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Differential and Integral Equations, Volume 5, Number 4, July 1992, pp. 721-745.

THE NONLINEAR SCHRODINGER LIMIT AND

THE INITIAL LAYER OF THE ZAKHAROV EQUATIONS

TOHRU OZAWA

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan

YOSHIO TSUTSUMI

Departament of Mathematics, University of Tokyo, Hongo, Tokyo 113, Japan

(Submitted by: H. Brezis)

Abstract. vVc discuss the formation of the convergence of solutions for the Zakharov equations as the ion sound speed A goes to infinity. We describe how the solutions tend to the corresponding solutions for the nonlinear Schrodinger equation and how the initial layer phenomena occur. A clear account is given of the relation between the convergence rate in A and the formation of the initial layers.

1. Introduction and theorems. A simple system of equations describing the propagation of Langmuir turbulence in an unmagnetized, completely ionized hydrogen plasma was first obtained by Zakharov [17] by means of a two-fiuid description of the plasma. The system consisting of the ion sound equation and the electric field propagation equation with nonlinear coupling terms is derived from l'v1axwell's equations and linearized hydrodynarnical equations. Another derivation from a Lagrangian formalism is given by Gibbons, Thornhill, ·wardrop, and ter Haar [5].

In suitably scaled coordinates, the Zakharov equations arc the following:

DE i--::;- + 6E = nE,

ut

. 82 n A-~-- 6n = 6IEI 2

8t2 '

( 1.1)

( 1.2)

where E and n are functions on the tirne-space lR x JRN with values in eN and IR, respectively, and A > 0 is a parameter. In these equations, E is the slowly varying complex amplitude of the electric field [of the Langmuir waves with plasma frequency Wp,

E(t, :r) = Re (E(:r, t) exp( -itwp)),

n is the deviation of the ion density from its equilibrium, A is the ion sound speed, the right hand side of (1.1) describes the shift of plasmon frequency caused by the slow density variation n, and the right hand side of (1.2) describes the driving force caused by the pressure of plasmon gas.

Received for publication September 1991. AiVIS Subject Classification: 35Q20, 76F99, 76X0.5.

An International Journal for Theory & Applications

721

722 TOHRU OZAWA AND YOSHIO TSUTSUMI

In the limit >.---+ oo in (1.2), we formally have the equation ~(n + IEI2 ) = 0, so

that n = -IEI2 if n + IEI 2 vanishes at infinity. Therefore, in the limit A---+ oo, the Zakharov equations (1.1)-(1.2) are reduced to the nonlinear Schrodinger equation

i8tE + ~E = -IEI2 E.

Thus, the Zakharov equations can be regarded as a natural extension of the nonlinear Schrodinger equation in order to take a finite response time of the nonlinear medium of the ion part of the plasma into account and the limit >. ---+ oo turns out to be related to an instant response of the medium.

We consider the Cauchy problem for the Zakharov equations and the nonlinear Schrodinger quation and investigate the rate of convergence of the solutions as A ---+ oo. The Cauchy problem for the Zakharov equations is written with the subscript A as follows:

i :tE:>. + ~E:>. = n;>..E;>.., t > 0, x E IR.N,

(Z:>.)

E:>.(O,x) = E0 (x), n:>.(O,x) = n0 (x), 8 at n).. ( 0, X) = n 1 (X) ,

where >. > 1 and (Eo, n0 , ni) are the given initial data. We consider the Cauchy problem for the nonlinear Schrodinger equation with the same initial condition as in (Z:>.):

i :tE + 6.E = -IEI2 E, t > 0, X E IR.N,} E(O, x) = E0 (x).

(NLS)

From now on, we assume that the initial data (Eo, n0 , nl) arc in the Schwartz space S for simplicity.

In H. Added and S. Added [1], Ozawa and Y. Tsutsumi [11], Schochet and Weinstein [13], and C. Sulem and P.L. Snlem [14], it is shown that if (Eo, n0 , nl) E S and 1 :::; N :::; 3, then (Z:>.) has the unique solutions (E:>., n:>.) in n~ 1 Ck([O, T]; Hk) for some T > 0 and, in addition, if n 1 E H- 1 and N = 1, then T can be chosen to be T = +oo, where ifm = { 1/J E S' : (- 6. )mfZ?j, E L 2 } for m E JR. On the other hand, it is well known that if Eo E S and 1 :::; N :::; 3, then (NLS) has a unique solution E in nk=l Ck ([0, T]; Hk) for some T > 0 (see, e.g., Kato [8]). Let T max > 0 be the maximal existence time of E(t). In [2], H. Added and S. Added have shown that if (Eo, n0 , n!) E S EB S EB (S n if-l) and 1 :::; N ::::; 3, then for any T with 0 < T < T max and any positive integer m, there exist two positive constants C0 and >.. 0 such that (E>.(t), n>.(t)) exist on [0, T] for>.. 2 >.. 0 and satisfy

sup [IIE:>.(t)- E(t)llum + lln:>.(t) + IE:>.(t)l 2

O<!,_t<::,_T

1/2 2 ] - cos(At(-6.) )(no+ lEal )IIHm-1

{

C0 >..- 1 / 2 ifno+IEoi2 =/'-0 andN=l,2

:::; CoA- 1/ 2 log>.. ifno+IEol2 =/'-0 andN=3

CoA- 1 ifno+IEol2 =0 andl:::;N:::;3

(1.3)

(1.4)

(1.5)

ZAKHAROV EQUATIONS 723

(see also Schochet and Weinstein [13]). There is a discrepancy between the noncompatible case no + lEo 12 oj. 0 and the compatible case n 0 + lEo 12 = 0 concerning the rate of convergence as >. --+ oo. This corresponds to an initial layer phenomenon and the term cos(>.t( -.6.) 112 )(no + IEol 2 ) represents the initial layer for (Z,\).

On the other hand, a formal perturbation argument suggests that the right hand side of (1.5) can be replaced by C0 >.- 2 (see Gibbons [4) and Ozawa andY. Tsutsumi [12]).

Our aim in the present paper is to investigate the precise rate of convergence of solutions for (Z-\) and the formation of the initial layer in detail. In fact, the rate of convergence of solutions for (Z-\) is closely related to the formation of the initial layers, as we see later. We use the standard Sobloev space Hm, m E JR., and the weighted Sobolev space Hm,s, m, s E JR., defined by

Hm = {1P E S': lltPIIJim = 11(1- .6.)m/2 tPIIL2 < oo},

Hm,s = {1P E S': lltPIIHm,s = 11(1 + lxl 2 ) 812 (1- .6.)m/2 tPIIu < 00 }.

We put w = ( -.6.) 112 throughout this paper. The main results in this paper are the following.

Theorem 1.1. Assume that 1 :::; N:::; 3 and (Eo, n 0 , n 1 ) E S EB S EB (S n .H-1 ). Let E and Tmax > 0 be the solution of (NLS) in n~1 Cj ([0, Tmax); Hj) and its maximal existence time, respectively.

(1) For any T with 0 < T < Trnax and any positive integer m, there exist two positive constants M and >.0 such that (Z-\) has unique solutions (E,\,n,\) in n~ 1 Cj([O,T);Hj) for>. 2 Ao satisfying

sup lln,\(t) + IE,\(tW- Q~l)(t)- Q~2 )(t)llum :S MA- 1 , A 2 Ao, (1.6) o<t<T

where Q~l)(t) = cos(>.tw)(n0 + IEol 2 ) and Q~2 )(t) = (>.w)- 1 sin(>.tw)(n 1 + 2Im(E0 ·

.6.E0 )). Especially for >. 2 Ao,

sup lln,\(t) + JE,\(tW- Q~2)(t)JJHm :S M>. - 1 . ( 1. 7) O<::,f'5_T

(2) Assume n 0 + JEoJ 2 oj. 0. Then E,\(t) satisfies

sup IIE,\(t)- E(t)JJHm::; M>..- 1 , >.. 2 >..o, (1.8) 0'5_t'5_T

for some M > 0, where T and >.0 are defined in Part (1).

(3) Assume no + lEo 12 = 0. In addition, let n1 = !}x ¢ for some ¢ E S when N = 1 and let n1 = V' ·¢for some¢= (¢1, ¢z) E S ffi S when N = 2. Then E"-(t) satisfies

sup JJE,\(t)- E(t)JJnm::; lv!>..- 2 , >.. 2 Ao, (1.9) 0'5_t'5_T

for some M > 0, where T and >.0 are defined in Part (1).

Remark 1.1. (1) S C .H- 1 for N = 3, whileS rt, .H-1 for N = 1, 2. Therefore, n 1 E .H- 1 is redundant for N = 3. The first fact follows from the usc of the Hardy inequality in the Fourier space. The second fact follows from the example e-/x/ 2

•

(2) Q\1) and Q\2) are the solutions of the wave equation %122 Q~) - >.. 2 !j.Q~) = 0,

j = 1, 2, with the initial conditions

and

( Q\2 ) (0), :t Q\2)(0)) = (0, n1 + 2 Im (Eo · fj.Eo)),

respectively. Q\1) ( t) is the first initial layer and Q\2 ) ( t) is the second initial layer. It

is important to investigate the effect of the second initial layer Q\2 ) when we study the convergence of E>.(t) to E(t) as>..----+ oo.

(3) If N = 1, the maximal existence time Tmax of E(t) can be chosen to be Tmax = +oo (see, e.g., [8]). But when N = 2 and 3, Tmax can be finite under certain circumstances (see Glassey [6] and M. Tsutsumi (15]). It is conjectured that when N = 2 and 3, (Z>.) also has blow-up solutions (see, e.g., [17]).

( 4) When N = 1 and 2, in Theorem 1.1 (3) we need the additional conditions that if N = 1, n 1 = 8x¢ for some ¢ E S and that if N = 2, n 1 = V' · ¢ for some ¢ = ( ¢1, cP2) E S ffiS. Such a class offunctions is dense in if-1. In fact, the gradient operator V' is defined on the whole H1 as an operator from H1 to tJJf= 1 £ 2 and

so the dual of -V', the divergence operator, is surjective from tJJf= 1 £ 2 onto iJ-1 .

Therefore, for any v E if-1 , there exists a 'ljJ E tJJf=1 £ 2 such that v = V' · '1/J. It remains only to note that Sis dense in£2 .

The rate of convergence in (1.6) of Theorem 1.1(1) turns out to be better when we consider the problem locally in space.

Theorem 1.2. Assume that 1 ~ N ~ 3 and (Eo, n 0 , nl) E S ffi S ffi (S n ii-1 ). Let E and Tmax > 0 be the solution of (NLS) in n~1 CJ ([0, Tmax); HJ) and its maximal existence time, respectively. For any T with 0 < T < Tmax, any positive integer m and any number s with s > N /2 + 2, there exist two positive constants M and >.. 0

such that (Z>.) has the unique solutions (E>., n>.) in n~1CJ ([0, T]; HJ), for>.. ~ >..0 ,

satisfying

(1.10)

where Q~1 ) and Q\2) are defined as in Theorem 1.1(1).

The results in Theorems 1.1 and 1.2 arc optimal concerning the rate of convergence with respect to >..

Theorem 1.3. Assume that 1 ~ N ~ 3 and (Eo, n0 , ni) E S ffi S ffi (S n if-1 ).

Let E and Tmax > 0 be the solutions of (NLS) in n~1 CJ([O,Tmax); HJ) and its maximal existence time, respectively. Let (E>., n>.) be the solutions of (Z>.).

(1) There exist a pair of initial data (E0 , n0 ) E S ffi S and n 1 = 0 such that for any T with 0 < T < Tmax, any nonnegative integer m and any s ~ 0,

(1.11)

where Q~l) and Q~2 ) are defined as in Theorem 1.1(1).

(2) Let n1 = 0. We assume that (Eo, no) E S E8 Sand (no+ IEoi2 )Eo -=f. 0. Then for any T with 0 < T < T max and any nonnegative integer m,

liminf[>. sup IIE,$t)- E(t)IIHm] > 0. (1.12) A-->oo O'Sf~T

(3) Let n1 = 0. We assume that (Eo, no) E S E8 S, no+ IEol2 = 0 and (Im (Eo· .6.E0 ))E0 -=f. 0. Then for any T with 0 < T < Tmax and any nonnegative integer m,

(1.13)

Remark 1.2. (1) Let 0 < T < Tmax· We note that under the assumptions of Theorem 1.3, there exist the solutions (E,\,n,$ of (Z,\) in n~1 Ci([O,T]; Hi) for sufficiently large>.> 0 (see Proposition 2.1 and Lemma 2.2 in Section 2).

(2) Theorem 1.3 implies that the results in Theorems 1.1 and 1.2 are generically optimal concerning the rate of convergence with respect to >..

The above three theorems give a detailed description of the formation of initial layers with almost optimal rate of convergence. Let T max > 0 be the maximal existence time of the solution E for (NLS) and let 0 < T < Tmax· Theorems 1.1 and 1.2 imply that for sufficiently large >. > 0, the solution E,\ ( t) of the Schrodinger part for (Z,\) behaves like E(t) on the interval [O,T] and the solution n"(t) of the acoustic wave part for (Z,\) behaves like -IE(t)i2 on the interval (O,T]. This difference between the time intervals for convergence of these solutions comes from the initial layer phenomena. The formation of initial layers also reflects the rate of convergence of (Z,\) to (NLS) with respect to >..

Our proofs of the above theorems depend essentially on the special propagation properties of the acoustic wave and of the Schrodingcr wave. An outline of the proofs is roughly illustrated as follows. By setting Q ,\ = n,\ + IE" 12 , (Z") is rewritten as the system of integral equations:

E,$t) = eit6. Eo+ i 1t ei(t-s)6.(1E,\I2 E,\- Q,\E,$(s) ds,

Q"(t) = Q~ll(t) + Q~2l(t) + 1t sin>.~w- s)w a;IE"I2(s) ds,

where Q~1 ) and Q~2 ) are defined as in Theorem 1.1(1). Therefore, we have

E"(t)- E(t) = i 1t ei(t-s)t1(1E"I2 E" -IEI2 E)(s) ds

- i 1t ei(t-s)6.Q,\(s)E,\(s) ds. (1.14)

Our main task is to evaluate the second integral in the right hand side of (1.14). In [2], H. Added and S. Added use the time decay estimate of the solution of the acoustic wave equation to show that Q" -+ 0 as >. -+ oo and so Q "E" -+ 0 as >. -+ oo.

This argument is the same as the one used for the problem of the incompressible limit of the compressible Euler equation (see Asano [3] and Ukai [16]). But this arguement is not sufficient for the optimal result. In fact, the product of Q >. and E>. tends to zero faster as A --+ oo than Q >. alone tends to zero as A --+ oo.

The integrand Q>.E>. in the right hand side of (1.14) corresponds to the interaction between the acoustic wave Q >. and the Schrodinger wave E>.. Q >. propagates according to the generalized Huygens principle and is localized in a neighborhood of the sphere lxl = At. Hence, Q >. propagates very fast as A --+ oo. On the other hand, E>. propagates with group velocity independent of A and is well localized in a bounded domain with radius independent of A. Therefore, the main part of the support of Q>.(t), say {x: IQ>.(t,x)l ~ E} for some E > 0, and that of E>.(t) become almost disjoint and so the product Q >.E>. should be proved to converge to zero faster as A --+ oo than Q >. alone. This is a main idea of our proofs and a different point from the proof of [2].

Our plan in this paper is the following. In Section 2, we summarize one proposition and one lemma needed for Theorems 1.1, 1.2 and 1.3. In Section 3, we describe the proofs of Theorems 1.1 and 1.2. In Section 4, we give the proof of Theorem 1.3.

We conclude this section by giving several notations. Let w = ~' as already stated. We denote by U( t) the evolution operator of the free Schrodinger equation. ForsE IR?., let [s] be the greatest integer that is not larger than s. By(·,·) we denote the scalar product in L 2 . For a and b E IR?.N, we denote by a · b the N-dimensional scalar product between a and b. For a multi-index a: = ( o:1 , ... , o:N) with O:j ~ 0, we put lo:l = o:1 + · · · + o:N. We abbreviate 8j8xk, 1 :::; k :::; N, and 8/Dt toOk, 1 :::; k:::; N, and 81, respectively. For a multi-index a:, we put 8':; = 8':; 1 ••• o'ft. For two multi-indices a: and (3, the inequality a::::; (3 means that O:j :::; /3j for 1 :::; j :::; N. For 1 :::; p:::; oo, we abbreviate LP(IR.N) to LP. Let wm,p denote the Banach space

wm,p = N E LP: L IID':'IjJIILP < oo} lai:Sm

for a positive integer m and 1 :::; p :::; oo. Let H 00 = nm?_oHm. In the course of the calculations below, various positive constants are simply denoted by C.

2. Preliminary results. In this section we prepare one proposition and one lemma needed for proofs of Theorems 1.1, 1.2 and 1.3.

\Ve first state the proposition concerning the local existence and the regularity of solutions for (Z>.)-

Proposition 2.1. Assume that 1 :::; N:::; 3.

(1) Let m be an integer with m ~ 2 and let (Eo, n0 , nJ) E Hm ffi Hm-2 . vVe put m =min{ m, 3}. Then for some T > 0 there exist the unique solutions (E>., n>.) of (Z>.) such that

[m/2]

E>. E n CJ([O.T]; Hm-2j), j=O

[m/2]

E>. E n Wj,8/N (0, T; wm-2j,4), ;_n

(2.1)

(2.2)

rTt

n E n Cj([O.T]; Hm- 1-j), (2.3) j=O

and ifm:;::: 6, [m/2]+1

n E n Cj([O.T]; Hm+ 2 - 2 j). (2.4) j=4

Here the existence timeT depends only on N, A, JJEoJJHz, JJnoJIHI and JJn1JJu.

( 2) Especially let (Eo, n 0 , nl) E H= E9 H= E9 H=. Then the solutions ( E >-., n >-.) of (Z>-.) given by Part (1) belong to n~1 Cj([O, T]; Hj).

Remark 2.1. In addition, if N = 1,2, n1 E if- 1 and JJEoJJLz is small for N = 2, then (E>-.(t),n>-.(t)) given by Proposition 2.1 exist globally in time (see [1], [11] and [14]).

The local existence and the regularity of solutions for (Z>-.) were first shown by C. Sulem and P.L. Sulem [14] (see also Schochet and Weinstein [13]). However, in the previous papers [13] and [14], the existence timeT depends on the higher order Sobolev norms of the initial data when the solutions are regular. Proposition 2.1 is proved in [11]. For details, see [11, Theorem 1.1 and Remark 1.1 ( 4) ].

Next we state the following lemma concerning the a priori estimates independent of A for the solution of (Z>-.)·

Lemma 2.2. Assume that 1 ~ N ~ 3. Let k be an integer with k :;::: 3. Assume that (Eo, n 0 , nl) E Hk+ 2 E9 Hk+ 1 E9 (Hk n if- I) for N = 1 and (Eo, n 0 , nJ) E Hk+ 3 E9 W 1·k+3 E9 (Hk n if- 1) for N = 2, 3. Let E and Tmax be the solutions of (NLS) in n}=0 cj ([0, Trnax); Hk+ 1- 2j) and its maximal existence time, respectively.

(1) For any T with 0 < T < Tmax, there exist two positive constants J( and Ao such that the solutions (E>-., n>-.) of (Z>-.) given by Proposition 2.1 exist on [0, T] for A 2: Ao and satisfy

sup sup (JJE>-.(t)JJHk+I + JJn>-.(t)JJHk) ~ K :\~:Ao O:St:ST

(2.5)

(2) In addition, suppose that JxJjEo E Hk-j for 1 ~ j ~ k. Let Ao be defined as

in Part (1) for any T with 0 < T < Trnax· Then there exista a positive constant K such that

(2.6)

Remark 2.2. (1) Lemma 2.2(1) is due to H. Added and S. Added [2]. In [2], the related results to Lemma 2.2(1) are separately stated (see the proof of Theorems 3.1, 3.2 and 3.3, Lemma 3.1, and Corollary 3.2 in [2]). Although Theorems 3.1-3.3 in [2] are stated somewhat in a different style from Lemma 2.2(1), Lemma 3.1 and Corollary 3.1 in [2] imply Lemma 2.2(1).

(2) Lemma 2.2(2) shows that the solution E>-.(t) of the Schrodinger part for (Z>-.) is localized near the origin regardless of A. This fact plays an essential role in the proofs of Theorems 1.1 and 1.2.


Proof of Lemma 2.2. For the proof of Part (1), see [2, Theorems 3.1-3.3, Lemma 3.1 and Corollary 3.1]. We only describe the proof of Part (2).

We put gz(x) = (1 + lx/W)-kf2 for a positive integer l. A simple calculation gives us

(2.7)

for 1 ~ j ~ k, where c1 does not depend on l. \Ve first take the imaginary part of the scalar product in L2 between the first

equation of (Z.x) and (1 + lxl 2 )g[ E.x(t) to obtain by (2.7),

11(1 + lxl 2 ) 112gzE.x(t)III2 ~ 11(1 + lxl2 ) 112gzEo(t)lli2 (2.8)

t N

+C 1 (LIIgzY'E.xjiiLz)11(1+lxl 2 ) 112gzE.xlluds, O~t~T, >->>.0 .

0 j=1

By (2.5) in Part (1) and Gronwall's inequality,

11(1 + lxl 2 ) 1 1 2 gzE.x(t)ll~:z ~ C2 , 0 ~ t ~ T, ). ~ >-o, (2.9)

where C2 does not depend on l. Letting l --+ oo in (2.9), we obtain by Fatou's lemma,

2 1/2 11(1 + lxl ) E.x(t)ll p ~ C2 , 0 ~ t ~ T, >. ~ >.0 . (2.10)

\Ve next show that if we have

11(1 + lxl 2 )(p-lal)/ 2 8~ E.x(t)llu ~ C(p), 0 ~ t ~ T, A~ Ao, 0 ~ Ia: I ~ p,

(2.11) for some integer p with 1 ~ p ~ k- 1, then (2.11) also holds with p and C(p) replaced respectively by p + 1 and C(p + 1), where C(p) does not depend on >.. For a multi-index o: with lo:l = p, we act 8';; on the first equation of (Z.x) and take the imaginary part of the scalar product in L2 between the resulting equation and (1 + lxl2 )g[a:::E.x(t) to obtain by (2.7),

11(1 + lxl 2 ) 112gz8';;E.x(t)Wb::; 1[(1 + [x[ 2 ) 112.t]l8';;Eo[[Iz

·t N

+C 1 (L lfgz8';;V'E.xj[fu)11(1 + [:~f) 1 1 2 g,8~E.x(t)l[~:2 ds 0 j=1

+C .lt ll8';;n.xllull(1 + lxl 2 ) 112gzE.xll L'[[(1 + lxl2 ) 112g,8';; E.x(t)llu ds (2.12)

+C L lt lla:::-iJn.xiiL=II(l + l:r[ 2 ) 1 f:l_r718~E.xllull(l + [.r[ 2 ) 112gt8:::E.xll£2 ds 1~liJI~Ial-1 °

!]~a

::=;[[(1 + [x[ 2 ) 112gt8';; Eo[[]~z + C ([ [[E.x[[nlai+I + [[n.x[[JIIai+J[[(1 + [1:[ 2 ) 112 E.x[[Hl .fo + L lln.xllui,+IPI+ 2 11(I + [:I:[ 2 ) 112 8~E.xll~: 2 ][1(1 + [x[ 2 ) 112gt8';;E.xllv ds,

1~liJI~Ial-1 ,B~cx

0:::; t:::; T, ..\ ~ ..\o, [o:[ = p.

For the last inequality of (2.12), we have used the Sobolcv imbedding theorem. By (2.5) in Part (1), (2.11) and Gronwall's inequality,

where C3 does not depend on A and l. Letting l--+ oo in (2.13), by Fatou's lemma we obtain

For a multi-index o: with lal = p- 1, we act 8'; on the first equation of (Z.\) and take the imaginary part of the scalar product in L 2 between the resulting equation and (1 + lxl 2 ) 2 g[8';E-'(t) to obtain by (2.7),

(2.15)

t N

+C 1 (L 11(1 + lxi 2 ) 112 8~'Y'E-'illu )11(1 + l:rl 2 )9t8~E-'IItz ds 0 j=1

+C L lt 118~-iJn-'IIL=II(l + l:rl 2 )gt8~E-'IIull(l + lxl 2 )gi8~E-'IIu ds liJI~Ial-1

!)~a

~11(1 + lxl 2 )8~ Eolliz + C 11 [ L 11(1 + lxl 2 ) 112 8~ E-'llu

0 liJI=P

+ L lln-'IIHifrl-1!31+ 2 11(1 + l:1f)8~E-'II u] 11(1 + lxl 2 )g,8~ E-'llu ds, lfll~lnl-1

(J<a

0 ~ t ~ T, A 2: Ao, lal = p- 1.

For the last inequality of ( 2.15), we have used the So bolev imbedding theorem. By (2.5) in Part (1), (2.11) and Gronwall's inequality,

where C1 docs not depend on.\ and l. Letting l--+ oo in (2.16), by Faton's lemma we obtain

If p 2: 2, we repeat this procedure to obtain (2.11) with p and C(p) replaced respectively by p + 1 and C (p + 1).

Thus, (2.5) in Part (1), (2.10) and an induction argument give us (2.6), which completes the proof of Part ( 2).

3. Proofs of Theorems 1.1 and 1.2. In this section, we describe the proofs of Theorems 1.1 and 1.2. Let E and Tmax > 0 be the solutions of (NLS) in

nj;': 1Ci([O,Tmax); Hi) and its maximal existence time, respectively. We note that

Tmax is the same as the maximal existence time ofthe H 1 solution of (NLS) (see, e.g., [8]). Let T be an arbitrary constant with 0 < T < Tmax· By using Proposition 2.1 and Lemma 2.2 with k = m + 6, we can choose three positive constants Ao, K, and K such that for A~ Ao there exist solutions (E>-., n>-.) of (Z>-.) in n~1 Ci([O, T]; Hi) satisfying

(3.1)

(3.2)

We put Q>-.(t) = n>-.(t) + IE>-.(t)j 2 • Then for A ~ Ao, (Z>-.) can be rewritten as follows:

(3.3)

8lq>-.-A2 ~Q>-.=8iiE>-.I, O~t~T, xEIRN, (3.4)

E>-.(0) =Eo, Q>-.(0) =no+ IEol 2 , BtQ>-.(0) = n1 + 2Im(Eo ·~Eo). (3.5)

Since E(t) satisfies (NLS) fortE [O,T], we substract (NLS) from (3.3) and rewrite the resulting equation in the integral form

E>-.(t)- E(t) = i 1t U(t- s)(IE>-.1 2 E>-. -IEI2 E)(s) ds

- i 1t U(t- s)(Q>-.E>-.)(s) ds, 0 ~ t ~ T, A~ Ao.

We take the Hm norm of (3.6) to obtain by the Sobolev imbedding theorem ,

IIE>-.(t)- E(t)ilnm ~ C 1t (IIE>-.II][m+2 + IIEII~m+2)jjE)..- Ellnm ds

+ 1T IIQ>-.E>-.IInm ds, 0 ~ t ~ T, A~ Ao.

By (3.7), (3.1) and Gronwall's inequality,

(3.6)

(3.7)

where C5 and C 6 do not depend on A. Therefore, we have only to investigate how fast the right hand side of (3.8) converges to zero as A -+ oo.

We first state the proof of Theorem 1.1.

Proof of Theorem 1.1. By Duhamel's principle, we represent Q>-.(t) as follows:


where

Q~1 \t) = cos(.Atw)(no + JEoJ 2 ), (3.10)

Q~2)(t) = (.Aw)- 1 sin(.Awt)(n1 + 2Im(Eo · !::J.Eo)), (3.11)

Q~3l(t) = t sin .Aw(t- s) a;JE>.(sW ds. (3.12) } 0 .Aw

As stated in Remark 1.1(2), we note that Q~1 ) and Q~2 ) are the solutions of the

wave equations ofQ~) - .\2 t:J.Q~) = 0, j = 1, 2, with the initial conditions

and (2) (2) -(Q>. (0),81Q.\ (0)) = (O,n1 +2Im(Eo ·.6.Eo)),

respectively, and that Q~3 ) is the solution of the inhomogeneous acoustic wave equation

with the initial condition

By the first equation of (Z>.), we have

N

8zJE>.J 2 =8t(2ImE>.. t:J.E>.) = Bt[2Irn Lv. (E>.jY"E>.j)] j=l

N N

=2\7 · [Im L81(E>.jY"E>.j)] = 2\7 · [Re L{-t:J.E>.jY"E>.J (3.13)

j=l j=l

By (3.13), Lemma 2.2(1) and the Sobolev imbedding theorem, we have

IIQ~3 )(t)IIH= :::; ~ t II sin >.w(t- s) a;JE>.(sWII ds (3.14) A Jo w II=

:::; ~ 11' llw-]o;IE>.(sWIIHm ds:::; ~ lt (IIE>.IIII=+2IIE>.IInm+3

+ JJn>.JJHm+2JJE>.JJnm+2JJE>.JJHm+J + JJE>.JJH=+2JJE>.JJnm+ 3

+ JJE>.JJHm+2JJn>.JJum+3JJE>.JJHm+J)ds:::; C7.A- 1 , 0:::; t:::; T, A 2: Ao,

where C7 docs not depend on >.. Furthermore, we easily sec that


where C8 does not depend on A. Therefore, (3.1), (3.14) and (3.15) show Theorem 1.1(1). Furthermore, (3.1), (3.14), (3.15) and the Sobolev imbedding theorem yield

1T ii(Q>.E>.)(s)iiHm ds

~ 1T IIQ~1 ) E>.iiHm ds + 1T(IIQ~2)IInm + IIQ~3)IInm)II£.\11Hm+2 ds (3.16)

T

~ 1 IIQ~1 ) E>.llnm ds + CgA - 1 , 0 ~ t ~ T, A 2: Ao,

where C9 docs not depend on A. Now we divide the proof into three cases according to the spatial dimensions. (Case 1). We assume N = 1. We first show Theorem 1.1(2). By the explicit

representation formula of a solution for the wave equation, we have

Q~1 )(t,x) = ~[f(x +At)+ f(x- At)], (3.17)

where f(x) =(no+ IEol 2 )(x) (see, e.g., [7, Chapter II]). Since n 0 , Eo E S, we have by (3.17),

A > 0, 0 ~ j ~ m, where C10 docs not depend on A. Therefore, (3.2) and (3.18) yield

IIQ~1 l ( t)E>- ( t) II nm m

~C L {11(1 + lx + Ati 2 )- 1 8~E>.(t)ilu + 11(1 + lx- Ati 2 )- 1 8~E>.(t)iiu} j=O

m

~c2.:)11(1 + lx + AW)- 1 (1 + lxi 2 )- 1 IIL=II(l + lxi 2 )8~E.xllu j=O

+ 11(1 + ix- Ail 2 )- 1 (1 + lxi 2 )- 1 IIL= 11(1 + lxi 2 )8~E>-IIu}

~C11 (1+At)- 2 , O~t~T, A2:Ao,

where C11 docs not depend on >... By (3.19),

T T 1 II(Q~1 ) E>.)(s)ilnm ds ~ Cu 1 (1 + >..s)-2 ds

= Cu>..- 1 [1- (1 + >..T)- 1] ~ C 11 >..- 1 , 0 ~ t ~ T, A 2: Ao.

Estimates (3.8), (3.16) and (3.20) show Theorem 1.1(2) for N = 1.

(3.19)

(3.20)


\Ve next prove Theorem 1.1(3). For the proof of Theorem 1.1(3), we need more

precise estimates of Q~2 ) and Q~3 ) than (3.14) and (3.15). By the explicit representation formula of a solution for the wave equation, we have

(3.21)

(3.22)

(sec, e.g., [7, Chapter III]). Since n1 = OxrP and Im (Eoa;,Eo) = Bx[Im (EoaxEo)J, we have

Q~2 )(t, x) = 2\ [f(x + >.t) + f(x- >.t)], (3.23)

where f(x) = (¢ + 2 Im (EoaxEo))(x). Since¢, Eo E S, (3.23) yields

). > 0, 0 ~ j ~ rn, where C 12 does not depend on >..Proceeding in the same way as in (3.19) and (3.20), we obtain by (3.24),

(3.25)

where C13 does not depend on>.. Since we have, by (3.13),

for N = 1, we obtain by (3.22),

1 r Q~3 )(t,x)= 2>-.Jo [f(s,x+>.s)a+f(s,x->.s)]ds,. (3.26)

where f(x) = 2Rc(-a;.E;..axE;.. + n;..E;..axE;.. + E;..a~E;..- E;..ax(n;..E;..))(.r). Since we have, by (3.2) and the Sobolev imbedding theorem,

laU(s, x ± ..\s)l:::; (1 + 1.T ± ..\sn-1 {(1 + lx ± ..\sl 2 )laU(s, x ± >.s)l}

~ C(1 + lx ± >.sl 2 )- 1 ll(l + lxl 2 )8~f(s)llu'

~ C(1 + lx ± >.xl 2 )- 1 , 0:::; t ~ T, >. 2: >-o, 0 ~ j:::; m,

formula (3.26) yields

\o~Q~) (t, x)l :::; c14r1 1t [(1 + lx + >.sl 2)- 1 + (1 + l:r:- >.si 2)-1] ds, (3.27)

O~t~T, >.;:::>.0 , O~j~rn,

where C 14 does not depend on >.. Proceeding in the same way as in (3.19) and (3.20), we obtain by (3.27),

T

111(Q~3 ) E;..)(s)IIHm ds ~ C1sr2, >. ~ >-o, (3.28)

where C15 does not depend on>.. Estimates (3.25) and (3.28) show Theorem 1.1(3) for N = 1.

(Case 2). We assume N = 2. We first show Theorem 1.1(2). We may assume >.0 T > 2 without loss of generality. For ). and t with >.t > 2, we define tp >.t E c= (IR \ { 0}) as follows:

tp>.t(x) = tp(lxl- ~t + 1), (3.29)

where tp E c=(IR), tp(s) = 1 for s ~ 0 and tp(s) = 0 for X ~ 1. We put h(x) = tp>.t(x)(no + IEol 2)(x) and fz(x) = (1- tp;..t)(x))(no + IEol 2)(x). Then we write

Q~1 )(t) = cos(>.tw)h + cos(>.tw)fz. (3.30)

A direct calculation gives us

II cos(>.tw)fziiJJm+2 ~ c1611fzi1Hm+2, t > 0, ). > 0, (3.31)

where C 16 does not depend on>.. Since fz E Sand suppfz C {x E IR: lxl > ~1 }, we obtain by the Sobolev imbedding theorem and (3.31),

118~ cos(>.tw)fzllu"' ~Gil cos(>.tw)fziiHI<>I+2 ~Gil cos(>.tw)fzi1Hm+2

~ CIIJziiJ[m+2 ~ C L 118~/zll£2 li3I:Sm+2

~ C L 11(1 + lxl)- 2{(1 + lxi) 2 8~fz}IIL2 li3I:Sm+2

2 t > ~' ). ~ >.0 , lal ~ m,

where cl7 does not depend on >.. By the explicit representation formula of a solution for the wave equation, we

have

1 1 ~(~) (cos(>.tw)~)(x) = 2IrA 8t lx-~I:S>.t ((>.t)2 -lx- ~12)1/2 d~, ~ E S

(see, e.g., [7, Chapter II]). If supp'ljJ C {~: I~ I ~ ~~},we obtain

(cos(>.tw)~)(x) = 2 ~). 8t lx-~I:S";>.t ((>.t)2 -~~~~ ~~ 2 )1; 2 d~ (3.33)

ZAKHAROV EQUATIONS

Therefore, noting that lx- ~~ ~ ~At for lxl ~ ~~ and 1~1 ~ ~~ and that suppfi C

{~: I~ I ~ ~~},we obtain by (3.33),

(3.34) where C18 does not depend on A.

On the other hand, by the Sobolev imbedding theorem, we have

II8~Q\1 )(t)IIL= ~ CII8~Q\1 )(t)IIH1al+2 (3.36)

~ Cllno + IEol 2 llu=+2 ~ C2o, t > 0, A> 0, lal ~ m,

where C19 and C20 do not depend on A. By (3.2), (3.30), (3.32) and (3.34)-(3.36),

1T II(Q\1) E>.)(t)llu= dt

~ {T ll(cos(>.tw)h)E>-IIH'"(Ixl<.}!l dt + {T ll(cos(>.tw)h)E>-II 11 =(ixl>.}!l dt }2/>- 4 }2/A 4

l T 12/>-+ ll(cos(Atw)h)E>-IIu= dt + IIQ\1) E>-IIH= dt

2/>- 0

~ C {T II cos(Atw)hllw=.=(lxi<.}!)IIE>-IIH=(ixl<.}!) dt }2/A 4 4

+ C {T II cos(Atw )!I llw=.=(lxl>.}!) IIE>-11 H=(ixl>.}!) dt (3.37) }2/>- 4 4

1T 1

2/>-+ C II cos(Atw)h llw=.= IIE>-IIH'" dt + C IIQ\1)IIwm,= IIE>-IIH= dt

2/ >. 0

where C21 does not depend on>.. Estimates (3.16) and (3.17) show Theorem 1.1(2) for N = 2.

We next prove Theorem 1.1(3). For the proof of Theorem 1.1(3), we need more

precise estimates of Q~2) and Q~3 ) than (3.14) and (3.15).

We first evaluate Q~2 ). We put

2

f = ¢+ LEoj'vEoj, fi(x) = 'P>.t(x)f(x) j=l

and h(x) = (1- 'P>.t(x))f(x) for >.t > 2, where 'P>.t is defined as (3.29). Then we have

Q(2)( ) _ sin >.tw" f sin >.tw" f >. t - -- v . 1 + v . 2· >.w >.w

(3.38)

By the explicit representation formula of a solution for the wave equation, we have

( sin >.tw " f ) ( ) 1 1 V' c !I ( 0 v·1 x=- d~ >.w 27r.X lx-el$>-.t ((>.t)2 -lx- ~12)1/2

(3.39)

(see, e.g., [7, Chapter II]). Since supp !I C {~ : 1~1 ~ ~t} and lx- ~~ ~ ~>.t for lxl ~ ~t and 1~1 ~ :t, by (3.39) and integration by parts we obtain

la"'(sin>.twV'·il)(x)l=-1 laa1 (x-~)·h(O d~~ x >.w 27r.X x lx-el9t ((>.t)2 -lx- ~12)3/2

~ c.x-11 ((>.t)2 -lx- ~12)-(2+lnl)/21h(~)ld~ lx-el9t

(3.40)

>.t 2 lxl ~ 4' t > -:\' >. ~ >.o, lal ~ m,

where c22 does not depend on >.. On the other hand, the Sobolev imbedding theorem and a direct calculation yield

lla~si:~twV'. fit=~ cllsi:~twV'. JIIIHI<>I+2 ~ c>.-lllfiiiHI<>I+2

2 ~ C.X-1IIfiiiHm+2 ~ C23A-1, t >~'A~ Ao, lal ~ m.

(3.41)

Since supp h C { x : \x \ ~ ~~ - 1}, we have by the Sobolev imbedding theorem,

11 8': si~ >.tw V' . h II 5o ell si~ >.tw V' . fz II 5o C >. -IIIfz II H~+2 AW £oo AW IJI<>I+2

5o c>.-1 L ll8~fz\l£2 = Cr1 L 11(1 + lx\)-2{(1 + lx\)- 2 8~!2}11£2

5o c>.-1 L ( 1 +I ~t- 11) - 2\1(1 + lx\)- 2 8~!211£2 (3.42)

I.BI~m+2

where c24 does not depend on >.. Furthermore, the Sobolev imbedding theorem and a direct calculation yield

II8':Q~2 )(t)\\£oo 5o CIIQ~2 )(t)IIHI<>I+2 5o c>.-1 llfllulnl+2

5o c>.- 1 \\fl\u~+2 5o C2s>-- 1 , t > 0, >. > 0, Ia\ 5o m,

where C25 does not depend on >..

(3.43)

Proceeding in the same way as in (3.37), we obtain by (3.2), (3.38) and (3.40)(3.43),

(3.44)

where c26 does not depend on >.. We next evaluate Q~3)(t). ForT~ t > s ~ t and>.~ >.o, we put

2

f = 2Re L[-~E>.jV' E>.j + n>,E>.jV' E>.j- E>.jV' ~E>.j- E>.jV'(n>.E>.j)], j=l

ft(s,x) = .t(x)f(s,x), h(s, x) = (1- .t(x))f(s, x),

where 'f!>.t(x) = tf>.(t-s)(x) and tf>.(t-s)(x) is defined in (3.29). Then, noting (3.13), we write

(3.45)

where

(l)( ) _ sin>.(t- s)w....,. f ( ) R>. t, s - >.w v 1 s , R (2)( )-sin>.(t-s)w ( >. t,s- >.w Y'·fz s).

Proceeding in the same way as in (3.40), we have by (3.1) ,

IIR~l\ t, S) ll£oo(lxl< .\(t;s))

__ 1 118a:1 (x-~)·ft(s,~) d~~~ -27rA X lx-(19(t-s) {(>.(t- s))2 -IX- ~~2}3/2 £OO(Ixl<.\{ .. -s))

(3.46)

5oC>.-1 (.\(t- s))-2-lalllh(s)llu 5o C21>.- 1 (>.(t- s))-2,

2 t-s>-:\' T~t>s~O, >.~>.0 , \a\'5om,

where C27 does not depend on >.. On the other hand, the Sobolev imbedding theorem and a direct calculation yield

118~ R~1 )(t, s)llu>O ::::; IIR~1 )(t, s)IIJiial+2 ::::; c>.-1llh (s)IIHI<>I+2

2 ::::; CA- 1 IIh(s)IIH~+2::::; C2sA-1, t- s > :\' T 2:: t > s 2::0, A 2:: Ao, lal::::; m,

(3.4 7) where C28 does not depend on A.

Since supp h C {(s,x): lxl2:: ~s -1}, we have by (3.1), (3.2) and the Sobolev imbedding theorem,

II8~R~2 )(t,s)IIL=::::; IIR~2)(t,s)IIHI<>I+2::::; CA- 1 IIh(s)IIH~+2

::::; cA-1 2:: llo~h(s)ll£2::::; cr1 2:: 11(1 + lxl)-2 {(1 + lxl) 2 o~f2(s)}ll£2

::::; CA-1 L ( 1 +I >.(t; s) - 11) -2 11(1 + lxl) 2 8~h(s)ll£2 li31~m+2

(3.48) where C29 does not depend on A.

Furthermore, (3.1), the Sobolev imbedding theorem and a direct calculation yield

II8~(R~1 ) (t, s) + R~2 ) ( t, s)) II L'"' ::::; CIIR~1 ) (t, s) + R~2 ) ( t, s )llul<>l+2

::::; c>.-1IIJ(s)IHI"I+2::::; c>.- 1 llf(s)IIII~+2::::; C3orl,

T2::t>s2::0, A>Ao, lal::::;m,

II8~Q~3 \t)IIL=::::; CIIQ~3 )(t)IIHI<>I+2::::; CIIJIIL=(o,T;Jim+2)::::; c31,

T2::t:::;O, A2::Ao, lal::::;m,

where C30 and C31 do not depend on A.

(3.49)

(3.50)

Proceeding in the same way as in (3.37), we obtain by (3.1), (3.2) and (3.46)(3.50),

+ \\R~1 )(t, s)E.\(t)I\H~(Ixl> "(t4-s)) + \IR~2 )(t, s)E.\(t)IIHm] ds dt

+ {T jt II(R~1 )(t, s) + R~2 )(t, s))E.\(t)iiHmds dt }2/ .\ t-2/ .\

12/.\ 1T lt-2/ .\ + IIQ~3)(t)E.\(t)11Hm dt::::; CA-1 (A(t- s))-2dsdt

0 2/.\ 0

(3.51)

where c32 does not depend on >.. Estimates (3.44) and (3.51) show Theorem 1.1(3) for N = 3. (Case 3) We assume N = 3. The proof for N = 3 is almost the same as that for

N = 2. We only describe how we have to change the proof of Case 2 for N = 3. The crucial difference between the cases N = 2 and N = 3 consists in the form of the fundamental solution of the wave equation.

We first consider the proof of Theorem 1.1(2). As in (3.30) of Case 1, we write

Q~1 )(t) = cos(>.tw)JI + cos(>.tw)h,

where fi and h are the same as in (3.30). Since cos(>.tw)h has a different form from (3.33), we can not use the same argument as in Case 2 to obtain (3.34). All the other estimates in the proof of Theorem 1.1(2) hold even when N = 3. However, a better estimate than (3.34) holds when N = 3. Since the Huygens principle holds in a strict sense for N = 3, we obtain

(cos(>.tw)JI)(t, x) = 0, >.t

lxl ::=; 2' 2

t > ~· (3.52)

which implies that (3.34) also holds for N = 3. We do not need any other important changes for the proof of Theorem 1.1(2) when N = 3.

We next consider the proof of Theorem 1.1(3). For the evaluation of Q~2 )(t), we 3 -

put f = n1 +Y'·O=j=1 EajY'Eoj), JI(x) = 'P>-.t(x)f(x) and !I(x) = (1--.t(x))f(x), where tp>-.t(x) is defined as in (3.29). Then we write

Q(2 ) ( ) _ sin >.tw f sin >.tw f >-. t - >.w 1 + >.w 2 ·

Since ( >.w) -l sin >.tw fi has a different form from ( 3.39), we can not use the same argument as in Case 2 to obtain (3.40). Even when N = 3, all the other estimates on Q~2 ) in Case 2 hold with some minor changes which come from the difference between the definitions off, !I and h in Cases 2 and 3. Since the Huygens principle holds in a strict sense for N = 3, we obtain

>.t lxl ::=; 2' (3.53)

which implies that (3.40) also holds for N = 3. We do not need any other important changes.

For the evaluation of Q~3)(t), we define f(t,x) to be the right hand side of (3.13) and we put fi(s, x) = -.t(x)f(s, x) and h(s,x) = (1- -.t(x))f(s, x), where -.t(x) = ltJ>-.(t-s)(x) and tp>-.(t-s)(x) is defined in (3.29). Then we write

where

R(1)( ) = sin>.(t- s)wf ( ) >-. t, s >.w 1 s , R(2)( )=sin>.(t-s)wf()

>-. t, s >.w 2 s .


Since the form of the fundamental solution of the wave equation for N = 3 is different from that for N = 2, we can not use the same argument as in Case 2 to obtain (3.46). Even when N = 3, all the other estimates on Q~3 ) in Case 2 hold with some minor changes which come from the difference between the definitions of f, fi, and h in Cases 2 and 3. Since the Huygens principle holds in a strict sense for N = 3, we obtain

II >.(t-s) X :S 2 '

2 t - s > - T >_ t > s 2': 0,

- )..' ).. 2': Ao, (3.54)

which implies that (3.46) also holds for N = 3. We do not need any other important changes.

Thus, the proof of Theorem 1.1 is completed.

Remark 3.1. (1) Estimates (3.34) and (3.40) imply the time decay of the local L 00 norm of the solution for the wave equation and they are closely related to the decay estimate (7 f) in Klainerman [9]. But it is important that the right hand sides of (3.34) and (3.40) do not contain the derivatives in t of the data at t = 0 of the solution because those quantities depend on >..

(2) Let N = 2. If we drop the assumption that n 1 = V' ·¢for some¢= ( ¢ 1 , ¢ 2 ) E S E9 S, then we have (1.9) with ).. - 2 replaced by ).. - 2 log >..

Theorem 1.2 is an immediate consequence of the estimates obtained in the proof of Theorem 1.1.

Proof of Theorem 1.2. We have only to evaluate the remainder Q~3\t), which is defined in (3.9) and (3.12). We note that (1 + lxl 2 )-s E H 0 '2 for s > 1¥- + 2. Therefore, we usc (3.27) for N = 1, (3.46)-(3.50) for N = 2 and (3.54) instead of (3.46) for N = 3 to obtain Theorem 1.2.

4. Proof of Theorem 1.3. In this section, we describe the proof of Theorem 1.3.

Proof of Theorem 1.3. We first state the proof of Theorem 1.3(1). For that purpose, it is sufficient to show that there exist a pair of initial data ( E0 , n 0 ) E S EElS and n 1 = 0 such that for a> 0,

( 4.1)

where Q~3)(t) is defined in (3.9) and (3.12). Let a> 0. We choose (Eo, n 0 ) E S CDS such that n 0 + IEol 2 = 0 and

N

w- 2 (1- cos aw)V' · [2Re L { -~EojY' Eoj + noEojY' Eoj j=l ( 4.2)

+ Eoj V' ~Eoj - Eoj V' ( noEoj)} J -=/:- 0.

By the mean value theorem, we have

Hence, the left hand side of (4.2) is in L2. Since w-2(1- cosaw)v =f. 0 for any nonzero v E L2, we easily sec that there actually exists a pair of initial data satisfying (4.2) and n 0 + IEol 2 = 0.

By making the change of variables As= Tin (3.12), we have

A2Q\3) (~) =A 1af>-. sin( a: As)w (o;IE;...I2)(s) ds

= 1a sin(a: tau)w (D;IE;...I 2 )(~) dT.

On the other hand, by (Z;...), we have

rj).. E;...(TjA) = U(TjA)Eo- i .fo U(TjA- s)(n;...E;...)(s)ds, 0 ~ T ~a,

1rf>-. sin( T- As)w n;...(T/A) = cos(Tw)no +A lliE;...I 2 (s)ds,

0 w 0 ~ T ~a.

By (2.5) and (4.5),

E;...(~)-----+ E0 in H3 as A-----+ oo,

uniformly for 0 ~ T ~ a. By ( 4.6) and integration by parts, we have

n;...(T/A) = cos(Tw)no -IE;...I 2 (T/A) + cos(Tw)IEol 2

rj).. +.1

0 cos((T-As)w)DsiE;...I 2 (s)ds, O<T<a.

By (4.8), (4.7), (2.5) and the Sobolcv imbedding theorem,

uniformly for 0 ~ T ~ a. Since no + lEo 12 = 0, we have

n;...(T/A)-----+ n0 in H 3 as A-----+ oo,

( 4.4)

(4.5)

(4.6)

(4.7)

(4.8)

(4.9)

uniformly for 0 ~ T ~a. Therefore, (3.13), (4.7), (4.9) and the Sobolev imbedding theorem imply that

(4.10)

uniformly for 0 ~ T ~ a, where

N

f = \7 · [ 2 Re L { -llEoj \7 Eoj + noEoj \7 Eoj + Eoj \76Eoj - Eoj\7(noEoj)}]. j=l

Since ia sin(a- T)w ., _.o___...:.._ dT f = w-~(1- cos aw )j,

• 0 w

by (4.4) and (4.10), we obtain

for a> 0. Since (4.11) and (4.2) show (4.1), this completes the proof of Theorem 1.3(1).

We next prove Theorem 1.3(2). We choose (Eo, n0 ) E S E9 S such that (no+ IEoi2)Eo -/= 0. We suppose that for some T with 0 < T < Tmax 1

liminf sup IIE-'(t)- E(t)llu = 0, A->oo o.:;;t.:;;T

and we derive a contradiction. By (3.6), (3.9) and (2.5), we have

0 :::; t :::; T, where

F1 (t) =lilt U(t-s)(Q~1 )E-\)(s)dst 2 ,

F2(t) = 1t II(Q~2 ) E-')(s)llu ds, F3(t) = 1t II(Q~3 ) E-')(s)llu ds,

(4.12)

and Lis a positive constant independent of>.. By the proof of Theorem 1.1(3), we obtain

( 4.14)

(see, e.g., (3.25), (3.28), (3.44), and (3.51)). It follows from (4.12) that there exists a sequence { >.k} such that >.k --7 oo as k --7 oo and

Ak sup IIE-'k(t)- E(t)llu --7 0 as k --7 oo. (4.15) O<t<T

On the other hand, let a be a positive constant to be determined later. By the change of variables >.s = T, we have

By (3.10) and (4.7), we obtain

in L2 as >. --7 oo, uniformly for 0 :::; T :::; a. Since

l a . s1naw cos(Tw)(no + IEol2 )dT =--(no+ IEol 2 ),

0 w

by (4.16) and (4.17),

Since (no+ [Eo[ 2 )Eo =J 0, we conclude that for some a> 0,

{ s1naw 2 } -w-(no +[Eo[ ) Eo# 0. (4.19)

Otherwise, we have

{ smaw I l2 } -w-(no + Eo ) Eo = 0 ( 4.20)

for any a > 0. We differentiate ( 4.20) in a to obtain

{cos(aw)(no + [Eo[ 2 )}Eo = 0 (4.21)

for any a > O.Letting a -+ 0 in ( 4.21 ), we have ( n 0 + [Eo [2 )Eo = 0, which is a contradiction. Now we choose a> 0 so that (4.19) holds. We multiply (4.13) with A= Ak and t =a/A by Ak and let k-+ oo to obtain by (4.14), (4.15) and (4.18),

II smaw 2 II 0 2: Eo-w-(no +[Eo[ ) £2,

which is a contradiction to (4.19). This completes the proof of Theorem 1.3(2). Finally, we prove Theorem 1.3(3). We choose (h 0 , Eo) E S CDS such that n0 +

[E0 [2 = 0 and (Im Eo·!::lEo)Eo =J 0. We suppose that for some T with 0 < T < Trnax,

liminf A2 sup [[E>.(t)- E(t)[[u = 0, >--oo o:::;t..:;T

(4.21)

and we derive a contradiction. By (3.6), (3.9) and (2.5), we have

[[E>.(t)- E(t)[[£2 2: F1(t)- F2(t)- L 1t [[E>.(s)- E(s)[[uds, 0:::; t:::; T, (4.22)

where

and L is a positive constant independent of A. Let a be a positive constant to be determined later. By (2.5) and (3.13), we have

A2 F2(a/A):::; A21af.\ 1s II sin( sA= T)w a;[E>-12(7)11£2 dTds

:::; CA2 (a/A) 2 A- 1 = Ca2 A- 1 -+ 0 as A-+ oo.

Hence, we obtain (4.23)


It follows from ( 4.21) that there exists a sequence {Ak} such that Ak ~ oo and

,\~ sup IIE>.k(t)- E(t)llu ~ 0 as ,\ ~ oo. 09~T

On the other hand, by the change of variables .\s = T, we have

(4.24)

1af>.U(aj>.- s)(Q\2 ) E>.)(s)ds = ,\-11aU((a- T)j>.)Q\2 )(Tj.\)E>.(Tj.\)ds. (4.25)

By (3.11) and (4.7), we obtain

in L 2 as ,\ ~ oo, uniformly for 0 ::; T ::; a. Since

1a . Sln TW - ---(2Im(E0 · .6.E0 ))dT = w- 2 (1- cosaw)(2Im(E0 · .6.E0 )),

0 w

( 4.25) and ( 4.26) show that

.\2 F 1 (af>.) ~ II { 1 - :~saw (2 Im (Eo· .6.Eo)) }Eo t 2 as>.~ oo. ( 4.27)

From (4.3), we see that the limit in (4.27) is finite. Since (2Im(E0 · .6.E0 ))E0 ::f 0, we conclude that for some a > 0,

{ 1 - :~saw (2 Im (Eo· .6.Eo)) }Eo ::f 0. ( 4.28)

Otherwise, we have

{ 1- cosaw - } w2 (2Im(Eo · .6.E0 )) Eo= 0 ( 4.29)

for any a > 0. We differentiate ( 4.29) twice in a to obtain

{cos(aw)(2Im(Eo · .6.Eo))}E0 = 0 ( 4.30)

for a> 0. Letting a~ 0 in (4.30), we have (2Im(E0 · .6.E0 ))E0 = 0, which is a contradiction. Now we choose a > 0 so that ( 4.28) holds. We choose ,\ = Ak and t = aj .A in ( 4.22), multiply the resulting inequality by ,\~ and let k ~ CXJ to obtain by (4.23), (4.24) and (4.27),

11{ 1- cosaw - } II 0 :2: w 2 (2 lm (Eo· .6.E0 )) Eo £2,

which is a contradiction to (4.28). This completes the proof of Theorem 1.3(3).

Acknowledgments. The authors would like to express their deep gratitude to Professors M. Tsutsumi and S. Ukai for their valuable comments.


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