数理解析研究所講究録 1411 巻 2005 年 37-47

37
Global existence for the Klein-Gordon-Zakharov equations

北海道大学・理学研究科・数学専攻

津田谷公利 (Kimitoshi Tsutaya)

Department of Mathematics,
Hokkaido University

Introduction and Results.

1

In this note we present results on the Klein-Gordon-Zakharov equations, which are

based on [15, 16, 20]. We consider the Cauchy problem of the Klein-Gordon-Zakharov
equations in three space dimensions:$\partial_{t}^{2}u-$

Au $+u=-nu$ ,$t>0,$$t>0,$$\partial_{t}^{2}n-c^{2}\Delta n=\Delta|u|^{2},$,$x\in \mathrm{R}^{3}$$x\in \mathrm{R}^{3}$$u(0, x)=u_{0}(x)$ ,$\partial_{t}u(0, x)$$=u_{1}(x)$,$n(0,x)=n_{0}(x)$ ,$\mathrm{d}n(0, x)$$=n_{1}(x)$

,

(1)

,

,

(2)
(3)

The propagation speed in equation (1) is normahzed as unit, while that in equation (2)is denoted by . Equations (1) and (2) describe the interaction of the Langmuir wave$c$and the ion acoustic wave in a plasma (see Dendy [4] and Zakharov [21]). The function
tz denotes the fast time scale component of electric field raised by electrons and thefunction
and

$n$

$n$

denotes the deviation of ion density from its equilibrium. The functions$u$are real vector valued and real scalar valued, respectively. We introduce known

results on $(1)-(3)$ dividing into two cases (i) $c=1$ and (ii) $c\neq 1,$ that is, whether or
not the propagation speeds are the same.

Before proceeding, we give notation. For
let $IP$ and$W^{m,p}$$(-5)^{1/2}$

and a nonnegative integer $m$ ,denote the standard $IP$ and Sobolev spaces on

put $H^{m}=$ lym,2. For $m\in$ R, we let$\omega_{0}=$$1\leq p\leq \mathrm{o}\mathrm{o}$. We put$\partial_{j}=\partial/\partial x_{j}$$(-\Delta)$ $-m/2L^{2}$$\dot{H}^{m}=$$\mathrm{R}^{3}$. We put$\omega$, respectively. We$=(1-))^{1/2}$ andfor $j=1,2,3$ . Let $\Gamma=(\Gamma_{j};j= 1, \cdots, 10)$ denotethe generators of the Poincare group$(\partial_{t},\partial_{1},\partial_{2},\partial_{3}, L_{1},L_{2}, L_{3},\Omega_{12}, \Omega_{23}, \Omega_{13})$,$L_{\mathrm{j}}$$=x_{j}\partial_{t}+t\partial_{j}$$\Omega_{ij}$$=x:\partial_{j}-x_{j}\partial.\cdot$$j=1,2,3$ ,,$1\leq i<j\leq 3,$, where38
and we puta
For a multi-i dex$\alpha=$$(\alpha_{1}, \cdots, \alpha_{10})$$=(\partial_{t}, \partial_{1}, \partial_{2}, \partial_{3})$.

, we put$\Gamma^{\alpha}=\Gamma_{1}^{\alpha_{1}}\cdots\Gamma_{10}^{\alpha_{10}}$.

For $m\geq 0$ and $s\geq 0,$ we define the weighted Sobolev space$H^{m,s}=${$v\in L^{2}$;( $1+|x|^{2}$ ) $s/2(1$$H^{m,s}$A) $)^{m/2}v\in L^{2}$-on$\mathrm{R}^{3}$

as follows:}

We put $H^{m}\equiv H^{m,0}$ for $m\geq 0.$

For a function $u(t,x)$ , we denote byof . For ,
$u$

$s$

$b\in$R, we define the spaces$X_{b,s}^{\pm}$

$=$

{

.

$=$

(

$||u||_{X5}$$\mathrm{Y}_{b,s}^{\pm}$$||v||\mathrm{r}5$

We note that

1.1

$X_{b,s}^{\pm}$

$=$

$=$

and

$u\in$$5’(\mathrm{R}^{4})$;$\tilde{u}(\tau, \xi)$$X_{b,s}^{\pm}$the Fourier transform in both and

and

$t$

,

$\mathrm{Z}_{\epsilon}^{\pm}$as follows:$||\mathrm{t}\mathrm{Z}||_{X_{b}\mathrm{t}_{s}}<+\mathrm{o}\mathrm{o}1$),$\int_{R}\int_{R^{3}}(1+|\xi|)^{2s}(1 +|\tau\pm |4| |)^{2b}|\tilde{u}(\tau,\xi)|^{2}d\xi d\tau$$\{v\in 5’(\mathrm{R}^{4}); ||\mathrm{t} ||_{\mathrm{Y}_{b,\epsilon}^{\pm<+}}\mathrm{o}\mathrm{p}\}$

(

$x$

,

$\int_{R}4_{3}^{(1+|4|)^{2s}(1+|\tau\pm c|\xi|}|)2b|$$\mathrm{Y}_{b,s}^{\pm}$a$\mathrm{e}$Banach spaces for ,$s$$b\in$

i&, )
$\xi$

$|^{2}\not\in d\tau)^{1/2}$

R.

Case c $=1.$

When the propagation speeds are the same, many results have been obtained concerning the global existence of small amplitude solutions for the coupled systems of

the Klein-Gordon and wave equations with quadratic nonlinearity. Two methods are

known to be applicable to solve those systems.
(i) First one is based on the theory of normal forms introduced by Shatah [18], which isan extension of Poincare’s theory. The idea of this method is to transform the original
system with quadratic nonlinearity into a new system with cubic nonlnearity. We state

our results obtained by applying the argument of normal forms in Ozawa, Tsutaya andTsutsumi [15]. We have the following theorem :3
Let

Theorem 1$H^{51}\cap W^{28,6/+}’$”,$0<\epsilon$$\delta>0$

Eo$||_{H^{\mathrm{s}1}}$$u_{0}\in H^{52}\cap W^{29,6/(5+2\epsilon)}$and $n_{1}\in H^{50}\cap W^{27,220/217}$,$u_{1}\in$$\cap\dot{H}^{-2}$.

such that if$||u_{0}$$(\mathit{1})-(\mathit{3})$. Assume that$n_{0}\in H^{51}\cap W^{28,220/217}\cap\dot{H}^{-1}$

Then there exists a$+$$\leq 10^{-2}$I

11y29,6/(S $+2e$ )$+||u_{1}||_{H}51\mathrm{n}\mathrm{W}^{28,6/(\mathrm{s}+}\mathrm{S}\mathrm{n})$$H^{52}\cap$

1W28,22Or217$\cap\dot{H}^{-1}$$+$$\delta$$\leq$$\mathrm{E}1||_{H}5\mathrm{Q}\mathrm{n}\mathrm{W}^{27,220/217}\mathrm{n}H-2$(4),

has the unique global solutions $(u,n)$ satisfying$u \in\bigcap_{j=0}^{2}C^{j}([0, \infty)_{1}.H^{52-j})$,Jl,$n \in[\bigcap_{j=0}^{2}C^{j}([0, \infty);H^{51-j})]\cap[\bigcap_{j=0}^{1}C^{j}([0,\infty);\dot{H}^{-1-}$$\sum 1||\partial_{t}^{j}$u(t)$||_{W}2\mathit{0}-i,at(1-2\epsilon)$$=O(t^{-(1+}$ ’$)$$(t arrow\infty)$

,

$j=0$

1
$\sum$$||g_{n}o)$$||_{W^{2\mathit{5}-j,220\mathit{7}^{S}}}$$=O(t^{-107/110})$$(t arrow\infty)$,$\mathrm{j}=0$

there$\delta$depends only on . Furthermore, the above solutions $(u,n)$ of

free profiles$\epsilon$$u_{+0}\in H^{52}$$(\mathit{1})-(\mathit{3})$have the, $u_{+1}\in H^{51}$ , $n_{+0}\in H^{51}$ and $n_{+1}\in H^{50}$ such that$\sum_{j=0}^{1}||\partial_{t}^{j}(u(t)-u_{+}(t))||_{H^{52-j}}$$+ \sum||1(\mathrm{N}(n(t)-n_{+}(t))||_{H^{51-\mathrm{j}}}arrow 0$$(tarrow\infty)$

,

$j=0$

there

,

$u_{+}(t)$$=$$(\cos\omega t)u_{+0}+(\omega^{-1}\sin\omega t)u_{+1}$$n_{+}(t)$$=$$(\cos\omega_{0}t)n_{+0}+(\omega_{0}^{-1}\sin\omega_{0}t)n_{+1}$.

Remark 1.
(1) $u_{+}(t)$ and $n_{+}(t)$ are the solutions of the free Klein-Gordon equation and the freewave equation with the initial conditions, respectively.

$(u_{+}(0), \partial tu+(0))=(u+0, u+1)\mathrm{m}\mathrm{d}$ $(n+(0)$,$\partial_{t}n_{+}(0))=(n_{+0},n_{+1})$(2) In the case of one or two space dimensions, the global existence result for small

initial data can be proved more easily than the case of three space dimensions. We do$\theta$

40
not need the time decay estimates to show the global existence of solutions in the one
and two dimensional cases.
(3) Recently, M. Ohta has proved the blow-up in a finite time for $(1)-(3)$ with large

initial data.
The following corollary is an immediate consequence of Theorem 1.

Corollary 2

Let $0<\epsilon\leq 10^{-2}$ and let m be a positive integer with m $\geq 52.$

Assume that$H^{m}\cap W_{f}^{29,6/(5+2e)}u_{1}\in H^{m-1}\cap W_{1}^{28,6/(5+2e)}70\mathrm{E}$ $H^{m-1}\cap W^{28,220/217}\cap$$\dot{H}^{-1}$$u_{0}\mathrm{B}$, $n_{1}\in H^{m-2}\cap W^{27,220/217}\cap\dot{H}^{-2}$ and$(u_{0},u_{1}, n_{0},n_{1})$satisfy (4). Then the solutions$(u,n)$ given by Theorem 1 satisfy$u\in\cap C^{j}([0, \infty);H^{m-\mathrm{j}})j=0m$

In addition, if$u_{0},u_{1}$,$n_{0},n_{1} \in\bigcap_{j=1}^{\infty}H^{\mathrm{j}}$$u(t, x)$ ,,$n\in m1\overline{\bigcap_{j=0}}C^{j}([0, \infty);H^{m-1-j})$.

, then we have$n(t, x)\in C^{\infty}([0, \infty)\mathrm{x}\mathrm{R}^{3})$.

The existence and uniqueness of local solutions for $(1)-(3)$ folows from the standard
iteration argument. The crucial part of the proofs of Theorem 1 and Corolary 2 is
to establish the a priori estimates of the solutions for $(1)-(3)$ . We use the argumentof normal forms of Shatah [18] to transform the quadratic nonlinearity into the cubic

one. However, in our problem the transformed cubic nonlinearity is represented interms of the integral operator with singular kernels. The singularity of the integral
kernels makes it difficult to solve $(1)-(3)$ . This is different ffom the case of the system

containing only the Klein-Gordon equations, where the integral kernels of the resulting
integral operators are regular (see [18]). Therefore, our main task in the proof of

Theorem 1 is to evaluhte the singularity of the integral kernels of the transformed
cubic nonlinearity. Then we can apply the usual $If-L^{q}$ estimate to the transformed
system.
(ii) Another method to solve $(1)-(3)$ is to use the invariant Sobolev space with respect

to the generators of the Lorentz group. This was developed by Klainerman [11]. He also

41
introduced the notion of the null condition to prove the existence of global solutions

for the wave equations with quadratic nonlnearity. We note that the null condition
technique is based on the Lorentz invariance of the equations.
In Theorem 1, one needs the high regularity assumptions on the data to ensure
the global existence. Moreover, the global solution

must belong to the homogeneous Sobolev space$n$$\dot{H}^{-1}$of $(1)-(3)$ given by Theorem 1of negative index. In this partwe show that there exist the global solutions of $(1)-(3)$ for small initial data using
the invariant Sobolev space but without applying the null condition technique andimprove the regularity requirements on the initial data. We do not need the nullcondition technique due to the nonlinearity of $(1)-(2)$ . The nonlinear terms in $(1)-(2)$do not seem to satisfy the$\mathrm{n}\mathrm{u}\mathrm{U}$condition as in [1] or [5].

We have the following theorem concerning the global existence of solutions to $(1)-$$(3)$.
Let $0<\epsilon<1/6$ and $k\geq 4.$ Assume that $u_{0}\in H^{k+5,k+4}$ ,Theorem 3$H^{k+4,k+4}$, $n_{0}\in H^{k+4,k+4}$ and $n_{1}\in H^{k+3,k+4}$ . Then there exists a$||u_{0}||H^{k+}\mathrm{s}$then$(\mathit{1})-(\mathit{3})$, $k+$ t $+||u1||Hk+4,k+4$$+||\mathrm{v}\mathrm{r}_{0}||H^{\mathrm{h}}$$\mathrm{t}4,k+<$$\delta>0$$+||n_{1}$ $||_{H^{k+3,k+4}}$$u_{1}\in$such that if$\leq\delta$,

has the unique global solutions $(u, n)$ satisfying$u \in\bigcap_{\mathrm{j}=0}^{k+5}C^{j}([0, \infty);H^{k+4-j})$,$n \in\bigcap_{j=0}^{k+4}C^{j}([0,\infty);H^{k+4-j})$,$\sum$$\sup(1+t)^{-\epsilon}\{||\partial_{t}\Gamma^{a}u(t)||_{L^{2}}+||\mathrm{C}\mathrm{J}[ \alpha u(t)||_{L^{2}}\}$$t\geq 0$$|\mathrm{a}|=k44$$\sup||\Gamma^{\alpha}n(t)||_{L^{2}}$$\sum$$+ \sum_{|\alpha|\leq k+4}\sup_{t\geq 0}(1+t)^{-\epsilon}||\Gamma^{\alpha}u(t)||_{L^{2}}+$$|a|\leq k$l4$t\geq 0$$\{|(1+t+|x|)" 2-2\epsilon_{\Gamma^{\alpha}u(t,x)|}$$+ \sum_{|\alpha|\leq k}$$x \in R^{3}\sup_{t\geq 0},$$+|(1+t+|x|)$ I”n(t, ) $|\}<\infty$ .
$x$

(5)

Remark 2.
(1) We see that the solutions $(u, n)$ of $(1)-(3)$ given by Theorem 3 asymptoticaly

approach the ffee solutions as

in time by (5).

$tarrow\infty$

since the right hand sides of $(1)-(2)$ are integrable

42(2) Compared to Theorem 1, the regularity assumptions on the initial data has improved significantly. Instead, we need some spatial decay on the data.
The following corollary follows easily ffom the proof of Theorem3.

In addition to all the assumptions in Theorem 3, if

Corollary 4$n_{1} \in\bigcap_{m\geq 1}H^{m}$$u_{0}$,$u_{1}$,$n_{0}$,, then the solutions $(u, n)$ given by Theorem 3 satisfy$u(t, x)$ ,

$n$

(t, )
$x$

$\in C^{\infty}([0, \infty)\cross \mathrm{R}^{3})$.

We can prove Theorem 3 by using two methods :the decay estimate of the inh0-

mogeneous linear Klein-Gordon equation by Georgiev [6] and the Sobolev inequality
in the Minkowski space by Klainerman $[10, 12]$ and H\"ormander [8]. See for details in

[20].

1.2

Case c$\neq 1.$

We may assume that $0<c<1$ without loss of generality. In fact, this condition isnatural ffom a physical point of view since the propagation speed in (1) is about one
thousand times as large as that in (2).
It seems impossible to use the method of normal forms because the integral kernels

have stronger singularity that in the case $c=1.$ Those kernels are difficult to handle.

The Lorentz invariance method does not seem useful, either since the operator

not commute with the d’Alembertian$\partial_{t}^{2}-c^{2}\Delta$$L_{j}$

doeswith $c\neq 1.$ We use another method toshow the global existence for the case $c\neq 1.$
We note that the solutions

$u$

and

$n$

of equations (1) and (2) formally satisfy thefollowing energy identity:$E(u(t),\partial_{t}u(t),n(t)$, $\partial_{t}n(t))=E(u_{0},u_{1},n_{0},n_{1})$ ,

where
$E$

$=$

(u,

$\partial_{t}u,n,\partial_{t}n$)$\frac{1}{2}||\nabla u||_{L^{2}}^{2}+\frac{1}{2}||u||_{L^{2}}^{2}+\frac{1}{2}||\partial_{t}u||_{L^{2}}^{2}$$+\mathrm{c}$$||n||_{L^{2}}^{2}+||$

cxrz$||_{\dot{H}^{-1}}^{2}$$+{\rm Re} \int_{R^{3}}n(t, x)|\mathrm{n}(\mathrm{t}, x)|^{2}dx$

.

(6)

43
Here and hereafter,$\hat{v}(4)$denotes the Fourier transform of $v(x)$ in the spatial variables.

The following theorem is about the time local well-posedness in$H^{1}\oplus L^{2}\oplus L^{2}\oplus H^{-1}$of $(1)-(3).\cdot$
Theorem 5[resp.

$H^{-1}$

Assume that $0<c<1$ holds. Let$H^{1}\oplus L^{2}\oplus L^{2}\oplus\dot{H}^{-1}$$(u_{0},u_{1}, n_{0}, n_{1})\in H^{1}\oplus L^{2}\oplus L^{2}\oplus$

f. Assume that $1/2<b<1$ andto 1/2. Then there exists a $T>0$ such that the problem$(u, n)$$(\mathit{1})-(\mathit{3})$$b$is close enoughhas unique solutions

on the time interval $[-T,T]$ satisfying$u\in C([-T, T]; H^{1})$ $\cap C^{1}([-T,T];L^{2})$ ,(7)$n\in C([-T,T];L^{2})\cap C^{1}([-T,T];H^{-1})$(8)[resp.
$u\mathrm{i}$

$i($

1-

$C([-T,T];L^{2})\cap C^{1}([-T,T];\dot{H}^{-1}$$\Delta)$ $-1/2\partial_{t}u\in

X_{b,1}^{\pm}$)],,$nc$ $i(1-c^{2}\Delta)^{-1/2}\partial_{t}n\in \mathrm{Y}_{b,0}^{\pm}$[resp.

where$T$

depends only on

addition, if$n_{1}\in\dot{H}^{-1}$$||\mathrm{f}\mathrm{j}_{0}||H^{1}$,$||u_{1}||L^{2}$,$||n_{0}||L^{2}$

and

],

[resp.

$||\mathrm{t}\mathrm{t}_{1}||H^{-1}$$||n_{1}||_{\dot{H}^{-}}1\mathrm{j}$. In, then the solutions $(u, n)$ satisfy the energy identity:$E(u(t), \partial_{t}u(t)$$h\hslash hermore_{J}$$n\pm$ $i(\alpha v_{0})^{-1}gn\in \mathrm{Y}_{b,0}^{\pm}$, $n(t),\partial_{t}n(t))=E(u_{0},u_{1},n_{0},n_{1})$ ,$t\in[-T,T]$.

(9)

the solutions as above depend continuously on the initial data in the

topology of (7) and (8) on the time interval $[-T’,T’]$ for $0<T’<T$

Corollary 6$\dot{H}^{-1}$

Assume that $0<c<1$ holds. Let. Assume that $1/2<b<1$ and$\eta>0$$b$$(u_{0},u_{1}, n_{0},n_{1})\in H^{1}\oplus L^{2}\oplus L^{2}\oplus$

is close enough to 1/2. Then there

such that if$||u$

0 $||H1+||u1||_{L^{2}}+||n0||_{L^{2}}+||n1||_{\dot{H}^{-1}}$$\leq\eta$,the solutions $(u, n)$ piven by Theorem 5 extends globally in time.

Remark 3.
(i) We note that$\dot{H}^{-1}\subset H^{-1}$.

$e$$\dot{m}t\mathit{8}$an44Since the energy functional$n_{1}\in\dot{H}^{-1}$$E$defined in (6) includes the$\dot{H}^{-1}$

norm of$\partial_{t}n$, we needfor the proof of Corollary 6.

(ii) Corollary 6 is an immediate consequence of Theorem 5 and the Sobolev embedding

theorem. In fact, we can obtain the a priori estimate of$(u,\partial_{t}u,n,\partial_{t}n)\in H^{1}\oplus L^{2}\oplus$

om the energy identity (9) and the Sobolev embedding theorem for small
initial data. This leads to the global existence result. The constant in Corollary 6$L^{2}\oplus\dot{H}^{-1}$$\mathrm{f}$$\eta$

depends only on and the best constant relevant to the Sobolev embedding$c$but not on$b$$H^{1}arrow\neq L^{4}$,.

Outline of Proof of Theorem 5.
We suppose that$(u_{0},u_{1},n_{0},n_{1})\in H^{1}\oplus L^{2}\oplus L^{2}\oplus H^{-1}$.

We first put,$u_{\pm}=u\pm i\omega^{-1}\mathit{8}gu$$n_{\pm}=n\pm i(c\omega)^{-1}\mathit{8}gn$

where$\omega$$=(1-\Delta)^{1/2}$,. Then $(1)-(3)$ are rewritten as follows:$(i\partial_{t}\mp\omega)u_{\pm}=\pm(4\omega)^{-1}(n_{+}+n_{-})(u_{+}+u_{-})$$(i\partial_{t}\mathrm{F}w)n_{\pm}=\pm(4c)^{-1})0\mathrm{c}\mathrm{u}$$-1|$$u\pm(0)=u\pm 0$,$n\pm(0)=n\pm$,

tt $++u_{-}|^{2}\mp c(2\omega)^{-1}(n_{+}+n_{-})$ ,

(10)
(11)

(h(12)

where$u_{\pm 0}=u_{0}\pm i\omega^{-1}u_{1}$,$n_{\pm 0}=n_{0}\pm i(c\omega)^{-1_{\mathrm{t}7_{1}}}$.

We note that$(u_{\pm 0}, n_{\pm 0})\in H^{1}\oplus L^{2}$.

We try to solve (10)-(12) locally in time. For that purpose, we consider the following
integral equations associated with (10)-(12):$u_{\pm}(t)$$=\varphi_{T}(t)W_{\pm}(t)u\pm 0$

45\mbox{\boldmath $\varphi$}T(t)$\mp i(4\omega)^{-1}$t$\int_{0}$&-(s))ds,($W_{\pm}(t-s)(n_{+}(s)+n_{-}(s))$ $u_{+}(s)+u$$t\in$$n\pm(t)$

$=$

(13)

$\varphi_{T}(t)W_{c\pm}(t)n_{\pm 0}$$\mathrm{F}i\varphi\tau(t)$$)7^{t}W_{\mathrm{c}\pm}(t-s)[(4c)^{-1}cJ\omega)$ $-1|\mathrm{i}\mathrm{J}+(S)$ $+$ $u_{-}(s)|^{2}$$-c(2\omega)$ $-1(n_{+}(s)+n_{-}(s))]$

where

R,

$W_{\pm}(t)=e^{\mp it\omega}$,$W_{\epsilon\pm}(t)=e^{\mp\cdot \mathrm{c}t\omega}.$

the process of the proof and,$T$$t\in$(14)

R,

is a positive constant to be chosen small in

is a function in$\varphi_{T}$ds,$C^{\infty}(\mathrm{R})$such that$\varphi_{T}(t)=0$

for $|t|\geq 2T$

and $\varphi\tau(t)=1$ for $|t|\leq Tr$ We note that the solutions of (13)-(14) in a suitable class isa solution of (10)-(12) on the time interval$[-T, \mathrm{I}]$.

We use the Fourier restriction norm method to show the well-posedness of (13)-(14)for small $T>0.$ For the scheme of the Fourier restriction norm method, see Bourgain$[2, 3]$(see also Kenig, Ponce and Vega [9] and Ginibre, Tsutsumi and Velo [7]).

If we try to apply the Fourier restriction norm method to (13)-(14), we have only
to prove the following proposition:

Proposition 7$b$

There exist two positive constants$a_{0}$

and$b_{0}$

such that for

$a$

and

with $a_{0}\leq a<1/2<b\leq b_{0}$ ,$|\langle v, - \rangle|\leq C||v||_{X_{a}}||w||_{\mathrm{Y}_{b}}||\omega u||X_{b}$

$|$

where

$X_{b}$

and$\mathrm{Y}_{b}$

(

$w$

,

$uv\rangle|\leq C||w||_{\mathrm{Y}_{a}}||\omega u||_{X_{b}}||v||_{X_{b}}$denote either of$\langle$,$X_{b,0}^{\pm}$and either of$\mathrm{Y}_{b,0}^{\pm}$

(15)

,

(16)

, respectively, and$f,g$ $)= \int_{R^{4}}f(t,x)\overline{g(t,x)}$ dtdx.

Remark 4.
The duality argument with (15)-(16) implies that
$||$

(17)

’Lt $||X-a\leq C||n||_{\mathrm{Y}_{b}}||\omega u||_{X_{b}}$ ,(18)$||\omega|u|^{2}||_{\mathrm{Y}_{-a}}\leq C||\omega u||_{X_{b}}||’ u||_{X_{\acute{b}}}$

for
and

$a$

and

$\mathrm{Y}_{b}$$b$with $a_{0}\leq a<1/2<b\leq b_{0}$ , where

denotes either of$\mathrm{Y}_{b,0}^{\pm}$$X_{b}$

and$X_{b}’$

denote either of$X_{b,0}^{\pm}$. Estimates (17) and (18) enable us to apply the Fourier

restriction norm method to (13)-(14). See [16] for the proof of Proposition 7.

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