数理解析研究所講究録
第 898 巻 1995 年 94-103

94

Normal form and global solutions
for the Klein-Gordon-Zakharov equations
Tohru Ozawa
Kiimtoshi Tsutaya
Yoshio Tsutsumi(/re ffi )$\rfloor\backslash$$(F$ ffl$($fff

Hokkaido University
Hokkaido University
University of Tokyo$’$$\grave{6}^{\prime\backslash \ovalbox{\tt\small REJECT}_{\mathfrak{l}j)}}L$$a_{J}\pm\Xi C_{4}\backslash \mathbb{R})$Introduction and main results

1We consider the Cauchy problem of the Klein-Gordon-Zakharov

equations in three space dimensions:
,

(1.1)

$\partial_{t}^{2}c\iota-\triangle u+u=-nu$(1.2)$\partial_{\iota}^{2}n-\triangle n=\triangle|u|^{2}$(1.3)$u(0, x)=u_{0}(x),$$\partial_{t}u(0, x)=u_{1}(x)$,$n(0, x)=n_{0}(x),$$\partial_{t}n(0, x)=n_{1}(x)$,,$t>0,$$x\in R^{3}$,$t>0,$ $x\in R^{3}$ ,, and $u(t, x)$ and $n(t, x)$ are functions fromwhere$\partial_{t}=\partial/\partial t$

toand from $R+\cross R^{3}$ to $R$ , respectively. The system $(1.1)-(1.2)$$C^{3}$$R_{+}\cross R^{3}$

describes the propagation of strong turbulence of the Langmuir wave

in a high frequency plasma (see [15]).
The usual Zakharov system
,

(1.4)

$i\partial_{t}u+\triangle u=nu$(1.5)$\partial_{t}^{2}n-\triangle n=\triangle|u|^{2}$$t>0$ ,,

$t>0$ ,

$x\in R^{3}$,$x\in R^{3}$is derived from $(1.1)-(1.2)$ through the physical approximation proce-dure.

95

In the present paper we consider solving $(1.1)-(1.3)$ around the zerosolutions. There are many papers concerning the global existence ofsmall solutions for the coupled systems of the Klein-Gordon and wave
equations with quadratic nonlinearity (see, e.g., [1], $[5]-[7],$ $[9],$ $[10]$ ,[12] and [13] . The methods to solve those systems can be classified$)$into two groups (for a good review of this matter, see Strauss [14]).
One is to use the Sobolev space with weight related to the genera-tors of the Lorentz group. This was developed by Klainerman [9] and[10]. The combination of this method and the null condition technique

has produced several nice applications to the hyperbolic systems ofphysical importance (see, e.g., Bachelot [1] and Georgiev [6]). However, this method does not seem to be directly applicable to $(1.1)-(1.3)$ .
In fact, since the system $(1.1)-(1.2)$ consists of the Klein-Gordon andwave equations with quadratic nonlinearity in three space dimensions,
we need to use not only the Sobolev norms with weights related to thegenerators of the Lorentz group but also the null condition technique(see, e.g., Georgiev [5] and [6]), while the nonlinear terms in (1.1) and(1.2) do not seem to satisfy the null condition as they are. Anothermethod is based on the theory of normal forms introduced by Shatah[12], which is an extension of Poincar\’e’s theory of normal forms to thepartial differential equations. In [16] the authors have applied the ar-gument of normal form to $(1.1)-(1.2)$ and proved the global existenceof solutions to $(1.1)-(1.3)$ for small initial data. In [16] the authors havealso shown that these global solutions to $(1.1)-(1.3)$ with small initial
data approach the free solutions asymptotically as $tarrow+\infty$ . In thisnote we briefly describe the results obtained in [16].
Before we state the main results in this paper, we give several no-tations. For $1\leq p\leq\infty$ and a nonnegative integer $m$ , letdenote the standard$L^{p}$and Sobolev spaces on$R^{3}$$L^{p}$

and$W^{m_{l}p}$, respectively. We

96

put $H^{m}=W^{m_{1}2}$ . For $m\in R$ , we let$\omega=(1-\triangle)^{1/2}$

and$\omega_{0}=(-\triangle)^{1/2}$$\dot{H}^{m}=(-\triangle)^{-m/2}L^{2}$. We put.

We have the following theorem concerning the global existence andasymptotic behavior of solutions to $(1.1)-(1.3)$ for small initial data.

Theorem 1.1$W^{15,6/(5+6\epsilon)},$

Let $0<\epsilon\leq 10^{-4}$ . Assume that $u_{0}\in H^{25}\cap$$u_{1}\in H^{24}\cap W^{14_{J}6/(5+6\epsilon)},$$n_{1}\in H^{23}\cap W^{13,28/27}\cap\dot{H}^{-2}$(1.6)$n_{0}\in H^{24}\cap W^{1428/27})\cap$. Then, there exists a$\delta>0$

fi$-1$

andsuch that if$||u_{0}||_{H^{25}\cap W^{15,6/(5+6e)}}+||u_{1}||_{H^{24}\cap W^{14,6/(5+6e)}}$$+||n_{0}||_{H^{24}\cap W^{14,28}/2\tau_{\cap\dot{H}^{-1}}}+||n_{1}||_{H^{23}\cap W^{13,28}/2\tau_{\cap\dot{H}^{-2}}}$$\leq$$\delta$,$(1.1)-(1.3)$ have the unique global soluiions $(u, n)$ satisfying

,

(1.7)

$u \in\bigcap_{j=0}^{2}C^{j}([0, \infty);H^{25-J}.)$(1.8)$n \in[\bigcap_{j=0}^{2}C^{j}([0, \infty);H^{24-j})]\cap[\bigcap_{j=0}^{1}C^{j}([0, \infty);\dot{H}^{-1-j})]$(1.9)$\sum_{j=0}^{1}||\theta iu(t)||_{W^{13-j_{l}\text{\’{e}}/(1-6\epsilon)}}=O(t^{-(1+3\epsilon)})$(1.10)$\sum_{j=0}^{1}||\partial in(t)||_{W^{12-j_{t}2\epsilon}}=O(t^{-13/14})$

where$\delta$depends only on .
$\epsilon$$Furthermore_{f}$of $(1,1)-(1.3)$ have the asymptotic sates$H^{11},$$n_{+1}\in H^{10}$(1.11)$(tarrow\infty)$$(tarrow\infty)$,,,

the above solutions $(u, n)$$u_{+0}\in H^{12},$ $u_{+1}\in H^{11},$ $n_{+0}\in$such that$\sum_{j=0}^{1}||\theta_{t}^{l}(u(t)-u_{+}(t))||_{H^{12-j}}$$+ \sum_{j=0}^{1}||y_{t}(n(t)-n_{+}(t))||_{H^{11-j}}arrow 0$$(tarrow\infty)$,

97

where,$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(1) In three space dimensions,

the Schwartz space onspace(2)$\dot{H}^{m}$$u_{+}(t)$$R^{3}$

but$S\subseteq\dot{H}^{-1}$$S\subseteq\dot{H}^{-2}$, where$S$is. For the details of the homogeneous Sobolev, see [2, \S 6.3 in Chapter 6].
and$n_{+}(t)$are the solutions of the free Klein-Gordon equationand the free wave equation with the initial conditions$=(u_{+0}, u_{+1})$$(u_{+}(0), \partial_{t}u_{+}(0))$

and $(n_{+}(0), \partial_{t}n_{+}(0))=(n_{+0}, n_{+1})$ , respectively. The rela-tion (1.11) implies that the solutions of $(1.1)-(1.3)$ given by Theorem

1.1 behave like the free solutions as$tarrow\infty$.

(3) In connection with the usual Zakharov system $(1.4)-(1.5)$ for threespace dimensions, it is conjectured that if the initial data are large, thesolutions of $(1.1)-(1.3)$ may not necessarily exist globally in time.
(4) 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 not need the time decay estimates toshow the global existence of solutions in the one and two dimensional
cases.
The following corollary is an immediate consequence of Theorem
1.1.

Corollary 1.2

Let $0<\epsilon\leq 10^{-4}$ and let $m$ be a$posit\dot{i}ve$

integerwith $m\geq 25$ . Assume that $u_{0}\in H^{m}\cap W^{15_{J}6/(5+6\epsilon)},$ $u_{1}\in H^{m-1}\cap$$W^{14_{2}6/(5+6\epsilon)},$and$n_{0}\in H^{m-1}\cap W^{14_{r}28/27}\cap\dot{H}^{-1},$$(u_{0}, u_{1}, n_{0}, n_{1})$Theorem 1.1 satisfy

satisfy$(1,\theta)$ ,$n_{1}\in H^{m-2}\cap W^{13,28/27_{\cap\dot{H}^{-2}}}$

Then, the solutions $(u, n)$ given by

98,(1.12)$u \in\bigcap_{j=0}^{m}C^{j}([0, \infty);H^{m-j})$(1.13)$n\in m1\overline{\bigcap_{j=0}}C^{j}([0, \infty);H^{m-1-j})$

In addition. if$f$(1.14)$u_{0},$$u_{1},$$n_{0},$$u(t, x),$$n_{1} \in\bigcap_{j=1}^{\infty}H^{j}$.

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

The unique existence and regularity of local solutions for $(1.1)-(1.3)$

follows from the standard iteration argument. The crucial part of proofsof Theorem 1.1 and Corollary 1.2 is to establish the a priori estimatesof the solutions for $(1.1)-(1.3)$ in order to extend the local solutions

globffiy in time. The global behavior of local solutions for $(1.1)-(1.3)$
can not be controlled directly, since the quadratic nonlinear term in(1.1) does not provide a sufficient decay property for the three dimen-sional case. Here we use the argument of normal forms of Shatah [12]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 kernel (see $(2.4)-(2.7)$ inSection 2). The singularity of the integral kernel makes it difficult tosolve $(1.1)-(1.3)$ . This is different form the case of the system contain-ing only the Klein-Gordon equations, where the integral kernels of theresulting integral operators are regular (see [12]). Therefore, our maintask in the proof of Theorem 1.1 is to evaluate the singularity of theintegral kernel of the transformed cubic nonlinearity. This enables us
to apply the usual $L^{p}-L^{q}$ estimate to the transformed system, which
provides us with the sufficient decay properties of solutions to $(1.1)-$(1.3) for the proof of Theorem 1.1. In Section 2 we apply the method

99of normal forms to our problem in order to transform the quadratic

nonlinearity into the cubic one. Detailed proof of Theorem 1.1 will begiven somewhere.

Normal form

2In this section we show that the transformation exists for the system$(1.1)-(1.2)$ .

We write $(1.1)-(1.2)$ as a first order system(2.1)$\frac{dU}{dt}=AU+F(U)$ ,where$U=(\begin{array}{l}un\partial_{t}u\partial_{t}n\end{array})$,$A=( \Delta\frac{00}{0}1$$\triangle 000$

$0001$$0001$,$F(U)=(\begin{array}{l}00-nu\triangle|u|^{2}\end{array})$

We consider the transformation with the following form:$(\begin{array}{l}vm\partial_{t}v\partial_{t}m\end{array})=V=U-(\begin{array}{l}[U,B_{1},U][U,B_{2},U][U,B_{3},U][U,B_{4},U]\end{array})$

where $B_{j}(j=1, \cdots, 4)$ are(2.2)$4\cross 4$,

matrices and$[U, B_{j}, U]$$=$.

$\int_{R^{3}xR^{3}}(\overline{u}, n,\overline{\partial_{t}u}, \partial_{t}n)(y)B_{j}(x-y, x-z)(\begin{array}{l}un\partial_{t}u\partial_{t}n\end{array})(z)dydz$

Let us consider the case $j=1$ . We put$B_{1}=(\begin{array}{llll}0 0 0 0G_{1} 0 0 00 0 0 00 0 G_{2} 0\end{array})$Here$G_{1}$

and$G_{2}$

are to be thought of as distributions and the integral

representation (2.2) is interpreted in an appropriate way. Then, we100have$=$$v$$u-[n, G_{1}, u]-[\partial_{t}n, G_{2}, \partial_{t}u]$$=u- \sum_{j=1}^{3}\int_{R^{3}xR^{3}}n(y)G_{1}(x-y, x-z)u_{j}(z)dydz$$- \sum_{j=1}^{3}\int_{R^{3}\cross R^{3}}\partial_{t}n(y)G_{2}(x-y, x-z)\partial_{t}u_{j}(z)dydz$

,

where
$u=(\begin{array}{l}u_{1}u_{2}u_{3}\end{array})$

We now compute$\partial_{t}v$$=$.

$\partial_{t}^{2}v=\partial_{t}^{2}(u-[n, G_{1}, u]-[\partial_{t}n, G_{2}, \partial_{t}u])$:$\partial_{t}u-[\partial_{t}n, G_{1}, u]-[n, G_{1}, \partial_{t}u]$$-[\partial_{t}^{2}n, G_{2}, \partial_{t}u]-[\partial_{t}n, G_{2}, \partial_{t}^{2}u]$$=$$\partial_{t}u-[\partial_{t}n, G_{1}, u]-[n, G_{1}, \partial_{t}u]$$-[\triangle n+\triangle|u|^{2}, G_{2}, \partial_{t}u]-[\partial_{t}n, G_{2}, \triangle u-u+nu]$$\partial_{t}^{2}v$$=$,$\partial_{t}^{2}u-[\triangle n, G_{1}, u]-2[\partial_{t}n, G_{1}, \partial_{t}u]$$-[n, G_{1}, \triangle u-u]-[\triangle\partial_{t}n, G_{2}, \partial_{t}u]-2[\triangle n, G_{2}, \triangle u-u]$$-[\partial_{t}n, G_{2}, \triangle\partial_{t}u-\partial_{t}u]+$(cubic terms).

Moreover,$(-\triangle+1)v$$=$$- \triangle u+u+[\triangle n, G_{1}, u]+\sum_{j=1}^{3}2[\partial_{j}n, G_{1}, \partial_{j}u]$$+[n, G_{1}, \triangle u]-[n, G_{1}, u]+[\triangle\partial_{t}n, G_{2}, \partial_{t}u]$$+ \sum_{j=1}^{3}2[\partial_{j}\partial_{t}n, G_{2}, \partial_{j}\partial_{t}u]+[\partial_{t}n, G_{2}, \triangle\partial_{t}u]$$-[\partial_{t}n, G_{2}, \partial_{t}u]$.

Therefore(2.3)$\partial_{t}^{2}v-\triangle v+v$$=$$-nu-2[\triangle n, G_{2}, \triangle u]+2[\Delta n, G_{2}, u]$$+ \sum_{j}2[\partial_{j}n, G_{1}, \partial_{j}u]-2[\partial_{t}n, G_{1}, \partial_{t}u]$$+ \sum_{j}2[\partial_{j}\partial_{t}n, G_{2}, \partial_{j}\partial_{t}u]+$(cubic terms).

101

We choose the distributions$G_{1}$

and$G_{2}$

so that all quadratic terms in(2.3) cancel out:
(2.4) $-nu-2[ \triangle n, G_{2}, \Delta u]+2[\triangle n, G_{2}, u]+\sum 2[\partial_{j}n, G_{1}, \partial_{j}u]$.

$-2[ \partial_{t}n, G_{1}, \partial_{t}u]+\sum_{j}2[\partial_{j}\partial_{t}n, G_{2}, \partial_{j}\partial_{t}u]J=0$

Here we define the Fourier transform of$G_{j}$

by.

$\hat{G_{j}}(p, q)=\int_{R^{3}xR^{3}}e^{-i(p\cdot y+q\cdot z)}G_{j}(y, z)dydz$

Then equation (2.4) becomes$-1-2|p|^{2}|q|^{2}\hat{G_{2}}(p, q)-2|p|^{2}\hat{G_{2}}(p, q)-2p\cdot q\hat{G_{1}}(p, q)$$=$$0$,$-\overline{G_{1}}(p, q)-p\cdot q\hat{G_{2}}(p, q)$

$=$

$0$

.

Thus, we obtain(2.5)$\hat{G_{1}}(p, q)$

$=$

(2.6)

$\hat{G_{2}}(p, q)$$=$$\frac{p\cdot q}{2\{|p|^{2}|q|^{2}-(p\cdot q)^{2}+|p|^{2}\}}$$-1$$2\{|p|^{2}|q|^{2}-(p\cdot q)^{2}+|p|^{2}\}$,.

We next consider the case $j=2$ . Similarly we put$m=n-[u, H_{1}, u]-[\partial_{t}u, H_{2}, \partial_{t}u]$.

As before, we obtain(2.7)$\hat{H_{1}}(p, q)$$=$$\frac{(p\cdot q-1)|p+q|^{2}}{2\{|p|^{2}|q|^{2}-(p\cdot q)^{2}+|p+q|^{2}\}}$

,

(2.8)

$\hat{H_{2}}(p, q)$$=$$\frac{-|p+q|^{2}}{2\{|p|^{2}|q|^{2}-(p\cdot q)^{2}+|p+q|^{2}\}}$.

We have thus completed the construction of the normal form, as re-quired.

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