The Cauchy Problem for the Coupled Schroedinger-Klein-Gordon Equations

G.M. de La Penha, L.A. Medeiros (eds.) Contemporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations @North-Holland Publishing Company (1978)

THE CAUCHY PROBLEM FOR THE COUPLED

SCHROEDINGE R-KLE IN- GORDON E QUATI0N S

and

J.B. Baillon

John M .

Department of Mathematics

Dept. de Math6matiques &ole

Chadam*f

Pontificia Universidade

Polytechnique

Cat6lica do Rio de Janeiro Paris, France

Rio de Janeiro, Brasil

Introduction. The classical versions of the non-linear equations o f relativistic quantum physics have for a long time been a subject of great interest.

One o f the major unsolved problems

in the area has been the global existence o f the solution t h e Cauchy problem in

R3

to

o f certain important equations

-

having quadratic non-linearities

the coupled Dirac-Klein-

Gordon equations,

P (-iY au + ~ ) =t g $ @ ,

-

( A - a tt

2

m ) @=

6t1

and the coupled Maxwell-Dirac (-i?

)r

+

a,,

+

O n leave from Mathematics Department, Indiana University, Bloomington, Indiana 47401.

Supported in part by NSF grant MPS 73-08567

J.B. BAILLON and J.M. CHADAM

38

Here we consider the Cauchy problem for the closely related coupled Schoredinger-Klein-Gordon ( S K G ) equations which are a semi-relativistic version o f equations (1):

with nucleon field

8 : R 3X R

-b

C

and meson field

@:

R'xR

4

R

I n previous works the Cauchy problem has been solved for equations (1) and (2) i n one space dimension [l] and for special cases in higher dimensions [2,31.

Equations ( 3 ) have

recently been treated i n bounded regions of

R3[

4,5]

.

The Results. We begin by writing equations ( 3 ) i n vector form,

from

R

JI

(g

) are considered as maps t to complex and R2-valued functions respectively. The

where the components

and

conventional solution spaces in which one wishes to solve equations

(4) for the components

"escalated energy" spaces

C m = Hm

( $ y(:t)) @

(HmCBHm'')

are the so-called [l].

Indeed,

following the example of quantum mechanics, it is really the exponentiated o r , in the non-linear case, the integrated form of equation

(k),

-

SC HR O E D IN GE R K LE IPJ - GO R D 0N E QUA T I0 N S

which a r e of

interest.

S(t) = e

Here

i A t

and

a r e t h e f r e e S c h r o e d i n g e r and Klein-Gordon respectively. of

(+ ,(:

Specifically then,

t h e SKG e q u a t i o n s o v e r t h e i n t e r v a l

($,(:

t

)):

(0,T)

4

Cm

i s continuous

t

39

propagators

))

is a

(0,T)

Cm solution

i f t h e map

and s a t i s f i e s e q u a t i o n s

( 5 ) where t h e i n t e g r a l i s i n t e r p r e t e d i n t h e s t r o n g Riemann

zm.

sense i n

-

Theorem

solution i n Proof:

zm

if

problem.

of

+

S ( t ) : tim

a r e continuous, uniformly

( a , (m 0 12 ) )

rn 2 2 .

-+

boundedness,

[ 6 1 c a n be a p p l i e d t o this

and

-+

K( t ):

bounded o n e - p a r a m e t e r

, ( O

IJI

I

2)): Zm + Cm

H

~

~

g r o u p s and

i s l o c a l l y Lipschitz

t h i s f o l l o w s from t h e l o c a l

o f t h e map v i a some t r i v i a l a l g e b r a .

The l o c a l

on t h e o t h e r h a n d , f o l l o w s f r o m t h e f a c t t h a t

i s an a l g e b r a i f

dimension

(@,$

H~

One c a n s h o w t h a t

boundedness

Hm

Segal

The l o c a l e x i s t e n c e f o l l o w s i n t h i s way b e c a u s e t h e

f r e e propagators

if

m)

2.

m 2

The a b s t r a c t r e s u l t

t h e map

T =

The SKG e q u a t i o n s h a v e a u n i q u e g l o b a l ( i . e .

m > d/2

where

d

i s the s p a t i a l

"71.

The main c o n t r i b u t i o n o f t h i s n o t e i s t o show t h a t this s o l u t i o n can b e e x t e n d e d t o

T =

p r e v i o u s l y mentioned r e s u l t

of

+m.

This o b t a i n s f r o m t h e

Segal i f t h e

Zm

-

norm of

s o l u t i o n c a n be s h o w n t o be f i n i t e f o r e a c h f i n i t e t i m e .

the We

H

~

-

J.B.

40

B A I L L O N and J . M .

CHADAM

b e g i n with t h e p h y s i c a l l y r e l e v a n t conserved q u a n t i t i e s o f c h a r g e and e n e r g y and t h e n s y s t e m a t i c a l l y b o o t - s t r a p o u r way

t o the xm-norm.

Specifically

[4,

p.4031

one h a s

3x = c o n s t a n t . I $ ( x , t ) 1 2 @ -b( x , t ) d 4

F r o m t h i s one h a s

t h e second l i n e f o l l o w i n g f r o m H s l d e r ' s

i n e q u a l i t y and t h e

l a s t l i n e from S o b o l e v i n e q u a l i t i e s and t h e f a c t t h a t li$(t)ii 2 i s conserved.

Now

Combining t h e above t w o e s t i m a t e s , one o b t a i n s

Adding e q u a t i o n ( 6 a ) t o i n e q u a l i t y

(7),

one o b t a i n s on t h e

l e f t h a n d s i d e t h e s q u a r e o f a norm which i s e q u i v a l e n t t o t h e

Cl-norm

t h u s p r o v i n g i t i s u n i f o r m l y bounded i n s p i t e o f

non-definiteness

of

t h e i n t e r a c t i o n energy.

the

SCHROEDINGER-K LE I N - GORDON EQUATIONS

and m o s t c r u c i a l ,

The n e x t ,

Proof:

s t e p i s t o show

Here we f o l l o w c l o s e l y t h e t e c h n i q u e of B a i l l o n e t .

C 8 , 9 I by showing t h a t

a1

~ ~ $ ( t ) ~i s~ fl i,n4i t e and t h e n u s i n g

111

t h e Sobolev i n e q u a l i t y

( t ) l l mS

( t ) l ( z / 4 119 (t)ll

C11 vp

to

To b e g i n

establish the desired result.

A l l t h a t r e m a i n s t h e n i s t o e s t a b l i s h t h e f i n i t e n e s s of

IlVh(t)114.

Let

D

d e n o t e a n a r b i t r a r y weak s p a t i a l d e r i v a t i v e .

D i f f e r e n t i a t i n g e q u a t i o n ( 5 a ) , u s i n g t h e L e i b n i t z f o r m u l a and t a k i n g norms

D

since

one o b t a i n s

commutes w i t h

S(t).

U s i n g t h e well-known

e s t i m a t e f o r t h e Schroedinger p r o p a g a t o r [lo, IlS(t)fIlp s

for a l l

f

E Lq(Rd)

where

decay

p.601

1 1 -d(-- -) C't

+

l/p

l/q

= 1

and

1

2;

q S

2,

one o b t a i n s f r o m i n e q u a l i t y ( 8 )

rt IIS(t)$0112,2 +

IID$(t)l14 s

t s

co

+

lglc

S

Co

+

Igl

c

Igl C i ,

( t - s ) - 3 / 4 I I D 0 ' ~ + @ D $ l l ~ / 3d s

c

(t-s)-3/4 c I I D 0 ~ s ~ l 1 2 1 1 ~ ~ s ~ l 1 4 + l l s ~ s ~ l l , I l ~ d s~ ~ s ~ l l , 1 C{C,

(t-s)-3/4 ds

+

C2

( t - ~ ) - ' / I~I D + ( S ) ( ~d~s ] .

The r e s u l t n o w f o l l o w s f r o m a v e r s i o n o f G r o n w a l l ' s sketched i n reference Lemma

3

-

[91.

The C2-norm of

f i n i t e f o r each

t <

lemma as

m.

the solution

($(t),(st(t)

42

J.B.

Proof:

But,

B A I L L O N and J . M .

CHADAM

From e q u a t i o n ( 5 b ) one h a s

o f Lemma 2 ,

f r o m t h e proof

one h a s t h a t

/IIII(S)//~,~ is

a

l o c a l l y i n t e g r a b l e f u n c t i o n g i v i n g t h e f i n i t e n e s s o f t h e meson p a r t of

t h e norm.

The r e s u l t now f o l l o w s from t h e s t a n d a r d v e r s i o n o f Gronwall’s lemma b e c a u s e e v e r y t h i n g e x c e p t

/ID2$

(.)I\

has previously

b e e n shown t o b e l o c a l l y i n t e g r a b l e .

To t h i s p o i n t we h a v e shown t h e e x i s t e n c e of a u n i q u e g l o b a l s o l u t i o n t o t h e Cauchy problem f o r t h e S K G e q u a t i o n i n t h e space

Z2.

A l l t h a t remains i s t o prove t h e r e g u l a r i t y

p a r t of t h e r e s u l t .

-

Lemma

4

than

2,

have

114 ( t ) l l m , 2 11@(t)11m,2 and

Suppose

m

i s an a r b i t r a r y p o s i t i v e i n t e g e r l a r g e r

t h e n t h e a b o v e , g l o b a l s o l u t i o n of e q u a t i o n s

II@,(t)ll

m-l,2

locally

(5))

-

47

S C H R O E D I N G E R - K LE I N GORDON E Q U A T I O N S

f o l l o w s f r o m L e m m a 3 by g i v i n g t h e i n d u c t i v e

Proof:

The p r o o f

step.

Suppose then that

some

But

m 2

H

3,

m- 1

the r e s u l t

is known in

Then

m 2

for

the left-hand

~

m- 1

for

i

3

i s an algebra g i v i n g t h e f i n i t e n e s s of

side f r o m the inductive hypothesis.

and t h e r e s u l t f o l l o w s

from Gronwall's

Similarly

l e m m a b e c a u s e of t h e

11 a ( s)II m , 2 .

l o c a l i n t e g r a b i l i t y of

References

[l] C h a d a m , J . M .

-

G l o b a l Solutions of

the ( C l a s s i c a l )

Coupled Maxwell-Dirac

Space D i m e n s i o n ,

[2]

Chadam,

J.M.

J. Functional A n a l . ,

and G l a s s e y ,

Solutions of

-

R.T.

Equations

3,1 7 3

in One

(1973).

On C e r t a i n G l o b a l

the C a u c h y P r o b l e m for the ( C l a s s i c a l ) E q u a t i o n s i n O n e and T h r e e

Coupled Klein-Gordon-Dirac Space D i m e n s i o n s ,

[ 3 ] C h a d a m , J.M.

the C a u c h y P r o b l e m f o r

Arch.

and G l a s s e y ,

Rat. R.T.

Mech.

-

Anal.,

54

(1974).

O n the M a x w e l l - D i r a c

E q u a t i o n s w i t h Z e r o M a g n e t i c F i e l d and T h e i r S o l u t i o n

i n T w o Space D i m e n s i o n s ,

53,

495 ( 1 9 7 6 ) .

J. M a t h .

Anal.

and A p p l i c .

44

C 41

J.B. BAILLON and J.M. Fukuda, I. and Tsutsumi, M.

-

CHADAM

On the Yukawa-Coupled

Klein-Gordon-Schroedinger Equations i n Three Space

Dimensions, Proc. Japan Acad., 5 1 ,

C 51

Fukuda, I. and Tsutsumi, M.

-

Schr8dinger Equations, 11,

C 61

Segal, I . E .

-

402, (1975).

O n Coupled Klein-Gordonto appear.

Non-linear Semi-groups, Ann. Math. 7 7 ,

339 (1963).

C 73

Grisvard P,

-

Contribution i n Proceedings o f Evolution

Equation Seminar, Nice, France

1974-5.

[ 81 Baillon, J.B., Cazenave, T. and Figueira, M.

-

Equation

de Schr8dinger Non-lingaire I, t o appear C.R. Acad. Sci

[ 91

.

Baillon, J.B., Cazenave, T. and Figueira, M.

-

Equations

de Schradinger Non-lin&aire, 11, to appear CR. Acad. Sci. [lo] Reed, M. and Simon, B.

-

Methods o f Modern Mathematical

Physics 11, Academic Press,

1975.