness in the c h o i c e of the m a p i or in the c h o i c e of the family {~,} d i s a p p e a r s at o n c e in the q u o t i e n t ~(~) so t h a t t h i s inclusion is n o t r e a l l y d i f f e r e n t from the apparently simpler case ~=E n .

19

(c)

If

fEC" (~)

we

get,

then,

two

elements

of

:

RI (~,x)

-- f(x)

and R2, g i v e n by ( 6 ' ) . We prove that R I - R 2 E ~ [ ~ ] , that is, R l and R 2 p r o d u c e the same g e n e r a l i z e d f u n c t i o n in ~ : g i v e n KCc~ and ~6~ 0 (E=),by (7) , there is n>0 such that for x6K and 0<E
i-x I

= f(x)

f(A)

) dk

~(

E n

= f (x)

- f

f ( x + ~ ) ~ (g) d~ E n

= -I

If (x+~g)-f (x)]

~ (g)dg

E n

(we u s e d

in the

last

equality

that I

~(g) d#=l).

If ~ 6 ~ q ( E n ) ,

E n

q~n

,applying

x, we

get

uniformly

we

get

that for

Taylor's xEK.

a similar

1.2.18

formula

I (Rl-R2)(~,x)

-

If

aEM n a n d

bound

for

up

I has N

to

order

a bound is

an

q

of

to

the

orthonormal

f

at

the

point

kind

c~ q+l,

basis

of

E n,

I D ~ ( R I - R ~ ) < ~ , ,x) I •

Remarks.

If we d e n o t e by Cr(~) the linear space of all p i e c e w i s e c o n t i n u o u s f u n c t i o n s on ~, i d e n t i f y i n g two f u n c t i o n s if they are equal e x c e p t p o s s i b l y at their p o i n t s of d i s c o n t i n u i t i e s , then, u s i n g the same f o r m u l a (6)or (6'), we have the i n c l u s i o n C~(O)

into

We can • (~) .

prove

in the

same

c ~(~) way

the

inclusion

of ~' (~)

20

1.2.19 We

Remark

-

have

imposed

in

the

definition

1.2.4

property of ~i to be c o n s t a n t in a O - n e i g h b o r h o o d , defined in 1 . 2 . 8 w o u l d be C ® o n E~. H o w e v e r , but non trivial, proof, shows that this condition necessary. More precisely , if t ~ ~0 (t) belongs ~o a n e v e n f u n c t i o n , then the map x ~ ~o ( ] x I E C®(En ) . Before

the

proof

we

Lemma

1

-

Let

f E C®(~),

Then

there

are

polynomials

state

~ E ~ n, Pj

two

lemmas,

i~l:m

in En,

and

~

of ~ 0 (~)

the

in o r d e r t h a t an elementary, is n o t really to C® (~), w i t h ) belongs to n

an

o.n.

deg(Pj)=m

for

basis all

of

E n-

j,

such

as

r -* O.

that Pj(x) D~(x

~

f(Ixlz

))(x)=

~

n

f~J)(IxlE

)

j:l

n

ix[2m-J E n

for

x~

2 - Let

Lemma If

O.

~ E ~ n,

fEC®([R)

]ff~=m~M

D~(x

and

and

~

~

assume

is

f(Ixls

))

that

o.n.

an

f(J)(r)=O

basis

= o(Ixlz

)

(r s - j )

of

E n then

as

Ixl~

n

Proof

of

the

remainder

to

asser%ion

:

Applying

~0

0,

we h a v e

around

r 2~ - - ~o~ k

r 2

~o ( 0 ) + . . . +

~o ( r ) = ~ o (0)+ 2

Setting

f(r)

-

constant

C.

follows D,(x

from ~

i

lemma

f(Ixls

)) n

with

integral

fr ( r - Z ) ~ k ~ o ~ 2 k + 1 ) (z)dz

(2k) !

(z)dz,

0

we h a v e

that

= C

i

r

(r_z)2k-j

~2k~i)

(z)dz,

0

Thus

f(~> (r) It

o.

0

f(J) (r) some

formula

(0)+'

(r-z)2kW0c2~+l) (2k) !

for

Taylor's

(2k) !

!

~ n

= O(r ~-.~*~)

2 that = O(Ixl~

if

as

l~Ixm~2k,

r~O. then

z~+i-m)~
as

Ixlz n~

O.

21

Then

IxIE2 k O~(x ~ if

~0(Ixl~

)))

= D~(~0(O)+...~

"~2k)(O))+O(IxlE

),

Iqi~2k. But

arbitrary,

the

first

term

is

infinitely

differentiable

and

k

is

hence lim ~xl

D~(x

w

~0 ( X l E

--0

)) n

E n

exists

for

all

continuously belongs

to

Detailed

1.3

C® ( E n \ ( O } )

proofs

LOCAL

-

Let obtain

We

can

have to

, we be

-

get

~ ~nd

~"

be a

that

origin the

in

two

all

and

derivatives

since

x *

Schwartz

[]

) n

[2].

AND C O M P O S I T I O N S .

of

generalized

open

subsets

restriction

are

W0 (Ixiz

result.

RESTRICTIONS

Restrictions

canonically

proved

the

found

PROPERTIES,

1.3.1

We

~ E ~ ~.

extendable

of

functions

to

open

sets.

E n such

th&t

~'

~.

subset

~"

c

map

G E S(Q) *-~ G ~. E S(~') by

restriction

1.3.2 We if

the

of

-

representatives

Support

say

that

restriction

of G E

GI~,

a

generalized

~(fl)

is

is

equal

nulZ to

function. ~n

zero

an in

open @(~').

We

say

in

~"

if

of that

i

two

generalized

difference

is

Let

functions

null

in

{~i}ie~

be

a

compact

subset

of

compact

subsets

GI~

is

null

in

in

U

on

coincide

~

their

~' . be

of

a

U

family

~i

of

We

open

may

subsets

write

K

of

as

a

~ and

let

K

finite

union

such

that

i~I of

~i,

~i,

for

iEI.

all

i 6

Then I,

, if it

G

E ~(~)

follows

is

that

i null

~i - In

other

in

i~I which a generalized

in

which

G

is

null.

words,

the

function

G

union is

null

of is

G I u o i~I i

is of

all

open

subsets

an

open

subset

of

Q

22

the is

complement null.

It

of a g e n e r g Z i z e d function G E @(f) l a r g e s t o p e n s u b s e t of ~ in which

The supporZ in f of the

Definition

is

1.3.3

denoted

-

by

supp

G.

Theorem.

L e t {fit} be a f a m i l y of o p e n s u b s e t s of E,. F o r e a c h i n d e x I, let G t E @(fit). S u p p o s e t h a t , for e v e r y p a i r of i n d i c e s E I w h e r e fit N fj ~ ¢, the r e s t r i c t i o n s Gt[ ~ na a n d Gjl ~ nn

i E i,j

t

j

i

coincide (we s a y t h a t the f a m ~ l ~ { G i} is c o h e r e n t ) . T h e n exists a unique generalized function G on f = U fit w h o s e tion

Proof

to

each :

(*)

set

If

hypothesis

Rtln

nQ

equal

@M[fi]

to

a representative

t

E X[f~

=

[

~i (x)

s u m , for each locally finite

fl b y

remains

to

functionsin prove

to

prove

(~,x) ( R is

that

6

@.[flt], I

= Gt,

G[~

I.

R i

)x~ i

~0(En) X defined in

f , the

since whole

in ~ 0 ( E n) x ft and, if x ~ ft R is a f i n i t e s u m of p r o d u c t s

that

n

E

by

R i (~,x).

RI~Q(z

i

I,

N f)].

i

each

G t, i E

t h e r e is a p a r t i t i o n of u n i t y in C ' ( f ) , { ~ t } t e l , the c o v e r i n g {ft} and such that {supp ~t)tel is L e t us c o n s i d e r the s u m

of C ' f u n c t i o n s o n

for

of

j

~ 0 ( E n ) X f s i n c e e a c h R t is d e f i n e d we h a v e ~ t ( x ) = 0) . S i n c e , locally,

equivalent

there restric-

G t.

is

- Rj[ a nQ

is a finite ~t}t~[ is

It

J

that

j

R(~,x)

This {supp

is

E

have

We know that subordinated to locally finite. (8)

ft

Ri

we

t

is

is G

E

~/[fli]

we

have for

that

each

R E @M[fl].

i E

I.

This

23

Given an

open

that,

K a

compact

neighborhood

for

each

(~,x)

R(~,x)

subset

~ of

=

K.

~i,

there

x

we

E ~0(En)

~

of

Then ~,

~j(x) Rj(~,x)

the is

sum

(8)

Ii c

1

is

finite

finite

in such

have

+ ~(x)

R,(~,x)

jEIi

Since

~

~j (x)

=

1,

for

all

x E

~,

we

may

write

jEIi

If

x

E K,

(R-Ri)(W,x)

:

the

j which

indices

for which x E supp result, b y (*) .

~ ~) (x) jEIi j#i

q) . S i n c e

[R) ( ~ , x ) - R i ( ~ , x ) ]

contribute

E N

supp

~j

to cc

.

this

~i

N

sum

~j,

we

are

those

have

the

[] 1.3.4

Notations. respectively,

-

If

(a)

Composition

~ and ~' let us denote

@[~,Em]

, the R

which

are

C®

(b) @ M [ ~ , E m]

in

the

~ME~,Em]

=
of

are open by :

set

of

all

: ~o (E.)

variable

, the

E @ [ ~ , E m]

there exist

generalized

X

x E

functions.

subsets

of

E~

and

E~

functions 0---~

~,

Em

for

each

fixed

~ E ~0(E~)

set

such

that,

for

all

is N E ~ s u c h t h a t , f o r c>0 and w>0 satisfying [ (D~R) (~,x)[z

~< c

I
all

(~ E ~n

~ E ~N(En),

there

~-N

m

for

all

x E K and

0<~<~}

(9

is

an

o.n.

basis

of

E n)

;

24

(c) analogously (d)

~[~,Em] ' to ~ [ ~ ] , for ~(~2,Em)

the

the subset of functions taking

@u[~,Em] constructed values in Em ;

quotient

as

a

(e) ~(~,~'), the set of all G E ~ ( ~ , E m ) which have representative R E @~[~,Em] w i t h the f o l l o w i n g property :

a

family

Obviously an element of of m e l e m e n t s of ~ ( ~ ) (by

~ ( ~ , E 2 m ) m a y be c o n s i d e r e d identifying E~ w i t h ~) ;

for e v e r y K c c ~ t h e r e is N E ~ s u c h t h a t , if ~ E ~ ( E , ) t h e r e is n > 0 s u c h t h a t the set { R ( ~ , x ) : x E K, 0 < ~ < W } is c o n t a i n e d in a c o m p a c t s u b s e t of ~" , ~ n d e ~ e n d e n t of ~.

(9)

N o t i c e t h a t , then, all r e p r e s e n t a t i v e s h a v e t h i s p r o p e r t y . In g e n e r a l we h a v e ~ ( ~ , E m ) example 8 ~(~,~) .

of G E % ( ~ , ~ ' ) will / ~(~,Em) , for

Example

: If ~ and ~" are open subsets of ~ a n d f is a continuous function f r o m ~ i n t o ~' , t h e n f £ ~ ( ~ , ~ ' ) (of course if we c o n s i d e r the s e t s ~ q (~) m a d e of real v a l u e d f u n c t i o n s ) . In f a c t , let R r : ~ 0 ( ~ ) X ~ --~ ~' be d e f i n e d as in (6) or (6') . G i v e n K C c Q a n d ~ E ~0 (~), t h e r e is c(~) > 0 s u c h t h a t

lRf(~,,x)

f(x)]

~ c(~)

sup

lf(y+~g)

f

yEE

~E~Upp~

for

all

x E K and

~>O

fx,, (~)

small

=

enough.

sup

Setting,

[ f(Y+~g)

-

f(Y)

I

,

yEN ~Esupp~

we

have

that

lira f x , , ( ~ )

= 0.

f(K)

+

If

W>0

[-n,n]

is

c

~2"

such

that

(y) I

25

there

is

~" ( 9 ) > 0

such

that

fx, ~ (£)

< c(~)

if

0 < ~ < ~ ' (9).

Then


f(K)

Definition

~M[~" ] • [~] N

(i0) for

that

is

0 < a < ~ " (~)}

independent

theorem

.

If

R~ E @ ~ [ ~ , E m] a s

with

E ~ such

have

: x E K,

[-~,W]

and

and

be

+

we

the

that,

following if

c

of

f(K)

~"

a n d G~ E ~ ( ~ , ~ ' ) representatives,

respective

9 E ~N(E.)

cc

9.

G 1 E ~¢(~')

property

+ [-~,w]

:

there

for

is

~>0

every

KCC~

such

have

Rz E

let

R

there

E is

that

R(9~,x) = RI(In(9~),R2(9~,x)) all

x E I< a n d

0(~(~. Then

R

is

an

element

of

~[~]

and

we

define

G I o G 2 E ~(~) as of

the c l a s s of R l and R2).

Proof to the

-

the

The

R

(this

proof

reader

that

Let

representatives

RI,RIE~M[Q' ]

and

us

class

RE@M[~ ] prove Rt

set,

is

now

and

not

depend

R 2 of

for

~ ~[~'],

9 E ~N(E~),

on

the

straightforward that

G]

Gz

I%2,R~E@M[O,Em] be such R,-R;

We

does

R~-~ o<~<~

and

o G 2 does

and

G~,

choices

not

it

is

respectively.

that

~

~[~,~o].

and

x E K,

left

depend

d (9, ,x)=R t (I~(~,),R2(9~,x))-R'~ (In(9,) ,R'2(9,,x))

=[R~(I n (9,),R2(9 ,,x~)-R~ (I~(9,),R'2C~o,,x))] -JR' I (I~(~),l%'~(~o,,x))-R~ (In(9,%R'~(.~,,x))] .

on Let

26

Using between we

have

since

the

brackets that

this

(a)

then Gof then, it

If

6 ~;(~) belongs (b)

then

fog

fl" w e

natural

term

f

is

If

1.3.5

is

theorem that

in E

C®

function

f E

@~[O,Em]

be

-

with

C(fl')

and

1.6.9

defined,

Restrictions

g

g

difference E ~[fl,Em],

The

second

has

the

is

is

a

a

one

in

C®

fl" a n d

function C®

~(fl)

G

continuous

~9(~') and,

from

fl i n t o fog

function

functions

on

from

associated,

function

generalized

E

(9)

function

is

g[fl']

(9).

property

continuous E

is

property

fl i n t o

usual

fog

the

of

first R~-R~

from

the

that to

the

@~[fl' ],

trivially

C®(fl" ) a n d

in

to

R I E

gift'].

R2

coincides

see

to

a

f 6

~(fl)

will

way

fact

since f satisfies to ~ 9 , ( ~ , ~ ' ) If

6

(c) into

value

the

R ~ - R I ~ g[fl" ] a n d

Remarks.

fl" fl

mean

and

in

fl a

fog.

%o s u b s p s o e s .

Notations : Let F = E., for some m E M We will denote by ~i : F n --4 F t h e i th p r o j e c t i o n [ ( x l , . . . , x n) E F ~ --~ x i E F ] , i = 1,...,n, and by Jl : F --* F n t h e i n c l u s i o n [x E F --~ ( x , O . . . . o) E F ~ ] Definition ~Z by

and the

: If

fl is

G 6 ~(~), we composition

an

open

define

subset

its

of

i

that

is,

a

representative

of

F ~ with

restriction

to

~i ~I,

= G[~

~1(flNJl (F)) E

~(~i),

I

G[~

is I

R,(~,x) for of

all G,

Remarks usual

~

and : case

generalized 1.6.11) .

E ~0(F),

I TMm n

is The

and

x

the

E

map

=

ill, w h e r e defined

restriction the

restriction

function

R(Immn~,(x,O,...,O)) R in

of

a of

"associated"

E

@MIni

l •2

Ii

"

function

a

continuous the

a

representative

•

C® to

is

usual

coincides

with

function restriction

the is

a

(see

27

Theorem : The

Proof G I E

: Let

restriction

us

~(~i),

map

consider,

let

G

= G i o

r

: Y(O)

first,

the

(~lla)

E

---, ~ ( ~ )

case

~(~).

~

=

is

~;

Since

X

surjective.

F "-~

Given

(Kil~)o(Jll

~ )

=

idQ

1

we

have

GIQ

= G i. 1

For

an

G I E

~(~I(~i))

that

a

E

of

-

~11(~i).

=

G 1 .

~

Gig

i

in

arbitrary be a

its

~,

if

extension,

neighborhood If

we

G i 6

set

as

of G

=

~(~)

~i

a

in

G I we

is

above. ~

and

have

given,

let

Let

a E

a

0

in

G

E

m

that

C'(D) a

be

such

neighborhood

~(~)

and

1

1.4

[]

-

NONLINEAR

PROPERTIES

Here multiplication functions.

we will which may

1.4.1

We

at

that,

for

-

if

all

GENERALIZED

FUNCTIONS.

see more general "operations" than be d o n e w i t h Colombeau' s generalized

Definition.

say

infinity

OF

a

that for

x E

function

every

ff E

M.

f E

C'(En)

there

exist

slowly

is c>O

and

increasing

r E

~

such

E.,

ID~ f ( x ) l

~ o(l+lxlE

)r n

(from

lemma

orthonormal functions

1.2.13, basis

slowly 1.4.2

this

~) . W e increasing

-

Definition

If

f

E ~M(~2p),

E ~(~),

let

R

: ~0(E~)

(11)

R(~,x)

=

definition by

at

infinity.

and

fl ×

f(R l (~,x)

is

denote

is

~ --~ ....

independent

~M(E.)

the

space

of

the

of

C"

theorem.

an ~

open be

,

subset

defined

Rp ( ~ , x ) )

of by

E~

and

GI,...,G

, 1

p

28

(we are identifying ~2p with gP), where each R i E ~[~] is a representative of Gi, l~i~p. Then R E @s[~] and its class in ~(~) will be d e n o t e d by f ( G ~ , . . . , G ~ ) (this class does not depend on t h e c h o i c e s of R~,...,Rp).

Proo@ are

:

It

e'>0

(12)

is

and

clear

r E ~

If(x)l

that

such

R

E @

that,

< c'(l+lxl

[~] . By

for

all

definition

1.4.1,

there

~ E ~n(E~)

there

x E ~2p

~p)~ m

Given are

KCc~

e>0

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<$<~)

and

{R(~,x){

and

l~
Then,

for

each

~ E ~(En),

we

(13) , =

[f(RI (~o~,x) . . . . . R p ( ~ , x ) ) [

m

c" ( I + p c s - N ) r ~< C' ( 1 + p c ) r ~ - N r for

all

chain

x E K and

0
For

first

rule

the

order

derivatives

of R

8 (15)

~

: ~ 3j

f(R~(~,x)

....

,Rp(~,x))

j:1

by

the

8jf(zl,...,zp)

(~,x) 3x i

denotes

the

partial

respect

to

the

complex

variable

zj.

Again,

8Rj --

we

get

a bound

of

and

have,

8Rj R(~,x)

8x i where

we

:

derivative using

of

bounds

f with

for

8jf

8

8x i uniformly x-derivative

the

kind

c~ -N

for

I

R(@~,x)I

8x~ for

x

of R

E

K.

implies

The

same

that

R

reasoning

E @~i[~].

for

any

partial

29 Let,

for

representative

of

each

~(~,x) then

R

- ~

i,

G~ . We

E ~[~].

In

will :

1~i4p, prove

R~

E

that,

if

be

another

f(R~(~o,x) ..... R~(~o,x))

fact,

by

the

mean

f in the i n t e r v a l with end points (R~ (~,x) ..... R ~ ( ~ , x ) ) , we h a v e

(16)

@s[D]

v~lue

theorem

applied

(R1(~,x),...,R~(~,x))

(R-~)(~,x)-- ~
to ~nd

,(0Rp÷(1-e)R~)(~,x)).

j = 1

(R.~ -R~ ) ( ~ , x ) } for s o m e t h a t , if

(iV)

0, 0 < 0 < i . If E c C ~ t h e r e are N E M, r E ~ a n d ¥ E r s u c h ~ E ~q(E~), q)N, t h e r e a r e c " > 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 <~

C"

~I (q)-N

for all ¥ E ~P, x E K, 0 < ~ < ~ a n d l~<j~
(18)

j=l

m

we

have,

i

[ (R-R)

(~o ,x)) [ ~<

0,0,1

i(Rj-Rj)(~,,x)i

~ c" [l+p max(](Rt-R~)(~,x)i+[R~(~,x)i)

]r

l~i~p

[(Rj_R~)(~,X)

i ~

(o,,)~

£~(q)-N

j=l D ~(q)-(r+l)N

p[l+p(c"

~

(q)

-N

"~" C "

~-N)]

r

30

where h&ve,

D = (c") ~ p ( l + 2 p c " ) ~ f r o m (lS) :

(R-~)(~,x)

= ~

3x i

For

the

first

order

x-derivatives

[ 3 ~ f ( R ~ ( ~ , x ) ..... R p ( ~ , x ) )

j:1

R~(~,x)

we

-

8X i

3 a ) f ( R ~ ( ~ , x ) ..... R~(~o,x)) ~

-

R~(~,x)

=

~x~

a = ~ < [ ~ j f ( R 1 ( ~ , x ) ..... R p ( ~ , x ) ) - 3 j f ( R ~ ( ~ , x ) j~1

..... R ~ ( ~ , x ) ) ] . ~ R j ( ~ , x ) + aX i

8

+ 0jf(R~(v,x)

..... R ~ ( v , x ) )

~

(Rj-R~)(v,x)}

Ox~ Using derivatives,

similar we

~D~(~-~)($~,x)l of the

kind

bounds

to t h o s e of (17), w i t h h i g h e r o r d e r 8 get b o u n d s for I -(R-~)(~,x)I ~ n d for 8x~ (where ~ E ~ and ~ is an o r t h o n o r m a l b ~ s i s of E~)

D

c~ ~ ( q ) - ~

1.4.3

-

Remark,

If g l , ' ' ' , g ~ E C'(~) t h e n f(gl, .... gp) 6 ~(~) c o i n c i d e s w i t h the c l & s s i c a l C ® f u n c t i o n If the functions gi are only continuous functions,lgi~p, then there is an "association" b e t w e e n f(gl .... ,g~) a n d f(g~ ..... gp), where gi is gi when c o n s i d e r e d in @ ( Q ) , l~i4p, see l 6.7.

1,5

-

POINTVALUE$

AND

INTEGRATION

THEORY.

We &re g o i n g to give a m e a n i n g to the v a l u e at a n y p o i n t x E ~ of a g e n e r a l i z e d f u n c t i o n G E @(~) and, if KccG, we w i l l also

define

~KG(X) )

to an a l g e b r a complex

that

numbers.

dx.

Both

we

define

"objects", below,

G(x)

n~mely

~nd ~ J K G(x) the

~igebra

dx,

belong

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

31

denote

1.5.1

-

Let by gM

@o

gM

Construction numbers.

the

algebra

~

of

be t h e s e t of a l l f u n c t i o n s from the following subalgebra of g0 :

= {Z

E go s u c h t h a t t h e r e is N E ~N(~) there are c>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<~<~}

by

= {Z E @M such that there are N E ~ and ~ E F such that if ~ E ~q (~), q ~ N , t h e r e a r e c > 0 a n d ~ > 0 s a t i s f y i n g IZ(~+)I

~< e ~

algebra

The

of

(q)-N

for

generalized

all

0<~<~}.

com?lex

numbers

will

be

the

quotient

There

is

a

natural

z E ~ ~

1.5.2

those

of

As an

-

inclusion [~

Nonlinear

theorem bF

:

Z(~)

~

properties

in §1.4 we algebra :

Definition and E go b e d e f i n e d

E ~o(~)

of

have

If

:

f

more

z E ~]

of

(~) ....

~,

through

proof

is

similar

to

that

map

E ~M

and

operations

Zz,...,Zp

in

E

~

~,

than

let

Z

,Z~(~))

for all ~ E ~o(~)j w h e r e e a c h Z i E @M i s a r e p r e s e n t a t i v e l~i~p. T h e n Z E gm a n d i t s class in ~ is independent choices of ZI,...jZ p ; it will be denoted by f(Zz,...,~p). The

the

~.

general

E ~m(~ 2p)

f(Zl

C into

of

1.4.2.

of of

~l, the

32

1.5.3-

define of

If G(x)

G and

Pointvalues

~ is E ~ as

let

R(x)

of

all

E go

~ E ~0(~).

is a n o t h e r

be

defined

It

is

and

at

it

is

Notice coincides will see

with in

"associated"

Examples

of

G,

that

it

is

- R" (x)

that,

if

g E C®(~),

with

the

classical

E @u-

clear

If

R"

E @u[fl]

that

E ~¢.

x E ~ o 5 G is d e f i n e d , denoted by G(x).

the usual complex I .6.3 that the

R(x)

also

then,

its

by

value

the

g(x)

n u m b e r g ( x ) . It new pointvalue complex

number

class

of

E ~ at

g E g(x)

R(x)

x E

C(~) E ~

we is

g(x).

: l)

Let

H

denote

the

H(x)

(H is ~9(~) is

R(I~,x)

immediate

representative

va~ue

by

(~o) =

R(x) The in ~

functions.

an o p e n s u b s e t of E~, x E ~ a n d G E ~(~}), w e follows : let R E @ ~ [ Q ] be a representative

JR(x)]

for

generalized

called

the

function

in C r ( ~ )

0

if

x<0

I

if

x>0

defined

by

= <

Hea~iside

RH(~,X)

function)

=

. A

~(A-x)

representative

of

H

dA.

0

Let

us

evaluate

H

in

0

: the

function

Z

: ~0(~)

+0~

z(~)

:

X 0

~(A) d~

--¢ ~ g i v e n

by

in

33 is

a representative 1 that Z (~)=--for all

of

H(0)

By

~ E ~ 0 (~)

i.e.

definition 1 H(0) = _

2

2)

Let

representative

$

denote

I~6 g i v e n

Note

~E~ o (~) & n d The

value

of

the

3) &

generalized

function

If

one

$ at

0 has

as

a

representative

= R~(~,0)

considers

complex

on

~ with

XE~,

in

~qi(~)

generalized

the

function

that

~(O)fO,

= ~(0),

only

number

such

real

valued

functions,

e x p ( i S (0)) ,

which

has

representative Z (~)

is w e l l

defined

1.5.4

If of

this

in d e f i n i t i o n of ~q; general s e t s ~q in

and, since, for all qEM we m a y o b t a i n ~E~(~) 8(O)E~ is not a classical complex number.

the

that

have

= ~(-x)

z(~)

then,

we

by

R s (~,x) all

~ 0 (~)

2

value for H(0) follows f r o m an a s s u m p t i o n it d o e s n o t h o l d w h e n o n e a d o p t s the m o r e Colombeau [3] a n d R o s i n g e r [4].

for

of

G.

Let

and

-

KCC~ I E @o

is

not

a

Integration sets. and be

classical

of

G E ~(~), defined

I(~)

= e x p (i~ (0)) complex

generalized

let

,

functions

R E @.[Q]

be

follows

its

class

immediately in

~.

that

a

on

compact

representative

by

= f R(I~,x)

dx

xEK

It

number.

I E $,j

by_JKG(x) f

and

we

denote

dx

34

We usual

one

have in

continuous

also

the

the

case

function,

l)

Examples

coincidence

G E see

C®(~). 1.6.4

I 8 (x) J 0

dx

of In

this the

definition case

G

with

is

the

only

a

("association").

: --

if

b>0

if

0~[a,b] .

2

b

2)

I

~ (x)

dx

= 0

&

1.6

-

ASSOCIATION

The

PROCESSES.

equality

in

@(~)

is

very

strict

(note

that

it

coincides with the equality of C ® f u n c t i o n s ) . H o w e v e r , there s o m e w e a k e r k i n d of e q u a l i t y , which also generalizes exactly

is the

equality of d i s t r i b u t i o n s and reconciles the new product @(~)) of continuous functions w i t h the c l a s s i c a l product.

(in We

call "association" t h i s w e a k e r k i n d of e q u a l i t y . As a faithful generalization of e q u a l i t y of d i s t r i b u t i o n s , it is n o t c o h e r e n t with the multiplication. However it p l a y s a b a s i c role in all the theory a n d its a p p l i c a t i o n s . This dissociation between the equality in @ ( ~ ) (coherent with the multiplication) and the association (incoherent with the multiplication) is a k e y of Colombeau's theory and is a main originality relatively to classical analysis. Although the association h a s of c o u r s e to be denoted by a symbol h a s to k e e p in m i n d equality

of

functions

1.6.1

We complez such

-

say

numbe~

that,

different f r o m t h e s y m b o l of equality, one t h a t it is a g e n e r a l i z a t i o n of t h e c l a s s i c a l

if

and

distributions.

Definition.

that Z E ~

an if

~ E ~q(~) lira

element

there with R(~)

is q

Z

large =

E

C

admits

a representative

Z

enough, in

an R E

~ssoc~eted @~

of

35

1.6.2

-

~(0)

and

1.6.3

-

value

If f(x)

Proof

- A

Examples

exp(i~(0))

have

no

associated

complex

numbers.

Proposition.

f E C(Q) and x 6 Q then f(x) 6 C admits E ~ as associated complex number. representative

R r of

R~(~,x)

:

~

the

classical

f satisfies

f(x+~)

~(~)

d~

.

~Esupp~

for and

all ~,all s u p p ~ is

xEQ and compact,

for ~ > 0 we h a v e

lim

small that

enough. Since

f is

= f(x).

Rr(~,x)

continuous

[]

¢40

1.6.4

If

eal

-

Proposition.

f 6 C(~)

f

integral

Proof

-

f(x)

and

dx

KCC~

then

E ~ as

A representative

of

~ f(x) JK associated

r f(x) J g

xEK Since

f is

uniformly

uniformly

for

x 6 K,

f(x+t~) as

as

£ --~ 0

on

~(~)

~ --~ 0

I(~)

6 ~

complex

dx

E ~

is

K,

we

have

d~--~

f(x)

g

continuous

I

dx

--+ r J

Then f(x)

dx

6

admits

number.

the

classi-

36

1.6.5-

We associated

Definition.

say wi~h

that each

two g e n e r a l i z e d f u n c t i o n s G I , G z E ~9(~) ~ e o t h e r iff, for a n y ~ 6 @ ( ~ ) we h a v e t h a t

[GI (x)

f

is a s s o c i a t e d

with

1.6.6-

-

If

0 E ~ and

we

~(x)

denote

dx

this

E

property

by

product f®g f.g E C(fl).

in

GI ~ Gs

Proposition.

If f,g associated with Proof

- G~ (x)]

E the

C(~) then their classical product

~b E

@(~),

I(~)

= I

~(~)

is

let

[Rr®:(~'x)

- Rr.:(~,x

)]

~b(x)

dx.

Then

I(~o,)--f [ f f(x+~/~)¢p(/-~)d/-~] . [ f g(x+8"r/)¢p(rt)d'O ] ~(x) dx

when

~*0. 1.6.7

[] -

Proposition.

If f E @ , ( R 2p) a n d g , , ' ' ' J g p E C(fl), w h e r e fl C En is open, then f(gl,'--,g~) E ~(fl) is a s s o c i a t e d with f(gx,---,gp) w h e r e gi E ~(fl) is the f u n c t i o n gi v i e w e d as an e l e m e n t of ~(fl), 1~i~p.

37

Proof Let, for e ~ c h i = l , . . . p , Ri be the r e p r e s e n t a t i v e of given by (6)or (6'). If R denotes the representative f(gi,-'',gp) given by (ii) and R the r e p r e s e n t a t i v e of continuous f u n c t i o n x ~-* f ( g i ( x ) , . . . , g ~ ( x ) ) given by (6'), have R(~,x)

~ of the we

= f ( R i ( ~ , , x ) ..... R p ( ~ , x ) )

and

R(~,x)

=

r

/

f(g1(x+~)

.... , g p ( x + ~ # ) )

~(~)

d#.

2 Let,

for

each

~ E @(~),

l(~e) = f [R(~,,x) - R(~,x)]

~(x) dx.

Since, for each x E ~ and ~ E ~q(En), q large e n o u g h , b o t h R ( ~ , , x ) a n d R ( ~ e , x ) t e n d to f ( g l ( x ) , . . . , g p ( x ) ) as ~ 0 and since has compact support, from the theorem of dominated convergence, we h a v e that I ( ~ ) t e n d s to 0 as ~ ~ 0. []

1.6.8-

map

A

Exercise,

For t h o s e w h o : @" (~) --* @[~]

know Distribution d e f i n e d by

A(T)(~,x)

=



Theory,

if

prove

that

the

~_-~n

or A(T) (~,x)

--
>,

where X± (,) is as in 1 . 2 . 1 7 (b) for ~ a r b i t r a r y , for all ~ 6 ~ 0 ( ~ n ) , x E ~, has r a n g e in @MIni a n d that if T E ~" (~) is such that A(T) E ~[Q] then T = 0. This map defines a natural i n c l u s i o n f r o m ~' (~) into ~(~).

38

1.6.S

-

Exercise,

(Continuation the

integral

of 1.8.8)

of ~ . T E ~(Q),

If T 6 ~" (~) and ~ 6 @(Q)

r (~.T) (x) dx Jn

(this

m e a n i n g since the support of ~.T is compact), number w h i c h is equal to the value of T in ~,

1.6.10

-

integral

is & usual .

then has a complex

Proposition.

(Continuation of 1.6.8 and 1.6.9) If ~ E C®(~) and ~" ( ~ ) t h e n ~ . T E ~ ( ~ ) i s a s s o c i a t e d with the distribution (classically defined by <~T,~> = for all ~ 6 @(~)).

Proof

A representative

-

of ~.T is given

T E sT

by

R~.T(~,X ) = ~(x). We have

for WE@(~)

~ R~.T(~,X)

By the

exercise

f(~T.~)

~(X)

1.6,9

(x)

<~T,~>

of the

b-~ [

T h e n we h a v e

1.6.11

E~(~) 1


last

:

integral



~(x)

~(x)

dx

of sT, we o b t a i n

: f(T.~)

(x) dx

J

is

~(x)

dx

~(x)

result.

[]

Proposition

With ~1

the

= f

and the d e f i n i t i o n

dx

and a r e p r e s e n t a t i v e

dx

is

the

notations

associated

with

of

1.3.5 the

and

classical

1.4.3

,

if

restriction

f6C(~) of

then f

to

~.

39

-

Proof

Then

a

Let

RfEgM[fl ]

representative

be

the

of ~I ~

representative is,

by

of

f given

by

(6').

1 .3.5,

1

R1 (~'X)

: f

f (~) ~( Fn

with

¢ 6 ~ o (F) , Given (x)

m

i (I

ran

(2L) ( I m m ~ ) (2t- (X, 0

..... 0))dh

~)

xEfi~

~E~(fll)

~ ( x ) dx

,

6 ~

if

$6@ M

given

is

the

representative

of

by

F = [

S(~)

R: ( ~ , x ) ¢ ( x )

dx

wF then,

by

(7) , f o r

S(~O~)

~ E ~ o (F)

each

= I

f F

nf(A)

there

W>0

(Imn~O~m) ( A - ( x , O , .

such

that

..,O))~(x)dAdx

F

---- ~ ~ n f (2t)(Immn~)e F F =

f (A) F

is

1 /n (/[- (x,O ..... , - -

m w) (Iron

F

v1

= ~ ~

O))T(x)dkdx

, ...,

~(x)dkdx

/n

f (x+~l /n~il ' ~i /nl£2 ' .... ~i /n~In) (Iron,O) m (~)~(x)dgdx. n

F

When

8~0

we

have

F

that

S(~e)

f f(x,o..... F

1.6.12

tends

to

dxf n (I~n~)~

(g) d g = ~

F

-

f (x,0 ..... 0 ) ~ ( x ) dx . []

F

Proposition.

I f fl a n d fl" a r e o p e n s u b s e t s o f E n a n d E~ r e s p e c t i v e l y f is a continuous function f r o m Q i n t o fl' a n d g E C ( f l ' ) then 6 @(fl) i s a s s o c i a t e d with gof 6 C(fl). Proof

m=n=l.

-To simplify We h a v e a l r e a d y

~(fl,fl'). (i0) :

Let

Rgof

the notation seen in the and

R ~ o r be

we g i v e example

the in

respectively

proof 1.3.4

in the that

given

by

(6')

, gof

case ? 6 and

40

R, or(~o,x)

where

R~

and

Since ~ and convergence,

I(~)

tend

R~

~re

as

=

in

(6') . If

~ have compact we obtain I(~)

to [ g o f ( x )

fCgof)CA)~.)CA) ~(A-x)d A

~(x)

~ E @(~),

support, by --* 0 as g *

let

the theorem of d o m i n a t e d O, s i n c e b o t h t e r m s of []

dx.

J 1.6.13

Colombeau 1.7

-

-

Remark.

Lots of [2,3].

other

TOPOLOGIES

ON ~

results

-

Topology

Now we then prove that structure. Definition

1 Q~

the

We values

enough, have :

=

found

in

be d r o p p e d w i t h o u t significant loss where we use these topologies of t h e C a u c h y p r o b l e m .

in to

and

uniform

define a natural this topology is

If

#>0

we

~

such

that

with

]x(~)]

E

note x(~),

depending

kind

c~n

structure

on

~.

topology on the algebra ~ and the topology of a uniform

there

is

< ~ for

x

E

all

@M,

a

representative

~ E ~0(~)}.

t h a t , b y a c h a n g e of r e p r e s e n t a t i v e ~ E ~N(~), with N large enough and on

be

set

of ~,

<~

this

~(fl).

AND

This section may the sequel,except in § 3 . 5 express the wellposedness

1.7.1

of

~,

are

significant

for

the

(since only ~>0 small quotient)

we

41

Q~

Ix(~) Cle~rly

Q~

~>0

is

-

The

set

~

the topology neighborhoods

+

U

Q~.

b&lanced ; therefore a neighborhood of ~

~+Q~ there

Definition

E ~ we

3

-

If ~ /

II~II

From some

0<~<~}.

s a y t h a t ~ C ~ is o p e n if for all ~ E ~ t h e r e ~ + Q ~ c ~. T h e s e o p e n s e t s d e f i n e a t o p o l o g y on

that

1 - For a base of

Proposition

Proof

if

2 - We such

g>O}

I < ~

= ~Q~.

Definition

is

t h e r e a r e x E @M, a r e p r e s e n t a t i v e depending on x, s u c h t h a t for all is ~>0 s a t i s f y i n g

= <~ E ~ such that of ~, a n d N E ~ E ~(~) there

definition

\

inf

<#>0

+~

if

l,

it

is is

is

defined of ~.

open,

above,

since

a neighborhood g>0 such that

Q~

the

is

family

<~+Q~

convex

of ~. N o w ~ + Q , C V.

and

if V

is

set

: ~

E Q~}

x ~ Q~

follows

for

if

~

all

immediately

E Q~

for

some

#>0

~>0.

that,

if ~

E ~,

for

~>0,

II~II =

inf

{ g > o s u c h t h a t t h e r e a r e x E g~, a r e p r e s e n t a t i v e of ~, a n d N E ~ d e p e n d i n g on x s u c h t h a t for e a c h ~ E ~ N ( ~ ) t h e r e is W > 0 s a t i s f y i n g

Ix(~o~) I

Examples

-

induces

the

I) If z E C, t h e n u s u a l one on ~.

< /~ if

o<~<~}.

Uzfi=Izl.

Hence

the

topology

on

42

2)

l[~(0) ll=+m ; c o n s e q u e n t l y ,

3)

[[exp(i$ (0)) I[=l,

if

C

is

~ q (~)

not

dense

has

only

in ~

real

;

valued

functions. Definition

the

4

-

Let

us

consider

x ~

in ~

the

filter

U

spanned

by

sets

u. if

g>O.

The

= <(x,~)

filter

U

is

the

topology

on

structure

(see

Bourbaki

The

topology

Remark

1

-

suffices

to

~

notice

neighborhoods

1,7.2 We on ~

in

on ~ the

on

is

II,

a

structure

not

and

uniform

on C

from

§i,

this

space

: g>O}

and i) .

:

it

of

0

topology

form

a

the

uniform

proposition

linear

{Q~

are

~9(fl) a

< ~}

derived

not

sets

Ilx-~ll

uniform

[+l], c h a p t e r

which

define

a

topology

Topology

-

may

defines

that

in ~

e ~ x ~" :

base

absorbing.

structure

topology

@(~).

on

simil&r

to

that

defined

1.7.1.

Definition

1

-

QK,k,~

If

KCC~,

= {G

k

E

E ~(~)

~

and

such

sentative

of

~>0

that G,

we

set

there

such

I (D~ 1%) (?,,x) I ~

sup

is

R

E SM[~],

a

repre-

that

/4}

Ic+l~k

~ E ~ 0 (E.) x6K

Definition each

G

2

£ X

-

there

We are

say

that

KCC~,

k

G These

open

sets

Proposition is

a base

I of

define

a

- The family neighborhoods

a 6

subset ~

and

+ Qx,k,.~

topology

of of

C

on

sets G.

X g>0

of

@(~)

such

is

open

if,

for

that

K. ~(~).

{G + Q x , k , ,

: KCC~,

k E ~,

#>0}

43

The

Definition

IIGIIK, k

Remark

proof

3

-

/

inf

\ I

constant

similar

{~>0 if

that

G ~' Q x , k , ~

If

we

generalized

for

induces 2 -

Remark

the by

is,

function

its

P~,~

3 -

If

Q,

it

for

some

~>0

,~>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,.

: <(G,H)

with KCC~, k E ~ structure on ~(~) and uniform structure.

Remark 5 via

X ~(~)

This

-

the

topology

inclusion instance

in

of

space

topology say

weakl~

the the

space

C®(~).

classic~l

~ 6 ~(Q),

the

x ~(~)

is of

the the

Its Since

to J G ( x )

exactly

the

$(x)

sequence

U

spanned

IIG-HI[K,k

U

defines a uniform arises from this

natural

generalization

classical

use we

looks the

Fr~chet more

use

have

[G~(x) J

This of

concept

of

< ~}

a

limited of

the

topology

weak

on ~ we

G n E ~(~)

~(x)

dx

convergence

tends

tends generalizes

distributions.

between non associated also Hausdorff topologies

(strong)

sense

of

the

equality

example of a natural, Hausdorff topology on ~(Q) ,which however does C® (~) i t s u s u a l t o p o l o g y .

elements of ~(~) . However s u i t e d to o b t a i n r e s u l t s in in •

. The

following

, translation not induce on

is

invariant the space

Definition 5 If K is a c o m p a c t s u b s e t of ~ ,p a n d q p o s i t i v e integers, we denote by V(K,p,q) t h e s e t of a l l G E • (~) which have a representative K such that -

]DR(~,,x) [ < ~q

for

,

space

fact that some natural topologies on ~(~) are not not a drawback to o b t a i n e x i s t e n c e results in the of t h e a s s o c i a t i o n : then the important p o i n t is to

Hausdorff is (weak) s e n s e distinguish there are

dx

convergence

Remark 6 .The

the

filter

of g e n e r a l i z e d f u n c t i o n s

/

in

the

:

analysis,

~" (~).

that a sequence to G 6 ~ ( ~ ) iff:

all

~(~)

C ~(~),

of

for

consider

~>0. The filter topology on ~(~)

on

C'(~)

topology for

£ ~(~)

and the

than, may

we

sets

~ E ~0(En) ' ~>0

small

enough

, x E K and

order

of

D ~

p

an

45

These sets invariant this

are a basis topology on •

topology

is

A well

as

its

of 0 - n e i g h b o r h o o d s for a translation (~) . F r o m the definition of • (~),

Hausdorff.

C ® function derivatives

is of

in V ( K , p , q )

order

4

iff

it

is

null

in t h e

7 - Up to n o w , t o p o l o g i e s on ~(~) have not olayed applications. T h e y m i g h t be u s e f u l f o r subsequent

Note

that

Remark

in

§3.5

well-posedness

1.8

-

THE

of

SUBSPACE

a

one

will

Cauchy

on

K

,as

p

use

a

topology

to

a role studies

express

the

problem.

~,(~).

We define here a subspace ~ s ( 2 ) of ~ ( £ ) , in w h i c h t h e subscript "s" stands for "simplified", due to its simpler definition . We still have a canonical inclusion of C'(2) into • s(D) b u t w e do n o t h a v e a p r i v i l e g e d natural inclusion of C(~) into ~(~), a n d t h u s of ~" (~) i n t o ~ ( ~ ) . To a n y d i s t r i b u t i o n on Q, several elements of ~(~), without any privilege, are associated.

for

Working in t h i s several developments

applications. extend at once can usually of n o t a t i o n s .

Let

-

considered

as

nothing

the

, for

N=0,1,2,

is a b i j e c t i o n •

more

than

a

that

diam(supp~)=l}.

~s

simplification

... such

map T

and

cause trouble in physical

Remark.

~N,I (En)=(~E~N(En) Then

does not natural

Anyway all results p r o v e d in t h i s b o o k w i t h to • ; in p r a c t i c e , working in ~s in p l a c e of

be

1.8.1

subspace of ~ ( ~ ) a n d is e v e n m o r e

= ~i/~

(given to

have

: ~ o , 1 (E~)X]O,+~o E ~ ~o (En) (~,~) ~ ~ ~pE~o (En)

T(~,~)=~)

it

suffices

. We

could

such

to

take

also

that

~=diam(supp~)

define

r

~ , , i (En)

IW(x) 1 2 d x = l E n

}

by

46

in i=~

o r d e r to h a v e t h e 0 T-~ f r o m ~0 (En)

of

onto

With on

above bijection . into ]0,+~[, where

each

line

the

1.8.2

of

all

definition

of ~ .

(8W~+(1-8)~}0<0<

open

R

subset

]0,+w[

:

of

are

C * in

s in @ ~ [ ~ ] point is replaced

X

An @[Q] is,

(see there

the

element is

variable

R

of

x E ~,

@~[~]

which

is

is,

KCC~

may

be

:

~[~]

~iI

We

an and

denote

by

@M,s[O]

e l e m e n t R of ~ E Mn t h e r e

x E K and (c)

@~ [fl] t h e

R(¢,x)

for

each

~>0.

The

subscript

the significant c o u l d as w e l l be

considered of

&s

W E

an

element

~0,~(En),

We

[(~,x)

the

~

subspace

R(~,x)]

of

@s[fl],

~s[fl] is in @ M , s [ ~ ] if, a n d a r e N E ~, c > 0 a n d W > 0 s u c h

<

c~

only that

-N

denote

by

gs[~]

the

subsp&ce

= M-I(X'[fl])

of

~[0]

defined

there

are

c>O

and

~>0

such

that

(N,)

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

~iI

x E K and

O<,Z<%.

by

,

i.e. a n e l e m e n t R of ~s[fl] is in gs[fl] if, a n d o n l y if, given K and ~ there are N E ~ ~nd ¥ E F such that, for all q ) N

if,

O<~<w.

.~' [ ~ ]

with

of that

--* ~ [ ~ ]

I(O~ R ) ( e , x ) l for

~

independent

R ~.-,

that

by

& map M

given

i is C ®

1.

1.2.14.&)

(b)

map

fl--*

s t a n d s for "simplified".Note that the behaviour of R w h e n ~ * 0 ; t h u s +~

by

the

denote

we

E n'

(~,x)

which

~ (En),

~

Notations.

-

If ~ is a n functions

(a)

set

gives & map the projection

.

second

segment

This H~ is

~< oe ~ < q > - ~

q E

47

Notice

that

(d)

denote

We

~s[~]

is

by

Ns(O)

~(~)

1.8.3

map

M,

-

an

ide&l the

of

quotient

(a)

The

algebr~

~s(~)

that

is,

if M

defined

R E @s,~[~] (b) . More

following

is

given such

is a

There is precisely,

one

is

~ sub~lgebr~

Proof

-

It

~(~),

via

the

--~ ~ ( ~ ) of of

[M(R)] G,

~ simplification property (N~) a b o v e

then of is

~

is

injective.

the definition equivalent to

of the

: K CC

~,

~£~n

and

q E ~,

there

~re

c>0

and

n>0

that

all

x E K ~nd

is

obvious

(~,x)

[ 4

c£ q

0
that

this

assume, t h e n , t h a t (N s) is t r u e E F be g i v e n b y h y p o t h e s i s . is q" then

of

by

representative

[ (D~R)

for

sp~ce

Remarks.

G ~-~ c l ~ s s

~[~]

.

: ~,~[~]I~[~]

: ~(~)

where

@s,,[~]

property

implies

: given K,~ and S i n c e ~(n) * +~

E ~ ( w h i c h m a y be & s s u m e d ~ N) ~(r) ~ N + q . S i n c e r)N, b y (N~

q, ~s

(N,).

Let

us

let N E ~ a n d n * +~, t h e r e

s u c h t h a t if r E ~, r ) q' there ~re c>0 and W>O such

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

all

x E K and

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

on

0<~
We h~ve [ (~,x) G there

in ~ , ( Q ) :

4

c~ ~ r>-~

c~q

= rain { l , n }

a n~tur~l injection ~-* f ( x ) ] E @ ~ , , [ ~ ] ) corresponds if

g

an

f E C(~),to

from but,

infinite e&ch

C'(~) into @M.s[Q] to e & c h c o n t i n u o u s

number

~ E @(~)

of

with

generalized I~(A)dA=I, )

48

and (T~},>0, 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 ~, 0<~<w}

by

c>0 ;

that

for

n~1~ural

each

JZ(~)I

q 6 ~,

there

~< c~ q for

all

are

c>0

0<~<W}~

inclusion

[(~,x)~ Z(~)]

a subalgebra

If

-

with us

@~,~,

~nd

A($)CX[~

].

Notice that @ M . s = M-i ( A ( $ M ) ) a n d @= g i v e n in 1 . 8 . 2 ) . T h e n the q u o t i e n t ~,,

map

Remark

of

such

~, is

~ for

s~tisfying

the

ZEto~ clearly

of

above

= {Z : ] 0 , + ~ [ P-* ~: s u c h t h a t t h e r e are a n d W>0 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 ) I<+~. e>0 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

x<-A(£)

and

x>A(~)

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 "~


RH(~,X)

the

1.9.3

where

all

that

-A(~ )

satisfying

any

If

for

such

Remark.

RH,R x

is,

=

[R~(~,x)-Rz(¢,x)]

f that

0

~*0,

and K are two Heaviside generalized functions, then if ~ E ~ ( ~ ) is g i v e n , from Definition 1.9.1., we

respectively,

aspect

as

A (¢)

/

(iii) -

=

R(¢,x) f

1.9.2

A(~)*0

E ~(~)

is

a

Heaviside

~

of

the

formula

H

generalized

function,

we

have

I (19)

H"H"

~

H" . n+l

1.9.4

-

We have Hn=H would imply it w e w o u l d have

Remark.

Hn#H in ~ ( R ) if n > l . Otherwise, the equality n H ~ - I H ' = H • . If w e m u l t i p l y by H both members of

52

nH~H"

From ( 1 9 ) ,

= HH'.

we o b t a i n n

1 H'

~

HH'

~

-- H'

n+l which

is

equivalent

impossible

-

there

n+l

2

(b) [ R s ( ~ , x ) J (c)

1,9.6

= 0

if

dx

= I,

A(~)*0

as

lxl>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 <+~

sup ~>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 b<0. The class of Rl~ in ~9, (~) is a Heaviside generalized function H, and H' z$. According to the g e n e r a l d e f i n i t i o n of c o m p o s i t i o n p r o d u c t in 1.3.4, H" ( x ~ - a ~) is the class in ~ , ( R ) of the m a p (~,x) ~ R~(~,x~-a'). Let ~ 6 ~(~) be given. Then

(*) same remark as in the f o o t n o t e of p . 5 0 behaviours of R~ ( g , x ) ( s u c h as a number N(~) between -A(~) and +A(~), with N(~)~+~ if eliminated

in a p h y s i c a l

context.

;very complicated of oscillations ~*0) h a v e to be

53

<0(~X+a ~) + ~

i+mR~(~,x'-a')~(x)dx

=- i

-co

i+m

2

(-~X+a 2) dX =

R'K(~,X)

-a ~

~X+a ~

2

-a 2 dX+a ~

- I +®

RH

[.~' (~X+a2)

(~,X)

-a 2

_

-A(~)

such that A(g)
2 ( X + a 2)

2(X+a2)

_

2 (X+a ~ ) ~

2 (X+a ~)

There is go>O above expression

2

(~X+a~)+ ~ (-X + ~ 2)

- ~ ' (-~X+a ~)

~

Then for 0<£<£ o the

(-iX+a~) ] -----|R~ 2 (X+a 2 ) ~ ]

2 ( X + a 2)

(~ , X ) dX

A(¢)

Since

/ 2

IR(~,x)ldx*0 -A

as

~*0

and

~ has

compact

support,

we

(¢)

have that the above integral from -A(~) to A(~) tends to 0 as ~*0. Let us consider now the integral from A(~) to +m :

2

A(~)

~' (~X+aU)

~ (IX+a~)

2 ( X + a =)

2 ( X + a =) ~

~' (-~X+a2) 2 ( X + a 2)

2 (X+a 2 ) ~

-I dX .I

p

54

siI I

Ulu 2a

2~ A (~) + a 2

Thus

we

have

proved

that

6(x~-a

2)

is

associated

1 with (of

--

[8(x+a)

2a course

with

1.9.7

Proposition with

where

a

+ a

-

-

$(x-a)], "formal"

The

If

classical

equality

T

association

8 E ~(~)

representative

~ E @(~)

a

satisfies

is by

p~0

supp

~ C

Let

~

{xE~

E ~(~)

be

[

a}.

IxI~

f[

given We

~

of

by

1+8.

~(h)dA=l,

and

let

a>O

then

be

such

that

have

l[.]] p2 (x/~)

- -- ~

--

Dir&c

the association).

1+~ ~ ~ 1+8

i i+--

2

place

written

a Dirac generalized function R:(~,x)--*(1/~) ~(x/~)

and [ J

1 Proof

in

~1+~

given

formula

~(x)

dx

55

I~l~
I~[~a

~l..
which

tends

-

Remark

n~l+Sn

IRW(x)

to

Under

~

Ixl~>aa

the

1+8

dx,

same

for

as

~0.

hypothesis

all

n)2.

on For

8, odd

as n

above,

we

have

the

assumption

8

0

that

o~O is

unecessary.

1.9.8

For

1.10

-

-

8

in

can

be

order

they the

sin

DIFFERENTIAL

to

~(E~)

identical

"wild

constants".

"wild in

given to

1.10.1

a

setting

classical

eliminated

the -

do

boundary

or

physical

below ~

in

functions

constants"

by ~

.

ALGEBRAIC

of

have

~

EQUATIONS

SOLUTIONS.

as

hold

extendable

OF

that

elements

on

results

in

context. the

which could not

an

element

They

were

all

operations

be

cause

initial We

simplified

reproduced. any

trouble

conditions

insist ~s

of

on

case

~

not

introduced and In since which

the

fact

that

are

at

once

case.

Proposition.

G E ~ s ( E 1) b e s u c h t h a t there i s x o E E1 f o r w h i c h a classical complex number. Then G'=O in ~,(E1) if, and G is the constant function identical to this number.

Let

G(xo) is only if,

SOLUTIONS

these

are

always

prove

considered to

automatisms practice

valued,

CLASSICAL

The ~

real

GENERALIZED AND

in

Exercise.

56

Proof

-

the m e a n

If R G E @ ~ , , [ E , ] is a r e p r e s e n t a t i v e v a l u e t h e o r e m g i v e s that the m a p [(~,x)

I-* R Q ( ~ , x )

-

then

- G(xo)l

is in $ , [ E I ] . T h u s the c o n s t a n t g e n e r a l i z e d is a l s o a r e p r e s e n t a t i v e of G. The c o n v e r s e

1.10.2

of G a n d G ' = O

f u n c t i o n (~,x) --*G(x 0) is o b v i o u s .

Proposition.

If G E ~ , ( E z ) is s u c h that G" ~ 0 in ~ , ( E ~ ) t h e n G a s s o c i a t e d w i t h a c o n s t a n t g e n e r a l i z e d f u n c t i o n (the c o n v e r s e o b v i o u s l y true).

JEt f

- Let

Proof

may

~o

be w r i t t e n

E @(El)

be

such

in a u n i q u e (x)

where

f

[ ]((x)dx JE I

= 0

(take

RQC~,x)

~(x)

dx

that

~o(X)

dx

= I. A n y

~ 6 @ ( E I)

way = A~ o(x)

F

A = |

JE

Let ~ E @ ( E z) be s u c h t a t i v e of G, we h a v e

~

that

is is

~"

~(x)dx

and ~((x)=

~(x)-

A~o(X)) •

z

= ~.

= A IRQ(~,x)

+ ~(x)

If R G E ~ H , , [ E z ]

~o(X)

dx

+ fRv(~,x)

is a r e p r e s e n -

#" (x) dx

Then

I[Ra(S,x)

which

-

tends

IRe(S,Y)

to

zero

as

~o(Y)

~ *

dY]

0.

~(x)

dx

= - IR~(~,x)

T h e n G is a s s o c i a t e d

tant g e n e r a l i z e d function ~G(y) ~o(Y) dy E ~ .

g(x)

with

the

dx

cons[]

57

Remarks

(a)

-

If

G,

in

proposition

i.i0.2,

associated with a classical function for some nonvoid o p e n s u b s e t ~ of E, classical classical @' (Q),

constant constant

implies

by

T

is

a classical

Propositions

a higher

1.10.3

Now a~b

are

order

-

Wild

let

us

two

real

G E ~,(EI)

be

i.i0.i

solutions

of

consider numbers

defined

i R(~,x)

and

1.10.2

algebraic let

by

algebraic

=

us

also

hold

a

if

b

if

G

; G

b

is

a is

One roots Then,

equations.

the

cl~ss

and

E

]O,w[

V~

the

~(E~) G" is

the

For

instance,

polynomial

1

2n+1

< ~ ~ -2-n

=

solution some

1 < ~ ~

of

P(G)=0

mixing

of

at

once

gets

W>0

in

of

in ~ ( E

a and ; let

,

n

E ~

2n+1 I) w i t h o u t

being

either

a

b. Q be

a nonzero

a l , . . . , a n a n d let X E Cs be solution if Z E @ M , s is a r e p r e s e n t a t i v e of X,

c>0

of case

(y-a)(y-b).

2n+2 Then

is

equations.

consider

1

or

GIQ

constant).

1

are

that

derivative.

P(y)

Let

such

a distribution) associated with a

of ~ ( E I ) . They are at once extended to t h e case G" r e p l a c e d by a partial derivative, a n d to t h e

replaced

if

is

even G is

(the g e n e r a l i z e d constant of 1.10.2 is a from the classical property that T'~0, T E

that

(b) case with

(or then

of for

polynomial

with

Q(X)=0 in ~s' each q E N there

satisfying

Hi(~)

E
such

that

IZ(E)-ai(~)

~ ~

c~ q

(when X result). Q(G) =

is a solution of Q ( X ) ~ 0 in ~ one gets Thus any G E ~s(E1) solution of a n algebraic 0 (or Q(G)~O) appears as a " m i x i n g " of t h e

a similar equation classical

roots

Q,

of

of

but

it

is

not

necessarily

a classical

root

Q.

58

Such might

be

~(~)

so

as

calculus

on

not

to

is

a

reason

are

all As

0 ~

in

the

to

(20) a

the

number

will

to

a

not

and

thus

this

our

enough

these

new

of the

any

g

longer

mixing

classical

root

But

of

solutions

occur.

of

~(~)

Indeed

if

small

8

classical

; so

This so

we

enough

on

Q

polynomial

solutions

depend

of

the

automatism

classical

(for

but

constructed

classical

definition of

context

have

construction.

have

incorporated.

does

w~

seen,

identical

function

ai(~)

theorem)

just

the

be

continuous

in

fact,

~utomatisms

have

that

modify

natural

In

present

to

is

is

considered

enough R

is

m~in

we

G

drawback

that

Q(G)=O

undesirable,

serious

automatism

value

of

as have

=

valid

equations it

solution

functions.

Q(G)

(20)

is

a

considered

is

that

postulate ~>0)

then

(intermediate solutions

is

avoided. Recall dropped

when

occuring can To

be

of this

a mistake

must

differentiability

1.10.4

-

Denote

by

§l.l

differentiability

introduced.

undesirable

as

we

in

was

these

considered

repair

some

that @~(U)

The

keep in

the

its

definitions

is

that

C"

enough,

the

in 0<e
8,

~ and

origin

: this

the

consequences of

of

8

was the

simplification appear ~(~)

and

now. ~,(~)

~.

and of

~9~(~).

all

(~,x) such

in

the

and

~z(~) set

is

solutions

W and

spaces

@i[~]

in

This

I

functions

,) R ( ~ , x )

map

Z and and

x 6

x

for

~) .

any

given

~i,

~2

6 ~0(E~)

(~>0

small

59

Define by

imposing

e(~-91), .

imposing not

all

the

results

in

proved

the

construction of

Now

in

Let

we

If

-

R

Let

a

and

only

Taking

into of

C"

C®

in

~ pair

C

(~,x)

~(~)

and main

true

with

~(~)

6,

has or

This

we

= by

all

in

+

(~)

just

the

one

and

~

that

result. thus

set

~(Q)

since

are

@[Q]

check

remain

this

we

reduce the

have to

proofs

to

to

prefer

or

check modify

book

has to

specific

results

classical

the give

showing

solutions.

in

the

simplified

in

one

variable.

identical

to

3~>0

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
~[~]/#i[~] and

@~[Q]

that

and

Then

classical

P(G)=O,

root

roots

of

P,

G

then

by

hypothesis

a i ~ a j if

x E

K

of

the

for

have

q6~

all

multiplicity

and

of

the

0<E<W. roots

of

P

and

the

get

[ ~< c'~ q

for

all

x E

K

and

0<6<w"

o

for

some

others

c'

and

n" , w h e r e

aL

is o

For

the

x-derivatives

of

R(~,x)

one

has

independent

of

P.

i~j. we

that

ce q

we

of

a

g and

x.

60

8

8R

--

(P(R(8,x))

= P' (R(g,x))

--

8x so t h a t , 8R [--

if

(8,x)[

P" (a i ) ¢ 0 , 0

. If

(g,x)

8x one

P' (a i ) = 0

~X

has

and

at

if

once

the

required

P"(a i )~0,

the

bound

formula

0

0

8 ~

8

(P(R(e,x)))

for

= P"(R(g,x))(

8 3

--

8x 2

R(g,x))

a(e,x)

2 + P" ( R ( 8 , x ) )

8x

8x 2

8R gives

also

the

required

bound

for

I--

(8,x)l

• By

induction

on

the

8x order

of

multiplicity

of

ai

, we

get

0

8R 1-8x for

some

order

(e,x)l

c">0

<

and

n">0.

x-derivatives

of

R

all

for

c"~ ~

x

6

Similarlyone

and

then

R

O
K,

gets

bounds

for

higher

E $~[Ex].

[]

1.10.6

In is

x~y"

to

in

~(E

1)

or

~(E~))

the

equation

abbreviated

notation

at

0

= R(~,I)

xG = 0

G=0.

We have the following implications : xy = 0 ~ = 0 = ~ x S ( y ' ) 3 = 0. L e t R b e a r e p r e s e n t a t i v e

integrable R(~,x)

Proposition.

~(E1)(even

equivalent

Proof In

-

one

we

gets

÷

have

that

R" ( ~ , ~ )

d~

}x2R" 3 ( ~ , x ) IR(~,x)l
if

x~O

q

I ~ c ~ q. (if

replace

x)0

xy'+y of G E

Since

[xl - 2 / 3

use

1 by

= 0 ~(EI).

the

-1)

.

1

from

In x~y'=0

the

fact

following

order .to ; setting that

get a similar bound z=y' , w e g e t x 3 ( z ' ) 4 = 0

Ix[ - S / 4

implications

is

integrable

at

f o r R" ( ~ , x ) w e s t a r t and now make u s e of 0.

For

R"

we

is

formula

use

the

61

x~y" and

then

start

In function

solutions

Proof

~(E1) +x o r

equality 1.10.6

we y

=

been

I

be

the

Since we

the

give

Sketoh

there

y

since

a

the

a

+i

that

]sp,+m[)

-x+c~

{aj}0¢j<

k

such

to G to classical

y2y'=yx we

and

have

to

the cI

gets

x~y'=yx y=×y'

x(y'-l)x(y'+l)=0.

and

e2

result

.

be

i.e. This

Again

1.10.S

constant

c1=c~=0.

from

gives

-I.

y"

Integration

are

generalized

D

including

Marzouk-Perrot

[I],

polynomial of ~. Then

@~(I)

is

very

very

proof

on

equivalent the two

proposition

where

one

a nonzero interval

finite

there

yy'=x,

or

< long

rough

-

set

~

First of

each is

a

(this

one

that

in

a

a

is

of

all

these

Marzouk

[I]"

two

variables

classical

results

perhaps

C" a

x

and

y

;

solution.

"hard"

result),

proof

proves

singular

interval family

in

G is

sketch

sI < s2 such

0

Theorem.

in

proof

of

is

0

x ~ y '' =

x2y'=0.

i.i0.6, and

general

in

P be open

only

=

y2=x~

more

-

=

of

imply

constant

obtained

P(x,G)

0 ~

G2=x 2 is -x (i.e.

implies

(y-x)(y+×)=0

Let an

place

proposition

(y'-l)(y'+l)=0

much

x ~ y '' =

dropped).

and

1.10.8

Let

are

~(EI)

x + c I or

A has

in

have

But

+

equation function

-JxJ

in

to

numbers.

xSy"=0

the the

From

identical gives

and

y2=x2

-

x(xy'-y)=0.

2xy'

Proposition.

-

[xJ

0 ~

from

1.10.7

the

=

the

classical

result

that

points

<

...

< sp

]si,si÷,[

of(some

kind

neighborhood

of

(as

of

well

as

meromorphic)

each

si

]-m,s1[

and

functions

62

k

P(x,y) with

tP:x-st

functions ]si,si+:[ One

for

a t are

some such

n

:

a o (x)

U j:1

pE~

and

a i a

that

at(x)

(y-aj

(t))

meromorphic

¢

am(X)

as

an

inequality

of

the

]

x6]si,si+i all

q

is

6

~.

n

long

continuity with as

W

of

small small

is

small

,X)

= ~ _ nj J

such

(Cq)i/n8

(~>0

x

any

x

8

6

-

aj (X))

J I ~

Cq 8q

that

for

j =i

[

If

j(x,8)

As

8

t.The

~m,

n

(g(~

H

this

implies

aj<~,~)(x)l

as

the

q/n

do

g

~

in

and

that

Ig(8,x)

width

of

as

kind

k

sup

there

function soon

•

considers

for

J

strips

no and

x

enough,

x E

8).

j(x,~)

Set

~< ( C q ) l ~ along

the

intersect,

it

that

does

Is i + =

j(x,8)

n, J0.

si+ I On

~q/~ curves

y

follows not

~[,

=

aj(X)

from

depend

on

x and

n arbitrarily

]si,si+n[

one

gets,

of the

small as

soon

enough

[ g (8 , X)

-

aj

(X) [ ~

1/n

n

8q/n

eq

o

(if

two

they

go

strips on

around

a t and

intersecting

Thus

we

get

in

the

requested

sup

lg(~,x)

xE]st,si+i

Similar Being aj

bounds uniform

obtained

for

all

limits on

both

of

am

[si,y])

intersect

bound

si
, then

-

aj

(x)l. o

derivatives the

sides

y,

for

[

x

at

.

C® of

&re

functions the

singular

obtained

inductively.

g ( 8 , .) , t h e points

functions

patch

like

together

0 and

give

a

C"

function

on

I.

O

63

The equations

classical

as

let

P

(in

$I(EI)

be

a

One

has

nonzero or

~

differential

ADE

is

and

an

with

the

is

a nonzero

The

class

study

results

concept

of Cm

the

f

is

function

; then

a

P(x,f)

classical

so

equations

of

of

the

y' (x) .....

ADEs

exposed

....

in

in

of

an

y(m> (x))

~

classical of

1.10.9

Let

P ( x , G (m))

that

(ADE) are

the and ~

0

solution.

one

gets

also

in m + 2

(x))

a

setting

together.

0

=

variables.

Colombeau's

ADE

provide

put

form

y(m)

Rubel

=

For

0

be

0

sense

theorem

-

P

3x0EI such are classical

more

formulating

theory,

[1,2,3,]

in

would

; note

that

if

@ <

> P(x,y(x)

y

conjonction clarify

is

a

the

function

of

then

consequence

be

variables

if,

1.6.10)

polynomial

solution

P(x,y(x),

in

by

continuous

derivation

equation

P(x,y(x),

P

a

two

only

(proposition

multiplication An

where

in

and

recovered

f be

solutions.

Algebraic which

if,

0

are

: let

polynomial

$(E~))

x~

distribution

in

solutions

association

in

(see

proposition

1.10.8

is

1.10.8

and

....

1.6.6)

,y(m)

(x)):

0

. An

immediate

Corollary.

as

in

in

let

m E ~.

Then

@~(I)

] ¢=~ G

that G(xo), G' (x 0) .... , G ( m - 1 ) ( x o ) complex numbers

general

complicated

ADE : the 2xy

p(x,G,G"

, . . . , G (m))

ADE +

(x2-1)

~ y'

=

0

=

0

the

is

a

C ® function

situation

can

B4

has a solution

R(~,x)

i

namely the class of

= -- e [x-~'1~] if

x E ]-1,+1[

1 R(~,x)

which

in ~ ( R ) ,

0

if

x ~ -1

or

x

)

1

is null on (-~,i[ U ]i,+~) and "infinite"

on ]-i,+1[

2.

Anyway this section shows that, provided ~ suitable definition of ~(Q) would be adopted, one can have strong results stating that generalized solutions are automatically olassic~l solutions.

APPENDIX

A

In of

SURVEY

this

ON

PRODUCTS

appendix

distributions

and

we

expose

~(~) . The reader of knowledge of d i s t r i b u t i o n

survey

their

1

-

THE

If product

the

insignificant

DUALITY

all

denote all

~E@.

SEX[o c

X)

if

<,>

and

denotes

on

the

(4E~,

X

and

distribution by

set

TEXIo

of

-

THE

of

the

proofs) but more notations

and in complete). we

the

is

if

X

the

and

=

are

m)

and

@

a

normal

dense T

if

ST

is

in

such

X ~ o c is

product

OF

first

space

power

consider

TEC

then

TM

space X) ,

that

defined

let T~EX,

from

defined

X'

of us for as

by

<> @

arbitrary

~.Examples

is

and

distributions

between

an

order

and

@'

and

element

given

by

of

the

<<,>> @

the

duality

identical

Sobolev

spaces

'

such

that

LOCALIZATION

let

8~ (of

us C®

D T E L p if

to

one

X = W z'p

AND

recall

FOURIER

that

functions

the

O~
Fourier

decreasing

at

transform infinity

is faster

1 any

in

to h a v e a good surveys can be

where

METHOD

At on

in

products

product

<S,T~>

~CX'C~'

o then

duality

; f

=

continuously

have

W¢'P=(TE@ 2

(with proofs,

of

generally,

(we

the X'

support l~p<m)

known the

is assumed detailed

Ill (without

<ST,b>

between

widely with

simplification

~CXC~'

X1o o the

If

X1o o from

More

(i.e.

by

~E@.

most

METHOD

S E ~ 'm (i.e. a ST is d e f i n e d

distributions

the

relations

<ST,~> for

DISTRIBUTIONS

this appendix theory. More

found in 0berguggenberger Colombeau-0berguggenberger [I] As a minor here ~=~n

OF

of

, as

well

~(~)

as

all

partial

= [

e-2

~i~

J

derivatives)

× ~(x)

dx

by

defined than

66

and extended formula holds

Now

let

g'

S,TE~'

neighborhood called • E~(fx)

to

by

transposition

and

fix of

the ~'we define

x

suppose and

It coinciding This

can with

defines

important front

be the the &

S

is

the

~nx(~-{0))

such

that

for

some

W F (S) does

@

not

with

W F (T)

=

by H~rmander) have disjoint 3

-

THE

METHOD

A

net

The lower

index

concerns

the

~(fS)

fml

on

an

is a

OF

element

in


the of

(a)

support

(b)

f~

(c)

~ 0

(x)dx=l for

index

~ has

~

1 = -~n

special

(x) net

of

W~ for

all to

~

a

WF(S)

(x,w)EWF

then is

THE

delta ~ *

of

a

(x,~)

cone

S and

definition

TO

An wave E

around

If

case

as

of

all

in

(S) ,

(x,0)

strict (0}

For

Ambrose.

set

of

x.

(x,~)EWF

PASSAGE

a

a is

exists

of

front

of

particular

called

are

what

concept

set

Ambrose's

AND

there

that

sense

the

wave

form

exchange

"

decreasing

that

of

xE~ n g(fxT)

the

the

rapidly

case

is

in

the

a unique element of @' STE@' ( f x ) d e f i n e d above.

from

neighborhood

MOLLIFICATION

(~}0<~

T

of

such

particular

and

is

the

:

. An interesting singular supports.

upper ~

S and

obtained

TE~'

such

[g(fxS),g(f×T)]>

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

=<~,~-i

shown local

distribution ~,

with

convolution

particular

set

that

fx6@

<ST,~>

; 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~]:
~ ~

such

that

E.,

~

is

R[

its

closure

E@s[~]

~nd

~

is

~nd

]O,÷~[x

I the m a p x E ~ ~ R ( $ , x ) a n d all extended to ~, f o r e a c h 8 > 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)l
E

c~n

,compact that

-N

be

continuously

subset

of ~,

and

70

for

all

all

q6~,

x6K

and

0<~
{R6@M,s[-~] there

;

such

are

c>0

that,

and

for

D>0

I (D~R)

for

xEK

all

and

O<~
all

K

and

(~ as

above,

and

for

satisfying

;and

(~,x)

I~
finally

we

set

~ (~):~, ~[~]I~[~]. 2.1.2

-

Remark

This

definition

is

also

applicable

with ~CXC~, for any QcE~ open (we continued to X a n d K is a c o m p a c t that,

in

the

case

X=~,

we

state subset

have

back

2.1.2,

we

for

a

subset

X

of

E~

that D~R(~,.) may be of X) . It is t h e n c l e a r

the

definition

of

~s(Q)

in

18.2. 2.1.3

If f:X*~

is

XCE n

of

is

class

derivatives such

Definition

-

can

functions

be

is

Notice

as C®

in in

if

fiQ

continuously

also

that

X

denoted

say

that

EC®(~) extended

by

C®(X)C~(X),

a

and

classical

if

to

X

f[~

and

.The

function all

space

its

of

all

C®(X). by

setting

R(~,x)=f(x)

for

each

f6C'(X)

2.1.4

If and in

-

XC~ n

GE~s (X),we 1.8.4,

2.2

-

Definition

is

may

with

~

as

there

ESSENTIAL

in

2.1.2,

define

the

being

FACTS

x6X,

K

is

generalized replaced

CONCERNING

by

a

compact

complex

subset

number

of

X

G(x)

as

SENSE

OF

X.

C = FUNCTIONS

IN

THE

WHITNEY

family

2.2.1

-

Let

X

{f~}

Definition

be of

n

an

arbitrary

complex

functions

subset

of

defined

E~. on

X

We

say

is

of

that c~ass

a C®

71

on

X

if,

for

neighborhood x,

x'EVnX

every of

we

mE~,

x o,

aE~ ~ with

and

A>O

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

an
I

function

for

all

equal

to

n~2, i -- o n n

a~*l each

and

b,~l

]a~,bn[.

°

72

Then f a n d all its d e r i v a t i v e s c a n be e x t e n d e d to X (f a n d a l l its d e r i v a t i v e s v a n i s h in 1), t h a t is, f E C ® ( X ) . H o w e v e r f cannot be the restriction of a C ® f u n c t i o n on ~, b e c a u s e t h e s l o p e of the

segment 1

(ba,

--) a n d n

n*~.

2.3

joining

Then

-

(an÷l,

the

-

) is

(f(a))ne~

2.3.1

of

pairs equal

If X a map

3

which

tends

to

-~

as

n(n+l)

~ C~(X).

FUNCTIONS

-

n

to

n+l

GENERALIZED

consider

the I

ON

AN

ARBITRARY

SUBSET

OF

En

Definition

is a non void f,:]O,+~[×X~. We

families

(f~)

as

n

subset denote

above

such

of E~, f o r e a c h ffEM n we by @ H , w , ~ [ X ] the space

that

ffEM

i) means of

each

f~

has

that,

for

each

NEM,

c>O

and

xo,

locally

ffEM~ a n d W>0

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,)

i
i --','+~

B!

x,x'EVnX.

constant c(E) in (ii) is locally bounded above precisely, for e v e r y m E M , ~ E M ~ w i t h I ~ I ~ m there a r e V, a n e i g h b o r h o o d of x o, NE~, c>O and w)O inequality (2) holds with c(~)=c/~ ~ for each pair

e/~ N w h e n

and such

(x_x,)

X

If~(~,x)-

and

the

~0.

More

0<8<~.

Y3

Then ~w,s[X]:

we

define

<(f~)

an

ideal

nE@~,w,sKX]

~w,~[X]

of

such

that

V

x o,

@s,w,s[X]

for

all

by

(~E~n,

:

xoEX

and

~EM

qE~

there

for

all

are

a neighborhood

xEVflX a n d

Finally

we

c>0

and

W>0

satisfying

O<8
set

~w,~(X)

2.3.2

-

If X of To

of

: g~,w,s[X]/~w,~[X],

Remark

is

a

locally

Definition 2.3.1 is p r o v e t h i s , let ( f ~

convex

closed

set,

a consequence of n be ~ f a m i l y

then

condition

(iii)

(i) a n d (ii) t h e r e . satisfying (i) a n d (ii).

If m E ~ , ~ E M ~, ] ~ [ g m and × o E X a r e g i v e n , let V be a c o n v e x compact neighborhood of x 0 in E~ s u c h t h a t V N X is a c o n v e x set. By c o n d i t i o n (ii) , for e a c h ~>0, ( f ~ ( ~ , . ) ) . belongs to C ~ ( X ) . The classic&l Whitney's extension theorem gives H,EC'(E~) such that D~Hc (x)=f~(~,x) for all x E X and ~E~ ~ If x,x'6VNX, the segment [x,x']: {tx+(l-t)x'}0
algebr&

have

Taylor

2.3.3

-

(a)

It

~s(X)

s formula

to

D~H~,

we

obtain

(iii) .

Remarks

is

obvious

coincides

(b)

If

~CXc~,

(c)

If

~cXc~

that,

with ~ an

and

X

open is

if

X

is

any

open

set,

the

~w,~(X) set,

locally

then

~w,~(X)C~(X)

convex

then,

from

2.3.2,

we

~w,s(X)=~(X). (d)

C~(X)

j

into (e)

restriction

If X

is

arbitrary,

there

is

a natural

inclusion

from

~ w , s (X) If map

~

is

any

open

set

containing

X,

we

have

a natural

74

r

:

~(2)*~,~(X)

G ~

GIX

where is

a

is

the

class

representative

of

G

~E.A"w,s[X],

{ ( D ~ R ) IX}

of

the (if

GIX

{(D~R) IX}

family RE#s[~],

therefore

the

it map

is r

if RE@.,s[~]

n

immediate

is w e l l

that

defined)

eEM

2.4

-

WHITNEY'S

We classical can

be

EXTENSION

are

just

theorem's found

Theorem.

in

Let

X

THEOREM

giving proof

in

a closed

of

the

FUNCTIONS

proof,

Hestenes[l]

Biagioni-Colombeau

be

GENERALIZED

idea

given

based

on

. A detailed

the

proof

[2].

subset

~ is

an

FOR

of

En.Then

the

restriction

map

(E~)o~w , ~ (X)

surjective

Idea

of

the

proof:

s e t of s p h e r e s some technical Let

Hestenes

$I,82,..., whose conditions.

GE~w,s(X)

representative

proves

of

~(~,x)

G.

be We

:(

given define

fo(~,x)

the

existence

reunion

and

let

is

E~-X,

R=(f~)

~:]0,+~[×En*C

if

if

by

x~X

where f~(~,a Qi ( ~ , x )

=

I o'1

i) (x-a

Z <m

o:'!

a countable

which

satisfies

~E@M,W,~[X]

xEX

E Qi(e,x)Hi(x)

of

i )

be

a

75 ( a i 6 X is a n e a r e s t p o i n t to the on the d i s t a n c e f r o m a i to xi), every

pEM,

HI+...+Hp=I (S~

c e n t e r x i of S i a n d m i 6 ~ d e p e n d s and HI6C~(E~) satisfies, for

is

Both

a

on

sphere

Q~

~ESM,s[En]

Sp,

with

and

H~

.

obtain

To

Hi=0 the

(if

same

satisfy

i>p)

on

Sp,

center

as

Sp

suitable

the

H~=0 and

on

SpCS~CEn-X)

inequalities

moderation

property

En-S ~

in of

~,

.

order

we

that

have

to

consider two cases, due to the relative positions of K and X : (a) if K : K ~ U K ~ , K~ a c o m p a c t s u b s e t of X a n d K 2 C C E a - X , and (b) if the closure in E~ of K D ( E , - X ) intersects K. In t h i s s e c o n d c a s e , it s u f f i c e s to p r o v e t h a t , for all m 6 ~ , ff6~ n with [ffi~m and aEDX there are V, a n e i g h b o r h o o d of a, NEM, c>O and ~>0 such

that I ~<

ID~O(8,x) for

all

xEV-X

We h a v e

and

With

the

also

that

0<~<~.

properties

of

Hi

and

D~(8,x)=f~ for

all

E>O In

and

xEX.

the

last

two particular closed half

cases space

different

much

and

c $ -N

Qi

we

get

that

SEgM,~[E~].

(~,x)

[] two

paragraphs

of this ; as in simpler

we

give

theorem, when the classical than

the

complete

X:
general

proofs

of

when X is a proofs are

proof

sketched

above.

2.5

-

BOREL'S

THEOREM

Theorem. Given, and ~ =(el,...,e GE~s(E~)

such

n) that

for an

FOR

GENERALIZED

FUNCTIONS

each ~=(ffl .... ,an) orthonormal basis

DIG(O)=

c~

6M~' of

an element E n, there

c~

in ~s exists

76 C B

Proof.

Let,

each

BE~ n,

xEE n and

mE~,

for

c~E@~l,~ be a r e p r e s e n t a t i v e

of

B~ For

each

Tm(8,x)

~>0,

= Z c~ ( ~ ) x n

(xZ=xl

let

us

define

i • . . x , , ~ if

X= Z x i e i ,

B=(BI,...,B,))

IBt=m

Then

(~)

D~[x~T m+l (~'x)]

for

a

all

with

[alUm

(0):0

.

Assertion. For each mE~ t h e r e are N m E ~ a n d q m > 0 s u c h t h a t if 0<~<~ m there exists g~EC®(En)(~ depending on m a n d a) vanishing in a n e i g h b o r h o o d of z e r o s u c h t h a t 1 (4)

Z I~l<m

Proof.

Since

1

-a!

cjEgM,s,

sup xEE~

IDa[TIn+ i ( £ , x ) - g ~ (X) ] [< 2m

there

are

NmE~,

Dm>0

and

Wm>0

such

that

-N

l;j(~)l< for

if

all

~

m

{~]=m+l

Let fEC®(E~) be s u c h t h a t f ( x ) ~ 0 1 < --, f ( x ) = l if I x l ~1 For each

Ixl En

(5)

0<~<~ m and

Dm

2

for

all

aE~ ~ let

E n

M~

=

max

i~[
sup

~

[D~ f(x) I
xEE~

let

N m

(6)

8=

~(m,~)=

rain (

,1), Cm2m+

I

x

and

f(x)=0

77

where

(7)

C.

=

max

D m

Z i II

,D m Iim

ll~l=m÷l

If

we

~ I ~') Iim I ~1

Y. i ,li l IiI

~!

(B-#.l.)

!

=m+l

g~ b y

define

X

f(--)

g~(x)=

Tm+I(E,X)

8 then,

clearly,

gs6C'(E.)

and

it

vanishes

gS(X)--Tm÷l(~)X

Ixl

if

)

near

zero.

Also

)8

E;, It

is

therefore

<xEE~:lxl

over

sufficient

~8}.

to

consider,

in

(4),

the

supremum

We h a v e

E.

~.I sup

Ixl

I D-~ ( ~ C~ ( ~ ) x ~ ) ~<8 '~'=~+~

1<

SUp Ixl

E,~

~

~8

,~,,om+~

(,8-~)!

E.

E.

~!

D m

N

if

[BI-I~[

Ixl

IC~ ( ~ ) [

0<~<W m and

[ffl~<m. T h e n

8 m+l-'e[

Z

from

(6)

and

(7)

we

have

1 (8)

--

sup

•

E

IDa(

e~ ( £ ) x ~) [

Z

It

m

lq)gm

m

2m+l

I ~ I =m+l

if

0 < E < W m.

Now

we

have,

from

the

[o] (9)

I D~g~ ( x ) I <

:~

definition

of

g~

and

(5),

x

8-' ~ ' ID~f(--) I IDa(

~:

e~ ( e ) x ~ ) [

78

B~

m+l-

I /~1

Ixl (B-~) !

IBl=m+l

M~ D~

E~

B ! Z

~<

$-,~I

m+l-I~l

Z

Ixl E.

if

From

o <~ <9

and

(7)

(9)

m

Z

1 --

sup

lal~m

a!

I×lEn<$

(io)

if

ID~g~(x)l

Inequalities

0<8
Cm$ N ~

~

m

8

(8)

and

(10)

prove

1 2m+I

the

assertion.

[]

m

Let

F:]O,+~[x

F(8,x)=

En*~

be

To(S,x)

+

defined

by

[Tm÷l (~,x)-g~

Z

(x)]

m=0

Clearly a

FE@s[En];

compact

subset

let of

us

En

prove .and

that

F

ff6~ n w i t h

is

moderate.

lff[=r,

Given

we

have

I (D~T~+I)

(e,x)

K

r - 1

I (D~F) (8 , x ) I<1 (D~To) (8 , x ) I+

Z

[ +

m=O

r - 1

m=0

As

in

(9),

r-

if

m=r

xE~, r - 1

1

z m=O

m=O

u+~=~

.< M~ ~ m=0

J B I =m+l

N

~ m

6:

max

~ l g ~ l

where L~

=

sup I~l~lal :<EK

<

~ I~l=m+l

~~

ID~x~l}

m

,1

.

79 Let

N

max

=

Nm

min { l , n m } .

n=

and

0~m~r-I

r-

(ll)

0~m~r-

~-i

1

E

ID~g~ (x)]~

Then

I

Lm

Dm

max

~

M~

m=O

6N(l+r)l~{
m=O

,1)

<(Cm 2m+I)

I

C ~

~N(l+r)

where

c'>O

is

independent

of

r-i

(12)

x6K

r-

E ID~Tm+I(~,x)I=

~6]O,~[.

Now,

1

Z

m=0

and

IDa(

~

c~(~)x~)l

I /{ I = m + l

m=O

r- 1

Dm

Z

Z

--F-

m=O

~

ID~x~l

m

I#l=m+l

D m Lm

r-i

e"

~< m:O

if

0<8<7) a n d

and

~o>0

I

m

IN

x6K.

If r=0, ID~To(e,x)l=16o(t)l, such that

then

there

are

N'E~,

and

the

~o>0

C o

~

Ico(E)l

.<-

o<e <~o

if

IN"

If

r>O,

DsTo(I,x)=O.

assertion,

we

Let D~[x

~

have

G6~s(E

With proved

R)

be

F(t,x)](0)=D~[x

the that

the

above,

(ii),

(12)

F6$M, ~[En] .

class

of

F.

If (T~+l

~* T o ( 8 , x ) +

t>O

we

(~,x)-g~

have (x))]

(0).

enough,

we

m=O

By have,

if

(3) k6~,

and

since

g~

is

zero

for

{x{

small

80

D~[x ~

E

(Tin+ I ( E , x ) - g ~ (x))] ( 0 ) = 0 ,

m=k

for

all

with

ff

Then,

[ul4k. lul
if

k

D~[x~F(8,x)]

(O)=D~[x~

~ Tm(~,x)]

(O)=Da[xe

m=O

E

cs(8)xJ]

(0)

t~l~k

B~

= Ix ~

E

~(~)

2.6

-

EXTENSION 2.6.1

there

is

E(G){S÷

If a

-

xB-~](O)=a!~+(~).

D

(B-a)!

i ~ i ~ l ~ 1 , k

OF A G E N E R A L I Z E D

FUNCTION

DEFINED

IN

A HALF

Theorem

S+ denotes the closed linear extension operator

=G,

for

all

The

proof

2.6.2

-

There

are

half space S+=EnX{tE~:t)O} E:$s(S+)+$s(E~×~) such that

GE~(S+).

is

b~sed

on

the

following

lemma

{ b k}

that

:

Lemma

(i)

sequences

{ak},

such

bk
m

(ii)

E

lakllbkl~
for

n:O,l,2,...,

k=O

m

(iii)

E

ak ( b k ) ~ = l

for

n=O,l,2,...,

k=O

(iv)

Proof.Set

SPACE

bk-+ - co &s

bk=-2 k

The

k-~+co

solutions

ak~

of

~nd

81 N

E

Xk. ( b k ) n : l

(n=O,l,...N)

k:O

by

are,

Cromer's

rule

and

V~ndermonde's

expansion,

akN=AkBkN,

where k-1

Ak =

~

( l + 2 J ) / ( 2 J - 2 k)

J=0

&nd N BkN

=

H J=k÷1

(1÷2J)/(2J-2k)

•

Then k-

[Ak[

~

II

1

2

2j÷~-k

= 2-(k

-3h)/~

|=0

~nd N

log

Bks

=

N

log

E

[l+(l+2k)/(2J-2k)]<

E

j=k+l

to

(l+2k)/(2J--2k)<4

j=k+1

Then Bks is m o n o t o n o u s Bk~e~Setting a k = A k B k we h ~ v e

incre&sing

with

N,

~nd

2

[ak[<

~nd

e4

2-ck

-3k)/2

so

[a k[

]bk[~<~o

k

for

e&ch

n.

Setting akN

we

= AkBks

for

kEN,

aks=0

for

k~N,

have E

aks(bk)

~ =

1

k=0

for

N~n.

Since

[akNllbkl~g

[akllbkl n a n d

Z k

[akllbk[~<~

converges

82

we

have

finally E

ak (bk) ~ :

lim

k=O

N

Proof of 2.6.1 Given representative of G a n d let

R(~, (x,t))=

~

aks (bk)n=l.

G6~(S÷),

let

R:]0,+~[XEnX~*C

r($,(x,t))

where and

¢EC®(~) (ak),(b k) For

the

is such are the

each

8>0,

for given

R(8,.)EC~(E~×~)

be

rE@~,~[S+] be

defined if

if

that ¢(t):l sequences

[]

k=0

a k ¢ (bl, t ) r (~ , ( x , b k t ) )

Z k:o

<

~®

tE[0,1], by the

by

t<0 t)0

@(t)=0 for lemma 2.6.2.

: indeed,

because

t~2

bk*

-

sum

Z

ak¢(bkt)r

(8, (x,bkt))

is f i n i t e for e a c h t
for

t-,0 +

Let compact

us set

those

any

property

for

prove now that RE$M,~KEn×~] . For this, let K be in E n × ~ . It s u f f i c e s to prove the moderation

K-S+.

[R(~,

For

(x,t))

a=0

and

if

[(

Z

[ ak [ ][¢]l ® [ r ( ~ ,

k: 0
(x,t)EK-S+,

we

have

(X,bkt))

I

t <2 k

Then

In the there are

above

NEN,

sum,

c>O

( x , b k t ) r a n g e s in a and ~>0 such that

Jr(8, for all inequality

that

(x,t)EK-S+. for R ,' for

Elak[[bk[n<~

for

(X,bkt))

I~< c~

compact

set

of

S..

-N

obtain the desired Since Zlak[<~, we we use the fact D~R, with aEM n , [ ~ [ ~ i [] n=l,2 ....

CHAPTER

GENERALIZED

SOLUTIONS

OF N O N L I N E A R

DIFFERENTIAL

3,1

-

3

PARTIAL

EQUATIONS

EXPLICIT C O M P U T A T I O N S OF SHOCK WAVE S O L U T I O N S I N N O N C O N S E R V A T I V E FORM

3.1.1.

The

used by appendix

following

engineers l) :

is for

a

system

the

of

numerical

elasticity

OF SYSTEMS

in

simulation

one

of

dimension

collisions

(see

Ot+(0U) x=o (0u) t + ( 0 u 2 ) ~ + ( p - S ) ~ = o (I)

(Oe) t+ (Peu) ~+[ ( p - S ) u ] x = o 4 St+uSx_

~Ux= O

I

3 p:F(O,I) where

o

the

is

stress

internal given that

energy

by ~

the

an

is

for of

derivation mass,

first

from the

of

continuous

of

and

make

velocity

total

mass

(thus as

more

a

)

system

energy

quantity and sense

solutions.But

from ; the

g and

=

I+

--)

physics

it

on

on

seek

mass,

; the

one

p

p and

Note

be

a

the

in

of

Appendix

are

The as

The

Hooke's

state

wave

knowledge

i.

conservations

expresses

analysis

I.

as

solutions

of

law

of

obtained

symbols

long

is

function

material

sketched

of

of I the

pressure

,some

material. shock

of

the

is

equation the

component

can

respectively

classical

a unit

2 P(p,I)

function

fourth

we

is

per u2

deformations

the

depend when

e

depending

express

within

, S

energy

generally

elastoplastic

material

equations

The

the the

state (

this

measurements system

unit

large the

three

momentum

elasticity.

per

u is

constant

including, history

e

equation

a

the The

density,

deviator,

"=" we (of

in

seek ha-

84

sic importance in n u m e r i c a l simulations of collisions) the f u n c tions u and S are discontinuous at t h e s a m e p o i n t , s o t h e t e r m uS x does not make sense classically. Therefore the s y m b o l s ..... h a v e classically to be c o n s i d e r e d as h e u r i s t i c in c a s e of s h o c k s . Indeed, working replaced by the association generalization of e q u a l i t y

with our theory, they are naturally symbol (which, a n y w a y , is the natural between distributions).

Engineers model also elasticity by t h e s y s t e m (9) b e l o w in 0 , u , ~ , w h e r e ~ is t h e s t r e s s . Assuming n o w D to be c o n s t a n t (D- Do) for simplification,we explain b e l o w h o w (9) g i v e s h e u r i s t i c a l l y the simplified system (2) in u and ~, which presents the main difficulty i.e. t h e f a c t t h a t the term u~ x does not make sense classically in c a s e of s h o c k w a v e s . In this section and the following one we study the simplified system (2) in a r i g o r o u s mathematical way, which was not possible before.

3.1.2.

Consider

the

system

of

two

nonlinear

I

equations

1 ut÷

u ux ~ - Ox DO

(2) ~t+U~x

~ k2 ux

where k-o represents t h e s p e e d of p r o p a g a t i o n of t h e e l a s t i c waves and u,aE~.(~2); u x and u t denote the partial derivatives in the variables xE~ and tE~, r e s p e c t i v e l y . We seek solutions which are constant except on the shock in which they have a jump. Mathematically, this kind of solution (steady shock) is represented by t h e f o r m u l a .

(S)

I

u(x,t)=

Au

H

(x-vt)+u~

a(x,t)=

Ao

K

(x-vt)+a t

where H , K E ~ , (R) are Heaviside generalized functions, Aa=~r-~ t . The aspect of u , ~ is g i v e n by t h e f i g u r e :

Au=ur-u

t

85

~U U~ U r

)

x

vt figure

3.1.3.

:

graph

of

system of partial if e a c h e q u a t i o n

partial derivatives. conservative form.

the

function

u(x,t)

)

differential in the system

Otherwise

the

first

equations is s a i d t o is a finite sum

system

The system (2) is in n o n c o n s e r v a t i v e equation may be written &s 1

second

important to conservative

3.1.5.

~

is

said

one

+

Solutions

- v Au

of

that have

(2) (3)

in

non-

since,

the (2)

form

we

H' + ( A u H + u e) ( A u H ' )

- v Ag

although

0, conservative

numerical attempts to failed (Appendix 2).

of in

form

form.

replace

(3).

have A~ ~ --

: K'

Do K' + ( A u H + u ~ ) ( A c K ' )

%

k2Au

H'

equivalently, (v-u~)AuH'

- (Au) 2 H H ' + - -

K'~

0

Do (4)

be

1

( -- u 2 ) x - -- ~x ~ 2 Do not written in

is

notice system

Replacing

or,

to

be of

Remark.

ut the

x

Definition.

A conservative

3.1.4.

the

I (v-u~)AgK'

- (Au) (A(~)HK' + k 2 A u H ' %

0.

(2)

It by

is a

86

Replacing formula first l i n e o f (4)

(19) o f C h a p t e r i is e q u i v a l e n t to

Au

if

AU

/

an + - - ]

2 to

0,

v

from

since

H'~

K' , t h e

i'~ 0

A~

= u t + - 2

However,

and

Do Au

(5)

(4)

A~

[(v-u,---) or,

in

the

(6)

second H

K'

~

where $ is a n y Dirac depending on k 2 , Au, gives, if A g / o ,

DoAu line

of

(4)

we

only

know

that

A

generalized A~ and v.

function a n d A is Then, the second

a

real line

number o f (4)

~u

(7)

v

= u~+A

Au-k 2 -Aa

Comparing

(5)

and

(7)

Au

(8)

This

1 --)Au

-(ADoAu

result

.

2

case

Rankine-Hugoniot different

The

jump

The following

shows

case

conservative

3.1.7.

:

Remark.

conservative

these

obtain

A~

ks Ag

3.1.6.

we

(an

(a u n i q u e jump

great

jump

formulas).

of In

numerical

conditions

for

e system

(2) one

is

an

dimensional

between

conditions) , given

Appendix

from

of

difference jump

condition

solutions

system

system

the

infinity

2 we

the and

nonthe

by

the

classical

put

in

evidence

tests.

of

heuristic

3 equations. simplification

elasticity:

of

the

87

+

n t

(9)

(nu)~

(nu) t C t +

In

fact,

the

[~,

line

(~u)~]u

Engineers constant, first with

since

line the

+

t

does

(2)

not

(ou) ~ ~

In engineers ~quality

+

law).

written

as

~ ox

(nu) x

by

0

and,

incompressible,

n=no)

cannot

conservation)

. If

do

the

that,

+

first

our

context

usual

H,K (9)

we

we

and

L

get

from

equation

this

can

precisely seek

be

interpreted

the

first

solutions

of

by

line

(9)

of

the

= Au

H(x-vt)

+

u,

a(x,t)

: Aa

K(x-vt)

+

O~

n(x,t)

-- A n

L(x-vt)

+

O~ .

are

He~viside the

first

generalized line

of

saying

of

u(x,t)

L'H

+

L'

LH'=

-

Au

(LH)'

(L'H

~

+

LH')

H' , t h i s

-

is Au

a)

is

p the

stated

because

(9)

functions.

(9) , a s s u m i n g

v-u~

= Au

+

n~

-An

n t -An

H'

equivalent

~

that

with

the

form

0.

to

Putting AP~o

An ( v - u t)

(ll

obtain

(nu) ~]u ~ 0.

(10)

Since

considering

they

0

write more in N(~) . As

into

be

+ ~ u~]

almost

(setting we

(Hooke's may

n t

are

(momentum

imply

[~t

where

(9)

conservation)

~x

k 2 ux

of

replace

association

(mass ~

+ ~ [u,

solids

of

0

(nu2)~

uU x ~

second

+

+

~

:

:

(10)

88

which

is

equation have,

the of

classical

(9).

using

Rankine-Hugoniot

Replacing

the

now

associations

(i0)

in

(L HS) '-~ - H' ,

u, (ii

b)

v-u,

: Au

(ii

replacing

(i0)

c)

in

the

v-u e : A

for

second

line

(H~)'~

H'

the

of

and

first

(9)

we

(L H)'~- H'

~e (au)- (A~)

+ u,

Finally,

condition

the

(ap)

+

third

~,

line

(Au) of

+

(9)

(Am) (au) :

~U - k s --

Au

A~ where

A

is

given

in

(6) .

Elimination

of

v-u e gives

hu k s --

(ii')

1 1 1 - -- ( - - + ~

In

order they

Comparison

to

Thus this

3.1.8.

we

shape,

(8)

The

two jump

the

Do

2

A

rid

statement

of

conditions. Pt

+

(9)

(ou)

(Ou) t + (Yt +

U(Yx

x

1

De

Dr

is

a

simplified

conditions

proved

form

this

the (9)

=

form

(in

that

this

of

that

(2)

(9),

is

is,

provided

k2

ambiguous, ambiguity

first

a

equation

of

u,~

(9)

and

we v) .

equivalent

for

(12)

the

equality

may

be

in

written

which of

~ s (E s) more

to

shock

waves

holds.

Ux

ax

or

(9")

not

physically

is

to

one

of

gives

precisely P t

=

is

technique and

o

(0us), ~

2

Au

gives

1

remains

of

with

Thus

(A - --) A U .

A.

get in

(2)

=

(~+--).

rigorously

of

relation i

)~

jump

(ii')

simplified

quantity To

equations

(9') I

and 1

h&ve a

that

same

1 --=-

Computation

acceptable. that

the

between

(12)

of

express

give

the

A~

Pt ~)r

2

Ac;

postulate

easily

+

(ou)

(ou) t + ~t

+ U(Y,

notice

the

last

nonambiguous as

x

=

o

( p u z)~ =

~

k ~ ux

a,

89

(9') can be justified by the f a c t t h a t t h e t w o f i r s t equations e x p r e s s b a s i c l a w s of p h y s i c s w h i l e t h e t h i r d o n e c o m e s only from specific observations (that have not been done within the very s m a l l w i d t h of t h e s h o c k ) . (9") w o u l d e x p r e s s that within the width of t h e s h o c k the t h i r d e q u a t i o n w o u l d be m o r e v a l i d t h a n t h e s e c o n d one (which could perhaps be j u s t i f i e d by the fact that viscosity and thermal effects are neglected in the second equation). The shock solutions of (9') a n d (9") w i l l be c o m p u t e d l a t e r in § 3 . 4 a n d there one will find precise v a l u e s of A . T h e s e v a l u e s w i l l be different

for

(9')

and

(9")

but

in

elasticity

in w h i c h

is quite small (less than I0 -~) t h i s difference s m a l l to be p u t in e v i d e n c e experimentally. Thus both a n d (9") a r e in a g r e e m e n t with experiments.

3.2

.

.. p

will be models

too (9')

D I S C O N T I N U O U S S O L U T I O N S OF THE CAUCHY PROBLEM FOR A S Y S T E M I N N O N C O N S E R V A T I V E FORM.

The shock solutions are represented by f u n c t i o n s having a f i n i t e n u m b e r of d i s c o n t i n u i t i e s , for i n s t a n c e , classical Heaviside functions. Let us consider the system (2) and let us s e e k solutions representing a shock which propagates with a constant velocity v ( s t e a d y s h o c k ) (see (3) a n d the f i g u r e t h e r e ) . T h e n the t e r m u~ x a p p e a r s heuristically under the form of a classically meaningless p r o d u c t ¥ . 8 (Y = c l a s s i c a l Heaviside function, 8 =classical Dirac delta distribution). T h i s m e a n s t h a t it is q u i t e i m p o s s i b l e to find a solution within classical a n a l y s i s : the i r r e m e d i a b l y ambiguous t e r m Y8 w o u l d t h e n be e q u a l to ~ t - k ~ u x which is a well defined distribution. In C o l o m b e a u ' s theory, however, seeking solutions (U, ~) in (~ (~+×~))~ one finds solutions. Then the classical discontinuous functions (u,O) a r e o n l y the m a c r o s c o p i c a s p e c t of the genuine solutions (U,Z) . The a b o v e a m b i g u i t y in the term ug~ comes only f r o m t h e l o s s of i n f o r m a t i o n in t h e p a s s a g e f r o m (U,~) to (u,g) . In t h i s s e c t i o n w e u s e t h i s p a t t e r n to s o l v e the Cauchy problem for the system (2) with discontinuous Cauchy data, representing shock waves.

3.2.1.

The

numerical

Consider conditions uo, go functions

on

scheme.

the system E BV(R) (the

~) .

Let

f i x e d r>o, be t h e t i m e setting, f o r all iE2,

h>o

be

(2). space the

meshsize.

Let of

space We

be all

meshsize

discretize

given two initial bounded variation and the

At=rh,

with

x-axis,

a by

90

I i

at

the

:

[Xi_ ~ , Xi÷~[

We

seek

an

time

t,

= n(At)

(respectively g~)

on

the

0t)

In follows

uo

and

we

shall

x i :

approximation and

the

i h.

(u~,

at

the

~)

of

point

function

which

is

(n+]d)At[

for

i62,

the

x i . We equal

to

solution denote

u~

(u,~) by

uh

(respectively

rectangle

Ii x

as

where

,

[(n-~)At, order

to

: let ~o,

u~

construct

and

~V

respectively,

deduce

(uT ÷I,

W~n

(13)

v~

= -

the

be

the

on

each

gT+1) , by

(u~ u i

-

n:0,1,2

quantities

mean

values

I, . F r o m t h e

induction.

+ u in _ 1 ) / 2

,

r

-

w~

uT of

.... and

gT

the

knowledge

we

proceed

initial

data

of

~)

(uT,

Let

u~_

(14) Z .i - . O i. - . r w i

Then

we

-

(~

(Yl~

I).

compute

(is)

ui.+1

: v i~ +

r

(16)

ff~÷1

= Z~

+

r k S (v,?÷1

that

the

Theorem.

Assume

(z~+1

initial

- z~_1)/2

+

_ v~_In ) / 2

rk(v~+1-2v +

conditions

and

[~o(X)[~
(171

fo," a l l

Let l

(lS)

M

=

sup

( U o < X > ÷ l O o < Xi) I / k > .

XER

Further (19)

if

r

max

(M,k)

~

1

l

i

x6[R.

n

~ + vi_~)/2 n@

n

rk(z~, I - 2z i Z i_I)/2.

(u o,

~o)

satisfy

uo~

o

91

then the above scheme is stable for the L®-norm, variation in space and for the total variation Tonnelli-Cesari' s sense, which means that there are B>O 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-

<.

i+

1

+ -o

~

n

n

I+Io,-R,I+IQ,

n+

n

~- (r~l.o?+.,-o?l+,'.w?ip+'t-Rn

~<

n

n

£k 1

"

211

~

"-

++

2K

+ rwTIQ':-Q':-,.l> 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

I
R. ( ~ , x , t ) ~ ( x , t )

dx

dt.

8x for

each

horizontal

and

and

~

the

functions

vertical

R u and

R@

are

on

vertical

strips

Ix- (i-~)E I<~ 3 8

n:0,1,2

.... ,i{2,

Then,

since

--

R~

is

constant

except

8x strips,

we

m~y

write

Q~

as 8

i , ~

where

8x

8 i , n

ax

the

97

and

Vi,.=((x,t)

Ix-(i-M)~l~<~",

It- (n-M) r~ I~
Ri,.=((x,t)

Ix- (i-~) ~ I.<~ ,

( n - ~ ) r ~ + r ~ " ~ t ~ (n+~4) r ~ - r ~

where

the

Vi,R For

each

summations

N supp

~ /

(i,n)

let

pair

are

g,

taken

K i ,.

on

the

[3 s u p p

indices

~ /

,

i and

~} ,

n such

that

gf.

8 (35)

I~,i,n

= II JJ

K. (~,x,t)

-8x

R. (¢,x,t)

-8x

R~ ( ~ , x , t ) ~ ( x , t )

dx

dt

Vi,n and

8 (36)

Je,i,l

= II JJ

E~ ( e , x , t ) ) ~ ( x , t )

dx

dr.

Ri,n By

a

change

of

variables,

K~ ( ~ , x , t )

= [ J

Then

a E

R~(~,x,t)=

we

~c ( X , T ) ~

'I

~

~

'r

the

L"

a (x-X)~ (~)

= --

By

have

uniform

(X,T)~'

3 (x-X) o

dT.

3 (t-T)

O, ( x - ~ 3 ~ ,

t-r~3~)O ' (~)O(z)d(

bounds

u,

of

and

g,,

t h e k i n d c~ 3, f o r I I c , i , n l . S i n c e the n u m b e r intersect supp ~ has a b o u n d of t h e k i n d (for s i m p l i c i t y in different constants,

notations, u n t i l the

(37)

I,.,,.l~

I z

dX

3 (t-T) r (e)

we use e n d of the

one

dXdT

gets

d1~.

a bound

of

of sets Vi, n which c/~ i, we h a v e in (34')

the same proof),

letter

c

for

c

i,n

where

c>o

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 _ I)H e (x-

values

8 --

of

have

R a and

= I J

(i-~)

e-e

-

i

.e

Ui

- i

- i( o ~ - o '~'i -

1

)+~(u~

-Ul

~-

]

.

K,,i,n

= -(u~+u?_~)

2

dx

i-1) ( ( ~ - ~ i - 1 ) H ~

) ( ~ i - ~ ~ ~)

1

(39)

from

~

H~ (x- (i-~)~)]

----

obtained

(x- (i-~) ~)] [ ( ~ - ~ _ i)

[u ~ - 1 + ( u ~ - u ~ - 1 ) H ~ x

R a,

ax

r"x: (i-~) e+~ ~ K.,,,.

(i-~)£)

(~l_c~n. ~ )

-

H~ dx

99 Let ~

Then

($6)

: ~ ((i-~)~, (n-~)r~) .

may

be w r i t t e n

~s 8 dt

+

dx JJ

t

dt.

8x

Ri,n From

(39),

~? ft Let

us b o u n d

K~,i. ~ dt : ~

8 f( [JJ K u - - R o 8x

r(e-Z~3)~(ui+uin ~-~) (~?-a9~-i) -

(~-~?)dx

dt[b y .:

the m e a n

value

theorem,

Ri,n

we

get ,~,J~(x't)-~?I~c~

for

some

c>o

if

(x,t)ERi, n. We

have

also

IRu (x,t) l~c. for some we have

c>o.Since

ff,

8 -- Rqldx 8x

the map

x i

~ R~ (~,x,t)

is m o n o t o n o u s ,

t= (n+~)rg-r~3 dt=±[ !E~ (~, ( i - ~ ) ~ + ~ , t ) - K ~ ~t= (n-~)r£+r~

for

Therefore u~+u~_~ n

2

This

implies

that U n+kln i -]

2

t,

(~, ( i - ~ ) ~ - ~ s , t ) ] d t

Ri,n

n

each

n_(~n

] D

i

100

Then

we

have U~+Ui-

(40)

I ~ J, ,i,n-

~

bounded

~i-~i

n

I

~

( ~ i-- ~ni n 1 ) I ~ O ~

i,n

i,n

since, by the x variable,

~

2

variation

property

of

the

function

~h

in

the

~/~

is

less

-

i

and since than o/~.

the

From

number

of

indices

(34') ,

(37)

n

such

that

Ri, n A

SUpp

(40) ,

and

n+un Ui i-I

i,n

The

two

other

integrals

in

(33)

are

treated

n÷u? Ul

(42) and,

order

to

prove

that n÷

So

when

~--*0,

we

have

to

( ~ - ~ _ l~) - r k

~--~0.

~

( u ~ - u i=_ I

)]--,o

Setting n n Ui+Ui - i n

n

E?=Gt + I-¢? +r we

have,

from

(13) ,

(14)

rk ~ E in- _

n

n

n

((~i-dl - i) - r k 2 ( u i - u i - i) , 2 and

(16)

:

rk (v~÷1-v

n

i - i)+--

2

(z~+~-2z"+z~

n1 - 1 ) - r k ~ (

u =i - u ~i - i )

=

2

rk z :--

CUn+I

-r

W ln +1

(u~

÷i

- u ? ) - - U in-

i+ r

W in -1

2 rk

+-- (z~+1-2~?+z~_1)-r 2

get

prove

n

Ui - i

i ,n when

we

(a~-~_1)-rk~(u~-u~_~)]l~c~

Q~--~O

Ui

Q~

-i

IQ,- ~ ~ , ~ [ ~ , i . -~+o ~ in

like

k~(u~-u~_~)=

(u~ -1--Ui =

2)]+

that

101

rk

rz k ~

[w~,~(u~,i-u~)-~, ~ ~(u~.i-~_~)], 2 We

have

rk

z i

,n

¢?

~ --

[(k

u in , 1 + r ? + ~ ) - Z ( k

u ? + z ~ )n + ( k

n u~_

1 .i.,~ ni- 1 ) l ~ IJ

i,n

rz k z

-

~

[wt

,

,

(Lli° + l - U i )°- W i

°

. I

(u~ _ l - U i _ ° ~ ] I

rk i ,n

2

r 2 k z n

n

n

n

n

J

rk

--

~

[(~,~° ,-~,)"

-(~-~.,~]

(k

u~o+~o~_,. I

r 2 k 2

+I ~ - -

~

(¢~+~-¢~_~)

w~°

( u ~° - u i n_ I) I .

i,n

Applying

the

mean

sums,

have

:

we

value

theorem

rk

I $ ~. n *~

E~I~ i , n

--~

r z

k 2

2

2

c~+c~.

(twice

in

the

first

sum)

8~

I aM 2 I L"

(~X~

+)

~lku~+~l+

8¢

8x

L " ( ~ x R ÷)

t ,n

in

both

102

Therefore

I = We prove

~ E~I"-'O

as

~--,0.

that P~--~O &s ~---~0 in & s i m i l a r

way.

We now prove that UI :=o has u o as m a c r o s c o p i c a s p e c t (the proof for EI~:o is a n a l o g o u s ) . A r e p r e s e n t a t i v e of ult= o is given by K0 (e,x)=Ru (~i/~,x,0) in (32) . If ~6~(~) we have

f

[Ro (~2,x)_u o (x)]~0(x)dx=~ X

[u~ ( x - ~ 3 ~ , - ~ r z ) ~ (@)~ (z)@ (x)dxd@dz-

X,~,'~ - I U° (X)~ ( x ) d x x

f = | U~ ( X , - ~ 3 r z ) < 0 ( X + a ~ ) O ( ~ ) O ( z ) d z J X,~,z

= ~

f

d~ dX- | Uo (x)~ (x)dx Jx

u, (X,-e3rr) [ $ ( X + e 3 ~ ) - ~ ( X ) ] P ( ~ ) ~ ( z ) d z

d~ dX+

X,~,z

+f

[U, (x,-e~rz)-uo(x)]~0(x);)(z)dzdx. x,'~

The first integral in the last e q u a l i t y Let us take the s e c o n d one :

f

clearly

tends

[U, (x,-~3rz)-uo(x)]~(x)p(z)dz dx x,'~

[ | u~ ( x , - T ) p s (T)~ (x)dTdxJ x,T r(~)

I uo ( x ) ~ ( x ) d x x

-- I [(u¢*~r ( ~ S) ~ (X'0)-U° (x)]~(x)dx'

to 0 as ~---~0.

103

Let

us

notice

that

(u~*o

~) ~ (x,O)=u~ (x,O)

for

all

x6~,

since

r (~)

(u~*o.(~)a

~

)~ (x,O)=

O(z)dz=l

f u~ ( x , - ~ r g ~ ) o ( ~ ) d ~ ,

supp

;)C[-l,l],

and

u~ (x,t)--u i on e a c h

lixE-~rg,~g

step

L e t ~i be the m e a n v a l u e of ~ on e a c h I i a n d f u n c t i o n w h i c h is e q u a l to ~i on e a c h I i . T h e n

let

~

[.

be

the

I Eu, <x,0)-uo <x)]~ <x)~x: I[~, ~,o)-uo cx)] [~ <x)-~. <x)]dx + IEu~(x,O)-Uo(X)]~(x)dx The

last 1

integral

u~=-

I

uo (x)dx

°

then

to

zero,

it s u f f i c e s

_

since

to

o

u~ ( x , O ) - u i on

prove

each

11 a n d

that

jEu.<x,0)-Uo <x)] [~<x)-~, <x)]dx F

(43) By

is e q u a l

the

mean

value

theorem

2

Given

xE~

let

we

h~ve,

for

e~ch

~ o

o.

as

xEIi,

A

JoE2

be

such

that

xEl o . T h e n

if °

~

I~ o

=

l*(Ao)-*(ioe)

I for

some

X,61 t , a n d o

I* (x)-~. (x)l.
Since

furthermore,

lU~ ( x , O ) - U o (x)l~<2iluoll

for

all

xE~,

we

get

(43).£3

104 Proposition. macroscopic

Proof.

The

classical

aspects

Let

of

~E~(~x]o,+~[

functions

U and

) be

u

and

0

are

the

respective

E.

given.

Let

I,= ff[Ru(~,x,t)-u(x,t)]W(x,t)dxdt. Since

~

the

(x)dx:l,

function we

~ appearing

in

the

definition

of

R u satisfies

have

[u, ( x - ~ S ~ , t - ~ r ~ ) - u ( x , t ) ] D ( ~ ) s ( ~ ) ~ 0 ( x , t ) d ~

d~

dx

= I R 4 [u, ( x , t ) ~ ( x + e S ~ , t + ~ 3 r ~ ) - u ( x , t ) ~ ( x , t ) ] O ( ~ ) p ( ~ ) d ~

f )~4 u,

= |

(x,t) [ ~ ( x + ~ S ~ , t + ~ S r z ) - ~ ( x , t ) ] ~ ( ~ ) p ( z ) d ~

+ I [u~ (x , t) -u (x , t)] ~0 ( x , t ) d x JR2

Since

3.2.3.

each

integral

Nonlinear

We compatible going to macroscopic

tends

Functions

to

of

0 as

the

dt

d~

d~

dx

dx

dt

dr+

dt.

~--*0,

we

solution

have

the

result.

[]

(U,E)

know the basic fact that the multiplication is n o t with the association but, in t h i s p a r t i c u l a r c a s e we a r e prove that nonlinear functions of u a n d g a r e s t i l l the aspects of the s a m e n o n l i n e a r functions of U and E,

105

respectively. This is very important from the physical viewpoint, for instance, to know some quantities like the energy which is a nonlinear function of u, in case u is a velocity. Its macroscopic aspect may thus be obtained at once from the macroscopic aspects of u and ~.

L e t P be a n y p o l y n o m i a l (with complex two v a r i a b l e s . Then P(u,~) is the macroscopic generalized function P(U,Z)£ Ss(~x]O,+~[ ).

coefficients) aspect of

Proposition.

Proof. For simplification tOE~ ( ~ x ] O , ~ [ ) and let

I,= In order

we w r i t e

the

f[R2u(~,x,t)-u 2 (x,t)]~(x,t)dx

to shorten

the notations,

proof

for

P(x,y):x

z.

dt.

we drop the t-variable.

x Assume ~)0.

Then

x

x

( f

u~ ( x ) ~ ( x + ~ @ ) p

(@)d~+u(x)~ (x))dx

=

= f J. (x) .< I u. (x)~o~(x+,~')~ (')d'+u(x)'~ (x)> dx. x

where

Let

Then

in the

106

J~ ( x ) = f

=~

Since

[u, (x), ~(x+~)-u(x)~

[u. ( x ) - u ( x ) ] ~

J~ (x)--*0

For non-negative

3.3.

as

~'(x)]o(~)d~

(x+£3~)o (~)d~+ I

~--*0,

we

have

the

u(x) [~o~ (x+g3~)-~o~ (x) ]o (~)d~.

result,

an arbitrary ~, we set continuous functions with

A NEW F O R M U L A T I O N

OF THE

for

~=~+-~-, compact

EQUATIONS

~)0. w h e r e ~+ support.

and

~

are

OF H Y D R O D Y N A M I C S .

We show in t h i s s e c t i o n t h a t we m a y o b t a i n in h y d r o d y n a m i c s a very simple system in n o n c o n s e r v a t i v e form which is e q u i v a l e n t to the c l a s s i c a l (conservative) formulation. The m i c r o s c o p i c aspect of the s h o c k s for t h i s s y s t e m is very simple. This nonconservative formulation gives rise to new and efficient numerical schemes see De L u c a [I]; s o m e n u m e r i c a l s c h e m e s w i l l be d e s c r i b e d in A p o e n d i x 3.

3.3.1,

The

system

of

conveniently of physics)

written

(use

(44) (46) (46)

hydrodynamics

of

in

equalities

one

space

i n ~ to

dimension

express

basic

can

be

laws

Or+ (~)u) ×=0 (0u) t + ( P u 2 + p )

~=0

@ ~ + (@u+pu) x = 0

where o=denslty, u=velocity, p=pressure, volume. If I z i n t e r n a l energy per unit (definition of t h e i n t e r n a l energy) u~ ~:Pl+O--; 2

@=oe=total mass, one

energy per u n i t has of course

107

this we

set

of

write

equations

with

been

done

valid

there).

which

may

inside

be

is

complemented

association the

locally

to

by

express

width

written

of

an

equation

that

the

shock(

of

state,

measurements

have

thus

it

is

that not

possibly

not

as I ~ ~(p,p).

Here

I,

0

functions are

and

"bounded

subset

p

and K

have

~

is

elements"

of

~

sentative

of

I,~

almost

(x,t) ,it

all

association

and

the

of

~s (~ ~)

then and

the

macroscopic

a C ~ function. in

aspects

We the

sense

sup [R(~,x,t) I<+~ (x,t)E~

p.

Then,since

R(~,x,t)

follows

at

once

theorem

of

dominated

pI

~

of

assume

from

piecewise

I,P

that : for

every

if

is

R

converges the

p

compact

any

repre-

when

definition

convergence

C®

and

~*0

for

of

the

that

p~(D,p)

Therefore (47)

@ Equations

physical that

null.

so

it

(44),

principles

they

be

are

The has

conditions).

get,

PO(D,p)+DU2/2

(45)

of

mass,

obtained

from

system

(44) ,

nonambiguous In

this

Multiplying since 0 / 0

and

(1)

by

we

express energy

assuming (47)

the is

(classical shall

variables

by

and

(46) ,

shocks

(44)

(46)

momentum

(45),

section

of 3 equations in the nonconservative form.

we

~

u and

u

replacing

stress

an

and

the

- Hugoniot

value

to

system

equivalent v=I/~,

the Note

deviator

conservative

Rankine

obtain

p,

a

respectively

conservation.

jump system

which

has

a

u 0 t in

(45),

by

(same

: I

(48)

ut+-- p x + u . u

=0.

D

Replacing justification

(47) as

in above)

(46) , we

have

and

multiplying

(47)

u

108

U 2

[~z~

+ ootDI~+Pp~Dz~+O

t --+~uut]+[Oxu~+Oux~+pOxu. 2 u~ 3 +~up~,~+o~ -- ÷ ~ u s u~]÷[p~ 2

which

implies

u~]

-- 0

(pt+upx) D2~+ (Ot+o~u+oux)~+ us

3

+ ~

(Dt+UDx)+DUU~+

-- D U 2 U x + P x U + P U x

2

~ O.

2

(44), u2 oZuxD1~+o(pt+up~)D2~_~ux

Replacing

the

(49)

We

u+p

2

that (p~+o~u) D~+p

From

D1~+

shall

volume).

value - 02

Therefore transform Let

us

ouu~

3

-- +~uut+ 2

o~tained

from

(48)

uxD1~+p(pt+up~)Dz~+pux we it

~

have obtained the by s e t t i n g vzl/D

define

a

function

~

by

p)

:

-- p u 2 u x + p x u + p U x 2 :

O.

system (thus

1 (v,p)

=

~(-,

~(o,p).

V

Then

a~ - -

-i ( v , p )

(~,p)

8v

v z 8p

a~

a~ (v,p)

:

~

ap Replacing

these

values

8~

-

in

(o,p).

ap (49),

we

have

:

~ 0

(44), (48) v is t h e

a n d (49) . specific

109

Pt

8~ U x

far

we

have

the

+pux~0.

8p

v system

a)

v~+u

b)

u t + u u~

c)

8~

-- _ _

8p

V

obtained

(50)

u

p×

+

8V So

8~

+

vx - v Ux = 0 + v

p~

: 0

1 8~

u 8~

8~

v 8p

(v,p)+p~-v 8p

(v, p) + [ p + - 8v

Pt

w h e r e ~ is d e f i n e d by I ~ ( o , p ) - - ~ ( v , p ) . Note that the (46) in p l a c e of t h e e q u a l i t y w o u l d a l s o g i v e (50).

In

the

particular

case

of

Aerodynamics, P

I

~

,

%'>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 ~<-i or ~>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+-- <0

1

H (x)

1

(*) = - ~

d~ = ~ H(A) -co

(I+

H(~) ) ~

f X

i+

) -®

d~} = I14 8, ( ~ )

-co

H(x) l + ~

H(x)

H (x)

b (,~)

exp{- I

1 =-£

= ~(~

] =~ (14

, ,) • (I

H(x)

113

From

(89),

we

invertible.

get

M

=

H.

Now,

M

3.3.3.

(58')

gives

N'=

H' , s i n c e

H+~

is

Then

The A

Mie

Gruneisen

classical

=

N

-- H.

law.

equation

of

state

is

the

Mie

Gruneisen

equation

p~(o,I)=~oI+G(o) where

¥>0

and

G

p

(v, p)

is

a

function

G (~)

.

.

vp

.

¥o

o.

of

Then

we

state

it

as

G (-)

v

.

¥o

¥

¥

Since

1 G (--) v

8~ --

(v,p)=

8p association

and

--

~ (50c)

(60)

p~

p

8~

--

v

1

(v,p)=

1

+ --

8v

~

~

G' (--) ,

~v

v

might

be

becomes +

u

Px

+

[(¥+I)p+F(v)]ux

~

O,

where 1 F(v)=

1

y Thus

the

3,3.4.

1

-- G' (--) - G (--) = D v

system

is

G' ( o ) - G ( o )

.

v

particularly

simple.

Remark. Similarly

equalities

in

as

in

equations

3.3.1. I,

the 2

and

system 3

and

(1)

association

in

written

with

equations

4

1 and

8.

The

above

proof

with

(p-S)

in

place

of

p

gives

that

v=

--, o

u

114

and p-S are r e p r e s e n t e d by exactly the same H e a v i s i d e functions since the proof uses only the e q u a l i t i e s in mass and m o m e n t u m conservation. But this f o r m u l a t i o n does not fix the r e l a t i v e a s p e c t s of the H e a v i s i d e f u n c t i o n s r e p r e s e n t i n g p and S 3.3.5.

Numerical

applications.

These results were at the o r i g i n of new n u m e r i c a l s c h e m e s in C c l o m b e a u - L e Roux [1,3] ; in aerodynamics these schemes gave good results ; in metallic collisions these schemes are p a r t i c u l a r l y e f f i c i e n t . See A p p e n d i x 3.

3.4. JUMP FORMULAS ELASTOPLASTIClTY 3.4.1.

Uniqueness

Consider I (9')

of

the ~+

FOR

shock

SHOCK

relations

WAVES

for

(9')

IN

ELASTICITY

AND

(9").

system

(~u) ×=0

(~u) ~+ (Du 2) x=~x ~t +u ~x ~ k2 U×

Let

us

seek

solutions I

(6i)

of the

u(x,t)=Au

form

H(x-vt)+u~

g (x, t) =A(r K (x-vt) +g~ p (x,t) =A~

L (x-vt) +p~

where H,K and L are H e a v i s i d e g e n e r a l i z e d f u n c t i o n s . Of course formulas (lla,b,c) hold. But due to the e q u a l i t i e s in the two first lines we have much more. The proof in §3.3 a p p l i e s (replacing p by-g) a n d gives that g and u are r e p r e s e n t e d by the same Heaviside function. Thus the q u a n t i t y A in (llc) is equal to ~ : the s y s t e m (9') has the n o n a m b i g u o u s jump c o n d i t i o n s (lla,b,c) w i t h A=~.

115

Now

let

us

consider

I

the

P~+ (#u) x=0 (~u) ~+ (~u 2)

(9")

Gt+u The

system

third

line -Ag

~, O' x

&x=k 2 u X .

gives

v K'

+

(Au H + u t) (Aa K ' ) = k 2 A u

H'

or

if A g / 0 ,

~u [(v-u~)-&u.H]K'

Replacing

the

value

of

v-u~

from

=-k 2 -Aa (lla)

we

H' .

have

Au (Au+ D

--

Au -

Au.H)

K'

=-k

2

--

Ap or,

:

H'

A~

if A u ¢ 0 , -k2

Pt

(l+-AO

-H) K' =

H' . A~

~t

since

--

>>

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~
---- t~ i

in

~s(I)

Proof For each (t,x), extending from the point ( 0 , x i) of L i with the figure).

R~ °) ( ~ , t , x )

(73.a)

(73.b)

l~
let C,(t,x) (t,x) to interval

for

n:0,1,..,

R~÷

be the

t h e p a r t of point of

I on

If r i E @x, s[ I] is a r e a l : ]0,+~[ × X --* ~ b e t h e m a p

R~ °)

and,

,

the

x-axis

representative defined by

of

the curve L L intersection

(see

the

~i,

let

above

= r i (e2,xi) let

R~ ~ + z )

I) ( ~ , t , x ) = r l

: ]0,+~[

X

X --~ ~

be

given

by

F i f , ( t , x , R ~ " ) ( 8 , t , x ) . . . . . R~") ( 8 , t , x ) )

(~2,xL)+

Ci(t,x) where the below.

precise

meaning

of

this

curve

integral

is

given

in

(78)

If x = xi ( t , t o , X o ) is the equation of the i th characteristic passing through (t0,x0), denoting by Ji(to,X0) the open interval of existence of x L ( r e l a t i v e to ~), w e h a v e t h a t : (i) t h e d o m a i n of d e f i n i t i o n of x i is t h e o p e n s e t

Di =

(ii) x i is derivatives For

(74)

U { ( t , t o , X o) : t E Ji ( t o , X o ) } (to,Xo)E~

continuous u p to a n y each

R[°) (8'')

on D i ; and (iii) order with respect

fixed

~>0,

+ t

[R~n÷z)

n:0

the

x i has continuous to x o.

expression

(8'')

;

- R[ ~) ( ~ ' ' ) ]

partial

123

defines ~ derivatives,

(78)

C® function on Q, Ri ( ~ , . ) , continuous l i m i t s as t --* 0*

Ri (~,t,x) = ri (~,xi)

w h i c h h a s , w i t h all and which satisfies

its

+ f fi (t,x,Rt (8,t,x),-..,RN(~,t,x)) C~ ( t , x )

This

assertion

result

(see,

subset

We K of

follows for

from

instance,

will X let

prove now ql(K) d e n o t e

containing K, with contains ~ii s e g m e n t s

c>0

and

(76) for

all

(77) for

Since

rj

0<~
such

(t,x) on

ft

classical

existence

[i,

that R( 6 ~ M , ~ [ X ] . G i v e n a c o m p a c t the s m a l l e s t compact subset of X

E @~, s[I]

and

xj (<0}

x ~(K))

(78)

(t,x)

CC

I we

: 6

Ir~(82,xj(0,t,x))l

~(K),

there

0<~
is

6 ~z(K),

c" >0

g>0

The

expression

R~*

I> ( s , t , x ) = r i

obtain

and

, such

all

(80) for

have,

IR~>(~,t,x)l

(79) for

we

(t,x)

and

I~<j~
the

boundedness

~< c"

l~
(73.b)

may

be

written,

(~2,xi (0,t,x))+

6 ~(K),

for

more

precisely,

fi ( ; , x i ( t , t , x )

(t,x)

6 ~z(K)

n=O,

~< ( c + c ' t ) ~ 0<6
I(R~')-lt~°))(~,t,x)l all

M 6

that

(8 ,'c,x i (~, t,x)) ..... R~ ~> (e,~ ,x i ( ~ , t , x ) ) ) d ~

Then

~(K)

~< c~ "~

0

R~ ~

then

that

If, (t,x,]%~°) ( ~ , t , x ) ..... R~(°) ( £ , t , x ) [ all

- uniqueness

§i0])

the property that, if ( t , x ) 6 K , of c u r v e Cj ( t , x ) , j=l .... ,N.

IRJ°>(S,t,x)I

condition

the

Fetrovsky

~nd

"~' From

~< c ~ - " some

ci>0

(76)

and

(79)

we

have

as

124

8f i If A I =

sup

(t,x,u:,...,uN)l

I

l~i,j~N (t,x)£~(K)

, it

follows

from

8uj

ui6~ (80)

and

(8:)

the

mean

value

I (R~S)-R~:))

for all have

(t,x)

E ~(K),

theorem

that

(g,t,x) I ~A:Ntc:g 0<8
and

l~i~N.

-s For

an

arbitrary

n we

(A:Nt)" (82)

I (R~'+ :)-R~n))(e,

t,x)I

~

e:~ -s n!

for and

all (t,x) (82) ,

E ~(K),

0<~
and

l~
from

(74) ,

(76)

= o A

~< c~ -M for

all

(t,x)

6 K,

0<~<~

If w e

differentiate

+ e

and

Nt

:

some (78)

. c:~ -s ~< D~ -H D>0. with

respect

to

ax

(~,t,x)

we

have

i

(0,t,x)

= r~(~2,x~(0,t,x))

8x

x,

t[afi +

axi ~,xt(z,t,x),R~")(e,z,x~(z,t,x))

.... ,R~)(a,z,xL(z,t,x))]

0 8x

+)-

(~,t,x)+

8x

8R~ ~ )

8f~

j = 18U

+

8x

Cz,x~,R~"'

( ~ , z , x , ( ~ , t , x ) ) . . . . . R~( ~

(~j~#X

(~,z,x, (z,t,x))] 8x i

j

~x i

(~:, t, x)]dz , 8x

i (z,t,x))

125

8R i and

following

the

sequence

(76)

to

(83) , w i t h

we

obtain

[

for

replacing

Ri,

8x

8R t (g,t,x)[

a bound

similar

to

(83).

8x Denoting 8 and

by

the

by

aR, . .

we

RI

arc

length

along

in

direction

the

the

i th of

characteristic, the

i th

have

[SR~ .

.

8s,

Then

the

differentiation

88 i characteristic,

(84)

s~

8Ri

1

t i

8t

(1

+ t ~ ) -~

8x

satisfies

8R i (1

+ ~)~

(~,t,x)

= fi(t,x,R1(~,t,x),...,RN(~,t,x)).

as i By

(77)

we

obtain

, we

a

have

bound

8Rj ] (E,t,x)] 8si for

D o i n g the s a m e that R i E @M,s[X], and have that ui is a condition (72) s i n c e a g i v e n by

Ri and,

from

(75),

:

]0,+~[ we

have

~c'

xj ( 0 , 0 , x )

= x and

thus

, from

the

kind

C$ -N

8R L ] ( ~ , t , x ) [ of 3t

(84)

for h i g h e r o r d e r d e r i v a t i v e s of R i , we get s e t t i n g u i 6 ~ , ( X ) its c l a s s in $,(X) we solution of (71) . It s a t i s f i e s the i n i t i a l representative of u i [ t = 0 is, from 1.8.6,

X I ~

~,

(a,g)

~

Ri(81/2,0,X)

that

Ri(~l/2,0jX) since

,and

C i (0,×)

= ri(~,X has

length

) zero.

126

In

order

u i E ~s(X)

be

to

two

prove

uniqueness

solutions

a

of

(71) ,

solution,

(72) . T h e n

let we

u~,

have

3 (ui-ui)

[Ui--Ui] [t=O =

Let and

of

R~,

R t E @~, ~[X]

u i . There

are

-

~xx]

8

~s ( I )

in

0

:

be

N i E $/~[X]

and

fi (t,x,ul,..,us)-fi

(t,x,ul,..,U~)

.

respective

representatives

n i E ~ s [ I]

such

of

u i

that

8

[~t

At ( t ' x )

(Ri-Ri)(~'t'x)

-

for

~11

~>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

(0R~+(I-0)R

theorem,

= ni(e~,xi(0,t,x))

+

Dr, . (~,x i (z,t,x),(OR~+(1-e)R~)(¢,z,x

i

i (¢,t,x)),..,

~ Duj

N)(~,~,x~

(z,t,x)),(R~-Rj)

If g is a c o m p a c t

subset

S(8)

(~,z,x~

of X,

=

(z,t,x))÷N~

(~,z,x~

(z,t,x))

let

I (Rj-Rj)

sup

(~,t,x)

[ .

(t,x)E~(~) I~j~N Since N i E ~s[X] and W>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

0<8<~.

(85)

From

I ~< c¢ '* + A~Nt

Applying (87) to each (85) we get, for each

I (R~-Rl)(8,t,x)]

~< (e/2)e q

Rj-Rj (t,x)

~< ce ~ + A~N

we have

S(¢) .

a p p e a r i n g in the r i g h t - h a n d E ~(K) and 0
i

t (c~ q +A~N S ( 8 ) ~ ) 0 (AINt)

= c8 q ( I + A 1 N t )

+

S(~) 2

d~

c>0

)

dz

128

Repeating n ~

~

we

this

operation

n

times

and

taking

limits

as

get

(A~ N t ) ~ I(R,-R~)(e,t,x)l

(88)

~< ce q n=o

for

(t,x)

all

We

6 ql(K)

obtain

3.5.2

-

When the

Coherence

with

the

[i,

the

§lO])

. If

m~croscopio

as

¢i

function

a as

E ~s(X) be solutions of initial conditions. Then u I [~

(b)

for

has

every

Ui [ t= t

r e s t r i c t i o n

u i [~

-

It m

is :

is continuous usual topology

enough (Cl(I))

derivatives

solution

as

of

end

from

functions

~

C *

Petrovsky

(see ¢i,

having

(71) ,

(71) ,

are

(72) . T h e

we

¢i

¢l

t o such as

eL

as

respective

aspect.

that

( t = t o)

macroscopic

N X ~,

the

aspect.

0

to N --~

prove (C1(~))

that N

when both spaces have of the spaces C~.

the ,

map

(¢i)~=1

the

~-*

product

(ui)~=l

topology

as

following

:

and

C I

obtain

¢i E ~s(I) be a generalized aspect. L e t u~ 6 C I ( X ) and

having

t

follows,

coincides

of

solutions

it

solution

function

macroscopic

ui[t=

the

of

two

C®

class

condition

(71)

O

Proof

if of

solutions

value has

our

generalized

these

u I

(a)

class

is

- Given @i E CZ ( I ) , let having ~i as macroscopic

Proposition

of that

However,

solution

"reconciles"

partial

generalized

are

above,

initial

functions

proposition

the

the

C ~ solution.

one.

consider

aspect,

generalized

between

proved

classical we

for

functions

result

classical

functions,

bounds

classical

the

uniqueness

0
and

similar

the

n!

of

the

129

In

fact,

ulI ~ o b t a i n e d ¢i,

we

have

in to

Ri

if

(75)

prove

as

~ ~

0.

of

the

map

Since m

The

of

m

enough

(~)

ft(t,x,ul

is

the

.....

as

for

as

a

each

representative

of

representative

of

~

£ ~(~),

used

give

=

in

~(t,x)

C~(I),

the

dt dx continuity

(89) . to

the

system

u~)

0

~ ~

prove

we

case

that,

¢i

~

argument

the

the

dt dx --~ f u, ( t , x )

to

simplicity, in

denotes

r i 6 $ M , , [ I]

~(t,x)

r~(~,.)

is same

For nuity

with for

I R,(~,t,x)

(89)

SM,~[~]

6

prove detailed

(71)

~

(b) .

is

aij(t,x

proof

linear,

) uj

of

the

that

conti-

is,

where

+ bi(t,x)

J=l

with

al j,

case

is

b i E

C®(X),

L e t @ i , ~i 6 (vi)~= I = m((~i)~1), Pz

the

of

U,

f

E

seminorms with

U

=

proof

in

the

general

nonlinear

an

on open

Ci(1), l~i~N, (ui)~= I = m((¢i)~=1), n, = ~ i - ~ i and zi=ui-v i . We will denote CI(U),

when

subset

of

max

KCC0

let

p ~ ( x ) (al j)

L i =

K

ranges

~,

and

xi({0} B

~2

or

satisfy

the

x ~(K)),

= max

l~i,j~N

(90)

(the

over ~

the

(pK(f)

compact =

sup xEK

by

subsets [Dcf(x)[,

c~(U)).

If A

l~i,j~N

similar).

n = m a x PL ( n j ) I~j~N J

PS(K) (Zj) • The

functions

zi

I~j~N following

zl ( t , x ) = ~ i

integral

equations

a~j(~,xi(~,t,x))zj(~,xi(~,t,x))

(xi (0, t , x ) ) + 0

j=l

]

dz.

130

Then (911

of

(t,x) E ~ ( K )

if

~z~(t,x)l

~

we n

+ ABNt.

Applying (91) (90) , we h ~ v e , f o r

Iz~(t,x)

Repeating

l

.< n

this

have

to e a c h zj e a c h (t,x)

+ AN

f

operation

appeering E ~(K),

.< ~

[i

(~+ANBz)dz

:

to

the

limit

Iz~(t,x)

(92)

l

as

right-hand

A~N~t 2 + - B

~(I+ANt)

side

.

0 n

times,

we

have

+ ANt

that "" 1

+...+ (n-l)

Passing

the

t

(ANt) Iz~(t,x)l

in

!

1+

B(ANt)

"

n!

~,

n ~

~< 19e T M az

for

all

(t,x)

E ~(K).

For

--

we

have

:

(0,t,x)

+

a×

ax

~z t

(t,x)

= ~(x~(o,t,×)>

3x

3x

+

s t [ ~_ 0

J~

Oaij

(z,x i (z,t,x)) azj

aij(z,xi(z,t,x)) J:~

If

Ox~ (z,xi(z,t,x))

o = {(z,t,x)

the

following :

] (z,t,x) dz

8x

3x i

(~(K)) 0 denotes (~(K))

(z,t,x) z j ( z , x i ( z , t , x ) ) +

8x

3x,

i +

8x~

compact

O~<%~
subset

6 ~(I<)}

,

of

Di :

131

let

P

:

max l~i~N

P(~(x))

(x~).

Then

we

have

o

~z i

(93)

(t,x)l

~

6

~(K)

P{~

+

P(~+2ANtB)

8x for

all

we

have

[

8z~

(t,x)

(t,x)[

~

.

ft

3x

Similarly

to

[NA(~+ANBz)

what

we

have

done

to

+ NAP(~+2ABNz)]

d~}

get

(92)

0 N2A~t 2 = P[v+NA~t(I+P)

+ -

B(l+2P)]

-

N~A 2t ~ (l+P)2 P{~[I+ANt(I+P)]

for

all

[

(t,x)[

(t,x)

E

~(K)

B}

+

. Repeating

this

operation

(AN(I+P)

~< P { ~ [ I + A N ( I + P )

(94)

[

~

!

~z i ~

P~e s~(l+v)t

8x for

all

(t,x)

6 ~(K).

For

we

have

8t ~Z i

~z

i

Ai at Then,

(95)

from

(92) az i .... (t,x)

8t

and

[ ~

(94)

m i

8x

j=!

atj

zj

,

NABe Ast

[AN(I+P)

+ P~

Ce AN(1 +r) t

we

get

t] ~ B}

n!

~,

(t,x)]

times,

]+ (n-l)

n

t) n-I

t+...d

8x Letting

n

.

,

132

for

all

(t,x)

E ~(K),

where

C =

I~,1

max

qz(g) 1~< i ~
from

(92) ,

(94)

pK(zi)

max

and

~ C'

l~i~
some

constant

3.5.3 We

P is

prove

:

that

(~,(I))

continuous,

topology

:

the

~ -~

where

@i,

max PL l~
then

that

of

the

(~,(~))~ u~

6

@,(1)

by

@(1)

@nd

6 ~,(I),

,

@~(X)

both

~i

we

have

(~i)

the

m~ p m i s

Cauchy

continuous.

problem.

map

and

induced Let

(v~)~.

proving

The wellposedness

-

representative,

Li

C'>O,

(95)

and

@(O), l~i~N,

I = P ( ( ~ i ) ~ , I)' ~i = ¢ i - ~ i xi ( { 0 } × ~ ( K ) ) . Assume that

(¢~)~=1 has

the

@~(~)

-~

(u,[~)~=~

expression ~re

respectively let

(74)

endowed (see

as

its

with

the

§1.7) .

(ul):= I = P((@i)~=1),

and z i = ut-v i . Given nL 6 QL , o , , f o r e a c h

KCC~ let i=l,..,N.

i

Then

(96)

sup

[ri(e,x)

[ ~

xELi

for

each

i=l,...,N

and

If R i , R L E @ M . , [ f l ] have that

(Ri-Ri)(~,t,x)

+ [f~(t,x,R~(~,t,x),

some

are

= r~(eZ,xi)

....

representative

respective

r i E

@H,,[I]

representatives

of

of

n t.

u i , vi,

+

,R~(e,t,x))-f,(t,x,~(E,t,x)

.....

RN(e,t,x))]+

.JC i ( t , x ) +N i (~,t,x) ]dt

we

133

for

some

(88)

Ni

from

E ~s[~]

. From

(85) , w e

the

have,

same

from

reasoning

? I(Ri-RL)(e,t,x)l

sup

we

have

done

to

obtain

k=0,

KCC~

(96) , t h a t

<

c~

(t,x)EK for

all

i=l,...,N

and

5>0

there

~l

and are

E QL

~

,o,,,

some =

constant and

5/c

i=l,...,N

c>0.

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

<0}

i or

over

8~

u X

(x,O)

=

×

[O,T].

(uo)

x

i

and

Since

(x)

on

u t

~,

is

it

null

over

remains

to

8~

x

[0,T]

estimate

u t

and in

t=O

i

u X

in

8~

X

[0,T].

For

all

ff s u c h

that

i

(103)

a

[IAuoll

>

. 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

~,

0<~<~,

lim r

~(x)

~ E C2[Q]

g(~,x)dx

null

on

= 0

for

8~×[0,T],

[Rt-AR+(R~+R1R2+R~)R]Y

by p a r t s

we g e t

we

have

:

dx d t

:

solutions

representatives of u I g E @M,s[~] satisfies

~ E ~(~) .

;

f . Y dx d t Q

:

{ ~Q{-R[~t+A~-(R~+R1R2+R~)~]-f~}dx (Iii)

dt

+ f R(x,T)~(x,T)dx

8~

= I R(x,O)X(x,O)dx

they have following

i=1,2.

all

Q

By i n t e g r a t i o n

that the

corresponding

If RI, R~ E @ M , 5 [ Q ] are respective a n d u~ t h e n R = R I - R 2 s a t i s f i e s (109) w h e r e g~0, t h a t is,

I

f,

that

Proof

each

on

[]

Then Uo, ] ~ uo, 2 E ~ ( Q ) implies that the u 1, u2 E ~ ( Q ) satisfy u l ~ u 2 in ~ s ( Q ) "

For

hypothesis

R E ~s[Q].

Theorem.

there

(110)

Now that

- I R(x,t) 8~×]0,T[

"~V ( x , t )

d~ d t

140

(an

~ was

choose

~

omited as

(ii2)

where

~

the

f

each

function

of

the

= 0

in

Q

]((x,t)

= 0

in

8QX]O,T[

we

have

(RW-f~)dx

dt

from

=

Q

I

in

the

backward

]((x,T)

E @(Q),

(Ii3)

from

solution

(iii)

g(x)

and

above

equality),

parabolic

(112)

~(x,O)dx

-

~

If w e

problem

:

f

h(x,t)

--(x,t)dg

dt.

8~

8~X]O,T[

Let us write (113) m o r e p r e c i s e l y , ¢, a s w e l l a s f, g, h a n d R :

f ]

JQ[R(8,x,t)

~b(x,t)

-

remembering

f($,x,t)

that

Z~ ( x , t ) ] d x

~

dt

depends

on

=

(i13")

'l/~,x)

~,(x,0)dx

-

h(~÷~/~,x,t)

.

(x,t)

8n 8fix] O,T[

Will

prove

that

R(¢,x,t)

~(x,t)

dx

dt

--*

0

as

e ---* 0

.

Q Using {Z,}¢,

again 0 has

(ii4)

f

lemma 3.6.3 a moderate

f(8,x,t)

for ~, growth.

Ze ( x , t )

dx

solution Since f

dt

---~ 0

o f ( 1 1 2 ) , we E $,[Q] then

as

8 -* 0

get

that

d~

141

For

the

same

re~son,

that

is,

La-CJ

(115)

since

h E ~s[8~X[0,T]]

,.,e have

~

h(~n+i/n,x,t)

(x,t)

dd

and

th*

dt

--* 0

as

~ -~ 0

8n 8~X]0,T[

Remembering that g E @M,s[Q] is a r e p r e s e n t a t i v e of U o , l - U o , ~, it satisfies (iI0) Furthermore , since (R~+RIR~+R~)(~,.) ~ 0 in q, for e&ch ~, from (ll2) , this implies that {~}, is & b o u n d e d s u b s e t of the H S l d e r space C~(Q), 0<~
I1~,11

,, c

(it s u f f i c e s , Ch&pter III, a n d we h & v e

(116)

f

~< ell~ll (5)

f o r t h & t , to r e t a k e I0]) . T h e n ~ (fl,0)

g(a(n+1)

zn

J

x)

~(~ (x,0)

.

(q)

L

the is

p r o o f of relatively

dx --* 0

as

~ ~

L&dyzenskay& [i , compact in C ° ( U )

0

f~ Now

(i14) ,

(115)

and

(116)

imply

the

result []

A

SYSTEMS

OF

P

P

PARTIAL

ENGINEERS

E

N

D

I

X

1

DIFFERENTIAL

FOR

THE

EQUATIONS

MODELLING

OF

USED

BY

COLLISIONS

Engineers model collisions by systems of partial differential equations coming from elasticity, fluid dynamics (=plasticity) and elastoplasticity which encompasses both theories as w e l l as the elastic-plastic transition. In very strong collisions the elasticity of materials can o f t e n be neglected and these collisions are valuably modelled by the classical system of fluid dynamics. But, for many collisions, elasticity h a s to be t a k e n i n t o a c c o u n t ( s e e the example of plane accidents in Appendix 4) . E l a s t i c H o o k e a n m a t e r i a l s obey Hooke's law, w h i c h r o u g h l y states that the deformation of an elastic s o l i d is p r o p o r t i o n a l to t h e f o r c e a c t i n g o n it. T h e b a s i c s y s t e m of p e r f e c t elasticity c a n be f o u n d in F u n g [i] .The p r o b l e m is a 3-dimensional one and its r e d u c t i o n to s m a l l e r dimension is in general not realistic.However,in o r d e r to s o l v e n u m e r i c a l l y such systems, one uses "splitting methods" which justify t h e s t u d y of one dimensional models s u c h as t h o s e g i v e n in C h a p t e r 3 But the domain of strong collisions lies far beyond perfect elasticity and the system in F u n g If] c a n n o t be used. Instead of stating Hooke's law as a linear stress-strain relationship, engineers who work on numerical simulations of collisions express it in differential f o r m in a ( l a g r a n g i a n ) f r a m e of r e f e r e n c e following the

deformations

frame,this

of

gives

the

&t

Note that

the

only chapter of gx

the are

three

that

considered 3 can

be

+

this

linearized

be

medium.

equations

u

elaboration

of

&~

=

(in

as

an

In

their

k 2

to

a

the

large

term

u&~

approximation a

simplified

one

systems

of

elasticity,

sequel

of

this

of

fixed

(eulerian)

u~

allows

which as

the

system

back

kind:

formulation

considered

large.

dimensional

the

form

multi-dimensional rather

Coming

of

elasticity

elasto-plastic

dropped)

The

system

by

and can (9)

dimensional even

appendix

used

codes.

deformations is

when

we

engineers

of

form u and

sketch for

a the

143

NOTATIONS.

Eulerian ~=(ui)t~i~ p

=

-~

stress

3

=

coordinates velocity

tr(~)

=

pressure

deviation

x

; ~

tensor

=

=

(xi),~i~

3

(gl,j)1~i,j~3

; ~

=

; ~

=

identity vi,~

; =

3x3

= ~

0

=

~

e

=

total

tensor,

~

specific

=

~i,j =

energy

;

I

½

+

=

; ~

;

= ~+p~

+

= 8x~

..

8x~ internal

;

tensor

matrix

8xj deformation

density

stress

=

=

rate

of

~,j spin

tensor

8xj ~,j specific energy

=

d e-~

0.0

If

f

is

~

d derivative

: ~

f

= ~

dt

and

we

The thermal

set

of

x

and

t,

~

(div

equations effects)

~

denotes

the

eulerian

dt + O.grad

8t ~

-

tensor,

function

8f f~.If

~

aaij

=

(ai, j) ~, j

a

5XS

>

A) i =

Z - J 8xj

of are

continuum

+

~

div

0

=

div

momentum

e

=

div(Y.D)

energy

0

is

=

; div

~

is

mechanics

0

mass

a

vector

(without

viscosity

conservation

dt d ~

conservation

dt d (I)

--

conservation

dt p

=

~m(O,l)

equation from

d

dt

of

state

experiments

obtained

;

144

The last equation is a constitutive e q u a t i o n in the p u r e l y e l a s t i c case : G and h are Lam@ c o n s t a n t s d e p e n d i n g on the medium and o b t a i n e d f r o m e x p e r i m e n t s . This e q u a t i o n f o l l o w s f r o m H o o k e ' s law w r i t t e n in a frame of r e f e r e n c e f o l l o w i n g the medium, and w i t h the h y p o t h e s i s of small d e f o r m a t i o n s in this l a g r a n g i a n frame (*). The d e c o m p o s i t i o n of the constitutive equation into the i s o t r o p i c part and the d e v i a t i o n part gives I

d a)

p = -(3A+2G)

~

div

dt (2) b)

d ~ ~ dt

: 2G[V-~

tr(V) 1]

+ sym (~.~)

Equation (2a) and the e q u a t i o n of state are r e d u n d a n t so that one of t h e m is dropped. In the case of fluids (plasticity) ~=0, and one uses only the first four e q u a t i o n s of (1) . In the case of e l a s t i c solids one has to take H o o k e ' s law into account. In two space dimensions the first four e q u a t i o n s of (I) and equation (2b) give the s y s t e m (Cauret [i], p . 1 0 7 - i 0 8 ; n o t a t i o n : coordinates x,y ; (u,v) = velocity components in the x,y directions ; ~ : G,~ : (Si,j)1<,,j~ ~ , ~ : ( ~ i , j ) 1 ~ i , j ~ ) : ~t + ( ~ u ) ~ + ( ~ v ) y

=

mass

0

(~u) t + ( ~ u ~ ) ~ + ( o u v ) y + ( p - S , 1 )

-(s1~)y

= 0 ~

(Ov) t + ( o u v ) ~ + ( ~ v ~ ) y + ( p - S ~ 2 )

-(S~)~

0

momentum

/

conservation

(me) t + (meu) x + ( o e v ) y+[ ( p - S 1 1 ) u ] ~+[ ( p - S 2 ~ ) v ] y -(Sl~u)

(s)

y

;

0

conservation

(S1 ~v) x

energy

conservation

4

(S11)t+u(S1*)x+v(S11)~=

-s g u x - ~ v y - ( u y - v ~ ) S 1 ~ 4

(S22)t+u(g~)x+v(S2~)y=-~gux+

-- gv - ( u y - v x ) S 1 ~ 3 S,t_S22

1 Hooke's

(S~2) t + u ( s l , ) ~ + v ( s , ~ ) y = ~ v ~ + ~ u y - ( u y - v ~ ) 2 p = g(s,l)

equaticn

law

of state.

(*) Strong c o l l i s i o n s create " i n s t a n t a n e o u s " large d e f o r m a t i o n s of the m a t e r i a l (breakings) but then the m e d i u m becomes plastic before the large d e f o r m a t i o n s and it is no longer r u l e d by this equation.

145

A used

by

simplified

one

engineers

dimensional

(Cauret Pt

(4)

+

[i]

mathematical

p.69)

(~u) ~ =

model

of

and

uy

(3)

is

0

(Su) t +

(~u~) x +

(pe) t +

[~eu+

(p-S) ~ =

(p-S)u] ~

0

-- 0

4 S t

where One

S

stands

gets

(4)

and

+

u

S x

--

p for

= ~(m,I) $I~ : one

the

independent

(4')

~

u~.

sets

Elastoplasticity Using

by (4)

we

~(S)=0

can

if

],show

ISI

set,in

an

(thus

curves

that,in

stress-strain

relationship

domain;

g

that

I

depend

(usually

the

these

projectiles

does

not

let

domain

consideration

ones

under ;

they

Defourneaux The basic

us

above

assumed

by

physicists

in

the

relations

that

domain.

above

forces The They

u~

by

a

=

0.

express

like that

>

if in

classical in

more

form

a

depend

also

on

has

received

"virgin"

several

;

to

experiments

quite

different

from

avoid in

the

introduction

, of

"history"

numerical ; an

tables

the

one

[i of

form

of

the

waves

Fung

restricted

which a

ISIgS o

form

complex

the

Lam@

or

So) . T y p i c a l

reported

, except in

the S

g(S)=~o6~

its

much

on

values

those in

there

depend

case,

hold

in

armour

shock

0

replacing

reach

law

not S

are

equations

(mass,

however external

g

=

the

classical

is

given

in

)

equations

neglected.

s

and

mention

involve

[i]

or

behave

confusions-,

$12

: G,A,~

cannot

does ~

an

=

-~

lal

coefficients :

v

metals,as

obtain

material

0

idealized

ISI Hooke'

on

experimentalists

numbers of

and

or

for

fact,

=

obtained

of

ISI=So

load-elongation §9.7

is

functions

:

equation

(S~2) t + u ( S ~ ) ×

constants and

-

to

be

are

are needed

different and

of valid to

true note

viscosity state for

and a

complement

physical

energy

perfectly

formulation

(gravity), equation

have

momentum

in

and given the

all

that heat

Hooke's

natures.

conservations)

law

circumstances

the

presence

effects are

material basic

The are

have

; of been

only

ad

in

certain

a

equations.

hoc

146

These equations are written formally as if t h e y c o n c e r n e d smooth flows. In t h e p r e s e n c e of d i s c o n t i n u i t i e s t h e y h a v e to be considered as reasonable since, via Colombeau's theory of generalized functions (in which one can compute explicit solutions of

in

engineers,

experiments. due

to

that

the

particular they Till

lack

mass,

give now

of

a

cases)

physicists suitable

momentum

and

of

nonlinear

via

heuristic

results

could

not

mathematical energy

conservative form as done equations t h i s is n o t at all system

or

reasonable

elasticity,

codes

agreement

clarify

their

setting.

conservation

in (3) clear

numerical

in

It

can

is

be

with premise clear

written

in

a n d (4). F o r t h e c o n s t i t u t i v e ( as already noticed, the as

stated

in

Fung

[l

§12.1],

c a n n o t be u s e d ) : the constitutive equations follow only from experiments on each specific material ; their validity in c a s e of discontinuities c a n o n l y be p r o v e d b y a p o s t e r i o r i verifications. The equations given here are considered as valid for most materials collisions

in

A,g,G

those

and

stated in variables).

system

the used

domain requested in the design in

the

the

fourth

equation

Mie-Gruneisen

The figure below (3) ; o t h e r o n e s c a n

for of

the numerical armour (then of

(1),which

form,depend

reproduces be f o u n d in

simulation of the quantities

on

a numerical C a u r e t [1] .

is the

usually physical

solution

of

147

t I

•

IOLJ~

* 5~s.

)D;r.,~(~/ rj ' l! l i :'

--+

G

6

Figure coming

from

collides layer

at

:

a

t=O; the

mathematicians

with

target The

that numerical ?

a

solution velocity

made

lines

figures

pretend in

A numerical

left,

copper.

could

involved by

with

of

begins Who

1 the

of

are

are the

given

of of

one

layer

at do

:

of

per

lines. of

deserve

copper second,

aluminium

t=5,10,15,20 not

A bar

meters of

isopressure

multiplications

solution

(3)

3000

and

one

Collision

microseconds. distributions to

be

studied

A

P

P

E

NUMERICAL

IN

It

has

been

X

IN

2

A

SYSTEM

in

§3.1

FORM

that

u t + uu~

~

o~

C~ t

~

k~Ux

the

system

\ an

with

infinite

H,K

depend

on

number

UO

solutions

O(x,t)

= A(~ K ( x - v t )

A

f

n

n

form

+ u~ + (~t

functions.

The

jump

formulas

in ~

A~

in e v i d e n c e these solutions [I] We consider the scheme n

ui+ I = m i - rm i (ui+1-u~_1)/2 "+ ( ~.i - i.) / 4. ( a t~+ 1 + 2(~i

~ a i~+~ +1

the

H(x-vt)

generalized

In this appendix we put tests, see also NoussaJr n

of

= Au

quantity

n+l

x

u(x,t)

Heaviside the

of

+

HK'

( S )o

I

NONCONSERVATIVE

/

has

D

TESTS

proved

(E)

N

n

by numerical (So)

n

+r(Vi+1-~i_1)/2

.rmi. ( ~. i + l

oi_i)/2

+ r k 2 ( u i +~1

- u ~l- i ) / 2

! m ni

where For

AE2

we

(S A)

-

consider

--_

Ui

_

n n (ut÷,+2u~+uL_1)/4

a

n

.

modified

scheme

(S A)

defined

by

n

Vi÷A

~ ÷ ¢~

( o ' ~ + x + 2 a ~ + on' i _ x ) / 4 - r ~ L ( o ' ~ + ~ - o ' ,n_ ~ ) / 2 ~ + r k S ( v nt , 1 - v l _ l ) / 2 "

with

~

=

(uT÷,+2u~+uT_1)14

m I =

(v~÷,+2v~+v~_1)/4

i_I)/2

+ r ( a i n+ 1 - a i - n1 ) / 2

(v~+1+2v~+v~.1)/4

=

-rm~(v~+l-v

n

v~+IL÷I =

149

REMARE.

-

numerical This

In

order

results

modification

scheme wing

but . We

it

to

we

improve

introduce

does

makes

not

the

introduce

a

(a)

the

a

play

shocks

quality

classical a

role

in

neater.

It

correction

only

in

n

>

>

>

n

Ui - ~

n

Lli - 1

and

neatness

of

antidiffusion the

convergence

consists the

two

in

the

method. of

the

the

follo-

cases

n

Ui

Ui + i

or

n

(b)

Ul

<

n

- 2

Ui

<

n

- 1

<

n

Ui

Ul

+ I

Let 1 n

c"

=

-

(u~

-u

i

)

"

4

set

we

cni = with

+

in

the

the

cases

(a)

by

U i +nC i -~n - C i - 1 ".

the

prove one

pattern

estimates -which

obtain

-provided of

One

(E).

values

of

HK' . W e

choose

A.

=

u t =

r

=

the dt/h

value is

proof.

in

Theorem

the

3

0,

k=l.

chosen

has

for from

A

the

can

for(S o ) by been

scheme

able

(S~)

numerical 3.2.2

all

are

not

the

Theorem

hold-

proved

for

correspond

tests-

(see

schemes

showed

Noussa~r

(S~) A=0,

to But

give

A=-3

and

to

different estimating

be

recovered

by

and

u,=l

in

Fig.2,3,4

, u~

=

0

0

The equal

be

conditions

Fig.l

Or =

can one

in

w e a r e n o t in u ni is r e p l a c e d

~ i"

A~0

obtained of

scheme,

§3.2

3.2.1

results

values

in

in For

estimates

initial

(a) . W h e n

the

for

result

curves

lu~+1-u~[/2)

case In

reasonable

the

Numerical

These

u,

and

those

of

the

the

the

cT=0.

as

looks

ascertains

in

n

i[/2,

replacement

again

[1]) . Thus

-

set

Similar

solutions A=3.

and

we

results

them

can

, lu~_2-ui_

(b) (b)

'

same

the

admiring

case and

Same using

n

±min([c~[

space to

0.2

step (dt

is

is the

h=I/100, time

the

step)

quotient and

I00

150

iterations A=0,

~re

A=~

The

pe&ks

A=-3

curves

sent

the

~re

,,

.o.sol oo

,,

the

limited

v&lue

horizontal

of

We

present

three

successively.

represent

these the

computed.

and

A

~xis

if

~re

The

curve

representing

curve

representing

function by

the

the

K'

~nd

function

~reas

taken

,

figures

,

to

the

c~ses

~re

u

the HK'

limited

equal

in

conventions

d~rk

surf&ces

; their

by

the

~re&s

under repre-

function

K'

and

I.

,

,

,,,,,

27

...........

,,

........

'i.W •2.00..

,

,

,

FIGURE

1

corresponding different.

~

,..

,

, .... ,

: Superposition

to

the

three

.,

~

,

,

,

,

,

,

of t h e n u m e r i c a l s o Z u t i o n s a b o v e v ~ l u e s ol I : t h e y are q u i t e

)

• s~oo~s

~l$oq

aol

--

=~

asf-)O

:

~

~llIlgI

;~

T

OOS

~0- -

;4

~"~,i . . . . . . . . . . . . . . i'

/

x :,"x ",';: × x'," x × ;: ×~ : ~ : : :~x x × "

/

/

-

I

I

OoS Io M

=============================== ~-F-

"'Z:-

rC_;L

152

u__

c~

7~m

{ :.__

~.

5~D

~__

v,.

~ca

V--

~)...i~

%555

y--

FIGURE

3

: Case

and

~=3 A=0,98

;

one

for

finds the

A=0,89

shock

on

for the

the right.

Zeft

shock

153

? °-- -

ll!

-

o, &~O

FIGURE

4

: Case

One

h=-3

and

A=0,06

checks

that

4<

; one f i n d s A=0,05 f o r the for t h e s h o c k on the right.

the

decentered

scheme

/

Zeft

considered

shock

in

§3.2

1

gives from

the (S o)

value :

A=

-2

by

superposition

with

the

curves

obtained

154

, , , ~

.

Scheme

-~-+-+-+ -o

.

,

(S0)

decentered

.

scheme

5

F I G U R E

:

decentered

The

scheme

(without

antidiffusion)

g i v e s the c u r v e s w i t h the l e s s (as n a t u r a l d u e to t h e a b s e n c e in

REMARK.

-

this

Comparison

shocks antidiffusion

s~ee?

of

scheme).

a

with

conservative

form.

Introducing

u 2 qzg

+

--

the

system

(E)

-if

written

with

equalities

in

~(~2)

, or

2 on

the

level

of

"form&l

comput&tions"

I

-can

be

written

in

the

u t + (u2-q)~ = 0 u 3

(E') qt

+

(

k2u)

~

=

0

Z Aq The

quotient

=

--

of

the

jumps

is

solution

Au

z~-(u,+u,)z+(u~+u,2+u,u,) = 0

of

the

equation

form

155

which

has

no

real

Lax-Friedrichs

solution

and

singularities

on

if

the

explained.

One

acceptable

mathematical

deduces

12k~-

(Au) 2 < 0

Lax-Wendroff shocks,

that

the

Using

schemes which

the

observes

be

physically

cannot

conservative

classical

one form

(E')

is

not

an

model.

m

0

::: i

T--

r--

(~

:.

.:

i

i

?

/<~i

i.....

i I:-

~e...o-.e-..N..%-..o--.o.-.o...~...~..-o--~..-~-..o-. Q.--O.-.O..-O..-D-.o--.o...e...e."~'-'o'-'e'"o'"e"

7/\ \

".,

c

FIGURE

6

:

A

sAock

of

(E')

obtained

from

tAe

Lax-FriedricAs

scAeme.

o

B I

t~

o

~+

0

0 0

o°

...~. ........

::.:.:.:.:.;.:.~.:,)):,~,:::::7.".',..." . . . . . . . . . . . . .

f~....

~ ' ~

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

F ~ N f 1 ~

........... ~ , . . ..................

! .........................

X~l.

~a

f~

m

~" f ~ o

J J J lJ

f I ] I I r i I I I I I I

~~T~Y~ ~ ~ Y7~1 ~] ~

A

P

NUMERICAL

We §5.3

(for

with

compare the

classical

applied schemes

of

-

SCHEMES

(a)

The

FOR

I

IN

X

3

FLUID

DYNAMICS

constructed

dynamics

in

from

the

Kichtmeyer, formul~tion. to B. P e r r o t

THE

in A p p e n d i x 2 for (NC2) d u e to A . Y . First we describe [l]). IN

SYSTEM

NCI

step.

it

:

We

NON

of

Kusanov

VI+I

+

two

FOKM.

steps.

-I

__

UI+I

-1

+I

-i

n Zi

We

compute

--

iteratively r n + I

vi

~

xi-

R

:

Yi-

r n

n

n

n

n

n

n

i)-

_

n

Zi-

_U

~

n

n

_

n

I-i)

-- X i (Pl +I

Pl-

)

I

2

r + 1

n

r

-- Y i (Ui + 1 - U l -

2

p~

n

-- Yi ( v ~ + 1 - v i - 1 ) + -- xi (ui+1 2 2 r

n+l

Ui

r n

n

3L

-- Yt ( P i + i - P i - 1 ) 2

)

These two (NC1), w h i c h

set

n X i

in

form),

the system there, and Le R o u x , w h i c h h & s b e t t e r the schemes (see a l s o

CONSEKVATIVE

is m a d e

results

non-conserv&tive

(Lax-Wendroff,

scheme

F~rst

D

classic~l conservative very simple scheme due

h&s alre&dy been used more el&borate scheme stability properties. De L u c a [I], N o u s s a i r 1

N

schemes, fluid

schemes

to t h e are a

E

TESTS

two

system

P

n

n

-- g Zi ( U i + l 2

~un

I -1) "

158

Second

(b)

A Godunov

and

a propagation

type

This

scheme

step

I:

tratment

step

2:

treatment

which

step

the

3:

(due

flux

the

scheme part:

to

A.

of

the

of

is

with

the

Y.

of

step

solves

Burgers'

(Ib),

shock.

On

(Ic)

each

(j=i+1/2) if

in

part

by

the

from

by

an

into

Godunov

three

steps:

scheme.

a Godunov

type

approximate

Riemann

scheme

in

solver.

The

convection

part

consists

system

v t

Pt

x=jh

as

a convection

decomposed

I (convection).

I

treats

into

same

is

propagation

u t

One

Roux)

calculated

the

scheme

convection

the

is

splitting

NC2

Le

It

antidiffusion.

Description in

: antidiffusion. Appendix 2,

step

u t 1+~

mesh

0

u

u~,

.~

0

(la)

+

u

v×

"~

0

(lb)

+

u

p~

,v, 0

equation in

such one

; this >

+

~nd

(la)

a way

by

the

that

computes

(]c)

the

Godunov

scheme

u,v,p

are

in

fluxes

through

phase the

and on

one a

interfaces

gives u'~+u~+

1

>

0

:

u~

u L

v~~ = v~ Pj

if

~nd

u ni < 0

n

Ui+Ui÷

n

I

<

0

:

n

n

=

It

=

Pj

=

Uj

Vj n

if

u~+~

>

0

&nd

U ~ ~.

<

0

:

Uj n

=

Pi

=

n

;

Ui@ 1 n

Vi+ 1 n Pi41

0

v~'=(u~÷~v~-u~v~ ~)/(u~,~-u~) n_

n

n

n

pj-(U,+lPi-UiPl+l)/(u~,

n

I _ U in)

-

159

the

speeds

of

waves

are

CL,j = ( u ~ + U ~ ) / 2 CRj = (u~+ui. ~ n )/2 Then the

one

projects

L 2 norm.

w =

each

rCR,j_lw)_ 1

v,u,p. The

mesh

m=n+I/2

n

WTM

for

on

Setting

+

by

one

for

the

left

for

the

right

the

usual

side

hand

projection

wave

side

wave

operator

of

has

(1 + rCL, J • rCR,.7_l)u,i

stability

hand

condition

rl

- rCL,jWr~

is

i r m ~ x [u~[ 2

Description in

the

of

step

2

propagation).

The

propagation

part

consists

system

Vt -- VUz ~ 0

(2)

Ut + vp~. ~ 0 Pt + 7pu~ ,~

0

One c o m p u t e s an a p p r o x z m a t e solution of (2) on the s t r i p ff~x [tn,tn+l] f r o m the v a l u e s v~m,um,P~T M c o m p u t e d at s t e p I. At f i r s t one c o m p u t e s the fluxes through the c e l l i n t e r f a c e s . T h i s is d o n e t h r o u g h an approximate Riemann solver; in this c a s e the s o l u t i o n of the R i e m a n n problem is m a d e w i t h two w a v e s ( s h o c k or r a r e f a c t i o n ) and a contact rn discontinuity. It g i v e s a p r e s s u r e pj, a v e l o c i t y u~ a n d two values

t,m and 1,2

m=n+I/2).

One

2j

of

sets

m

The

four numbers conditions)

(recall

and

ZLj.

p ~ , u ~ , v TM 1,3 a n d

v 2,j TM

are

m

rn. p.,

_

linked

by

U 2r,nj )

j

pjm = - ( z . , j )m m

j=i+I/2

and

3 q- Pi

71m= - Z R r,nj ( v , ~m 1 - ra

P~

that

impedance)

"t'vS+ , ~ v.2,~

lt,~+ 1

(3)

volume

Pi"~l 4 p~

Z.,~ =

(jump

specific (acoustics

Pj = ~

2

m (~,+,

_

yrt

( Z m "12[vrn

L,jI ~ ,

_

Vm

~j)"

~,j)

Vm

the

four

relations

160 The

solver

iterative an

error

consists method

~

10 4

in

solving

this

(convergence

is

).

speeds

The

sound

m + v TM m ~ m Yi+I 2,j CR,] = ~R,j 2 '

One to

obtains the

easily

scalar

:

m + l

-rC

m

in

obtained

system

in

3 or

by

4

Newton's

iterations,

with

are Z ~ v~ + v7 't'J - - 2

m eL5 =

following

product

U n+l vi

the

nonlinear

values

of

the

projections

according

L 2 ( ~ ):

u TM + r C ~ ) _ , u ~ _ , + (1 + r C ~ j - rCR~,j_,)u'~

L,j j m

rn

r~

m

= - - r C L j V L i + rCR,j_lV.2,j_l +

rn

m

(1 + rC~,j - rCn,i_a)vi

(4)

p~

This

method

. , j _ l p j _ a + (1 + r C ~ , j _ r C Rms _ I ) p ,~ i • 1 = _ r C L ., j p j. ÷. r C R

is

stable

provided

1 + rCL, j -- r C R , j _ I ~ O.

Therefore

stability

for

both

steps

is

ensured

under

the

condition

]

Description

of

the

step

3

(antidiffusion).

sjm

One

computes

A~

_-

m (ui÷1-ui)/2,

b~

=

r(1-r(um+~+uT)/2)((uT+~)2-(uT)=)/4

aim = sjm M a x ( 0 , r n i n ( A ~ +

a n d one r e p l a c e s u mi by v o l u m e a n d the p r e s s u r e ,

U m

= s g n ( A TM j)

successively

(j=i+1/2,

m=n+I/2)

m [bTI)) I s T , A T _ i sj,

i - ajm + a jm _ ~ . S i m i l a r l y o n e sets, for w = v or p,

for

the

specific

161 m

A~-

m

wi+12- wi

,

sj~=

sign of Aj

r m + m b? = ~ ( 1 - r U i + ' 2 u, )(ui'~, +ui"~)(wi+lm _ w y ) same formula as above for ajm

~nd

The

one

NC2

repl&ces

scheme

The

w~

with

and

the

become

noteworthy

constraints,

(4)

in

projections.

(L 2 p r o j e c t i o n s

pressure)

see

u'~ - a 3~+ a ~ l .

conservative

formulas

volume

by

can

the

be

case

Noussair

at

of

of

the

solid

[I]):

one

the

origin

velocity, of

materials

can

the

defects

(that

submitted

observe

specific

defects

to in

large the l

conservationof

mass,

instead

of v, e n s u r e s

prefers

to

u

and

place

p. of

To v)

project

momentum mass

the

the new

for

v one

energy.

conservation.

momentum

get

1

and

formula

and

But

For

the

projection

the

same

energy,

by

projection

sets

~

-

+ _ _A+ _ _ fl v2,~_ 1 v ~ v'~j '

-

P'~ then

m~

aA

(I)1,2 ~ Pi V m - m) \ V2,3 l'Vt

*-=,r( #i--=]. \ ~'i "t:1,7 /

The

(5 a)

formula

giving

v~+l i

the :

specific

~ ~

volume

U2,j- 1 + •U1, 3" @ ~ m

-- , , 3 ( v 2 , j _ ,

-

reason

total

(obtained

is

-- ¢1,2(y~,3_ I

vrn ~2 ,,jj - ¢2,3(v,~

-- v ? ) ~.

yirn)2

of

in

p = one

place of

of 1 p = - in

162

projection

of m o m e n t u m

(5 b)

C ~' =

and

~_,

total

+ ~7 + ~

-~l,~(~,j-i

(~ c)

K"'

:

remainings

formulas

(4).

given

the

the

in

NC2

this

2

-

We

The

book in

scheme

metallic

CLASSICAL

the

the are

- ~,2) ~ + ~, .~(C~_,

same

as

in

results

since

they

cases given

NC2:

scheme

uT)~

-

,

cannot

+

~

%,~(.?

formulas

concerning be

'~

-

last

here. [I],

from

Numerical De

Luca

just

scheme

distinguished

considered

.7)~].

(5 a , b , c )

this

in N o u s s a i r

SCHEMES

FOR

TKE

m

= Du

W

=

(o,m,e)

F =

(m,m2/~

SYSTEN

IN

[I]

in

CONSERVATIVE

those

case

FORM.

is w r i t t e n

:

are

results

+p, (m/~) (e+p))

W t + F(W~) C I

-,

collisions.

set

system

~ -~j)-¢',3(':lj

2

Numerical

scheme

last

strong

are

- ~, ~ ( ~ % , - ~', )(~J-,- ~ )

" 1'~, ~ ) ( % - ~

-

give

~g'-, + ~p7 + A~C + ~ -___2

[¢,.~(~L, The

energy

= 0

Lax-Wendro~f

predictor

:

W "+~ i÷~

=

(W~+

corrector

:

W ~ +~

= W ~i- ( A t / A x ) ~ rF~+~ ÷~ i +~- F ~i-~"

t

+W~)/2

-(At/2Ax)(F~÷:-F~

"

t)

replace not of for of

163 C 2 scheme

:

Richtmeyer

predictor

:

W ~i÷~ ÷~

=

corrector

:

W ~ +i

= W~- (At/Ax)

C~

:

scheme

(W~ +I + W i'~ )/2-(At/Ax)

(F~+i-F?

I

-

1

)

(F~-F~+~) ._

Rusenov

W~ +l

: W?-(gt/2gx)

(F~+1-F~_1)+l/4(a~÷i+(~?)

(W~+z-W?)

-i/4(~+(~?_

i)

(W~-W'~_ i)

with ~ where basis

3

-

NUMERICAL

apply

conditions

these

;

air

air

The rarefaction a

is the

tube

in

the

and

in

a

two

p=0.1

uz0

u=0

disappears

travelling

wave

to

travelling

slightly

different

obtain

rather

neat figures

results 5

to

at the

to are

chosen

given

for

all

9 below.

the

of

on

the

recall

(of of

the

shock

the

infinite

tube

initial

length).

different

A

pressures

air

t=O

left,

the

from

us

tube

zones

o=0.125

membrane

problem

[l] . L e t

pzl

scale

in

to

Sod

confined

experiments

given

,~6~

arbitrary

p=l

wave

shock

by

numerical

is

our

schemes

described

membrane separates and densities :

of

is

(u+c)~

RESULTS.

classically

and

~(at/ax)

c is t h e s o u n d speed;~ of n u m e r i c a l tests.

We as

:

a

; this contact

right.

The

gives

computation

below.

We

have

one

Sod

[I]

schemes;

in

their

rise

to

a

discontinuity

in

times

chosen

a

order

to

time-evolution

164

The

scheme

solution

(see

roughly

to

NC I is

Sod the

precisely

[l]) same

All

superpos&ble

the

curves

computation

time.

to

given

the

below

Finally

explicit

correspond

it

appears

from

these results that the nonconservative form gives results of good quality, and can compete successfully

numerical with the

classical

schemes

viewpoint

of

book

this

Colombeau's

theory

verification COMPUTATION

on

a

PC

Number

using the

of

the

fact give

the

conservative that

the

validity

the

correct of

From

schemes

solution

this

obtained is

and

hundredths

of

meshes

: I00,

time

without

step

: At=0.28,

antidiffusion

i00

with

1.30.07

2.23.83

5.15.24

6.47.72

C I (Richtmeyer) f)

2.01,04

C s (Rusanov) that time these

3.00.87

antidiffusion of

in

computation

the

C i schemes

without

correcting

would

have

increased

the

defects

shown

curves. Time

schemes

SO i t e r a t i o n s

antidiffusion

1.34.35

C I (Richtmeyer) C 2 (Lax-Wendro

At=0.25,

with

antidiffusion

explodes

Roux)

C s (Rusanov)

step

without

NCI ( P e r r o t ) NC s (Le

seconds,

1.53.17

C~ ( L a x - W e n d r o f

in

of

antidiffusion

N C 2 (Le

Roux)

explodes f f)

explodes explodes

from

numerical

iterations.

NC I (Perrot)

the

a

computer):

schemes

Note

the

theory.

(in m i n u t e s , s e c o n d s

TIME

form.

new

2.02.14

165

Perspectives from

the

for

oscillations

in

aerodynamics

there

method

pressure already

from

the

classical

that

the

nonconservative

efficient a

new

domain

that

of

the

of

to

they a

far

conservative pressure

even

a

large

as

we

form

give

on

contact

The below;the

amount

NC 2

curve

curves

NC 1 c u r v e

is

the

know,

large of

produce all of

schemes a

poor

of

computations;

obtained the

one

from one

with

with

the

has

tested

computation

computation

great

of

fluid

this

variations from

(oscillations

instance) been

[I]) that the nonconservative results.This domain of of armours and projectiles.

is

in

obtained

quality it

a

time

the

system

large

is

revealed

of

state);

of

there

have

the

very

discontinuities,for

expect

origin

constraints,

by

of

obtained

could

the

immediately

to

equation

; one at

of

case

schemes

tests

modelled

density

results

De Luca [I], N o u s s a i r give better numerical encompasses the design

is

the

specialists.But

numerical

method

obtained

absence

In

be

by

submitted

in

curves the

numerical

will

which

be

the

formulation

in

Mie-Gruneisen

variations

pressure.As

good

developed

may

In

observes

velocity

be

are

solids

(with

small

very

exist

nonconservative

behaviour dynamics

and

method

applications

when

use. one

conservative

schemes

interest:

very

industrial

noneonservative

or

case in the in demand

observed

(see

method applications

can

schemes time

are

2.23.83;

2.02.14

given the

166 ,,

,,i,,

densi~

0.?~4 0.¢74 0,qq.I 0.334 0.224 0.1i.I O,OOl -t0,0

I

0,?~i ;i;scheMe ttCt 0,67 O.S6,

1

I

l

I

~w

/

~~"

/

/ . ~

~ "

0.33

022: -10,0

0,e~

-7.ti:b

"~,D

-3,68

-1.~

i

... sche,eH

i

i

i

prellurt

00" 0~0'~'0;P7,q$q.33 ***sche,eHC2 ~

============================================== .....................

0,~2

0AI O,OOl~i

~

• ~o,o

.7,8~

Figure

1

~

,

.s',~ :

.£.

Comparison

,

i

<'

-l,~ of

the

NC1

and

NC2

schemes.

167

sch~ NCl

O,f7

.L 0.3~

-iO,O

IS

4',74 ~chCMeNC! ;;;5¢h~Hl Cl

°I -IO,O

0.SD

'

'

i

.i .

512~

,

,

,

i~. i

/

/ -I,7~

,53

5,79

!

pressure 0,$jI

"'~ChaM~C! X

4 ,! o -iO,O

4,74

I

~ '

I

I

I

~

Figure 2 : Comparison of the C1 (Lax-Wendroff) NC1 scheme .

I

I

I

and the

)

168

.,~!.':"

::

,,:

,. ........

:

, ................

;

;

:

,

,

~

te,s ;ty

o,~71 0.13

O . ¢ N ,, ~ • 10,0

i

'",

i

't

~

l

' w

"~',7~ i

~'

i

-10,0

21 ,

: _L::.,.LUJ~I,~

i

'~ i

i

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

I' "

I

I

S,7~

O,S)

!;.7~

I

' '"'

I I'

'

I

, , ,

i,

-I

,i

L. . . .

NC1

Y"

,

i

Figure the

'

sche,e IICl

O'O0"t i • 10,0 i

I

/

-~

,

I"

H,,

sch~,eC~ ,cl ;;;iche,e

,,~ ........

I

O,S3

.~'.74

I

i

3

scheme.

:

Comparison

, .... 1 i

i I

~ O,S~

i

iiiiiii

of

the

C2

iiiii

, i i

iiii

z i i

i i

(Richtmeyer)

~ i

t

i ~ ~ ,

S,79 scheme

and

O) 1

169 i

~lensi ty

O.S¢ O,~q 0,33 0,2!

O,ll 0,0~

,

,

~

,

,,

,

,

0,!13

v

v

v

2,11~

~

~,?q

v

~

~,lq

v

~

)[

............... ! !

--IL ,,

t,~l!i ! O"

A

0,1t.

II

/-

S¢I~eM~ IIC!

i~'"'"

I

•

I

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

uelOt:it~,

0.7t,

0,117, 0, !;tl, 0 ,qq,

0,22, 0,11 O.O0 .10,0 II

I

I I

I

':

i

'

.?.tI~

¢

~

-S,?~

.

,

' i

-),~ J

v -l,!it

IIIIII

!

I

V

r

I

I

!

0,~3

L II

I

I

I

pressure 0 711'

;;;sch~,e CI N

o.s, 2

°'"i

0 42,

O,LI,

o.,I,

J'|

"lO,O ' "?',8)

| I

Figure NCI

scheme.

'|

|

• !1,7~ 4

" |

-),~II I

I

|'u

'1 j

"l !it gll

: Comparison

III

of

,u.

I

the

C3

(Rusanov)

scheme

and

the

A

170

r, c l

6 e n s i t_~

•.

"l

sc he~,e

j

",~.~-~-~,. ii,t.J.-7 t t i~. ~! i|

ncI

velocit~

~

scheme--

m m ' ,

///I / l/ill

'

!

,,,'//I.I

I

ncl

~cheMe

i!~ Figure iterations).

5

:

Time-evolution

of

the

NCI

scheme

(every

I

20

171

clen_¢ i ty

no2

s v.. h e M e

I

.

'hi,

Jl

I

ii ~

i "I

velocity

n~Z

•

~ e

"' ,

'II

li~

Figure

6

'11

no2

s s u:~.e

:

Time-evolution

of

the

NC2

m'r :"

• L

scheme.

zc})e Me--

~o~eMe--

172

ci

d e n s i ty

scheme

x',x "~,.. '.. ~.,, ii

,,.-.,-... , ~.%~,,...,,. _

!.

ve I oc i t~

cJ. s c h e m e ~

i

// /

/.//// / / mn

m

i

u

! i

cl. s c h e m e

Figure

7 : Time

evolution

of the C1

(Lax W e n d r o f f )

scheme

173

c2

"t

"',

s c}~e~e

ill

'

&

%~%

',,

"% "%.

.,

"'~

hl 5

I

II

veioci,

t!

ty

c2

/>./i/

rI

scalene

I{ c2

>~e s s ~ e

sc}~e~e

\\i: Figure

8

:

Time-evolution

of

the

C2

(RichtmeFer)

scheme.

174

clens i t ~

c3

.%

"%',%~ ; "tl,I

scheme

•

itiii

'~

',

~

:: I.

'

c3

velocit_v

"t

,!

so:heMe~

/%~":} i

:,1t4! /

.I

i

c3

~ e s s u ~ e

~ t Figure

9

:

Time-evolution

of

the

C3

scheme-----

~ ~ (Rusanov)

scheme.

A

NUMERICAL

As consider

a

TESTS

simplified

the

P

P

IN

E

N

D

I

MODELS

form

of

X

OF

the

4

ELASTOPLASTICITY

system(l'

)(70)of

chapter

3 we

; here

k2 ((i)

system I

LI t

+

U

Ux

~

(it

+

U

(~x

m

(7 x

(i)

which is

a

is

a

modification

function

might

have

critical

of

k2 ( ~ ) = k 2 0 o

value

elastic-plastic We

of

the

(i ( o b t a i n e d if of

system

from

in

Appendix

experiments)

]~l~(io the

k 2 ((~)u x

and

k~ ( ~ ) = 0

stress,

at

1

; for if

instance

one

I~]=(io,

for

some

there

is

an

which

transition.

consider

the

following

numerical

scheme

:

we

set

1

(mu) ni

=

--

[ Uni + l +2 Ui"+ U~L - I ]

4 i

(mo) n i

:

--

[

n

+2

Ui+I

~+

Oi

~

].

Gi-:

4

We

define

the

values

and

u:

a:n

n ~i+l

n n Ui÷l--Ui-1 U n i+1

_-

(mu)?-r[

inductively _fin

1-1

(mu)?

2

2 n

n+l

~i

Gi+l

=

(mZ)~-r[

Ui+l--Ui-

-1

we

]

k2(

= ~n i -Gn i+ t-1

the

antidiffusion shocks

2

set

(l~~ ) ~i

An

1

- (k 2 ) ~

(mu) 2

where

by

heater.

similar

to

the

Gn

i-1

one

in

.)dx 2

Appendix

2

is

used

to

make

176 Admitting

that

this

v a r i a t i o n p r o p e r t i e s in § 3 . 2 the numerical tests) then convergent

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

Setting we

get

the

(a)

k 2 (0)=1

following

Case

scheme

has

the

L"

and

bounded

( w h i c h l o o k s r e a s o n n a b l e in v i e w one c a n p r o v e as in § 3 . 2 t h a t it

of is

solution.

Io1<~0=o,3

if

numerical

and

k ~ (0)=0

otherwise

results

u t

--

1

,

u r

=

0

O' r

--

0

,

O" r

--

0

O,t2~

0,|$4

0 ,)4~ O,tt4 m

O.l~l

-10,0

"7,0!

I,I

-S,7$

1

ut+uu its

-l,SJ m

Figure equation reaches

-3.¢!

Is

: ~ ~

critical

one O)

,

, m

0

observes surrounded

value.

St

! ~3

et,21

middle

shock

~,$~

t,)S t !

e

a

by

two

(solution

shocks

in

of which

the 0

177

Relatively Appendix left

2 we

and

its

right

critical

Ut+UUx~

to

observe shocks

value

the

case

k2 ( u ) = l

one

more

shock

raise

the

G0;the

shock

2

=

for (the

stress in

all

considered

in

(in

absolute

a

the

as

G

one

middle

the

is

in

middle)

a

value

.The )

solution

to of

O.

(b)

Case

u,

=

, ur

0

1.70 l .SO !.30 l.lO O,JO 0,70 0.50 0,30

°'°

r'"I

.............................. ~.........................................................

LO'''!'

'

/ -lo.o

-7,~,

Figure importance increases.

I

'

l

:

one

observes

the

shock

-s.7,

2 of

I

.3.,

I

'

'

.......ii ............X )i ' l

.l.st

o'.,,

a in

!

~'.~3

similar the

middle

!

,'.,,

f

ii ii l

,.~,~

situation; increases

'

the

si,s i

~.~

relative when

u t

178

When the

one

of

u ~

the

representing is

g

associated

its

values

-lO,O

.7,~)

Figure

3

result ut+uu x ~

to

show

a

zero

(its

support

between

.S.TJ

observed

equation

tends with

lie

the

single

0 and

-3,¢~

: observe

0

closer

.Note

generalized is

located

and

that at

closer

0,~)

continuous

~.~3

'[.7't

evolution

curve

function

which

a

point,

~.~1

in

to

the

-a0) :

-1,~

the

nonzero

gets

~,~

fig.1,2,3.

and

179

Now

we

consider

I

the

system

u t + uu×

+

(p-S)~

pt

+

(ap+b)u,

+ up×

S t + uS~<

~0

#; (S)u~

~0 ~0

which is a simplified f o r m of the s y s t e m (1') (70) of c h a p t e r 3, w h e n the d e n s i t y varies very slightly in the n e i g h b o r h o o d of the value 1. We adopt a numerical scheme similar to the one at the beginning of this appendix : it is very simple and give qualitative results in a g r e e m e n t with those expected) .We c h o o s e g as f o l l o w s : #2(S)=2

S0:0,30 initial

if

ISl<S 0

,and

~(S)=O

if

ISl=S0

;

In figures 4 , 5 , 6 we c h o o s e s u c c e s s i v e l y the v a l u e s S o = l , and S0=0,02 .We choose a=l and b=0.We consider the values (Kiemann) problem: urn2 p~=O,l S~=O

Ur=O pr=O,l Sr=O

Let us observe the left h a n d s i d e s h o c k s . The s h o c k of the p l a s t i c system (S=O a n d ~=0 in the s y s t e m ) s p l i t s into two shocks:the more rapid one raises ISI to the v a l u e S0; it is called an elastic precursor ; it is f o l l o w e d by a plastic shock solution of the p l a s t i c s y s t e m . As a b o v e , the r e l a t i v e importance of the e l a s t i c precursor decreases when S O decreases ; at the limit S0=O one observes the s h o c k s of the p l a s t i c system. For fixed S o one could also observe the s a m e p h e n o m e n o n w h e n u~ * ~. In figures 7 to 12 we present numerical results for various o t h e r v a l u e s of the p a r a m e t e r s and initial conditions. They provide a good description of the p h e n o m e n a observed by physicists: elastic precursors and their agglomeration in the plastic shocks for i n c r e a s i n g velocities of the p r o j e c t i l e s . More complex examples of e l a s t i c precursors in the c a s e of a n u m e r i c a l simulation of a c o l l i s i o n iron-iron are given in Cauret [I p . 1 4 9 - 1 5 2 ] .

180 m

1JS. I,SO.

I

1,2S, ! ,00,

_~plastic

shock

. . . ~elocily

O,?g.

~._

O,CO, O,ZS, 0,0~ I

. . . . . . . . . .

'0,7 ,

I

.7',.

' .s',7,

precursor

L_lll . . . . . . . . .

'O,2St ,0 ,S~ 'l Oe| "lO,O

-~elastic

I

I

'

I, I .... ~f

I

I~ ~-.

! I

I

q,Tq

I

I

S,H

~,95 iO"1 !

" )

preSSUre ; ; ; stress

plastic

shock

--)

~elastic

= _c9=.

-10.0

-7.~$ Figure

-S,TJ

-).S!

4 : elastic

-I,S~

0,S3

precursors

precursor

A..

2,S3

1,71

and p l a s t i c

(,H shocks.

~,~S i0"1

181

plastic

shock

:~ e l a s t i c

i

;;~ ~%t'~S~

precursor

~, k.~'~ S =-So=-%~,o

"?D

"10,0

"~'.D

'

-3.,

-i.,'

"

o'.~3

~'.+~

,+.7,+

~.+,+

~..+~ J¢,'~

& i 25.

I,SO~ i,lS, m

..o

i,OO4 0.75,

_~ p l a s t i c

pre~s~Jre

shock

0.50', 0,2~

-.,-

-~ e l a s t i c

precursor

o,oot._'' __2~.__~_2:_~...... _"_-2"_'.-.__--2:,_+~ ,o.+s+

% % =-

,O.SO

~

=

-,%3o

~

i

'O.TS. 'I.001..

,.

• !0,0

,

,

-7.%~

Figure relatively diminishes.

,

'r'

-~.D

5 to

fig.4

,"

,

',

.i

-3,6%

:

elastic the

!

i

..... I

i

~

i

' w

v

,~0

precursors

importance

and of

the

plastic elastic

1.1 0

shocks precursors

;

182

t .?S.I i.SO,i t,2S,I

.~plastic

l,O0,i

;;; velocity ~;ress

shock

O.?S4 0,50,1

~ elastic

precursor

0,2S,I

,1.00/ , -lO,O

, , ,,

!

,

• ~i?$

-?,~

r ")IG8

" -i.S~''

'

0.~3' ' ....2.~3' '

~i?,1 '

glS~

'

~,9~ iO"1)

i.?3J i,S04

PFeSIVre

1.2S4

; ; ; stress i.OOJ

plastic

O.?SJ

shock

O,SOJ 0.25J

o,OOL.

,~elastic

,

..... j,_ .....

,-~__ '___._...~__~__~-

\

°°::;t I

.10,0

-?.:B!

|'

I

-S 7~,

I

1

"3,~

precursor

I

2,f3

'i,?~

$',~

Figure 6 : Observe the decay of the elastic precursors and the continuous evolution in figures 4,5,6 .The elastic precursors still exist but they are very small.

~.~ iO"i

183

),Oq.I !,qll 1,921 !,3N

0,~01

0.2~11 I

J

'0,324 "~ '0,~$~ ,i.Ut4

•l,O01 i i ! "~,0~ "3100 -I,91

ii~'

J

-o,~t J

J

o,1~

]

I

I

1,21

I

[

2,27

* ,.,.,~,

,

-~

] ,f(~

i.32,

1,)~ 0 ,OO,

,,,fl

o,~: /'

°"t .O,t~ \ '~,00t -~,OS

,

,,

,,

.3100 , -t,. I

Figure

i

~ -o,. * l

7 : Plastic

/

,

o',l,

1121'2',27'

shocks and e l a s t i c

precursors

~re~f-

184

0 .it

0,7{ 0,|1

~.

0,11

,t* stre~s . . . U~lo¢ity

ILl| O.lt ,0.0|

I .......................................

i....

,O.i~l ,O.ti '0.3!

,O,S', -lO,O

-7,.

-s',7~

-),.

.,',.'o',.

'

'i.7~

2'.¢3

¢.$~

l i

__

' .7:. Figure precursors

_ ..........

' -s:7, 8

~'.gS I

it* stress . . . pressure

L_

' "3.1;! ' :

The

plastic

shocks

tend

to

i

catch

the

elastic

185

' , ' st

i

|

I

J

$|

,

,

d

-IO,O

-?.t~

-S,7~

-3.GS

0.~3

.I,5~

t,~3

'1,7'I

¢.~1

¢.55 lO"!

"'

m.a

~J;[||Ire

/ -I0.0

-?.$J

Figure any

-S.TJ 9

:

"3.¢!

The

plastic

longer:

there

not

exist

two

elastoplastic

shock

waves

-L.50

0.53

shocks is

a

and

unique

appear

in

2.¢3 the

,t.?~ elastic

elastoplastie the

figure).

C.~14 precursors shock

O.~S I0"I do (indeed

186

10-' A

4

i .04 0,7~ 0.S~ 1.¢04

O,fN

0 . 2 ~ ~

................ '

O,l~~,~f..

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

.

.

.

.

.

.

.

.

0,S~

.

q

0.7~ 4 !,00]

0.4M ,

,

,

)

i.OOI

,

r )

i'll. '

Figure an

enlargement

side

shock

in

"microscopic" with

the the

10

.Top

of

the

on

in

the

i.27

the

left:

right

profile

on

!.It' '

hand

Fig.9.Observe

figure

Top of

i'll, '

their

and

compare

3.4.3.

right:

generalized

representation functions

KH"

H' (see 3.4.3);KH'~B ~ and r e p r e s e n t e d by t h e q u o t i e n t of areas

limited

and

the

situation close

H'K to

On

by

these

horizontal ~A

two

curves

axis;in 8

with

A

and

B is the this

=

1-B

0. the

two

different

that

there

left:the times: is

a

unique

same

shock

the

evidence

shock.

at

187

it+

1°'i9^ 0.22

I_ IJ

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

t .

.

.~2+

.

.

.

.

.

+

.

.

.

.

.

.

.

.

.

.

03 07 O~

03

Figure

11: Two other

of the p r o j e c t i l e

elastoplastic

is stronger

than

in fig.

shock 9).

waves

(the velocity

188

il0"l 'i 2.34

i. ~

4 .t;O

,~,04

l.d

3.48

¢,~ O,

2.~2

14: 1.30

!.24

2.0~ . . . . . . . . . . . . . 'l.OO

1.22

1,24

1.2~

1,2~

for

0.~3

2.~)

I~,?~

~.$~

.

0

.

1'~4

Figure Fig.9 ;

between

m

.

t.~

12 . here

and

0,5

.

,

1~

1:,

Same captions the value A

, )

1.3o

is

as in

189 Comments.

These

engineers, case

who

of

a

instance

plane

tail

of

the

tail

thus

the

concrete

have

certain

caused

a mountain)

by it

been

observed

phenomena

a

frontal

has

been

:

in

collision often

in t h e t a i l of t h e p l a n e c o u l d engineers r e l y in the elastic

plane

and tail

explain

by the (for

observed

survive. The precursors:

in t h e p l a n e w o u l d be d i v i d e d into a plastic shock precursor. W h e n the e l a s t i c precursor reaches the

precursor

the

precursors

to

accident with

passengers given by

shock wave an elastic

elastic

them

collision

that a few explanation the and

elastic

use

it

meets

then

the

escapes

situation

is

reflected

and

plastic

shock

the tail

from

puts

is

the

in

goes

dissociated effect

evidence

of the

back.

wave the

from

the

plastic

great

The

reflected

somewhere

before

front

part

shook.

practical

;

This

interest

of a g o o d u n d e r s t a n d i n g of the solutions of these equations involving "multiplications of d i s t r i b u t i o n s " An intensive study is p r e s e n t l y being done by engineers in specialized (military) research centers,In the next decade this study will enter into current

industrial A

Various Perrot

practice.

theoretical

numerical [7],

study

studies

Noussair

[1].

(physics)

are

reported

can in

be

found

in C o l o m b e a u

Colombeau-Le

[13].

Roux-Noussair-

APPENDIX

5

S E M I L I N E A R HYPERBOLIC SYSTEMS WITH IRREGULAR C O E F F I C I E N T S AND SYSTEMS OF EQUATIONS OF A C O U S T I C S

We perbolic

consider

system

in

the two

initial

MI ( x , t ) D t u ( x , t ) + M 2

(I) where

( x , t ) E ~ 2,

value

problem

u:~ 2 *

~

(x,t)Dxu(x,t)

the for

all

(2) for

following compact

the

semilinear

hy-

, M~

and

M~

-- F ( x , t , u ( x , t ) )

are

smooth

t h a t t h e s y s t e m is d i a g o n a l i z a b l e and the e x i s t f o r a l l t i m e , a n d F : ~z × ~n , ~n is satisfies

for

variables.

bounded subset

gradient K of

~2

n×n

condition there

matrices

such

characteristic curves a smooth function which

is

:

c>0

such

that

]a u F ( x , t , u ) I ~ c J all

(x,t)EK,

u=(uj)

6~ n j:l...,n

we

may

From the above assumption that restrict ourselves (by a l i n e a r

to

the

diagonalized

/

the s y s t e m is d i a g o n a l i z a b l e c h a n g e of u n k n o w n functions)

problem

:

ut (x , t) + A (x , t) Ux (x,t)

F(x,t,u(x,t))

(3) u(x,0)=u

0 (x) .

Let us assume t h a t A is a s m o o t h r e a l s u c h t h & t A or 8 ~ A is g l o b a l l y bounded that characteristic curves exist for extension of 1 . 4 . 2 o n e d e f i n e s F ( x , t , u ) FE~(~ compact is

an

n)

in

the

subset extension

THEOREM : solution of respectively.

For (3)

of of

variable ~z.

The

Theorem

u,

uniformly

following 3.5.1.

valued diagonal nxn matrix (this assumption implies all time). Using an easy if u6 (~s (~2)) n, provided for

theorem

(x,t)

ranging

(0berguggenberger

in a [4,5]

:

any u0E(~(~)) n there in the sense of equalities

is

a in

unique

uE ( ~ ( ~ ~)) n

~(~)

and

~(~) •

191

In many

the

points,

case

the

of existence of genberger [3]) . In the

or

even

the

of the

We satisfy,

in

estimates

:

several

whose

formal the

lie

of

the

~he

on

the

the

systems

they

shows

case

with

discontinuous

since

This

to

fluids

systems

are

statement

sense

results

immiscible

singularities

the

finitely

Ill , O b e r g u g -

hyperbolic

coefficients

see

results

(Kauch-Reed

of

at

particular

in

involve

need

of

coefficients

the

and

F

functions. that

which

the

terms

*

after

diagonal of

diagonalization coefficients

for

all

compact

W>0

such

our

~k ( ~ , x , t )

subset

system

AkE~s (~),

representatives

and

sup

(4)

(these

support

more

linear

distributions).

assume

in

m~de are

medium),

above

generalized

(3)

a medium

a

with

with

solution

acoustics

(in

multiplications of

distribution

distributions

of

[1,2,3,4]

extension

a

consistent

coefficients

functions

form

of

linear

discontinuities

are

is

distribution

case

of

irregular

Poir@e

~

the

systems

u 0 is

above

K

of

~

1 g the

there

has

the

~

n,

k

following

are

c>0

that

] ~ (~,x,t)I~
(x,~)EK

if *

0

for

< ~

<

all

compact

"0

differential are

c>0

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

< ~

<

imply so

R(~,x)

that

special

= I

the

k'ks a r e

: using

f (x+~;~)~(g)dg

special

elements

of

192

with

£ replaced

1 -i (log --) g

by

(for

any

~ E ~ 0 (~2))

piecewise continuous funotion f on ~ is G E $ ~ ( ~ 2) s a t i s f y i n g (8) (in p l a c e copic

we

get

that

for

(even any distribution) of A k) a n d h a v i n g f as

any

there macros-

~spect.

We generalized

assume function

F j E ~ s ( ~ n+~ )

that the nonlinearity F=(Fj) i~ ~ is a f r o m ~ n + 2 i n t o ~n such that each component

satisfies

representatives

the

fj ( g , x , t , y ) , i f

(el

following

* for

all

compact

and

W>0

such

sup

estimates

yz(y~) l~

~ E ~n

subset

K of

in

terms

of

:

~2

there

are

c>0

that

la~ fj(~,x,t,y) I~

o

(x,t)EK n yEm

if for

0<~<W all

compact

subset

differential operator there are c>0 and ~>0

K

of

~z

and

of o r d e r o n e such that

D a in

partial (x,t)

1

(7)

ida

sup (X

, t)

EK

f~ (~,x,t,y)I~< Y

o lo~ -

g

n yE~

if Of course these properties. F(x,t,u)=a(x,t) .u this

extension

extends

to

0<~<~

the smooth function F of (I) a b o v e s a t i s f i e s In linear acoustics one may have w h e r e a is a p i e c e w i s e continuous function; thus nonsmooth

F

is

really

needed.

The

theorem

of

(1)

:

THEOREM : ( Lafon [1]) - Under the conditions any initial data u0E(~Cs(~)) n, the system (3) solution uE(~ s (~2))n

(4),(5),(6),(7), admits a unique

for global

193

This

solution

0berguggenberger a

function

or

importance

to

a

a

generalized

that

,even

in

distribution.

check

physics this physicists.

The

is

[6,8]

that

in

abstract

simplest

case

and

linear

case,

course of

solution

system

{

Of

the

function

the

linear

at

xzxo~0

is

systems

has

of

it

the

it

is

it

of

shown

may

a

be

in not

particular

of

equations

of

aspect

expected

by

acoustics

is

(Poir6e

[1,2,3,4])

St+DoUx:O DoUt+Px:O

(8)

pzc~p s,u,p

where x=0,

so

is

c o are

discontinuous

density it

and

given

and

the

perturbed

lagrangian u might form

of

Ill],

lead

by

speed

acoustic

product to

at of

interpret

functions

the

abstract

having

represent,

in

medium,

the

; p,u

and x=0

assumed

and

pressure,

so

that

the

formal

[6,8] solution

exactly

the

has

x,

C®

p

at

are

term

the

system

(8)

proved

that,

s 0 u×

to

with in

a by

the before

acoustic . A

priori

appears

arguments,

expected

outside

rest

respectively

the

reduces

aspect

of

respectively,

distributions.Various

~s (~2) . 0 b e r g u g g e n b e r g e r system,

functions

(they waves)

velocity

discontinuous a

positive

x=0

sound

density,

be

strictly

at

in

Colombeau

equalities the

set

case of

in

of

this

classical

physicists

:

THEOREM : (a) existence : There exist generalized functions ~o C o E d s ( ~ ) , associated with s o and e o respectively, such that for any given s I .... ulx:× EC®(~) the system (8) (written with o

~0,c0,~,u,p) has sense of equalities

o

unique in ~

global solution (~2) (and ~ s (~)

(~,u,p)6 (~s ( ~ 2 ) ) ~ i n for the condition

the at

X=Xo). (b) (o,u,p)E

uniqueness

(~s (~2)) 3 w i t h

:

So,C o having

As

soon

the

macroscopic

as

(8)

has aspect

a

solution of

So,C o

(and also a physically realistic condition a t x=O e x p r e s s i n g that the representatives are bounded there, as g*O), then this solution (p,u,p) has the macroscopic aspect of a set (o,u,p) of elassical functions C® outside x=0, with u and p continuous at x=0 (and usually ~ discontinuous at x=0).

194

This last theorem reflects exactly the r e s u l t e x p e c t e d by physicists ; the k n o w l e d g e of the classical functions c0 ~ 0 a n d (p,u,p) Ix=x determines exactly the macroscopic aspect of the 0

solution ( p , u , p ) E (~s ( ~ ) ) 3 Further they satisfy the jump conditions w h i c h a r e the c o r r e c t o n e s f r o m the p h y s i c a l viewpoint. The important p o i n t is t h a t we h a v e o b t a i n e d these jump conditions as a c o n s e q u e n c e of the e q u a t i o n s written globally. The existence of a s o l u t i o n in the s e n s e of e q u a l i t i e s in ~s (~ 2) j u s t i f i e s formal calculations of p h y s i c i s t s (Poir@e [1,2,3,4]) . 0berguggenberger's proof uses exactly the explicit form of (8) . It would be interesting to obtain the same results for the other more complicated systems of acoustics (Poirge [i,2,3,4]) . A similar "concrete" result has been obtained in L a f o n Ill for an electron transport problem in a medium m a d e of the j u x t a p o s i t i o n of two different

metals.

Remark : System(8) c a n be t r a n s f o r m e d i n t o one w h i c h is c l a s s i c a l l y solvable in the s e t t i n g of the S o b o l e v s p a c e H ~ o c (~ 2) w h e r e one h a s existence and uniqueness of a s o l u t i o n ; u is c o n t i n u o u s f r o m the c h o i c e of the space. However this setting appears not to be adequate for linear acoustics in media with piecewise C® characteristics Using (acoustic) eulerian variables in eulerian coordinates, the s y s t e m (8) a p p e a r s in the f o r m ( P o i r g e [1,2,3,4])

I

(9)

pt+(P0u)= DoU~+Px=O

0

p=c~ st+u(s0) ~ = 0 p=c~ + ~os -

~.~×

(where s is the acoustic eulerian entropy, not mentioned in (8) because it is d i s s o c i a t e d f r o m the o t h e r e q u a t i o n s ) , w h e r e ~ is the acoustics displacement ; c ~ , ~ 0 a n d ~x a r e c h a r a c t e r i s t i c s of the fluid, discontinuous on the d i o p t r a . It is w e l l k n o w n f r o m physics that p and s have a singularity of the k i n d of a D i r a c m a s s on the dioptra at x=0. T h u s the t e r m s c~o a n d dos a p p e a r in the form of nontrivial products of d i s t r i b u t i o n s , even a posteriori. One cannot e x p e c t to s o l v e (9)in a classical setting s u c h as the one of the Sobolev spaces, a n d to j u s t i f y the p a s s a g e b e t w e e n (8) a n d (9).

195 The uniqueness to

derive

multidimensional result

an

[I] . See

in

dimensions

two

from

this

numerical last

(the

Laurens

lies

on

the

Lafon-Oberguggenberger

original

Perrot

extracted

in

case

paper

acoustic

method and

[I],

described

Laurens

diedra).

theoretical

The

[I]

which in for

has

existencebeen

numerical

following

figures

[I].

--7 ~o: Co:±

~o= Co= &

used

Barka-Colombeauresults are

il

tl

II

li

0 '!

197

.5

In the above visualization (isobaric curves) one considers a double diedra (3 media) of right angle (one represents only the upper part, to be completed by simmetry relative to the bissectrix of the diedra). An acoustic wave in the left medium moves towards the diedra and meets its edge at the time t = 0. The evolution of the phenomenom is shown at larger and larger times. There are successively reflection and refraction on both diedra. Finally one remarks a pressure peak, followed by a pressure hollow moving along the fwst dioptra, as well as a reflected wave (to the left) and a refracted one (to the right).

BIBLIOGRAPHIC

NOTES

---000---

Colombeau

These [5],

new generalized functions were t h e n in C o l o m b e a u [6,7] a n d in

introduced the book

in 1 9 8 1 in Colombeau

[2] , w r i t t e n in 1982. This introduction u s e d as b a s i c f r a m e w o r k differential calculus and holomorphy over locally convex spaces, see Colombeau Ill , [ 2 , c h a p t e r 5] and Colombeau [ 5 , 6 , 7 , 8 , 9 ] . It became rapidly c l e a r t h a t the c o n t r i b u t i o n of d i f f e r e n t i a l calculus over locally precisely the

convex original

spaces bounds

was to g i v e t h e in the d e f i n i t i o n

basic idea and more of @ ~ [ ~ ] a n d X [ ~ ] .

Already in 1982 (Aragona-Colombeau El] , Colombeau [2, 7]) , d i f f e r e n t i a b i l i t y over infinite dimensional spaces was a n d o n l y t h e g e n e r a l m e t h o d of a q u o t i e n t and the original

Chapter dropped

b o u n d s w e r e k e p t . T h e d e f e c t of t h i s m u c h m o r e e l e m e n t a r y presentaLion is that the original idea for the construcion of ~(~) is concealed. However this simplification allows the introduction of this theory to students with the knowledge differentiation and integration t h e o r y on ~n Appendix 7]) . differentiability

In

of Chap.l straight

to

get

nicer

of [3,

we return to s o m e k i n d of line segments (Gateaux

results

on

solutions

of

algebraic

The basic concepts of a s s o c i a t i o n was already introduced in first papers, Colombeau [ 5 , 6 , 7 ] , to s h o w t h e c o h e r e n c e between n e w a n d the c l a s s i c a l operations ( multiplication of c o n t i n u o u s

functions, coherence many other proved with

i0 on

differentiability) equations. the the

§

of s o m e r u d i m e n t s (see Colombeau

multiplication of a C ~ f u n c t i o n and a distribution). The with the multiplication of H i r a t a , 0gata, Mikusinski and authors (obtained with convolution by "~-sequences")was

directly

the

(Colombeau

multiplication

through a work multiplications

of is

[2], of

0berguggenberger

Hormander

Tysk [I] . proved in

and

[1,2])

Ambrose

Coherence

was

The coherence with Oberguggenberger

consistency problem with the solutions of the M o n g e - A m p @ r e is c o n s i d e r e d in O b e r g u g g e n b e r g e r [2] , i m p r o v i n g K i s e l m a n Jelinek

E l]

for

another

characterization

of distributions in t e r m s of L o j a s i e w i c z Chap.1 for a summing u p on m u l t i p l i c a t i o n s

of

the

Colombeau

obtained

many other [1,2] . The equation Ill . S e e multiplication

section. See Appendix of d i s t r i b u t i o n s .

of

199

The theory of generalized functions thus obtained is a n extension of the c l a s s i c a l analysis of C ® a n d h o l o m o r p h i c functions and has all its richness, see Colombeau [2,3], A r a g o n a [1,2] , AragonaColombeau [1,2], C o l o m b e a u - G a l ~ [1,2], Biagioni-Colombeau

[l, 2,3]. see

Connections with Nonstandard Todorov [1,2] , 0 b e r g u g g e n b e r g e r [ 9 ] . Although

riori

ascertain

this that

theory many

considered particular

by various we m a y q u o t e

limit

defining

(a) for

particularly

looks

of

its

Analysis

very

original,

ideas

and

method

of [1]

of for

a

(c) the numbers

studied,

can

a

poste-

were

already

of e a c h o t h e r . methods :

regularization

multiplications

0berguggenberger

large

one

authors, independently the f o l l o w i n g ideas and

the

being

methods

and

passage

distributions summing

(b) W a e l b r o e c k ' s a n a l y s i s in bornological vector spaces (see Colombeau Biagioni-Colombeau [3]).

infinitely

are

up

to

(see on

this

the more

method)

quotient of n o r m e d [3, A p p e n d i x 3]

idea of considering (Nonstandard Analysis).

In

infinitesimal

and and

and

(d) the i d e a of e m b e d d i n g the d i s t r i b u t i o n s into a more general setting : Rosinger's algebras containing the s p a c e of distributions, see Rosinger [1,2], see also KSnig [i] . B e f o r e Colombeau discovered the differential algebra ~(~), Rosinger constructed a general setting of c h a i n s of a l g e b r a s containing the s p a c e ~' (~) ; for a d e t a i l e d account of his e n c o m p a s s i n g theory see Rosinger [4] .It is a k i n d of g e n e r a l background of a l a r g e c l a s s of possible nonlinear theories of generalized solutions of partial differential equations. (e) derivatives,

so

as

the to

families

of

ordinary

Colombeau

[4] .

physicists

to

distinguish

physicists

to

obtain

(f)

(g)

method

replace

of

partial

differential

the

the

use the so

numerical

regularization

equations,

of

called

jump

in

Rosinger

laws

as

by

[3]

and

done

by

conditions.

"Renormalization

results

partial

equations

see

conservation

correct

of

differential

Quantum

Theory" Field

used

Theory.

by

200

(h) the c o n c e p t of " m e a s u r e v a l u e d s o l u t i o n s " of s y s t e m s of c o n s e r v a t i o n laws (Di P e r n a [I]) is indeed e q u i v a l e n t to our s o l u t i o n s in the a s s o c i a t i o n sense, see C o l o m b e a u - O b e r g u g g e n b e r g e r [3]. (i) a c o n c e p t of d i s c o n t i n u o u s s o l u t i o n s of s y s t e m s in ~onconservative f o r m i n t r o d u c e d r e c e n t l y by Dal M a s o - L e F l o c h - M u r a t [I] is a p a r t i c u l a r case of our c o n c e p t of s o l u t i o n , see C o l o m b e a u H e i b i g [I].

Some

But C o l o m b e a u ' s t h e o r y also p e r m i t s to o b t a i n new r e s u l t s . v e r y p r o m i s s i n g d o m a i n s of a p p l i c a t i o n s in full d e v e l o p m e n t s

are: (a) general results of existence-uniqueness for nonlinear partial differential equations with initial data distrib u t i o n s : C o l o m b e a u [3,10] , C o l o m b e a u - L a n g l a i s Ill , O b e r g u g g e n b e r g e r [3,4,5,] , Colombeau-Oberguggenbergerr2], L a n g l a i s [I] ;see § 3.5, §

3.6

and Appendix

5 of

Chap.3

(b) general results linear partial differential equations Oberguggenberger [8,7,8,] , L a f o n [i],

of existence-uniqueness for with irregular coefficients: Lafon-Oberguggenberger [l] .

(c) e x i s t e n c e of d i s c o n t i n u o u s s o l u t i o n s 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: C a u r e t - C o l o m b e a u - L e Roux [1,2], C o l o m b e a u Heibig

[1],

§3.2.

(d) p h y s i c a l a p p l i c a t i o n s are g i v e n in C o l o m b e a u Le Roux [1,4], C o l o m b e a u - L e R o u x - P e r r o t [I], C o l o m b e a u [11,13] Barka-Colombeau-Perrot [I]. N u m e r i c a l a p p l i c a t i o n s are g i v e n in C o l o m b e a u - L e R o u x [1,3], A d a m c z e w s k i - C o l o m b e a u - L e Roux [I], Co lombeau-Le Roux-Noussair-Perrot [I], C o l o m b e a u [14], and in the e n g i n e e r i n g w o r k s C a u r e t [I], A d a m c z e w s k i [I], N o u s s a i r [I], De Luca [I], L a u r e n s [I].

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Numerical

Schemes,

Partial Equa-

Pitman,

1982.

E.E.ROSINGER

[4]

- Generalized

Differential 1987.

Equations.

Solutions

of

North-Holland

Nonlinear

Partial

Math. Studies

144,

210

L.A.RUBEL

[I]

- Solutions

of a l g e b r a i c

J.Diff.Equations

L.A.RUBEL

[2]

- Some

rential

49

(1983)

research

equations.

differential

equations.

p.441-452.

problems

about

algebraic

T r a n s . A m e r . M a t h . Soc.

280

diffe-

(1983)

p.43-

52.

L.A.RUBEL

[3] - G e n e r a l i z e d

J.Diff.Equations

62,2(1986)

L.SCHWARTZ

[I]

- Theorie

L.SCHWARTZ

[2] - S e m i n a i r e

de P a r i s , Expose

R.T.SEELEY

[I]

[I]

[I]

- A survey

of

27

C=

Hermann,

la F a c u l t e aux

Paris,

des

derivees

functions

[I]

standard

1966.

Sciences

partielles.

a

no

(1988)

new

and

p.625-626.

methods

conservation

laws.

contexto

fungSes

for J.

of m u l t i p l y i n g

93,1

(1985)

generalized

Generalized

their

p.23-27.

das

1985.

In:"B.Stankovic,

(editors)

Structures

in a h a l f

difference

Sao P a u l o ,

two m e t h o d s

Colombeau's

defined

p.I-31.

P r o c . A m e r . M a t h . Soc.

Analysis.

(1964)

hyperbolic

Thesis,

of

15

finite

- A equacao

V.S.Vladimirov

New York

several

(1978)

- Comparison

gence

of

of n o n l i n e a r

s I L V A [I]

butions.

T.D.TODOROV

de

Equations

P r o c . A m e r . M a t h . Soc.

generalizadas.

J.TYSK

Schwartz

- Extension

systems

DA

equations.

p.242-251.

Distributions,

1954/1955.

Comput.Phys.

J.TOLEDO

des

of a l g e b r a i c

n97.

space.

G.A.SOD

solutions

distri-

p.35-39.

functions E.Pap,

and

S.Pilipovic,

Functions,

Applications.

Non-

Plenum

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211

T.D.TODOROV

[2]

- Sequential

generalized tional

H.WHITNEY

[I]

Trans.

approach

functions,

Center

for

- Analytical Amer.Math.

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to C o l o m b e a u ' s

36

IC/87/126.

Physics,

of

(1934)

theory

of

Interna-

Trieste.

differentiable p.63-89.

functions.

ALPHABETICAL

Algebraic

INDEX

differential

equation,63

Associated complex number, Association,

36

Antidiffusion method, Acoustics,

Jump formulas,

114,194

Lax-Friedrichs

scheme,155

34 Lax-Wendroff

149,160

133,190

scheme,

157,162 Linear hyperbolic

Borel's Coherent

theorem,

75

191,193

family,

22

Macroscopic

Composition,

23

Conservative

systems,85,106,154

Dirac generalized

functions,

Elasticity,

systems,

aspect,

49

Mie Gruneisen law,

113

Numerical

89,148,

schemes,

157,175

52

Numerical 83,86,114,118,142

Elastoplasticity,

155,

tests,

150,166,176,

195 Nonconservative

~19.142

systems,

83,

87,109,114,148 Fluid dynamics,

Nonlinear parabolic equation,

157

133 Generalized complex numbers,

31

Generalized functions on open

Pointvalues,

sets,

Parabolic

16

Generalized

32

equation,133

functions on arbitrary

sets, 69

Real valued generalized

Godunov scheme,

functions,

158

16

Restriction of generalized Heaviside generalized functions, Hydrodynamics, Hyperbolic

49

systems,

functions

121,190

subspaces,49 Richtmeyer

Integral on compact

sets, 33

Initial value problems, ~90

to

open sets, 21

106

94,121,133,

scheme,163

Rusanov scheme,163 Rankine-Hugoniot 86,88,114

conditions,

213

Support

of

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

a generalized

functions,

21

Semilinear

hyperbolic

17

Cf (~) . . . . . . . . . . . . . . . . 19 systems,

121,190 Seeley's

c ° (x)

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

~o

c~

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

71

theorem,80

Simplified

generalized

functions,

45

(x)

c" (~(fO)

Shock

waves,

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

C'(~"

Topology on

~ ,40

on

~(~) , 4 2 , 4 4

(0)) ..............

on

2

1

@ (Q)

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

@, ~

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

65

D~ ...................

Uniform

2

84,189,114,148

t5

structure ~

~(~(~))

, 42

on ~ ( ~ ) ,

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

4

44 * (U) .................

Wave

front

set,

Weak

convergence

Wild

constants,

66

~9(~)

on

,

44 ~[~]

Whitney's

problem,

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

15

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

58

55 theorem,

extension

Wellposedness

5

of

74

~i[~]

Cauchy

the

g[~,Em]

132

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

23

@M . . . . . . . . . . . . . . . . . . . #M ( ~ ( f l ) )

SYMBOLS

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

gM(~(~)) ,gq .....................

31

.

.

.

.

.

.

3

.

.

.

.

.

.

.

~ q (E n) . . . . . . . . . . . . . . . . .

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

16

~[~]

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

59

11

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

.

.

.

.

.

.

.

.

.

.

31

.

.

.

.

16

45

"~ . . . . . . . . . . . . . . . . . . . . . . .

-~s

4

gM[~]

@M,~[~ ] ...............

i (E~)

~q,

.

3

.

.

.

.

.

.

.

48

gM[~,Em]

~

, s

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

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

23

48

214

,~,

s[~]

@M , ~

[~]

Ws[fl ] ...................

46

72

~ ' w , ~[X] . . . . . . . . . . . . . . . . .

73

~

27

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

46

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

69

~'~,w, siX]

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

~[~]

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

46

g~[~]

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

69

(En)

(mEn) Q.

E n ......................

~(~)

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

• (~,E,~)

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

~' ) .................

~(~,

6

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

~4

~

49

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

I~ . . . . . . . . . . . . . . . . . . . . .

f ............................

16

3

~o

,x ..........................

~

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

13

II

II ..........................

4 l

15

47

16

~[~]

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

58

~=[~]

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

16

] ................

NOTATIONS

70

•~ C f l ] . . . . . . . . . . . . . . . . . . .

~[fl,Em

65

17

Qn .....................

42

z x ...........................

U [l~,k

~

40

65

OTHER

59

~

6

~4

.~ (~) .................. (0) ..................

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

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

Q~,k,~

4,16

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

24

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

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

43

ii