数理解析研究所講究録
第 824 巻 1993 年 11-18

11Global Existence and Asymptotic Behavior of Solutions
for the Zakharov Equations

TOHRU OZAWA\dagger AND YOSHIO TSUTSUMI\ddagger(堤 誉志雄 東大・数理)(小澤 徹 北大・理)\dagger Department of Mathematics
Hokkaido University
Sapporo 060, Japan\ddagger Department of Mathematical Sciences
University of Tokyo

Hongo, Tokyo 113, Japan

In the present paper we consider the global existence and asymptotic behavior in timeof solutions for the Zakharov equations:(1)$\dot{\iota}\frac{\partial}{\partial t}E+\triangle E=nE$(2)$\frac{\partial^{2}}{\partial t^{2}}n-\triangle n=\triangle|E|^{2}$(3)$E(O, x)=E_{0}(x)$ ,,$t>0$ ,,$x\in R^{N}$$t>0$ ,,$x\in R^{N}$$n(O, x)=n_{0}(x)$ ,where $E$ is a function from $R^{+}\cross R^{N}$ to,$\frac{\partial}{\partial t}n(0, x)=n_{1}(x)$,$x\in R^{N}$,is a function from $R^{+}\cross R^{N}$ to $R$ and$1\leq N\leq 3$ . Equations (1) $-(2)$ describe the propagation of Langmuir turbulence in an
unmagnetized, completely ionized hydrogen plasma (see Zakharov [20]). $E(t, x)$ is theslowly varying complex amplitude of the electric field of the Langmuir wave with plasma$C^{N},$$n$$\mathcal{E}$frequency$\omega_{p}>0$:$\mathcal{E}(t, x)=\Re(E(t, x)\exp(-it\omega_{p}))$.

$n(t, x)$ is the deviation of the ion density from its equilibrium.

The right hand side of (1)represents the shift of plasmon frequency caused by the slow density variation $n(t, x)$ , and12
the right hand side of (2) represents the driving force caused by the pressure of plasmon

gas.
There are many papers concerning the global existence in time of solutions for (1) $-(3)$ .
In [17] C. Sulem and P.L. Sulem proved by using the Glerkin method that if $N=1$ and$(E_{0}, n_{0}, n_{1})\in H^{m}\oplus H^{m-1}\oplus(H^{m-2}\cap\dot{H}^{-1}),$ $m\geq 3$, then (1) $-(3)$ have the unique global

solutions $(E, n)\in L^{\infty}(O, \infty;H^{m})\oplus L^{\infty}$ ( , oo; $H^{m-1}$ ) and that if
norm and thenorm ofare small for $N=2$ and $N=3$ , respectively,and thethen (1) $-(3)$ have the global weak solutions $(E, n)\in L^{\infty}(O, \infty;H^{1})\oplus L^{\infty}(0, \infty;L^{2})$ for$(E_{0}, n_{0}, n_{1})\in H^{1}\oplus L^{2}\oplus$$0$$\dot{H}^{-1}$$1\leq N\leq 3$$H^{1}$$L^{2}$$E_{0}$. But the uniqueness of weak solutions in$H^{1}\oplus L^{2}$is not yet known. Heredenotes the standard Sobolev space $H^{m}(R^{N}).\dot{H}^{m}$ denotes the homogeneous Sobolev, where isspace consisting of all tempered distributions withthe Fourier transform of . In [15] Schochet and Weinstein showed a similar result by the$H^{m}$$u$$|\xi|^{m}\hat{u}\in L^{2}\equiv L^{2}(R^{N})$$\hat{u}$$u$different method. When $N=2$ , H. Added and S. Added [1] improved the global existence
results due to C. Sulem and P.L. Sulem [17] and showed that if $N=2$ , $(E_{0}, n_{0}, n_{1})\in$

is small, then (1) $-(3)$ have thenorm ofunique global solutions $(E, n)\in H^{m}\oplus H^{m-1}$ . Recently in [11] the authors have improved
the local existence and regularity results due to C. Sulem and P.L. Sulem [17] and Schochetand Weinstein [15] and have brought down the lower bound $m=3$ of regularity of thesolutions to $m=2$ concerning the unique local existence of solutions for (1) $-(3)$ . However,the uniqueness of the weak solutions’ infor $1\leq N\leq 3$ and the global existenceof strong solutions for $N=3$ are still open. On the other hand, it is conjectured that for$N=2,3$ , there exist solutions blowing up in finite time (see [20]). In the present paper,$H^{m}\oplus H^{m-1}\oplus(H^{m-2}\cap\dot{H}^{-1}),$ $m\geq 3$

and the$L^{2}$$E_{0}$$H^{1}\oplus L^{2}$for $N=3$ we consider solving (1) $-(2)$ with the final data given at $t=+\infty$ instead of theinitial value problem (1) $-(3)$ . This leads to the construction of the wave operator.
The difficulty of constructing the global solutions of (1) $-(2)$ consists in the quadratic nonlinearity of (1) $-(2)$ . In [9] Klainerman introduced the notion of the null condition to show
the global existence of small amplitude solutions for the wave equation with quadratic nonlinearity in three space dimensions. Recently, Bachelot [3] and Georgiev [6] have improved
the null condition technique to show the global existence of small amplitude solutions forthe Dirac-Klein-Gordon equations and the Maxwell-Dirac equations, respectively. However, the null condition technique does not seem to be directly applicable to (1) $-(3)$ , because
the null condition thechnique is based on the Lorentz invariance of the equations. But theSchr\"odinger equation does not necessarily have the same invariance as the wave equation,

13and especially the Schr\"odinger equation is not invariant under the Lorentz transformation.

On the other hand, in [5] Flato, Simon and Taflin study the global existence and as ymptotic behavior of solutions for the Maxwell-Dirac equations with the final data given at$t=+\infty$ . This corresponds to the construction of the wave operator, more precisely, themodified wave operator.
In our problem, the wave operator $W+is$ difined as follows. For the free solutionsof the Schr\"odinger and wave equations, we find the solutions $(E(t),n(t))$ of(1) $-(2)$ such that $(E, n)$ exist on $[0, +\infty$ ) and satisfy$(E_{+}(t),n+(t))$(4)$\Vert E(t)-E_{+}(t)\Vert_{L^{2}}+\Vert\nabla n(t)-\nabla n+(t)\Vert_{L^{2}}$$(tarrow+\infty)$$+ \Vert\frac{\partial}{\partial t}n(t)-\frac{\partial}{\partial t}n+(t)\Vert_{L^{2}}arrow 0$.

Then, we consider the wave operator $W+as$ a mapping from the scattered states $(E_{+}(0)$ ,. However, all the solutionsof (1) $-(2)$ may not exist globally in time and so we also consider the pseudo wave operator with the reference time $t=0$ of the interacting states replaced by some $T>0$ .
That is, the pseudo wave operator
defined as a mapping from the scattered states$(E_{+}(0), n_{+}(0),$for some $T>0$ ,to the interacting states $(E(T), n(T),$where $(E, n)$ exist on $[T, +\infty$ ) and satisfy (4), and may change for each scattered states.
In the present paper, we announce that when $N=3$ , we can prove the existence of thewave operatorof (1) $-(2)$ for small scattered data and the existence of the pseudo wave
operator $\overline{W}+of(1)-(2)$ for the scattered data (not necessarily small) with the support ofincluded in the unit ball centered at the origin. Accordingly, it can be also shown(1) $-(3)$ have thethat for the initial data belonging to the range of $W+or$ the range ofunique global solutions.
$n_{+}(0),$ $\frac{\partial}{\partial t}n+(0))$

to the interacting states $(E(O), n(O),$$\frac{\partial}{\partial t}n(0))$$\overline{W}+is$$\frac{\partial}{\partial t}n(T))$$\frac{\partial}{\partial t}n_{+}(0))$$T$$W_{+}$$\hat{E}_{+}(0)$$\overline{W}+,$and let $U(t)=$Before we state the theorems we define several notations. Le
be the evolution operator of the free Schr\"odinger equation. For nonnegative integers$m$ and, we define $H^{m,s}$ as follows:$\omega=\sqrt{-\triangle}$$e^{\frac{:}{2}t\Delta}$$s$$H^{m,s}=\{v\in S’(R^{N});\Vert(1+|x|^{2})^{\frac{s}{2}}(1-\triangle)^{\frac{m}{2}}v\Vert_{L^{2}}<+\infty\}$with the norms$\Vert v\Vert_{H^{m,*}}=\Vert(1+|x|^{2})^{\frac{s}{2}}(1-\triangle)^{\frac{m}{2}}v\Vert_{L^{2}}$.

14
For a multi-index$\alpha=(\alpha_{1}, \cdots, \alpha_{N})$with nonnegative integers$|\alpha|=\alpha_{1}+\cdots+\alpha_{N}$$\alpha_{j}$, we put,$( \frac{\partial}{\partial x})^{\alpha}=(\frac{\partial}{\partial x_{1}})^{\alpha_{1}}\cdots(\frac{\partial}{\partial x_{1}})^{\alpha_{N}}$.

For $p\geq 1$ and a nonnegative integer , we let$k$$W^{k,p}= \{u\in L^{p}(R^{N});(\frac{\partial}{\partial x})^{\alpha}u\in L^{p}(R^{N}), |\alpha|\leq k\}$with the norm$\Vert u\Vert_{W^{k,p}}=\sum_{|\alpha|\leq k}\Vert(\frac{\partial}{\partial x})^{\alpha}u\Vert_{L^{p}}$.

We now state the theorems.
THEOREM 1. Let $N=3$ . Assume that $E+0\in H^{6,7}$ . We put $E_{+}(t)=U(t)E+0$ . Assumethat $n+0\in H^{5,2}$ and $n+1\in H^{4,2}$ . We put $n_{+}(t)=(\cos\omega t)n+0+(\omega^{-1}\sin\omega t)n_{+1}$ . Then,
there exists $\eta>0$ such that if(5)$\Vert E+0\Vert_{H^{6,7}}+\Vert n+0\Vert_{H^{5,2}}+\Vert n_{+1}\Vert_{H^{4,2}}\leq\eta$,(1) $-(2)$ have th unique solutions $(E(t), n(t))$ sa tisfying
$e$

(6)

$E(t) \in\bigcap_{j=0}^{1}C^{\dot{J}}([0, \infty);H^{3-2}$(7)$n(t) \in\bigcap_{j=0}^{2}C^{j}([0, \infty);H^{2-j})$(8)$\Vert E(t)-E_{+}(t)||_{H^{3}}+\Vert n(t)-n_{+}(t)\Vert_{H^{2}}$り,,$\frac{\partial}{\partial t}n(t)\in C([0, \infty);\dot{H}^{-1})$$+ \Vert\frac{\partial}{\partial t}n(t)-\frac{\partial}{\partial t}n_{+}(t)\Vert_{H^{1}\cap\dot{H}^{-1}}=O(t^{-1/2})$(9)$( \int^{+\infty}\Vert E(s)-E_{+}(s)||_{W^{24}}^{8/3},ds)^{3/8}$,$(tarrow+\infty)$,

15$+( \int^{+\infty}\Vert n(s)-n+(s)\Vert_{W}^{8/3_{14}},ds)^{3/8}=O(t^{-1/2})$$(tarrow+\infty)$,. Therefor,andbelong tounderRemark 1. (i) For $N=3,$
the assumptions in Theorem 1.
(ii) Theorem 1 implies that for any scattered data satisfying the assumptions in Theorem 1, there exist the solutions of (1) $-(2)$ approaching asymptotically the free solutionsprescribed by those scattered data as $tarrow+\infty$ . This is quite different from the case of theMaxwell-Schr\"odinger equations, although (1) $-(2)$ are the system of the Schr\"odinger equation and the wave equation with quadratic nonlinear coupling like the Maxwell-Schr\"odingersystem. In the case of the Maxwell-Schr\"odinger equations, the solutions do not approach
the free solutions in the sense of (8) (see [18]).
$H^{1,1}\subset\dot{H}^{-1}$

THEOREM 2. Let $N=3,$ $\delta>0$ and$\dot{H}^{-1}$$n_{+1}$$n_{+0}$$0< \epsilon<\frac{1}{41}$Assume that $E_{+0}\in H^{6,7}$ and$supp\hat{E}_{+0}\subset$. We. We put $E_{+}(t)=U(t)E_{+0}$ . Assume that $n+0\in H^{5,4}$ and
put $n_{+}(t)=(\cos\omega t)n+0+(\omega^{-1}\sin\omega t)n+1$ . Then, there exists a $T>0$ such that (1) $-(2)$$n_{+1}\in H^{4,4}$$\{\xi;|\xi|\leq 1-\delta\}$$h$

ave the unique solutions $(E(t), n(t))$ satisfyin$g$,(10)$E(t) \in\bigcap_{j=0}^{1}C^{j}([T, \infty);H^{3-2j})$(11)$n(t) \in\bigcap_{j=0}^{2}C^{j}([T, \infty);H^{2-j})$(12)$\Vert E(t)-E_{+}(t)\Vert_{L^{2}}=O(t^{-2+2\epsilon})$(13)$\Vert E(t)-E_{+}(t)\Vert_{H^{1}}+\Vert n(t)-n_{+}(t)\Vert_{L^{2}}$,,$\frac{\partial}{\partial t}n(t)\in C([T, \infty);\dot{H}^{-1})$$(tarrow+\infty)$,$+ \Vert\frac{\partial}{\partial t}n(t)-\frac{\partial}{\partial t}n_{+}(t)\Vert_{\dot{H}^{-1}}=O(t^{-3/2+\epsilon})$(14)$\Vert E(t)-E_{+}(t)\Vert_{H^{2}}+\Vert n(t)-n+(t)\Vert_{H^{1}}$$+ \Vert\frac{\partial}{\partial t}n(t)-\frac{\partial}{\partial t}n_{\dagger}(t)\Vert_{L^{2}}=O(t^{-2+3\epsilon})$(15)$(tarrow+\infty)$$(tarrow+\infty)$,$\Vert E(t)-E_{+}(t)\Vert_{H^{3}}+\Vert n(t)-n_{+}(t)\Vert_{H^{2}}$$+ \Vert\frac{\partial}{\partial t}n(t)-\frac{\partial}{\partial t}n+(t)\Vert_{H^{1}}=O(t^{-2+4\epsilon})$$(tarrow+\infty)$,,

16
where$T$

depends only on$\delta,$$\epsilon,$$\Vert E_{+0}\Vert_{H^{6,7}},$$\Vert n_{+0}\Vert_{H^{5,4}}$and$\Vert n_{+1}\Vert_{H^{4,4}}$.

is indispensable toRemark 2. (i) The support condition on the Fourier transform of
the proof of Theorem 2. However, we can replace this support condition by the following:$E_{+0}$$supp\hat{E}+0\subset\{\xi;|\xi|\leq 1-\delta\}\cup\{\xi;|\xi|\geq 1+\delta\}$(ii) The support condition on the Fourier transform of$E_{+0}$seems to have some physicalmeaning. This condition implies that most part of the Schr\"odinger wave propagates atspeed less than one and so the ion sound speed is faster than that of the Schr\"odinger wave.
This situation seems important from a physical point of view (see Zakharov [20], H. Addedand S. Added [2], Shochet and Weinstein [15] and Ozawa and Tsutsumi [13]).
The following corollary is an immediate consequence of Theorems 1 and 2.

COROLLARY 3. $AssumeN=3$ .
(i) By $D+we$ denote the set of all scattered states $(E, n, n)$ satistying (5). Then,is well definedfor (1) $-(2)$ the wave operator $W+:(En, n.)\mapsto(E(0), n(O),$on(ii) Let $\delta>0$ . By
denote the set of all scattered stat es $(E_{+0}, n_{+0}, n_{+1})\in$
such that $supp\hat{E}+0\subset\{\xi;|\xi|\leq 1-\delta\}$ . Then, for (1) $-(2)$ the pseudo$\frac{\partial}{\partial t}n(0))$$D+\cdot$$\tilde{D}+we$$H^{6,7}\oplus H^{5,4}\oplus H^{4,4}$

wave operator $\overline{W}+:(E_{+0}, n_{+0}, n_{+1})rightarrow(E(T), n(T),$$\frac{\partial}{\partial t}n(T))$

is well defined on$\tilde{D}+\cdot$

belonging to the range of $W+or$ th range ofthere exist the uniq $ue$ global solutions $(E(t), n(t))$ of (1) $-(3)$ such that(iii) For any in itial data$(E_{0}, n_{0}, n_{1})$$e$$E(t) \in\bigcap_{j=0}^{1}C^{j}([0, \infty);H^{3-2j})$$n(t) \in\bigcap_{j=0}^{2}C^{j}([0, \infty);H^{2-j})$,$\overline{W}+$,,$\frac{\partial}{\partial t}n(t)\in C([0, \infty);\dot{H}^{-1})$.

Here $W+and\overline{W}+are$ defined in (i) and (ii), respectively.

Remark 3. The range of $\overline{W}+includes$ some large data, while the range of $W+includes$
only small data. Therefore, Corollary 3 (iii) implies that for some large initial data, there17exist the unique global solutions of (1) $-(3)$ . However, it is not clear what initial data belong
to the range of$\overline{W}+\cdot$Theorems 1 and 2 follow from the combination of the various energy and decay estimates.
The energy estimates are the standard ones and the decay estimat es are crucial. The decay
estimat es needed for the proof of Theorem 1 are based on the $L^{p}-L^{q}$ estimate and theStricahrtz type estimate (see, e.g., [7] and [19] for the Schr\"odinger equation and [14] for thewave equation). The decay estimates needed for the proof of Theorem 1 are based on thespecial properties of the nonlinear coupling for the Zakharov equations and the propagation
properties of the Schr\"odinger wave and the acoustic wave. The support condition on theFourier transform of $E+0$ ensures that the nonlinear coupling term of (1) decays fast enoughas $tarrow+\infty$ (see Ozawa and Tsutsumi [13] and Tsutsumi [18]). The details of the proofs
will appear elsewhere.

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