数理解析研究所講究録
1059 巻 1998 年 74-88

74

TIME LOCAL WELL-POSEDNESS FOR THE ZAKHAROVSYSTEM WITH THE PERIODIC BOUNDARY CONDITION

東京大学大学院数理科学研究科
高岡秀夫 (Hideo TAKAOKA)

1. INTRODUCTION AND RESULTIn this paper we consider the time local well-posedness for the one dimen-

sional Zakharov system with the periodic boundary condition:(1.1)$i\partial_{t}u+\alpha\partial_{x}^{2}u=un,$(1.2)$\frac{1}{\beta^{2}}\partial_{t}^{2}n-\partial_{x}^{2}n=\partial_{x}^{2}(|u|^{2}),$

(1.3)

where

$u(\mathrm{O}, x)=u_{0}(x),$$\alpha$and$\beta$$[-T,T]\cross \mathbb{T}$,$(t, x)\in[-T, T]\cross \mathbb{T}$$n(\mathrm{O}, x)=n_{0}(x),$are real constants withon the time-space$(t,x)\in[-T,T]\cross \mathbb{T}$$\alpha\neq 0$,$\partial_{t}n(0, X)=n_{1}(x),$and $\beta>0,$

with values in$\mathbb{C}$

and

$u$

$\mathbb{R}$

and

$n$

$x\in \mathrm{T}$,are functions, respectively,$\mathbb{T}$is aone dimensional torus which implies the periodic boundary condition and$T$

is a positive constant to be determined later.
The equations of $(1.1)-(1.2)$ was presented by V. E. Zakharov [15] tounderstand the the propagation of Langmuir turbulence waves in an un-

magnetized, completely ionized hydrogen plasma. In these equations,

$u$

is

75
the slowly varying complex envelope of the electric field$\omega$with frequency,$E(t, x)={\rm Re}(u(t, x)e^{-})it\omega$

$n$

$E$

,

denotes the deviation of the ion density from the equilibrium,$\alpha$is pro-portional to the differentiation of the group velocity in the wave number
and$\beta$is the speed of ion acoustic wave in a plasma.

Our purpose in this paper is to show the time local well-posedness results
in a weak class for the initial value problem $(1.1)-(1.3)$ . The difficulty tosolve the initial value problem $(1.1)-(1.3)$ by the integral equations is thatthe equation of (1.2) has space derivatives in the nonlinear term. One
derivative can be regained because of the second order hyperbolicity, but it

is a hard task to regain the remaining one derivative in that term. Thenthe usual contraction argument seems to meet with the so-called “loss ofderivative”
There are a large amount of papers concerning the well-posedness of theinitial value problem $(1.1)-(1.3)$ for the$\mathbb{R}^{d}$case$(\mathrm{S}\mathrm{e}\mathrm{e}[1,5,6,10,11,14])$. How-

ever, to our knowledge, there are only a few papers concerning for theperiodic boundary condition problem (see, e.g., [4]).
Before precisely stating our results, we prepare the following notations.

76
Definition 1.1. Let$\mathcal{V}$be the space of functions$f$$f(\cdot, x)\in S(\mathbb{R})$

For

$s,$

$b\in \mathbb{R}$for each:$\mathbb{R}\cross \mathrm{T}arrow \mathbb{C}$$x\in \mathrm{T}$, we define the spaces

and$X_{s}$$f$such that,$f(t, \cdot)\in C^{\infty}(\mathrm{T})$and$\mathrm{Y}_{s}$for each$t\in \mathbb{R}$to be the completion of.

$\mathcal{V}$withrespect to the norms:$||f||_{X_{s}}=||f||_{(1,s},1/2)$ ’$||f||_{\mathrm{Y}_{S}}=||f||_{(/)}2,s,12$

,

where

$||f||_{(1,s,b})=( \sum_{n\in \mathbb{Z}}(1+|n|)^{2}s\int_{-}^{\infty}\infty(1+|\mathcal{T}+\alpha n|2)2b|f(\mathcal{T}n\wedge,)|2d\mathcal{T})^{1}/2$,$||f||_{(2,s},b)=( \sum_{n\in \mathbb{Z}}(1+|n|)^{2s}\int^{\infty}-\infty(1+||\mathcal{T}|-\beta|n||)2b\wedge)^{1}|f(\mathcal{T}, n)|^{2}d\mathcal{T}/2$

Let$\psi\in C_{0}^{\infty}(\mathbb{R})$$\psi(t/\delta)$

with $\psi=1$ on [-1, 1] and$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}_{0}\mathrm{r}\mathrm{t}\psi\subseteq(-2,2)$

and$\psi_{\delta}(t)=$.

We now state our theorem.
Theorem 1.1. Assume that$\beta/\alpha$$s-l\leq 1$ and $0\leq l+1/2\leq 2s$ .

is not integer. Given

$s,$

$l$

satisfying$0\leq$

For any $(u_{0}, n_{0},n1)\in H^{s}\cross H^{l}\cross H^{l-1}f$there exist $T=T(||u0||L2, ||n_{0}||_{H}-1/2, ||n_{1}||_{H^{-3/2}})>0$ and a unique solution

77$(u, n, \partial_{t}n)$of the initial value problem$(\mathit{1}.\mathit{1})-(\mathit{1}.\mathit{3})$in the time interval $[-T, T]$such that$u\in C([-T,\tau] :HS(\mathrm{T})),$$n\in C([-\tau, T] : H^{l}(\mathrm{T})),$$\psi_{T}n\in \mathrm{Y}_{l}$$\partial_{t}n\in C([-\tau,T] : H^{l-1}(\mathrm{T})),$For any $T’\in(0, T)_{f}$ there exists$(\tilde{u},\tilde{n}, \partial_{t}\tilde{n})$is Lipschitz from {$||\tilde{n}_{1}-n_{1}||_{H}l-1<\epsilon\}$to,$\psi_{\tau u}\in X_{s}$,$\psi\tau\partial_{t}n\in \mathrm{Y}_{l-1}$.

such that the map$\epsilon>0$:$(\tilde{u}0,\tilde{n}0,\tilde{n}1)$$(\tilde{u}_{0},\tilde{n}_{0},\tilde{n}1)-*$$||\tilde{u}_{0}-u_{0}||_{H^{\epsilon}}+||\tilde{n}_{0}-n_{0}||_{H^{l}}+$$||\tilde{u}-u||L^{\infty}t([-^{\tau\prime},\tau’]:H^{S})+||\tilde{n}-n||L\infty t([-T^{;})T’]:H^{l})+$$||\partial_{\tau^{\tilde{n}-}}\partial tn||L_{\ell}\infty([-^{\tau}/,T’]:H^{\iota+||}-1)\psi\tau;(\tilde{u}-u)||X_{S}+||\psi\tau’(\tilde{n}-n)||\mathrm{Y}\iota+||\psi\tau^{r(}\partial_{t}\tilde{n}-$$\partial_{t}n)||_{\mathrm{Y}\iota}-1^{\cdot}$

Remark 1.1. In [4], J. Bourgain showed that when $\alpha=\beta=1$ the initialvalue problem $(1.1)-(1.3)$ is time locally well-posed in a certain class. Moreprecisely, he showed that for any ($u_{0},$

no,$u_{0}\in H^{\mathrm{s}}$(1.4)$n_{1}$) such that,$\sup_{k\in \mathbb{Z}}(1+|k|)s_{1}|\overline{u0}(k)|,$ $\sup_{k\in \mathbb{Z}}(1+|k|)\sigma|\overline{n_{0}}(k)|,\sup_{\mathbb{Z}k\in}(1+|k|)\sigma-1|\overline{n_{1}}(k)|<\infty$with $\sigma<0<s<1/2<s_{1}<1$ where$\sigma,$$s$

are sufficiently close to $0,1/2$ ,respectively. There exist $T>0$ and a unique solution$u\in C([-\tau,T] : Hs),$,$(u, n,\partial_{t}n)$$\sup_{k\in \mathbb{Z}}(1+|k|)s_{1}|F_{x}u(t,k)|<\infty$,

satisfying

78$n\in C([-T,T] : H^{s-1}),$$\partial_{t}n\in C([-T,T]:Hs-2),$$\sup_{k\in \mathbb{Z}}(1+|k|)^{\sigma}|F_{x}n(t, k)|<\infty$,$\sup_{k\in \mathbb{Z}}(1+|k|)^{\sigma-1}|F_{x}\partial_{t}n(t, k)|<\infty$,

for $t\in[-T, T]$ and$|||u|||= \inf${$||v||x_{S}$: $v(t)=u(t)$ on

Under the non resonance condition that$[-T,T]\cross \mathrm{T}$$\beta/\alpha$}$<\infty$.

is not integer, Theorem 1.1

implies the time local well-posedness with the data in the usual Sobolev

spaces$H^{s}\cross H^{l}\cross H^{l-1}$. We do not assume the weighted$L^{\infty}$

condition in

the Fourier space such as (1.4) and we shall give a slightly simpler proof
than in [4].
Remark 1.2. By Theorem$L^{2}\cross H^{-1/2}\cross H^{-3/2}$$1.\dot{1}$, we have the time local well-posedness in. Of course, the result in Theorem 1.1 also contain the

Hamiltonian class, that is, the energy space.

Remark 1.3. When $0\leq s-l\leq 1$ and $1/2\leq l+1/2\leq 2s$ , the result in
Theorem 1.1 hold with$\beta/\alpha\in \mathbb{R}..’$.

The proof of Theorem 1.1 is based on the method introduced by Bourgain[3] and$\mathrm{K}\mathrm{e}\mathrm{n}\mathrm{i}\mathrm{g}^{\mathrm{p}_{0}-}- \mathrm{n}\mathrm{c}\mathrm{e}\mathrm{V}\mathrm{e}\mathrm{g}\mathrm{a}[9]$to treat the$\mathrm{K}\mathrm{d}\mathrm{V}$equation. We use respectively

the two different Fourier restriction norms for the equations $(1.1)-(1.2)$ under
the non resonance condition that$\beta/\alpha$is not integer.

79
2. PROOF OF THEOREM 1.1
First, we rewrite equations $(1.1)-(1.3)$ as following integral equations:$u(t)=S(t)u_{0}-i \int_{0}^{t}S(t-\mathcal{T})(un)(\mathcal{T})d_{\mathcal{T}}$,$n(t)= \partial_{t}V(t)n0+V(t)n_{1}+\int_{0}^{t}V(t-\mathcal{T})\partial^{2}(x|u|^{2})(\mathcal{T})d_{\mathcal{T}}$where$S(t)=e^{it\partial_{x}^{2}}$and$V(t)=\sin\beta t(-\partial^{2})^{1}x/2/\beta(-\partial^{2}x)^{1/}2$,.

We first state the two lemmas concerning the estimates of the linear andnonlinear part of the Schr\"odinger and Wave equations on the function spaces

we consider.
Lemma 2.1. For any$s\in \mathbb{R}$, we have$||S(t)u_{0}||X_{s}\leq c||u_{0}||_{H}\theta$

Lemma 2.2. For any

(2.1)

,

$||\partial_{t}V(t)n_{0}||_{\mathrm{Y}}s\leq c||n_{0}||Hs$,$||V(t)n_{1}||_{\mathrm{Y}}s\leq c||n_{1}||_{H^{s}}-1$.

$s\in \mathbb{R}$, we have$|| \psi(t)\int_{0}^{t}S(t-\mathcal{T})F(\tau)d\tau||x_{S}$$\leq c||F||_{(}1,s,-1/2)+c(_{k\in \mathbb{Z}}\sum(1+|k|)^{2}s(\int\frac{|\overline{F(\tau,k})|}{1+|_{\mathcal{T}+}\alpha k2|}d\tau))21/2$

,

80

(2.2)

$||\psi(t)I^{t}\mathrm{o}V(t-\tau)\partial xF(\mathcal{T})d_{\mathcal{T}}||_{\mathrm{Y}}s$$\leq c||F||_{(2,\delta},-1/2)+c(_{k\in \mathbb{Z}}\sum(1+|k|)2s(\int\frac{|\overline{F(\tau,k})|}{1+||\tau|-\beta|k||}d\tau)2\mathrm{I}1/2$

Lemma 2.1 and 2.2 follow from a direct calculation. For the proof ofthat lemmas, see $[3,9]$ . The following lemma plays to use the contraction

argument in the proof of Theorem 1.1.
Lemma 2.3. For any$s\in \mathbb{R},$and $0<b<1/2$ , it$0<\epsilon\ll 1,$ $\delta\in(0,1]$

follows$||\psi_{\delta}F||_{(/2}i,S,1)\leq c_{\epsilon}\delta^{-\epsilon}||F||_{(}i,s,1/2)$,$||\psi_{\delta}F||(i_{S,b)},\leq c\delta^{1/2b}-+\epsilon||F||_{(}i,s,1/2)$,

for $i=1$ and 2.
Lemma 2.3 is proved by using the Leibniz rules for fractional derivatives[8, Theorem A.12], H\"older and Sobolev inequality with respect to time vari-able. We give the following lemma needed to estimate the nonlinear terms

appearing in the right hand side of (2.1) and (2.2). The following lemma
plays a key role in our proof.
Lemma 2.4. Assume$l+1/2\leq 2s_{f}$(2.3)$\beta/\alpha$is not integer. For $0\leq s-l\leq 1$ and $0\leq$

we have$|^{1} \infty\frac{(1+|k|)^{s}}{(1+|\mathcal{T}+\alpha k^{2}|)^{1/}2}\sum_{k_{1}\in \mathbb{Z}}\int_{-}^{\infty}\frac{f(\tau_{1},k_{1})}{(1+|k_{1}|)^{s}(1+|\mathcal{T}1+\alpha k_{1}^{2}|)^{1/}2}$

81$\cross\frac{g(\tau-\mathcal{T}_{1},k-k1)}{(1+|k-k_{1}|)^{l}(1+||\tau-\tau_{1}|-\beta|k-k_{1}||)1/2}d\tau_{1}||_{L_{r}^{2}l_{k}^{2}}$$\leq c||f||_{Ll_{k}}22|r|g||_{L_{r}^{2}l^{2}}k$

(2.4)

’

$\mathrm{i}\frac{k(1+|k|)^{l}}{(1+||_{\mathcal{T}}|-\beta|k||)^{1/2}}\sum_{\mathbb{Z}k_{1}\in}\int_{-\infty}\infty\frac{f(\tau_{1},k_{1})}{(1+|k_{1}|)s(1+|\mathcal{T}1+\alpha k_{1}^{2}|)^{1/}2}$$\cross\frac{h(\tau-\mathcal{T}_{1},k-k1)}{(1+|k-k1|)^{s}(1+|\tau-\tau_{1^{-}}\alpha(k-k1)^{2}|)^{1/}2}d\tau_{1}||_{L^{2}l^{2}}rk$$\leq c||f||_{Ll_{k}}2r2||h||_{L}2\tau l_{k}^{2}$.

Proof of Lemma 2.4. The proof uses the folowing two identities:$( \tau+\alpha k^{2})-(\tau-\tau_{1}\pm\beta(k-k_{1}))-(\tau_{1}+\alpha k_{1}^{2})=\alpha(k-k1)(k+k_{1}\mp\frac{\beta}{\alpha})$,,$( \tau\pm\beta k)-(\tau-\tau_{1}-\alpha(k-k_{1})^{2})-(\tau_{1}+\alpha k_{1}^{2})=\alpha k(k-2k1\pm\frac{\beta}{\alpha})$which implies(2.5)$\max\{|_{\mathcal{T}}+\alpha k2|, |\tau-\tau_{1^{\pm}}\beta(k-k1)|, |\tau_{1}+\alpha k^{2}|1\}\geq\frac{|\alpha|}{3}|k-k_{1}||k+k_{1}\mp\frac{\beta}{\alpha}|$

,

(2.6)
$\max\{|\tau\pm\beta k|, |\tau-\tau_{1}-\alpha(k-k1)^{2}|, |\mathcal{T}1+\alpha k^{2}|1\}\geq\frac{|\alpha|}{3}|k||k-2k_{1}\pm\frac{\beta}{\alpha}|$.

Using these inequalities, we are able to overcome the difficulty of derivative
lose. In this paper, we prove only (2.3) and (2.4) in the regions that
(2.7)$|\tau+\alpha k^{2}|\geq|\tau_{1}+\alpha k_{1}^{2}|,$ $||\tau-\mathcal{T}_{1}|-\beta|k-k_{1}||$

,

82

(2.8)

$||\tau|-\beta|k||\geq|\tau-\tau_{1}-\alpha(k-k1)^{2}|,$ $|\tau 1+\alpha k_{1}2|$,respectively.
We first prove (2.3) restricted to the region of (2.7). In that region by

Schwarz inequality we have that the left-hand side of (2.3) is bounded by$\sup_{\tau,k}(\frac{(1+|k|)2S}{(1+|\tau+\alpha k^{2}|)}\sum_{k_{1}}\int\frac{1}{(1+|\tau_{1}+\alpha k_{1}^{2}|)(1+|k_{1}|)2s(1+|k-k1|)^{2}l}$$\cross\frac{1}{(1+||\mathcal{T}-\mathcal{T}_{1}|-\beta|k-k1||)}d\tau_{1})^{1/2}||f||_{L_{\tau}}2l_{k}2||g||L_{r}2l_{k}2$$= \sup_{\tau,k}I_{\tau,k}||f||_{Ll_{k}}22|\tau|g||_{L_{\tau}^{2}l^{2}}k$.

In order to prove (2.3) it suffices to show that $\sup_{\tau,k}I\tau,k<\infty$ . Integratingwith respect to$k_{1}=k$$I_{\tau,k}^{2}$and$\tau_{1}$$k\neq k_{1}$and dividing the integral region into two regions that

. In $k_{1}=k$ , we have that the contribution of $k_{1}=k$ tois bounded by$\frac{1}{(1+|\tau+\alpha k^{2}|)}<C$

We next consider the region of$k_{1}\neq k$.

. Then by (2.5), (2.7) and$\beta/\alpha$is notinteger, we first note that$|\tau+\alpha k2|\geq c(1+|k-k1|)(1+|k+k_{1}\pm\beta/\alpha|)$First in the regions$k_{1}\pm\beta/\alpha|\geq c|k|$$\mathrm{o}\mathrm{f}|k_{1}|<|k|/2$.

or $|k|/3<|k_{1}|$ it follows that $|k-k_{1}|,$ $|k+$, then we have that the contribution of the region$k_{1}\neq k$83to$I_{\tau,k}^{2}$

is bounded by$c \sum_{k_{1}}\frac{1}{(1+|k1|)2s(1+|k|)2l-2s+2(1+|\tau+\alpha k_{1^{\pm\beta(k_{1}}}2k-)|)^{1-\epsilon}}$$\leq c\sum_{k_{\iota}}\frac{1}{(1+|\mathcal{T}+\alpha k^{2}1^{\pm\beta(k)|)^{1-}}k-1\epsilon}$$\leq c$for $\epsilon\in(0,1/2),$ $s\geq 0,$ $l\geq-1/2$ and $s-l\leq 1$ . Next in the region of$|k|/2\leq|k_{1}|\leq 3\}k|/2$$I_{\tau,k}^{2}$, we have that the contribution of the region$k_{1}\neq k$tois bounded by$c \sum_{k_{1}}\frac{1}{(1+|k-k1|)^{2l}+1(1+|\mathcal{T}+\alpha k_{1^{\pm\beta(kk)}}2-1|)^{1}-\epsilon}$$\leq c$,for $l\geq-1/2$ .

Next we prove (2.4) restricted to the region of (2.8). In the same argument

as in the proof of (2.6), it suffices to show that$\sup_{\tau,k}II_{\mathcal{T}},k$$= \sup_{\tau,k}(\frac{(1+|k|)2l|k|2}{(1+||\tau|-\beta|k||)}\sum_{k_{1}}\int\frac{1}{(1+|\mathcal{T}1+\alpha k_{1}^{2}|)(1+|k1|)2s(1+|k-k1|)^{2}s}$$\cross\frac{1}{(1+|_{\mathcal{T}-}\mathcal{T}_{1}-\alpha(k-k_{1})^{2}|)}d_{\mathcal{T}}1)^{1/2}$$<\infty$.

We can assume $k\neq 0$ . First in.the regions that $|k_{1}|\leq 2|k|/5$ or $2|k|/3\leq$$|k_{1}|$, integrating with respect to$\tau_{1}$and using $(2.6),(2.8)$ we have that the

84contribution of the region $k\neq 0$ to$II_{\tau,k}^{2}$

is bounded by$c \sum_{||k_{1}|\leq 2|k|/5\mathrm{o}\mathrm{r}2|k|/3\leq k_{1}|}\frac{(1+|k|)^{2}l+1}{(1+|k1|)2s(1+|k-k1|)^{2}s(1+|k-2k_{1}\pm\beta/\alpha|)}$$\cross\frac{1}{(1+|\tau-\alpha k^{2}+2\alpha kk_{1}|)1-\epsilon}$$\leq c+C\sum_{\leq|k_{1}|\leq 2|k|/50\Gamma 2|k|/3|k_{1}|}\frac{1}{(1+|k_{1}|)2s(1+|k|)2S-2l+1-\epsilon}$$\cross\frac{1}{(1+|k_{1}-\gamma(\mathcal{T},k)|)1-\epsilon}$,

for $\epsilon\in(0,1/2),$ $s\geq 0$ and $s-l\geq 0$ where
following linear equation with respect to$\tau-\alpha k^{2}+2\alpha kk_{1}=0$$\gamma=\gamma(\tau, k)$$k_{1}$, i.e.,

Then, we divide the sum with respect to$k_{1}$

is the solution of the,$\gamma=\frac{k}{2}-\frac{\tau}{2\alpha k}$.

into the following three regions:$|k_{1}|\leq 2|k|/5,2|k|/3\leq|k_{1}|<3|k|/2$

and $3|k|/2\leq|k_{1}|$ .

In the regions that $|k_{1}|\leq 2|k|/5$ or $2|k|/3\leq|k_{1}|<3|k|/2,$$II\tau,k2$

restricted

to that regions is bounded by$c \sum_{k_{1}}\frac{1}{(1+|k_{1}|)4S-2l+1-\epsilon(1+|k_{1}-\gamma(\mathcal{T},k)|)1-\epsilon}$$\leq c$,for $s\geq 0,$ $s-l\geq-1/2+\epsilon/2$ and $2s-l>-1/2+\epsilon$ . In the same way as
above, in the region of $3|k|/2\leq|k_{1}|,$$II^{\frac{9}{\tau}},k$restricted to the above region isbounded by$c \sum_{k_{1}}\frac{1}{(1+|k_{1}|)4S-2l-\epsilon(1+|k-2k_{1^{\pm\beta/\alpha}}|)(1+|k_{1}-\gamma(\mathcal{T},k)|)^{1-\epsilon}}$$\leq c$,85
for $s\geq 0,2s-l>-1/2+\epsilon$ and $2s-l\geq-1/2+\epsilon/2$ . Next, in the regionof $2|k|/5<|k_{1}|<..2|k|/3$ , we have that $|k|/3<|k-k_{1}|<5|k|/3$ . Then we

have that$II_{\tau,k}^{2}$

restricted to the above$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}.\mathrm{r}\mathrm{a}\mathrm{l}$region is bounded by$C \sum_{/2|k|/5<|k_{1}|\leq 2|k|3}\frac{1}{(1+|k_{1}|)4S-2l-1(1+|k-2k_{1}\pm\beta/\alpha|)}$$\cross\frac{1}{(1+(1+|k|)|k1-\gamma(\tau,k)|)1-\epsilon}$$\leq c$,for $\epsilon\in(0,1/2)$ and $2s-l\geq 1/2$ .

$\square$Remark 2.1. When $0\leq s-l\leq 1$ andholds with for any$\beta/\alpha\in \mathbb{R}$$1_{\iota_{\dagger}}/\backslash \tilde{2}\leq l+1/2\leq 2s$, Lemma 2.4.

The following proposition is an immediate consequence of Lemma 2.4.

Proposition 2.5. Assume that$\beta/\alpha$is not integer. For any $0\leq s-l\leq 1$

and $0\leq l+1/2\leq 2s_{f}$ we have
(2.9)$||un||_{(S}1,,-1/2)\leq c||u||_{(1/)}1,S,2||n||(2,l,1/2)$(2.10)$||\partial x(|u|^{2})||_{(2},l,-1/2)\leq c||u||_{(}^{2}1,s,1/2)$,.

Proof of Proposition 2.5. We define$f(\tau, k)=(1+|k|)^{S}(1+|\tau+\alpha k^{2}|)^{1/}2|u\wedge(\mathcal{T}, k)|$,$g(\tau,k)=(1+|k|)^{l}(1+||\tau|-\beta|k||)^{1/2}|\wedge n(\mathcal{T}, k)|$,86$h(\tau, k)=(1+|k|)^{s}(1+|\tau-\alpha k^{2}|)^{1}/2|u(\wedge-\mathcal{T}, -k)|$.

From Lemma 2.4, the proof of the proposition follows.

Remark 2.2. Proposition 2.5 holds when the right hand side of $(2.9)-(2.10)$

are replaced by$||u||_{(}1,s,b)||n||_{(}2,l,1/2)+||u||(1,s,1/2)||n||_{(l,b}2,)$$||u||(1,s,b)||u||_{(1}1,S,/2)$,,

respectively, with $1/2-1/8<b<1/2$ .

Remark 2.3. For the proof of the second terms of the right hand side of(2.1) and (2.2), we obtain the similar result to Proposition 2.5.

Proof of Theorem 1.1. For$T\in(\mathrm{O}, 1)$, we define,$\Phi(u, n)=S(t)u0-i\psi(t)\int_{0}^{t}S(t-\tau)(\psi_{\tau}u\psi Tn)(\tau)d_{\mathcal{T}}$$\Psi(u)=\partial_{t}V(t)n0+V(t)n_{1}+\psi(t)\int_{0}^{t}V(t-\mathcal{T})\partial^{2}(x|\psi\tau u|2)(\tau)d_{\mathcal{T}}$.

From Lemma 2.1-2.3, Proposition 2.5 and Remark 2.2, we have that$||\Phi(u, n)||x_{S}\leq c||u_{0}||H^{s}+cT^{\mu}||u||\mathrm{x}_{s}||n||\mathrm{Y}_{l}$,$||\Psi(u)||_{\mathrm{Y}_{l}}\sim||\partial_{t}\Psi(u)||_{\mathrm{Y}_{l-}}1\leq c(||n0||_{H}1+||n_{1}||Hl-1)+C\tau^{\mu}||u||_{x_{s}}^{2}$.

87
for some $\mu>0$ . Similar argument to above, we have that$||\Phi(\tilde{u},\tilde{n})-\Phi(u, n)||x_{s}\leq cT^{\mu}(||u||_{X_{s}}+||n||_{\mathrm{Y}_{l}}+||\tilde{u}||_{X_{s}}+||\tilde{n}||_{\mathrm{Y}\iota})$$\cross(||\tilde{u}-u||xs+||\tilde{n}-n||_{\mathrm{Y}_{l}})$,$||\Psi(\tilde{u})-\Psi(u)||_{\mathrm{Y}_{l}}\sim||\partial_{t}\Psi(\tilde{u})-\partial t\Psi(u)||\mathrm{Y}_{l-1}\leq c\tau^{\mu}(||u||Xs+||\tilde{u}||_{X}s)||\tilde{u}-u||_{X_{s}}$Thus, we have that$\Phi\cross\Psi\cross\partial_{t}\Psi$is a contraction map. Then, we obtainthe unique local existence results inargument.

.

$X_{s}\cross \mathrm{Y}_{l}\cross \mathrm{Y}_{l-1}$

by the contraction$\square$

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[1]$Schr\overline{\mathit{0}}dinger$$Schor\overline{\mathit{0}}dinger$$V$$\mathrm{T}\mathrm{s}\mathrm{u}\mathrm{t}_{\mathrm{S}}\mathrm{u}\dot{\mathrm{m}}$88limit and the initial layer of the Zakharov equa[12] –, The nonlinear
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$Schr\overline{\mathit{0}}dinger$$Schr\overline{\mathit{0}}dinger$