A remark on the nonrelativistic limit for semilinear dirac equations

NonhnearAnalvsl~s, l'heorv, Methods & Applications, Vol. 25, No. 11, pp. 1139 1146, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Greal Britain. All rights reserved 0362 546X/95 $9.50 + .00

Pergamon 0362-546X(94)00235-5

A

REMARK

ON THE NONRELATIVISTIC LIMIT SEMILINEAR DIRAC EQUATIONS

FOR

TOKIO MATSUYAMA G e n e r a l E d u c a t i o n . H a k o d a t e N a t i o n a l College of Technology. T o k u r a - c h o , 14 1, H a k o d a t e . H o k k a i d o 042. J a p a n

( Received 15 Nooember 1993; rece,Jed for publication 26 August 1994) KO, words and phrases: S e m i l i n e a r D i r a c equation, n o n l i n e a r S c h r 6 d i n g e r equation, nonrelativistic limit.

I. INTRODUCTION We consider the initial value problem

i3,~b=

icadJ~tp+c2/3~+2A(~b,~b)/3~, 4,(0, x) = ~,b0,(x),

t > 0 , x ~ R 3,

x ~ R 3,

(1.1) (1.2)

where c is the speed of light, a is a positive constant and q, is a wave function from R × R 3 to C 4. We denote ;//at and a/axs by ,9, and ~,, respectively, o~c~x denotes a differential operator E)_lajO:. The ass and /3 are 4 × 4 Hermitian matrices satisfying a j a k + a k a j = 2 6 j k I , aj/3 + /3% = 0 ( l , k = 1. . . . . 3) and /32 = I (I is a 4 × 4 unit matrix). We are interested in the nonrelativistic limit of the initial value problem (1.1), (1.2), that is, the behavior of the solutions ~b, of the initial value problem (1.1), (1.2) as c ~ m. In [1] Najman proved H 2 well posedness and strong H ~ convergence for the solutions of the nonlinear Dirac equations when the initial data strongly converge to the initial data of nonlinear Schr6dinger equations in H ~ as c ---, z, where y satisfies 0 < y < 1. His method is based on the representation of the solutions of the nonlinear hyperbolic equations of second order. There exist H2 solutions of the nonlinear Dirac and Schr6dinger equations; nevertheless only H ~ convergence arises. This is because we meet the derivative loss, which comes from the nonlinear part of the nonlinear Dirac equations when they are transformed into the nonlinear hyperbolic equations of second order. In [2] the author discussed the existence of solutions and nonrelativistic limit of (1.1), (1.2) in the space of the rapidly decreasing functions and weighted Sobolev spaces. Our aim is to prove the nonrelativistic limit of the nonlinear Dirac equations in H 2. Our method is the standard density argument. That is, we investigate the solutions of the initial value problem (I.1), (1.2) approximated by regularized solutions in the space of rapidly decreasing solutions. We briefly survey the nonrelativistic limit of the nonlinear Dirac equations following Najman [1]. We put c z = 1 / 2 e . Then the initial value problem (1.1), (1.2) is equivalent to

;1

1

iO,~b= -/~, ~ad,~b+,~S-~/3~b+2a(/3~,b,6)/3~b, 6(0, x ) = tb0~(x). 1139

t > 0 , x ~ R 3,

(1.3) (1.4)

1140

T. MATSUYAMA

Now we introduce a new function ~ = 2e'~'/2~/3~ and insert it into (1.3). Then we obtain the following initial value problem "1 iA ~, q~ = ]//~-TE e'~t/" o~c~~ - ~-- (/3q~, q~)/3q~,

t > 0 , x E R 3,

q~(0, x) = q~0,(0, x).

(1.5) (1.6)

Furthermore, differentiating (1.5) with respect to t we obtain the initial value problem 1

a~°q~-~[Aq~+2itSO, q~-A(~qv, q~)¢]=F,(¢), q:(O,x) = q%,(x),

t>0,

x ~ R 3,

4~(O,x) = q h , ( x ) ,

(1.7) (1.8)

where F,(q~) = - TA2 (/3•,

~;)z q~ _ ~ ( 2 s )

'/:{e '¢''j~ , [a~(/3~p, q~)]a/3q~ + 2Re(/3~p, eie'/'aax q~)/3q~}, (1.9) s%~ = 2/3q*0~,

(1.10)

/1 iA ~1~ = V 2 - • c~c~q~0,- 2 - ( flq~0~, q%,)/3q%~.

(1.11)

This suggests that the solutions q~, of the initial value problem (1.7), (1.8) formally converge to the solutions ~0 of the nonlinear Schr6dinger type equations iA = 2/3A ~0 - ~-(/3~P0, ¢0)/3~0, •

,9t

t > 0, x ~ R 3,

q~0(0, x) = ¢00(x)

(1.12) (1.13)

as • --+ 0 (c --, ~). T h e n we have the following result. THEOREM. Assume that 0 < • < 1. Let ~#0, and q%0 be lying in H e such that q~,, ~ ~00

strongly in H 2

( • --+ 0).

Then there exists some T* > 0, independent of •, such that q~ --, ~,

*-weakly in C([O,T*];H 2)

(E-+O).

We conclude this section by stating several notations. ~p* is transposed conjugate of ft. We denote by J ( R 3) the Schwartz space of the rapidly decreasing functions on R 3. The Sobolev space H m ( R 3) is introduced as usual with the norm Hg, llu~ = 11(1 + [ sclz)m/2~llL e, where ^ is the Fourier transform. We briefly denote ~'(R3) 4 by ~', H'~(R3) 4 by H m and L2(R3) 4 by L 2. Let H be a Hilbert space with the norm 11' [[H and the inner product (., ")H. For an interval I we denote Ct(I; H) (l >_1) the space of /-times continuously differentiable functions from I to H. In the calculations below, various constants are denoted by C and change from line to line. Especially, C(* . . . . . * ) denotes a constant depending only on the quantities appearing in the parentheses.

A remark on the nonrelativistic limit for semilinear Dirac equations

1141

2. P R O O F O F T H E O R E M

In the sequel we suppress the space and time variables x, t when no confusion arises. It is well known (see [1]) that the initial value problem (1.7), (1.8) (respectively, (1.12), (1.13)) has a unique solution % (respectively, q~0) such that q~ E C ( [ 0 , T ] ; H 2 ) ("1C' ([0, T ] ; H 1) 0 C 2 ( [ 0 , T]; L 2)

[respectively, ¢0 e C([0, T]; H 2 ) r-'l c ' ( [ o , T]; L2)] if the initial data ~o~ are lying in H e (respectively, q~o, e H2). T h e n the following result is obtained by Najman [1]. PROPOSITION 2.1. Let s~o, and q~oo be lying in H 2. Assume that Furthermore, assume that ~,,~ --' ~oo

strongly in H ~

SUpo<,<,ll~ooollH:
( E --, 0).

T h e n there exists some T > 0, independent of E, such that % -+ ¢,~

strongly in C([O, T]; H l )

( e - + 0).

Let ~I;',~ and ~{'ff be in the space of rapidly decreasing infinitely differentiable functions such that '#I,'~' -~ ~q)~

strongly in H 2

( j -+ ~c)

~I~'1' --" ~o,)

strongly in H :

( j ~ :~).

Here it is easy to see that the convergence is uniform in e by the boundedness in H 2 of {q~0E}We now consider the initial value problems a, zsc

1

977[Asc+2tfl,),¢-a(t3q:,,g)sc]=F,(q~), ,¢(0, x) = ~l,{~(x),

t>0,

x e R 3,

a , ¢ ( 0 , x) = ~ { ) ( x )

(2.1) (2.2)

and i a~ scc)= W/3~ ~,0

ih ~(13,p(~,s%)[3q~ o ,

¢~( O, x ) = ,~:'~,,)(~, x ).

t>O,x~R

3,

(2.3) (2.4)

It is shown in [2] and [3] that for each e and j ~ N there exist unique solutions ~ol = q~o)(t ' x) and q~0j~= p I / ) ( t , x ) of the initial value problems (2.1), (2.2) and (2.3), (2.4), respectively, satisfying q;(,J)~C~([O,T~,:];,~) and ~ / ) e C~([0, T~,j];Y). Let T~*j be a maximal existence time of the initial value problems (2.1), (2.2) and (2.3), (2.4). That is, we define T~*: by the supremum of T such that ~ J ) ~ C~[(0, T]; f ) and q~/) ~ C~([0, T ] ; 5 °) are the solutions of (2.1), (2.2) and (2.3), (2.4), respectively. Here we can choose the maximal time intervals independent of e and j so that an energy estimate can be obtained in H 2. In fact we have the following lemma.

1142

T. MATSUYAMA

LEMMA 2.2. A s s u m e that 0 < e < 1 and j ~ N. T h e n T,*j is b o u n d e d from below. That is to say, there exists some T* > 0 such that T* > T* for any e, j. Furthermore, it holds that

sup

II¢;(j) (t)IIH ~"_< C(T*)ll¢~oJ)ll,2,

tE[0, T*]

for any E, j. Proof. Let K be any real n u m b e r with 3 / 2 < K < 2. W e apply A K = (1 - A) K/2 to both sides of equation (1.3), take the L2 inner product of them with AK~0~J)(t) and add the complex conjugate of the p r o d u c e to obtain

+ 4 A l t o ( ( / 3 ~ j', 6~ J))/3~ j), 6~J}), ~ = 4Aim ((/36~ j', ~J,(J))/34J~j), 6~J)) n~.

(2.5)

H e r e we need the following estimate II(/36, ~,)/3611, o, _< C l l 6 l l ~ . l t 6 l l , '

(2.6)

for any K with 3 / 2 < K < 1 and any m with 0 _< m < l. Using (2.6) we integrate (2.5) with respect to t to obtain

o + C ll6~J)(t)ll~ ~ _< 1160~)117~.

i t1162Z>(s)ll~z, ds,

t e [0, T,*j),

(2.7)

where T*,,~ satisfies 0 < T,*s < 1/CII6{~11~. ° _ T,,j * for Put a = sup{ll%,~J)IIH~; 0' < ~ < 1, ]' ~ N} and 7~ = 1~Ca. T h e n we suppose that 7~ > some e, j and lead to a contradiction. We note that [Iq,,~J)(t)ll~. is b o u n d e d by a solution y ( t ) of an integral equation y( t ) = a + C

f

. s ) 2 ds. 'v(

(2.8)

T h e solution y ( t ) of equation (2.8) exists on [0, {). Therefore, tp~J)(t) satisfies lim supll~J)(t)llH~ _< C,

(2.9)

t /" T*,

where C is a positive constant independent of E, j. Let 3' be any real n u m b e r with K < 3' _< 2. In the same way as above we apply A ~ to the both sides of equation (1.3) to obtain

4- 4/~/m (( /36E(j), t[I;J))/31/J(j), ~(J))H ~, -<

CIl~Y)ll~ql'"~s)l z,~,, H

•

(2.10)

A remark on the nonrelativistic limit for semilinear Dirac equations

1143

We integrate (2.10) with respect to t to obtain

114,2"(t)11},. _< Ila."',.0.,,Hz ÷ c

fO t 11&2,,(s)llZH.IIq~J)(s)ll2~

ds.

(2.11)

By Gronwall's inequality we have 11~2j (t)ll~,_
(f

t

Cll4,2J)(s)ll2n, ds .

(2.12)

/

*'(I

Hence by (2.9) and (2.12) we obtain for K < y _< 2

lim sup II qJ2J'( t)ll~,~ <_ C(Td,)ll

~,0(~)II2~.

(2.13)

On the other hand, since ~h):'(t) satisfies the equation

OJe'(t) = ,',<')-

7[ i 1

~OU)(s) +

i

¢l%(J)(s)]] ds

f

fO

- 2iA

(~qt~)(s), 44U)(s))~qJ~J)(s)ds,

lim, ~ 7-, q~J)(t) = 4~J)(T~*j) exists and lies in H ~. Then the initial value problem (1.3)with the initial "~ondition 0(0, x)'= +~J)(T~*j) has a unique solution ~[J)(t, x) for 0 < t < 7~. We s e t t~(t,x)= 4,~(J)(t,x) for O <_t <_ T~*j and 8 ( t , x ) = qJ~J'(t- T~*j,x) for T*j <_t <_ T~j + 7". Then ~ ( t , x ) is a solution of the initial value problem (1.3), (1.4) for 0 < t _< T*j + T and lies in C~([0, T *,J + 7~]; H ~) (and, hence, in C~([0, T~*,J + T ] ; ~ ) ) , which contradicts the definition of

%

Therefore, we obtain 0 < 7~ < T~*j for 0 < e < 1 and j ~ N. Furthermore, we can chose T* with 0 < T* < T~*: so that .,.u) 116~)(t)llH : _< C ( T * ) %,,,ne-

sup

(2.14)

te[(I,T*]

Inserting

~;(j) (t) = ~e'lJ,/'2p%~),t,

(2.15)

and ~0~ --

t-"~' 0 ~

into (2.14) we obtain the desired estimate

sup

II ÷i(J' (t)ll. _
t~[0, T*]

This completes the proof of lemma 2.2.

•

We now consider the convergence of {q~J)(t)}. First we have the following lemma.

(2.16)

1144

T. MATSUYAMA

LEMMA 2.3. Assume that 0 < e < 1. Then {~p~(t)} i is a Cauchy sequence in C([0, T*]; H 2) for any e. Moreover, it holds that

sup

ll~2J~(t) - ¢2k'(t)ll,~z < C(T*)ll~p¢0~ ~ - ~(k) IIH2 --

q"O~

t~[0, T*]

for j, k = 1, 2 , . . . .

Proof. It is sufficient to prove that II~(:)(t) - ~0~k)(t)llH- ~ _< C(T* )tl4,o('. ~ - q'o,¢k)ll'=,

sup t~[O,T*]

since we have the equality q:,c~)(t) = 2ei#t/2~[3t~J)(t). 4,,°(t) satisfies an integral equation

~(t)

=D(t) ,t,~l ~'"~ - 7iA ~ D(t -s)[(/3~,("(s), 4,~J)(s))/3g,~)(s)] ds,

(2.17)

where D(t) is the evolution operator of a free Oirac equation and expressed by

FT it D(t ) = exp - ~¢,'2 e taa, - ~

)

"

The Dirac operator is a skew-adjoint on H 2 from which it follows that D(t) is a one-parameter continuous unitary group on H 2. Hence by (2.17)we have for t ~ [0, T*] --

"Ft}~-

+("

--

'4~I)E

H 2

11(/34,y'(s),4~J'(s))~t0,~J'(s)

-(/3~o(k)(s), q#k'(S))/3tt,~k)(s)llH-~ ds

+(7

II~,F'(s)ll~t~ + II 4,,~<'(s)llT,=)ll ~ F ) ( s ) - ~0,~k)(s)lI. ~-ds. (2.18)

H e r e we have used the following estimate [l(/3g,, g,)/3g~- (/3~#, ~)/3~#llH-" < C(II~IIH-" + II,;ll~=)llq,- ,#ll,-" for g,, ~#~ H 2, which is an immediate consequence of the G a g l i a r d o - N i r e n b e r g inequality. F r o m lemma 2.2, (2.18) and the uniform boundedness in H 2 o f {~0(~)}, it follows that

114~,°(t) I 4,,(k~(t)llH- " -< I1~'o, (J~ - ~'o,""k)lllIH-'÷Cf' 11~,3]'(S)--~,~k)(s)IIH=, A

t~[0, T*].

(2.19)

1145

A r c m a r k on the nonrelativistic limit for s e m i l i n e a r D i r a c e q u a t i o n s

By Gronwall's inequality we obtain the desired estimate sup

I t g ( ~ ' ( t ) - 4¢(~'(t)llH-" < C(T*)II ,*,°~--

"P' 0 e

4,o~.)J[.2 '

:e[o,g*]

This completes the proof of lemma 2.3.

•

LEMMA 2.4. { ~ ~(t)} iS a Cauchy sequence in C([0, T* ]; H z). Moreover, it holds that

II "~(:)(t)~o

sup

-~- -< C( T* )ll~o~)ll- ~-

~[.1'*]

and sup t~

119~I/)(t) - ~l,kqt)ll,: -< C(T*)II "~(j>~-00- 9~l~qlH-~

I0,T*]

for j, k = 1, 2 . . . . .

Proof The evolution o p e r a t o r e " ~ : of a free Schr6dinger equation is a 1-parameter continuous unitary group on H2. Therefore. in the same way as lemmas 2.2 and 2.3 we can prove the assertion. • LEMMA 2.5. Assume that 0 < E < 1 and j ¢ N. Then we have

¢~" -~ ¢{~

*-weakly in ('([O,T* ], H 2)

( E ~ O),

where the convergence is uniform in j ~ N.

Proof We note that ~ J q t ) - @:'~(t)e C([0, T*]: H 2 ) for each e, j. By l e m m a 2.2 and the boundedness in H : of ,f'~0~ _~n~, /,.~jq ,,v00 J we have for an}' E, j sup

I!:~"(I)-~#I/'(t)llH:<_:

telt~l*]

sup

I!,~:~(t)ll.=+

: , [0,T* ]

sup

119/oJ~(t)liu--

t e [0,T*]

<" 7 ~

H e n c e the sequence {sc~J~(t!- q:I/~(t)} is b o u n d e d in C([0, T*]; H2). By proposition 2.1 we have for any ~ e , / sup

~(sc:')(t)- ¢{/)(t),O): I

t ¢ [(). T* !

_<

sup

11,~2~)(t)- ~/)(t)llL:llOIIz:

! e [0. T* ]

--, 0

(2.20)

uniformly in ) as e -, 0. Therefore from the boundedness in C([0, T* ]; H 2) of {q~J~(t) - q~c0J)(t)}, (2.20) and the fact that ~': is dense in H : it follows that

~ : ' ( t ) -~ @~J~(t )

* -weakly in C([0, T* ]; H 2)

uniformly in ] as e -~ 0. This completes the p r o o f of lemma 2.5.

•

Proof of theorem completed. From lemmas 2.2 and 2.3 for each e there exists a q~¢= q~(t, x) such that for any t ~ [0, T*] q:~:~--, ¢~ strongly in C([0, T*]; H 2) as j --+ ~c. F r o m l e m m a 2.4 there exists a ~,,, = ,#.,( t, x) such that for any t ~ [0, T* ] ~ --+ q:0 strongly in C([0, T* ]; H 2) as

T. MATSUYAMA

1146

j ~ ~. It is easily seen that ~, and q~0 solve equations (1.7) and (1.12), respectively. H e n c e by the uniform b o u n d e d n e s s in H 2 of {q~oE} we obtain

~J~ -~ q~,

strongly in C([0,T* ]; H 2)

( j --, o~),

(2.21)

q~J~ ~ ~0

strongly in C([0, T* ]; H 2 )

( j --~ ~c)

(2.22)

uniformly in E. We d e c o m p o s e the difference q~,(t) - q~0(t) into

~,(,)- ~o~,~ = [ ~,(,~- ~,'J'(,)] + [,,'J'(t)- ~,,k,(t)] + [ ~"(t)-

~p(ok)(t)] + [ q~(ok'(t)- q~o(t)].

T h e n from (2.21), (2.22) and l e m m a 2.5 we conclude that q~ ~ o o

.-weaklyinC([O,T*];H

This completes the p r o o f of theorem.

2)

(e--*O).

• REFERENCES

1. NAJMAN B., The nonrelativistic limit of the nonlinear Dirac equation, Ann. Inst. H. Poincar~ Analyse non Lin~aire 9, 3-12 (1992). 2. MATSUYAMAT., Rapidly decreasing solutions and nonrelativistic limit of semilinear Dirac equations, Reu. math. Phys. (to appear). 3. TSUTSUMI M., Weighted Sobolev spaces and rapidly decreasing solutions of some nonlinear dispersive wave equations, J. diff. Eqns 42, 260-281 (1981).