Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
507
Michael Reed
Abstract Non Linear Wave Equations
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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)
=
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
(
=
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
=
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 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.
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))
~ 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)
)
in part
=
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( 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~
_
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,
=
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,
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
~ _
l
e
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
$(t),
(6) is global
in t.
I I~(t) I I is bounded. Further,
Then the solution
if J satisfies
m times strongly differentiable
interval of existence
of
~(t) of
condition Jm then
~(t)
is
for all t and satisfies
dd--•j-•(t)
( D(A m-j)
Proof
The idea is the same as in Theorem
existence showed
I I~(t) I I
is apriori bounded.
in T h e o r e m 2 that
IIA~(t) II ~
2.
On any finite
interval of
On this same interval we
IIA~oll
e K~t.
Now for
A2~(t)
we have the estimate
IIA~(t) II < IIA2%If § It I IA2j((~(S))
I Ids
o
u(t) uniformly on ~
u
of the
F
a.e. n
in
~
T,T] we have
L 2 ( [-T,T] X Rn)>
u
(again denoted by u n) so that
T,T] X Rn.
It
follows
Fn(Un(t,x))
a.e.
in
[- T,T] X R n.
~
)u
Since
u(x,t) p
IFn(X ) I< 1 + G(x)
IFn(Un(X,t)) I dxdt < 2T Vol [ S ( r ~ + T)] + -T
un
i m m e d i a t e l y from p r o p e r t i e s
that
(46)
pointwise
be a fixed finite time interval.
n
so we can choose a subsequence
pointwise
argument and clever
n
we have
Gn(Un(X,t))dxdt -T
Rn
< 2T Vol [S(r O + T)] + 2TC
where we have used the finite propagation FatoU's
i!
-T
sO
speed and
(43).
Thus, by
lemma,
lu(x,t)
n
p
dxdt
o
2~ o.
global
continuous
where
and J a non-
(iii) hold. with
Then
] I~ I I ~ ~o'
e -iA (t-s) J(~(s))ds
-valued
For each t, ~(t) E [scat
(b)
ll~(t)
(c)
In case
The basic
X(n,~
so that
(ii), and
~ _ L [scat
(a)
we employed Let
~
space ~
that there exist norms
solution
(6O)
~(-) with
I I I~(') I I I
Moreover,
- e-itA#
- ~_llscat
§ o
) denote
1 except with
~ is chosen
initial
the set of continuous
- e-itA~_l I I ~ ~.
~
Assume
so that hypothesis
to:
as t § -~
idea is to use the contraction
in Section
I I l$(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 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 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)
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
_ +~
~
-
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
~, ~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 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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