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The initial value problem for the derivative nonlinear Schrödinger equation in the energy space
Nonl,near Analysrs, Theory, Methods Printed in Great Britain.
& Applrcarrons,
Vol. 20, No. 7, pp. 823-833,
1993. 0
0362-546X/93 $6.00+ .OO 1993 Pergamon Press Ltd
THE INITIAL VALUE PROBLEM FOR THE DERIVATIVE NONLINEAR SCHRODINGER EQUATION IN THE ENERGY SPACE NAKAO HAYASHI Department of Mathematics, Faculty of Engineering, Gunma University, Kiryu 376, Japan (Received 1 October 1991; received for publication 2 June 1992) Key words and phrases: The derivative nonlinear SchrGdinger equation, the energy space.
1. INTRODUCTION THIS PAPER
is concerned with the Cauchy problem for the derivative nonlinear Schrodinger
equation iy/, + wXX+ 2iS(ly/lzyl), = 0, XEL? i Y(O, x) = %9,
(t, x) E IRx m,
where 6 = +i. This equation was first derived by [l, 21 to study the propagation polarized nonlinear Alfvtn waves in plasma. Our purpose in this paper is to show the following theorem.
(1.1) of circular
1.1. We assume that 4(x) E H’( E?)and ]I4 )ILzis sufficiently small. Then there exists a unique global solution I+Y(~, x) such that
THEOREM
l#Y(l,x) E C(lR; H’(R)) f-l L&$(E?;P3(lR)).
(1.2)
In [3,4] Tsutsumi and Fukuda proved that the local existence of solutions to (1.1) under the condition such that 4 E H”(R) (s > 3/2) and the global existence of solutions to (1.1) under the conditions such that 4 E H2(Q and 1)411 Hl are sufficiently small. Hence our result is an improvement of [3,4]. Remark 1.1. In [5], Kenig et al. established interesting results for the Korteweg-de equation + UU, = 0, (t, x) E IRx IR, u, + UXXX XE IR. u(O,x) = 6(x), i
Vries
(1.3)
They proved that if 4 E H”(E) (s > 3/4), then (1.3) is locally well-posed in time and if 4 E H’(R), then (1.3) is globally well-posed in time. Their method depends on the smoothing property of solutions to the Airy equation and their results are improvements of the previous papers (refer to [5] and references therein). Our result is analogous to theorem 1.2 in [5]. However, our method is different from that in [5]. It seems that the “L* - Lq estimate”, which works well for the usual nonlinear Schrddinger equation, is not directly applicable to (l.l), since (1.1) has a space derivative in the nonlinear term which causes the so-called derivative loss. To overcome this difficulty, we make use of the gauge transformations stated in Section 2 823
824
N. HAYASHI
which enable us to treat the problem in a way similar to the usual equation. We give notation and function spaces used in this paper.
nonlinear
Schrodinger
Notation and function spaces We let Lp(lR) = (f(x); f(x) is measurable
on R, 11 f llLp< ~01, where 1)f ll$ = (SIR)f(x)Ip dx) if E R) ifp = 05, and,;e let ZPP(R) = (f(x) E P(R); we put H ’ (R) = P(R). We denote by H,,!.P= cj”=, IIa:f lip < a). For simplicity (a, *)Lz the inner product in Z’(R). For any interval Z of R and a Banach space B with the norm functions from Z to B, and we let P(Z; B) be II* llLIl we let C(Z; B) be the space of continuous the space consisting of strongly measurable B valued functions u(t) defined on Z such that \Iu(t)lIB E G’(Z). Different positive constants will be denoted by the same letter C. If necessary, by C(*, . . . . *) we denote constants depending on the quantities appearing in parentheses.
;If7,p < COand Ilf lILw= ess*sup(l f(x)l;x
2. PROOF
In order to prove theorem
i
OF THEOREM
1.1 we consider
i(wj), +
(WjLx+ 2i41Wj12Vj)x = 0,
Wj(O9
=
x,
1.1
x
4j(x),
E
(t,
x)
E
m
x
R, (2.1)
I?,
where ~j(X) E H”(R) and rPj + 4 strongly in H’(R) as j + co. In [4, theorem 31 it was shown that for each j E ~PJthere exists a unique global solution wj of small. We remark (2.1) satisfying wj(t, X) E C”(R; H”(R)) provided that Il~jIlN I is sufficiently here that if we take the Sobolev inequality (lemma 2.1) into account, then the result for wj mentioned above holds valid under the condition such that ll~jIlL2 is sufficiently small. For the nonlinear Schrodinger equation of the form ily, + wXX+
m, 4 = @(x),
Id4w = 0,
(NW
Weinstein [6, theorem A] proved that a sufficient condition for global existence space is \(@llrz < 6 by using the “best constant” for the Sobolev inequality. We first recall the Gagliardo-Nirenberg inequality and the space-time estimate of free Schrodinger equations. LEMMA
integers
2.1. Let q, r be any real numbers such that j I m. Then
satisfying
in the energy of a solution
1 5 q, r s co and let j and m be nonnegative
ll~l4lLP5 ~llm41aL’I14L~“‘, where 1 -=j+a
i-m
P
(
+(1-a): >
for all a in the interval j/m 5 a 5 1, and M is a positive constant r and a. For the proof
see, e.g. Friedman
[7].
depending
only on m, j, q,
825
Initial value problem
LEMMA 2.2. For any $ E L2(lR) we have
where 2 5 r 5 00, 2/q = (l/2)
- (l/r)
u(t)4
and
= &
IKe
-(X-v)*/4ri 4Q) dye
.i For the proof
see, e.g. Ginibre
Remark 2.1. Lemma
and Velo [S].
2.2 was first shown by [9] in the case r = p = 6 and improved
We next recall lemma 1 in [4] which is derived by the infinite covered by [lo] (see also the Appendix). LEMMA2.3. Let wj be the solution
family of conservation
by PI. laws dis-
of (2.1). Then
lIVj(t)lIL2= ll~jllLz~ ll~xVj(f>llZZ + 2~211Vj(f)lli6 + 3~Im(~xVj(f),lVj(t)12Vj(f))L2 = Ila,4jllL2+ 2~2114jllP6 + 36Im(~,djj 14j124j)Lz. By lemma
2.1 and lemma
LEMMA2.4. We assume
2.3 we easily have lemma
that I,Y~is the solution
2.4.
of (2.1) and Il~jIlrz is sufficiently
small.
Then
Ilu/j(OllH1 s c(l14jllH1). We are now in a position
to prove the theorem.
Proof of theorem 1 .l. We only consider analogously.
the case t > 0, since the case t < 0 can be treated
We shall prove that ~wj) is a Cauchy
sequence
in C([O, T]; H’(R))
II L”([O,
T]; H’,‘(R))
for any T > 0. For that purpose we generate some new nonlinear equations from (2.1) through the gauge transformations. Following the paper by Kundu [l 11, we first multiply both sides of (2.1) by
Ej = E,(t, x) = exp ia (
X i-,
IVj(t, W)I’dw * >
Then we have i(Vj),
+
(Vj),
+
2iSlVj12(Vj),
= I.7
Vj(0,
where
X)
=
@j(X)
id
exp (!
i Vj(t,X)
=
Wj(t,X)exp
-m
0,
(t, X) E IR x I?,
I+j(W)I’dw 3 1
XE R,
/wj(t, w)12 dw >
(see also the Appendix
in [12]).
(2.2)
N. HAYASHI
826
both sides of (2.2) by Ej and Ejd, to obtain
We again multiply i(U!“) J i(d2’)
J
f f
+ (ZP) J +
(ZP’) J
zz F!”
xx
=
xx
J
’
Fc2’ J
’
I+j(W)I’ dw
(2.3)
3
i
where F!”
= 2~4~!1))2~!2)
Ff2’ J
=
J
u,“)(t,
(see the Appendix).
x)
=
vj(t,
J
J
x)
-2i&~!~‘)~u!”
’
J
J
IVj(t, w)12 dw 3 )
exp
We note here that (Uj”),
= j&lUj(t)(2+)
+ Uj’z).
(2.4)
The proof of theorem 1.1 will be preceded by a series of lemmas. For the sake of brevity in lemma 2.5 and lemma 2.6 we suppress the subscriptj of uj, t//j and Fje We first prove lemma 2.5. LEMMA
2.5. Let u(t) and u (2) be solutions SUP
Odf57 Proof.
By the definition
by lemmas
h4~~ll,~
5 c.
suP
suP
this inequality
LEMMA 2 . 6 . Let uc2)
Il”‘l’(t)llff~c c(ll+jllH’)IlU(2)(t)llL2 4 c(ll~jllH1)~
and (2.5) the lemma be the solution ” (1.O
where C(T)
II Iv/IW~II~~+ /l~,mll,4.
2.1 and (2.5) to (2.4) to obtain
ostr7 From
+
2.4 and 2.1 we have ost57
We apply lemma
Il”‘2’(t)llL2 5 c(l14jlIH’)*
suP
O~fzzT
of u(l),
Il~“)(&l Hence,
II~(l)mP +
of (2.3). Then we have for any T > 0
is a positive
constant
follows.
n
of (2.3). Then we have for any T > 0
IIu’2’Ct)IIsdt5 C(T) * C(II+jIIHI), >
independent
of j.
(2.5)
827
Initial value problem
Proof. By the integral
equations
with (2.3) we have
associated
f
d2’(t) = u(t)d2’(o) +
U(t - s)P’(.s)
ds.
.i 0 The Lp-Lq estimate
yields
llLP(&3 We use lemma
ds. = llU(t)u’2’(0)IlL~+ c f (t - s)-“~IIF(~)(s)II~~/~ i .O
2.1 to obtain l(P(.s)llp
From
(2.6)
(2.6), (2.7) and lemma
5 CJlU(1)(s)l(~~llU’2’(s)ll~3.
2.5 it follows
(2.7)
that
IId2)(t)llL3 I IIu(t>d2)(o)ll~3 + c(II~~II~~) (t - S)-“611U’2’(S)llt3dS. (2.8) In what follows, inequality “IIu(‘)(s)ll;: (1.0 The second
we let C, = C((J4jjllH~). By (2.8) and the singular
for simplicity
” )IU(s)~‘~‘(O)112 ds ) “12 ds “I2 5 ) (i 0
term of the right-hand
&(I;
+
side of (2.9) is estimated
(Iu(2)(s)11~‘11 dsji1’i2.
integral
(2.9)
by (2.10)
“6t3”. )
We let
(iIf
lld2’(s)l/$ ds “12.
Y(f)=
0
Then we have by lemma
)
2.2, (2.9) and (2.10)
y(t) I clp(o)~~L* Since by lemma
(2.11)
+ C,t3’4yZ(t).
2.1
Il”‘2’(o)IlL2 5 c
from (2.11)
y(t) I C&l Now we choose
+ Yy2(t)).
(2.12)
r, > 0 so small that Ti <
(2C&4’3,
sup
y(t) 5 2CH.
then OSlST,
By using lemma
2.5, we repeat
Remark 2.2. The same argument
the above argument as in lemma
to obtain
the lemma.
n
2.6 has been used in [13, 141.
828
N.
HAYASHI
We next prove that (~7’) is a Cauchy sequence in C([O, T]; L’(R)) fl L12([0, T]; L3(lR)) for any T > 0 and I = 1,2. LEMMA2.7. Let uj” and uj’2’be solutions of (2.3). Then we have for any T > 0 and I = 1,2 I I(uj”‘(t) - ~ei(i)lI::d1)l/12
5 C(T) . C,ll~j
- ~kllH1,
(i sup
IIu,“‘(t) -
uf’(t)IILz 5 C(T) *CHII$j-
where C(T) is a positive constant independent
of j and k,
cff = c(l14jllH’ Proof.
$kllHl,
9 l14kllH1).
We let W”‘(t) = u!‘)(t) - z#@). J
Then we see that i(W(‘)) I + (WC’))xx = F!‘) _ F(‘) k .
(2.13)
J
From the integral equation associated with (2.13) it follows that II~(WllL3
5 Ilw)(~vm3
+ c
’ (t - S)-qFqS)
- Ff)(S)(lL3/Z
J
ds.
(2.14)
0
By Holder’s inequality we obtain
IIFj”’ - FpyL3/* I IIUj1)llLrnllUj1)llL311Uj(2) - U~2qr3 + (llUjl)llp
+ Il~6’~Il~~)ll~~~II~3II~j(~~ -
4’)11~3,
(2.15)
and IIFi’z’- F$2$‘/Z I (]IUj’z’II L3 + IIu~2)IIL3)lluj(1)llL~IIuj(2) - u~2)llL3 +
Ilup’y53~~4)
-
(2.16)
zd/y’II,-.
Lemma 2.1 and (2.4) imply I]Uj”’- U:“]lL~ 5 C(]&(u,cl) - u/$“)l]#]#) I C(IJUi(2)I
From (2.15)-(2.17),
IIF”’ J
c(I2.p
-
Uf)llL3
f4pI(L3
+
+
- u/$‘)Il;P
(Iluy’ll:cc
+
~~up&#.4j’1~
-
up)~L3)1’311z4j1)
(1 + IIuj(“)];G + IIu~1)](~~3))Iz4j1) - upll~3.
-
z&‘ll~~’
(2.17)
lemmas 2.1 and 2.5 it follows that
- Ff’(s)llL3/2 4
c,
.
1 +
c n=j,k
11~~2’(~)112,3 ,i, >
lI~“‘(4llL3.
(2.18)
Initial value problem
By (2.14), (2.18), lemmas
2.2, 2.6 and the singular
829
integral
inequality
we have
‘j IIW”‘(s)JI$ d&s “12 5 cI( W(‘)(O)IILZ (1
0
> +
C(T)&
* (t’l”2
+
P)
i
I=1
t IlW(‘)(s)(I;: (.r
0
ds “12. >
(2.19)
We next prove that
Ilw”‘(o)llL25 cffI14j - 4kllH1. By the mean value theorem U>“(O) - ZAj”(O) =
(~j(X)
we see that there exists a constant B (0 < 0 < 1) such that ‘1. - 4k(X)) exp 2id _m 14j(w)12dw ) ( i
+ &(x) =
(+j(X)
-
i
exp
23
(
@k(X))
+ 2iS&(x)
(I--
w
x
x exp 23 e ((1 . We apply lemma
(2.20)
2.1 and Holder’s
X --
‘X
ui --m (
2s
I@j(W)12d w) - ew(Zid x i
_-m
IQj(w)I’dw
jlw bkCw)12 dw)]
)
I+j(W)I’ - Id/c(W)12dW ) I$j(W)I’dw + (1 - 0)
inequality
b,h912 dw
>>
.
(2.21)
to (2.21) to obtain
II”j’l’(o)- u!‘)(“)llLz5 ll4j - 4kllL’ + cl14kIILzlll4jl’ - 14k1211L1 5 ll4j - 4klILZ + cffl14j - 4kllL2. By a direct calculation
(2.22)
we have
zp(O) - u12’(0)
x 14j(W)12 dw --m ) dw)] . x 14ji(W)12 dw) - ew(2id I:_ 14Awl12 -m
830
N. HAYASHI
In the same way as in the proof
of (2.22)
IIu,‘2’(0) - U/$2’(o)ll,z 5
llax(4j
4k)llL2+ cII l4jl’+j - 14k124kllLZ
-
l+jl’ - l~k1211L~ + (IIMIIL~ + cl1l+k124kllL2)lI 5 cHll+j - GkllH’.
(2.23)
By (2.22) and (2.23) we have (2.20). From (2.20) we see that the first part of the lemma holds valid for small T > 0. Hence we have the first part of the lemma for any T > 0 by the standard argument. We next prove the second part of the lemma. Multiplying both sides of (2.13) by IV(‘), integrating in x, taking the imaginary part, we obtain $ (lIV”‘(t)112,z = 2 Im(Fj”‘(t) 5 CjIFj”(t) We apply (2.18) to the above inequality
IlW”‘(t)ll;Zs c,
$
- F$“(t),
W”)(t))rz
- F~l)(t)llL2/~((W(l)(f)llL1.
to see that
* (1 +
ll~%l1:-) f: lw”‘(ollt~,
c
I=
n=j,k
from which it follows
I
that
lIw’“(t)ll:~ I (1 W”)(O)l($ +
c,
” I(
c
1+
+
5 IIw”‘(o)(I:*
cH
Ibf2’(dlt3 ,i, 1lW”‘Wll:3 ds
n=j,k
,O
t5/6
l t2/3
where we have used Holder’s inequality. Lemma W imply the second part of the lemma. The following LEMMA 2.8.
lemma
is important
Let wj be the solution
(I
,::
dsy’6]
IIu;2)(s)((;:.
in the proof
d)ll’,
2.6, (2.20) and the first part of the lemma
of the theorem.
of (2.1). Then we have for any T > 0
II’Yj(t)- WkCt)IIG~~’ di>“125 C(T) ’ cHI16j SUP
(\‘: 11 W”‘(s)((s
j,
IIY’j(t)-
vk(f)llH1
5
C’(T)
’ cHll$j
-
(DkllH’r
-
6kllH1,
05157
where C(T) Proof.
is a positive
constant
independent
of j and k.
We have vj(t, x) = uj”‘(t, x) exp
luj”(t,
.
W)12dw >
(2.24)
831
Initial value problem
By a simple calculation a,wj(t, X) = lJ,uj”(t,
_~ lu,!“(t, w)12dw
x) - 2i8(~4j(~)(‘~,!‘)(t, x)] exp
.
Since we have .
(2.25)
In the same way as in the proof of (2.20) we have by (2.24), (2.25) and lemma 2.5 IIVj(t3-x) - ‘Yk(f*x)II~l 5 Cm 1 + i
C
l=l
n=j,k
(
Ilu!i”(t)llLz ‘Ii, lIuj”(t)- ~i’)(t)llL~ (2-W >
and
II’J’j(fv X) -
Wk(f,
4 x)IIHI.35 c’ 1 + ; c (Il@(t& + I\@(t)\\& I=1 n=j,k > ( x
,il
(IIq(0
-
uf’(t)JJ,3
From (2.26), (2.27) and lemma 2.7 the lemma follows.
+
))u’[‘(t) J
-
z@(f)JIp).
(2.27)
n
Proof of theorem 1.1 completed. From lemma 2.8 we conclude that there exists a t,~= t+v(t,x) satisfying (1.2) such that for any T > 0 SUP
IIV’j(t)
-
W(f)l/H1
+
‘IIV’j(t)- W(t)IIS,jdt+ 0
1 0
OStST
as j -+ 00. It is easily seen that y is a unique solution of (1 .l>. This completes the proof of theorem 1.1. Remark 2.3. We note here that global existence in time of Schwartz class solutions of (1.1) has been shown by Lee [15] and smoothing property of global solutions of (1.1) is investigated by [12] under some conditions on data. Acknow/edgement-The for remarks.
author
would like to thank
Professor
Y. Tsutsumi
for valuable
Note added in proof-T. Ozawa and the author proved that a sufficient condition space for (1.1) is ]l$llr~ < &% by making use of the function E, defined in Section number X& is the same as that of (NLS) obtained in [6].
comments,
and to the referee
for global existence in the energy 2 and the method used in [6]. The
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832
N. HAYASHI
4. TSUTSUMIM. & FUKUDA I., On solutions
of the derivative
nonlinear
SchrGdinger
equation
II, Funkciuluj Ekvacioj
24, 85-94 (1981). 5. KENIG C. E., PONCE G. & VEGA L., Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Am. math. Sot. 4, 323-347 (1991). 6. WEINSTEIN M. I., Nonlinear SchrGdinger equations and sharp interpolation estimates, Communs Math. Phys 87,
567-576 (1983). I. FRIEDMAN A., Partial Differential Equations. Holt (Rinehart & Winston), New York (1969). 8. GINIBRE J. & VELO G., Scattering theory in the energy space for a class of nonlinear Schradinger
equations,
J. Math. pure appl. 64, 363-401 (1985). 9. STRICHARTZ R. S., Restriction of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke math. J. 44, 705-714 (1977). 10. KAUP D. J. & NEWELL A. C., An exact solution for a derivative nonlinear SchrBdinger equation, J. Math. Phys. 19,789-801(1978).
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107-118 (1989).
APPENDIX PROPOSITION A. 1. We assume
that u = u(t, x) is a solution
of
iv, + u,, + 2iS(o(zux = 0 and LJE C”(lR; H”(m)).
Then
(A.1)
V, = Eu and V, = Eu, satisfy
i(V,), + (VI), = 3idV*T 1 2, i(V2), + (V,), = -2idV*F * 1, where *x
Iv(t, w)l* dw
E = E(t, x) = exp id (
Proof. Multiplying
!-_
>
both sides of (A.l) by E, we obtain
i(V,), + (V,), + 2iSlV,12Vz = iE,v + 2E,u, + E,,u. We again
apply
(A.3
both sides of (A.1) by E?J, to have
I(V,), + (V,),, + 2iSlV, 12uxxE= -2iB(V,K By a simple calculation
+ V,y)V,
+ iE, v, + 2E,u,, + E,,u,
(A.3)
we see that
iE, = (i&V, Vz - V,q) + d21V,14)E, E,, = (i&V,% + V,V,) - SzIV1“)E, and 2E, = 2iS/ L’l*E. Therefore,
it follows
that
iEt + 2E xxu + E xxu = 2iSlVI’V 1 21 I 2 + 2i&‘*v i&v, From
(A.2)-(A.4)
the result follows.
+ 2E,u,, n
+ E,,u,
= 2i~lV,j’u,,E
+ 2iS[V212V,.
(A.4) (A.5)
833
Initial value problem For the convenience
of the reader
we give the proof
Proof of lemma 2.3. For simplicity we suppress the real part and integrating in x, we obtain
of lemma 2.3 which is different the subscript
j. Multiplying
from that in [4].
both
sides of (2.2) by q,
taking
By a direct calculation
(A.7) From (A.6) and (A.7) it follows
that (A.8)
/WV,w)l’dw m
>
we have
(A.9) and -tic5 By (A.8)-(A.lO)
IulZ(u,V - lIG)dx
I .m
we have the lemma.
n
= -*is
I ,R
IwIZ(w,P
- ViK)dx
+
~211wlli~.
(A.lO)
& Applrcarrons,
Vol. 20, No. 7, pp. 823-833,
1993. 0
0362-546X/93 $6.00+ .OO 1993 Pergamon Press Ltd
THE INITIAL VALUE PROBLEM FOR THE DERIVATIVE NONLINEAR SCHRODINGER EQUATION IN THE ENERGY SPACE NAKAO HAYASHI Department of Mathematics, Faculty of Engineering, Gunma University, Kiryu 376, Japan (Received 1 October 1991; received for publication 2 June 1992) Key words and phrases: The derivative nonlinear SchrGdinger equation, the energy space.
1. INTRODUCTION THIS PAPER
is concerned with the Cauchy problem for the derivative nonlinear Schrodinger
equation iy/, + wXX+ 2iS(ly/lzyl), = 0, XEL? i Y(O, x) = %9,
(t, x) E IRx m,
where 6 = +i. This equation was first derived by [l, 21 to study the propagation polarized nonlinear Alfvtn waves in plasma. Our purpose in this paper is to show the following theorem.
(1.1) of circular
1.1. We assume that 4(x) E H’( E?)and ]I4 )ILzis sufficiently small. Then there exists a unique global solution I+Y(~, x) such that
THEOREM
l#Y(l,x) E C(lR; H’(R)) f-l L&$(E?;P3(lR)).
(1.2)
In [3,4] Tsutsumi and Fukuda proved that the local existence of solutions to (1.1) under the condition such that 4 E H”(R) (s > 3/2) and the global existence of solutions to (1.1) under the conditions such that 4 E H2(Q and 1)411 Hl are sufficiently small. Hence our result is an improvement of [3,4]. Remark 1.1. In [5], Kenig et al. established interesting results for the Korteweg-de equation + UU, = 0, (t, x) E IRx IR, u, + UXXX XE IR. u(O,x) = 6(x), i
Vries
(1.3)
They proved that if 4 E H”(E) (s > 3/4), then (1.3) is locally well-posed in time and if 4 E H’(R), then (1.3) is globally well-posed in time. Their method depends on the smoothing property of solutions to the Airy equation and their results are improvements of the previous papers (refer to [5] and references therein). Our result is analogous to theorem 1.2 in [5]. However, our method is different from that in [5]. It seems that the “L* - Lq estimate”, which works well for the usual nonlinear Schrddinger equation, is not directly applicable to (l.l), since (1.1) has a space derivative in the nonlinear term which causes the so-called derivative loss. To overcome this difficulty, we make use of the gauge transformations stated in Section 2 823
824
N. HAYASHI
which enable us to treat the problem in a way similar to the usual equation. We give notation and function spaces used in this paper.
nonlinear
Schrodinger
Notation and function spaces We let Lp(lR) = (f(x); f(x) is measurable
on R, 11 f llLp< ~01, where 1)f ll$ = (SIR)f(x)Ip dx) if E R) ifp = 05, and,;e let ZPP(R) = (f(x) E P(R); we put H ’ (R) = P(R). We denote by H,,!.P= cj”=, IIa:f lip < a). For simplicity (a, *)Lz the inner product in Z’(R). For any interval Z of R and a Banach space B with the norm functions from Z to B, and we let P(Z; B) be II* llLIl we let C(Z; B) be the space of continuous the space consisting of strongly measurable B valued functions u(t) defined on Z such that \Iu(t)lIB E G’(Z). Different positive constants will be denoted by the same letter C. If necessary, by C(*, . . . . *) we denote constants depending on the quantities appearing in parentheses.
;If7,p < COand Ilf lILw= ess*sup(l f(x)l;x
2. PROOF
In order to prove theorem
i
OF THEOREM
1.1 we consider
i(wj), +
(WjLx+ 2i41Wj12Vj)x = 0,
Wj(O9
=
x,
1.1
x
4j(x),
E
(t,
x)
E
m
x
R, (2.1)
I?,
where ~j(X) E H”(R) and rPj + 4 strongly in H’(R) as j + co. In [4, theorem 31 it was shown that for each j E ~PJthere exists a unique global solution wj of small. We remark (2.1) satisfying wj(t, X) E C”(R; H”(R)) provided that Il~jIlN I is sufficiently here that if we take the Sobolev inequality (lemma 2.1) into account, then the result for wj mentioned above holds valid under the condition such that ll~jIlL2 is sufficiently small. For the nonlinear Schrodinger equation of the form ily, + wXX+
m, 4 = @(x),
Id4w = 0,
(NW
Weinstein [6, theorem A] proved that a sufficient condition for global existence space is \(@llrz < 6 by using the “best constant” for the Sobolev inequality. We first recall the Gagliardo-Nirenberg inequality and the space-time estimate of free Schrodinger equations. LEMMA
integers
2.1. Let q, r be any real numbers such that j I m. Then
satisfying
in the energy of a solution
1 5 q, r s co and let j and m be nonnegative
ll~l4lLP5 ~llm41aL’I14L~“‘, where 1 -=j+a
i-m
P
(
+(1-a): >
for all a in the interval j/m 5 a 5 1, and M is a positive constant r and a. For the proof
see, e.g. Friedman
[7].
depending
only on m, j, q,
825
Initial value problem
LEMMA 2.2. For any $ E L2(lR) we have
where 2 5 r 5 00, 2/q = (l/2)
- (l/r)
u(t)4
and
= &
IKe
-(X-v)*/4ri 4Q) dye
.i For the proof
see, e.g. Ginibre
Remark 2.1. Lemma
and Velo [S].
2.2 was first shown by [9] in the case r = p = 6 and improved
We next recall lemma 1 in [4] which is derived by the infinite covered by [lo] (see also the Appendix). LEMMA2.3. Let wj be the solution
family of conservation
by PI. laws dis-
of (2.1). Then
lIVj(t)lIL2= ll~jllLz~ ll~xVj(f>llZZ + 2~211Vj(f)lli6 + 3~Im(~xVj(f),lVj(t)12Vj(f))L2 = Ila,4jllL2+ 2~2114jllP6 + 36Im(~,djj 14j124j)Lz. By lemma
2.1 and lemma
LEMMA2.4. We assume
2.3 we easily have lemma
that I,Y~is the solution
2.4.
of (2.1) and Il~jIlrz is sufficiently
small.
Then
Ilu/j(OllH1 s c(l14jllH1). We are now in a position
to prove the theorem.
Proof of theorem 1 .l. We only consider analogously.
the case t > 0, since the case t < 0 can be treated
We shall prove that ~wj) is a Cauchy
sequence
in C([O, T]; H’(R))
II L”([O,
T]; H’,‘(R))
for any T > 0. For that purpose we generate some new nonlinear equations from (2.1) through the gauge transformations. Following the paper by Kundu [l 11, we first multiply both sides of (2.1) by
Ej = E,(t, x) = exp ia (
X i-,
IVj(t, W)I’dw * >
Then we have i(Vj),
+
(Vj),
+
2iSlVj12(Vj),
= I.7
Vj(0,
where
X)
=
@j(X)
id
exp (!
i Vj(t,X)
=
Wj(t,X)exp
-m
0,
(t, X) E IR x I?,
I+j(W)I’dw 3 1
XE R,
/wj(t, w)12 dw >
(see also the Appendix
in [12]).
(2.2)
N. HAYASHI
826
both sides of (2.2) by Ej and Ejd, to obtain
We again multiply i(U!“) J i(d2’)
J
f f
+ (ZP) J +
(ZP’) J
zz F!”
xx
=
xx
J
’
Fc2’ J
’
I+j(W)I’ dw
(2.3)
3
i
where F!”
= 2~4~!1))2~!2)
Ff2’ J
=
J
u,“)(t,
(see the Appendix).
x)
=
vj(t,
J
J
x)
-2i&~!~‘)~u!”
’
J
J
IVj(t, w)12 dw 3 )
exp
We note here that (Uj”),
= j&lUj(t)(2+)
+ Uj’z).
(2.4)
The proof of theorem 1.1 will be preceded by a series of lemmas. For the sake of brevity in lemma 2.5 and lemma 2.6 we suppress the subscriptj of uj, t//j and Fje We first prove lemma 2.5. LEMMA
2.5. Let u(t) and u (2) be solutions SUP
Odf57 Proof.
By the definition
by lemmas
h4~~ll,~
5 c.
suP
suP
this inequality
LEMMA 2 . 6 . Let uc2)
Il”‘l’(t)llff~c c(ll+jllH’)IlU(2)(t)llL2 4 c(ll~jllH1)~
and (2.5) the lemma be the solution ” (1.O
where C(T)
II Iv/IW~II~~+ /l~,mll,4.
2.1 and (2.5) to (2.4) to obtain
ostr7 From
+
2.4 and 2.1 we have ost57
We apply lemma
Il”‘2’(t)llL2 5 c(l14jlIH’)*
suP
O~fzzT
of u(l),
Il~“)(&l Hence,
II~(l)mP +
of (2.3). Then we have for any T > 0
is a positive
constant
follows.
n
of (2.3). Then we have for any T > 0
IIu’2’Ct)IIsdt5 C(T) * C(II+jIIHI), >
independent
of j.
(2.5)
827
Initial value problem
Proof. By the integral
equations
with (2.3) we have
associated
f
d2’(t) = u(t)d2’(o) +
U(t - s)P’(.s)
ds.
.i 0 The Lp-Lq estimate
yields
llLP(&3 We use lemma
ds. = llU(t)u’2’(0)IlL~+ c f (t - s)-“~IIF(~)(s)II~~/~ i .O
2.1 to obtain l(P(.s)llp
From
(2.6)
(2.6), (2.7) and lemma
5 CJlU(1)(s)l(~~llU’2’(s)ll~3.
2.5 it follows
(2.7)
that
IId2)(t)llL3 I IIu(t>d2)(o)ll~3 + c(II~~II~~) (t - S)-“611U’2’(S)llt3dS. (2.8) In what follows, inequality “IIu(‘)(s)ll;: (1.0 The second
we let C, = C((J4jjllH~). By (2.8) and the singular
for simplicity
” )IU(s)~‘~‘(O)112 ds ) “12 ds “I2 5 ) (i 0
term of the right-hand
&(I;
+
side of (2.9) is estimated
(Iu(2)(s)11~‘11 dsji1’i2.
integral
(2.9)
by (2.10)
“6t3”. )
We let
(iIf
lld2’(s)l/$ ds “12.
Y(f)=
0
Then we have by lemma
)
2.2, (2.9) and (2.10)
y(t) I clp(o)~~L* Since by lemma
(2.11)
+ C,t3’4yZ(t).
2.1
Il”‘2’(o)IlL2 5 c
from (2.11)
y(t) I C&l Now we choose
+ Yy2(t)).
(2.12)
r, > 0 so small that Ti <
(2C&4’3,
sup
y(t) 5 2CH.
then OSlST,
By using lemma
2.5, we repeat
Remark 2.2. The same argument
the above argument as in lemma
to obtain
the lemma.
n
2.6 has been used in [13, 141.
828
N.
HAYASHI
We next prove that (~7’) is a Cauchy sequence in C([O, T]; L’(R)) fl L12([0, T]; L3(lR)) for any T > 0 and I = 1,2. LEMMA2.7. Let uj” and uj’2’be solutions of (2.3). Then we have for any T > 0 and I = 1,2 I I(uj”‘(t) - ~ei(i)lI::d1)l/12
5 C(T) . C,ll~j
- ~kllH1,
(i sup
IIu,“‘(t) -
uf’(t)IILz 5 C(T) *CHII$j-
where C(T) is a positive constant independent
of j and k,
cff = c(l14jllH’ Proof.
$kllHl,
9 l14kllH1).
We let W”‘(t) = u!‘)(t) - z#@). J
Then we see that i(W(‘)) I + (WC’))xx = F!‘) _ F(‘) k .
(2.13)
J
From the integral equation associated with (2.13) it follows that II~(WllL3
5 Ilw)(~vm3
+ c
’ (t - S)-qFqS)
- Ff)(S)(lL3/Z
J
ds.
(2.14)
0
By Holder’s inequality we obtain
IIFj”’ - FpyL3/* I IIUj1)llLrnllUj1)llL311Uj(2) - U~2qr3 + (llUjl)llp
+ Il~6’~Il~~)ll~~~II~3II~j(~~ -
4’)11~3,
(2.15)
and IIFi’z’- F$2$‘/Z I (]IUj’z’II L3 + IIu~2)IIL3)lluj(1)llL~IIuj(2) - u~2)llL3 +
Ilup’y53~~4)
-
(2.16)
zd/y’II,-.
Lemma 2.1 and (2.4) imply I]Uj”’- U:“]lL~ 5 C(]&(u,cl) - u/$“)l]#]#) I C(IJUi(2)I
From (2.15)-(2.17),
IIF”’ J
c(I2.p
-
Uf)llL3
f4pI(L3
+
+
- u/$‘)Il;P
(Iluy’ll:cc
+
~~up&#.4j’1~
-
up)~L3)1’311z4j1)
(1 + IIuj(“)];G + IIu~1)](~~3))Iz4j1) - upll~3.
-
z&‘ll~~’
(2.17)
lemmas 2.1 and 2.5 it follows that
- Ff’(s)llL3/2 4
c,
.
1 +
c n=j,k
11~~2’(~)112,3 ,i, >
lI~“‘(4llL3.
(2.18)
Initial value problem
By (2.14), (2.18), lemmas
2.2, 2.6 and the singular
829
integral
inequality
we have
‘j IIW”‘(s)JI$ d&s “12 5 cI( W(‘)(O)IILZ (1
0
> +
C(T)&
* (t’l”2
+
P)
i
I=1
t IlW(‘)(s)(I;: (.r
0
ds “12. >
(2.19)
We next prove that
Ilw”‘(o)llL25 cffI14j - 4kllH1. By the mean value theorem U>“(O) - ZAj”(O) =
(~j(X)
we see that there exists a constant B (0 < 0 < 1) such that ‘1. - 4k(X)) exp 2id _m 14j(w)12dw ) ( i
+ &(x) =
(+j(X)
-
i
exp
23
(
@k(X))
+ 2iS&(x)
(I--
w
x
x exp 23 e ((1 . We apply lemma
(2.20)
2.1 and Holder’s
X --
‘X
ui --m (
2s
I@j(W)12d w) - ew(Zid x i
_-m
IQj(w)I’dw
jlw bkCw)12 dw)]
)
I+j(W)I’ - Id/c(W)12dW ) I$j(W)I’dw + (1 - 0)
inequality
b,h912 dw
>>
.
(2.21)
to (2.21) to obtain
II”j’l’(o)- u!‘)(“)llLz5 ll4j - 4kllL’ + cl14kIILzlll4jl’ - 14k1211L1 5 ll4j - 4klILZ + cffl14j - 4kllL2. By a direct calculation
(2.22)
we have
zp(O) - u12’(0)
x 14j(W)12 dw --m ) dw)] . x 14ji(W)12 dw) - ew(2id I:_ 14Awl12 -m
830
N. HAYASHI
In the same way as in the proof
of (2.22)
IIu,‘2’(0) - U/$2’(o)ll,z 5
llax(4j
4k)llL2+ cII l4jl’+j - 14k124kllLZ
-
l+jl’ - l~k1211L~ + (IIMIIL~ + cl1l+k124kllL2)lI 5 cHll+j - GkllH’.
(2.23)
By (2.22) and (2.23) we have (2.20). From (2.20) we see that the first part of the lemma holds valid for small T > 0. Hence we have the first part of the lemma for any T > 0 by the standard argument. We next prove the second part of the lemma. Multiplying both sides of (2.13) by IV(‘), integrating in x, taking the imaginary part, we obtain $ (lIV”‘(t)112,z = 2 Im(Fj”‘(t) 5 CjIFj”(t) We apply (2.18) to the above inequality
IlW”‘(t)ll;Zs c,
$
- F$“(t),
W”)(t))rz
- F~l)(t)llL2/~((W(l)(f)llL1.
to see that
* (1 +
ll~%l1:-) f: lw”‘(ollt~,
c
I=
n=j,k
from which it follows
I
that
lIw’“(t)ll:~ I (1 W”)(O)l($ +
c,
” I(
c
1+
+
5 IIw”‘(o)(I:*
cH
Ibf2’(dlt3 ,i, 1lW”‘Wll:3 ds
n=j,k
,O
t5/6
l t2/3
where we have used Holder’s inequality. Lemma W imply the second part of the lemma. The following LEMMA 2.8.
lemma
is important
Let wj be the solution
(I
,::
dsy’6]
IIu;2)(s)((;:.
in the proof
d)ll’,
2.6, (2.20) and the first part of the lemma
of the theorem.
of (2.1). Then we have for any T > 0
II’Yj(t)- WkCt)IIG~~’ di>“125 C(T) ’ cHI16j SUP
(\‘: 11 W”‘(s)((s
j,
IIY’j(t)-
vk(f)llH1
5
C’(T)
’ cHll$j
-
(DkllH’r
-
6kllH1,
05157
where C(T) Proof.
is a positive
constant
independent
of j and k.
We have vj(t, x) = uj”‘(t, x) exp
luj”(t,
.
W)12dw >
(2.24)
831
Initial value problem
By a simple calculation a,wj(t, X) = lJ,uj”(t,
_~ lu,!“(t, w)12dw
x) - 2i8(~4j(~)(‘~,!‘)(t, x)] exp
.
Since we have .
(2.25)
In the same way as in the proof of (2.20) we have by (2.24), (2.25) and lemma 2.5 IIVj(t3-x) - ‘Yk(f*x)II~l 5 Cm 1 + i
C
l=l
n=j,k
(
Ilu!i”(t)llLz ‘Ii, lIuj”(t)- ~i’)(t)llL~ (2-W >
and
II’J’j(fv X) -
Wk(f,
4 x)IIHI.35 c’ 1 + ; c (Il@(t& + I\@(t)\\& I=1 n=j,k > ( x
,il
(IIq(0
-
uf’(t)JJ,3
From (2.26), (2.27) and lemma 2.7 the lemma follows.
+
))u’[‘(t) J
-
z@(f)JIp).
(2.27)
n
Proof of theorem 1.1 completed. From lemma 2.8 we conclude that there exists a t,~= t+v(t,x) satisfying (1.2) such that for any T > 0 SUP
IIV’j(t)
-
W(f)l/H1
+
‘IIV’j(t)- W(t)IIS,jdt+ 0
1 0
OStST
as j -+ 00. It is easily seen that y is a unique solution of (1 .l>. This completes the proof of theorem 1.1. Remark 2.3. We note here that global existence in time of Schwartz class solutions of (1.1) has been shown by Lee [15] and smoothing property of global solutions of (1.1) is investigated by [12] under some conditions on data. Acknow/edgement-The for remarks.
author
would like to thank
Professor
Y. Tsutsumi
for valuable
Note added in proof-T. Ozawa and the author proved that a sufficient condition space for (1.1) is ]l$llr~ < &% by making use of the function E, defined in Section number X& is the same as that of (NLS) obtained in [6].
comments,
and to the referee
for global existence in the energy 2 and the method used in [6]. The
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832
N. HAYASHI
4. TSUTSUMIM. & FUKUDA I., On solutions
of the derivative
nonlinear
SchrGdinger
equation
II, Funkciuluj Ekvacioj
24, 85-94 (1981). 5. KENIG C. E., PONCE G. & VEGA L., Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Am. math. Sot. 4, 323-347 (1991). 6. WEINSTEIN M. I., Nonlinear SchrGdinger equations and sharp interpolation estimates, Communs Math. Phys 87,
567-576 (1983). I. FRIEDMAN A., Partial Differential Equations. Holt (Rinehart & Winston), New York (1969). 8. GINIBRE J. & VELO G., Scattering theory in the energy space for a class of nonlinear Schradinger
equations,
J. Math. pure appl. 64, 363-401 (1985). 9. STRICHARTZ R. S., Restriction of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke math. J. 44, 705-714 (1977). 10. KAUP D. J. & NEWELL A. C., An exact solution for a derivative nonlinear SchrBdinger equation, J. Math. Phys. 19,789-801(1978).
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115-125 (1987). 14. HAYASHI N.,
Schrbdinger 15. LEE J. H.,
NAKAMITSU K. & TSUUUMI M., On solutions of the initial value problem for the nonlinear equations in one space dimension, Math. Z. 192, 637-650 (1986). Global solvability of the derivative nonlinear SchrGdinger equation, Trans. Am. math. Sot. 314,
107-118 (1989).
APPENDIX PROPOSITION A. 1. We assume
that u = u(t, x) is a solution
of
iv, + u,, + 2iS(o(zux = 0 and LJE C”(lR; H”(m)).
Then
(A.1)
V, = Eu and V, = Eu, satisfy
i(V,), + (VI), = 3idV*T 1 2, i(V2), + (V,), = -2idV*F * 1, where *x
Iv(t, w)l* dw
E = E(t, x) = exp id (
Proof. Multiplying
!-_
>
both sides of (A.l) by E, we obtain
i(V,), + (V,), + 2iSlV,12Vz = iE,v + 2E,u, + E,,u. We again
apply
(A.3
both sides of (A.1) by E?J, to have
I(V,), + (V,),, + 2iSlV, 12uxxE= -2iB(V,K By a simple calculation
+ V,y)V,
+ iE, v, + 2E,u,, + E,,u,
(A.3)
we see that
iE, = (i&V, Vz - V,q) + d21V,14)E, E,, = (i&V,% + V,V,) - SzIV1“)E, and 2E, = 2iS/ L’l*E. Therefore,
it follows
that
iEt + 2E xxu + E xxu = 2iSlVI’V 1 21 I 2 + 2i&‘*v i&v, From
(A.2)-(A.4)
the result follows.
+ 2E,u,, n
+ E,,u,
= 2i~lV,j’u,,E
+ 2iS[V212V,.
(A.4) (A.5)
833
Initial value problem For the convenience
of the reader
we give the proof
Proof of lemma 2.3. For simplicity we suppress the real part and integrating in x, we obtain
of lemma 2.3 which is different the subscript
j. Multiplying
from that in [4].
both
sides of (2.2) by q,
taking
By a direct calculation
(A.7) From (A.6) and (A.7) it follows
that (A.8)
/WV,w)l’dw m
>
we have
(A.9) and -tic5 By (A.8)-(A.lO)
IulZ(u,V - lIG)dx
I .m
we have the lemma.
n
= -*is
I ,R
IwIZ(w,P
- ViK)dx
+
~211wlli~.
(A.lO)