Abstract Non Linear Wave Equations

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

507

Michael Reed

Abstract Non Linear Wave Equations

Springer-Verlag Berlin. Heidelberg. New York 197 6

Author Michael C. Reed Department of Mathematics Duke University Durham, North Carolina 27706 USA

Library of Congress Cataloging in Publieation Data

Reed, Michael. Abstract non-linear wave equations. (Lecture notes in mathematics ; 507) Based on lectures delivered at the Zentz~m f~r interdisziplin~me Forsehung in 1975. Bibliography: p. 1. Wave equation. I. Title. II. Series : Lecture notes in mathematics (Berlin) ; 507. QA3.La8 no. 507 [QCI74.26.W3] 510'.8s [530.1'24] 76-2551

AMS Subject Classifications (1970): 35L60, 47H15 ISBN 3-540-07617-4 ISBN 0-387-07617-4

Springer-Verlag Berlin 9 Heidelberg 9 New York Springer-Verlag New Y o r k . Heidelberg 9 Berlin

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 9 by Springer-Verlag Berlin 9 Heidelberg 1976 Printed in Germany Printing and binding: Beltz, Offsetdruck, Hemsbach/Bergstr.

Preface

These notes cover a set of eighteen lectures delivered at th e Zentrum f~r interdisziplin~re Forschung of the University of Bielefeld in the summer of 1975 as part of the year long project "Mathematical Problems of Quantum Dynamics".

It is a pleasure to thank the Zentrum

for the opportunity to give these lectures and the Physics faculty of the University of Bielefeld for their warmth and hospitality. people deserve special %hanks:L. C. Pfister,

Streit,

Three

for extending the invitation,

for help in the preparation of the man~scrlpt,

and M. K~mper

for her excellent typing.

Mike Reed Bielefeld, August,

1975

Table

Introduction

Chapter

1

I. L o c a l

of Contents

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

Existence

and global

and properties

existence

1

of s o l u t i o n s

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

5

2. A p p l i c a t i o n s : A.

utt

-

B.

The

case

C.

Other

D.

The

E.

An example

F.

The

~u + m 2 u

p,

5. W e a k

Chapter

7.

2

8. S c a t t e r i n g

9. G l o b a l

.

.

.

.

.

.

.

.

.

II

equation global

. . . . . . . . . . . . . . . . . . existence

fails

and K l e i n - G o r d o n

. . . . . . . . . . .

equations

. . . . . . . .

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

speed

and continuous

25 29 33

40

dependence 49

56

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

64

theory

of scattering

for s m a l l

existence

10.Existence

3 .

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

Scattering

Formulation

=

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

solutions

6. D i s c u s s i o n

D

20

Dirac

propagation

,

A . . . . . . . . . . . . . . . . . . . . . .

of solutions

on the data

3

=

19

where

coupled

n

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

sine-Gordon

3. S m o o t h n e s s

4. F i n i t e

m = o n and

,

= -~lulP-lu

data

for s m a l l

of t h e W a v e

problems

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

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

67

71

data

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

88

operators

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

90

Vi

II.

Applications: A.

The

non-linear

B.

utt

- Uxx

C.

utt

- Au

D.

The

coupled

12.

Asymptotic

13.

Discussion

Bibliography

Schr~dinger

+ m2u + m2u

= =

lu p Au p

Dirac

completeness

, n = , n =

and

equation I 3

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

~94

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

96

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

102

Kleln-Gordon

equations

. . . . .

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

105

IIO

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

121

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

125

Introduction

During the past year there has been a great deal of interest, both in applied physics and in q u a n t u m field theory, equations.

Like all n o n - l i n e a r problems,

in n o n - l i n e a r w a v e

these equations must to some

extent be dealt w i t h individually because each equation has its own special properties. treated separately;

Thus,

in the literature these equations are often

the proofs of existence and properties of solutions

often seem to depend on special properties of the p a r t i c u l a r equation studied.

In fact, these equations have in common certain basic problems

in abstract n o n - l i n e a r

functional analysis. Using just the standard

tools of linear functional analysis and the c o n t r a c t i o n m a p p i n g principle, one can go q u i t e far on the abstract level, thus p r o v i d i n g a unified approach to these n o n - l i n e a r equations.

Furthermore,

the ab-

stract a p p r o a c h makes it clear which p r o p e r t i e s of the solutions are general and w h i c h depend on special p r o p e r t i e s of the equations themselves.

The abstract approach, which o r i g i n a t e d in the w o r k of Segal

[~|], has been n e g l e c t e d p a r t i a l l y because one can always push further in a p a r t i c u l a r case using special properties.

Nevertheless,

the ab-

stract methods and ideas form the core of much of the recent work, although this is somewhat obscured

in the literature,

Furthermore,

there are m a @ y b e a u t i f u l and important unsolved p r o b l e m s both on the abstract and the p a r t i c u l a r

level.

For all of these reasons,

it seems

an a p p r o p r i a t e time to pull together what is known about the abstract theory and how it is applied. To see how the abstract q u e s t i o n s arise n a t u r a l l y from an example, consider the n o n - l i n e a r K l e i n - G o r d o n equation:

(I)

utt(x,t)

Basically,

- Au(x,t)

+ m2u(x,t)

= - glu(x,t) IP-lu(x,t)

u(x,o)

= f(x)

ut(x,o)

= g(x)

x ~ Rn

one w o u l d like information about local e x i s t e n c e of solutions,

global existence,

smoothness,

finite p r o p a g a t i o n speed, continuous de-

p e n d e n c e on the initial data and e v e n t u a l l y we want a s c a t t e r i n g theory, As we will see, the results and the t e c h n i q u e s depend c r i t i c a l l y on the size of p and n and the sign of g.

To treat

(i) in a general setting

we r e f o r m u l a t e it as a first order system as follows:

-

Let

v(x,t)

2

-

= ut(x,t) , t h e n v t - du 9 m 2 u = - g l u l P - l u

(2)

ut = v

Now,

for e a c h

(3)

u(x,o)

= f(x)

v(x,o)

= g(x)

t, d e f i n e

~(t)

=
v(x,t)>

we

can r e w r i t e

(2) as

Then, (4)

#' (t) -

J(~(t))

(0 ~-m 2

Io)~(t)

~(0)

The

operator

with

domain

where

^ always

que p o s i t i v e calculus. space

(
the

transform.

inner

product

W B = D(B)

,
, v>

root

)B =

(Bu

of

positive (Bu

We d e n o t e

-4 + m 2 g { v e n and

closed,

, BU)L2

~ L 2 ( R n)

on

L 2 ( R n)

with

, BU)L2

+

We can the

( v

inner

by B the uniby the

functional

D(B)

is a H i l b e r t

thus

define

the

Droduct

, v )L 2

let

Then, D(A)

the F o u r i e r square

operator

so t h a t

(k) G L 2 (R n)

B is s t r i c t l y

>

=

self-adjoint

f G L 2 ( R n)

- glulP-lu

= J(~(t))

(k 2 + m 2) f

self-adjoint

space

, v>

by t h o s e

denotes

Since

under

Hilbert

is a p o s i t i v e

- A + m2 given

= <0,

one

can

check

= D(B 2) ~ ) D ( B )

that

A is a s y m m e t r i c

and A is c l o s e d

since

operator B and B 2

on

~B

are

with closed.

domain

-

3

-

Now define for each t, W(t)

=

B-isin(tB) 1

Ic~

\ -B

sin (tB)

cos(tB)

where each of the entries is defined by the functional calculus for B

on L~(Rn).

One can now compute directly that

continuous o n e - p a r a m e t e r group and that for tive of itself,

W(t)~

exists and equals

- iA~.

a corollary of Stone's t h e o r e m

self-adjoint in D(A)

and generates

our original p r o b l e m

(i) as follows.

function

W(t)

~ fD(A) Since

the strong deriva-

W(t)

(see~,p.26~)

W(t).

is a strongly

takes

D(A)

into

says that A is

We have thus reformulated

We must find an

~B

- valued

@(-) on R w h i c h satisfies:

d~(t) dt

r

= _ iAr

=

r

+ J(@(t))

-

w h e r e the self-adjoint operator A is given by This shows that

(5) and J is given by

(3).

(I) is really a special case of a very general class

of Hilbert space problems. Hilbert space ~ , a vector

Namely, given a self-adjoint operator on a r in~ , and a n o n - l i n e a r m a p p i n g J of ~

into itself, when can one find an

~-valued

function

~(-) on R w h i c h

solves the initial-value problem:

(6)

@' (t) = - iAr

r

+ J(r

= r

It is this abstract p r o b l e m w h i c h is the main subject of these lectures, but the main m o t i v a t i o n classical properties equations.

for studying the abstract problem is to prove

(existence, smoothness,

etc.)

of n o n - l i n e a r wave

So, we will always return to the equation

n o n - l i n e a r equations like the s~ne-Gordon equation, K l e i n - G o r d o n equations,

the n o n - l i n e a r

(i) and to other

the coupled Dirac-

S c h r ~ d i n g e r equation.

Our treat-

ment of applications is uneven because there is no attempt to be complete - the applications are used to show the d i f f e r e n t ways the abstract theory can be applied.

Thus we sometimes provide most of the details

and sometimes just sketch the important features or say how one appli-

-

cation d i f f e r s

from other.

cations w h i c h we don't listed

The m a t e r i a l

of the

Here the a b s t r a c t

problems,

well

either

ding the t h e o r y

lectures

In C h a p t e r

theory

understood.

there

but w h i c h

are many b e a u t i f u l can be found

falls n a t u r a l l y theory

appli-

in the DaDers

is quite

complete

2 we d e v e l o p

various

roughly means c o m p a r i n g

and the method

applications

to cover new applications. clear

In

of solutions

of

of

there are many u n s o l v e d

of s p e c i f i c

will become

into two parts.

and p r o p e r t i e s

Nevertheless,

in the details

are very d i f f i c u l t

(6) w h i c h "free"

Furthermore,

even m e n t i o n

I we treat the e x i s t e n c e

application

which

-

in the bibliography.

Chapter (6).

4

or in exten-

T h e s e problems,

some of

as we proceed.

aspects

solutions

of a s c a t t e r i n g of

theory

(6) to solutions

for

of the

equation

~' (t) = - iA~(t)

(o) = ~o for large p o s i t i v e

and n e g a t i v e

times.

Here the a b s t r a c t

satisfactory

in that many more h y p o t h e s e s

applications

are

less well

understood,

required

in order to have a c o m p l e t e

interest

in these many u n s o l v e d

main purpose.

is less

on A and J are required,

and more a b s t r a c t

theory.

problems

theory

then

results

the

are

If I can arouse your I will

have

achieved

my

Chapter

i

E x i s t e n c e and P r o p e r t i e s of Solutions

I. Local and Global E x i s t e n c e In this section we prove a local e x i s t e n c e t h e o r e m for give various applications.

existence of solutions of o r d i n a r y d i f f e r e n t i a l equations. late

(6) and then

The basic idea is the same as the proof of We reformu-

(6) as an integral equation problem:

(7)

f~

~(t) = e-iAt~o +

and then solve

Theorem 1

e-iA(t-s)J(~(s))ds

(7) by the contraction m a p p i n g principle.

(local existence).

Let A be a self-adjoint operator on a Hil-

bert space ~ and J a m a p p i n q from D(A) to D(A) w h i c h satisfies:

(Ho) (H i )

TIAJC~)II ~ c ( I T ~ ] I ,

(HLo)

llJr

(H~)

llA(Jr

- J(~)ll < c ( l l ~ t l ,

I1~1t)11~- ~11

J(~))ll <_c(ll~li,

for all # , ~ D ( A ) where

I IA~LI~IIA~T!

finite)

IIA~II, f l a i l ,

w h e r e each constant~C

IIA~II)

is a m o n o t o n e

function of the norms indicated. Then,

there is a T > O so that solution for tE[O,T).

IIA~-A~II

increasing

(every-

for each ~o6D(A)

(6) has a unique c o n t i n u o u s l y d i f f e r e n t i a b l e

For each set of the form

T can be chosen u n i f o r m l y for all #o in the set. Proof. Let X(T) be the set of D(A) ~(t) and A~(t)

-valued functions on

are continuous and

[I*( )I[ T = sup 11~(t) In + sup

l lA~(t)[l

<

[O,T)

for w h i c h

-

Since A is a closed operator, Banach

space.

Choose

X(T,u,~ O) consist

-

X(T) with the norm

some fixed ~ > O.

of those

II~( ) - eiAt ~ollT ~ ~(8)

6

~( ) in X(T) with

~(0) = #o and

+ ;~ e -iA(t-s)

J(~(s))ds

is a contraction

on X ( T , u , ~ ) if T is small enough.

of the constants

in the hypotheses that

with arguments

~(. ) s

IIJ(~(s+h))

-J(~(s))

- e-iA(t-s)J(~(s))

and

II

II + II(e -iAh -I) J(~(s)) II

is a continuous

proof shows that Ae-iA(t-s)d(9(s)) hand side of

+ u

-~(s)l] + l](e -tAb-z) J(~(s))il

_< %ll~(s+h)

so e-iA(t-s)J(~(s))

We denote by C~ any II#o[J

then

I le-iA(t-(s+h))J(~(s+h))

<

(8) can be defined

n n n (t)

~-valued

function

is also continuous.

using the Riemann

of s. A similaE Thus,

integral,

the rightand if

~l_-i (t- (m/n) t) A Jf~(~m t ))

--

m= 1 and n(t)

then qn(t)--~n(t) ,n(t)~D(A),

-= st e -i(t-s)A

as n - - ~

J(~(s))ds

. Now, by the hypotheses

on J, each

so

n Ann(t)

=

I -i(t-(m/n) t)A A; (~ ( ~i~ ,mt )) E~

;

m=l

--~

is a

be given and let

We will show that the map

(S~) (t) = e-iAt#o

I IA~oi I + u . Suppose

II~( ) J]T

Let ~o 4 D(A)

t o

e

- i,(t-s) A

AJ(#(S) )ds

-

Therefore,

(9)

n(t)~D(A)

7

-

and

A/te -i(t-s)A J(~(s))ds ~o

= Ite -i(t-s)A AJ(#(s))ds ao

Further, I IA~(t+h)

- A~(t) I I <

[ t+h e -iA(t-s)

e-iAhAj(~(s))ds

1

Jt +

[t(e-ihA -i) e-iA(t-s) ~o

<_ hC~II~IIT

+

Aj(4)(s))dsll

I( e-ihA -I)AJ(#(s))Ilds

The integrand in the second term converges to zero as h-90 for each s and by the hypotheses on J, the integrand is uniformly bounded. Thus, by the dominated convergence theorem the right-hand side converges to zero as h--90, so An(t) is continuous and similarly, n(t) is continuous. Further, exactly the same kind of estimates ~(.) and ~(.)~X(T,O,~o) , we have

l l(s~)(t)

- e-iA%oll

< C T sup IIr --

I IA(S~) (t) - Ae-iAt~oll

as above show that for any

~

t

~

II

[o,T)

<_ CoT sup I IA~ (t) I i t 9 [o,T)

II (S%) (t) - (S~) (t) ll <_ CoT sup I Ir t ~[o,T)

- ~(t) ll

l l~(~s~) (t) - (s~) (t)) i l _< C T sup liAr(t) - A~(t) ll tG [o,T)

Th~s, for T small enough, S is a contraction on X(T,o,~0) so, by the ~ontraction mapping principle (see[~&],p,|5|), S has a unique fixed point 4(.) in X(T,O,~o) which satisfies (7). Now, suppose that ~ is a continuoulsy differentiable D(A) -valued solution of (6) on the inter-

-

val

[0,@) w i t h

tinuous, obeys

if

(7)

T

so

9(0)

= #o"

=

~(t)

for

t

< T, l i m t § T,~(t)

-

By the d i f f e r e n t i a l

~(t) 9 X ( T o , ~ , ~ O)

~(t)

8

for t in some

< TO .

Let

T

= ~(T ) and

1

I

equation,

interval be

the

sun

limt+ T,A~(t)

A~t)

[O,To). of

such

= A~(T

1

~(T ) = ~(T ) G D(A) 1

is conSince TO.

Then,

) exist

so

!

and the same a r g u m e n t

as a b o v e

shows that

~(t)

= ~(t)

1

for some small

~nterval

T

< t < T < T which contradicts 1 -2 reality of T . T h u s T > T, so ~(t) = ~(t) for t E [O,T). 1 1 -any s t r o n g s o l u t i o n of (6) on [O,T) e q u a l s ~(t). To p r o v e the s t r o n g d i f f e r e n t i a b i l i t y

(t+h)

- ~ (t) =

( e - i A h -I ) e - i A t ~ o

h

0oED(A)

and s i n c e t h e to J ( # ( t ) ) .

is,

of ~(t), we w r i t e

+ ~i Jt It+h e - i A ( t - S ) e - i h A j ( ~ ( s ) ) d s

h

the first t e r m c o n v e r g e s

integrand

-ihA II e

of the t h i r d t e r m c o n v e r g e s

as h - - 4 0 converges

to

for e a c h s and

-I

h

so the i n t e g r a n d gence theorem,

to - i A e - i A t ~ ~

of the s e c o n d t e r m is c o n t i n u o u s , i t

The integrand

e - i A ( t-s) ( - i A J ( ~ ( s ) ) )

is u n i f o r m l y

the third

I

bounded.

term converges

te-iA(t-s) o

~which b y

That

h

-0

Since

the m a x i -

Thus,

by the d o m i n a t e d

as h--~O to

( - iAJ(~(s)))ds

(91 e q u a l s - i A [ t e -iA(t-s) -o

J(~(s))ds

conver-

-

Therefore

#(t)

Corollar~ 1

-

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

for

t~[O,T)

Let A be a self-ad~oint o p e r a t o r o n e ,

and n o n - l i n e a r m a p p i n g J takes ~ for each #o e ~ lution ~(t) on

Proof

9

and satisfies(6).|

and suppose that

into itself and satisfies

' there is a T > o such that

(H~). Then

(7) has a continuous so-

[o,T).T can be chosen u n i f o r m l y on balls

in ~ .

This is really a c o r o l l a r y of the proof of T h e o r e m i. The proof is

similar, only much easier since we don't have to d i f f e r e n t i a t e the integral equation.

Just define

X(T,~,~ o) = { ~ (t) I supl I $ (t) - e-iAt~ol I ! ~, ~ (o) = ~o } t~ [o,T) and proceed as before. ~ We remarkT that the hypotheses

(Ho) and

(H I ) follow from h y p o t h e s e s

T

(HE) and

(H~). We state them separately

for easy comparison with the

hypotheses T h e o r e m 2 below. Notice also that the same proof shows local existence on an interval

(-T,T)

since e -iAt is a group.

It is w e l l known from freshman calculus that n o n - l i n e a r ordinary d i f f e r e n t i a l equations may not have global solutions in time (for exdx 2 ample: ~ = x ). As in that case, one can prove global e x i s t e n c e if one has as apriori estimate which guarantees that the solution remains bounded.

T h e o r e m 2 (global existence) T h e o r e m 1 except that

Let A, J, and ~

(H I ) is r e p l a c e d by

(i.e. C does not depend on

I IA~I I ). Suppose that on every finite in-

terval on w h i c h a strong solution of Then

satisfy the hyDotheses of

(6) exists,

I l~(t) I I is bounded.

(6) has a unique global solution for all time.

Proof:

By T h e o r e m 1 we know that a solution exists on

T > o small enough. solution of

for some

Let T be the s u p r e m u m of the numbers T so that a

(6) exists on

that ~ < ~ and

[o,T)

[o,T) with ~(t)

I IA~(t) I I is bounded on

and A~(t)

continuous.

SupDose

[o,T). Now in the local existence

proof in T h e o r e m 1 the length of the interval of e x i s t e n c e d e p e n d e d only on the numbers

ll#ol I , I IA~ol I. Therefore,

if we choose a point T 1 close

-

iO

-

enough to 5 we can construct a solution of T 1 < 5 < T 2. solution on

(6) on IT1, T 2) where

By local uniqueness this solution extends our previous [o,T)

and thus violates the m a x i m a l i t y of 5.

= ~ , so the solution exists on that it exists on the interval

It remains to show that

[o,~)

and a similar proof shows

( -~,o].

I IA~(t) I I is apriori bounded on any finite

interval where the solution exists. The solution of on any such interval,

I IA~(t) If <

Therefore

(6) satisfies

(7)

so

I IA~oll +

I

t Ie_iA (t_s) I AJ(~(s)) lids

o

<_ r l~,orr +

by

(Hi). By hypothesis,

o

c(ll,cslrl~

I FA,(s) Ids

I l~(s) I I is bounded on any finite interval

w h e r e the solution exists, say by CI(T). Since C(-)

[o,T)

is locally bounded

we have

r1A~(t) ll <_ I IA~oll + C(Cl(T)~

f

t

o

IIA~(s) llds

for all t < T. By iteration this implies that

l i A r ( t ) II ~ I I A ~ o l l e t C ( C l ( T ) )

w h i c h shows t h a t

J l A ~ ( t ) ll

The original

i s b o u n d e d on [ o , T )

idea of using this abstract formulation to prove

e x i s t e n c e for non linear w a v e equations I and 2 and also the theorems his ideas

if T is finite.

is due to I.Segal~l]. T h e o r e m s

in Section 3 are simplified v e r s i o n s of

(see also R e e d [ ~ ] a n d Reed and Simon[~6]).

This is further dis-

cussed in the notes to Section 3. Other h y p o t h e s e s on J and A guaranteeing local and global existence can be found in Browder [~]. Wahl

[40~ treats the case where A depends on t.

W. yon

-

I i -

2. A p p l i c a t i q n s

In t h i s Usually, estimates general,

Finally,

and sometimes

one passes

Therefore,

first

that

usual

calculations.

nature,

In

vectors

of A b y a l i m i t i n g

for

argument.

functions

inequalities.

some extra

Although

it is i m p o r t a n t

we will do most

the requisite

many

of

to under-

and

Sobolev

say that

of t h e t e d i o u s

estimate.

After

the technical

tech-

that we

details

are

example.

we restate

I u,v~L2c~'~,

ll'II

on Sobolev

set of n i c e

vector-valued

example

estimates

B = ~- A + m 2. F o r

Notice

may be applied.

through.

-__~u + m 2 u = - l l u l P - l u

{

where

the

to the

known

convenience

~=

full d o m a i n with

tedious

on a d e n s e

to u s e e n e r g y

first

and prove

just quote

utt

them

in o u r

nical details

For

to the

theorems

, i = O,l,depends

long and rather

are o f a " t e c h n i c a l "

stand how to carry

(A)

on

(H i ) , (H~)

are v e r i f i e d

are necessary

details

similar

show how the existence

since we are dealing

arguments

will

we

hypotheses

the estimates

A and then

these

section

proving

always

,

I > o,

the basic

m > o,

n = 3,

definitions:

11ll 2 ~ llBull~2 § iivll2 % = e~, denotes

we define

the norm on~

L p n o r m o n R 3 . As p o i n t e d

D = 3

out

in t h e

J(~)

and

< | }

=

If-If D w i l l d e n o t e

introduction,

Io I) A = i

is s e l f - a d j o i n t

Sobolev

f

~

by

:

{ ~

of t h e e s t i m a t e s

simple

o

on

D(A)

All

-B 2

in t h i s

inequality.

and various

I u(D(B2),

example

We denote

universal

v~ D(B)}

are based

the Fourier

constants

will

on t h e

transform

be denoted

following of a function b y K.

-

Lemma

I

Let

Proof: Denote calculus,

uEC~

(R').

Then

~u(x)/~x i by

12 -

Ilull,

~ KIIBulI,

~i u. Then by the fundamental theorem of

lucx) l ~ ~ 411u'aiuldxi where the integral is taken over the llne where x. is held fixed for 3 j # i. Thus,

,o,x,,~_,~Cf,u,~u,,~ >" ([,u,~u,,x )~' r

u,,~ ) "

By integrating both sides (by iterating the integrals) and using the Schwarz inequality, one obtains

fR, lUt~O x --<

K(fR. lU.~.uldx )~= ([R.!u., uld ~ ).~ C/R. fU~ utdx R3

R3{%X u]=dx )

From this one easily obtalns

s,[Ul'dx

_< Kr

= KClIklQ]I,

,

+ Ilk

= ~llBull, |

all , * Ilk

,

QII , ~

)lr~

-

1 3 -

To e~tend this e s t i m a t e to D(B),we need:

Lemma 2: B is e s s e n t i a l l y s e l f - a d j o i n t on C o

Proof:

(R ~).

By using the Fourier t r a n s f o r m it is clear that both B and B 2

are e s s e n t i a l l y self-adjoint on the Schwarz s p a c e ~ ( R n ) .

Now let u s

n)

be given.

Since C~(R 3) is dense in ~ ( R n) (in the ~ ( R n) topology) we o LZ L2 can find UnEC (R S) so that u n , * u and B2u n ) B2u. But,

llB(u n - u) ll ! I I ( B~

+

! ]IB2(Un

I)(u n

-

u) ll

-

u) lT + flun

-

urJ

) o

T h e r e f o r e B r e s t r i c t e d t o ~ ( R n) is in the closure of B r e s t r i c t e d to C~

(Rn). Thus, B is e s s e n t i a l l y self-adjoint on C~(R').

Lemma 3

Suppose that u

!

, u , u ~D(B). 2 3

I

Then

lluu=u 11 2 -<- KrlSu, rl 2 11Su 11 2 lIBu 11 2 1 Pr___ooof Let u s

Since B is e s s e n t i a l l y self-adjoint ~n Co(R3), we

can find a sequence of C ~ ( R 3) functions u n so that Un L _ ~ u, and L2 Bu n > Bu, and by passing to a subsequence if necessary, we may assume u n converges p o i n t w i s e to u also.

I lu~

-

u m~

II 2

=

II

(u~ + UnU m +

(u n

-

u m)

! Kllun

-

umll61](u~

<

--

But

um) ll

+ UnU m

+

2

Um)]l 3

~IIu n- Umll (llUnll 26 + llUnll 6

_< K [ [ B U n

_ BUmll2(lIBUnll,

2 +

ilBunl

I[Uml[

6

+ llUm[l 2) 6

I 2 iiBUml

[~ +

TlBUml]~) ,

-

so {u~} u3eL~.

is C a u c h y

in L 2 a n d s i n c e

it c o n v e r g e s

pointwise

to u 3 we h a v e

T a k i n g t h e l i m i t in t h e i n e q u a l i t y w e o b t a i n

IIull6 ' = The statement twice.

Lemma

For all @,,r

4

flu'If ~ _< K I I B u I I '

of the lemma n o w f o l l o w s by a p p l y i n g

equality

~,

I IJ(~,,) Proof:

Let

(by the c a l c u l a t i o n

- J(,~)ll

ll~u~

Kilr

u

l

p r o v e s the lemma.

Lemma

5

1

II 2

3,

< ~llBu 1 I

--

' < Kl1~

2 --

1

113

3)

I II = t l ~ , ( u,~ , - u 2~ 2 2 711 2

>11 2 (~lBu ; ] 2~ +

- r II(II~

which

Let @ ,r i 2

2

in-

_< c(ll~,

by L e m m a

in L e m m a

I IJ(r ) - J(r

< K]/B(u-

H61der's

J satisfies

~i = < m i ' v i > ' T h e n '

llJ(~,)II = and

14-

B~,il

2

IIDu 2 II 2 + It~u~ll 2~)

II ~ + II~ II l l ~ ' J

+ IIr

~)

, then

(11)

IIACJ(~) - J ( ~ ) ) I I _ < Proof

Let

r

ccII~, I , I I % I

=

where

,IIA~II,IIA~

uiGD(B2),

viED(B).

II)IIA~-A(hll We c o m p u t e

- 15-

IIBaiull

so, by Lemma

]]ai(u~)

~ = II ( [ k [

+ m ~)

2 = l{B2ul12

kiQ{{~

3,

]} ~ =

12uuai~ + uaai~][ ~ _< KllBull=llBaiul[2

Thus,

~ KIIBul[211B2uil ~ 2

3

§ ~m2ttu:%tI: 2

<

--

which

proves

II~i(u:U

K(llSu

the first

- u~U )ll:

1

~= I

{l~{{B2u, tl ~2 + m~llBu 2

inequality.

l{ ~) 6

To prove the second,

:. Ilu:~i(~ ~ - u~)lI:

we compute:

§ t t (u~ - u:)ai(%)t

i~

+ {12(tu ,.{" - lu t~) aiu, ll ~, .,- {121u l'ai(u,. -- u2)l[=

'{{s'(u

< K(I{Bu --

1

- u ){

2

1

2riB(u-

_< K(II~,I ~{IA(r

+ {IA~ll~(lIr

2

u ) { I ~ ]B~(u

- u ) l { ~)

- ~2) lI 2

+ II~

I)IIA(r

- r

~

-

16

-

Therefore,

l la(d(~ 1 )

d(~ 2 ))li 2

A211B(u2~u2~)I12 1 1 2 2 2 3

= ~ 2 _ [ I la i ( u : ~

+ 12m211u~,-

- %u:) I I ~

u2~

I I~

<_c(li~ li,II~211,11A~ II)lIA(~ ~- ~)II 2

+C(II~,II,II~II)IIA(~,

- ~)11 ~

w h i c h proves the lemma. We have several times used the inequality

IIBull~ _< KIIB~uil

J

The last two lemmas show that A, J, and ~ satisfy the h y p o t h e s e s of T h e o r e m 1 so ~ocal solutions exist. Notice that in Lemma 5 (formula (11)) we have a c t u a l l y shown the stronger hypotheses

(H') of T h e o r e m 2. 1 Thus, to prove global existence we need just show that I l$(t) II is uniformly bounded.

Let $(t) =

initial d a t a ~o = ~ D(A). D(BJ

T h e n u ( x , t ) ~ D(B 2) and ut(x,t)

as functions of x for each fixed t.

E(t)

Since

be a local solution of

if

- ~

{IBu(x,t) 12 +

u E L~(R3)~L6(R3)

= v(x,t)

We now define the "energy"

lut(x,t ) Ix + ~ l u ( x , t ) l ~ } d ' x

we have u E L ~ ( R 3) and since Bu and u t are also

in L 2 the integral on the right side makes sense. Now, we know that $(t)

(6) w i t h

from T h e o r e m 1

is strongly d i f f e r e n t i a b l e as a n ~ - v a l u e d

function

w h i c h means that u(t+h)

liB(

h

- u(t)

- st(t)) II

2

) o

(12)

-

17-

I I ( utct+h> - u tct~ ) - utt(t) If 2

as h

)

o. T h u s the first t w o t e r m s

t h a t the t h i r d t e r m

u(t+h)

in E(t)

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

are d i f f e r e n t i a b l e .

- ut,,:,),,.._

follows

(u(t) 2,u(t) 2) entiable

2

To see

n o t e that

,,u,,..,.,,,,u,,...,,,,C 2

(12), the r i g h t h a n d side goes to z e r o as

easily

o

- u(t)

I lu(t) <

By

;

that

h--~o and

u(t) 2 is s t r o n g l y d i f f e r e n t i a b l e .

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

so the t h i r d t e r m

'':+''-~ h

from t h i s

Z

it

Therefore

in E(t)

is d i f f e r -

and

E' (t) = ~ ( B u t ,

Bu) + ~ ( u t t , u t) +

1 + ~(Bu,

1 Bu t ) + ~ ( u t ,

(uut,u2)

utt)

~(u2,

+

uut)

1

= T ( u t, B2u + utt + llul2u) +

=

o

since u satisfies

B2u + utt

the s o l u t i o n

+

xlui2u

II~(t) I1 is

N o w w e can s h o w t h a t where

(B2u + utt + AIuI~u,ut )

exists.

Since

=

o

bounded

on any f i n i t e

interval

~ > o,

-~II~ct~II ~ < ~II~ct~II ~ + ~

f Iu~x,tll ~d~

R 3

= E(t)

T h u s we have v e r i f i e d following

theorem.

= E(o)

the h y p o t h e s e s

of T h e o r e m

2 and so we have the

-

Theorem

3

Suppose

I > o,

m > o

g ~ D((

- ~ + m 2) ua)

u(x,t)

, t & R,x E R 3, so t h a t

strongly for

in

La(R3).

differentiable

all t, u(x,o)

several

both

u(x,t)

= f(x),

real

and

remarks.

and u(x,t)

and

feD(

Then

there

t~----*

u(.,t)

L a ( R 3) - v a l u e d ut(x,o)

utt We m a k e

18-

Notice

that

satisfy

(12)

the

I > o

linearity

global

so that

existence the t e r m

is p o s i t i v e .

that

if this

even

is not

though

positive

Notice

that

assumption equation fied.

E D(-A

+m a)

and

(13) valued

u = ~.

then

Thus

u is

know

it is t r u e

makes

for

also can

we

cannot

that

(7)

remains

to this

that

(Theorems

the rest

hand

We w o u l d

sense.

1 and

of t h e p r o o f

On the o t h e r

For

equation

the p r o o f

just

like

to k n o w if we

one

if w e start

C~

(13)

since

gained

we

and

framework.

Given

3 is not

~(t)

some

in S e c t i o n

the general

Sobolev

functional

can s o l v e with

it since

bounded

(non-trivial)

technical

of T h e o r e m

The reason

we n e e d

abstract

easy.

is s a t i s -

(H~)

has b o t h

in the quite

2) and the was

the r e s u l t

example

was

integral

question

to r e a l i z e

term

on the

the

Thus

it is i m p o r t a n t

is t h a t

exis-

differentiate

E(t)

We r e t u r n

the K l e i n - G o r d o n

global

non-linear

is global.

on the data.

gained

hold.

depended solve

ex-

doesn't

the h y p o t h e s i s of

fact

to t h e non-

prove

of the

differentiable.

to show set.

result

solution

sense,

a dense

energy

1 since

the

be s t r o n g l y

argument

sometimes

~oE~,we

due

existence

dependence

results

satisfactory.

can

on the

in an a n a l o g o u s

continuous

lost by t r e a t i n g

classical

by Corollary

energy

global

to the

a general

show that

(in general)

we d i s c u s s

details.

For

E(t)

of c o n t i n u i t y

equality,

one

contribution

global

locally

kind

abstract

then

strongly

(part E)

existence

although

What we have

conserved later

fails then

depended

the

not

Finally,

see

= - lu 3

D and F below)

will

4 when

of t, u ( , t )

= - l]u[2u

(see D a r t s

#o ~ D(A). (7)

in the

is m i l d

the

But we c a n n o t

is that

continuously

if f and g are r e a l

result

We w i l l

condition

If the n o n - l i n e a r i t y tence

function

satisfies

Secondly,

ample

function

so by u n i q u e n e s s

utt - ~u + m ~ u

that

a unique

is a t w i c e

= g(x)

- Au + m 2 u

- A + m2), exists

in-

analysis

entirely in a c o m p l e t e l y

data,will

the

solution

-

remain

C ~ ? The

quired

(see s e c t i o n The

[I~].

and

proof

The

case

m = O

The

case

m = o

mates teed

on J

(except

as b e f o r e

in p a r t

Thus,

E(t)

as

v>)

in part

=
linear

(13)

was

to d e v e l o p e

which

we have

are re-

d u e to J ~ r g e n s the a b s t r a c t

presented

are t a k e n

not

changed

A, w h e n = ~(o)

bounded

remains

bounded.

on

A but

term

device.

We

choose

i. the

conserved

~ o E D(A), and

E(t)

f r o m this

as

+ moU>

constants) Of c o u r s e

+

(14)

J is n o w d e f i n e d

in J d o e s

+ lute

]Vul ~

in o r d e r

remains

following

as

- l[ul2u

by c h a n g i n g

by T h e o r e m

!! However,

of

Segal

by the

the e q u a t i o n

E(t) - ~

As

(and e s t i m a t e s )

- Au + m o U = - lu ~ + m o U

of the

(14) we h a v e

details

can be h a n d l e d

J(
addition

existence

stimulated

technical

and A are d e f i n e d

The

of g l o b a l that

and w r i t e

utt ~

but more work

Simon[~7].

The

some m ~ > o

is yes

3).

his w o r k

in~l].

from Reed

B.

first

It w a s

approach

answer

1 9 -

not

affect

so local

a n y of t h e e s t i -

existence

by r e w r i t i n g energy

which

the

in g u a r a n -

equation

is:

Iui }dx

is d i f f e r e n t i a b l e it f o l l o w s

and E' (t) = o.

that

(15)

+ Jut 12} dx ~ 2E(o)

to c o n c l u d e finite

By t h e

that

intervals

fundamental

as

I l~(t) I I 2 = we

also

theorem

need

11Bu(t) I I 2 +I lut(t) I 1 2 2 2 that

of calculus,

I lu(t) I [

2

-

u(t)

since

u(t)

so from

= u(o)

is a s t r o n g l y

(15) w e g e t

20

i

+

-

t

Us(S)ds o

continuously

differentiable

L2-valued

function,

that

Ilu(t) II~

_<

11u(o) I I

+ r 2E(o)

t

Thus,

I1 (t ir = !!r ul + m;rul + lutl l dx is a p r i o r i global case

bounded

existence.

on

Thus,

intervals.

all the

By T h e o r e m

statements

2, w e t h e r e f o r e

of T h e o r e m

3 hold

get

in t h e

m = o. The case

because

rates

m = o

(-A + m2) -I

difficulties

is o f t e n

(in t h e

Because

cases

can be handled

tence

theory

treated

is u n b o u n d e d

and because

of decay

different.

C.

finite

the

separately

in t h a t c a s e c a u s i n g

scattering

s u p norm)

theory

of solution

of the above

device

on an e q u a l

in t h e

literature,

some technical

is d i f f e r e n t

of the

at

least

since the

free equation

(see S t r a u s s [ ~ 7 ] ) ,

footing

are

the two

as far as t h e

exis-

is c o n c e r n e d .

O t h e r ~,

n,

In o r d e r

to discuss

and

the

equation

u t t - Au + m 2 u = - ~ l u l P - l u

for v a r i o u s

both

p a n d n, w e

lev estimates.

state

The proofs

u(x,o)

= f(x)

ut(x,o)

= g(x)

the

following

generally

consist

x GR n

(I)

special

cases

of t h e s a m e

of t h e sort

Sobo-

of t r i c k s

-

we

used

See,

above

in

lemma

for example

Theorem

4

Friedmann

(Sobolev

o < a < 1

1 plus

appropriate

, and

Let

for

all

m and

SUDpOSe

1 p =

Then,

use

of

interpolation

theorems.

[IOB

Estimates)

, 1 < p < ~

21-

1 2

u E C ~o ( R n ), t h e r e

n be

positive

integers,

that

am n

is a c o n s t a n t

K so

that

Ilollp _< KIIDmul[allull x-a2 2 except

when

both

and

a =

integer

of the I.

[IDmu[[2 2

Let same

us

see when

analysis

as

these

A and

carry

through

our

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

permit

us t o

carry

through

the

Let

~

L 2 (R n)

as b e f o r e .

same

m ~ n/2

n m ; ]I (~-~.) u1[ 22 j~l j

estimates A.

= D(B)

B defined

hold:

above,

denotes

in p a r t

~o

with

following

In the

(16)

methods

Now, we

J(~)

need

=
an e s t i m a t e

- llulP-lu> of

the

so to

form

11uEl2p ~ KIIBulI~ As

indicated

in p a r t

A, w e

can

r

always

estimate

I IDu] I

and

I lull

2

I IBui I 2.

So

it f o l l o w s

from

(16)

that

(17)

will

hold

by 2

in t h e

following

cases:

n =

1

,

2 < p < ~

n = 2

,

2 < p

n = 3

9

2 < p < 3

n = 4

,

p--

< ~

(18)

2

-

In a l l o f t h e s e hypotheses data

cases

Er

Thus, er

if

~ > o

Theorem the

5

clvul Rn

we g e t

Let

of T h e o r e m if

This

theorem

is r e a s o n a b l y

will

to hold

can become

class

later

of e x a m p l e s

L e t us d i s c u s s

where

as

(19)

it w i l l

E(t)

if t h e

uI~

dx

Since

the

in t h a t

strong-

case.

The

(18).

Then

w e have:

in t i m e

in

if

i < o

and

(i).

since we do not

case

I < o, s i n c e

both

still

conserve

energy.

the

that global

expect

lie(t) I 12 a n d

existence

In fact, does

not

the

Notice,

equation

that

not

initial

we

in

- Au + m ~ u = - lu p

if p is odd b u t be

initial

I < o.

for a m o m e n t

p is an i n t e g e r .

ces t o

Thus,

locally

(part E) when

the

for n i c e

ll.

existence

B.

satisfactory

and

utt

and

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

in t h e

large

in p r o v i n g

+

II~(t)

I > o for t h e e q u a t i o n

show explicitly

a whole

lutl')dx

+

bound on

3 hold

in t i m e

~+l[lUl p

lul '

in p a r t

p and n s a t i s f y

existence

in p a r t A,

we get global as

globally

global

as

involved

energy:

an apriori

is h a n d l e d

conclusions

same

+ m

(H i) h o l d s

zero case

details

a conserved

I

=

hypothesis

mass

1 are t h e

we have

= Ect

-

the technical

of T h e o r e m

~o e D ( A ) '

22

if u is r e a l - v a l u e d

i~ p is even.

data

(19)

are real,

= I ((Vu) 2 + m 2 u 2 + u ~ ) d x

+

Assuming (19) h a s

~1- ~

then

(I) r e d u -

u is r e a l - v a l u e d ,

a conserved

energy

!uP+Idx

Rn

We can

only

insure

a n d p is odd. these

cases

by the

that

So we

it is g i v e n

same methods

as

be a simple

conserved

pect

existence

shown

global

in p a r t E t h a t

the term

only

expect

on t h e r i g h t w i l l global

by Theorem in p a r t A.

energy

but

anyway. global

existence

5.

When

For

complex

it d o e s n ' t

In f a c t

existence

be positive when

p is e v e n w e valued

matter

if I > o

D is o d d

u there will

since we don't

in t h e c a s e m = o, n = i, doesn't

hold.

and

can treat

in (19) not

exit is

-

We now

consider

the

Since we do not have begin

to try to prove

However,

we

consequences integer

strong

we work.

when we

k so t h a t

local

holds.

t r y to p r o v e

a Sobolev

By T h e o r e m

b e done;

~k

with

=

depend

{

on o u r

see

global

! xl

covered

form

if w e

change

shortly,

(18). even

the Hilbert

this has

existence.

Bkul[

in

(17) w e c a n n o t

s p a c e ~ O = D(B) ~ L 2 ( R n ) .

of the

Choose

unfortunate a positive

form

(20) 2

techniques on n.

not

of t h e

existence

inequality

4 and t h e

k will

cases

As you will

ilull

ways

in t h e estimate

local existence

can p r o v e

space with which

problem

a Sobolev

23

of

lemmas

2 and

3 this

can

al-

Now we define

I u eD(B2k+l)

, vs

the norm

ILll 2 = ILB2k+luli 2 + lIB2kvli2 We

let A b e t h e

same operator

D(A)

A is

self-adjoint

introduction. that the

= {

J($)

So,

=
following

and generates

-luP>

the

to get

which

of l e m m a

it r i g h t .

we will

Let

L IJ(~)I[~

D(B2k)}

group

One need

just

side

discussed

we need

B as t h o u g h

just

of T h e o r e m it a c t s

in t h e show i.

Then

= i~i l lB2k(u p) I[

is l e s s t h a n

Kli(Bk~u)

2 or equal

to a sum of terms

of the

,,, (Bk'u)[l 2

where

2k = I k. i=l i

In

on u p b y

use the technique

$ = ~k"

k and the right

W(t)

existence

the hypotheses

treat

it d o e s n ' t .

v

same

local

and A satisfy

B ( u p) = p u P - i B u ,

but now

I u GD(B2k+I),

in o r d e r

calculation

4 to d o

as b e f o r e

and the k i are non-negative

integers

less

than

or

form

-

equal are

t o 2k.

less

Let k I be the

than or equal

-

largest

t o k.

k i.

Thus,

9 9 (Bkpu)[[

KI [ ( B k l u ) .

24

Then

a l l t h e k i for

i > l

we can estimate:

2 ~ ~llBk'ul 12 ~

llBk~ull

! KllB2k+lull,~llB~+k ull

< KIIB2k+IulIP 2

KII~IIp

w

where

we have used

(20)

in t h e

second

step.

Thus,

<

The

other

estimates

Therefore,

Theorem Then g s

we can

6

Let

there

of Theorem

integers

is a n

n > o,

integer

- A + m2)k+T),

differentiable

1 are proven

then

(in t)

p > 2

k so t h a t there

function,

utt(x,t)

- Au(x,t)

if

is a

possible one would

Further,

t g ( - T,T). - A + m2) k + ~

The problem

with

this

for ourselves expect

s a m e way.

which

+ m2u(x,t)

it

)

u(t)s

for e a c h

result

(for o d d p).

and

twice

strongly

satisfies

= - Au(x,t) p

=

f(x)

= g(x)

- A + m2) k+l)

t ~ ( - T,T).

is t h a t w e h a v e n o w m a d e

to prove

~ be given.

~ A + m2) k+l)

T > O and a unique

u(t,x),

ut(x,o)

ut(t) e D((

the

and any m > o and f 6 D((

u(x,o)

for each

in e x a c t l y

state

global

existence

The difficulty

it a l m o s t

im-

in t h e c a s e s w h e r e is t h a t w e m u s t

show

-

25-

that the n o r m In ~ k

ll~(t) ll ~ = lIB2k§

2 § ilB2kutll 2

does not go to infinity in finite time.

From the energy inequality

(for odd p) one only gets that

llBuli~2 § 11utI[22 stays finite. One might try to use this and some higher order energy inequalities to prove that has been able to do this. weak

I [~(t) il~.~ stays finite, but so far no one It is known however that for odd p global

(in the sense of distributions)

solutions exist

(see Section 5)

and also that global solutions exist if the data is small enough. we have the following intriguing situation:

Thus,

e x i s t e n c e of strong so-

lutions locally, e x i s t e n c e of w e a k solutions globally, but no strong global e x i s t e n c e proof.

The use of the space

"~k' so called "escalated energy spaces",

has been r e p e a t e d l y e m p h a s i z e d by C h a d a m [3 ], [ ~ ], [ ~ ], [ 6 ] . In p a r t i c u l a r C h a d a m has used t h e m to prove local e x i s t e n c e for the coupled M a x w e l l - D i r a c e q u a t i o n s

in three dimensions.

We discuss this

further in Section 6.

D.

The s l n e - G o r d o n e q u a t i o n

We can easily apply the e x i s t e n c e theory to the equation

utt - Au + m2u =

when the number of space d i m e n s i o n s

g sin(Re{u}

is

(21)

)

n = 1,2,3,

or 4.

If the in-

itial data are real then the solution w i l l be real and thus w i l l satisfy

utt - ~u + m2u =

which

g sin

is known as the s l n e - G o r d o n equation.

(u)

(22)

We treat the real solutions

-

of

(22) by studying

(21) because

Re{u}

instead of and u

u

is bounded

does not affect the technical details

functions.

complex case,

for all complex

in the imaginary directions.

have the same d i f f e r e n t i a b i l i t y

also treat the real solutions of real-valued

-

sin(Re{z})

z while sin z grows exponentially Re{u}

26

But,

properties.

space of

since most of our terminology (21).

since One could

(22) by using a Hilbert

it is easier to treat

Using

is from the

For ease of notation we will

write Re{u} = a from now on.

~=

To begin with, we suppose D(B) ~ L2(Rn), and

A=

i

(o -B 2

I) o

just as in part A.

D(A)

The estimates of T h e o r e m

=

Ilsin

and define

= D(B 2) ~

B=

D(B)

~ll~

=

g

_< I l u l l

~ _<

II~ll

(sin ~,B2sin u) 2

n = ~ (~isin 0,~isin ~) + m 2 (sin ~,sin 0) i=l *

<

2

1 are proven as follows:

2 = IIAJ(#)II 2 = liB sin ~II 2

--

~

But, now we have

J(#)

IIJc~)ll

m > o

llvull:

KIIBulI~

+m21Jull

~ 2

_< KII+II ~

(23)

- 27 -

{{J(q~) - J(*) {{ = [{sin 5

- sin 5 {{ {

2

_< {{u - 5~ {

w h e r e ~ = 1

and

~ =

1

2

2

Finally,

I[A(J(#)

-

J(*)){l

=

JiB(sin

5

-

I

= II ( B 5 ) C O S

5

1

--

1

1

_< I{ B2u,II2

11+-,{{

sin

1

5 ) { 2

2

-(B5 )cOs 5211 2

2

2

2

1

lIB(u| - u2)I12

2

+ .l'{B(ul - U2 ) I12

ILA+I{ § {{+-.11

In this last computation we have again for ease of exposition treated B as though

it acts by differentiation.

Note that~the next to last

step is the only place where we use the fact that the dimension because we needed the Sobolev

is < 4,

inequality

< KKIBu I{,llBu~ll~ Since the extra hypothesis show that tence.

[l~(t) l] is bounded

Unfortunately,

(for all g,m).

on finite

intervals to show global exis-

(21) does not have a positive

But the fact that the n o n - l l n e a r i t y

us to show apriori boundedness equation we get

(Hi) of T h e o r e m 2 holds, we need only

of

II~(t) 11 anyway.

conserved energy is

mild

allows

From the integral

-

II~(t)

ll < I

Ie- iAt @~

28

fl + II !

-

[te-iA (t-s) J(r

I

t 0

t

Io l IJ(r

_< I I%f

+

_< I(%l

+

(~(s) IIds

Ir

~ IIr

so by iteration

Thus

lie(t) II

is b o u n d e d on bounded

solution exists globally. part B.

intervals so by T h e o r e m 2 the

The mass zero case is handled just as in

We summarize:

Theorem 7 given.

et

Let

n = 1,2,3, or 4 and let

Then for each

m E [o, ~) and ,g geD((-

f 9 D(- A + m2),

e

(-~,~) be

A + m2)/~') the initial

value p r o b l e m

utt - Au + m2u = g sin u

u(x,o)

ut(x,o)

= f(x)

=

g(x)

has a unique global solution u such that u is twice c o n t i n u o u s l y d i f f e r entiable as an L2(R n) -valued function of

t, u ~ D ( -

A + m 2)

and

1

utED((-

A + m2) T)

for each t.

N o t i c e that if we just want to solve the integral e q u a t i o n

~(t) = e -iAt + Ite -iA(t-s) i "O instead of

~' (t) = - iA~(t) + J(r

J(~(s))ds

-

then, by C o r o l l a r y

C.Lo

29

-

1 of T h e o r e m i, we only need the h y p o t h e s i s

IIJcr

- J(r

<_c~IIr

II~- ~I

Since this h y p o t h e s i s holds for the sine - Gordon equatlon in all dimensions

(we only used the Sobolev inequality

IIA(J(r

- J(~)) I [) we conclude that

all dimensions.

(21) has w e a k global solutions

By weak, we mean here that the c o r r e s p o n d i n g

e q u a t i o n has a continuous Y - v a l u e d

E.

in our e s t i m a t e for in

integral

global solution.

A__nnexample when global e x i s t e n c e fails

Let us consider the equation

(24)

utt - Uxx = u p

u(x,o)

ut(x,o)

=

x eR

Uo(X)

= Vo(X)

w h e r e p is an integer ~ 2. We know that local e x i s t e n c e holds and, antic i p a t i n g Section in

CO

(i.e.

3, we can g u a r a n t e e that if the solution u starts out

uO ~Co(R),

vo&C

(R))

then on some finite t interval

the solution w i l l be twice c o n t i n u o u s l y differentiable.

Further,

another result of section 4, it w i l l have compact support.

using

These

r e g u l a r i t y statements don't affect the ideas below, they just a l l o w us to integrate by parts w i t h impunity.

If

uo

and v o are r e a l - v a l u e d

then u will be r e a l - v a l u e d and

E(t) =

{ (Vu) 2 + m2u 2 + ut2

2 u p+l } dx p+l

R

is the c o n s e r v e d energy. any

Thus we do not expect global existence

p ~ 2. We will show that if

u ~ and v o are chosen correctly,

F(,t) = IRU(X,t) 2dx

for then

-

goes to i n f i n i t y initial data

Then

30

in f i n i t e time.

u~

and

vo

-

S u p p o s e that we can find an ~ > o and

so t h a t

(A)

(F(t) -e) " < O

for all

(B)

(F(t)-u) ' < O

at

F(t) -~

w i l l go to z e r o

(B) is a u t o m a t i c a l l y same sign on

(-~,~)

(F(O)-U)'=

It r e m a i n s

satisfied

Q(t)

t = O

in f i n i t e time; by choosing

uo

see the Figure. and

vo

Condition

tO h a v e the

since

- u F ( O ) - I - a F ' (O) = -2eF(O) - I - ~ I U o V o d X

to a r r a n g e

same s h o w i n g t h a t

t > O

for

Q(t)

(A) to hold. > O

Since

F(t)

> O

this

where

- (-u)-IF~+2(F-~)"

= F"F-

(e+l) (F') 2

But, F'(t)

= 2fuutdx

F"(t)

= 2f(UUtt + u~)dx

= 4,o .,-,,f,.,,~dx-,-~f,u,.,t,=-,~o-,-,,,.,,~,oSO~

o,~,.,,o~,,

{Cf'."dx)Cfu,~d-t-{f,.,u, dx)']

is the

-

31-

2F,t{fuutt0x f,2o§ The

first

we need

The

term

only

on t h e r i g h t

arrange

conserved

That

is,

2(2~

+ I) = P

E(t)

that

energy

if

E(O)

= !(n-l) we

scale

is i n d e p e n d e n t

> O

by the

Schwarz

inequality,

so

where

iS

of t.

Thus,

if w e c h o o s e

e so t h a t

+ i, w e h a v e

< O,

then

> O. Now,

= -

~+l)E(t)

+ 2~ lu~dx

=

~+I)E(O)

+

-

H %s a l w a y s

choosing

eventually

F(t)

If we

(241

u ~ by m u l t i p l y i n g

(this w i l l data,

H(t)

for

H(t)

Thus,

is p o s i t i v e

goes

to

consider

u o > O,

infinity

since

v

P+I

in f i n i t e

- Uxx

=-

> O

positive

For

(B)

until any

time.

U p

since

so t h a t

constant

> 2).

the d i f f e r e n t i a l

utt

lu~dx

strictly

by a nositive

happen

instead

2~

(24)

equation

such

is s a t i s f i e d

E(O)

< O

initial

-

then

H(t)

again satisfies

E(t)

= ~

(24),

(u~ +

If P is even then by c h o o s i n g (B)) w i t h

but now the c o n s e r v e d energy is

Uo(X) ~ O,

Vo(X) ~ 0

u ~ s u f f i c i e n t l y large we can obtain

E(t)

E(O)

(thus s a t i s f y i n g < O

and thus the

If, on the other hand, p

is odd,

is always greater than or equal to zero so the above argument

does not work. existence

-

I

solution blows up in finite time. then

32

This

is not s u r p r i s i n g since we have proven global

in this case in parts A, B, and C. Notice that we always had

to choose the initial data large in order to get the solution to b l o w up. Later we will see that if the initial data are small enough then global solutions exist independent of w h e t h e r p is even or odd or the sign of i.

The author learned this simple example from H.Levine ~8]. details are taken from R e e d - S i m o n [~7]. Examples of n o n - e x i s t e n c e of global solutions have been known for a long time. Keller

[|6]

or Glassey[||].

See for example

The

-

F.

The Coupled In part

riate

saw that

A much

proof~5~

and K l e i n - G o r d o n

t w o b y two

T h e n we

can w r i t e

the

(F i)

free

~(t)

We w o u l d

= <~i (x,t)

like

to

solve

g is a r e a l

function with

8~(t).

bounded to

coupling

below

lose by

These and

coupled

constant,

equations

Since

(F

4)

(F 2) w i t h

8 is H e r m e t i a n

As u s u a l

we r e w r i t e

u(t) (F3)

d

(for one

u(t)

denotes

have

Let ~ and 8 be

space

dimension)

function

= u(x,t)

as

on R 2.

is an R - v a l u e d

the dot p r o d u c t

a conserved

not much

use.

energy

in C 2 o f

but

~(t)

it is not

So w e d o n ' t

have

anythimg

defining

= Re(u(t))

u(t)

on the r i g h t

will

remain

as a first

d--t ~(t)

coupled

- igSu(t)~(t)

be c o m p l e x - v a l u e d ,

u(t)

and r e w r i t i n g

ex-

= ~(t). 8~(t)

~(t)'8~(t)

u(t)

.

CZ-valued

- me8 )~(t)

is t h e r e f o r e

letting

of g l o b a l

for t h e Y u k a w a

= -i ( i e ~ x - me 8 )~(t)

= -i ( i e ~ x

on R 2 , and

equation

system

utt - Uxx + m o U

conserved

satisfy

is a

(F 3)

the

8e + e8 = o

p

Dirac

in an a p p r o p -

though

example

existence

, ~2(x,t)>

d ~(t)

where

striking

the

(F 2)

is "mild" even

in o n e - d i m e n s i o n

which

I = 82

d ~-6 ~(t)

~here

more

equations

:

existence

of g l o b a l

Hermetian matrices

e2

Equations

if the n o n - l i n e a r i t y

can get g l o b a l

positive.

is C h a d a m ' s

-

and K l e i n - G o r d o n

t h e n we

is not

istence Dirac

D we

sense,

energy

Dirac

33

real

order

= -iAo~(t)

side if its

system

+ Jo (~(t))

instead

of

initial for

~(t)

u(t).

data

are real.

=

-

where

BO = ~

, Jo(~(t))

34

-

=

(o I1

AO =

-B~

Letting coupled

D e : ia~ first

- me8

order

and

and,

o

combining

(F2)

and

(F4) w e h a v e

the

system"

~' (t) = -iDe~(t)

+ Je (r ( t ) )

$' (t) = -iAor

+ Jo

(F 5)

where

Je (@(t))

Finally,

if we

: -igBu(t)~(t)

set

=(t)

J(=(t))

then w e can w r i t e

(F 6)

where

(F5)

:


= <J

e

($(t)),

is the

four b y

J

o

(@(t))>

as

-' (t} = - i A - ( t )

A

(~(t))

four

+ J(-(t))

operator

matrix

A(Oe o) o

Thus apply take

we h a v e

rewritten

the e x i s t e n c e as our H i l b e r t

our

theory

Ao

coupled

equations

of S e c t i o n

I.

Bet

in a f o r m so t h a t w e can B

e

space

~-~= D(Be) (~ D(Be) ~)D(Bo) ~} L2(R)

=v~-m

r . e

Then we

-

where the norm of ~ = <@i'42'

35

-

u,v>

is

= I IBe@, 1122 + [ IBe@2 I [22 + llBo u1122 + [ I V l ] 22 A is self-adjoint

on

D(A) = D(B~) O D(B~) (~) D(B O) ~) D(Bo)

To see that J is a well-defined equality

mapping on ~ we use the Sobolev

in-

If(x) I I~ < cl If (x) [ I2*/2 llf(x) i [1/2 2

(F 7)

,

valid in dimenslon

n = 1 (see

part

C).

This implies

[ Ifi]~ ! C] IBef]l~ 2

[f]]~2

I Ifl I~ ~ cl[Bofl I ~

Ifll ~

and 2

Now I [J(~(t)) I I 2 has terms of the form IBe(U~i) I I2 and of the form [ [~.8~I I 2 and we ca~ estimate (treating the B's like differentiation 2

as u s u a l ) I IBe(U~i) I12 ~

II@il I. I IBeU112 + IIBe@i1121 lul 1~

< c[ IBe~i112

_

I IBoUl I 2 + I IBe~el

211BoU112

< c11~If 2

and similarly for the other term. We have used the fact that BelBo and B-IB are bounded (by the functional calculus). In fact, using o e this same idea one can easily check that J satisfies hypotheses (H~) and (H~) for m = o,i. Thus we get a local solution of (F6) by Theorem I. Our problem is to get a global solution. Since (H I) is satisfied we need just show that I l~It) I I is apriori bounded. But we cannot use the conserved energy to do this since it is not bounded below so

-

36

-

our only hope is to use tee integral equations directly (as we did for the sine-Gordon equation). The two integral equations are ~(t) = e-iDet~(o)

+ It e_iDt (t_s) je (#(s))ds o

~(t) -- e-iA~

+

it e_iAo(t_s) jo (~(s))ds o

so

II*(t>l

e

<__ c e

+

I*]lJer

lledS

o

(F 8)

[[~(t) l ~ < C 0 + where the

[[" [ I e

and

ll:

I ' ]1 0

12=

I*o

norms are defined by

I,II:+

=

ll,lro 2 + JiBe,2112

I[~11~ = IIBoull ~ Using

llJo(~J(s)) llodS

+ Ilvll ~

(F7) , we have 2 I IJe(~(s))

l e <-- ~ i =Il

lIB e(~u, i>ll

2

2 (F 9)

!c

[ {ITBoOlI

i=l

2

TIBe*ilr~z[l*ill~ +

IBoUl i~ 21 l~I l~al } 2IBe*il I2

and IJ ~ (~ (s))

(F lO)

Io = I1~,~*11

< c{llBe**ll~2][r

I~ 2 § I I B e , z l i ~ l l , 2 1 1 ~ }

-

So far, the situation (F9) and

(FIO)

37

-

looks pretty hopeless

are quadratic

expressions

since the right side of

in the norms

so we can't expect to get anywhere with

we can't get anywhere unless we use more information system

(F2),

(F3).

I I. I Ie and

(F8) by interation.

about the coupled

So far, the only property of the matrices

which we have used is that they are Hermetian Using the other properties

(so that

De

e and 8

is Hermetian).

of e and 8 one can easily show" (see for ex-

ample Bjorken and Drell I t ] o r Chadam IS])

that the total electric

l~1(x,t) 12dx +

Q (t) = [ L=

I I" I 10

In fact,

charge

(x,t) 12dx

J

Q(t)

is a conserved quantity;

= Q.

This

is an immediate held for

(FIO)

can now be written

I IJoC~Cs)) 1 !o _< c {11 Be~, I1~'=+

(F iI)

< cll~r

conservation

}

w~

--

However,

IBe~ll~ ~

e

of charge as it stands

is only a small help for

(F9) which can now be written

(F

2 12)

I 1JeC~Cs)) I le

-< c I i=l

{llBo~ll ~ IIBe~ill

Our only hope to make this better Fortunately,

u(t)

Uo(t)

propagation

is to get an estimate

we can do this by using special properties

gator and conservation

where

of charge(again!),

= Uo(t)

is the first component preserves

Thus,

u(t)

ll~llBehll on

the norm

of

e-iA~

of the propa-

satisfies

the equation

Since the free

I IBoUI I2 + I lutl [22 we have

I1~~lluor

ll[ ~ _< c

}

I ]u(t) I I~.

+ st [B -i O sin Bot ] Jo(~(s))ds 0

1 lUo(t) I1= <_ cl IBoUor for all t.

~= + II~(t)

2

-

(F i31 lluCt) ll. < c

+ st lIFBT~oi

38

sin

-

BoCt-s)l. Jo(~(sllll

Now P B-Isin Bot acts by convolution by a function o is the inverse Fourier transform of (k2+m~) - ~ sin Since

Jo(~(s))

the L* Thus,

is quadratic

norm of if

Jo(~(s))

I I (Bol

for all s and t. properties

~i(s)

H(x,t-s)

which

(~r~t-s))

and charge is conserved,

is uniformly bounded

llH(x,t) nl, ~ c

(F 14)

in the

as

independently

of s.

we can conclude that

sin Bo(t-s))Jo(~(s)) II = _< c

One can compute the inverse Fourier

of B e s s e l functions

to conclude that

transform and use

I IH(x,t) ll~ ~ c

but

the easiest thing to do is to notice that the inverse Fourier transsin kt form of k is uniformly bounded for all t since it is just a fixed constant times the characteristic

function of an interval.

On

the other hand sin t ~

sin t k k

is an L~ function whose L* interval.

norm is uniformly bounded on any finite t

Thus

I IH(x,t-s) for s and t in an finite interval. Then for

CF ~5)

t 9 [- T,T]

I I%C~r

Therefore,

,

Choose such an interval

t lu(t) I I= ~ c,

so

< o{ll~r162

(FI2)

becomes

e,/z + I]~(s) l[e}

if we set f(t) =

g(t)

then

I I~ <__ c

(F8) becomes

the following

=

l~r

[Io e

set of coupled

inequalities

[- T,T]

-

39

-

(F 16a)

g(t) 2 < c e + a ft (f(s)g(s) o

(F 16b)

f (t)

where

into the

t o g(s)ds

s

< Co + b

a can depend

equality

o n T b u t n o t on t. first,

+ g(s) 2)ds

Substituting

the

g(t) 2 < c e + ac ~

(s)ds + a

g(s)2ds

+ ab

o Using

the

Schwartz

inequality

on the

g (t) 2 <_ Ce + d

where

d depends

bounded f(t)

on

on T b u t

[- T , T ]

.

is a l s o b o u n d e d

proven

that the

interval This

the method

Using

the

show the

existence

mensions

energy of

local

However,

is open.

in-

of

second

i

(r)dr ~o

and

fourth

terms,

Therefore, back

into

by

iteration

(Fl6b)

Since T ~as

arbitrar~

is a D r i o r i

bounded

by Theorem

2.

spaces strong

in o n e d i m e n s i o n coupled

solutions

the question

in D a r t

For more discussion,

on a n y

finite

equations.

6.

can easily

dimensions

existence

see S e c t i o n

we have

c, o n e

in t h r e e

of g l o b a l

is

that

is d u e t o C h a d a m [ 5 ]

Maxwell-Dirac

introduced

g(t) ~

we conclude

(F6)

to the

we have

t o g(s) ~ds

globally

existence

is a p p l i e d

escalated

(Chadam[~]).

H(t)

exists

of g l o b a l

where

o n t.

[- T , T ]

solution

and therefore proof

not

Substituting

on

second

we have

in t h r e e d i -

3. Smoothness

of Solutions

As we have remarked before, not completely ple we would smooth

satisfactory

the existence theorems

hypotheses

1 are

For exam-

like to know that if we choose the data ah time zero to be

(say C = ), then the solution of

the equation

in Section

from a classical point of view.

in the classical

are needed.

sense.

(i) will stay smooth and satisfy Essentially

In our examples

A = if~

the powers of A, Where

I 1 ~-m 2

act like powers of the Laplacian

two kinds of further

.

o

So,

if the solution

~(t)

remains

in the domain of high enough Dowers of A then we should get smoothness in the x variables.

To achieve this we require only higher order esti-

mates of the same kind as

(H I ) , (H~) in Theorem

i.

Secondly,

looking

at our equation

~' (t) = -iA~(t)

(6)

+ J(~(t))

we can see that to get higher differentiability to assume some "differentiability" ~(t)

must remain

(Hi) in part

the d e r i v a t i v e s

powers of A.

on J awkward to formulate;

it works out quite easily

hypotheses

of ~ in t, we will have

Further,

in the domain of appropriate

factors make our conditions see,

in J.

in applications.

(a) below follow from

for easy comparison with Theorem 9.

but as you will

As in Theorem (H~).

of

These I, the

We state them thus

We will refer to the hypothesis

in part b below as "condition J " m

Theorem 8 (a)

(local smoothness)

Let A be a self-adjoint

mapping which takes satisfies

D(A 9)

operator into

on a Hilbert

D(A j)

for all

space W a n d 1 ~ j ~ m

J a and which

(for j = O,l,...,m)

(Hj)

I IAJJ(~) I I < C(I l#II ..... I IAJ~I I) I IAJ~II

(H3)

I IAJ (J(~)

- J(~)) I I < c(II~II,II~ll ..... IIAJ~rI,IIAJ~II)IIAJ~

- AJ~II

-

for all ~,@ ~ D ( A j)

where each constant

function

of all its variables.

is a T m

so that

~(t) 9 D(A m )

4 1 -

Then for each ~o ~ D ( A m ) '

(6) has a unique

for all

C is a (everywhere

t&[O,Tm).

solution

~(t)

for

finite)

m ~ I,

t~[O,Tm)

there with

For each set of the form

{~ I I IAJ~I I ~ aj, j = 0 ..... m}, T can be chosen uniformly

for ~o in the

set. (b)

In addition

has the following

to the hypotheses property:

uously differentiable

in (a) assume that

If a solution

with

for each j < m,J

~ is j times

~(k) (t)~ D(A m-k)

strongly

contin-

and Am-k~(k) (t) is con-

tinuous for all k < j, then J(~(t)) is j times differentiable, dJJ(~(t))/dt j ~ D(A mrj-l) , and Am-j-ldJj(~(t))/dtJ is continuous. Then the solution in

t and

Proof

The proof of part

Theorem #(.)

given

in part

(a) is m times

strongly

differentiable

d j ~ (t)/dt j ~ D (Am-j ) . (a) is essentially

1 except that we take

on [O,T m)

so that

X(Tm,e,~ o)

~(t),...,Am~(t)

the same as the proof of to be the set of functions are strongly

continuous

and

m

[

sup l IAJ~(t)

_ e-iAtAJ ~o

II < --

j=o t, [O,•

Then one proves that S is a contraction Part

(b) is proven by induction.

is strongly

continuously

By the same arguments

A~(t)

= Ae-iAt~o

=

A~(t)

(a) that ~(t)

and ~' (t) = - iA~(t)

+ J(~(t)~.

i,

+ A[te -iA(t-s)J(~(s))ds ~o

9 t A ~ o + [te-iA (t-s) AJ (~ (s)) ds e-lA ~o

and from this it follows that

We know from part

differentiable

as in Theorem

as before.

is strongly

(using another continuously

argument

in Theorem

differentiable.

I: see

Therefore

hypotheses on J,J(~(t)) is strongly continuously differentiable, dJ(#(t))/dt G D ( A m-2) , and Am-2dj(~(t))/dt is continuous. Thus, is strongly

differentiable, ~"(t)

d = - A~' (t) + d-t J(~(t))

(IO~

by the #' (t)

-

= (-iA) 2~(t)

#"(t) E D(Am-2), gument again

and

#(t)

-

- iAJ(#(t))

Am-2#"(t)

(dJ(~(t))/dt

know that ~(t)

42

d J(~(t)) + ~-~

is continuous.

is differentiable

is twice continuously

We now repeat the ar-

by hypothesis

differentiable)

is three times strongly differentiable

since we now

to conclude that

and so forth. I

Notice that the interval on which the solution exists depends on m. In particular, have slightly

T m may go to zero as stronger

estimates

m

>

and smoothness.

Theorem 9

and smoothness)

(global existence of part

As in Section

and apriori boundedness

then we get global existence

the hypotheses

~ .

of

1 if we

I I~(t) I I,

Let A, J, and ~ satisfy

(a) of Theorem 8 except that

for j = 1,2,...,m

(Hj) is replaced by

(H~)

IIAJJ(~)II ~c(II~II ..... IIAJ-X~II)IIAJ~II

Let ~ o E D(A m) and suppose that on any finite the solution

$(t),

(6) is global

in t.

I I~(t) I I is bounded. Further,

Then the solution

if J satisfies

m times strongly differentiable

interval of existence

of

~(t) of

condition Jm then

~(t)

is

for all t and satisfies

dd--•j-•(t)

( D(A m-j)

Proof

The idea is the same as in Theorem

existence showed

I I~(t) I I

is apriori bounded.

in T h e o r e m 2 that

IIA~(t) II ~

2.

On any finite

interval of

On this same interval we

IIA~oll

e K~t.

Now for

A2~(t)

we have the estimate

IIA~(t) II < IIA2%If § It I IA2j((~(S))

I Ids

o

<

lIAr%If + It c(l l~(s) il,llA~(s)ll)llA2~(s)llds o

by

(H89

Since we already have

ll#(s) ll and

ilA#(s) II

apriori

-

bounded,

43

-

C(I l~(s) I I,I IA~(s) 11) is less than some constant K 2 on the

interval in question.

Thus,

lIA~Ct~ll ! ttA~o]l §

I]A~Cs~EI ds

20

SO.

iiA2~t) lli 11A2~oltJ ~t

Now we have

I iA2~(t) [ i

bounded so we can continue in the same fashion

to conclude that

i TAJ~(t) i I

of existence for

j = l,...,m.

is apriori bounded on any finite interval

The length of the interval T m in T h e o r e m 6 on w h i c h the solution exists depends only on the constants Therefore,

C(I i%oI I + e ..... 11Am~ol I + e).

as in T h e o r e m 2 we can extend the solution past the end of

any finite interval.

So, the solution is global

in t.

The other state-

ments of the t h e o r e m follow from T h e o r e m 6. D

Corollary each m.

Let A, J, and ~ s a t i s f y

m = 1,2,... Then for each

the h y p o t h e s e s of T h e o r e m 9 for

And suppose~ that J satisfies condition Jm for each ~oE

/~D(AJ),

the equation

(6) has a unique global

solution so that ~(t)

is infinitely often s t r o n g l y d i f f e r e n t i a b l e and

each d e r i v a t i v e

~

~

is in

D(AJ).

First we will apply these ideas to

utt - ~u + m 2 = - u 3

u(x,o)

= f(x)

ut(x,o)

= g(x)

(25)

We use exactly the same set up as in Dart A of Section 2.

We need just

verify the higher order estimates and the h y p o t h e s e s on J in part b of T h e o r e m 6. techniques

The higher order estimates are easy and use exactly the same (i.e. the same Sobolev estimate)

I TAmj(~) If =

I IBmu~ll

2

as in part A.

For example:

-

The

t e r m on t h e r i g h t

form

I I (Bmlu) (Bm2u) (Bmnu) If

largest

II

is less

of

the

m.. 1

2

4 4 -

than

or e q u a l

with

m

< c IlBm+lul[ 1

2, p a r t D ). Checking

(H~)

are

proof

of

condition

t a k e m = 2.

Suppose

a of T h e o r e m

6 in o u r

A@' (t) is c o n t i n u o u s differentiable. The difference

satisfied it w e r e

.

Let m

that

~(t)

case and in t.

Since

(J(~(t+h))

IBmull

as t h e

2

IIBmull

2

for a l l m.

(Note t h a t w e

just differentiation

by

the s a m e

easy.

To see what

= that

is i n v o l v e d

is t h e

solution

~(t) E D ( A 2 ) , ~ ' (t) ~ D(A)

We must

show that

~(t) 9 D ( A 2 ) ,

J(~(t))

u ( t ) E D ( B 3)

- J(~(t)))

= <0,-

sum of three

terms

and

of w h i c h

inequality

we have

I lu(tl~ Cuct+~hl "H- uCtl ) - u t r <_ el IBuCt) I1~ I

[ uCt+hl h - u(t)

of p a r t

and

ut(t) s D ( B 2 ) .

u(t+h) 3 _ u(t) h

one

let us

is s t r o n g l y

for J,

Sobolev

are

- see Section

>

is

<0, u(t)2(u,(,t+h). - u(t) )> h

Thus,

be the

l

(I]~) is s i m i l a r .

J m is a l s o

quotient

2

i

can be written

= m

1

B as t h o u g h

The

s

I]Am-l~ll2flAm~l]

< c --

treating

+ m

2

I I ~ <_ C[ ]Bm~+lul l~l IBm2+lul I~I IBms+lul I 2

--

so t h e h y p o t h e s e s

+ m

Of t h e

Then,

(Bm~u)(Bm2u)(Bm3u)

again

l

to a s u m o f t e r m s

the e s t i m a t e :

I - u ,
-

Since

~(t)

to zero. that

is s t r o n g l y

The

same

J(~(t))

differentiable,

argument

is s t r o n g l y

works

~o

d ~-~ J(~(t)) E D(A 2-I-I)

are

essentially

the

for the

right

other

hand

side

terms,

converges

so w e c o n c l u d e

differentiable,

d d-~ J(%(t))

Therefore,

4 5 -

=
= ~

- 3u(t)2ut(t)>

and

is c o n t i n u o u s .

All

other

cases

identical. the

corollary

to T h e o r e m

7 is a D D l i c a b l e .

Let

us

suppose

that

f(x) E C

Then has

G O =

E ~D(A

the p r o p e r t y

tiable

t.

The

the w e a k

tributions) tives

then

is a C

same.

Theorem cases ~

Co(R

n

I0

in the

).

Then

~(t)

of t and each of

u(t,x)

(i.e.

Further

Therefore,

in the t i m e

Dmu(t,x) we h a v e

by S o b o l e v ' s

strongly

derivative

derivatives

if

=

often

differen-

is in ~

D(B n)

direction

are

in the s e n s e

of d i s -

is one of t h e s e APD~u(t,x)(L2(R

lemma

(see [~7],p.

deriva~)

52)

locally ,

of x and t.

case

m = o and

for O t h e r

p satisfying

(18)

are

we have:

Let

(in

solution

D ~ u ( t , x ) , ~ID(BJ )

Thus,

(18)

g(x) ~ Co(R3)

is an i n f i n i t e l y

derivatives

function

The p r o o f s the

so the

function

u(t,x).

since

for all p and m. u(t,x)

,

L2-derivatives

time

of

j)

(R 3)

u(t,x)

L 2 ( R 3) - v a l u e d

for each just

that

O

m > o,

w there

I > o

and

suppose

D a r t C) w h e r e p i s is a u n i q u e

C

~

n and p are

an odd i n t e g e r .

function

satisfies

utt - Au + m 2 u = - lu p u(x,o)

= f(x)

Ut(X,O)

= g(x)

u(t,x)

in one

Supnose on R ~

of the

f,

which

g, ~

-

46

-

We remark that various bounds on the growth of

u(t,x)

and its d e r i v a t i v e s

(in time)

of the L2-norms

follow from our estimate and the energy

inequality.

Example 2

For the cases of high p in d i m e n s i o n

or the case where the non linear t~rm is

-lu p

n ~ 3 and for p even with

I
T h e o r e m 9 since there is no aDriori estimate on the norm d i s c u s s e d in Section 2 part C.

I I@(t) [ I as

(Notice that for high p when

mean the n o r m in an escalated energy space).

n ~ 3, we

N e v e r t h e l e s s the hypotheses

of T h e o r e m 6 are satisfied for each m as can be easily checked by m a k i n g similar calculations to those in Example

1 and in part C of Section 2.

Thus we can apply T h e o r e m 6 and Sobolev's

lemma as above to conclude

that if the initial data is smooth enough, then the solution continues to have the same degree of smoothness in the interval where it exists and satisfies the a p p r o p r i a t e d i f f e r e n t i a l equation in a classical sense.

One cannot get that

C~

data go into

C ~ data in these cases

b e c a u s e the higher m, the shorter the interval of e x i s t e n c e

(- Tm,T m)

may become. Example 3

(sine-Gordon equation)

For the s i n e - G o r d o n e q u a t i o n the situation is a little d i f f e r e n t in that for high m the term.

I IAmj(r

;I =

lIB m sin ull 2

will have more and more products of

(BmLu) m u l t i p l i e d together.

This

poses no d i f f i c u l t y in one or two d i m i n s i o n s since in those cases I lul Ip ~ el IBu[ I2 for all p < -. for

p ~ 6

and in four d i m e n s i o n s

But in three d i m e n s i o n ~ this only holds for

p ~ 4. However we can also use

the estimate

I IBn~

I

i C[ JBn~

I

(26) 2

which

is valid for

part C).

n m< 4 (see the Sobolev inequalities in Section 2,

In the following we will use just

I [Bn~

(26) and

I~ i el [Bn~

Thus the proof we give s i m u l t a n e o u s l y handles the cases

(27)

n = 1,2,3,4.

We have already proven the e s t i m a t e s of T h e o r e m 7 in Section 2, part D for the cases

m = o,I.

For a general m,

-

I IAm(j(~) We will

show

that

- J(~))ll

the

term

47

-

= l IBm( sin ~

on

the

right

c(llsmG, II2 [ I B m ~ l l 2 ) I {

- sin u ) I f

1

2

side

is

Bm+l (G

r

less

-~

1

2

(28) 2

than

)ll

or equal

to

2

(29)

< C(11Am-l~tl,llAm-l~lI)

This

will

hold

for

prove all

that

m.

the

Let

hyDotheses

~ =
B like

a number

of

and

using

, ~ =
Isin u

- sin

u

we

find

finite

that sum

the

right

of terms

1 < 2

~

- cos ~ 1

2

hand

of the

(taking

~ = o)

we write

out

(28)

2

and

subtracting

the

same

term

estimates:

]

Icos

(H'm )

,v >. F i r s t 2

Adding

the

- ~)ll

and

1

differentiation.

times

(H)

,v > 1

treating

IlAm(r

side

--

{U

- ~

l

I 2

[ s {~ 1 - ~ 2 I

of

(28)

is

less

than

or e q u a l

to

a

form

11 (Bm4w) .,. (Bm'Q)

(w)~t {

(30) 2

where or

m i = m,e i=l - d

~ 1

is a z e r o

. We may

always

or o n e ,

assume

and where

m I = maxj

w

mj.

stands Now,

for either

for m ~

5, w e m u s t

2

m - m. > 3 f o r ] -

all

m., ]

j #

I.

Thus,

we

can

just

use

(26)

P il(Bmlw)'''(Bm'w)

(Q)~i[~

!

IIBm1*ll

P

< c llBm§

in c a s e

u , , u 2,

m > 5.

I IBmwi

-5[ !IBml

to conclude

-

For (27),

m = 2,3,4

or both.

we

For

just

check

example,

1

each

when

of the p o s s i b i l i t i e s

m = 3, t h e r e

are t h r e e

: 2

If (B2w) (Bw)w] i

II

< 2

--

(B2w) (Bw) II

kinds

of terms:

by

2

I I (Bw) (Bw) (Bw)wr I 2 _<

l

(21)

i lw[l

< rlB'wll ]iB~w]r il~'wrl

=

(26),

2

--

m

using

< [IB~wil l[Bwlr

=

m

4 8 -

2

by C27)

2

I I (Bw) 211211Bwl I= I lwi [.

l

!Kir(Bw) 211~ iB~wIi~llB'wTi KIIB2wII~~llB~wll ~ llB~wll 2 The

cases

right

m = 2

hand

side

a term with Thus,

B n§

we h a v e

equation. mates,

global

we h a v e

technical

been

A.

Theorem

ii

equation

domains

assume) but

then

that

ing the

(22)

has

original

Notice

there

is a m o s t

stronger

estimates

smoothness to

for the

leaving

out

B like d i f f e r e n t i a t i o n

ways

that

one

on the

Dower

(H~)

of

hold.

sine-Gordon some

of t h e e s t i -

and we h a v e

of o p e r a t o r s ,

in s i m i l a r

of A j into the

same the

J takes

data

a unique paper

Namely,

estimates.

etc.

All

paid of t h e s e

as the d e t a i l s

each

as

corollary

to T h e o r e m

in

global

C~

solution

he p r o v e s

that

in S e c t i o n

(this

hold. are the

has

same

9 just

then by

difficult

by the

of A i n t o

the

result take

assuming

only way

(H~) ~

the and

we

result one

can

is by h a v -

if one has

can get

the

estimates

themselves

out t h a t

sine-

m = 1,2,3,4.

is an i n t e r e s t i n g since

one

if

the

its d e r i v a t i v e s

is i m p l i e d This

pointed

spaces

Co(R) , t h e n

a more

if J and

of D o w e r s

von W a h l energy

and

other

conclusions

the d o m a i n s W.

are real

[3~],

he shows

two results

in e s c a l a t e d

in the

and

like d o m a i n s

initial

existence large.

the

in a d d i t i o n

can be p r o v e n

8.

in p r a c t i c e

prove

existence that

similarly.

estimates

reason

treating

If the

Theorem

right

are c h e c k e d

We s u m m a r i z e .

In S e g a l ' s than

this

to q u e s t i o n s

details

2, p a r t

m = 4

For

We r e m a r k

no a t t e n t i o n

Gordon

and

of all of t h e s e

same (H') J

global conclusions for j

4. Fin&te propagation

speed and continuous dependence

Recovering classical

smoothness

from the abstract

on the data setting was re-

latively difficult

in that it took more work and estimates.

finite propagation

speed and continuous

dependence

In contrast,

on the data are easy.

Theorem 12 Let A be a self-adjoint operator on a Hilbert space ~ and J a non-linear mapping satisfying (H~). Let ( - T,T) be the interval of existence of the solution of T h e o r e m

I.

Suppose that

#(t) of

(7) guaranteed

{Pt}t r (-T,T)

by Corollary

1

is a family of closed sub-

spaces of ~ so that

(31)

e-iA(t2

- tl):Ptl

) Pt2

if

T > t2 --> tl --> o

(32)

e-iA(t2

- tl):pt1_____9 Pt2

if

-T < t2 --< t, _< o

and J : Pt

Then,

if

Proof

) Pt

for

~ o e Po' we have

#(t) E Pt for all

t ~ (-T,T)

t E (-T,T).

We just use the same proof as for Corollary

cept that we take for

~(T,e~o)

~(t) on

satisfy

and

all

(-T,T)

which

~(t) ~ Pt for each

1 of T h e o r e m

the set of continuous

W-valued

1 exfunctions

~(o) = ~0' sup I ]~(t) - e -It 9 A~o l [ ~ %e(-T,T) t ~ (-T,T). X(T,~,~ o) is again a complete metric

space and if we define

(S~) (t) = e -iAt~o + [te-iA (t-s) j (~ (s))ds -o then all the estimates

are as in the Corollary

just check that S takes ty is concernedl.

~(T,e,~o)

Then by

e

-iA (t-s)

J(4(s))

~ Pt

We must

Then ~(s) ~ P s

on J, J ( ~ ( s ) ) E Ps also.

(31), for each s satisfying

I.

(as far as the Pt proper~

Suppose that ~(.)E ~ ( T , s , ~ o) .

each s and by the hypothesis t > o.

into itself

to T h e o r e m

o < s < t

Now,

for

suppose

.we have

-

and as in the proof of Theorem

50

-

i, it is a continuous

function of s.

Therefore,

I

te-iA(t-s) o

since Pt is closed.

Since

J(~(s))ds

e - ~ A t ~ o ~ Pt we conclude that

for

t 9 [o,T) and a similar proof using

for

t G (-T,o].

so its unique t ~(

- T,T).

Example:

s Pt

Therefore

S is a contraction

fixed point lies in

(S~) (t) e Pt

(32) shows that

(S~) (t) e Pt

on ~(T,u,~ o) C X(T,U,~o)

~(T,~,~o).

Thus,

~(t) 9 Pt for each

|

Let ~ b e

one of the Hilbert

spaces discussed

in Section 2.

That is

where

B = ~- A + m ~

= ~

k/2 = D(Bk+I)

on

R n and

+ D(Bk)

k is any non-negative

Z be a compact set in R n and define Qt to be the set of that the support of u is contained exist

y 9 Z and

z ~ R n with

Izl ~

Pt = (D(Bk+I) ~ Qt ) ~

It is easy to check that conditions

(31) and

in the set

S(Z,t)

integer.

= {xG Rnl there

Itl and

x = y + z } and let

(D(Bk) ~

Qt )

{Pt} is a family of closed subspaces.

(32) are just the statement

Let

u E L 2 ( R n) so

The

that the linear equation

utt - Au + m~u = o

has propagation by parts

(see[@],p.&@~) theorem

speed equal to one.

This can be proven by integration

(for smooth solutions) 3 by the explicit

or by the Fourier transform and the Paley - Wiener

(see ~ 7 ~ p .

linear equation

form of the solution

309).

In any case this is a statement

so we won't reproduce the proof here.

that all the non-linear

terms

Now,

about the it is clear

J(.) which we have considered have the

property J : Pt for each #o e Po

t & ( - T,T). then

) Pt

Thus, we conclude

from Theorem

#,(t) G Pt for each t, i.e. the non-linear

12, that if equation has

-

propagation whether

speed one.

globally

This

51

-

is true w h e r e v e r

in t or only

in a finite

the s o l u t i o n

t interval.

exists,

We summarize

what we have proven:

Theorem

13

In all the examples

propagates remains

Notice

In particular,

in Section

for

2 the solution

C~(R n) data,

the solution

C~(Rn). o

that this

existence ting

at speed one.

in

discussed

shows that

example

large

the blow up of the L 2 - n o r m

in Section

locally,

2, part

not because

E, is caused

in the n o n - g l o b a l

by the function

it fails to d e c a y

sufficiently

get-

fast

at infinity.

Now, data.

we treat

Whenever

the q u e s t i o n

of c o n t i n u o u s

we have the h y p o t h e s i s

dependence

(H~)

on the initial

of T h e o r e m

i, the~,

v

according

to the corollary,

~Cr)

we can at least some

interval

non-linear

=

{~o~J11~oll <_ r

solve the ( - T(r),

on each ball

integral T(r)).

e q u a t i on

For

}

(7)

(for

t G ( - T(r),

~o E ~(r))

T(r))

on

we d e f i n e

a

mapping

M t : ~(r)

,~

by

~(o) Notice, since

that no M t T(r)

M t exists uniqueness

Mt

(except M o = I)

may get smaller

it is s t r o n g l y

as

~ is n e c e s s a r i l y

r ---9

continuous

~.

defined

on all of

We know however

as a function

that w h e r e

of t and by

local

satisfies

MtM s = Mt+ s

Now,

continuous

dependence

on the data

(33)

says that

for each

fixed

t,M t

-

is a continuous

mapping

52

-

(in the topology of ~ )

where

it is defined.

For general non-linear mappings boundedness is not enough to guarantee (H~) it is. Since we are for the continuity but in conjunction with moment

just using

(H~),by "solution exists" we mean a continuous

so-

lution of

#(t) = e-itA# ~ + Ite -iA(t-s) -o

J(@(s))ds

(7)

where we just assume ~o 6~4.

Theorem

14

Let A be a self-adjoinh

operator on a Hilbert space ~ a n d

let J be a non-linear mapping on ~ satisfying hypothesis rem I.

Let D be a {not necessarily

finite interval

so that for

M t iS uniformly

(i)

= Mtd~o.

(7) on

I r,~,. (t)

( - T,T)

Then,

since

continuous

each

<_.. 11~o~

-

Let ~o ' ~o

+

f

Then

satisfies

9 D and define

the

integral

equation

initial data, we have,

lL _< I I ~"o*" - ~"~'1 I +

+ ~11

~o"

on D.

~i(t)

( - T,T) with corresponding

- ~,_ r

a

_< z<(.z,,o)

The proof is almost trivial.

Oi(t)

(H~) of Theo-

and

t ~ ( - T,T) where K depends on T and D but not on

for each t Proof

set in ~ ,

~o s D, Mt~ ~ exists and

I IMt~oll for all

closed)

la(~,. (s))

- J(~:, (s)) 1 Ids

t

oC(I I,~,. (s)I I, I1~,

2

-

11~, ,. (s)

(s) ll)

r

-)

(s) llas

I Ids

SO

(34)

Corollary

,

i

iol(t)

- ~2 (t) ll --< I1~'~--0 O~JII e tc(~'K)

Assume hypotheses

(Hn).

interval of existence of solutions the Corollary to Theorem Then for each

1 for all

t E ( - Tr,Tr) ,

of

Let

( - Tr,T r) be the finite

(7) constructed

~o in ~ (r) = {~oI

M t is uniformly

II

by the method of I I~oI I ~ r}.

continuous

on

~ (r).

-

Proof

satisfied with

Corollary

2

for all D =

solutions

Proof

Let

v e r g e to

the h y p o t h e s e s

bounded

continuous

~n(t)

T h e n by the e s t i m a t e -valued

function

f o r m l y to

J(~(t))

~(t)

on

tion

~n(t)

n

Then

on

for all

( - T,T);

~o

these

- T,T).

of v e c t o r s

converge

in D w h i c h

con-

solutions

on

uniformly

to a c o n t i n u o u s

Since

J(~n(t))

(H~)), and each

> ~ and c o n c l u d e

Since the on

~n(t)

Now,

( - T,T).

converges

~n(t)

continuous

~(t)

bounded,

uni-

satisfies

on D for each

~(t)

blows

(7)

is b o u n -

14 to

t G ( - T,T). I

corollary

for all the e x a m p l e s

the s o l u t i o n

satisfies

applying Theorem

it is clear t h a t the first those where

that

are u n i f o r m l y

( - T,T).

for short t i m e s

2 including

14 are

w i t h the same b o u n d and

t s

( - T,T).

M t is u n i f o r m l y

First,

that

of T h e o r e m

-itA~ + it e _ i A ( t _ s ) J ( ~ n ( S ) ) d s ~o o

ded w i t h t h e same b o u n d

uous d e p e n d e n c e

( - T,T)

~o | be a s e q u e n c e

we can t a k e the limit as

Examples:

on

(by the h y p o t h e s i s

t e ( - T,T).

1 it follows

14.

(7) e x i s t s

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

(34) the

~n (t) = e

we get that

of T h e o r e m

of

on ~ for each

~ o G D and let ~o" Let

to T h e o r e m

(r) so the h y p o t h e s e s

D the s o l u t i o n

are u n i f o r m l y

M t is u n i f o r m l y

~oG~

~(r).

Suppose

in the c l 6 s u r e of

for

-

F r o m the p r o o f of the C o r o l l a r y

I IMt~ol I ~ K + ~

~

53

implies

considered

continin Sec-

up after a f i n i t e

amount

of time. The a p p l i c a t i o n

to the s i n e - G o r d o n

equation

is also easy.

In that

case

rr~(t)II ~ [l~o]le t for all

~o @ ~

.

T h u s we just c h o o s e D to be the

I [~oI I ~ r} and a p p l y T h e o r e m time

interval

( - T,T),

14 d i r e c t l y

M t is a u n i f o r m l y

ball

to c o n c l u d e

~(r)

= { #o

I

that on any finite

equicontinuous

f a m i l y of

mappings. T he a p p l i c a t i o n subtle

to the case

and uses C o r o l l a r y

existence

of s o l u t i o n s

2.

J(~)

= ,

Notice that

n = 3, is a l i t t l e m o r e

in this case we p r o v e d g l o b a l

of

~' (t) = - iA~(t)

+ J(~(t))

,

~(o)

= ~o

-

if

~oED(A).

In case

to hold,

lution b e c a u s e

(H~) holds.

bound gy

on

solution

I l~(t) If.

E(t)

Then

for

but the integral However,

equation

we don't

expect

the differ-

(7) has a local

know yet w h e t h e r

since we have no d i r e c t w a y of g e t t i n g The reason

is c o n s e r v e d

we can handle

-

~o is just in 94 then we don't

ential e q u a t i o n

has a g l o b a l

54

is that we can't

conclude

since we can't d i f f e r e n t i a t e

this d i f f i c u l t y

#o ~ D, energy

by u s i n g C o r o l l ~ r y

is conserved.

Let

(29)

an apriori

that

the ener-

Nevertheless

E(t).

2.

so-

D = ~(r) (% D(A).

U s in g the e s t i m a t e

(u(x,o)

=

u o)

luol'dx~ {luo]{~ rluolJ~ 3

! {fuol]~ K IrBuol] 3 ~ llBuo{{ ~ and c o n s e r v a t i o n

of e n e r g y , w e

have

(for all

~oG~(r))

~ l}~(t) l}2 + ii ~ }uCt) r~dx

{;r

3

=

2E(t)

=

2E(o)

,f

= {{,(o){F ~ + ~

lUo r~dx

i {{~(o){t' + K{{~io){[~ < r + Kr 2

Thus,

for every

are satisfied.

finite

Therefore,

tion has a solution Since

this

is true

the e s t i m a t e defined

(35)

This

for all

for all

solutions.

B of Section

Note

the h y p o t h e s e s

~o G D = ~

(r), the

of C o r o l l a r y integral

the solution

of T the family

t is u n i f o r m l y

is g l o b a l

2, we can c o n c l u d e

equa-

{Mr} w h i c h

and since is now

equicontinuous.

for all the cases w h e r e we got global that

2

interval w h i c h has the same bound.

( - T,T),

is i n d e p e n d e n t

same p r o o f works

( - T,T)

for every

on the w h o l e

on all of ~

of strong

interval

(35)

in the case that M t

m = o, considered

is u n i f o r m l y

continuous

existence in part for each

-

t, but only equicontinuous estimate corresponding

to

55

on finite

Finally, we make two remarks.

intervals because the

(H~),

By using the escalated energy spaces 14 applied successively

(H~) hold),one J

m

for

-~(t))ll

< C(t)

j=l

4o' # o G D(Am).

~ llAJ($(o)j=l

~(o))II

So, if a certain number of derivatives

of the ini-

tial data are close enough then the same number of derivatives solution will be close. constants

involved

Secondly,

all of this discussion

15

continuously

(which may depend on t).

in the following

In all the examples on initial data.

of the

since we can explicity estimate the

in all the problems we have discussed,

explicit modulus of continuity

Theorem

j = I,...~,

can conclude that

m

IIAJ($(t)

if

( - T,T)

(35) depends on T.

or by using the trick of T h e o r e m (in the cases where

-

(somewhat vague)

in Section

we have an We summarize theorem.

2, the solutions depend

5. W e a k

Solutions

In t h i s

section

we will

show

utt

has g l o b a l veral the

weak

proofs

same

solutions

problem.

One

global

One

then

(36)

using

the

in a w e a k

sense.

applies

most

tions.

Therefore,

= f(x)

ut(x,o)

= g(x)

the

the

regularized

inequality)

one

on

D(A)

= D ( B 2) ~

sections

r

We w i l l be clear Define

take

from Fn(X)

are

in the

go o v e r the

a way

that

approach

-uP

shows

that

u(t)

of S t r a u s s

satisfies which

to r e a l - v a l u e d

func-

onl[:

0

(B L 2 (R n)

D(B)

of r e a l - v a l u e d

and of c o u r s e

introduction.

initial

the a r g u m e n t

se-

A + ~~

spaces

to this

to be the

then

argument

section

are

of the r e g u l a r i z e d

side

attention

for this

There

all use b a s i c a l l y

{Un(t) } b y a c o m p a c t n e s s By a l i m i t i n g a r g u m e n t

our

L 2 ( R n)

past

hand

(36)

in such

Un(t)

the

and

given

side

follow

D(B)

= e -tA

all n. but

if w e r e s t r i c t

= D(B)

adjoint

right

s u b s e q u e n c e of the energy estimates.

B=

W(t)

hand

solution

B2

where

x eR n

p > 0 and

right

for the

We w i l l

naturally

= - up ,

u(x,o)

regularizes

energy

equation

L i o n s [|gJ, S t r a u s s [ S ~ )

and e x t r a c t s a c o n v e r g e n t a r g u m e n t b a s e d on u n i f o r m (again

- Au + m 2 u

existence lets

the

for all odd

(Segal[30],

idea.

one o b t a i n s

that

case data

that

we

continuous

All

f and could

generates the

so we use

g to be

in

A is skew-

on ~ the g r o u p

considerations

them without

take

function

functions.

C~(Rn),

a much more

of the

comment. it w i l l

general

on R s a t i s f y i n g :

class.

-

xP F

n

(x)

=

57

-

ixL i n

linear

o

n <_ ixi < n + 1

I~I >_ n+1

and let x

G n(x) =

S

F n(y)dy

o

-

~+i}

-~

Finally,

define

Jn(~)

Then, since satisfies

Fn(X) (H~).

= Jn()

is a Lipschitz

=

function,

it is easy to see that

Thus, by Theorem 1 there is a local solution

which satisfies

~n(t) = e-At#o + [te-A(t-S)Jn(~n(S))ds

(37)

Jo

#n(O) =

The function

(38)

Un(t)

u (t) = cos n

satisfies

(Bt)f + sin(Bt) B

Jn

~n(t) =

g + it [B-Isin B(t-s)] Fn(U(s))ds o

-

and fqrmally

58-

satisfies:

(39)

utt - Au + m2u = - Fn(U)

Thus,

Un(t)

(40)

should have the conserved

EnCtl : ~I BUn (t) II~2 +

Notice ded,

that Gn(O)

(40) certainly = o, and C o~(Rn).

and g are

89

makes

Un(t)

[uJ(t) 2 n I I2

However,

f and g are nice,

of the sharp corners Let to

Fn(X),

obtained

(H~),

(H~),

~n(t).

by rounding

~

F

n

SO the solution

f

is coneven

(H~) because

differentiable

off the corners

E(t)

is that,

(H~), not

uniformly.

G n is boun-

show that

The reason

We avoid this difficulty

of continuously to

since

for each t because

we cannot directly

Fn(X).

~m) converges (H)

support

Jn only satisfies

in

~nl(X) be a sequence

and so that

+ !nGn(Un(t))dx

sense and is finite

has compact

stant because we can't differentiate though

energy

as follows.

approximations

so that

x~_%(x) ~ o

For each m, ~ *

n

satisfies

of

(n m) (t) = e-At~o + [te-A(t-S)~n)(#~}(s))ds -o

(41)

is strongly

continuously

differentiable.

Thus we can differentiate F

(42~

~i I IBuT(t) I I

~n'(t)

and p r o v e

that

it

is

is aprior&

bounded

by Theorem

2.

nm'(t ll § 2

conserved.

From t h i ~

it

)Rn

n ( n~ ' )dx

follows

that

tr~kt)

lf

~m1(x) > o) so the solution of (41) is global n [- T,T] be any subinterval of the interval on

solution

l l~mi(t) I I are uniformly trick

89

(since

Now let

which a continuous

2

+

of

(37) exists.

bounded

on

[- T,T]

Then both

(see for example,

Theoreml4

) we can prove that

) o uniformly

[- T , T ~ .

From this

on

I l~n(t) ] I

so by our usual it easily

and

iteration

I I~nml(t) - ~n(t)l~ follows

that

~nl(t) > En(t) for t 9 [- T,T] and thus, since each ~nlt) is constant we conclude that E (t) is constant. Finally, this implies as n usual that I I~n(t) I I is apriori bounded (since Gn(X) > o) and so the solution

~n(t)

of

(37) is global

and

En(t)

is constant

Now we come to the main part of the argument.

Since

for all t. f is nice,

-

59

-

E n(O) = }I IBfll z. + +I Ig]122 + IGn(f)dx

converges

as

n

> ~ to a number

E{O)

Therefore

K }l IBfll 2

the numbers

is constant

2

n

{En(O) } are uniformly hounded.

But since

En(t)

in t for each n, this means that there is a constant

C so

that (43) Let

E (t) < C n

for all t and n.

S(r) be the ball in R n of radius r and choose r o so that the sup-

ports of f and g lie in Then by

(43) and

S(ro).

Let

~

T,T] be a given finite interval.

(40) I fUn(t) I I2 _< / 2C

SO

Un(t)

are a uniformly

with values

in

equicontinuous

L~(S(r ~ + T)).

But

family of functions

(again by

(40) and

on

~

T,T]

(43)) the values

lie in

{v

.(scr o + T ) )

Since this set is compact (see [ ~ , p .

@n(t)) trick

that we have a subsequence Let v be in by

C~(R n)

_<

{Un(t) }

so that

L2 (S(ro + T))-valued

usual d i a g o n a l i z a t i o n

llvll

, llBvll 2 <

in L2(S(r ~ + T)), the A s c o l i - A r z e l a

155 ) tells us that

(which we also call continuous

I

has a convergent

Un(t)

function

converges u(t)

(Un(t),v)

+

o

theorem

subsequence

uniformly

on [- T , T ] .

to a By the

(for larger aBd larger T) we can assume so that this statement holds

for each T.

(again, this is stronger than necessary).

(38),

(44)

}

= (cos

(Bt)f,v)

+ (B-isin(Bt)g,v)

(-B-Isin B(t-s)] F (Un(S)),v)ds n

Then

-

60

-

It (cos(Bt)f,v)

+ (B-isin(Bt)g,v)

+

(-

Fn(Un(S)),B-Isin

B (t-s)v) dS

o Suppose that we can show that

(45)

Fn (Un (x,t))

then since B - I s i n ( B ( t - s ) ) v can take the limit in

(46)

(u(t),v)

=

L' (Rnx [-T,T] )) u (x,t) p

is a

C=

function of all its variables we

(44) to conclude that:

(cos(Bt)f,v)

+ (B-isin(Bt)g,v)

+

( - u (s) P , [ B - l s i n

B (t-s)~v) dx

o Since f and g are nice and the integrand on the right left side is absolutely continuous

d

(u(t),v)

=-

is in L*

, the

and

(B sin(Bt)f,v)

+ (cos(Bt)g,v)

+

(-u(s) p, [cos B(t-s)]v)ds o

Again,

the right hand side is absolutely

d~

(u(t),v}

4

= (c0s(Bt)f,

( - u(t)P,v)

+

-B2v)

continuous,

so

+ (B-Isin(Bt)g,

( - u(s)P,[B-lsin

B(t-s)](

-B2v)

- Bav))ds

o = (u(t),-

for almost all t.

d2 ~

B2V)

+ (-

u(t)P,v)

Thus,

(u(t),v)

- (u(t),Av)

+ m2(u(t),v)

(u(o),v)

= (-

=

u(t)P,v)

(f,v)

-

6 1 -

~t(u(t) ,v) I

= (g,v) t=o

so

u(t,x)

is a weak global solution oZ

utt - Au + m2u = - u p

It remains to prove use of

(43)

Since

Un(t)

(45) by a real variables

(againZ). Let [- T,T] Lz > u(t) uniformly on ~

u

of the

F

a.e. n

in

~

T,T] we have

L 2 ( [-T,T] X Rn)>

u

(again denoted by u n) so that

T,T] X Rn.

It

follows

Fn(Un(t,x))

a.e.

in

[- T,T] X R n.

~

)u

Since

u(x,t) p

IFn(X ) I< 1 + G(x)

IFn(Un(X,t)) I dxdt < 2T Vol [ S ( r ~ + T)] + -T

un

i m m e d i a t e l y from p r o p e r t i e s

that

(46)

pointwise

be a fixed finite time interval.

n

so we can choose a subsequence

pointwise

argument and clever

n

we have

Gn(Un(X,t))dxdt -T

Rn

< 2T Vol [S(r O + T)] + 2TC

where we have used the finite propagation FatoU's

i!

-T

sO

speed and

(43).

Thus, by

lemma,

lu(x,t)

n

p

dxdt <

lim

IT!

-T

ul p E LI ( [- T , T ] X R n ) .

IFn(x,t) I dxdt

Now, by the finite propagation

Fn(U n) and u p have support

in

by E g o r o v ' s

Fn(Un(X,t))

theorem that

<

n

S(r O + T) for

Itl ~ T

converges to

so

speed,

(46) implies

u(x,t) p uniformly

-

62

-

except on a set of arbitrarily small measure 6 in

~ T,T] K S ( r ~ + T).

Therefore to conclude choose 6 so that

, given e, we can

I

(45) we need only show that

I IFn(Un(X't)!I dxdt < c

M

whenever the measure of M in

[- T , T ] K S ( r o + T) is less than 6.

Now,

Fn(X) is only large if Ixl is large so given 2TC/E we can find a constant K so that IFn(X) I ~ K implies Ixl ~ 2TC/E Choose 6 so that

6K ~ s

and for any M write

M'n -- { <x,t>

and

M"n is its complement

M = M~uM ~

where

I IFn(Un(X,t)) I _> K}

in M.

Then

I /IFn(Un) I dxdt = / IIFn(Un) J dxdt + / lJFn(Un) J dxdt M

M '

M"

n

< 4--~

n

lUn(~,t) I JFn(Un) I dxdt + K6

s I [ Gn (Un(X't)) dxdt +e/2 < 4-~

<

s

(by (43))

In the next to last step we have used the fact that for each n and all x.

This holds because

Fn(X)

IxIIFn(X) [ ~ Gn(X)

is monotone decreasing

to the right of n which is why we needed the sharp corner in the definition of F . n As we remarked before,this proof follows the outline in Strauss ~6]. Strauss a c t u a l l y t e n c e holds for

proves

the

more

general

result

that

global

weak

exis-

utt - Au + m2u = F(u) as long as xF(x) < o. m

F(x)

is a continuous real-valued

function satisfying

The general ideas come from Segal's paper Do]but Segal

-

chooses to r e g u l a r i z e tity.

u p by

63

-

j$(u~j) p w h e r e j is an a p p r o x i m a t e iden-

This makes it easy to h a n d l e the c o m p l e x - v a l u e d case but makes

the proof of the c o n v e r g e n c e of

(44) as

n---~ =

more difficult.

We

remark that in these proofs one loses u n i q u e n e s s b e c a u s e of the compaetness argument.

6.

Discussion

Before

going on to s c a t t e r i n g

we have p r e s e n t e d best

aspects

proofs were

of the abstract quite

differential theses -itA e

and to point

theory

equations)

and that

is sometimes

construct an abstract -itA on e We have

chosen

the abstract conditions

special

to compute

theory

which

(the a b s t r a c t

in that the hypo-

properties .

illustrate

of the grou~

As we will

easily

without

what

The

from o r d i n a r y

one needs much more

and its limitations f(x,u,u t)

to discuss

problems.

is simple ideas

it is quite general

scattering

on functions

research

followed

and not

difficult

examples

theory

it is w o r t h w h i l e

are that_it

easy and in general

involve A and J d i r e c t l y which

theory

out w o r t h w h i l e

see,

to

information

the a p p l i c a t i o n s

trying

to give

of

general

so that

utt - Au +m2u = f(x,u,u t)

has

local or global

in Strauss ~ ] f o r T here

are a w i d e

range

can be applied. listed

interest worked

Chu,

but w h o s e

in n o n - l i n e a r of course

these methods

our old

Essentially

as it now stands.

lead,

in my opinion, The

for high

odd p.

new idea

solve

We know that

The

smooth

is

solve this problem,

problems

interesting

where

there

x GR 3

local

to prove that the solution

If one could

of the lack of Sobolev second

strong

and we k n o w that g l o b a l weak

a host of other

because

problem

friend:

is r e q u i r e d

derivatives.

are

to great p r o g r e s s

specific

utt - ~u + m2u = - u p

the data are nice

of p h y s i c a l

theory have not been

and one general w h i c h

equations.

theory

in the paper

can be applied.

one s p e c i f i c

differential

this a b s t r a c t

are many e q u a t i o n s

of the theory

solution w o u l d

partial

may be found

in most of the papers

the r e f e r e n c e s

There

of the e x i s t e n c e

in a p p l i c a t i o n

are two problems

to w h i c h

can be found

See in p a r t i c u l a r

out and to w h i c h

conditions

in C h a d a m [~] for the case m > o.

equations

examples

the details

are p r o b l e m s

difficult

and

and M c L a u g h i n [~9],

for w h i c h

There

Such general

m = o

of other

Specific

in the index.

by Scott,

there

solutions.

the case

solutions

solutions

exist

exist.

does not k e e p then

if Some

losing

I'm sure one could

is no strong e x i s t e n c e

theory

estimates.

question

is to i n v e s t i g a t e

problems

where

-

65

-

either global e x i s t e n c e is false or where s

is unknown and to try to

prove global e x i s t e n c e for certain subclasses of initial data.

As one

example of this we will prove in Section iO that

utt - Au + m2u = lu p

has a global strong solution for high D in three d i m e n s i o n s tial data are small enough.

if the ini-

A more interesting example is p r o v i d e d by

the work of C h a d a m and G l a s s e y on the Yukawa coupled Dirac and KleinGordon e q u a t i o n in three dimensions:

u

(-iV ~u + M) 4 = gr

(48a)

(48b)

utt - ~u + m2u & g~y~ 4

where

M > 0,2 = <~t'

~x I' ~x 2' ~x3 >

, the y's are certain four by

four matrices, and

= <~o(X,t) ,4, (x,t) ,42 (x,t) '~3 (x,t)>

always denotes the complex @ o n j u g a t e . of the y's,

E(t) =

Because of special properties

there is a conserved energy

~ Yo 3

(

+ M) 4 + 1

(Vu) 2 + m2u2 + ut

g

3

but it is not positive definite. d e g r e e of the time d e r i v a t i v e in space to prove local existence.

~%/0 ~ R 3

Furthermore, because of the first (48a) one must use an e s c a l a t e d energy Therefore,

it is not clear how much

good the conserved energy would be even if it were positive. these equations have both the d i f f i c u l t y of

(47) for high odd p and in

addition the fact that the energy can be negative. tions goes back to Gross ~ 3 ] w h o

Work on these equa-

investigated the coupled M a x w e l l - D i r a c

equations where similar problems arise. lutions exist.

Thus,

Gross proved that local so-

C h a d a m [ ~ ] t h e n proved that solutions exist in an arbit-

rary region of space time if g or the initial data are small enough.Then, in [~], C h a d a m showed that the c o r r e s p o n d i n g equations have global solutions

for a r b i t r a r y initial data

in one d i m e n s i o n

(this is the proof

-

66

-

that we gave in Section 2, part F). In three d i m e n s i o n s the q u e s t i o n of global existence

for

(48) is

ppen, but the f o l l o w i n g partial result of C h a d a m and Glassey [ 7 ] i s very suggestive.

They show that w h e n e v e r a solution exists then

! is conserved. - ~3(x,o)

Thus,

{1~

- ~I ~

§ I~ + ~ l ~ } d x

(49)

3

if initially,

~l (X,O) = ~ (x,O) and ~2(x,o)

then the same will be true for all time.

follows that

(in the standard r e n r e s e n t a t i o n of the

~o~

for all time.

=

From this it y's)

= I~,I 2 § l~2J ~ - I~, 12 - l ~ r2 = o

Thus,

for such initial d a t a r satisfies the free Klei~-

Gordon e q u a t i o n and one can use this to show global e x i s t e n c e for

(48).

So, for a certain subsDace of initial data global solutions exist. T h e s e results suggest that in g e n e r a l for systems w i t h o u t p o s i t i v e energy

(and w h e r e global e x i s t e n c e does not or is not known to hold

for all data)

the~e may be w h o l e subspaces of initial data

by some a l g e b r a i c

conditions)

(determined

for w h i c h global e x i s t e n c e does hold.

Such proofs will require the d i s c o v e r y and clever use of new conserved quantities.

It is not clear, of course, what the p h y s i c a l s i g n i f i c a n c e

of these subspaces will be,

but this whole q u e s t i o n seems to me to be

very interesting and worthwhile.

Chapter 2

7.

Scatterin@ Theor[

F o r m u l a t i o n of S c a t t e r i n @ Problems in this lecture we provide b a c k g r o u n d and m o t i v a t i o n

in the rest of the chapter.

for d e v e l o p m e n t s

Let us begin by looking at the equation

utt - Au + m2u = -Xu p

(50)

whose e x i s t e n c e theory for some n > 1 we treated in Chapter d e v e l o p e a scattering theory for

I.

To

(50) one would show that for large

positive and negative times the solutions of

(50) look more and more

llke solutions of the c o r r e s D o n d i n g free equation:

utt - Au + m2u = o

(51)

It is not clear ~ priori why this should be true since the n o n - l i n e a r term

-Xu p not only creates an interaction t h r o u g h o u t all of space but

is clearly very large if u is large. of the free equation like

On the other hand, the solutions

(51) w i t h nice initial data decay

t ~n/2 in n dimensions.

(in the sup norm)

This raises of the p o s s i b i l i t y that the

same decay will hold for solutions of theory might be p o s s i b l e since

(50) in w h i c h case a s c a t t e r i n g

-Xu p is very small w h e n u is small.

Thus, we expect the e x i s t e n c e of a s c a t t e r i n g theory will depend on a d e l i c a t e interplay between the rate of decay, the degree of the nonlinear term, and the sign of the n o n - l i n e a r term since we know from Section 2, part E that some solutions b l o w up in finite time if X < o or if p is even. As in Chapter

1 our t e c h n i q u e is to r e f o r m u l a t e the s c a t t e r i n g

theory p r o b l e m as an abstract Hilbert space problem, prove abstract results, and then return to apnlications. o p e r a t o r on a Hilbert space ~ a n d

So let A be a s e l f - a d j o i n t

J a n o n - l i n e a r m a p p i n g from ~ t o

itself so that

#(t) = e-itA~ O + [ t e - i A ( t - s ) J ( ~ ( s ) ) d s -o has global solutions at least for some initial data in Chapter then

~(t)

(52)

r

I, if J satisfies certain a d d i t i o n a l conditions is d i f f e r e n t i a h ~ e and satisfies

As indicated and ~o ~ D(A)

-

68

-

r (t) = -iAr

However,

in this c h a p t e r we c o n c e n t r a t e

do not w o r r y equation

about

(52).

smoothness,

What we would

of i n i t i a l d a t a

Xscat~

(a)

r

For e a c h

- e-itA@

That

is, as t

(c)

~

r

is that t h e r e

as

r

t

- e -itA r

That

is,

)

~(t) with

(a) and

there

o

@(t)

of

~(t)

(52) so that

like t h e s o l u t i o n @_.

of

)

free s o l u t i o n

+~-

the w a v e o p e r a t o r s

r (o)

n+ : r

;

~(o)

(52) so that

+~.

like the

as t

)

~+ and R

by

like to p r o v e that

~+ = R a n g e

(this is c a l l e d the p r o b l e m scattering

~

~_ : r

Range

~+

t

~ +

(b) h o l d w e d e f i n e

F u r t h e r we w o u l d

also that

is a s o l u t i o n

looks m o r e and m o r e

initial data

is a nice set

)

l o o k s more and m o r e

as

and

integral

properties:

is a s o l u t i o n

o

-~

~Zsca t

~(t)

If

like to p r o v e

w i t h the f o l l o w i n g

)

behavior

just w o r k w i t h the

of the free e q u a t i o n w i t h i n i t i a l d a t a

For e a c h

e-itA~+

on the a s y m p t o t i c

so we w i l l

t there

~(t)

e-itA~_ (b)

+ J(r

and

~_

of a s y m p t o t i ~

~_ are one to one.

operator S : [scat

by s =

(~+)-i~_

)

[scat

completeness)

an~

T h e n we can d e f i n e the

-

(d)

Finally,

69

-

one would like to prove some properties of S.

ample, that S is a continuous m a p p i n g of [scat Basically,

For ex-

[scat onto itself if

has an a p p r o p r i a t e topology.

our a p p r o a c h to the p r o b l e m is to try to solve the initial

value p r o b l e m for

(52) with Cauchy data

~_ given at

t = -~

That

is, t o show that the e q u a t i o n

~(t) = e - i A t ~ _ + It e - i A ( t - s ) J ( # ( s ) ) d s

(53)

has a global solution, that and that there exists a

~(t) - e - i A t ~ _

~+ so that

9 )

o

as

~(t) - e-iAt~+ ~

As we m e n t i o n e d at the beginning,

t o

~

-~,

as

t

) +~.

carrying through this c o m p l e t e

p r o g r a m for highly .non-linear equations w i t h interactions t h r o u g h o u t space is a very d i f f i c u l t problem. a few cases.

However,

the initial data

~

C o m p l e t e results are known only in

two parts of the theory are easier.

at

t = -~ is small

First if

(or if the c o u p l i n g constlat

is small) then one can prove global e x i s t e n c e and for a scattering theory quite easily if the solutions

(53) and d e v e l o p

e-!At~_-

of the free

equation decay s u f f i c i e n t l y rapidly and J has high enough degree. E s s e n t i a l l y the reason in that the n o n - l i n e a r t e r m never gets large enough to d o m i n a t e the equation. for small data in Section

We d e v e l o p

this s c a t t e r i n g t h e o r y

8.

The second part of the p r o b l e m that is fairly easy is to show the e x i s t e n c e of the w a v e operators

~§

and

~_.

(52) have global solutions, that -e-ltA# and that J have high enough degree.

The proof w h i c h uses many of the

ideas of Section 8 is given in Section In Section and

i0

i0.

ii we give applications of the results in Sections 8,9,

What remains in most cases is the p r o b l e m of a s y m p t o t i c com-

pleteness. in Section

This is a really d i f f i c u l t p r o b l e m b e c a u s e

(as we discuss

12) its solution requires a priori decay estimates on solutions

of the non-linear equations. cases.

What is r e q u i r e d is that

decays s u f f i c i e n t l y rapidly,

However,

Such estimates are known in only a few

if one has such an aDriori estimate,

then one can carry

through the entire theory if J satisfies the right properties. we do in Section

This

12.

It is w o r t h m e n t i o n i n g before we start, two t e c h n i c a l d i f f i c u l t i e s which do not occur in q u a n t u m m e c h a n i c a l scattering. ~ave and s c a t t e r i n g operators are non-linear,

First,

since the

it is not sufficient to

pro~e that they are bounded on a dense set in order to conclude that

-

they are continuous. states,

Secondly,

so that

in most applications,

condition are known for such decay. [scat in

be closed

(for example,

-

it is natural to take as s c a t t e r i n g

~scat' the set of ~ in ~

Unfortunately,

70

e-itA~

decays appropriately.

only sufficient but not n e c e s s a r y Thus if we w a n t in a d d i t i o n that

to be a Banach space)

then the n o r m of a

[scat must involve e ~ p l i c i t l y the large time b e h a v i o r of

e-itA~.

In what follows these two d i f f i c u l t i e s will cause quite a bit of technical pain. Finally,

as you will see, we will need quite strong hypotheses on

J in order to d e v e l o p e as abstract s c a t t e r i n g theory, many more hypotheses than we needed for the e x i s t e n c e theory.

This is natural since

the e x i s t e n c e of a s c a t t e r i n g theory reflects a very close r e l a t i o n s h i p b e t w e e n the solutions of the free and interacting equations. requires quite strong h y p o t h e s e s on J. f o l l o w i n g situation sometimes occurs.

Thus,

We remark however, that the In a p a r t i c u l a r application, J

may not fulfill all the h y p o t h e s e s of the a p p r o p r i a t e t h e o r e m below. Nevertheless,

one can o~ten Drove the c o n c l u s i o n of the t h e o r e m by

using special p r o p e r t i e s of the p a r t i c u l a r equations and the idea of the general theorem.

it

We w i l l see an example of this in Section

IZc.

8.

S catterin~

In this directly idea

lecture

&f the

is t h a t

enough

for s m a l l

the d a t a

then

to the

and a s c a t t e r i n g no e n e r g y theorems global

show h o w to c o n s t r u c t at - ~ are

if the d a t a

degree

comparison

we

data

is s m a l l

the n o n - l i n e a r linear

terms.

theory.

The

inequalities provide

existence

fails

sufficiently

The basic

term

remain

From

should this

always

one

point

can

small

data;

in

existence

is that we w i l l

at all.

data

high

small

get g l o b a l

here

boundedness) for

for large

"small".

operator

t e r m has

important

existence

scattering

and the n o n - l i n e a r

(or a p r i o r i

global

the

Thus

in some

for e x a m p l e ,

use

these

cases

where

for

utt - du + m 2 u = u p for

large

p.

This

p a r t E of S e c t i o n get

the

solution

We r e m a r k that

is,

to b l o w

that

if w e

is c o n s i s t e n t

with

2 since we had

up in finite

if t h e r e

are d e a l i n g

the n o n - e x i s t e n c e

to m a k e

the

example

"large"

in

in o r d e r

to

time.

is a s m a l l with

the d a t a

coupling

integral

constant

equation

in front

that

of J,

corresponds

to #' (t) = - iA~(t)

then

the t h e o r e m s

necessarily larger methods

the

be two

small) data

the

of this

So,

in this

section

initial

data

smaller

section)

all d a t a

"norms"

satisfies

all the p r o p e r t i e s

value the

(i)

+~.

We

following

There

that

on ~

except

assume

A, J,

There

enough.

onW.

not

except

handle

I I I Ia and

the (by the

I I ] Ib

all the p r o p e r t i e s that

that

and the n o r m s

However,

(not

g.

Let

imply

for any

can't

fixed

satisfies

need

of a n o r m

existence

so we

for any one

: I [.I ]a

(54)

~ = o;

it m a y

I I .I Ia,

take I I -I [b

] [. ] Ib the satisfy

hypotheses:

is a

c > o

so t h a t

]l~II a _< cl I~[ I

(ii)

be c h o s e n

operator

I I$[ ]a = o

that

global

if g is s m a l l

let A be a s e l f - a d j o i n t auxiliary

imply

g must

of a n o r m

+ gJ(~(t))

are

constants

for all

c I > o, d > o

~ c~

So that

(55)

for

~ e

-

72

-

{{e-iAt~l{ a < c t-d{{~{Ib (iii)

There exist

IlJ(~) 1

if

It{ > z

B > O, 6 > o, and q > I with

(56)

dq > I, so that

- J ( ~ )2l l

(57) 8(I{~ilI a + Jl#x{la)q[{#x

- ~2{ I

{IJ(#x ) - J(#z) l{b (58)

<_ 8{(I[r

for all

a + ll@2[la)q-ljl~z - ~aIla + (II~zll a + II~211a)qll~z-~2[l}

~z'#2 E ~

satisfying

{{#i{l < ~.

In the case

q : 1

we

assume that 8 can be chosen arbitrarily small if ~ is chosen small. We can now define the scattering states and the scattering norm. First, for an 94-valued function

III~()rI![N,N)~

sup

ll~(t) ll +

N
In the case where

N

--

2

R we define

(l+]t{)dl l#(t) I{ a

s.p

N
= - =, N 1

Ill" I J I"

~(t) on

--

2

= + ~ we will denote the norm simply by 2

NOW we define

[scat E {~ e ~

III

e-itA~l I I < | }

and I I~I Iscat 5 }} {e-itA~l I}

That is, the scattering states are just those vectors in ~ w h i c h nicely under the free propagation.

Notice that if

{ le-itA~l la < c (l+[tl)-d(ll~[{ --

where

c

= max {2dcl,2dc 2

+ {{~I {b)

2

},

so

~ ~ ~scat and

decay

I I~Ilb < ~, then

for all t

(59)

-

7 3 -

il~llsc~t < (~+c)II~II --

+c

2

11~tlb

2

We can now state our first main theorem: Theorem

16

(g~obal existence

Let A be a self-adjoint linear mapping I I'll a,

of ~

II "lib

there is an the equation

for small data)

onerator

into itself.

so that hypotheses

no > o

2~ o.

global

continuous

where

and J a non-

(iii) hold. with

Then

] I~ I I ~ ~o'

e -iA (t-s) J(~(s))ds

-valued

For each t, ~(t) E [scat

(b)

ll~(t)

(c)

In case

The basic

X(n,~

so that

(ii), and

~ _ L [scat

(a)

we employed Let

~

space ~

that there exist norms

solution

(6O)

~(-) with

I I I~(') I I I

Moreover,

- e-itA#

- ~_llscat

§ o

) denote

1 except with

~ is chosen

initial

the set of continuous

- e-itA~_l I I ~ ~.

~

Assume

so that hypothesis

to:

as t § -~

idea is to use the contraction

in Section

I I l$(t)

II § o as t § --

q > I, (b) can be strengthened

lle-itA~(t)

Proof.

(i),

so that for all

#(t) = e-itA~_ + it

has a unique

on a Hilbert

Suppose

(iii)

mapping

conditions -valued

that

method which at

t = -=.

functions

~(t)

I I~ I Isca t ~ ~ ~

holds,and

for

~(.)tX(n,~_)

define ft (~)

(t) =

As in the proof of Theorem is a continuous

function

e-iA(t-s)J(~(s))ds

I, it is easy to check that e-iA(t-s)J(~(s))

of s for each t.

I II~(t) I {I ~

(61)

Further.

II le-itA~_I I I + ~ ~ 2n

since

-

74

-

we have,

l lJ(~(s))II <_B f i~(~)Ilaq Jl*(s){f

by (57)

< 8 ( 2 n ) q + l ( l + l s l ) -dq

so, the right hand side of (61) makes sense and

I1(~*~(t) ll <_ it I l e_iA (t-s)J(~(s~) I Ids < 8(2n)q+l [ t (1 + Isl)-aqds

(62)

J

since

dq > I.

Also,

iJ ( ~ )

(t) IIa < I~

IIe-iA(t-s)J(~(s))

lads

t

by (59)

< c ~' (l+ft-s)-a(lIJ(,Cs))flh + IfJ(*(s) ll)ds

--

<

2 J_~

c

2

B

ft

(l+It-s[)-a{ [lr

< C 8(2n)q(l+2n) < m

c 2 8(2n)q(l+2q)c

q I+ 21 I~(s) I I) } ds lla(

(l+It-sl)-d(l+]Sl) -dq ds

(63)

(l+Itl) -d 3

by the lemma proved after the completion of this proof. I I l ( ~ ) (t) IIi < -- We now define (M~) (t) = e - i A t ~

and choose

by (57)

no(and 6 in the case

+ (~)

Therefore

(t)

q = I) small enough so that

8(2~ q+l I_~ (1+Isi)-aqds <_ no/2

(64)

-

75

-

c 8(2no)q(l+2qo)C 2

< n /2

~

3

--

0

it is easy to check that (M~) (t) is continuous. Thus, for ~ ~ no, M maps X(q,~_) into itself. Furhher, it is easy to check using (58) that by choosing q O still smaller

(or 6 in the case q = I) we can guarantee

that M is a contraction.

Thus,

since

X(q,#

space, M has a unique fixed point

~(-)

of M, ~(.)

(60).

is a global

solution of

in

) is a complete metric X(q,~_).

By the definition

Notice also that

I I I~(') I ] I <

2~ O 9 To prove uniqueness,

let ~ (t) be another

solution of

(60) with

1

I I I~ ! (t)I I I < ~.

Then

~(t)-~, (t) T I <

t

<_

s

<

Bcllle(.)ll

B(ll~cs)

I

t l IJ(~(s)) - J(#, r

I a + lie (s) ]a )q lie(s) - e

+ Ille,(')l

t)q(

) l lds

(~)llds

s u p Ilecs)-e -~<s
st ~l+lsl)-dqds

r z

-

SO

sup -~<s
]~(s)

- ~

(s) tf 1

tt

<

{~r

I~(-)l

I + Ill~ r !

11, q I

J_

el§

dqds

sup il Cs,

But this gives a contradiction

for t sufficiently

~(s) = ~ ! (s)

for all

By local uniqueness

~(s) = ~ ! (s)

for all s.

To show that

sup r

s < t.

~(t)G[ scat

Cs,

--m<s
close to

1

-~ unless

(see T h e o r e m

I)

for each t; we fix t and compute:

i ie-irA~(t) [ i < supl le-irAe-itA~I i + it I 1e-iA (t+r-s) J(~(s)) I ads r -~

-

i supl]e-irA~-II + B

76

-

It I Ir

r

ll q I Ir a

-~

IIr

§

sup(1+IrT)Ul le-irAr

1 Ids

~0

I la _< sup(1+Irl)dl le-i(t+r)Ar II -

r

a

r

+ sup(i+lrl)dlt r

lle-iA(t+r-s)J(r eo

! sup {(l+Irl)d(l+It+r[)-dl[r r + sup{ (l+Ir I)d [t (l+It+r-sl)_d(l+iS I)_dqds } 8c2 (l+2no) (4no)g r -= no + ~-- )

<_ sup{(1+Irl)d(l+It+rl)-d}(IIr r Thus, r

& ~scat"

To prove (b) and (c),we estimate:

I Ie-irA (eitAr (t) -*_) I I s it I le_irAeiSAj( r

< ~ ,|

qa l l,Cs~llas

s 8(2no)qno

(l+Isll-dqds

--

) O

Taking

r = o, this proves (b)~

Ifds

Further,

as

t § -=

-

7 7

-

(l+lrl)dl le-irA(eitA~(t)-%_) I la

r

I~ Ile-iACr-s)jr

t c 2 (l+Ir-sl) -d {IIJc~Cs))II < (l+Irl)d I_. + I IJr

--< (l+r)dc2

< 8 C2(2~ O

SO, if

It -. (l+Ir-sl)-ds{2i I~(s)Ilqla l~(s) II

+

.

ds

+ I l~(s) l Iqa}

I)d(l+Ir-sl)-d(l+Isl) -dq ds

q > i, sup(l+Irl)dIIe-irA(eitA~(t)

- ~_)I[ a

as t § -= by part (b) of the lemma below. follows that

o

From this and the above it

lleitA#(t) - ~ IIsca t ~ o

as

t

§

-m

, I

Notice that the solution of (60) constructed above satisfies

~(t) = e-iAt~_ + It -e-iA(t-s)J(~(s))ds

= e -iAt { ~

+ I~ e-iAsj(, (s))ds}

+ [t e-iA(t-s)J(%(s))ds o

so

ds

~(t) satisfies

(52)with ~O = ~-- +

;~

e-iAsj(~(s))ds

-

The following

Lemma

(a)

lemma completes

Suppose that

|

(b)

-

the proof of Theorem

q > I,

d > o,

and

16.

dq 9 I. Then

-dq ds ~ c ( l + J t l ) -d

r162

Suppose that

sup

78

q > 1,

d > o,

and dq > i.

((1+l~l)d It2 (1+Ir-sl ) - d r

Then

~o

t]

as

t ,t 1

Proof.

~

+=

or

2

To prove

t ,t 1

>

(a) it is sufficient to consider the case where t is

large and positive.

The proof for t large and negative

We break the integral

is similar.

into two parts and estimate:

( ~+ I t-s I )-d ( ~+ I ~ t ) -dqds <_ (1+~) -d

and for

-~ .

2

! r 1+1 ~ I ) -dqds

d # 1 ,

I

3~(1+1t-

s

)-d(l+Isl)-dqds

t

(65)

t <

(l+~l-dq

{I

(l+(t-s))-dds t

! 2 (I+~)"aq (r

t + 13~ (l+s-t)-dds} t

r(I+~)-d+I+I) }

-

79

-

< c(l+t)-dq -d+l + c(l+t) -dq

< c (l+t)

since If

dq > 1 and

-d

q > I.

d = 1, the right hand side of (65) can be estimated

by

2(1§ -q log(1+~) < c (i+t) -I

since

q > I

if

d = I.

Combining

these estimates

proves

(a).

We will prove (b) in the case t ,t § +~. We assume that t, is ] 2 large and positive and divide the domain of integration into two parts.

(l+lrl)d

I

.

(l+lr-sl)-d(l+lsl)-dqds

r 3r

It I 't 2]~[~' T ]

r )-d ! (1+Irl)d(1+1~l !~\ (~+lsr)-dq~s [~, ~r [t , T] t

!

and assuming

c

I~t

(l+Isl)-dqds

(66)

!

d # I

(l+Irl)d

!

rr 3r](l+Ir-sl)-d(l+Isl)-dqds

It I t 2 ] n L 2 , 2

J

< (l+Irl)d(l+I~l)-dq

I

(l+Ir-s I)-dds

"

3r Lt t2].[~r] -

r

-

< (l+Ir[) d

80

-

r -dq [c+c(i+(r I)-d+i1 (i+I~()

< c (l+Irl) d(l-q) --

2

< c (l+It

since the integral

The case

d = 1

The fact that part case

q = 1

is actually

I) d(l-q)

is zero unless

Combining this estimate q > i.

special

+ c (l+Irl) l-dq

1

and

+ c (l+It,l) l-dq

r ~ ~ t I.

(66) we see that

(b) holds since

is handled as in the proof of (b) of the lemma holds only if

in all that

false in the case

q

follows.

(a). q > 1

dq > i,

l makes the

To see that the conclusion

= I, notice that

I

t*+l(l+It-sl)-d(l+Isl)-dds t! > (l+It +ll)_d --

!

It l(l+It-sl)-dds +I t

l

= (l+It +ii) -d Ii(l+Isl)-dds !

if

t = t . 1

0

Thus, sup t

rt +I (l+It )d ]ti(l+It-sl)-d(l+Isl)-dds !

l~lhil o(l+isl )_dds l+[ti+l

We now have global existence

and the right properties

To construct the scattering operator we must construct ~(t) - e-iAt~+ ) o as t ) +~.

at

a ~ +

-~ so that

-

81

-

Theorem 17 (the scattering operator Assume all the hypotheses of Theorem

for small data) 16 and let ~(t) be the solution

of (60) corresponding to ~o sufficiently small,

with

(a)

There exists

~ _ i [scat

~+6 [scat

' with

The map

topology)

~_ - - ~ #+

) o

as

(in the

II

{~ & [scatlIl~l Inca t ~ n O } into the ball

{~& [scatlIl~l Iscat ~ 2~o}"

Except in the case

q = i, the following

(c)

- ~+[}scat ---) o

(d)

I leitA~(t)

S is continuous

Proof.

also hold:

as t § +~

]l.llscat -topology.

in the

IIl~(t) lll

From Theorem

16 we know that

I leitIA% (t)

- eit2A~( t ) I I < I I It2eiSA J (~(s))dsl 1 2 Jtl

I

!2no.

Thus,

--

< I 'Bi1 (s) IIq II <s) IIds

--

a

! < 8(2no)q+II~2

(l+Isl)-dqds i

by

(57).

Thus

{eitA#(t)}

is Cauchy in ~ a s

t § +=

L~tting

~+

=

lim t§

eitA~(t)

we have ll~(t)

- e-itA%ll

§ o

for

t + +~

is a one to one and continuous

map of the ball

Then,

l~+llscat ! 2~ o, so that

I l#(t) - e-itA~+l I

(b)

I I~ I Iscat ~ n o .

as t§ +|

since

dq > i.

II

-

by the unitarity of

e -itA.

82

-

To show that

~+ E [scat' observe that

eitA~(t) = ~_ + it eiSAj(~(S))d s

Letting

t § +~

we conclude that

~+ = #_

+ I|174

NOW, by (59) and (57), I Ie-iA(t-s)J(~(s)) lid -< c 2 (l+It-s

< c

B

< for each s and t.

s(1+It-sl~'a~ll~r

R

C

--

~-d~{{ar162

+

{{ar162

q (I+211~cs) II)~

8(2no)q(l+no ) (l+It-sl)-d(l+Isl) -dq

2

Since for fixed t,

e-itA~+ = e-itA#_ + [~ e-iA(t-s)j(~(s))ds

we conclude that

I le-itA~+ IIa < ~

sup(l+ltl)dlle-itA~+lf

and

a

t

sup(l+itT)dlle-itA~ IIa t

+ c28(2no)q(l+4no)SUp{(l+Itl)

t

d I ~-

r162

-

83

-

n0 sup(l+Itl)dlle-itA~_lla + ~-t

by the lemma

(part a) and the choice of ~o

ll%llscat

This proves

Thus

no + ~- ~ 2no

DO

+ ~-

(a).

We can now define S: {# ~ s c a t [

~ ll~_llscat

in Theorem 16.

~_ § ~+

I l~IIscat ~ ~o } into

and it is clear that S takes {~ ~ [scat

[I@l Iscat ~ 2no} "

The proof that S is one to one is similar to the uniqueness proof in Theorem 16.

To prove that S is continuous

proceed as follows. and let ~n(t) and

]l~m(t)-4='(t)ll

< II ~ -~'~'I I_

< I I ~ * ' - ~ ~' II +

-

l~L~'(s) l 11)qlt

(x+Isl)-dql lr

[ I < ] l ~ (1~ "422 I lexp{

By part

[IJ(~'*'(s))

dq > I, interation of this inequality

I I~r

I I" I I-topo~ogy we

ft

J(~'~'(s))!Ids

(z+lsl) -dq] l~U'(s)-@~'(s) 11as

I las implies that

S(4no)q(z+[sl)-dqds}

(a) , eiAt (r

as

It

+ s(111~S"(s) lll + I qt + s(4n o) ;_|

Since

in the

Let ~ I and ~ be in {~I II@[Iscat ~ n O} @21(t) be the corresponding solutions of (&O),Then

- 42'{t))

t § ~ , so we conclude that

]I J l,9 ~i** _ ~+I T,+

-

84

-

(67)

which proves that S is continuous To prove

in the

II'll norm.

(c) we estimate:

(z+[rl)d] ]e-irA(eit~A~( t

) - eitIA~( t )I Ja 2

!

_< (l+Irl) d It2 [ le-iAr tl d

/t2

< (I+)r$> c B| --

2

II ds a

(1+}r-si>-a(li~(s>li q +2)l~r

"t,

i1~r

il>ds

a

_< c 8(2no)q(l+2~ o) 2

{(l+Irl) d ;t2 tl

(l+lr-sl)

-d( l+IsJ)-dqds}

By part (b) of the lemma,the sup of the right hand side goes to zero as t , t § +~ if q > I. It follows in this case that eitA~(t) is 1

2

eauchy in the To prove Let

II.mlscat (d) we use

4 .I (t),@ 2. (t)

norm so (c)

mleitA#(t)

- ~+IIscat + o

and the continuity already proven in (67)

be as in the proof of (b) and define:

ocs)

= r162

-

~'r

lla

and P(t) =

sup Q(s) -~<S
Then, Q Ct) <

(l+ltl)dl

le-itA(~u_~

+ (l+Itl) d It --co

_

as t + +-

~(2f )

II

II e-iA(t-s) cJ r162162)

-

J

(,'2'r

I [a as

-

<

I14 ~'-

~llscat

+ Bc ( 1 + I t l )

9I~|

85

-

d

I~'1'C~ Ila + I I~'~'(~)lla)q-~l

< II~"- ~'llscat + Sc (4no)q-l(l+Itl)dlt

+ 2Bc r

II~'*_' -

where

we

have

lemma

we

see

used

(67)

J~'*'Cs) - ~'(s) I lad~

(l+It-sl)-d(1+Isl)-dqQ(s)ds

~=_'IIscat(l+Itl)dlt(l+l

in t h e

last

step.

t-s )-d(l+Isl)-dqcl8

Thus,

uslng

part

(a)

of the

that

Qr

< c lJ~'- 4~'Ll

+ Bc r su~ Qr

< c I I~l' _ ~ I 4 -

+ ~c P(t)

--

~

--

--

scat

5 -~<s
or P(t)

where made

the

(68)

s

constants

arbitrarily

8 small

I scat

in t h e

c and c just depend on ~o a n d 8c may be 4 5 s small by choosing nO s m a l l in t h e c a s e q > 1 and

case

q =

i.

Thus,

for

no

small

enough

C P(t)

Combining

this

with

~

(67)

(i-8c) 5 we

I I~41~ - ~ f l l s c a t

conclude

that

there

is a c o n s t a n t

c

so t h a t : G

1[l~'(t) - ~x*(t)[tl < c I I~r --

Finally,

we

6

--

estimate

( l + I r l)d 1 le-irA(eiAt#l**(t)

_ eiAt~(21(t))I Ia

42,1[ --

s c a t

(69)

- 86 -

_ II~T <

~t2_'llscat+(l+lrl) d I t I I e-iA(r-s)

-

(J (*r

! c l l~'*J ~'~_'Trscat + c?l I l~m(t)-~'2'(t) I l ] (l+Irl)dI~

< c 8 I 1 ~-'-

usual

estimates

c depends

part

From

and

(69).

From this

- ~iAt~'(t)t Iscat

I ]eiAt~"(t) where

lads

(i+Ir-sl)'d(1+Isl)-dqds

~'~*[I -- scat

--

by the

)I

-J (r

on

no

(and

6 if

(67)

we h a v e

c l I~"_'- 42_'IIscat

<

q =

and

I) but

not

(70)

on t or #(,I, ~ !

(c),

) I I .I Iscat ~ ~L*)

eiAtr

-+

(71)

e

if

iA~

q > i, so in that

I1"11

~l(t )

~2) -+

scat.

case w e c o n c l u d e

that

I I~I'+ - ~'f Iscat -< c I ~'_ - ~='I_Iscat which

proves

Notice q > I, .+ ~Ln _

all

q ~

in the

tinuous

the

following

and

in

[scat

~2~ .+

norm case

in the

We g i v e going

that we have

1 we h a v e

I I .I Iscat only

(d). I

are

uniform

for all

interesting

by p a r t

continuity

finite

times

q > 1 t h a t we

can

of the m a p t

applications

on let me

16 and

17.

priori

estimates

of t h e s e

emphasize

Namely,

again

energy

inequalities

of the

linear

used.

equation

two v e r y

(70)).

f r o m this in this

important

requirement

But

that

case

%n S e c t i o n

non-linear

sufficiently

For and

~_ § eiAt~(t)

is just

of the t h e o r e m s

of the

The only

decay

only

theorems

the h y p o t h e s e s

on s o l u t i o n s

(this

conclude

I I .I Iscat n o r m b e c a u s e

situation.

(a) of the t h e o r e m

in the it is

S is con-

d o e s (71)

II.

hold.

But b e f o r e

aspects

of T h e o r e m s

did

require

not

equation

nor w e r e

was

the

rapidly

all for

that

and

that

any

solutions

the n o n -

-

87

-

linearity be of s u f f i c i e n t l y high degree.

In particular,

the method

w i l l w o r k for cases w h e r e the c o n s e r v e d e n e r g y is not b o u n d e d below. The idea that one can d e v e l o p a s c a t t e r i n g theory for "small data" goes back to the paper by Segal ~3~].

In this paper Segal concentrates

on applications to the K l e i n - G o r d o n equation with many special properties of the kernal of

e -iA(t-s)

up

interaction and are exploited.

S t r a u s s ~ $ ] simplified Segal's W o r k and formulated the p r o b l e m in terms of abstract hypotheses on A and J. of Strauss'

ideas in [%7].

We have followed the e l a b o r a t i o n

-

9.

Global existence There

88

-

for small data

is amother aspect of Theorems

that is deserves hypotheses

(i),

16 and 17 which

to be set out separately. (ii),

is so important

That is, if we have the

(iii), then the initial value problem at

t = o,

namely, ~(t) = e-iAt~o + I I t e -iA(t-s) j (~(s))ds o

(~)

$(o)

=

$o

has a global solution

]Ir

if

t

is small enough.

The proofs of

this and the other parts of the theorem below are almost exactly the same as the proofs of Theorem

16 and 17.

Theorem

for small data)

18

(global existence

operator on a Hilbert space ~ itself.

Suppose that there exist norms

hypotheses

(i),

in Section

8.

(a)

For each

(ii),

(iii) of Section

~o i [scat

Let A be fixed.

existence Further,

8 hold.

' the equation ~(t)

Let

]l~ol]scat

of part {al holds

for all

nO > o

into

~scat be as defined

I lr

- e-lAte II

II$(t)

- e-iAt$+ll

)

''>

(a)-1

~

I [~ol ]scat ! no"

and

~+ in ~scat so that

o

as

t

o

as

t ---~ + ~

(~+) -I ~(o) ~

in the

{c)

(iii)), then

q > 1 (in hypothesis

or ~ is small enough.

so that the global

#o satisfying

#o' there exist

are one to one and continuous If

~

II -II b so that the

(1) has a global continuous

if either

Then there exists

for each such

and the maps

ll- lla,

Then,

[scat -valued solution (b)

Let A be a self-adjoint

and J a non-linear mapping of

9 - ~

r

II'I I norm. (b) can be strengthened

] leiAt~(t)

- ~ I ]scat

~

o

as

t

I leiAt~(t)

- ~+I Isca t

> o

as

t

~

+~

to

-

and

(n+)-1,

(~_)-i

This t h e o r e m

89

are continuous_

-

in the

can be used to show that

utt - Au + mZu = Xu p has global

strong

the sign of X is

solutions

II

llscatnorm.

for small e n o u g h

initial

data

x ~R 3

as long as p is large e n o u g h no m a t t e r w h a t

(see Section

II, part c).

iO.

Existence

of the W a v e o p e r a t q r s

In the case w h e r e data,

global

solutions

we can use the ideas of the

operators

on all of

will denote

[scat'

not

of

~(t)

(63).

is the

)

local s o l u t i o n

estimates

of

a contraction

(52).

in the e x i s t e n c e

The

proof

idea ~s as follows.

of T h e o r e m

+

m a p p i n g w e had to m a k e the r i g h t

( -~,T o) w h e r e T O

(if

~(t)

to an i n t e r v a l

q > i) the r i g h t

are s m a l l e v e n

if

s m a l l enough.

~ is not small.

a l l o w one to e x t e n d

fine the w ~ v e o p e r a t o r 16 and

~(t)

(existence

operator

Suppose

that there

(ii)


t i o n 8)i

Suppose that

for e a c h

uniformly

bounded

If.If-norm)

(a)

For e a c h so t h a t

( - ~,O~ are s i m i l a r

on a H i l b e r t

(i),

(72)

to

on

and g l o b a l

and thus deto t h o s e

in

For c o n v e n i e n c e

we

(72)

of the w a v e o p e r a t o r s )

itself.

(in

local u n i q u e n e s s

+ it e -iA(t-s) j (~ ( s ) ) d s o

so t h a t the h y p o t h e s e s

o < It[ < T.

Then

solution

-~

the i n t e g r a l s

equation

= e-itA~(s)

Let A be a s e l f - a d j o i n t of ~ i n t o

this

just s k e t c h the proof.

integral

is close to

In this way we get a s o l u t i o n

S i n c e the d e t a i l s

17 w e w i l l

the b a s i c

19

~ .

If we are w i l l i n g

sides w i l l be small b e c a u s e

( - ~,T o) w i t h the r i g h t p r o p e r t i e s . existence

mapping

(62)

sides of t h e s e e s t i m a t e s

to r e s t r i c t

Theorem

16 w e r e

( # ~ ) (t)

n

Theorems

As b e f o r e we

~ (t)

s m a l l and we d i d this by c h o o s i n g

restate

initial the w a v e

In o r d e r to m a k e

(M~) (t) = e - i A t ~

then

for all

by M t the m a p

The c r u c i a l and

exist

just on a small ball.

M t : ~ (o)

where

(52)

last s e c t i o n to c o n s t r u c t

and ~,T

space'and exist n o r m s

hold w i t h

the s o l u t i o n s

for all

J a non-linear I I-IIa, I I .If b

q > I (see SecMt~(o)

I I~(o) II ~ n

of

(72) are

and all

Then,

~_ G [scat ~(t) ~ [scat

there

is a u n i q u e

for e a c h t, and

global

solution

~(t)

of

-

i I~(t)

(b)

The m a p p i n g

(c)

which

: r

~+

Proof. Let $--e [scat is not a s s u m e d small). $(-)

on

as t § -~

§ o

as t § -~

is a one to one map of

continuous

statements

and the map

functions

§ r

is u n i f o r m l y

Analogous

t + +~,

- ~ _ilsca t

~_

-

- e-iAt$_l i § o

lieiAt~(t)

[scat

91

as in

on b a l l s

(a) and

in

(b) hold

[scat

into

[scat" for

~+ ~ [scat'

: ~+ § $(o).

be given, w i t h li$_ilscat ~ n (note that n Let X(n,$_,T) d e n o t e the ~ - v a l u e d c o n t i n u o u s

( - ~,T]

so that

II i~(t) - e-itA~_l i I (-|

+

sup i I~ ( t ) - e - i t A ~ _ I I -~
sup ( l + I t t ) d l -~
l@(t)

- e-itA$_I la

< n

For each T, X ( n , ~ _ , T ) . i s

a complete

metric

space.

We d e f i n e

~

by

(61)

and M by (M~) (t) = e - i A t r

The e s t i m a t e s that

(63) of T h e o r e m

ill ~ (~) (t) aiD (_.,T ] can be made

close to same

(62) and

+

-~,

so in that case M w i l l

sort of e s t i m a t e s that we need

the smallness. for each

Just

t ~ ( -~,To]

By the ~ s t i m a t e s

(t)

16 and the f o l l o w i n g small by c h o o s i n g

take

X(~,#_,T)

show that M is a c o n t r a c t i o n

and so M has a fixed p o i n t Notice

(~)

~(.)

in

as in T h e o r e m and that the (62) and

into itself.

in

The -~

fixed T O .

(b) of the lemma to p r o v e

16 one can p r o v e that limits

show

for T close to

X ( n , ~ _ , T o) for some

q > 1 since we need part

lemma

T sufficiently

~(t) C[sca t

(a) hold.

(63) we can use the same T O for all

- 92

r

II~_IIo

with

~ n.

Thus, we can define the map T o 2+ :

on

~q

r

~

r (T o)

§

E {~ E [ s c a t l I l @ I l s c a t ~ q}.

and 17,

-

By the same proofs as in T h e o r e m

is a one to one uniformly

continuous map of ~ n

16

into

[scat" It can easily be checked

that for

t ~ TO

our solution

r

satisfies

r

= e-iA(t-To)~(To ) +

It T

Since J is Lipschitz locally

by hypothesis

in a neighborhood

boundedness uniqueness earlier

hypothesis t ~ To .

(57),(58),

of

r

implies by T h e o r e m 14 that, .

this equation

r

= Mt_ T r

in t.

coincides with the

imply that

r

r

prove that M t is a uniformly

~(t) defined

is in

(72)

[scat for all t.

of M t assumed in the hypothesis (58) and

continuous

Now,

By local

satisfies

for each t, M t is uniformly

This combined with

[scat for each fixed t.

can be solved

o) and by the

is global

It is easy to check that and (59)

J(r

o

(iii)

by

In fact, the uniform boundedness balls in ~

-iA (t-s)

on M t this solution

this definition

for

and that

of T O

e

continuous

on

(59) can be used to easily

one to one mad on b~lls

in

for each ~ we define T o (q)

~+ = That is,

~+ : ~_

By the properties uniformly takes

of

M_T

continuous

[scat

easy argument proves

M _ T o (~) ~ +

(a) and

into

[scat

shows that

#(o)

To(n) , we have that ~+

and

map of ~

)

into [scat"

and is uniformly ~+

~+ is a one to one

Since n was arbitrary, continuous

is one to one on all of

~b); the proof of

(c) is similar.

on balls. [~cat"

[scat" -

The existence

then we can construct

of the scattering operator

An

This

B

This theorem shows that when we have the hypotheses on J and global existence

~+

(i) , (ii),

(iii)

the wave operators on depends

on whether

-

range

93

-

~+ = range

~_

-i

so that we can d e f i n e is d i s c u s s e d in Section hypotheses

(i),

S = ~_. 12.

(ii), and

This

(much more difficult)

proble~

We remark that in a p p l i c a t i o n s not all the

(iii) may be satisfied.

Often one can con-

struct the w a v e o p e r a t o r s anyway by using special p r o p e r t i e s of the p a r t i c u l a r equations and the idea of T h e o r e m 19. of this in Section llc.

We w i l l see an example

ii.

Applications In this lecture we will sketch some a p p l i c a t i o n s of the theorems

in sections 8,9,10.

In most cases we will not give very many details.

Our main purpose is to show how one chooses the norms and

II. ll b

in a variety of situations.

I I" I I, I 1" I Ia,

The typical p r o c e d u r e is as

follows.

The n o r m

II. IIa

is a sup n o r m on some or all of the com-

ponents.

The norm

I I. I Ib

is then the best

which one can prove the decay estimate to be a Hilbert space norm which holds.

(i.e. lowest)

(ii).

Then

is as simple as possible so that

This is usually done by a Sobolev inequality.

norms

If" If,

II'll a,

p r o b l e m alone.

norm for

I Io I I is chosen (i)

Thus, the three

If" IIb are t y p i c a l l y d e t e r m i n e d from the linear

One then checks to see what p r o p e r t i e s the n o n - l i n e a r

term J must have so that

(iii) holds.

is interested doesn't satisfy change the norms

(iii)

if the J in w h i c h one

one may be able to go back and

I I I l,I I I la, l I I Ib

In all our examples we treat

Of course,

so that it does.

B = /- ~ + m 2 as though it acted on

powers of functions like differentiation.

The correct details can be

easily supplied as in part A of section 2. Part A.

The n o n - l i n e a r S c h r S d i n p e r e q u a t i o n

We begin w i t h an easy example,

the n o n - l i n e a r S c h r S d i n g e r e q u a t i o n

in one dimension, u t - iUxx + ku p = o u(x,o)

(73)

= f(x)

because it illustrates nicely the method for c h o o s i n g the norms described above.

The c o r r e s p o n d i n g

free equation

is

u t - iUxx = o u(x,o)

with

A

=

-

d 2 ~ .

The

= g(x)

solution can be written e x p l i c i t l y as

dx 2

u(x,t)

=

(4~it)'I/&I ei(X-Y)K/4tf(y)dy

-

95

-

and thus, t-~/2

I lu(x,t)i Therefore,

we

cannot

rem

flu[I|

thus

satisfied

choose

L2(R)

[ lU[ {~ ~

Ilfll

choose

11ul[a: we h a v e

I. <_ T-~.

C{IU{ I 2.

4 in part

,

llullb:

]Is[I,

hypothesis

(ii) w i t h

as our H i l b e r t

space

However,

I

d = ~

because

in one d i m e n s i o n

c of s e c t i o n

.

Notice

it is not

it is t r u e

that

that true

as u s u a l

2):

Thus,

we take

+

B = ~/':-A

(i)

is s a t i s f i e d .

2

To check

for w h i c h

we compute,

I ld(u 1 ) - d(u 2 )11 = KIIB(uP 1 = KP[I(Bu

uP) ll 2

-

1

)u p-].1

_< KPII(Bux)(u~

§ KPlI(Bu

< cllBu

(74)

m 2.

l]ull : IIBull so t h a t h y p o t h e s i s

that

(see T h e o -

I lulloo ! cl IBull 2V2 l[ulI2~ -< c i IIBull where

we

1

(Bu)uP-I'll 2

2

2

- u~)P(u,,u2)Jl2

- Bu)uP-Xll 2 2

1

II,llu,

+cilBCu

2

2

- u I[~lIPcu

- u )2 l l

2

2

Ilu

p-I

1

II ~o

,u)ll| 2

p

(iii)

holds,

--

< cllBu

--

1

I1

IIB(u

2

+cllBcu~ er

llu,ll and

for

with

holds

I IJr

1

lla + Ilulla

g = p-2.

= I luP1 - u~ll

< Bllu

we must

have

and

q

17 p r o v i d e

equation The p is d u ~

Part

B.

of

1

II

- u

the

first

hypothesis

in

r

)11

IIU2

- u 2 )11

data

dq

global

> i.

1

(llu, II 2 +llu, tl)(lilu,

2

II.

1

(iii) a l s o h o l d s

> 2 so t h a t small

1

- u 2 II(llu

1

application to S t r a u s s

of t h e s e

+ Ilu~ll2

with

if

I1| p-2

2

p-2

q = p-2.

Therefore

existence

ll.+ll'.

Now,

since

d =

p > 4, T h e o r e m s

and a s c a t t e r i n g

theory

16 for

small

utt - U x x + m Z u = lu p

need

data

techniques

to

(73)

for h i g h

[38].

to discuss

the

utt first

+ I1%.11=) p-2

(73).

In o r d e r

we

Ilu

Thus,

- u)Q(u 2

1

--

part

)p-2

small.

< cllBr

second

(tlu.,ll,=

2

Similarly,

= IIr

so the

- u 2 )11

- u~)ll~ Ilu~ll~ -~

llull

) - J(u 2 )lib

1

96--

a decay

equation

(75)

- U x x + m a u = lu p

estimate

utt

, one d i m e n s i o n

-

for the

Uxx

linear

+ m2u

= o

U(X,O)

= f

Ut(X,O ) = g

equation:

(76)

1

-

Although

this

is a fact about

vial so w e w i l l o r o v i d e Lemma with

1

Suppose

Ilu(x,t)[I.

Proof

.

a linear equation,

the p r o o f

is n o n - t r i -

and let

u(x,t) u(t)

u(x,t)

be the s o l u t i o n

of

(76)

Then z +

~ Ct-ua{]]f][

For each t,

-

a sketch.

f,g(~(R)

initial data

97

[If'll

x +

= u(t)

If"ll * + ]lg'l]

*+

llgll *}

(77)

is g l v e n by

= cos(Bt)f

+ sin(Bt) B

g

or

sin(/-~/~t) ~(k)

A

u(t)

Thus

u(t)

= cos(/--~t)

~R

= /~

(k 2 + m 2 ) ' ~ 2 s i n

The convolution There

follows

analyticity

of

Second,

of

that

differentiating

and s u b t r a c t i n g

since

R(x,t)

R 9 /'(R)

the

function

theorem

t h a t we w r i t e

1

r~

J-we

twice with respect

the r e s u l t s ,

R(x,t)

R(x,t)

Thus;

for

ik x

in the

that

in two d i m e n s i o n s .

--

g ~ ~(R). is.

Here

x 2 < t 2.

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

(k 2 + m2)-Zl2sin ~/~-~+~t

Suppose

and

is zero e x c e o t w h e n

we can c o m p u t e d i r e c t l y

H( t~-T~-x 2 ) - R(x,t)

x

dk

in the s e n s e of d i s t r i b u t i o n s . sense

transformations t 2 - x 2.

transform

sin~~t

to f i g u r e out w h a t

and g r o w t h

Rig

Fourier

from the P a y l e y - W i e n e r

under Lorentz function

makes

F i r s t we n o t i c e

k directions.

Then,

+

I ~_ e i k x

/k 2 + m2t

R~g

are m a n y w a y s

is one. This

= ~ f

for each t , R is the i n v e r s e

R(x,t)

of

+

can be w r i t t e n

u(t)

where

f(k)

and the

imaginary is i n v a r i a n t

R(x,t)

is a

x 2 < t 2,

sin /k 2+m2 t

Wt~'+m2

dk

to t and t w i c e w i t h r e s p e c t

one finds that

H(t2~--x2 ) satisfies:

to

-

m2H,,(

tZ_x 2) +

98

1 H'( t2-x 2

-

t2-x 2) + H(

tZ-x 2) : o

SO

H(

t2-x 2) : CJo(m

where J is the first Bessel function. 0 1 that c = ~ . Thus, 1 R(x,t) = ~ X

(78)

{xlx2!t 2 }

(x)

t2-x 2) Setting

x = o, one determine%

Jo(m ~/~-~-x %)

Therefore, we have the representation:

z/t

(79)

(R~g) (x,t) = 2

To analyse the decay of R i g

-t Jo(m ~ - y % )

we need the estimates

(2

(8o)

g (x-y) dy

Jo(~) : .V~- o

cos(~ - ~ )

(see, for example,

+ 0(p'~)

J 1 (.) = a(p -"~) as p ) ~ We write (79) as an integral over {Y[IYl ~ ~ } and an integral over {YI~ ~ IYl ~ t}. Using IJo(p) I ~ cp -*12, the first can easily be estimated by

ct' ,-t/2[t/=

fdy

_
l I.gl

There are two integrals left one of which is

1

(~)

1 (~)

%u2 rt ]t/2Jo(m ~---~-~) g(x-y)dy

V2 It

COS ( m / ~ - ~ - ~ )

t/2 (t2-y 2) V~

t

g (x-y) dy +

I

t/2

O( (t2-y 2)'~) g (x-y) dy

(8z)

-

99

-

For the second term we have

i

t O((t 2 - y2)'31~)g(x - y)dy t/2

! ct-'"llg11| t/2 (t

- y) "31~dy

< ct-"2Jlg, II --

I

To handle the first term, we integrate by parts obtaining:

(t2-~ 2) U~ -

] y=t sin (m ~

g (x-y)

+ ~ )

my ~

+

] y=t/2

I t sin(mt*~-~ + ~ ) ~y [(t2-~2,)~Wg(x-y) } dy

I

m2Vr- t/2

Y

ct-U2(11gll

Both terms may be easily estimated by Combining this with (81) we have

IIR-gl I. ! ct'"~ (I IgIl, To treat the

~R

~-~f

+

(82)

~R

(~f)

+ llg'll

1

).

IIg'II )

term, notice that

DR = 51 (6(x+t) + 6(x-t)) + m ~-t so

I

i[ f(x+t)

(x,t) = ~

+ f(x-t)

t

J (m t~/~T~-~)

]

+ 5m It_t ~-~t--L"- j, (mt~/~_y,}f(x_y)dy

First we estimate the integral over {yly 2 ~ t 2} as before. Then we integrate the remaining integral by parts, observe that the boundary terms at

y = +t cancel the first term in (82), and estimate the other

boundary terms and the remaining integral just as we did for

R ~g.

- I00-

The result

is

aR I I (-,~*

The extra

f) (x,t) I I|

derivative

tion by parts. For

c t - ~ ( I Ifll

i

on f o c c u r s

This

proves

f and g nice,

this

the

IIf'll

+

1

because

1

IIf"{I 1 )

+

there was

one extra

integra-

lemma.

lemma gives

us a d e c a y

estimate

and

leads us

to define

IIl[ a =

[lu][~

II {b = I l u l l , + We take

as o u r H i l b e r t

W o=

Then,

by

(74),

since we

~ ~

with

[1~

- ~nl Ib

and

linear

since

(i) holds.

,it

and

follows

this holds p

I l J ( r l)

J(~b )1

_< c l ~ l '{ I j"u '

J r 1 62 2 - u

[1~

holds.

!

Ixl

< --

that we don't quite have

on

D = ~(R) X~(R).

b

=

- ~nl I

) o.

Since

(ii) yet

However,

given

~ n G D so t h a t -itA e is c o n t i n u o u s

that

_< Ixl

I IJcr 1

Notice

for e a c h n.

(iii)

llv'll I

IfBult, + llvll ~ < |

I [~II b < " ,we can find a s e q u e n c e

for which

-

II1~11 ~ =

the estimate

) o

I +

space

{~ =

only know

+ flu II + llvll I

I{u'{{

Thus we have

Since

J(~)

Ilu p1 - uPl{ 2

{ x l ( l l u P 1- u P l f

2

It r e m a i n s

to check

2

Ilu, - u~ II ~ ( l l u l l l ~

clxl(ll~,lla+

(ii).

= < o , k u p>,

+

l u 11.) p-1

1~211a )p-1

lu I - u 2 II

1

+ l I D ( u P1 -

uP) I{ 1 ) 2

I~ +11,*'-, u'lll(llu~ ,11~+ Ilu~ I~)(llu, ll.+llu~ll,.) p-2

-

Since

16,

17, and

data and the e x i s t e n c e p > 4. N o t i c e p is either

that

In the case w h e r e

18, global

of the s c a t t e r i n g

this

even or odd

have global

result

Furthermore

energy

is b o u n d e d Mt

in

cases we have the e x i s t e n c e

Theorem

20

(a)

initial data consta n t

If

of T h e o r e m

solutions

ll~olIscat

Further,

p is odd and

19 are s a t i s f i e d operators.

of

norm

so

The com-

We summarize:

(75)

exist

if the

(or the c o u p l i n g

the s c a t t e r i n g

I is n e g a t i v e

(uniformly

of lemma

II. Ilscat

operator

the w a v e

exists

operators

continuous

1 we used the exnlicit

in terms of the Riemann

derivative.

This

are other

is the s t a n d a r d

fancier ways

function "old

to d e r i v e

exist

on balls)

fashioned"

However,

for c in terms

n o r m on the

of a t r a c t a b l e

to use the old methods. to the case

in von Wahl

of

if we want

n = i.

A large number and Costa

technique

[46].

(see[9

is the proof of d e c a y

then

])

llull~.

an e x p l i c i t

initial data

of the

and its time

ct -v2 d e c a y of

The proof we give

[42], [43]

representation

of the e q u a t i o n

the

See for e x a m p l e [ ~ ] , a n d [ , ~ ] .

found

bounded

itself.

In the proof

adapted

(like we did

is u n i f o r m l y

on such small data.

If in a d d i t i o n

There

~(t)

is an open question.

~o has small enough

and are one to one maps

into

)

of the wave

D > 4, then global

I is small enough).

and is c o n t i n u o u s

soluti o n

the h y p o t h e s e s

operators

then we

fact that the term

c I ul Ip+l to show

: ~(o)

in t he s e

of the wave

~ and w h e t h e r

dx

by

on balls

pleteness

Thus,

we can use the

u p+I

4) that the m a p p i n g ~o"

for small d a t a if

the sign

(or fractional).

~

in Section

operator

for small C a u c h y

p is odd and the sign of I is negative,

existence.

in the c o n s e r v e d

existence

holds w h a t e v e r

I

[scat

{I~ , - ~ 2 {l

the e s t i m a t e s in h y p o t h e s i s (iii)are satisfied w i t h q = p-2. 1 d = ~ and dq > 1 we must choose q > 2 so p > 4. We have thus

proven by T h e o r e m s

(b)

-

{I~ 2 lla )p-2

s (ll~,l{ a + Thus,

i O 1

estimate

it is easiest

in Strauss

estimates

[212

can be

-

Part

C~

u~t

- Au + m 2 u

In o r d e r dimensions

Lemma

to h a n d l e

we

2

first

Let

= lu D

, three

dimensions

the n o n - l i n e a r

need

Klein-Gordon

a lemma w h i c h

f,g G C = ( R 3)

'-

1 0 2 -

O

And

let

equation

is a n a l o g o u s

u(x,t)

be the

in t h r e e

to l e m m a

I.

solution

of

"

utt

- AU + m 2 u = o

u(x,o) = f(x) Ut(X,O)

Then,

there

is a u n i v e r s a l

constant

= g(X)

c so t h a t

(83)

luCx,t){[. ~ ct-"2llJlb where

{ {i . ,b I

is d e f i n e d

as the

sum of the L

norm

of all the

1

derivatives

The proof [~|]);

of f of o r d e r

of this

only

lemma

< 3 and

is s i m i l a r

the d e t e r m i n a t i o n

complicated.

The

cause

R(x,t)

itself

extra

twice

for the

all the d e r i v a t i v e s

to the

of the

derivative

involves

J

proof

f o r m of

on the . Thus,

of g of o r d e r

of L e m m a I

R(x,t)

initial

(see S t r a u s s

is a l i t t l e

data

One m u s t

< 2.

comes

integrate

more

about

be-

by p a r t s

1

So, we

g terms

choose

and t h r e e

, l~i I I ,b

times

for the

to be as d e f i n e d

f term.

in the

lemma

and

II~ la = llul]. Then,

(ii)

longer

use

is s a t i s f i e d

for n l c e

the H i l b e r t

snace

ci IBul I 2 in t h r e e so we

can use ~ .

~o

dimensions. That

is,

because

then

(i) holds.

with mension

As

I i~I] b < -.

if we

show that 7

in part

B one

that

However true

w e can no

that

I lul I,

i lul I~ _< cl IB2ul r 2

set

+

ii~ii

=~IIirl

Further,

it is not

It is__ t r u e

liii 2 = IfB2ulI, ~i

3 d = ~ .

data with

can n o w

similar

~

< | ] extend

calculations

(83)

to all

to t h o s e

~ 6~S

in one di-

- 103

-

l lJ(4~,) - J(~){{ = 1~1 [lB(u puP) ll 2 l 2

(84)

<_ Ix (II~ , [[ +.I[r , ll)(I[r T h e r e are m a n y t e r m s highest

order

llJ(#,) - J(~2)ll,.

in

in D (denote by

<

-

Let us look at one of

D i any p a r t i a l d e r i v a t i v e ) .

I{(D[lu

,u)II

-u))Plu

+ 21 [ (D i(ul

+

a + I I~ z l la)P-2lI~-, ~,II

- U2 ) )DiP(uI'u2) I I I

EIIu~- u~)D[P(u,u)ll,

<__C({[Bu 1[2+ ]]Bu2{[ )(]{u []=+ []uz{[=)P-2

IlB=(u,

- u)l{

(85)

+cc{{Bu ii + lIBu [121~cfiu E{.+ {lu~li~) D-3 L1u-u

i[.

<_ fl { ( [ l ~ x i l a + l l ~ 2 I i a ) P - 3 1 I # , - ~ 2 1 1 a + ([I~,II a + l l r

Thus,

(iii)

is s a t i s f i e d w i t h

t h a n in one d i m e n s i o n in all d i m e n s i o n s ) .

q = ~-2

I I~,- ~2iI

for s l i g h t l y d i f f e r e n t

(you s h o u l d not a s s u m e that Notice

)p-2

in the case

q = p-2

reasons

is all r i g h t

q = 1 , the c o n s t a n t

~ in

(iii)

if I I~ I i + I ]~ 11 is small as r e q u i r e d (see (84) and (85)). 3 i 2 d = ~ we n e e d o n l y c h o o s e q ~ i, so o _> 3. For all such p,

is s m all Since

T h e o r e m s 16 and

17 give a small d a t a s c a t t e r i n g

utt - Au + m 2 u = lu p

independent

of w h e t h e r

the r e s u l t s

ere s o m e w h a t

For of the

p > 3 integral

solutions c).

p ~s

Theorem equation

initial data existence

of g l o b a l w e a k

x e R3

(86)

in the two cases

18 g i v e s g l o b a l corresponding

strong

is s m a l l e n o u g h

for the e q u a t i o n

e v e n or odd and the sign of I.

different

are s t r o n g l y d i f f e r e n t i a b l e

Thus we h a v e g l o b a l

,

theory

existence to

(86).

of

and

that

p = 3.

strong solutions these

(see S e c t i o n

2, part

of 86 for all p ~ 3 if the

(or I is s m a ll enough).

solutions

Notice

Furthermore,

in t l o c a l l y

solutions

p > 3

for all d a t a

We a l s o k n o w the

(Section

5).

What

-

1 0 4

-

remains open is the d i f f i c u l t p r o b l e m of proving global strong solutions for large data. Notice that as the d i m e n s i o n is increased three)

(in our case from one to

the s c a t t e r i n g theory gets easier because the decay of free so -

lutions is better

(though

I I II b

is a little more complicated)

but

the global e x i s t e n c e theory gets harder because the Sobolev e s t i m a t e s are weaker.

This puts us in the following peculiar p o s i t i o n w i t h regard

to the wave operators.

To use T h e o r e m 19 to guarantee the e x i s t e n c e

of the wave operators we need to k n o w global e x i s t e n c e of strong solutions.

In three d i m e n s i o n s we only know this for

and I negative).

p ~ 3 (with p odd

On the other hand the hypotheses of T h e o r e m 19 ex-

clude the b o r d e r l i n e case

q = p-2 = 1 and thus require

p > 3.

There-

fore, as it stands, T h e o r e m 19 says n o t h i n g about the e x i s t e n c e of the wave operators

in three d i m e n s i o n s if D = 3.

This provides us with an o p p o r t u n i t y to illustrate a very important point.

Namely, the h y p o t h e s e s of the abstract theorems

in t h e s e lec-

tures are quite general in that they depend only on estimates r e l a t i n g A and J

(and energy inequalities).

Therefore,

the conclusions are some-

times not as strong as those one can obtain by e x p l o i t i n g special properties of the e q u a t i o n being studied. of proof as in T h e o r e m s

Typically,

one uses the same idea

16, 17, 18, 19, but the step which fails on the

general level is carried through using the special property. To see how this works

in the case

p = 3 (where we know global ex-

istence) notice that the r e a s o n that T h e o r e m 19 cannot handle the case q = 1 is that (l+Itl) d times the e x p r e s s i o n (63) on page 7 ~ does not go to zero as

t

case

Thus we need a new way to estimate:

q = i).

> - ~ (because part

I le-iA(t-s)J(~(s)) I la =

(b) of the lemma excludes the

I I ~ -Isin B(t-s)]u(s)'I I.

< Cll (k 2 + m2)- *12(sin / ~ ( t - s ) )

< CII (k 2 + m2)-l11211 (k2+ m 2 ) ~ 2 ( 9 )

<_ cl IB(ucs)')112
/% u"(k) II

(k) I12

_

105

_

Using this estimate

in place of the ones before

through

This how Segal originally

as before.

the wave o p e r a t o ~ for

(63) the proof goes

proved the existence

p = 3 in three dimensions

(see ~ ] ) .

of

We summa-

rize: Theorem

21

~

If

p > 3,then global

tial data has small enough is small). tinuous ~

I l#ollscat

Furthermore

-norm

the scattering

of

(86) exist

if the ini-

(or the coupling

operator

exists

constant

and is con-

on such small data.

In the case

p = 3 and I negative,

Part D. The Coupled

the wave operators

Dirac and Klein-Gordon

In order to discuss in three dimensions, itself.

solutions

the coupled

we will

first

Equations

Dirac and Klein-Gordon look at the non-linear

We can write the free Dirac equation

(x,t) = -i(i~.V

equations Dirac equation

as

(89)

+ iSM)~(x,t)

-;De~ (x,t)

=

(x,o)

exist.

= f(x)

where ~(x,t)

=

<~o(X't) ' ~z ( x , t ) , ~ 2 ( x , t ) , ~ 3 ( x , t ) >

f(x) =
The

e i s!

and

a formally

On

~

~

8 are certain

self-adjoint

f (x), f2(x), I

f (x)> 2

~

anti-Hermetian

operator

on

4M4 matrices.

i~oL3(RS).

L2(R ~) then the anti-commutation

properties

If

we

let

Thus,

D e is

Be = ~

of the ~i's and 8 (see

[ [ ]or [~ ]) imply that

2

D2e =

Further

De

I OBe

is self-adjoint

0 B2e B2e

~ ~2e

on the domain

(88)

-

106

D(D e) =

and since

D e commutes with

-

.e

D(D e)

--ZBe it commutes with all powers of

Thus, we can take for our Hilbert space any of the escalated

Be"

energy

spaces: 3

9~ : (*I II,II~ ~ [ liB,,ill ~ < | i=o

Further,

the group

into themselves operator on

e -itDr

and this

~k

generated by

implies

D e takes each of these spaces

(~2~,p.269)

that

D e is a self-adjoint

with domain _k+l. ~o D(B e )

D(D e ) =

By the properties

2

of the

u's and

satisfies the Klein-Gordon

8, each component

~i(x,t)

of

~(x,t)

equation ~2 I

2

~i + Be~ i = O

~t 2 ~i(x,o)

= fi(x)

(x,o) = gi(x)

where

gi is the

.th component of I

-iD f. e

II~i(~,t)]I| a ct"~ where

I fll b

of all the

denotes

fi's.

Thus, by lemma 2 in part c,

IIfIIb

the sum of the L I norms of all derivatives

Thus,

~ 3

if we define 3

II~IIa =_~

II~i(x,t) II|

i=o we have

II~lla _< ct -'~ Ilfllb We are now ready to treat the non-linear

~t + iDe~ = AJ(~)

(89)

Dirac equation (90)

-

107-

w h e r e we will assume that each component of ~i's

each of w h o s e terms has order p.

J(~)

is a p o l y n o m i a l in the

The a p p l i c a t i o n of T h e o r e m 16,

17, and 18 differs in one respect from the application to the n o n - l i n e a r K l e i n - G o r d o n equation in part c. J()

=

IIJ(~) il b

The n o n - l i n e a r term of part c ,

was zero in the first term so the worst terms

looked like

llB2uPi]z.

Here J looks like ~P(symbolicallv)

in each component so the worst terms in liB~Pi]z._ thesis

in

I IJ(~) I ib

will look like

Thus, we cannot hope to satisfy the second part of hypo-

(iii) unless we choose as our Hilbert space ~

so that the n o r m

3 3

2

rl~il' : 11,1123 : i -[- o lIBe~iIE has three d e r i v a t i v e s on each component. example, that This term

But, this means,

just as in part c but now we will have we must require

I IB3(~P) I I 2~

(iii)) can be e s t i m a t e d

q = p-3

(instead of

p - 2) so

p ~ 4 in order to apply T h e o r e m 16, 17, and 18.

Suppose that each component of

J(~)

~i each term of w h i c h has degree at least 4. equation

for

IIJ(~) II w i l l be a sum of terms of the form

(and the other terms in h y p o t h e s i s

T h e o r e m 22

2

is a p o l y n o m i a l in the

Then the n o n - l i n e a r Dirac

(90) in three d i m e n s i o n s has global strong solution

Qauchy data is s u f f i c i e n t l y small

(in the

if the

II.T Jscat norm). F u r t h e r m o r e

on these small data the scattering operator exists and is continuous. Now we can set up the spaces and norr~for the coupled D i r a c - K l e i n Gordon equations:

d J (~,u) T 0 + iDe~ = Xe e (91) utt- Au + m~u -- XoFo(~,u) w h i c h we rewrite as ~t + iDe~ = XeJe (~'~) (92) Ct + iAor = loJo (~'r

where

# = ,

B O = ~- A + m O ,

Jo(~,~)

Je (~,u) ,and A~

-B o

o

= , Je(~,~)

-

as usual.

1 0 8 -

We take as our Hilbert space

with the norm

on

~ = <~,~> given by 3

2 + liB2uti22

2

II II~- i=o [ lIB'~ill 2

IBsuII2

+

Then the operator A =

ID e

o

o is self~adjoint

1

A~

on

D(A) = 4i_~o D(B~)) (~ (D(B ~) (9 D(B3))

and if we set

J(-)

= then we can write

(92) as

-' (t) = -iA---(t) + J(-(t)) which

is in the right

We define of

I IHI Ib

~i' i=o,i,2,3,

(93)

form to apply the theorems of these

lectures.

to be the sum of the L* norms of all the derivatives and u of order

vatives of v of order

< 2.

~ 3 plus the L* norms of all deri-

And, we set

3 I---~Ia = ~. I I~il I~ + [lul I~ i=o The three norms with

I I I I, I I "I Ia , I I "I Ib satisfy hypotheses

d = 3/2, so it only remains

must have so that

(iii)

linear Dirac equation wer.

Theorem

23

(ii)

From the calculations

on the non-

like the calculations we have already done.

16, 17, 18, we have:

Suppose that

are polynamials Je

is satisfied.

(i) and

Je and Jo

and part c you should be able to guess the ans-

The proofs are exactly

So, from Theorems

to check what propert•

Fo(~,u)

and each component

of

Je(~,u)

in the ~i and u so that each term in the components

of

has order at least 4 and each term in F o has order at least 3. Then the coupled equations (91) have unique global solutions if the I I I Iscat

- 109norms of the initial data are small enough data in

[scat

if

1o

and

le

(or for auy fixed initial

are taken small enough).

Furthermore

the s c a t t e r i n g operator exists on these small initial data and is continuous. This t h e o r e m does not cover the case of a Yukawa i n t e r a c t i o n

Je(H)

= leU~u r

9

Jo (H) = lo~Yo~

because the degrees are too small, but it does cover g e n e r a l i z e d Yukawa interactions of the form

Je(- ) = leU(~7o~) k

for any

k ~ 2.

,

Jo (-) = lo(~Yo~) k

You can easily formulate for y o u r s e l f the analogous

t h e o r e m for two coupled Fermi~ns

mddt ~m _ i~2,e~m = IE2|e~e(~m,i,,~,)

w h e r e degree four is

required in both terms.

C h a d a m applied these small

data ideas to the classical v e r s i o n s of the equations of q u a n t u m field theory in [ 6 ] .

He uses more d e l i c a t e

(Lp) decay estimates and more

i

special p r o p e r t i e s of the Dir~c and K l e i n - G o r d o n p r o p a g a t o r s handle more cases.

I

For example, I

he only requires degree 3.

so he can

in the F e r m i o n - F e r m i o n case above,

12,

Asymptoti 9 Completeness

We come n o w to a r e a l l y h a r d q u e s t i o n , completeness. satisfied

Let us s u p p o s e

so that t h e w a v e o p e r a t o r s

to o n e m a p s of that is, if

~scat

i n t o itself.

~+ exist ~f

the s c a t t e r i n g

of T h e o r e m

19

are

and are c o n t i n u o u s

are a s y m p t o t i c a l l y

~

Range R ~ = Range

t h e n w e can d e f i n e

the p r o b l e m of a s y m p t o t i c

t h a t the h y p o t h e s e s

one

complete,

~_

operator

s = (~+)-i~_ T o see w h a t is a

is i n v o l v e d

~ _ E [scat

let

#o

be in the r a n g e of

and a s o l u t i o n

.

Thus there

of

r

= e - i A t ~ _ + it

r

=

e - i A (t-s) j (~ (s))ds

SO t h at

r

and

lib(t)

- e-itA~_Jl

W h a t we m u s t p r o v e

is t h a t t h e r e

(94)

l[~(t)

If

is true t h e n

(~)

)

- e-itA~+I[

o

as

is a )

t

)

~+ ~ [ s c a t

o

as

t

-~

so that )

+~

#+ = l i m eitA~(t)

since

e

-itA

show that

is u n i t a r y ,

elAte(t)

is a

so the n a t u r a l w a y to c o n s t r u c t

Cauchy s e q u e n c e in W .

~+

By h y p o t h e s e s

w e h a ve l leitlA*( t ) - e i t 2 A * ( t * z

< --

'I l=r t 2

<_, ,3

I 2

l laqr

I Ids

is to (iii)

-

i i i

-

In general we get an upper b o u n d on

II~(s) II from an energy inequality,

so what we need to know, to insure that the right hand side goes to zero as

t , t $

---)

+~, is that

2

l" For example,

II~(s) ll~,~s < -

in the case of the equation

(95)

utt - Au + mZu = -u 3

x~R 3

the required estimate is

e~

What is needed t h e r e f o r e is an apriori estimate on solutions

of the

n o n - l i n e a r equation w h i c h guarantees that solutions w i t h nice initial d a t a decay s u f f i c i e n t l y rapidly in the Furthermore,

I I -I Ia n o r m as

t

> +-

.

if we expect to be able to prove c o n t i n u i t y of the scatter-

ing operator we need to estimate the constants in this decay in terms of the decay of the c o r r e s p o n d i n g solution of the free equation.

For

most n o n - l i n e a r w a v e equations no such apriori e s t i m a t e s are known (or only very weak estimates)

so one cannot even b e g i n to attack the

p r o b l e m of asymptotic completeness.

However,

in the case of

three d i m e n s i o n s this p r o b l e m has been solved by M o r a w e t z [~t)].

(95) in

and Strauss

Their t h e o r e m states:

T h e o r e m Z ~ Let

~

denote the closure of

C o ( R ' ) X C o ( R 3) in the n o r m

II112scat = suPt {!I u~ + IVu[2 + m2u2)dx}

+ sup sup t

where

u(x,t)

u(x,o)

= f,ut(x,o)

(95) exists and is a continuous

Further, of

_~ J x

denotes the solution of the free e q u a t i o n u t t - A u + m 2 u = o

w i t h initial data for

lu(x,t) l 2 + | suplu(x,t) I2dt

x

if

e~

and

(95) w i t h initial data

Vf

= g.

Then the s c a t t e r i n g operator

one to one map of ~

onto itself

has finite energy, the solution



satisfies

~(x,t)

-

112-

II~(x,t) l l . <_cr

-3'~

The crucial fact w h i c h made this t h e o r e m possible was a new apriori estimate w h i c h was derived from a weak apriori estimate p r e v i o u s l y o b t a i n e d by M o r a w e t z [ ~ .

I r e c o m m e n d this argument to anyone who wants

to see how b e a u t i f u l and d i f f i c u l t these n o n - l i n e a r theories are in case you are not already convinced).

(just

The same arguments go through

for a fairly r e s t r i c t e d class of other n o n - l i n e a r terms d e s c r i b e d in an a p p e n d i x in [~|3. M o r a w e t z and Strauss prove further p r o p e r t i e s of S in [~]. Basically,

there are two kinds of arguments in [~|]. The arguments

used to derive the apr~ori e s t i m a t e and other n e c e s s a r y e s t i m a t e s use the e x p l i c i t r e p r e s e n t a t i o n of the Riemann function for the free equation and methods from partial d i f f e r e n t i a l equations.

These t e c h n i q u e s

are n e c e s s a r i l y d i r e c t l y related to the p a r t i c u l a r d i f f e r e n t i a l equation being studied and t h e r e f o r e do not g e n e r a l i z e the abstract

level.

However,

in a natural way to

the second part of the M o r a w e t z - S t r a u s s

argument is e s s e n t i a l l y a functional analysis argument w h i c h shows that, given the estimates,

one can c o n s t r u c t the s c a t t e r i n g operator.

This part of the argument can be formulated on the abstract

level and

it is the purpose of the rest of this section to outline how this may be done.

Since the most important thing is the general s t r u c t u r e of

the argument we will just outline the ideas.

Details may be found in

Reed [I~. Since there are quite a few hypotheses, we will collect t h e m all here rather than refer back to previous

sections.

The h y p o t h e s e s

fall

n a t u r a l l y into four parts: I (Existence) (with n o r m

I I" I I) and let J be a n o n - l i n e a r m a p p i n g on ~ w h i c h

[96~ Then,

Let A be a self-adjoint operator on a Hilbert space

IIJr for each

-Jr

~o & ~

I 1 ~ - ~11

' the c o r o l l a r y of T h e o r e m

I gives a local so-

lution of [97)

satisfies

~(t) = e-iAt ~o + I t e-iA (t-s) j (~ (s)) ds o

-

We denote

by

is a p r i o r i implies

Mt

that

: D

the

~(t)

is a d e n s e

of free D

I IJ(~)

for some

set of

§ #(t)

result

= Mt# ~

C~

D wou~d Cauchy

solutions)

that

: D

be of the

compact

there

which

Finally, )

we

Doo and

form

~

v%~4

D(A n )

support.

is an a u x i l i a r y

norm

so t h a t

r > o

D-valued

and all function

,t I

e -iAt

typically

We a s s u m e

inequality)

in t.

so that

IIMt~oJ I

that

of an e n e r g y

data with

- J(4) I I _< c(I I~II ,1141 I)

IIl~(')lllt

and a s s u m e

is g l o b a l

set D in ~

-

(Decay

norm)

: ~(o)

(typically

solution

there

to the

II.ll a on

(98)

the

Mt

) D. In a p p l i c a t i o n s

or e q u a l

II

bounded

that

assume

M t the m a p

113-

= 2

#,~ ~ D. on

sup t

I1~ - ~II

If

tl,t2,

~(-)

--

I1 ~(t) ll 'r +

IIII

(in the

sup llg~(t) ll r + [t211~(s) r t t

2


When

t = -- and t = -,we denote 1 2 E D, w e a s s u m e t h a t

--

ll

a

ds a

2

the n o r m

Ille-itA~ll

(99)

is a c o n t i n u o u s

we define i


(I 1411 ra + 11411 r)

1

simply

by

II1-III.

If

I < -

and d e f i n e

II~llscat The

Banach

closure

III

space

of s c a t t e r i n g

of D in the

(the a D r i o r i

IV.

f is a l o c a l l y

(Kernel

functions

on

Estimates) It I, t2].

that

function

~(.)

Define

1

the

for all

~ e D,

~ f(IT~Ilscat)

Let

[[,~(.)] ]rt

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

[scat

We a s s u m e

bounded

~eD

norm.

IIIMtr

'(100)

where

states

I l-llscat

estimate}

= Ille-itA~lll

,t

on

and

for each

= 2

t

s

t

(o,-).

4(')

be

pair

(tl, t2), r

~11~(t) I I a !

continuous

ds

D-valued

-

114

-

and Jt ! ,t 2

(~) ( t ) =

[~2 e-iA (t-s)

J(~(s))ds

i We assume that

Jtl,t 2 satisfies:

lllJt ,t (~) - Jt ,c (~)III ~cr162 i

2

I

(lol)

,t 'lll~(')lllt ,t)

2

I

"([[~]]t

,t 1

9 ( sup

+ [[~]]c 2

Ilk(t)

,t 1

- ~(t)

2

I

)o~, II I,~(t)

- ~(t)

ll It'

2

ll

2

2

')

t
(102)

IIIJt ,t r

_~ c(IIl*(-)ll It ,t ) [[*]]t ,t I

I

where the

~i (which are independent of

#(-) and

2

I

2

~(.)) satisfy

ui > o, u I + u3 -> I, ~2 + u3 -> i. (iOl)and (102)are further assumed to hold ~n the case where t 2 = t on the left and t2= = on the right and similarly for

If we let

t I = t and

Xtl,t 2

t I = -~.

denote the closure of D in the

norm, then what we are essentially assuming is that H61der continuous , norms~ from on and

and

Jtl,t 2 is locally

in both the

Xtl,t 2 to

I~l~tl,t2

Ill Ill tl,t2

I II I IItl,t 2 and the s u p II I I tl!t!t 2 X.~,~) where the constant depends appropriately

llI~]Iltl,t2, since

sup I I#() I I ! III~( )IIltl,t 2

tl<_t<_t1

e2 + ~3 Z I, Jtl,t 2 is actually Lipschitz in the

I I III Itl,t 2

norm. In applications, such estimates are typically proven by expressing e -iA(e-s) as an integral operator and using L p estimates on the kernel and non-llnear term J. A simple example of such an estimate appears in Section ii, part c. For other examples, see S e g a l [ ~ ] , ~ ] , Chadam [ ~ ] , [6], Morawetz and Stauss[~|], and Strauss[3F]. The proof of the existence of the scattering Qperator begins by investigating the properties of

{M t}

and

{e-iAt}.

-

Lemma 1

115

The families of mappings,

uniformly equicontinuous

on

-

{e -itA},

{M t}, and

{~itAM t}

, are

I I.I Iscat-balls in D.

The proof of Lenuna 1 proceeds by first showing that continuous by using the kernel estimates and iteration.

M t is I I ,I IThen this is

used to show that {eitAMt } is an equicontinuous family similarly to the proof of continuity in Theorem 17. Since e -itA is, for each t, linear, bounded and of norm one on shows in particular continuous map of

[scat

the integral equation to

into itself.

Eemma 1

extends uniquely to a uniformly Since the extension satisfies

(97) it coincides with the restriction of

M t on

[scat"

The ~ollowing to

[sc9 t the lemma follows.

that for each t) M t

lemma shows that the apriori estimate also extends

[scat"

Lemma 2

There exists a locally bounded function

I I IMt~l I I ~

~(I I#I Iscat )

for all

~

on

(o,~) so that

~ ~[scat"

Lemma 2 is proven by choosing a sequence

~n ~ D

so that

I I~n - #Ilsca t > o. First one shows that Mt$ n converges to Mt~ and then one takes the limit in the apriori estimate which gives lemma 2 with ~(x) = lira f(y) We can now prove the existence of asymptotic states: Theorem 25

The maps [scat Proof

For each

~

into

~ & [scat there exist

~+ and

~_ in [scat so that

I IMt~ - e-itA~+11

~ o

as

t

~ +~

llMt~ - e-itA~

) o

as

t

) --

~ ~+

9

~

II

~

#

are continuous

and one to one from

[scat"

We will give the first part of the proof since it shows how

the apriori estimate and the kernel estimates are used. be given and fix t I and t 2.

Let

~ ~ [scat

Then,

I leit~AMtl ~ - eit~AMt2~l Iscat = I I le-iUA(eit~AMtl~-eitzAMt2~) If I

-

=

116

rjt2 e_iA(u_s)

Ill

-

J(Ms~)dsl[[

tI

<_ I1 t [Jtl,t a (Ms~') ] (u) II 1 I+~ 4

! c(111%*11tt1,%)

[[Ms*33t,,t= i+~ 4

! C(I I~1 I scat ) [[Ms*]]tl,t

2 IIl~t~ll

By the apriori estimate we know that

I < - which implies that

l+u 4 o

[[Ms*]ltl,t 2 Thus as

eitAMt~ t

~ -~

[scat so we can define

The convergence e -ira

.

lim eitAMt ~ t++~

statement~

By lemma I, the

formly equicontinuous proof that

as

is Cauchy in

#+ =

as tl,t 2

,

t

)~

~ +~ and the same is true

~_ = lim

eitAMt~

t~--~

in the theorem follow from the unitarity of

.. maps

a21

a21

and

are pointwise

limits of uni-

families of maps and are thus continuous.

are one to one uses the ~ernel estimates

as in the one to one proof in Theorem In order to construct

The

and same trick

16.

the scattering operator we need to know that

the maps ~+I just constructed take [scat ont___~o [scat and that are continuous. To prove this we must solve the Cauchy p r o b l e m at

•

~.

Given

~+ E.;scat, the solution of

II~(t)

- e-iAt#+II

)

(97) which satisfies

o

as

t

should be the solution of

(103)

~(t) = e-itA~+ -

e-iA(t-s)J(~(s))ds t

> +~

~

-

There are two difficulties.

But, this

it as a lim•

closure

of D in the

~N(N)

= e-iAt~+

= e-iAN~+

In particular,

be proven

is that as

to a solution 3

(104).

< ~

To conclude

of vectors

,

(97) with Cauchy data

Lemma

]lle-itAr

that

in D since

I I " Isca t norm.

~N(t)

in I.

,

~(o)e[sca t

_[scat

we must

was defined

Thus, we define

as the to be the

#N(t)

of

(104)

Since

|

(|03) has a solution

that is,

is not quite enough.

exhibit solution

Ill<

-

First we must show that

with the right decay properties,

][1~(t)

117

Let Then,

of

- [Ne-iA(t-s)J(~(s))ds ~t

~N(t)

e-iNA#+ if

~+~D,

is just the unique at

t = N

then

N --> ~, ~N(t)

global

guaranteed

~N(t) E D converges

solution

of

by the hypotheses

for all t. (Dointwise

What must in

[scat )

(103).

@+ E D

and let

@N(t)

be the corresponding

solution

of

if T is large enough,

(a)

] l'l~N(t) I I IT ~ ~ 2[ I [e-itA~+l [ IT,~

(b)

[[~N(t)]]T, ~

< 2 [[e-itA]] --

T,~

The point of this lemma is that the right hand sides are independent of N and thus give us some control Lemma in

3 is proven

by defining

of the limit of the

the space

B(T)

as N § ~.

to be the set of

~(-)

XT, ~ so that

I ] l$(t)

- e -itA~+ I IT,~ -< 1!le-itA@+llIT,~

[[~(t)-

e-itA#+]]T,

< [[e-itA~ --

and then showing that lies in the

~N(t)

~f.

B(T) Next

for all we h a v e ,

for T large enough N > T.

L

]] +I~T,

the solution

~N(t!

The proof uses the strict

~

(104)

pg~$vity

of

-

Lemma

4

(a)

Let

~N(t)

as

N § = ,

which

be the solutions ~N(.)

satisfies

-

discussed

converges

in

(c)

~(t) 9 [scat

to a function

~(t)

for each

t E IT,=)

][]eitAo(t)

and

- #+)IIscat--9 o

t --~+~.

To prove

lemma

in ~emma

4 one first uses the kernel

3 to show that

to check that the pointwise (b) follows

statements

Then,

(103).

I I I~(t) I l IT,. ~ 211 le-itA~+l I IT,.

formity

above.

XT, =

(b)

as

in

118

#N(.)

limit

estimates

is Cauchy

~(t)

satisfies

from the uniform estimates

~+ ( D

and the uni-

XT, ~.

It is easy

(103) and the estimate

on the

in (c) also use the uniform estimates

For fixed

in ~N.

The proof of the

from Lemma

3.

we can now define

n+T : r

)

~+ : ~ +

) M_T~#

r (o) +

~+ is thus a map from D into [scat and (by Theorem 24 and its proof) ~ I ~ + ~+ = ~+. Similar definitionsand statements hold for ~T_ ,~_. What remains to be shown is that ~+ can be extended and that the extension is continuous. Theorem 26 fudction~(.) (b) into

[scat

(a) Let # + E [scat; then there is a T and a [scat-Valued which satisfies (103) and parts (b) and (c) of Lemma 4.

The map

T > M_T~+~ + is a continuous

~+ : ~+

map of

[scat

[scat"

If ~+ were uniformly proof of Theorem from D.

on

ll#+llsca t but also on

- ~Ilscat

)

o.

on T. Let

is a T so that

in D then the

just extend

First one chooses ~n(t)

balls

~+ directly

since the choice of T depends

[[e-itA~]]T, ~.

Thus,

a sequence

be the corresponding

Then one first shows that there

that there

ll.llscat

26 would be easy; we would

some local uniformity (103).

continuous

But, this is not at all obvious

not only on II~

to all of

one needs n ~+ 9 D so that

solutions

of

is an N so that n > N implies

- 119

If e

-itA. n

-

-itA n

~+I11 <_ 2k

r247174 <--~o

lie

for a c o n s t a n t k and a small constant 8 o. This u n i f o r m i t y and the fact that show that of

#n(.) are Cauchy in

#n(t)

satisfies

(103)

allow one to

XT, m and c o n v e r g e to a solution

#(t)

(103) w h i c h has the right properties. To prove c o n t i n u i t y one uses the same idea to choose a small

so that for all

qo

~+ in the ball

B(no'~+) = (~+r

II~+ - ~+llscat !no}

one can use the same T and u n i f o r m estimates

11 [~(t)I I IT,. _< [[~(t)~,.

hold.

21 l~+I Iscat

< 2 [[e-itA~+]]T,.

With these u n i f o r m e s t i m a t e s the proof of c o n t i n u i t y proceeds T ~+ is

by iteration much like the proof of T h e o r e m 17 to show that u n i f o r m l y c o n t i n u o u s on ~+ = M _ T ~ ~

B(no,~+).

is u n i f o r m l y continuous on

continuous on

Thus,

T h e o r e m ~7

B(no,~ +) since

M t is u n i f o r m l y

[scat balls for each t.

Of course, analogous .

F r o m this one concludes that

s t a t e m e n t s to those on T h e o r e m 25 hold for

c o m b i n i n g T h e o r e m 25

and T h e o r e m 26

The s c a t t e r i n g o p e r a t o r

tinuous map of

[scat onto

S =

we have:

~

is a one to one con-

[scat"

I have two remarks to m a k e about this theorem.

First, a l t h o u g h

it looks very nice, r e m e m b e r that we assumed the e x i s t e n c e of an apriori estimate

(hypothesis III).

If the e x p e r i e n c e of M o r a w e t z and Strauss

is any guide r the proof of such an aDriori e s t i m a t e is the h a r d e s t part of the problem.

Secondly,

there is no reason to think that our choice

of norms, spaces, and estimates

in h y p o t h e s e s

II and IV give the best

general a b s t r a c t result or are the only choices possible.

In fact,

in

a p a r t i c u l a r application, one may have an apriori e s t i m a t e but some of the h y p o t h e s e s

in II or IV may fail so that one can't apply T h e o r e m

-

27 directly.

1 2 0

-

In such a case one will have to choose ~hher norms and

spaces w h i c h are b e t t e r suited to the p r o b l e m at hand.

Some t e c h n i c a l

details of the c o n s t r u c t i o n of the s c a t t e r i n g o p e r a t o r w i l l then be d i f f e r e n t but the general idea should follow the o u t l i n e p r e s e n t e d here.

13. D i s c u s s i o n

It should be clear theory

for n o n - l i n e a r

partially

solved

and discuss

First of all,

problems.

there

since one must the norms

would

second

operators

general

(seeing])?

problem

but t r a c t a b l e

of the w a v e

as functions

consider

the coupled

(ul)tt - ~u

+ m2u 1

(u) 2

T hese

1

- ~u tt

equations

have

2

the wave

operators

investigate

right hand

symmetries Another has the tively, should

that

of various

1

operators

given by T h e o r e m

of the theory.

2

+ ~u )3_ 28u u 2 l

z

conserved

2

energy

properties

various

if

].

I > o,

in Section

of ~+ as f u n c t i o n s

internal

scattering

coupled

is to prove

of

c

operators

have the same

is broken.

non-linear scattering

such a d e c o m p o s i t i o n

equations

It w o u l d

that the s c a t t e r i n g

for the small data

to

I,~,~.

Dirac

symmetries.

the s v m m e t r v

19 is another

8 > o and

Ii, part

It should be i n t e r e s t i n g

form S = I + T w h e r e T is a "small"

Whether

com-

the b e h a v i o r

parameters

2

= -41e(u

a case w h e r e

this must be true

on [scat

+ ~u )3 _ 28u u 2

~

problem

groups)

equations

the small d a t a

interesting

scattering and to inves-

and i n v e s t i g a t e

1

display

or to e x h i b i t

be proven.

symmetry

in the case of n o n - l i n e a r

sides w h i c h

understan-

w o u l d be to take more

can be shown to exist.

imagine

be nice to prove

to exist

and u s i n g the t e c h n i q u e s

the a n a l y t i c i t y

One can easily

not advance

do they commute w i t h the n a t u r a l

terms

= -41(u

a positive

is any real number

and choose

greater

small data

(or other

1

+ m2u 2

provide

are known

question

non-linear

operators

For example,

that

group

but not trivial

Such work w o u l d

is to take the

interesting

and e n g i n e e r i n g

interest.

For example,

of the Lorentz

Another

we will

problems.

of them can be applied.

free e q u a t i o n

but w o ul d

physical

to harder

straightforward for the

correctly.

operators

properties.

representation

18 or v a r i a n t s

estimates

of direct

or the w a v e

their

plicated

in the physics

17,

theory very much,

ding of equations

tigate

equations 16,

or

to point out

For convenience,

from easier

I I I ] ,I I I]a,[ I I Ib

the m a t h e m a t i c a l

of u n s o l v e d

it is w o r t h w h i l e

explicitly.

be r e l a t i v e l y

prove d e c a y

that the s c a t t e r i n g

mostly

progressing

are many

to w h i c h T h e o r e m

sections

consists

Nevertheless,

four parts,

Such a p p l i c a t i o n s

The

equations

some of these problems

group them into

literature

from the p r e c e e d i n g wave

is true

more d i f f i c u l t

operator

operator. operator

Intuibut

for the wave question.

it

-

122

Ideally one would like to show that

S = I +

-

) can be e x p a n d e d as

S (or ~

~ InT n n=l

w h e r e I is,for e x a m p l e , a small coupling constant and the T n are least for low n) simple operators.

the scattering o p e r a t o r approximately. tations are w e l l - k n o w n

(at

This wo~id allow one to calculate Such expansions or represen-

in linear theories

(for example,

see[~in

the

q u a n t u m m e c h a n i c a l case a n d [ i T ] f o r the case of classical linear wave equations).

It is clear that w e could go on and on with this list of

q u e s t i o n s about S and ~+L but the above examples give the idea.

Theorems

17 and 19 guarantee the--existence of certain n o n - l i n e a r operators.

The

p r o b l e m is to investigate the p r o p e r t i e s of those n o n - l i n e a r o p e r a t o r s and how the properties reflect the structure of the n o n - l i n e a r i t i e s

in

the o r i g i n a l equation. The third general p r o b l e m is to d e v e l o p new techniques for h a n d l i n g the small data s c a t t e r i n g theory and the existence of the wave operators when

the n o n - l i n e a r i t y is not s u f f i c i e n t l y high or the decay is too

slow to allow a p p l i c a t i o n of the techniques we have presented. ample,

For ex-

consider the e q u a t i o n

(105)

utt - Uxx + m2u = -u 3

in one-dimension.

In order to prove the existence of the wave o p e r a t o r s

by the t e c h n i q u e s we have outlined,

one must have that

(see Section

Ii,

part c)| le-iA(t-slj(,(s)) . adS =

I T

I IB(u(s) s) I I ds 2

~

( --

S frBu(s) II 1lu(s) II~ds T

2

)

The t e r m

{ ]Bu(s) I {

o

as

T

~

)

~.

is of course b o u n d e d by the energy, but in one 2

d i m e n s i o n free solutions

u(s) only d e c a y like

so we can't expect this convergence to hold.

s -Vz in the sup n o r m Nevertheless,

it is clear

that there should be a s c a t t e r i n g theory for(lO5) i n t e r m s of solutions of the linear e q u a t i o n

-

(106)

- u utt

1 2 3 -

+ m2u = o xx

The rate of c o n v e r g e n c e solution

of(iO6)

will be slower;

use other norms b e s i d e s w here

the d i v e r g e n c e

cases w h e r e and

This

it really

example,

The best

this

fourth

pleteness.

but w h e r e

approach

class

fusion.

lately

wave

and there

My point that

I want

does

two cases

to consider.

not

in the Hilbert approach reason

that

spaces

non-zero

theory

case most n a t u r a l l y cussed.

they are not The more the Hilbert at

x = f~

data w h i c h

only

in one

be very

important.

is

there

of the

form

equation

u(x+t). much

literature

here

of the p r o b l e m

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

In this

for data w h i c h

the soliton

x

x = Z=,

are

are

) ~ to be solutions is no apriori

solutions

space m e t h o d s

solutions

There

solutions

as

case there

is small at

(see~).

of soliton

theory.

(many of the soliton

of the sol~ton

keeps

among physi-

is that the p r e s e n c e

that the soliton

+~).

which

Soliton s o l u t i o n s

interest

is no s c a t t e r i n g

suppose

There

seems to be some con ~

of a n o n - l i n e a r

h a n d l e d by the Hilbert

Essentially,

com-

estimates

is very difficult, are known

is they are not small enough

at

decay

would

to think that the p r e s e n c e

the s c a t t e r i n g

of a s y m p t o t i c

not

is a g r o w i n g

constants

on a p a r t i c u l a r

theory t h e r e ~ a n d

for any equations

have g e n e r a t e d

First,

in these

problem

not mean that there

"normalized",

first two b e c a u s e

on this

to e m p h a s i z e

solutions

should n e v e r t h e l e s s

apriori

this p r o b l e m results

17

techniques.

the q u e s t i o n

complete

a solution

equations

lots of other

to c o n c e n t r a t e

of d e r i v i n g

equations,

an i l l u s t r a t i o n

those d e s c r i b e d

about new m o r e g e n e r a l

before,

the a p p r o p r i a t e

of T h e o r e m s

than the

to get a s c a t t e r i n g

is a solution

for example

of n o n - l i n e a r cists

I think,

and S t ~ a u s s [ ~

soliton

are

theoty

go beyond

of this p r o b l e m ~bout w h i c h

A

its form,

is,

Any p r o g r e s s

by M o r a w e t z

one aspect

which

is, of course,

of n o n - l i n e a r

of examples.

handled

a scattering

of the n e c e s s i t y

and as we have m e n t i o n e d

There

in the methods

is much harder

is n e c e s s a r y

problem

Because

on solutions

area

suggests

We have picked

is borderline~

of integrals

techniques

do w h a t e v e r

then see what The

faster,

third p r o b l e m

requires

~ectures.

of(iO5)

of(lO5)to

in fact that one may have to

the energy norm.

the d i v e r g e n c e

19 is much

exist.

of solutions

so slow

should which

affect

is the

we have dis-

should play no role b e c a u s e

in the class of initial data under discussion. interesting space

or b e c a u s e are

case

is w h e r e

under d i s c u s s i o n we choose

large at infinity

the solitons

either

because

our Hilbert

solutions

they

are in

are small enough

space n o r m so that

initial

are allowed. If ~o is the initial

data

-

124-

for such a soliton solution, then we would not expect the soliton Mtr ~ to decay into free equations at

t = •

since the wave keeps its shape.

But this does not preclude a complete scattering theory, that we should expect that contained in [scat"

Range ~+

and Range ~_

it just says

w i l l be strictly

If one has asymptotic completeness,

Range ~_ = Range ~+

then one has the scattering operator and setting r

= Sr

then,

S = ~i~_.

the distant past, we will get out a free wave future.

Given a ~ _ ~ [ s c a t ,

if we send in a free w a v e

e-iAt~_

in

e-iAt~+ in the distant

This s i t u a t i o n is similar to the situa~cion in q u a n t u m m e c h a n i c s

where one expects that the ranges of the wave operators equal the part of the Hilbert space c o r r e s p o n d i n g to the absolutely continuous part of the s p e c t r u m of the interaction H a m i l t o n i a n

HI .

In general,

H I will

have bound states w h i c h will not decay to free solutions but this does not prevent the c o n s t r u c t i o n of

a

scattering theory.

Of course,

in

the q u a n t u m m e c h a n i c a l case one stays in the Hilbert space, the free and i n t e r a c t i n g d y n a m i c s are given by unitary groups, and the b o u n d states are n a t u r a l l y s e p a r a t e d from the s c a t t e r i n g states since they are orthogonal.

In the case of n o n - l i n e a r w a v e equations it is not

clear how to separate the initial data in [scat w h i c h c o r r e s p o n d to soliton solutions from the initial data which are s c a t t e r i n g states; that is part of the p r o b l e m of proving

Ran~+ = Ran~

, but

Ran~ C [scat"

To find an example of a n o n - l i n e a r w a v e equation w h i c h illustrates these points and to d e v e l o p a complete scattering theory for such an e q u a t i o n seems to me to be an e x t r e m e l y important and interesting problem.

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