# Infinite dimensional symplectic capacity and nonsqueezing property for the
Zakharov system on the $1$-dimensional torus

<EMAIL_ADDRESS>
###### Abstract.

We prove the invariant of the symplectic capacity for the Zakharov system on a
torus. If the Zakharov solution map is well-defined, then it can be regarded
as a symplectomorphism. Thus, we first show the global well-posedness via the
local well-posedness and the conservation law. The invariant of the symplectic
capacity can be obtained using an approximation method. Many authors use an
approximation method to obtain the nonsqueezing theorem, instead of an
invariant of the symplectic capacity. However, the conditions of the
Hamiltonian system introduced by Kuksin can be relaxed by a new modified
infinite dimensional Hamiltonian system. Thus we can back to the symplectic
capacity which contains the nonsqueezing property. Heuristically, we obtain
the invariant by using the Hamiltonian system which has linear flow at high
frequencies and nonlinear flow at low frequencies.

###### Key words and phrases:

Zakharov system, Hamiltonian system, symplectic capacity, nonsqueezing
property

###### 2010 Mathematics Subject Classification:

Primary 35Q53

Sunghyun Hong111Email address<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Notations and Function spaces
  3. 3 Symplectic capacity for Hilbert space
  4. 4 Basic estimates
  5. 5 Global well-posedness
  6. 6 Proof of Theorem 1.1
    1. 6.1 Local approximation
    2. 6.2 Proof of symplectic invariant

##  1\. Introduction

In this paper, we consider the Zakharov system

$\left\\{\begin{array}[]{ll}i\partial_{t}u+\alpha\partial_{x}^{2}u=un,&\left(t,x\right)\in\mathbb{R}\times\mathbb{T},\\\
\beta^{-2}\partial_{t}^{2}n-\partial_{x}^{2}n=\partial_{x}^{2}\left(\left|u\right|^{2}\right),&\left(t,x\right)\in\mathbb{R}\times\mathbb{T},\\\
\left.\left(u,n,\partial_{t}n\right)\right|_{t=0}=\left(u_{0}(x),n_{0}(x),n_{1}(x)\right)\in
L_{x}^{2}\times H_{x}^{-1/2}\times
H_{x}^{-3/2},&x\in\mathbb{T}\left(:=\mathbb{R}/{2\pi\mathbb{Z}}\right),\end{array}\right.$
(1.1)

where $\alpha$ and $\beta$ are positive real constants, respectively. The
functions $u$ and $n$ are complex valued and real valued, respectively. The
Zakharov system (1.1) enjoys conservation laws,

$M\left[u\right]\left(t\right)=\int_{\mathbb{T}}\left|u\left(t\right)\right|^{2}dx=\int_{\mathbb{T}}\left|u\left(0\right)\right|^{2}dx=M\left[u_{0}\right],$
(1.2)

and

$\displaystyle
H\left[u,n,\partial_{t}n\right]\left(t\right):=H\left[u,n,\dot{n}\right]\left(t\right)$
(1.3)
$\displaystyle=\int_{\mathbb{T}}\left(\alpha\left|\partial_{x}u\left(t\right)\right|^{2}+\frac{\left|n(t)\right|^{2}}{2}+\frac{\beta^{2}\left|i\partial_{x}^{-1}\dot{n}\right|^{2}}{2}+n\left(t\right)\left|u\left(t\right)\right|^{2}\right)dx=H\left[u_{0},n_{0},n_{1}\right].$

The first (1.2) is called the mass conservation law and the second (1.3) is
called the Hamiltonian. They are important tools for showing global well-
posedness and to define the symplectic capacity, respectively. From (1.1), we
have

$\frac{d^{2}}{dt^{2}}\int_{\mathbb{T}}n\left(t,x\right)dx=\beta^{2}\int_{\mathbb{T}}\partial^{2}_{x}\left(n+\left|u\right|^{2}\right)dx=0,$

and so

$\int_{\mathbb{T}}n\left(t,x\right)dx=c_{1}t+c_{0},$

where

$c_{0}=\int_{\mathbb{T}}n_{0}\left(x\right)dx~{}\text{and}~{}c_{1}=\int_{\mathbb{T}}n_{1}\left(x\right)dx.$

Hence, we denote

$u^{\prime}\left(t,x\right)=\exp\left[i\left(\frac{1}{4\pi}c_{1}t^{2}+c_{0}t\right)\right]u\left(t,x\right)~{}\text{and}~{}n^{\prime}\left(t,x\right)=n\left(t,x\right)-\frac{1}{2\pi}\left(c_{1}t+c_{0}\right),$
(1.4)

then $u^{\prime}$ and $n^{\prime}$ are also the solutions to (1.1), and

$\int
n^{\prime}\left(t,x\right)dx=0~{}\text{and}~{}\int\partial_{t}n^{\prime}\left(t,x\right)dx=0.$

If initial data $n_{0}\left(x\right)$, $n_{1}\left(x\right)$ have general
mean, then one can easily change the data into the mean zero data by (1.4).
Therefore, it will be convenient to work in the case when initial data
$n_{0}\left(x\right)$, $n_{1}\left(x\right)$ have mean zero.

The system (1.1) was introduced by Zakharov [Zakharov:1972tz]. It represents
the propagation of Langmuir turbulence waves in unmagnetized ionized plasma
[Zakharov:1972tz]. In the system, $u\left(t,x\right)$ expresses the slowly
varying envelope of the electric field and $n\left(t,x\right)$ describes the
deviation in ion density from its mean. The constant $\alpha$ is a dispersion
coefficient and the constant $\beta$ is the speed of an ion acoustic wave in
plasma.

There are many results for the symplectic capacity and the nonsqueezing
theorem for the infinite dimensional Hamiltonian system. The symplectic
capacity was introduced by Ekeland and Hofer [Ekeland:1989dh, Ekeland:1990is]
for $\mathbb{R}^{2n}$, and by Hofer and Zehnder [Hofer:1990ul, Hofer:2011vo]
for $2n$-dimensional general symplectic manifolds. It was developed from the
Darboux width, which was discovered by Gromov [Gromov:1985ww]. Kuksin
[Kuksin:1995ue] was the first contributor of the infinite dimensional
symplectic capacity for Hamiltonian Partial Differential Equations(PDEs).
Kuksin’s concept, of course, is based on the finite dimensional symplectic
capacity which was developed by Hofer and Zehnder. Indeed, Kuksin proved an
invariance in the symplectic capacity for particular Hamiltonian flow, and so
he also captured its nonsqueezing property. Furthermore, he introduced an
abstract method in which the Hamiltonian flow on the appropriate function
space can be regarded as a symplectic map. Although there are results which
have applied this condition [Kuksin:1995ue, Roumegoux:2010sn], Kuksin’s
condition for solution flow is somewhat strong. Thus, many contributors to
this issue have turned to the nonsqueezing theorem for specific equations.

To prove the nonsqueezing results for Haimtonian PDEs, one of the main steps
is to find a ‘good’ truncation. Besides, the given Hamiltonian system turns
out to be well-behaved with ‘good’ frequency truncations. There are two
techniques for the truncation, the methods of [Bourgain:1994tr] and
[Colliander:2005vv]. In [Bourgain:1994tr], Bourgain proved the nonsqueezing
theorem of the cubic nonlinear Schrödinger equation (NLS) in its phase space
$L^{2}_{x}\left(\mathbb{T}\right)$ space. A sharp frequency truncation and the
$X^{s,b}$ space were used to approximate the original solution. Later, this
argument was extended by Colliander et al. [Colliander:2005vv] for the KdV
equation in its phase space $H^{-1/2}_{x}\left(\mathbb{T}\right)$. The
argument in [Colliander:2005vv] is more complex than the one in
[Bourgain:1994tr]. They used a smooth truncation, and also used the Miura
transform which changes the KdV flow to a mKdV flow. Indeed, they showed an
approximation using truncated mKdV flow and used Miura transform and its
inverse. In this way, they obtained the estimate for the KdV flow. We use the
methods of Bourgain [Bourgain:1994tr] instead of the method of Colliander et
al. [Colliander:2005vv], because the modulation effects from the non-resonant
interaction of (1.1) is better than that of the KdV equation. In Section 4, we
show the bilinear estimates produced by these modulation effects and a similar
calculation in [Colliander:2008cq, Takaoka:1999uw]. Specifically, bilinear
estimates are needed to approximate the truncated solution flow, and this is
stronger than the estimates of [Takaoka:1999uw] to prove the local well-
posedness. Hong and Kwak [Hong:2016fn] extended the result to the higher-order
KdV equation, and Mendelson [Mendelson:2014vh] also showed the nonsqueezing of
the Klein-Gordon flow on $\mathbb{T}^{3}$ via a probabilistic approach.
Moreover, Kwak [Kwak:2017wb] proved the nonsqueezing and the local well-
posedness for the fourth-order cubic nonlinear Schrödinger equation on a
torus. Recently, Killip et al. [Killip:2016vd, Killip:2016wj] proved the
nonsqueezing theorem of the cubic NLS equation on a real line and a plane,
respectively. These results are the first nonsqueezing study for an unbounded
domain.

Nevertheless, we want to go back to the ‘capacity’ beyond ‘nonsqueezing.’
There are some results that are independent of the nonsqueezing theorem. For
example, Abbondandolo and Majer [Abbondandolo:2015cb] constructed the
symplectic capacity on a _convex set_ in the Hilbert space without the
approximation approach. However, we focus on the relaxation of Kuksin’s
condition. As a result, we obtain a symplectic capacity for the Zakharov
system flow which does not satisfy Kuksin’s condition. In particular, there is
no nonsqueezing result associated with the Zakharov flow. Moreover, we do not
even know the global well-posedness for the symplectic Hilbert space
$L_{x}^{2}\left(\mathbb{T}\right)\times
H_{x}^{-\frac{1}{2}}\left(\mathbb{T}\right)\times
H_{x}^{-\frac{3}{2}}\left(\mathbb{T}\right)$. We use the appropriate frequency
truncation and approximate the finite dimensional solution to the original
infinite dimensional solution, preserving the symplectic form. These are
nontrivial facts, because the nonlinear terms in the Zakharov system does not
satisfy Kuksin’s results. To overcome these obstacles, we need to prove that
the frequency truncated solution flow well-approximates to the original
solution flow. In addition, the truncated flow should be a Hamiltonian flow.
We now introduce the main result.

######  Theorem 1.1.

Assume that $\frac{\beta}{\alpha}$ is not an integer. Let $Z\left(T\right)$ be
the Zakharov flow map at time $T$. For any bounded domain $\mathcal{O}$ in
$L_{x}^{2}\times H_{x}^{-\frac{1}{2}}\times H^{-\frac{3}{2}}_{x}$, we have

${\rm cap}\left(\mathcal{O}\right)={\rm
cap}\left(Z\left(T\right)\left(\mathcal{O}\right)\right)$

where ${\rm cap}\left(\cdot\right)$ is the infinite dimensional symplectic
capacity.

For the solution map to exist as the symplectic map for any $T>0$, we should
have the global well-posedness in the phase space as follows.

######  Theorem 1.2.

Assume that $\frac{\beta}{\alpha}$ is not an integer. The initial value
problem (1.1) is globally well-posed for any
$\left(u_{0},n_{0},n_{1}\right)\in L_{x}^{2}\left(\mathbb{T}\right)\times
H_{x}^{-\frac{1}{2}}\left(\mathbb{T}\right)\times
H^{-\frac{3}{2}}_{x}\left(\mathbb{T}\right)$.

Theorem 1.2 can be proved by combining the local well-posedness with the mass
conservation of $u$ (1.2). The details are in Section 5. It is the Duhamel’s
formula for (1.1) which can be written as follows,

$\displaystyle u\left(t\right)$
$\displaystyle:=S\left(t\right)\left(u_{0},n_{0},n_{1}\right)=U\left(t\right)u_{0}-i\int_{0}^{t}U\left(t-s\right)\left[un\right]\left(s\right)ds,$
(1.5) $\displaystyle n\left(t\right)$
$\displaystyle:=W\left(t\right)\left(u_{0},n_{0},n_{1}\right)=\partial_{t}V\left(t\right)n_{0}+V\left(t\right)n_{1}+\beta^{2}\int_{0}^{t}V\left(t-s\right)\partial_{x}^{2}\left[\left|u\right|^{2}\right]\left(s\right)ds,$
(1.6)

where $U\left(t\right)=e^{i\alpha t\partial^{2}_{x}}$ and
$V\left(t\right)=\frac{\sin\left(\beta
t\left(-\partial_{x}^{2}\right)^{1/2}\right)}{\beta\left(-\partial_{x}^{2}\right)^{1/2}}=\frac{\sin\left(\beta
t\sqrt{-\Delta}\right)}{\beta\sqrt{-\Delta}}$. We denote the solution to (1.1)
by

$\displaystyle{\bf{z}}\left(t,x\right)$
$\displaystyle=\left(u\left(t,x\right),n\left(t,x\right),\partial_{t}n\left(t,x\right)\right)$
$\displaystyle:=Z\left(t\right)\left(u_{0}(x),n_{0}(x),n_{1}(x)\right)$
$\displaystyle=S\left(t\right)\left(u_{0},n_{0},n_{1}\right)\times
W\left(t\right)\left(u_{0},n_{0},n_{1}\right)\times\partial_{t}W\left(t\right)\left(u_{0},n_{0},n_{1}\right).$

Thus, we also have $Z\left(t\right)$ as the solution flow to (1.1). In the
same way as here, we will use bold fonts to present vectors in the appropriate
space. The spatial Sobolev space is given by

$\left\|u\right\|_{H_{x}^{s}}=\left\|\left<k\right>^{s}\widehat{u}\right\|_{\ell^{2}_{k}}:=\frac{1}{\left(2\pi\right)^{1/2}}\left(\sum_{k\in\mathbb{Z}}\left<k\right>^{2s}\left|\widehat{u}\right|^{2}\right)^{1/2}$

for $s\in\mathbb{R}$, where
$\left<k\right>=\left(1+\left|k\right|^{2}\right)^{1/2}$. Let $\mathcal{H}$ be
the symplectic Hilbert space $L_{x}^{2}\left(\mathbb{T}\right)\times
H_{x}^{-\frac{1}{2}}\left(\mathbb{T}\right)\times
H^{-\frac{3}{2}}_{x}\left(\mathbb{T}\right)$, and

$\left\|\left(u,v,w\right)\right\|_{\mathcal{H}}=\left\|u\right\|_{L^{2}_{x}}+\left\|v\right\|_{H^{-1/2}_{x}}+\left\|w\right\|_{H^{-3/2}_{x}}.$

We also define the absolute value in $\mathcal{H}$ by

$\left|\left(\widehat{u}_{k_{0}},\widehat{v}_{k_{0}},\widehat{w}_{k_{0}}\right)\right|_{\mathcal{H}}=\left|\widehat{u}_{k_{0}}\right|+\left|{k_{0}}\right|^{-\frac{1}{2}}\left|\widehat{v}_{k_{0}}\right|+\left|{k_{0}}\right|^{-\frac{3}{2}}\left|\widehat{w}_{k_{0}}\right|$

for fixed frequency component $k_{0}$.

From Theorem 1.1, we can consider the nonsqueezing theorem of the Zakharov
system as well. We first define a ball and a cylinder in the function space
$\mathcal{H}$.

######  Definition 1.3.

Let $B^{\infty}_{R}\left({\bf{v}}_{*}\right)$ be an infinite dimensional ball
in ${\mathcal{H}}$ which has the radius $R$ and is centered at
${\bf{v}}_{*}\in{\mathcal{H}}$. That is,

$B^{\infty}_{R}\left({\bf{v}}_{*}\right):=\left\\{{\bf{v}}\in{\mathcal{H}}:\left\|{\bf{v}}-{\bf{v}}_{*}\right\|_{\mathcal{H}}\leq
R\right\\}.$

For any
$k\in\mathbb{Z}\setminus\left\\{0\right\\}\left(:=\mathbb{Z}^{*}\right)$,
$C^{\infty}_{k,r}\left(\eta\right)$ is defined an infinite dimensional $k$-th
cylinder in ${\mathcal{H}}$ which has the radius $r$ and is centered at
$\eta\in\mathbb{C}^{3}$. That is,

$C^{\infty}_{k,r}\left(\eta\right):=\left\\{{\bf{v}}\in{\mathcal{H}}:\left|\widehat{{\bf{v}}}_{k}-\eta\right|_{\mathcal{H}}\leq
r\right\\}$

where
$\widehat{\bf{v}}_{k}=\left(\widehat{u}_{k},\widehat{v}_{k},\widehat{w}_{k}\right)\in\mathbb{C}^{3}$.

The nonsqueezing property of the Zakharov system is as follows,

######  Corollary 1.4.

Let $0<r<R$, $\left(u^{*},n^{*},n^{**}\right)\in L_{x}^{2}\times
H_{x}^{-\frac{1}{2}}\times H^{-\frac{3}{2}}_{x}$, $k\in\mathbb{Z}^{*}$,
$\left(z,w_{0},w_{1}\right)\in\mathbb{C}^{3}$ and $T>0$. Then

$Z\left(T\right)\left(B^{\infty}_{R}\left(u^{*},n^{*},n^{**}\right)\right)\not\subseteq
C^{\infty}_{k,r}\left(z,w_{0},w_{1}\right).$

In other words, there is a solution
$Z\left(T\right)\left(u_{0},n_{0},n_{1}\right)\in C_{t}L_{x}^{2}\times
C_{t}H_{x}^{-\frac{1}{2}}\times C_{t}H_{x}^{-\frac{3}{2}}$ to (1.1) and
$k_{0}\in\mathbb{Z^{*}}$ such that

$\left\|\left(u_{0},n_{0},n_{1}\right)-\left(u^{*},n^{*},n^{**}\right)\right\|_{\mathcal{H}}\leq
R,$

and

$\left|\mathcal{F}_{x}\left[\left(Z\left(T\right)\left(u_{0},n_{0},n_{1}\right)\right)\right]\left(k_{0}\right)-\left(z,w_{0},w_{1}\right)\right|_{\mathcal{H}}>r$

where $\mathcal{F}_{x}\left[\cdot\right]$ is the spatial Fourier transform.

######  Remark 1.5.

There are no smallness conditions imposed on
$\left(u^{*},n^{*},n^{**}\right)$, $\left(z,w_{0},w_{1}\right)$, ${R}$ or
${T}$ in Corollary 1.4.

Corollary 1.4, the symplectic nonsqueezing theorem, tells us the Zakharov flow
cannot squash a large ball into a narrow cylinder, despite the fact that the
cylinder has infinite volume.

##  2\. Notations and Function spaces

In this section, we introduce notations to discuss our argument. The spatial
Fourier transform, the space-time Fourier transform and the inverse Fourier
transform are defined as follows,

$\displaystyle\mathcal{F}\left(u\right)=\tilde{u}\left(k,\tau\right)=\iint_{\mathbb{R}\times\mathbb{T}}e^{-ikx}e^{-i\tau
t}u\left(t,x\right)dxdt,$
$\displaystyle\mathcal{F}\left(u,v,w\right)=\left(\tilde{u},\tilde{v},\tilde{w}\right),$
$\displaystyle\mathcal{F}_{x}\left(u\right)=\widehat{u}_{k}=\int_{\mathbb{T}}e^{-ikx}u\left(x\right)dx,$
$\displaystyle\mathcal{F}_{x}\left(u,v,w\right)=\left(\widehat{u}_{k},\widehat{v}_{k},\widehat{w}_{k}\right),$
$\displaystyle u\left(x\right)=\int
e^{ikx}\widehat{u}\left(k\right)dk:=\frac{1}{2\pi}\sum_{k\in\mathbb{Z}}\widehat{u}_{k}e^{ikx}.$

We also introduce function spaces. First of all, we use the $X^{s,b}$ spaces
(the Bourgain spaces) which are defined by the following norms,

$\displaystyle\left\|f\right\|_{X_{S}^{s,b}}$
$\displaystyle=\left\|\left<k\right>^{s}\left<\tau-\alpha
k^{2}\right>^{b}\tilde{f}\left(k,\tau\right)\right\|_{l^{2}_{k}L^{2}_{\tau}},$
$\displaystyle\left\|f\right\|_{X_{W}^{s,b}}$
$\displaystyle=\left\|\left<k\right>^{s}\left<\left|\tau\right|-\beta\left|k\right|\right>^{b}\tilde{f}\left(k,\tau\right)\right\|_{l^{2}_{k}L^{2}_{\tau}}.$

Note that the first and the second are associated with Schrödinger flow and
wave flow, respectively. Using the $X^{s,b}$ spaces, we define $Y_{S}^{s}$,
$Z_{S}^{s}$, $Y_{W}^{s}$ and $Z_{W}^{s}$ spaces for the solution and the
nonlinear terms,

$\displaystyle\left\|f\right\|_{Y_{S}^{s}}$
$\displaystyle=\left\|f\right\|_{X^{s,1/2}_{S}}+\left\|\left<k\right>^{s}\tilde{f}\left(k,\tau\right)\right\|_{l^{2}_{k}L^{1}_{\tau}},$
$\displaystyle\left\|f\right\|_{Z_{S}^{s}}$
$\displaystyle=\left\|f\right\|_{X^{s,-1/2}_{S}}+\left\|\frac{\left<k\right>^{s}}{\left<\tau-\alpha
k^{2}\right>}\tilde{f}\left(k,\tau\right)\right\|_{l^{2}_{k}L^{1}_{\tau}},$
$\displaystyle\left\|f\right\|_{Y_{W}^{s}}$
$\displaystyle=\left\|f\right\|_{X^{s,1/2}_{W}}+\left\|\left<k\right>^{s}\tilde{f}\left(k,\tau\right)\right\|_{l^{2}_{k}L^{1}_{\tau}},$
$\displaystyle\left\|f\right\|_{Z_{W}^{s}}$
$\displaystyle=\left\|f\right\|_{X^{s,-1/2}_{W}}+\left\|\frac{\left<k\right>^{s}}{\left<\left|\tau\right|-\beta\left|k\right|\right>}\tilde{f}\left(k,\tau\right)\right\|_{l^{2}_{k}L^{1}_{\tau}}.$

we give some embeddings for the $Y$ and $Z$ spaces

$\displaystyle Y^{s}_{S,W}\subseteq$ $\displaystyle C_{t}H^{s}_{x}\subseteq
L^{\infty}_{t}H^{s}_{x},$ (2.1) $\displaystyle L^{2}_{t}H^{s}$
$\displaystyle\subseteq Z^{s}_{S,W}.$

in a compact time interval $\left[0,T\right]$ by the Hölder inequality. To
simplify notations, $\mathcal{Y}$ spaces would be

$\left\|\left(u,v\right)\right\|_{\mathcal{Y}}=\left\|u\right\|_{Y^{0}_{S}}+\left\|v\right\|_{Y_{W}^{-\frac{1}{2}}}.$

For each dyadic number $N$, we denote the Littlewood-Paley projection by

$\begin{split}&\widehat{P_{N}u}\left(k\right):=1_{N\leq\left|k\right|<2N}\left(k\right)\widehat{u}_{k},\\\
&\widehat{P_{\leq N}u}\left(k\right):=1_{\left|k\right|\leq
N}\left(k\right)\widehat{u}_{k},\\\ &\widehat{P_{\geq
N}u}\left(k\right):=1_{\left|k\right|\geq
N}\left(k\right)\widehat{u}_{k},\end{split}$ (2.2)

where $1_{\Omega}$ is a characteristic function on $\Omega$.

We call a connected open set a domain. For an nonempty domain
$\mathcal{O}\subset\mathcal{H}$ and $n\geq 1$, we denote

$\displaystyle\mathcal{O}_{N}$ $\displaystyle=\mathcal{O}\cap P_{\leq
N}\mathcal{H}\left(=P_{\leq N}L_{x}^{2}\times P_{\leq
N}H_{x}^{-\frac{1}{2}}\times P_{\leq N}H^{-\frac{3}{2}}_{x}\right),$
$\displaystyle\mathcal{O}^{N}$ $\displaystyle=\mathcal{O}\cap\left(1-P_{\leq
N}\right)\mathcal{H},$

and observe that

$\partial\mathcal{O}_{N}\subset\partial\mathcal{O}\cap P_{\leq N}\mathcal{H}.$

For $x,y\in\mathbb{R}_{+}$, $x\lesssim y$ denotes $x\leq Cy$ for some $C>0$
and $x\sim y$ means $x\lesssim y$ and $y\lesssim x$. Using this, we denote
$f=O(g)$ by $f\lesssim g$ for positive real-valued functions $f$ and $g$.
Moreover, $x\ll y$ denotes $x\leq cy$ for small positive constant $c$. Let
$a_{1},a_{2},a_{3}\in\mathbb{R}$ and the quantities $a_{max}\geq a_{med}\geq
a_{min}$ can be defined to be the maximum, median and minimum values of
$a_{1},a_{2},a_{3}$, respectively.

##  3\. Symplectic capacity for Hilbert space

We begin with the definition of the symplectic Hilbert space $\mathcal{H}$.
Let $\omega$ be a symplectic form $\mathcal{H}$ as follows,

$\omega\left(\left(u,v,w\right),\left(\acute{u},\acute{v},\acute{w}\right)\right)=\omega_{i}\left(u,\acute{u}\right)+\omega_{-1/2}\left(v,\acute{v}\right)+\omega_{-3/2}\left(w,\acute{w}\right)$
(3.1)

where Hello there, When is its update? Is it LIve? why dic puaed? don’t have
mayu

$\omega_{i}\left(f,g\right)=\operatorname{Im}\int\overline{f}{g}dx,\quad\omega_{-1/2}\left(f,g\right)=\int
f\partial_{x}^{-1}gdx,~{}\text{and}~{}\omega_{-3/2}\left(f,g\right)=\int
f\partial_{x}^{-3}gdx.$

Let $J$ be an almost complex structure on $\mathcal{H}$ which is compatible
with the Hilbert space inner product $<\cdot,\cdot>$. In other words, a
bounded self adjoint operator with $J^{2}=-I$ such that $\omega\left({\bf
u},{\bf v}\right)=<{\bf u},J{\bf v}>$ for all ${\bf u},{\bf v}\in\mathcal{H}$.
One easily checks that the Zakharov system can be written in the form

$\partial_{t}{\bf u}\left(t\right):=\dot{\bf u}\left(t\right)=J\nabla
H\left[{\bf u}\left(t\right)\right]$ (3.2)

where ${\bf u}\in\mathcal{H}$. The notation $\nabla$ in (3.2) denotes the
usual gradient with respect to the Hilbert space inner product. Hence, we have

$\displaystyle\left<\begin{bmatrix}v_{1}\\\ v_{2}\\\
v_{3}\end{bmatrix},\begin{bmatrix}\nabla
H\left[u\left(t\right),\cdot,\cdot\right]\\\ \nabla
H\left[\cdot,n\left(t\right),\cdot\right]\\\ \nabla
H\left[\cdot,\cdot,\dot{n}\left(t\right)\right]\end{bmatrix}\right>$
$\displaystyle\equiv\left<\begin{bmatrix}1\\\ 1\\\
1\end{bmatrix},\begin{bmatrix}dH\left[u\left(t\right),\cdot,\cdot\right]\left(v_{1},\cdot,\cdot\right)\\\
dH\left[\cdot,n\left(t\right),\cdot\right]\left(\cdot,v_{2},\cdot\right)\\\
dH\left[\cdot,\cdot,\dot{n}\left(t\right)\right]\left(\cdot,\cdot,v_{3}\right)\end{bmatrix}\right>.$
(3.3) $\displaystyle\equiv\left<\begin{bmatrix}1\\\ 1\\\
1\end{bmatrix},\begin{bmatrix}\nabla_{\varepsilon_{1}}H\left[{u}+{\varepsilon_{1}v},\cdot,\cdot\right]\\\
\cdot,\nabla_{\varepsilon_{2}}H\left[{n}+{\varepsilon_{2}v},\cdot\right]\\\
\cdot,\cdot,\nabla_{\varepsilon_{3}}H\left[{\dot{n}}+{\varepsilon_{3}v}\right]\end{bmatrix}\right>.$

######  Definition 3.1.

Consider a pair $\left(\mathcal{H},\omega\right)$, where $\omega$ is a
symplectic form (3.1) on the Hilbert space
$\mathcal{H}\left(=L_{x}^{2}\left(\mathbb{T}\right)\times
H_{x}^{-\frac{1}{2}}\left(\mathbb{T}\right)\times
H^{-\frac{3}{2}}_{x}\left(\mathbb{T}\right)\right)$. We say that the pair
$\left(\mathcal{H},\omega\right)$ is the symplectic phase space for the
Zakharov system.

One easily check that an equivalent way to write the Zakharov system
corresponding to the Hamiltonian 1.3 in $\left(\mathcal{H},\omega\right)$ is

$\displaystyle\partial_{t}\begin{bmatrix}u\\\ n\\\
\dot{n}\end{bmatrix}=\begin{bmatrix}\nabla_{\omega_{i}}H\left[u\left(t\right),\cdot,\cdot\right]\\\
\nabla_{\omega_{-1/2}}H\left[\cdot,n\left(t\right),\cdot\right]\\\
\nabla_{\omega_{-3/2}}H\left[\cdot,\cdot,\dot{n}\left(t\right)\right]\end{bmatrix}$
(3.4)

where the symplectic gradient is defined an analogy with (3.3),

$\displaystyle\omega\left(\begin{bmatrix}v_{1}\\\ v_{2}\\\
v_{3}\end{bmatrix},\begin{bmatrix}\nabla_{\omega_{i}}H\left[u\left(t\right),\cdot,\cdot\right]\\\
\nabla_{\omega_{-1/2}}H\left[\cdot,n\left(t\right),\cdot\right]\\\
\nabla_{\omega_{-3/2}}H\left[\cdot,\cdot,\dot{n}\left(t\right)\right]\end{bmatrix}\right)=\left<\begin{bmatrix}1\\\
1\\\
1\end{bmatrix},\begin{bmatrix}dH\left[u\left(t\right),\cdot,\cdot\right]\left(v_{1},\cdot,\cdot\right)\\\
dH\left[\cdot,n\left(t\right),\cdot\right]\left(\cdot,v_{2},\cdot\right)\\\
dH\left[\cdot,\cdot,\dot{n}\left(t\right)\right]\left(\cdot,\cdot,v_{3}\right)\end{bmatrix}\right>.$

Therefore, we can consider $L_{x}^{2}\left(\mathbb{T}\right)\times
H_{x}^{-\frac{1}{2}}\left(\mathbb{T}\right)\times
H^{-\frac{3}{2}}_{x}\left(\mathbb{T}\right)$ as the phase space. In the phase
space, we can consider an invariance of the symplectic capacity by the
Zakharov solution flow.

In the following, we introduce the infinite dimensional symplectic capacity
which was introduced by Kuksin [Kuksin:1995ue]. The symplectic capacity was
first discovered by Ekeland and Hofer [Ekeland:1989dh, Ekeland:1990is] in
$\mathbb{R}^{2n}$ and was developed by Hofer and Zehnder [Hofer:2011vo].
Specifically, it is a symplectic invariant and the proof of existence is based
on the variational principle.

######  Definition 3.2 (Symplectic capacity).

A symplectic capacity on the phase space $\left(\mathcal{H},\omega\right)$
with respect to 1.1 is a function ${\rm cap}\left(\cdot\right)$ defined open
subset $\mathcal{O}\subset\mathcal{H}$ which takes values in
$\left[0,\infty\right]$ and has the following properties:

  1. i)

Translational invariant:

${\rm cap}\left(\mathcal{O}\right)={\rm
cap}\left(\mathcal{O}+\xi\right)\quad\text{for}~{}\xi\in\mathcal{H}.$

  2. ii)

Monotonicity: Let $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ be open sets in
$\mathcal{H}$.

${\rm cap}\left(\mathcal{O}_{1}\right)\leq{\rm
cap}\left(\mathcal{O}_{2}\right)\quad\text{if}~{}\mathcal{O}_{1}\subseteq\mathcal{O}_{2}.$

  3. iii)

2-homogeneity: For $\tau\in\mathbb{R}$,

${\rm cap}\left(\tau\mathcal{O}\right)=\tau^{2}{\rm
cap}\left(\mathcal{O}\right).$

  4. iv)

Nontriviality: For bounded nonempty set $\mathcal{O}$,

$0<{\rm cap}\left(\mathcal{O}\right)<\infty.$

  5. v)

For an $r$-ball $B^{\infty}_{r}$ in $\mathcal{H}$ and a $k$-th cylinder
$C^{\infty}_{k,r}$ which has radius $r$ in $\mathcal{H}$,

${\rm cap}\left(B^{\infty}_{r}\right)={\rm
cap}\left(C^{\infty}_{k,r}\right)=\pi r^{2}.$

We point out two notable remarks. Combining ii), v) and the invariant of the
symplectic capacity, we can get the nonsqueezing theorem for a Hamiltonian
flow. Moreover, we note that the symplectic capacity does not determine a
unique capacity function. We, thus, have many ways to construct capacity
functions, but we follow [Kuksin:1995ue]. To construct a capacity function, we
first introduce few definitions.

######  Definition 3.3 (Admissible function).

Let $\mathcal{O}$ be a simply connected open set in the phase space
$\mathcal{H}$ (we call it domain in the sequel). Assume that a function $f$ is
a smooth function in $\mathcal{O}$, and $m>0$. The function $f$ is called
$m$-admissible if

  1. i)

$0\leq f\leq m$ in $\mathcal{O}$,

  2. ii)

$f\equiv 0$ in a nonempty subdomain of $\mathcal{O}$,

  3. iii)

$f\big{|}_{\partial\mathcal{O}}\equiv m$,

  4. iv)

The set $\left\\{f<m\right\\}$ is bounded, and the distance from this set to
$\partial\mathcal{O}$ is strictly positive, i.e., ${\rm
dist}_{\mathcal{O}}\left(f\right)(:={\rm
dist}\left(\left\\{f<m\right\\},\partial\mathcal{O}\right))>0$.

For each $m$-admissible function, we denote

${\rm Supp}f=\left\\{u:0<f\left(u\right)<m\right\\},$

and so we have

$\displaystyle{\rm
dist}\left(f^{-1}\left(0\right),\partial\mathcal{O}\right)\geq{\rm
dist}\left(f\right),$ (3.5) $\displaystyle{\rm dist}\left({\rm
Supp}f,\partial\mathcal{O}\right)\geq{\rm dist}\left(f\right).$ (3.6)

Let $N\geq 1$ be integer. By using the Fourier basis and restricting the form
$\omega$, we see that the truncated phase space $\left(P_{\leq
N}\mathcal{H},\omega\right)$ is a $3\times 2N$-dimensional real symplectic
space, and so is symplectomorphic to the standard phase space
$\left(\mathbb{R}^{2N\times 3},\omega_{0}\right)$ by Darboux theorem.

######  Definition 3.4 (Fast function).

Let $f_{N}=f\big{|}_{\mathcal{O}_{N}}$ be a frequency truncated function. We
consider the corresponding symplectic Hamiltonian vector field $V_{f_{N}}$ in
$\mathcal{O}_{N}$. In other words, for $u,v\in\mathcal{O}_{N}$,

$\omega\left({\bf v},V_{f_{N}}\left({\bf u}\right)\right)=df_{N}\left({\bf
u}\right)\left({\bf v}\right).$

We call a periodic trajectory of $V_{f_{N}}$ a ‘fast trajectory’ if it does
not pass through a stationary point and the period $T\leq 1$. Furthermore, a
$m$-admissible function $f$ is called ‘fast’ if there exists
$N_{0}=N_{0}\left(f\right)$ such that for all $N\geq N_{0}$ the vector field
$V_{f_{N}}$ has a fast trajectory.

A fast $m$-admissible function has a notable property which comes from its
definition,

######  Lemma 3.5 (Kuksin [Kuksin:1995ue]).

All fast periodic trajectories in $V_{f_{N}}$ are contained in ${\rm
Supp}f\cap P_{\leq N}\mathcal{H}$

This lemma can be proved by the definition of the fast trajectory and the fact
that the derivative of $f$ is zero in the complementary set of ${\rm Supp}f$.
We next introduce a definition of the (infinite dimensional) symplectic
capacity.

######  Definition 3.6 (Infinite dimensional symplectic capacity, Kuksin
[Kuksin:1995ue]).

For a nonempty domain $\mathcal{O}\in\mathcal{H}$, its symplectic capacity
${\rm cap}\left(\mathcal{O}\right)$ equals

$\inf\left\\{m_{*}:\text{\rm each $m$-admissible function with $m>m_{*}$ is
fast}\right\\}$

From [Kuksin:1995ue], we already have that the symplectic capacity ${\rm
cap}\left(\cdot\right)$ satisfies Definition 3.2. Note that the capacity ${\rm
cap}\left(\mathcal{O}\right)$ depends on the consecutive subsets
$P_{1}\mathcal{H}\subset P_{2}\mathcal{H}\subset\cdots$ of the space
$\mathcal{H}$.

##  4\. Basic estimates

In this section, we show estimates to prove Theorem Theorem 1.1 and 1.2. First
of all, we recall the lemma regarding the linear estimates.

######  Lemma 4.1.

Let $\Psi\left(t\right)\in C^{\infty}_{0}\left(\mathbb{R}\right)$ such that
$\Psi\left(t\right)=1$ on $\left[-1,1\right]$ and $\Psi\left(t\right)=0$
outside of $\left[-2,2\right]$. We have

$\displaystyle\left\|\Psi\left(\frac{t}{T}\right)U\left(t\right)u_{0}\right\|_{Y_{S}^{0}}$
$\displaystyle\lesssim\left\|u_{0}\right\|_{L^{2}_{x}},$
$\displaystyle\left\|\Psi\left(\frac{t}{T}\right)\int_{0}^{t}U\left(t-s\right)F\left(s\right)ds\right\|_{Y_{S}^{0}}$
$\displaystyle\lesssim\left\|F\left(t\right)\right\|_{Z_{S}^{0}},$
$\displaystyle\left\|\Psi\left(\frac{t}{T}\right)\partial_{t}V\left(t\right)n_{0}\right\|_{Y_{W}^{-1/2}}$
$\displaystyle\lesssim\left\|n_{0}\right\|_{H^{-\frac{1}{2}}_{x}},$
$\displaystyle\left\|\Psi\left(\frac{t}{T}\right)V\left(t\right)n_{1}\right\|_{Y_{W}^{-1/2}}$
$\displaystyle\lesssim\left\|n_{1}\right\|_{H^{-\frac{3}{2}}_{x}},$
$\displaystyle\left\|\Psi\left(\frac{t}{T}\right)\int_{0}^{t}V\left(t-s\right)\partial_{x}F\left(s\right)ds\right\|_{Y_{W}^{-1/2}}$
$\displaystyle\lesssim\left\|F\left(t\right)\right\|_{Z_{W}^{-1/2}},$

where $U\left(t\right)=e^{i\alpha t\partial^{2}_{x}}$ and
$V\left(t\right)=\frac{\sin\left(\beta
t\left(-\partial_{x}^{2}\right)^{1/2}\right)}{\beta\left(-\partial_{x}^{2}\right)^{1/2}}$.

Lemma 4.1 will be used for the contraction mapping principle, and so it is
well-known estimates. For that reason, we omit the proof of Lemma 4.1, but it
can be found in [Bourgain:1993hz, Bourgain:1993cl, Bourgain:1994ej,
Colliander:2008cq, KENIG:1996tq, Kenig:1996bu].

From now on, we discuss bilinear estimates with respect to the Schrödinger and
the wave flow. Define the resonant set and nonresonant set as follows,

$\displaystyle N_{\mathcal{R}}$
$\displaystyle=\left\\{\left(k_{0},k_{1},k_{2}\right):k_{0}\sim k_{1}\sim
k_{2}\right\\}$ $\displaystyle N_{\mathcal{NR}}$
$\displaystyle=\left(N_{\mathcal{R}}\right)^{C}.$

######  Proposition 4.2.

Let $N_{i}$ be dyadic numbers, $N_{i}\sim\left|k_{i}\right|$ and $0<T<1$.
Assume that $\frac{\beta}{\alpha}$ is not an integer,

$\left\|P_{N_{0}}\left(P_{N_{1}}uP_{N_{2}}v\right)\right\|_{Z^{0}_{S}\left(\left[0,T\right]\times\mathbb{T}\right)}\lesssim
T^{\gamma}\left\|u\right\|_{Y^{0}_{S}}\left\|v\right\|_{Y^{-1/2}_{W}},$ (4.1)

and

$\left\|P_{N_{0}}\partial_{x}\left(P_{N_{1}}u\overline{P_{N_{2}}v}\right)\right\|_{Z^{-1/2}_{W}\left(\left[0,T\right]\times\mathbb{T}\right)}\lesssim
T^{\gamma}\left\|u\right\|_{Y^{0}_{S}}\left\|v\right\|_{Y^{0}_{S}}.$ (4.2)

for sufficiently small $\gamma>0$.

######  Proposition 4.3.

Let $N_{i}$ be dyadic numbers and $N_{i}\sim\left|k_{i}\right|$. Assume that
$\frac{\beta}{\alpha}$ is not an integer,

$\left\|P_{N_{0}}\left(P_{N_{1}}uP_{N_{2}}v\right)\right\|_{Z^{0}_{S}}\lesssim
N^{-\delta}_{max}\left\|u\right\|_{Y^{0}_{S}}\left\|v\right\|_{Y^{-1/2}_{W}},$
(4.3)

and

$\left\|P_{N_{0}}\partial_{x}\left(P_{N_{1}}u\overline{P_{N_{2}}v}\right)\right\|_{Z^{-1/2}_{W}}\lesssim
N^{-\delta}_{max}\left\|u\right\|_{Y^{0}_{S}}\left\|v\right\|_{Y^{0}_{S}}.$
(4.4)

for $\left(k_{0},k_{1},k_{2}\right)\subset N_{\mathcal{NR}}$ and sufficiently
small $\delta>0$.

Takaoka [Takaoka:1999uw] proved these type estimates (without dyadic
decompositions) using the arguments in [KENIG:1996tq, KENIG:1993ts], but we
have to show slightly stronger estimates for the approximation step in the
proof of Theorem 1.1. In addition, we prove the propositions using a little
simpler argument in [Tao:2001tu] than in [Takaoka:1999uw, KENIG:1996tq,
KENIG:1993ts]. Roughly, Proposition 4.3 can be deduced by observation of the
intersection of hyperspace $\tau=h\left(k\right)$ (a function $h$ be chosen by
each equation). We first have the following algebraic results for observation
of resonant relations.

######  Lemma 4.4.

Let ${\rm sgn}(x)$ be the sign function. Then
i) Let $\tau_{0}=\tau_{1}+\tau_{2}$, $k_{0}=k_{1}+k_{2}\neq 0$,

$\max\left\\{\left|\tau_{0}-\alpha k_{0}^{2}\right|,\left|\tau_{1}-\alpha
k_{1}^{2}\right|,\left|\left|\tau_{2}\right|-\beta\left|k_{2}\right|\right|\right\\}\gtrsim\left|\alpha\right|\left|k_{2}\right|\left|k_{0}+k_{1}-\frac{\beta}{\alpha}S_{1}\right|,$
(4.5)

for $S_{1}={\rm sgn}\left(\tau_{2}k_{2}\right)$.
ii) Let $\tau_{0}=\tau_{1}-\tau_{2}$, $k_{0}=k_{1}-k_{2}\not=0$,

$\max\left\\{\left|\left|\tau_{0}\right|-\beta\left|k_{0}\right|\right|,\left|\tau_{1}-\alpha
k_{1}^{2}\right|,\left|\tau_{2}-\alpha
k_{2}^{2}\right|\right\\}\gtrsim\left|\alpha\right|\left|k_{0}\right|\left|k_{1}+k_{2}-\frac{\beta}{\alpha}S_{2}\right|,$
(4.6)

for $S_{2}={\rm sgn}\left(\tau_{0}k_{0}\right)$.

###### Proof.

We first prove i). From the support of time and spatial frequencies,

$\displaystyle\tau_{0}-\alpha k_{0}^{2}-\tau_{1}+\alpha
k_{1}^{2}-\tau_{2}+\beta S_{1}k_{2}$
$\displaystyle=\tau_{0}-\tau_{1}-\tau_{2}+\alpha\left(k_{1}^{2}-k_{0}^{2}-\frac{\beta}{\alpha}S_{1}k_{2}\right)$
$\displaystyle=-\alpha
k_{2}\left(k_{0}+k_{1}-\frac{\beta}{\alpha}S_{1}\right).$

Similarly,

$\displaystyle\tau_{0}-\beta S_{2}k_{0}-\tau_{1}+\alpha
k_{1}^{2}+\tau_{2}-\alpha k_{2}^{2}$
$\displaystyle=\tau_{0}-\tau_{1}+\tau_{2}+\alpha\left(k_{1}^{2}-k_{2}^{2}-\frac{\beta}{\alpha}S_{2}k_{0}\right)$
$\displaystyle=\alpha
k_{0}\left(k_{1}+k_{2}-\frac{\beta}{\alpha}S_{1}\right).$

Therefore, we are done. ∎

We can obtain Proposition 4.2 by the similar argument of Proposition 4.3, so
we first prove the latter.

###### Proof of Proposition 4.3.

Let $L_{i}:=\left<\tau_{i}-\alpha k_{i}^{2}\right>$ and
$M_{i}=\left<\left|\tau_{i}\right|-\beta\left|k_{i}\right|\right>$. To prove
(4.3) and (4.4), we are going to show the $X^{s,b}$ part and
$l^{2}_{k}L^{1}_{\tau}$ part respectively. However, essential ideas are
similar. From [Bourgain:1993hz], we have the Strichartz estimate for the
Schrödinger equation,

$X_{S}^{0,3/8}\subset L^{4}_{t,x},$ (4.7)

which will be used many times in the following calculation. We first consider
the $X^{s,b}$ part of (4.3). The left hand side of (4.3) can be rewritten by
the Plancherel theorem, so our claim is

$\left\|\sum_{\begin{subarray}{c}k_{0}=k_{1}+k_{2}\\\ (k_{0},k_{1},k_{2})\in
N_{\mathcal{NR}}\end{subarray}}\int_{\tau_{0}=\tau_{1}+\tau_{2}}\frac{\left|k_{2}\right|^{1/2}}{L_{0}^{1/2}L_{1}^{1/2}M_{2}^{1/2}}\tilde{u}\tilde{v}d\tau_{1}\right\|_{l^{2}_{k_{0}}L^{2}_{\tau_{0}}}\lesssim
N_{max}^{-\delta}\left\|u\right\|_{L^{2}_{t,x}}\left\|v\right\|_{L^{2}_{t,x}}.$
(4.8)

  1. i)

$\max\left\\{L_{0},L_{1},M_{2}\right\\}=M_{2}$
By (4.5), the left hand side of (4.8) is bounded by

$\left\|\sum_{\begin{subarray}{c}k_{0}=k_{1}+k_{2}\\\ (k_{0},k_{1},k_{2})\in
N_{\mathcal{NR}}\end{subarray}}\int_{\tau_{0}=\tau_{1}+\tau_{2}}\frac{\left|k_{2}\right|^{1/2}}{L_{0}^{1/2}L_{1}^{1/2}\left|\alpha\right|^{1/2}\left|k_{2}\right|^{1/2}\left|k_{0}+k_{1}-\frac{\beta}{\alpha}S_{1}\right|^{1/2}}\tilde{u}\tilde{v}d\tau_{1}\right\|_{l^{2}_{k_{0}}L^{2}_{\tau_{0}}}.$

Hence, it is reduced to show that

$\left\|\sum_{\begin{subarray}{c}k_{0}=k_{1}+k_{2}\\\ (k_{0},k_{1},k_{2})\in
N_{\mathcal{NR}}\end{subarray}}\int_{\tau_{0}=\tau_{1}+\tau_{2}}\frac{N_{max}^{\delta}}{L_{0}^{1/2}L_{1}^{1/2}\left|k_{0}+k_{1}-\frac{\beta}{\alpha}S_{1}\right|^{1/2}}\tilde{u}\tilde{v}d\tau_{1}\right\|_{l^{2}_{k_{0}}L^{2}_{\tau_{0}}}\lesssim\left\|u\right\|_{L^{2}_{t,x}}\left\|v\right\|_{L^{2}_{t,x}}.$

Since $k_{0}=k_{1}+k_{2}$ and $(k_{0},k_{1},k_{2})\subset N_{\mathcal{NR}}$,
we have $k_{max}$ and $k_{med}$ such that $N_{max}\sim k_{max}\sim k_{med}$,
and they are same sign. Thus, we have

$\frac{N_{max}^{\delta}}{\left|k_{0}+k_{1}-\frac{\beta}{\alpha}S_{1}\right|^{1/2}}\lesssim
1$

for sufficiently small $\delta$. By duality, it is enough to show that

$\left|\iint_{\mathbb{R}\times\mathbb{T}}u_{0}u_{1}u_{2}dxdt\right|\lesssim\left\|u_{0}\right\|_{X_{S}^{0,1/2}}\left\|u_{1}\right\|_{X_{{S}}^{0,1/2}}\left\|u_{2}\right\|_{L^{2}_{t,x}},$

and then by the Hölder inequality and (4.7), we have

$\displaystyle\left|\iint_{\mathbb{R}\times\mathbb{T}}u_{0}u_{1}u_{2}dxdt\right|$
$\displaystyle\leq\left\|u_{0}\right\|_{L^{4}_{t,x}}\left\|u_{1}\right\|_{L^{4}_{t,x}}\left\|u_{2}\right\|_{L^{2}_{t,x}}$
$\displaystyle\lesssim\left\|u_{0}\right\|_{X_{S}^{0,\frac{1}{2}}}\left\|u_{1}\right\|_{X_{S}^{0,\frac{1}{2}}}\left\|u_{2}\right\|_{L^{2}_{t,x}}.$

  2. ii)

$\max\left\\{L_{0},L_{1},M_{2}\right\\}=L_{i}$
Without loss of generality, we may assume that
$\max\left\\{L_{0},L_{1},M_{2}\right\\}=L_{0}$. From (4.5) again, the left
hand side of (4.8) is bounded by

$\left\|\sum_{\begin{subarray}{c}k_{0}=k_{1}+k_{2}\\\ (k_{0},k_{1},k_{2})\in
N_{\mathcal{NR}}\end{subarray}}\int_{\tau_{0}=\tau_{1}+\tau_{2}}\frac{\left|k_{2}\right|^{1/2}}{\left|\alpha\right|^{1/2}\left|k_{2}\right|^{1/2}\left|k_{0}+k_{1}-\frac{\beta}{\alpha}S_{1}\right|^{1/2}L_{1}^{1/2}M_{2}^{1/2}}\tilde{u}\tilde{v}d\tau_{1}\right\|_{l^{2}_{k_{0}}L^{2}_{\tau_{0}}},$

so the claim is

$\left\|\sum_{\begin{subarray}{c}k_{0}=k_{1}+k_{2}\\\ (k_{0},k_{1},k_{2})\in
N_{\mathcal{NR}}\end{subarray}}\int_{\tau_{0}=\tau_{1}+\tau_{2}}\frac{N_{max}^{\delta}}{\left|k_{0}+k_{1}-\frac{\beta}{\alpha}S_{1}\right|^{1/2}L_{1}^{1/2}M_{2}^{1/2}}\tilde{u}\tilde{v}d\tau_{1}\right\|_{l^{2}_{k_{0}}L^{2}_{\tau_{0}}}\lesssim\left\|u\right\|_{L^{2}_{t,x}}\left\|v\right\|_{L^{2}_{t,x}}.$
(4.9)

Again, we have $k_{0}$ and $k_{1}$ in the former case, but we prove in a
different way from i). There are several subcases. We first consider
$\left|k_{0}\right|\sim\left|k_{1}\right|\gg\left|k_{2}\right|\sim N_{min}$.
Since $k_{0}=k_{1}+k_{2}$, we have that
$\left|k_{0}+k_{1}-\frac{\beta}{\alpha}S_{1}\right|$ is similar to $N_{max}$.
Thus, the left hand side of (4.9) is bounded by

$\left\|\sum_{\begin{subarray}{c}k_{0}=k_{1}+k_{2}\\\ (k_{0},k_{1},k_{2})\in
N_{\mathcal{NR}}\end{subarray}}\int_{\tau_{0}=\tau_{1}+\tau_{2}}\frac{1}{\left|k_{2}\right|^{1/2-\delta}L_{1}^{1/2}M_{2}^{1/2}}\tilde{u}\tilde{v}d\tau_{1}\right\|_{l^{2}_{k_{0}}L^{2}_{\tau_{0}}}.$

By duality, the claim equivalent to

$\left|\iint_{\mathbb{R}\times\mathbb{T}}u_{0}u_{1}u_{2}dxdt\right|\lesssim\left\|u_{0}\right\|_{L^{2}_{t,x}}\left\|u_{1}\right\|_{X_{S}^{0,1/2}}\left\|u_{2}\right\|_{X^{1/2-\delta,1/2}_{W}}.$

From the Hölder inequality, (4.7) and the Sobolev embedding with the time
translation,

$\displaystyle\left|\iint_{\mathbb{R}\times\mathbb{T}}u_{0}u_{1}u_{2}dxdt\right|$
$\displaystyle\leq\left\|u_{0}\right\|_{L^{2}_{t,x}}\left\|u_{1}\right\|_{L^{4}_{t,x}}\left\|u_{2}\right\|_{L^{4}_{t,x}}$
$\displaystyle\lesssim\left\|u_{0}\right\|_{X_{S}^{0,0}}\left\|u_{1}\right\|_{X_{S}^{0,\frac{1}{2}}}\left\|u_{2}\right\|_{X^{1/2-\delta,\frac{1}{2}}_{W}},$

for sufficiently small $\delta$. The remaining cases are
$\left|k_{1}\right|\sim\left|k_{2}\right|\gg\left|k_{0}\right|$ or
$\left|k_{0}\right|\sim\left|k_{2}\right|\gg\left|k_{1}\right|$. Then we have

$\frac{N^{\delta}_{max}}{\left|k_{0}+k_{1}-\frac{\beta}{\alpha}S_{1}\right|^{1/2}}\sim\frac{N^{\delta}_{max}}{\left|k_{1}\right|^{1/2}}\sim\frac{1}{\left|k_{1}\right|^{1/2-\delta}}\ll\frac{1}{\left|k_{0}\right|^{1/2-\delta}}$
(4.10)

or

$\frac{N^{\delta}_{max}}{\left|k_{0}+k_{1}-\frac{\beta}{\alpha}S_{1}\right|^{1/2}}\sim\frac{N^{\delta}_{max}}{\left|k_{0}\right|^{1/2}}\sim\frac{1}{\left|k_{0}\right|^{1/2-\delta}}$
(4.11)

Thus, the left hand side of (4.9) is bounded by

$\left\|\sum_{\begin{subarray}{c}k_{0}=k_{1}+k_{2}\\\ (k_{0},k_{1},k_{2})\in
N_{\mathcal{NR}}\end{subarray}}\int_{\tau_{0}=\tau_{1}+\tau_{2}}\frac{1}{\left|k_{0}\right|^{1/2-\delta}L_{1}^{1/2}M_{2}^{1/2}}\tilde{u}\tilde{v}d\tau_{1}\right\|_{l^{2}_{k_{0}}L^{2}_{\tau_{0}}}$
(4.12)

for both cases. By a similar calculation of the former case, we get the goal.
More precisely, we need to show that

$\left|\iint_{\mathbb{R}\times\mathbb{T}}u_{0}u_{1}u_{2}dxdt\right|\lesssim\left\|u_{0}\right\|_{L^{2}_{t}H^{1/2-\gamma}_{x}}\left\|u_{1}\right\|_{X_{S}^{0,1/2}}\left\|u_{2}\right\|_{X^{0,1/2}_{W}}.$

By the Hölder inequality, (4.7), and the Sobolev embedding,

$\displaystyle\left|\iint_{\mathbb{R}\times\mathbb{T}}u_{0}u_{1}u_{2}dxdt\right|$
$\displaystyle\leq\left\|u_{0}\right\|_{L^{2}_{t}L^{4}_{x}}\left\|u_{1}\right\|_{L^{4}_{t,x}}\left\|u_{2}\right\|_{L^{4}_{t}L^{2}_{x}}$
$\displaystyle\lesssim\left\|u_{0}\right\|_{L^{2}_{t}H^{1/2-\gamma}_{x}}\left\|u_{1}\right\|_{X_{S}^{0,1/2}}\left\|u_{2}\right\|_{X^{0,1/2}_{W}}.$

We now consider $l^{2}_{k}L^{1}_{\tau}$ part,

$\left\|\sum_{\begin{subarray}{c}k_{0}=k_{1}+k_{2}\\\ (k_{0},k_{1},k_{2})\in
N_{\mathcal{NR}}\end{subarray}}\int_{\tau_{0}=\tau_{1}+\tau_{2}}\frac{\left|k_{2}\right|^{1/2}}{L_{0}L_{1}^{1/2}M_{2}^{1/2}}\tilde{u}\tilde{v}d\tau_{1}\right\|_{l^{2}_{k_{0}}L^{1}_{\tau_{0}}}\lesssim
N^{-\delta}_{max}\left\|u\right\|_{Y^{0}_{S}}\left\|v\right\|_{Y^{-1/2}_{W}}.$
(4.13)

First of all, from the Cauchy-Schwarz inequality and the fact that
$L_{0}=\left<\tau_{0}-\alpha k_{0}^{2}\right>,$

$\displaystyle\left\|\sum_{\begin{subarray}{c}k_{0}=k_{1}+k_{2}\\\
(k_{0},k_{1},k_{2})\in
N_{\mathcal{NR}}\end{subarray}}\int_{\tau_{0}=\tau_{1}+\tau_{2}}\frac{\left|k_{2}\right|^{1/2}}{L_{0}L_{1}^{1/2}M_{2}^{1/2}}\tilde{u}\tilde{v}d\tau_{1}\right\|_{l^{2}_{k_{0}}L^{1}_{\tau_{0}}}$
$\displaystyle\lesssim\left\|\sum_{\begin{subarray}{c}k_{0}=k_{1}+k_{2}\\\
(k_{0},k_{1},k_{2})\in
N_{\mathcal{NR}}\end{subarray}}\left\|\int_{\tau_{0}=\tau_{1}+\tau_{2}}\frac{\left|k_{2}\right|^{1/2}}{L_{0}^{15/32}L_{1}^{1/2}M_{2}^{1/2}}\tilde{u}\tilde{v}d\tau_{1}\right\|_{L^{2}_{\tau_{0}}}\left|\int_{\mathbb{R}}\frac{1}{L_{0}^{17/16}}d\tau_{0}\right|^{1/2}\right\|_{l^{2}_{k_{0}}}$
$\displaystyle\lesssim\left\|\sum_{\begin{subarray}{c}k_{0}=k_{1}+k_{2}\\\
(k_{0},k_{1},k_{2})\in
N_{\mathcal{NR}}\end{subarray}}\int_{\tau_{0}=\tau_{1}+\tau_{2}}\frac{\left|k_{2}\right|^{1/2}}{L_{0}^{15/32}L_{1}^{1/2}M_{2}^{1/2}}\tilde{u}\tilde{v}d\tau_{1}\right\|_{l^{2}_{k_{0}}L^{2}_{\tau_{0}}}.$

We can use a similar calculation of $X^{s,b}$ part. More precisely, we can
write our claim is,

$\left\|\sum_{\begin{subarray}{c}k_{0}=k_{1}+k_{2}\\\ (k_{0},k_{1},k_{2})\in
N_{\mathcal{NR}}\end{subarray}}\int_{\tau_{0}=\tau_{1}+\tau_{2}}\frac{\left|k_{2}\right|^{1/2}}{L_{0}^{15/32}L_{1}^{1/2}M_{2}^{1/2}}\tilde{u}\tilde{v}d\tau_{1}\right\|_{l^{2}_{k_{0}}L^{2}_{\tau_{0}}}\lesssim
N_{max}^{-\delta}\left\|u\right\|_{L^{2}_{t,x}}\left\|v\right\|_{L^{2}_{t,x}}.$
(4.14)

The worst case is $\max\left\\{L_{0},L_{1},M_{2}\right\\}=L_{0}$ and
$\left|k_{2}\right|\sim N_{max}$. By Lemma (4.5), the left hand side of (4.14)
is bounded by

$\left\|\sum_{\begin{subarray}{c}k_{0}=k_{1}+k_{2}\\\ (k_{0},k_{1},k_{2})\in
N_{\mathcal{NR}}\end{subarray}}\int_{\tau_{0}=\tau_{1}+\tau_{2}}\frac{\left|k_{2}\right|^{1/2}}{\left|\alpha\right|^{15/32}\left|k_{2}\right|^{15/32}\left|k_{0}+k_{1}-\frac{\beta}{\alpha}S_{1}\right|^{15/32}L_{1}^{1/2}M_{2}^{1/2}}\tilde{u}\tilde{v}d\tau_{1}\right\|_{l^{2}_{k_{0}}L^{2}_{\tau_{0}}}.$

We use (4.10) or (4.11) again, so it is bounded by (4.12).From here, we can
use the Hölder inequality, 4.7, and the Sobolev embedding. Since
$\frac{15}{32}>\frac{3}{8}$, the remaining cases can be proved by the similar
process with $X^{s,b}$ part.

Next, we consider (4.4), but it can be proved by the similar calculation and
using (4.6) instead of (4.5). In other words, we can show that

$\displaystyle\left\|\sum_{k_{0}=k_{1}-k_{2}}\int_{\tau_{0}=\tau_{1}-\tau_{2}}\frac{\left|k_{0}\right|^{1/2}}{M_{0}^{1/2}L_{1}^{1/2}L_{2}^{1/2}}\tilde{u}\tilde{v}d\tau_{1}\right\|_{l^{2}_{k_{0}}L^{2}_{\tau_{0}}}$
$\displaystyle\lesssim
N_{max}^{-\delta}\left\|u\right\|_{L^{2}_{t,x}}\left\|v\right\|_{L^{2}_{t,x}},$
(4.15)
$\displaystyle\left\|\sum_{k_{0}=k_{1}-k_{2}}\int_{\tau_{0}=\tau_{1}-\tau_{2}}\frac{\left|k_{0}\right|^{1/2}}{M_{0}L_{1}^{1/2}L_{2}^{1/2}}\tilde{u}\tilde{v}d\tau_{1}\right\|_{l^{2}_{k_{0}}L^{1}_{\tau_{0}}}$
$\displaystyle\lesssim
N^{-\delta}_{max}\left\|u\right\|_{Y^{0}_{S}}\left\|v\right\|_{Y^{0}_{S}}.$

As shown in (4.15), it is similar to (4.8) and (4.13) except for indices.
Hence, we can obtain (4.4) by (4.6) and a similar argument for (4.4). ∎

###### Proof of Proposition 4.2.

Note that Proposition 4.2 has the time growth instead of the frequency decay.
Hence, we need the following lemma.

######  Lemma 4.5 (Lemma 2.11, [Tao:2006tn]).

Let $\eta$ be a Schwartz function in time and $I$ be a time interval. If
$-1/2<b^{\prime}\leq b<1/2$, then for any interval $\left[0,T\right]\subset I$
such that $0<T<1$, we have

$\left\|\eta\left(t/T\right)u\right\|_{X^{s,b^{\prime}}\left(I\times\mathbb{T}\right)}\lesssim
T^{b-b^{\prime}}\left\|u\right\|_{X^{s,b}\left(I\times\mathbb{T}\right)}$

###### Proof.

By duality, we may assume that $0<b^{\prime}\leq b<1/2$. From the _Christ-
Kiselev lemma_ , we suffices to show that

$\left\|\eta\left(t/T\right)u\right\|_{X_{S}^{s,b^{\prime}}\left(\mathbb{R}\times\mathbb{T}\right)}\lesssim
T^{b-b^{\prime}}\left\|u\right\|_{X_{S}^{s,b}\left(\mathbb{R}\times\mathbb{T}\right)}.$

From the following estimate,

$\left<\tau-\tau_{0}-k^{2}\right>^{b}\lesssim\left<\tau_{0}\right>^{\left|b\right|}\left<\tau-k^{2}\right>^{b},$

we have

$\left\|e^{it\tau_{0}}u\right\|_{X_{S}^{s,b}\left(\mathbb{R}\times\mathbb{T}\right)}\lesssim\left<\tau_{0}\right>^{\left|b\right|}\left\|u\right\|_{X_{S}^{s,b}\left(\mathbb{R}\times\mathbb{T}\right)}.$

Since $\eta\left(t\right)$ is a Schwartz function,

$\left\|\eta\left(t\right)u\right\|_{X_{S}^{s,b}\left(\mathbb{R}\times\mathbb{T}\right)}\lesssim\left(\int_{R}\left|\hat{\eta}\left(\tau_{0}\right)\right|\left<\tau_{0}\right>^{\left|b\right|}d\tau_{0}\right)\left\|u\right\|_{X_{S}^{s,b}\left(\mathbb{R}\times\mathbb{T}\right)}\lesssim\left\|u\right\|_{X_{S}^{s,b}\left(\mathbb{R}\times\mathbb{T}\right)}.$

To simplify our argument, we first assume that $s=0$. Moreover, we may assume
$b^{\prime}=0$ by the duality with $b^{\prime}=b$ case. Thus, our claim is

$\left\|\eta\left(t/T\right)u\right\|_{L_{t}^{2}L_{x}^{2}\left(\mathbb{R}\times\mathbb{T}\right)}\lesssim
T^{b}\left\|u\right\|_{X_{S}^{0,b}\left(\mathbb{R}\times\mathbb{T}\right)}$

for $0<b<1/2$. We now consider two cases separately,
$\left<\tau-k^{2}\right>\geq 1/T$ and $\left<\tau-k^{2}\right>\leq 1/T$. In
the former case, we have

$\left\|u\right\|_{X_{S}^{0,0}\left(\mathbb{R}\times\mathbb{T}\right)}\leq
T^{b}\left\|u\right\|_{X_{S}^{0,b}\left(\mathbb{R}\times\mathbb{T}\right)}$

by the fact that $\eta$ is a Schwartz function and $0<T<1$. In the latter
case, we have

$\displaystyle\left\|\eta\left(t/T\right)u\right\|_{L_{t}^{2}L_{x}^{2}}$
$\displaystyle\leq
T^{1/2}\left\|\hat{\eta}\left(\tau\right)\right\|_{L_{\tau}^{2}}\left\|\mathcal{F}_{t}{u\left(t\right)}\left(k\right)\right\|_{L_{t}^{\infty}\ell_{k}^{2}}$
$\displaystyle\lesssim
T^{1/2}\left\|\hat{\eta}\left(\tau\right)\right\|_{L_{\tau}^{2}}\left\|\int_{\left<\tau-k^{2}\right>\leq
1/T}\left|\widetilde{u}\left(\tau,k\right)\right|d\tau\right\|_{\ell_{k}^{2}}$
$\displaystyle\lesssim
T^{1/2}\left\|\hat{\eta}\left(\tau\right)\right\|_{L_{\tau}^{2}}T^{b-1/2}\left\|\left(\int\left<\tau-k^{2}\right>^{2b}\left|\widetilde{u}\left(\tau,k\right)\right|^{2}d\tau\right)^{1/2}\right\|_{\ell_{k}^{2}}$
$\displaystyle=T^{b}\left\|u\right\|_{X_{S}^{s,b}\left(\mathbb{R}\times\mathbb{T}\right)}.$

by the Plancherel and the triangle, the Cauchy-Schwarz, and the fact that
$\eta$ is a Schwartz function.

Since the above argument does not depend $s$, we can get the same result in
$s\in\mathbb{R}$. ∎

From Lemma 4.5, we have a modified embedding,

$\left\|\eta\left(\frac{t}{T}\right)u\right\|_{L^{4}_{t,x}}\lesssim\left\|\eta\left(\frac{t}{T}\right)u\right\|_{X_{S}^{0,\frac{3}{8}}}\lesssim
T^{\gamma}\left\|u\right\|_{X_{S}^{0,3/8+\gamma}}$ (4.16)

for sufficiently small $\gamma>0$. We now get Proposition 4.2 by the similar
argument in the proof of Proposition 4.3 and (4.16) instead of (4.7). ∎

######  Remark 4.6.

In fact, we can get the Proposition 4.2 without the time growth. Hence from
Proposition 4.2 and summation with respect to each of dyadic frequency
supports and the orthogonality of dyadic decomposition, we can obtain the full
frequency estimate as follows,

$\left\|uv\right\|_{Z^{0}_{S}}\lesssim\left\|u\right\|_{Y^{0}_{S}}\left\|v\right\|_{Y^{-1/2}_{W}},$

and

$\left\|\partial_{x}\left(u\overline{v}\right)\right\|_{Z^{-1/2}_{W}}\lesssim\left\|u\right\|_{Y^{0}_{S}}\left\|v\right\|_{Y^{0}_{S}}.$
(4.17)

##  5\. Global well-posedness

In this section, we prove Theorem 1.2, the global well-posedness for (1.1). It
can be easily proved by combining the local well-posedness and the
conservation law. More precisely, after splitting the time interval as a
finite union of intervals (obviously, the length of each interval depends on
(1.2)), we get the solution for each interval by the local well-posedness (see
[Takaoka:1999uw]) and glue each solution by using the mass conservation law of
$u$. In particular, the nonlinear term of the wave part only consists of $u$,
so the mass conservation of $u$ is sufficient to glue each solution222The
details of this argument are in [Colliander:2008cq]. In fact, Colliander et
al. [Colliander:2008cq] proved the global well-posedness of the Zakharov
system on $\mathbb{R}$, but we can apply a similar argument to a torus..

We briefly explain a sketch of the proof of Theorem 1.2. It is sufficient to
prove that

$\sup_{\left|t\right|\leq
T}\left\|W\left(t\right)\left(u_{0},n_{0},n_{1}\right)\right\|_{H^{-1/2}_{x}}\lesssim
C\left(T,\left\|u_{0}\right\|_{L^{2}_{x}},\left\|n_{0}\right\|_{H_{x}^{-1/2}},\left\|n_{1}\right\|_{H_{x}^{-3/2}}\right)$
(5.1)

and

$\sup_{\left|t\right|\leq
T}\left\|S\left(t\right)\left(u_{0},n_{0},n_{1}\right)\right\|_{L^{{2}}_{x}}\lesssim
C\left(T,\left\|u_{0}\right\|_{L^{2}_{x}},\left\|n_{0}\right\|_{H_{x}^{-1/2}},\left\|n_{1}\right\|_{H_{x}^{-3/2}}\right).$
(5.2)

for any $T>0$. To prove (5.1) and (5.2), we define the norm as follows,

$\left\|W\left(t\right)\left(u_{0},n_{0},n_{1}\right)\right\|_{\mathcal{W}}=\left\|(n,\partial_{t}n)\right\|_{\mathcal{W}}:=\left(\left\|n\right\|_{H^{-1/2}}^{2}+\left\|\partial_{t}n\right\|_{H^{-3/2}}^{2}\right)^{1/2}.$
(5.3)

Let $0<\alpha<1$ be a constant, we can choose a sufficiently small time
$T^{\prime}$ such that

$\displaystyle\left\|u\right\|_{Y^{0}_{S}\left[0,T^{\prime}\right]}$
$\displaystyle\lesssim\left\|u_{0}\right\|_{L^{2}_{x}},$ (5.4)
$\displaystyle\left\|u_{0}\right\|^{2}_{L^{2}_{x}}$
$\displaystyle\ll(T^{\prime})^{-\alpha}\left\|(n_{0},n_{1})\right\|_{\mathcal{W}},$
(5.5)

and

$\left\|(n_{0},n_{1})\right\|_{\mathcal{W}}\lesssim(T^{\prime})^{-1}$ (5.6)

by (1.2). From (5.3), (2.1), (1.6), the triangular inequality, Lemma 4.1,
(4.17), and (5.4), we estimate

$\displaystyle\sup_{\left|t\right|\leq
T^{\prime}}\left\|W\left(t\right)\left(u_{0},n_{0},n_{1}\right)\right\|_{\mathcal{W}}$
$\displaystyle\lesssim\left\|\Psi\left(\frac{t}{T^{\prime}}\right)W\left(t\right)\left(u_{0},n_{0},n_{1}\right)\right\|_{Y^{-1/2}_{W}}$
(5.7)
$\displaystyle\lesssim\left\|\Psi\left(\frac{t}{T^{\prime}}\right)\partial_{t}V\left(t\right)n_{0}\right\|_{Y^{-1/2}_{W}}+\left\|\Psi\left(\frac{t}{T^{\prime}}\right)V\left(t\right)n_{1}\right\|_{Y^{-1/2}_{W}}$
$\displaystyle+\left\|\Psi\left(\frac{t}{T^{\prime}}\right)\beta^{2}\int_{0}^{t}V\left(t-s\right)\left[\partial_{x}^{2}\left(\left|u\right|^{2}\right)\right]\left(s\right)ds\right\|_{Y^{-1/2}_{W}}$
$\displaystyle\lesssim\left\|n_{0}\right\|_{H^{-1/2}_{x}}+\left\|n_{1}\right\|_{H^{-3/2}_{x}}+\left\|\partial_{x}\left(\left|u\right|^{2}\right)\right\|_{Z^{-1/2}_{W}}$
$\displaystyle\lesssim\left\|n_{0}\right\|_{H^{-1/2}_{x}}+\left\|n_{1}\right\|_{H^{-3/2}_{x}}+\left\|u\right\|^{2}_{Y_{S}^{0}}$
$\displaystyle\lesssim\left\|(n_{0},n_{1})\right\|_{\mathcal{W}}+\left\|u_{0}\right\|^{2}_{L^{2}_{x}}.$

From (5.5), we have

$\sup_{\left|t\right|\leq
T^{\prime}}\left\|(W\left(t\right)\left(u_{0},n_{0},n_{1}\right)\right\|_{\mathcal{W}}\lesssim
C\left(T^{\prime},\left\|u_{0}\right\|_{L^{2}_{x}},\left\|n_{0}\right\|_{H_{x}^{-1/2}},\left\|n_{1}\right\|_{H_{x}^{-3/2}}\right)$
(5.8)

and by the similar calculation with (1.5),

$\sup_{\left|t\right|\leq
T^{\prime}}\left\|S\left(t\right)\left(u_{0},n_{0},n_{1}\right)\right\|_{L^{{2}}_{x}}\lesssim
C\left(T^{\prime},\left\|u_{0}\right\|_{L^{2}_{x}},\left\|n_{0}\right\|_{H_{x}^{-1/2}},\left\|n_{1}\right\|_{H_{x}^{-3/2}}\right)$
(5.9)

for sufficiently small $T^{\prime}$333In (5.9), the time $T^{\prime}$ is the
same as in (5.8) because it is obtained by the local well-posedness, the fact
that the nonlinear term of the Schrödinger part has $n\left(t,x\right)$, (5.4)
and (5.8)..

We now consider the gluing step. For any time $T$, we divide the total time
interval $\left[-T,T\right]$ into time intervals such that each interval
satisfies (5.8) and (5.9). In the first such interval
$\left[0,T^{\prime}\right]$, we directly obtain (5.1) and (5.2), and in the
next interval, we let $T^{\prime}$ be the initial time and then obtain the
claim by (1.2). Hence, we can use the same iteration up to
$\left\|W(\widetilde{T})\left(u_{0},n_{0},n_{1}\right)\right\|_{\mathcal{W}}\gg\left\|u_{0}\right\|_{L^{2}_{x}}^{2}$.
By taking this time as the initial time $\widetilde{T}=0$, we can repeat the
entire procedure again. To reach the given time $T$, we need to show that time
$\widetilde{T}$ is independent of
$W\left(t\right)\left(u_{0},n_{0},n_{1}\right)$. From the final term in (5.7)
and (5.5), we can iterate $m$-times such that

$m\sim\frac{\left\|\left(n_{0},n_{1}\right)\right\|_{\mathcal{W}}}{\left\|u_{0}\right\|^{2}_{L^{2}_{x}}},$

and from (5.6), we have

$\widetilde{T}=mT\lesssim\frac{1}{\left\|u_{0}\right\|^{2}_{L^{2}_{x}}}$

which is independent of $W\left(t\right)\left(n_{0},n_{1}\right)$. Therefore,
we are done.

##  6\. Proof of Theorem 1.1

In this section, we prove Theorem 1.1, the invariant of symplectic capacity
with respect to the Zakharov flow.

###  6.1. Local approximation

We introduce a new system as follows,

$\left\\{\begin{array}[]{ll}i\partial_{t}\left(P_{\leq
N}u\right)+\alpha\partial_{x}^{2}\left(P_{\leq N}u\right)=P_{\leq
N}\left[\left(P_{\leq N}u\right)\left(P_{\leq N}n\right)\right],\\\
\beta^{-2}\partial_{t}^{2}\left(P_{\leq
N}n\right)-\partial_{x}^{2}\left(P_{\leq N}n\right)=P_{\leq
N}\left[\partial_{x}^{2}\left(\left|P_{\leq N}u\right|^{2}\right)\right],\\\
i\partial_{t}\left(\left(1-P_{\leq
N}\right)u\right)+\alpha\partial_{x}^{2}\left(\left(1-P_{\leq
N}\right)u\right)=0,\\\ \beta^{-2}\partial_{t}^{2}\left(\left(1-P_{\leq
N}\right)n\right)-\partial_{x}^{2}\left(\left(1-P_{\leq
N}\right)n\right)=0\end{array}\right.$ (6.1)

for the initial data $\left(u_{0},n_{0},n_{1}\right)\in\mathcal{H}$. Let
${Z}^{N}\left(t\right)$ be a solution flow with respect to (6.1), and its
Hamiltonian is

${H}^{N}\left[u,n,\dot{n}\right]=\int_{\mathbb{T}}\left(\alpha\left|\partial_{x}u\right|^{2}+\frac{\left|n\right|^{2}}{2}+\beta^{2}\frac{\left|i\partial_{x}^{-1}\dot{n}\right|^{2}}{2}+P_{\leq
N}n\left|P_{\leq N}u\right|^{2}\right)dx.$

Note that the new system (6.1) has the same nonlinear operator in the low
frequencies, and a linear operator in the high frequencies. Hence, the
solution map ${Z}^{N}\left(t\right)$ is a smooth symplectomorphism with the
symplectic form (3.1) on the Hilbert space $\mathcal{H}$, and the new system
has the global well-posedness as well.

######  Proposition 6.1.

For a global-in-time $T>0$ and any large integer $N$. The initial data
$\left(u_{0},n_{0},n_{1}\right)$ is in $\mathcal{H}$. Assume that
$Z\left(t\right)$ and ${Z}^{N}\left(t\right)$ be the Zakharov flow and the
solution flow for (6.1), respectively. Then we have

$\sup_{\left|s\right|\leq
t}\left\|\left(Z\left(s\right)\left(u_{0},n_{0},n_{1}\right)-{Z}^{N}\left(s\right)\left(u_{0},n_{0},n_{1}\right)\right)\right\|_{\mathcal{H}}\leq
C\left(T,\left\|\left(u_{0},n_{0},n_{1}\right)\right\|_{\mathcal{H}}\right)N^{-\delta}$

for a local-in-time $0<t\ll 1$ and $\delta>0$.

###### Proof.

We denote that ${\bf{z}}_{0}:=\left(u_{0},n_{0},n_{1}\right)$,
${\bf{z}}\left(t\right)=\left(u\left(t\right),n\left(t\right),\partial_{t}u\left(t\right)\right):=Z\left(t\right){\bf{z}}_{0}$
and
${\bf{z}}^{N}\left(t\right)=\left(u^{N}\left(t\right),n^{N}\left(t\right),\partial_{t}n^{N}\left(t\right)\right):={Z}^{N}(t){\bf{z}}_{0}$.
From the global well-posedness, there exists constant
${C}\left(T,\left\|{\bf{z}}_{0}\right\|_{\mathcal{H}}\right)$ such that

$\left\|{\bf{z}}\left(t\right)\right\|_{\mathcal{Y}}+\left\|{\bf{z}}^{N}\left(t\right)\right\|_{\mathcal{Y}}\leq{C}\left(T,\left\|{\bf{z}}_{0}\right\|_{\mathcal{H}}\right):=\mathcal{R}.$
(6.2)

We split the solution into two portions as follows,

$\displaystyle{\bf{z}}\left(t\right)={\bf{z}}_{lo}+{\bf{z}}_{hi}$
$\displaystyle:=P_{\leq N}{\bf{z}}\left(t\right)+\left(1-P_{\leq
N}\right){\bf{z}}\left(t\right)$ $\displaystyle=P_{\leq
N}u\left(t\right)+P_{\leq N}n\left(t\right)+\left(1-P_{\leq
N}\right)u\left(t\right)+\left(1-P_{\leq N}\right)n\left(t\right)$
$\displaystyle=:u_{lo}+n_{lo}+u_{hi}+n_{hi}.$

By (6.2), we also have

$\left\|{\bf{z}}_{lo}\right\|_{\mathcal{Y}}\leq\mathcal{R}~{}\text{and}~{}\left\|{\bf{z}}_{hi}\right\|_{\mathcal{Y}}\leq\mathcal{R}.$

Likewise, ${\bf{z}}^{N}$ is also split, and is bounded by $\mathcal{R}$ for
each flow. Especially, $u^{N}_{hi}$ and $n^{N}_{hi}$ are linear flow by the
definition of the new Hamiltonian system flow. By the structure of the wave
part and (2.1),

$\sup_{\left|s\right|\leq
t}\left\|Z\left(t\right)\left(u_{0},n_{0},n_{1}\right)-{Z}^{N}\left(t\right)\left(u_{0},n_{0},n_{1}\right)\right\|_{\mathcal{H}}\lesssim\left\|Z\left(t\right)\left(u_{0},n_{0},n_{1}\right)-{Z}^{N}\left(t\right)\left(u_{0},n_{0},n_{1}\right)\right\|_{\mathcal{Y}}$

The right hand side is bounded by

$\left\|\int_{0}^{t}U\left(t-s\right)\left[un-P_{\leq
N}\left(u_{lo}n_{lo}\right)\right]\left(s\right)ds\right\|_{Y_{S}^{0}}+\left\|\int_{0}^{t}V\left(t-s\right)\partial_{x}^{2}\left[\left|u\right|^{2}-P_{\leq
N}\left|u_{lo}\right|^{2}\right]\left(s\right)ds\right\|_{Y_{S}^{-1/2}}$

by the Duhamel’s formula and the fact that the initial data is same. We first
estimate the Schrödinger part. By Lemma 4.1 and the Minkowski inequality, we
have

$\displaystyle\left\|\int_{0}^{t}U\left(t-s\right)\left[un-P_{\leq
N}\left(u_{lo}n_{lo}\right)\right]ds\right\|_{Y_{S}^{0}}\lesssim\left\|un-
P_{\leq N}\left(u_{lo}n_{lo}\right)\right\|_{Z^{0}_{S}}$ (6.3)
$\displaystyle=\left\|un-P_{\leq N}\left(un\right)+P_{\leq
N}\left(un\right)-P_{\leq N}\left(u_{lo}n\right)+P_{\leq
N}\left(u_{lo}n\right)-P_{\leq
N}\left(u_{lo}n_{lo}\right)\right\|_{Z_{S}^{0}}$
$\displaystyle\leq\left\|\left(1-P_{\leq
N}\right)\left(un\right)\right\|_{Z_{S}^{0}}+\left\|P_{\leq
N}\left(\left(u-u_{lo}\right)n\right)\right\|_{Z_{S}^{0}}+\left\|P_{\leq
N}\left(u_{lo}\left(n-n_{lo}\right)\right)\right\|_{Z_{S}^{0}}$

We apply (4.3) to the first term, and (4.1) to the second term and the third
term, thus the right hand side of (6.3) is bounded by

$\displaystyle
N^{-\delta}\left\|u\right\|_{Y_{S}^{0}}\left\|n\right\|_{Y_{W}^{-1/2}}+t^{\gamma}\left\|n\right\|_{Y_{W}^{-1/2}}\left\|u-u_{lo}\right\|_{Y_{S}^{0}}+t^{\gamma}\left\|u_{lo}\right\|_{Y_{S}^{0}}\left\|n-n_{lo}\right\|_{Y_{W}^{-1/2}}$
$\displaystyle=$ $\displaystyle
N^{-\delta}\left\|u\right\|_{Y_{S}^{0}}\left\|n\right\|_{Y_{W}^{-1/2}}+t^{\gamma}\left(\left\|n\right\|_{Y_{W}^{-1/2}}\left\|u-u_{lo}\right\|_{Y_{S}^{0}}+\left\|u_{lo}\right\|_{Y_{S}^{0}}\left\|n-n_{lo}\right\|_{Y_{W}^{-1/2}}\right).$

By the global well-posedness (6.2), we have the estimate for the Schrd̈inger
part as follows,

$\left\|\int_{0}^{t}U\left(t-s\right)\left[un-P_{\leq
N}\left(u_{lo}n_{lo}\right)\right]ds\right\|_{Y_{S}^{0}}\lesssim\mathcal{R}^{2}N^{-\delta}+t^{\gamma}\mathcal{R}\left\|{\bf{z}}\left(t\right)-{\bf{z}}^{N}\left(t\right)\right\|_{\mathcal{Y}}.$

By the similar calculation with (4.4), (4.2), and the global well-posedness,
the wave part is bounded as well. Indeed,

$\displaystyle\left\|\int_{0}^{t}V\left(t-s\right)\partial_{x}^{2}\left[\left|u\right|^{2}-P_{\leq
N}\left|u_{lo}\right|^{2}\right]\left(s\right)ds\right\|_{Y_{W}^{-1/2}}$
$\displaystyle\lesssim
N^{-\delta}\left\|u\right\|^{2}_{Y_{S}^{0}}+t^{\gamma}\left\|u\right\|_{Y_{S}^{0}}\left\|u-u_{lo}\right\|_{Y_{S}^{0}}$
$\displaystyle\lesssim\mathcal{R}^{2}N^{-\delta}+t^{\gamma}\mathcal{R}\left\|{\bf{z}}\left(t\right)-{\bf{z}}^{N}\left(t\right)\right\|_{\mathcal{Y}}.$

Therefore, we have

$\displaystyle\left\|Z\left(t\right)\left(u_{0},n_{0},n_{1}\right)-{Z}^{N}\left(t\right)\left(u_{0},n_{0},n_{1}\right)\right\|_{\mathcal{Y}}$
$\displaystyle\lesssim\left\|\int_{0}^{t}U\left(t-s\right)\left[un-P_{\leq
N}\left(u_{lo}n_{lo}\right)\right]\left(s\right)ds\right\|_{Y_{S}^{0}}+\left\|\int_{0}^{t}V\left(t-s\right)\partial_{x}^{2}\left[\left|u\right|^{2}-P_{\leq
N}\left|u_{lo}\right|^{2}\right]\left(s\right)ds\right\|_{Y_{W}^{-1/2}}$
$\displaystyle\lesssim\mathcal{R}^{2}N^{-\delta}+t^{\gamma}\mathcal{R}\left\|Z\left(t\right)\left(u_{0},n_{0},n_{1}\right)-{Z}^{N}\left(t\right)\left(u_{0},n_{0},n_{1}\right)\right\|_{\mathcal{Y}}.$

Thus, choosing local-in-time $t$ such that
$t<\left(\frac{1}{\mathcal{R}}\right)^{\frac{1}{\gamma}}$, we have

$\displaystyle\left\|Z\left(t\right)\left(u_{0},n_{0},n_{1}\right)-{Z}^{N}\left(t\right)\left(u_{0},n_{0},n_{1}\right)\right\|_{\mathcal{Y}}$
$\displaystyle\lesssim
RN^{-\delta}=C\left(T,\left\|\left(u_{0},n_{0},n_{1}\right)\right\|_{\mathcal{H}}\right)N^{-\delta}.$

∎

######  Remark 6.2.

The local-in-time $t$ in Proposition 6.1 does not depend on frequency $N$. We
thus conclude that the map $[Z-{Z}^{N}]\left(t\right)$ is regarded as a small
perturbation in a sufficiently short time interval.

###  6.2. Proof of symplectic invariant

We separate the solution flow, and use an iteration argument. There exists a
local time length
$\tau\left(T,\left\|\left(u_{0},n_{0},n_{1}\right)\right\|_{\mathcal{H}}\right)>0$
such that the Zakharov flow satisfies Proposition 6.1. The global time
interval $\left[0,T\right]$ is split to
$\left[0=t_{0},t_{1}\right]\cup\left[t_{1},t_{2}\right]\cup\cdots\cup\left[t_{n-1},t_{n}=T\right]$,
and length of each interval is the constant
$\tau\left(=\left|\tau_{i+1}-\tau_{i}\right|\right)$ that depends only on the
implicit constant in Proposition 6.1. Let $\Omega_{0}$ be a initial domain
which contains the initial data $u\left(x,0\right)$. Likewise, we denote that
$\Omega_{i}\left(:=Z\left(t_{i}\right)\left(\Omega_{0}\right)\right)$ is a
domain which has the solution $u\left(x,t_{i}\right)$.
FIRST STEP (Local-time symplectic invariant)
We first prove that

${\rm cap}\left(\Omega_{1}\right)={\rm
cap}\left(Z\left(t_{1}\right)\left(\Omega_{0}\right)\right)\leq{\rm
cap}\left(\Omega_{0}\right).$ (6.4)

Let $f_{1}$ be a $m$-admissible function in $\Omega_{1}$ such that $m>{\rm
cap}\left(\Omega_{0}\right)$. From the Definition 3.6, it suffices to show
that the function $f_{1}$ is a fast function in $\Omega_{1}$. Since the fact
that the initial domain $\Omega_{0}$ is bounded and the Zakharov system has
the global well-posedness, the domain $\Omega_{1}$ is a bounded domain as
well. Denoting
$\widetilde{\Omega}_{0}=\Omega_{0}\cap\left(Z\left(t_{1}\right)\right)^{-1}\left(\Omega_{1}\right)$,
we have ${\rm cap}\left(\widetilde{\Omega}_{0}\right)\leq{\rm
cap}\left(\Omega_{0}\right)$ by Definition 3.2. Define $\varepsilon_{1}={\rm
dist}_{\Omega_{1}}\left(f_{1}\right)$, we can get a sufficiently large integer
$N$ such that

$\frac{\varepsilon_{1}}{2}>N_{1}^{-\delta}$

where $\delta$ is the implicit constant in Proposition 6.1. The Zakharov flow
is decomposed to

$\displaystyle Z\left(t_{1}\right)$
$\displaystyle=Z\left(t_{1}\right)-Z^{N_{1}}\left(t_{1}\right)+Z^{N_{1}}\left(t_{1}\right)$
$\displaystyle=\left[I+\left(Z\left(t_{1}\right)-{Z}^{N_{1}}\left(t_{1}\right)\right)\circ\left({Z}^{N_{1}}\left(t_{1}\right)\right)^{-1}\right]\circ{Z}^{N_{1}}\left(t_{1}\right)$
$\displaystyle=:\left(I+Z_{\varepsilon_{1}}\left(t_{1}\right)\right)\circ{Z}^{N_{1}}\left(t_{1}\right),$

where $I$ is an identity map from $\mathcal{H}$ to $\mathcal{H}$, and the map
$Z^{N_{1}}\left(t_{1}\right)$ is the solution map for (6.1). Note that
$I+Z_{\varepsilon_{1}}\left(t_{1}\right)$ and $Z^{N_{1}}\left(t_{1}\right)$
are smooth symplectomorphisms. In the low frequencies, the solution map
$Z^{N_{1}}\left(t_{1}\right)$ is composite operator with linear and nonlinear
solution operators which is a finite dimensional symplectomorphism. In the
high frequencies, the map are linear solution operator only, and they are
isometries on the symplectic Hilbert space $\mathcal{H}\left(=L^{2}\times
H^{-1/2}\times H^{-3/2}\right)$. Hence, the classes of $m$-admissible
functions are preserved by $Z^{N_{1}}\left(t_{1}\right)$. We thus show that

${\rm
cap}\left(\left(I+Z_{\varepsilon_{1}}\left(t_{1}\right)\right)(\acute{\Omega}_{0})\right)\leq{\rm
cap}\left(\acute{\Omega}_{0}\right).$ (6.5)

where a domain
$\acute{\Omega}_{0}=Z^{N_{1}}\left(t_{1}\right)\left(\widetilde{\Omega}_{0}\right)$.
By the decomposition of the Zakharov flow, we have

$\left(I+Z_{\varepsilon_{1}}\left(t_{1}\right)\right)(\acute{\Omega}_{0})=\Omega_{1}.$

Since an inverse operator $\left(Z^{N_{1}}\left(t_{1}\right)\right)^{-1}$ is
also bounded, the operator
$\left(Z\left(t_{1}\right)-Z^{N_{1}}\left(t_{1}\right)\right)\circ\left(Z^{N_{1}}\left(t_{1}\right)\right)^{-1}$
has an estimate

$\left\|\left(Z\left(t_{1}\right)-Z^{N_{1}}\left(t_{1}\right)\right)\circ\left(Z^{N_{1}}\left(t_{1}\right)\right)^{-1}\right\|_{\acute{\Omega}_{0}\to\Omega_{1}}\lesssim
C\left(T,\acute{\Omega}_{0}\right)N\left(\varepsilon_{1}\right)^{-\delta}=:N_{1}\left(T,\acute{\Omega}_{0},\varepsilon_{1}\right)^{-\delta}$
(6.6)

for the constant $\delta>0$ by Proposition 6.1.
Let $V_{f_{j}^{1}}$ be a vector fields of the function $f_{1}$. It suffices to
show that the vector field $V_{f_{j}^{1}}$ have a fast trajectory in the
domain $\Omega_{1}$, for large integer $j$. The function $f_{1}$ is extended
as $m$ outside $\Omega_{1}$, and provides an extended smooth function $g$ in
$\mathcal{H}$. Moreover, let $h$ be a function which is restriction $g$ to
$\acute{\Omega}_{0}$. Since the operator
$Z_{{\varepsilon}_{1}}\left(t_{1}\right)$ has a estimate (6.6), the
$\varepsilon_{1}$-neighborhood of $\partial{\Omega}_{1}$ is enclosed in the
$\frac{\varepsilon_{1}}{2}$-neighborhood of $\acute{\Omega}_{0}$, where
$h\equiv m$. Furthermore, we have
$h^{-1}\left(0\right)=f_{1}^{-1}\left(0\right)\subset\acute{\Omega}_{0}\cap{\Omega}_{1}$
by (3.5). In other words, ${\rm Supp}h$ is equal to ${\rm Supp}f_{1}$. Hence,
the function $h$ is an $m$-admissible function in $\acute{\Omega}_{0}$. Since
$m>{\rm cap}\left(\acute{\Omega}_{0}\right)$, the vector field $V_{h_{j}}$ has
a fast trajectory in $\acute{\Omega}_{0}$ for all $j\gg 1$. By Lemma 3.5, this
trajectory lies in ${\rm Supp}h$, which equals ${\rm Supp}f$ by (3.6).
Therefore, the vector field $V_{f_{j}}$ has a fast trajectory in
${\Omega}_{1}$ for all $j\gg 1$. That is, the function $f_{1}$ is fast in
${\Omega}_{1}$.
The opposite case can be shown by the same argument for the inverse operator.
Therefore, we have

${\rm cap}\left(\Omega_{1}\right)={\rm
cap}\left(Z\left(t_{1}\right)\Omega_{0}\right)={\rm
cap}\left(\Omega_{0}\right)$

for the local time $t_{1}$.
SECOND STEP (Iteration step)
Fix a domain $\Omega_{i-1}$ for any time $t_{i}$, we can get an appropriate
constant $\varepsilon_{i}$ which is depended on ${\rm
dist}_{\Omega_{i}}\left(f_{i}\right)$. Thus we have
$N_{i}\left(T,\acute{\Omega}_{i-1},\varepsilon_{i}\right)$, and so we show
that the symplectic capacity is preserved for $\left[t_{i},t_{i+1}\right]$ by
the similar argument of the first step, since the constant
$N_{i}\left(T,\acute{\Omega}_{i-1},\varepsilon_{i}\right)$ is independent of
the local time length $\tau$. Repeating the process, we have that the Zakharov
flow preserves the symplectic capacity is its the phase space for the given
global-time $T$.

## References