\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 04, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/04\hfil Global well-posedness of the Cauchy problem]
{Global well-posedness of the Cauchy problem of a higher-order
Schr\"{o}dinger equation}

\author[H. Wang\hfil EJDE-2007/04\hfilneg]
{Hua Wang} 

\address{Hua Wang \newline
Department of Mathematics, Sun Yat-Sen University,
Guangzhou,  Guangdong 510275, China}
\email{wanghua\_math@126.com}

\thanks{Submitted September 7, 2006. Published January 2, 2007.}
\subjclass[2000]{35Q53, 35Q35}
\keywords{Higher order Schr\"{o}dinger equation;
 global well-posedness; I-method}
\thanks{Supported by grant 10471157 from the
China National Natural Science Foundation.}

\begin{abstract}
 We apply the I-method to prove that the Cauchy problem of
 a higher-order Schr\"{o}dinger equation is globally
 well-posed in the Sobolev space $H^{s}(\mathbb{R})$
 with $s>6/7$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

This paper concerns the Cauchy problem of the
higher order Schr\"{o}dinger equation
\begin{equation}
\begin{gathered}
  \partial_t u+ia\,\partial_{x}^{2}u+b \partial_{x}^{3}u+ic |u|^{2}u
  +d |u|^{2}\partial_{x}u+e u^{2}\partial_{x}\bar{u}=0
  \quad \text{in } \mathbb{R}^{2},
  \\
  u(x,0)=\varphi(x)  \quad {for }\quad  x\in \mathbb{R},
\end{gathered}
 \label{e1.1}
\end{equation}
where $a$, $b$, $c$, $d$ and $e$ are real constants with $be\neq 0$,
and the unknown function $u$ is a complex-valued function.

Hasegawa and Kodama \cite{h1,k2} proposed \eqref{e1.1} as the model for
propagation of pulse in optical fiber. It is easy to see that cubic
nonlinear Schr\"{o}dinger equation, nonlinear Schr\"{o}dinger
equation with derivative and complex modified KdV equation are
particular cases of \eqref{e1.1}. Therefore, in the literature, this model is
also called the Airy-Schr\"{o}dinger equation.

Well-posedness of the Cauchy problem of \eqref{e1.1} in Sobolev spaces has
been investigated by a few authors;
see for instance \cite{c1,h2,l1,s1,t1}.
 Laurey \cite{l1} proved that the Cauchy problem of \eqref{e1.1} is
locally well-posed in $H^{s}(\mathbb{R})$ for $s>3/4$. Laurey's
result was improved by Staffilani \cite{s1}, who obtained the local
well-posedness in $H^{s}(\mathbb{R})$ with $s\geq \frac{1}{4}$. This
local well-posedness combined with mass and energy conservation laws
naturally yields that \eqref{e1.1} is globally well-posed in
$H^{1}(\mathbb{R})$. Recently, using I-method introduced by Colliander, Kell,
Staffilani, Takaoka and Tao \cite{c2,c3,c4}, Carvajal \cite{c1}
 established global
well-posedness in $H^{s}(\mathbb{R})$ with $s> \frac{1}{4}$ under the
relation $c=\frac{(d-e)a}{3b}$. Our aim of this paper is to get
global well-posedness in $H^{s}(\mathbb{R})$ with $s> \frac{6}{7}$
without the above restriction condition.

Without loss of generality, we may assume that $a=0$ in the sequel. In
fact, when $a\neq 0$ we may utilize the  gauge transform \cite{c1}
$$
v(x,t)=\exp(i\frac{a}{3b}x+i\frac{a^{3}}{27b^{2}}t)u(x+\frac{a^{2}}{3b}t,t),
$$
then $u$ satisfies \eqref{e1.1} if and only if $v$ is such that
\begin{equation}
\begin{gathered}
  \partial_t v+b \partial_{x}^{3}v+i(c-\frac{(d-e)a}{3b}) |v|^{2}v
  +d |v|^{2}\partial_{x}v+e v^{2}\partial_{x}\bar{v}=0
   \quad\text{in } \mathbb{R}^{2},
  \\
  v(x,0)=e^{i\frac{ax}{3b}}\varphi(x)   \quad\text{for } x\in \mathbb{R}.
\end{gathered}
 \label{e1.2}
 \end{equation}
Note that when $c=\frac{(d-e)a}{3b}$, \eqref{e1.2} is the
complex mKdV equation satisfying a scaling invariant property. It is
well known that real mKdV possesses global well-posedness in
$H^{s}(\mathbb{R})$ with $s>1/4$ \cite{c5}. Using the same argument
as the one in \cite{c5} the same result was obtained for the complex case
\cite{c1}. Since in our case a scaling invariance disappears, thus we must
modify $I$-method suitably. Similar results as the one of this paper
were also obtained for some other nonlinear dispersive systems and
equations (e.g., \cite{p1,w1} and therein).

 To precisely state our main result, we first introduce some
notation. We use the notation $a+$ and $a-$ to respectively denote
expressions of the forms $a+\varepsilon$ and $a-\varepsilon$, where
$0<\varepsilon<<1$. We denote by $D_{x}^s$ the Riesz potential of
order $-s$, or the Fourier multiplier with symbol $|\xi|^s$ ($s>0$).
Recall that the Sobolev space $H^{s}(\mathbb{R})$ is defined by
$$
  f\in H^{s}({\mathbb{R}})\Leftrightarrow \|f\|_{H^{s}(\mathbb{R})}:=
  \|\langle\xi\rangle^{s}\hat{f}(\xi)\|_{L_{\xi}^{2}(\mathbb{R})}<\infty,
$$
where $\langle\xi\rangle^{s}:=(1+|\xi|^2)^{s/2}$, and  $\hat{f}$
represents the Fourier transformation in one variable of $f$. We define the
space $X_{s,\alpha} ({\mathbb{R}}^{2})$ (as in \cite{b1,k1}) by
$$
  u\in X_{s,\alpha}({\mathbb{R}}^{2})\Leftrightarrow
  \|u\|_{X_{s,\alpha}(\mathbb{R}^{2})}:=
  \|\langle\xi\rangle^{s}\langle\tau-\xi^{3}\rangle^{\alpha}
 \tilde{u}(\xi,\tau)\|_{L_{\tau}^{2}L_{\xi}^{2}}<\infty,
$$
where  $\tilde{u}$  represents the Fourier transformation in two
variables of $u$. For any given interval $L$, we define the space
$X_{s,\alpha}({L\times\mathbb{R}})$ to be the restriction of
$X_{s,\alpha}({\mathbb{R}}^{2})$ on ${L\times\mathbb{R}}$, with norm
$$
  \|u\|_{X_{s,\alpha}(L\times\mathbb{R})}={\rm{inf}}\{\|U\|_{X_{s,\alpha}({\mathbb{R}}^{2})}: U|_{L\times {\mathbb{R}}}=u\}.
$$
If $L=[0, T]$ (resp. $[0,\delta]$), we use $X_{s,\alpha}^{T}$ (resp.
$X_{s,\alpha}^{\delta}$) to abbreviate
$X_{s, \alpha}({L\times\mathbf{R}})$.

For given $N>>1$ and $s<1$, we define the multiplier operator
$I^s_N: H^{s} ({\mathbb{R}})\to H^{1}(\mathbb{R})$ by
$$
  (I^s_N u) \hat{} (\xi):=m_{s,N}(\xi)\hat{u}(\xi), \quad
  u\in H^{s}({\mathbb{R}}),
$$
where $m_{s,N}(\xi)$ is an even $C^{\infty}$ function,
non-increasing in $|\xi|$, and
$$
  m_{s,N}(\xi)=
  \begin{cases}
   1, & |\xi|\leq N ,\\
   \big(|\xi|/N\big)^{s-1}, & |\xi|> 2N.
 \end{cases}
$$
In the sequel, for simplicity of notation we shall omit the
superscripts and subscripts $s, N$ of the operator $I^s_N$ and the
multiplier $m_{s,N}(\xi)$.

It is obvious that for some positive constant $C$,
$$
C^{-1}\|u\|_{H^{s}(\mathbb{R})}\leq \|Iu\|_{H^{1}(\mathbb{R})}\leq
CN^{1-s}\|u\|_{H^{s}(\mathbb{R})}.
$$
We denote by $\|\cdot\|_{X_{s,\alpha,N}(\mathbb{R}^{2})}$ the
equivalent norm in $X_{s,\alpha}(\mathbb{R}^{2})$ defined by
$$
\|u\|_{X_{s,\alpha,N}(\mathbb{R}^{2})}:=\|I
u\|_{X_{1,\alpha}(\mathbb{R}^{2})}.
$$
The space $X_{s,\alpha}(\mathbb{R}^{2})$ endowed with this norm will
be re-denoted as $X_{s,\alpha,N}(\mathbb{R}^{2})$. Clearly, there also
hold the inequalities
$$
  C^{-1}\|u\|_{X_{s,\alpha}(\mathbb{R}^{2})}\leq
  \|Iu||_{X_{1,\alpha}(\mathbb{R}^{2})} \leq
  CN^{1-s}\|u\|_{X_{s,\alpha}(\mathbb{R}^{2})}.
$$
The notation $X_{s,\alpha,N}^\delta$ denotes the restriction of
$X_{s,\alpha,N}(\mathbb{R}^{2})$ on $\mathbb{R}\times [0,\delta]$.

Next we give a local well-posedness result. This local result is a
variant of that of \cite{h2,t1}, with precise estimates on the lifespan
and the norm of the solution and it can be established by the same
argument as \cite{h2,t1} and the interpolation lemma in \cite{c6}.

\begin{theorem} \label{thm1.1}
For  $\frac{6}{7}<s<1$, the initial
value problem of \eqref{e1.1} is locally well-posed in $H^s(\mathbb{R})$. More
precisely, for given $\varphi\in H^{s}(\mathbb{R})$ and $N>>1$, there
exists a corresponding $\delta>0$ such that \eqref{e1.1} has a unique
solution $u\in X_{s,\frac{1}{2}+, N}^{\delta}\subseteq
C([0,\delta],H^s(\mathbb{R}))$ satisfying the condition
$u(0,\cdot)=\varphi$. Moreover, the lifespan satisfies the estimate
\begin{equation}
\delta\sim \|I\varphi\|_{H^{1}(\mathbb{R})}^{-\theta},\quad \theta=12+
\label{e1.3}
\end{equation}
and the solution satisfies the estimate
\begin{equation}
\|Iu||_{X_{1,\frac{1}{2}+}^{\delta}}\leq
C\|I\varphi\|_{H^{1}(\mathbb{R})}. \label{e1.4}
\end{equation}
\end{theorem}

Finally, we state our main result of this paper as follows:

\begin{theorem} \label{thm1.2}
The Cauchy problem of the equation \eqref{e1.1}
is globally well-posed in $H^{s}(\mathbb{R})$ for $s>6/7$. More
precisely, let $\varphi\in H^{s}(\mathbb{R})$ with $s>6/7$. Then
for any $T>0$ the equation \eqref{e1.1} has a unique solution $u\in
X_{s,\frac{1}{2}+}^{T} \subseteq C([0,T],H^{s}(\mathbb{R}))$ satisfying
the initial condition $u(0,\cdot) =\varphi$, and the mapping
$\varphi\to u(t,\cdot)$ belongs to $C(H^s({\mathbb{R}}),
 X_{s,\frac{1}{2}+}^{T})\subseteq
 C(H^s(\mathbb{R}), C([0,T],H^s(\mathbb{R})))$.
\end{theorem}

We note that the improvement of $\theta$ in Theorem \ref{thm1.1} will
directly lead to a better Sobolev index $s$ in Theorem \ref{thm1.2}. Here we
do not pursue this although it is possible to get a smaller $\theta$
by more precise trilinear estimates of nonlinear terms in \eqref{e1.1}.


\section{The almost conserved energy}

 Laurey \cite{l1} showed that the Cauchy problem of \eqref{e1.1} has the
following two conserved quantities
\begin{gather}
M(u)=\int_{{\mathbb{R}}}|u(x,t)|^{2}dx :=M_{0}, \label{e2.1} \\
\begin{aligned}
E(u)&=k_{1}\int_{{\mathbb{R}}}|\partial_{x}u(x,t)|^{2}dx
+k_{2}\int_{{\mathbb{R}}}|u(x,t)|^{4}dx\\
&\quad +k_{3} \mathop{\rm Im}\int_{{\mathbb{R}}}u(x,t)
\overline{\partial_{x}u(x,t)}dx:=E_{0},
\end{aligned}\label{e2.2}
\end{gather}
where $k_{1}=3be$, $k_{2}=-\frac{e(e+d)}{2}$ and $k_{3}=3bc$.

Applying Gagliardo-Nirenberg inequality, Young inequality and
H\"{o}lder inequality, we have
\begin{gather}
\int_{{\mathbb{R}}}|u(x,t)|^{4}dx\leq
C\|\partial_{x}u\|_{L_{x}^{2}}\|u\|_{L_{x}^{2}}^{3} \leq \varepsilon
\|\partial_{x}u\|_{L_{x}^{2}}^{2}
+C(\varepsilon)\|\partial_{x}u\|_{L_{x}^{2}}^{6},
\label{e2.3}
\\
\int_{{\mathbb{R}}}u(x,t)\overline{\partial_{x}u(x,t)}dx
\leq C\|\partial_{x}u\|_{L_{x}^{2}}\|u\|_{L_{x}^{2}} \leq
\varepsilon
\|\partial_{x}u\|_{L_{x}^{2}}^{2}
+C(\varepsilon)\|\partial_{x}u\|_{L_{x}^{2}}^{2}.
\label{e2.4}
\end{gather}
By \eqref{e2.1}--\eqref{e2.4}, we obtain an a-priori bound of
the $H^{1}$-norm of
the solution $u$ and an upper bound of $E$
\begin{gather}
\|\partial_{x}u\|_{L_{x}^{2}}^{2}\leq C(E_{0}+M_{0}^{3}+M_{0}),
\label{e2.5} \\
|E(u)|\leq C(\|\partial_{x}u\|_{L_{x}^{2}}^{2}+M_{0}^{3}+M_{0}).
\label{e2.6}
\end{gather}
 From the local well-posedness and the a-priori bound \eqref{e2.5}, it
follows that the Cauchy problem of \eqref{e1.1} is globally well-posed in
$H^{1}(\mathbb{R})$. However, we are searching solutions in $C({\mathbb{R}},
H^s({\mathbb{R}}))$ with $s<1$, so we shall alteratively consider the
modified energy $E(Iu)$ as in Colliander et al \cite{c2,c3,c4,c5}.
We shall show
the modified energy $E(Iu)$ is almost conserved, that is, it has a
very slow increment in time if $N$ is sufficiently large. First we
give the precise expression of the increment of $E(Iu)$ in the
following lemma.

\begin{lemma} \label{lem2.1}
If $u$ is a solution of \eqref{e1.1} on
$[0,\delta]$ in the sense of Theorem \ref{thm1.1}, then
\begin{equation}
\begin{aligned}
E(Iu(\delta))- E(I\varphi)
&= 2k_{1}d
\mathop{\rm Re}\int_{0}^{\delta}\!\!\int_{{\mathbb{R}}}
\partial_{x}^{2}I\bar{u}\Big(I(|u|^{2}\partial_{x}u)-|Iu|^{2}
\partial_{x}Iu\Big)\,dx\,dt\\
&\quad + 2k_{1}e  \mathop{\rm Re}\int_{0}^{\delta}\!\!\int_{{\mathbb{R}}}
\partial_{x}^{2}I\bar{u}\Big(I(u^{2}\partial_{x}\bar{u})-(Iu)^{2}\partial_{x}I
\bar{u}\Big)\,dx\,dt\\
&\quad - 2k_{1}c  \mathop{\rm Im}\int_{0}^{\delta}\!\!\int_{{\mathbb{R}}}
\partial_{x}^{2}I\bar{u}\Big(I(|u|^{2}u)-|Iu|^{2}Iu\Big)\,dx\,dt\\
&\quad - 2k_{3}e  \mathop{\rm Im}\int_{0}^{\delta}\!\!\int_{{\mathbb{R}}}
\partial_{x}I\bar{u}\Big(I(u^{2}\partial_{x}\bar{u})-(Iu)^{2}\partial_{x}I
\bar{u}\Big)\,dx\,dt\\
&\quad - 2k_{3}d  \mathop{\rm Im}\int_{0}^{\delta}\!\!\int_{{\mathbb{R}}}
\partial_{x}I\bar{u}\Big(I(|u|^{2}\partial_{x}u)-|Iu|^{2}\partial_{x}Iu\Big)
\,dx\,dt\\
&\quad - 2k_{3}c  \mathop{\rm Re}\int_{0}^{\delta}\!\!\int_{{\mathbb{R}}}
\partial_{x}I\bar{u}\Big(I(|u|^{2}u)-|Iu|^{2}Iu\Big)\,dx\,dt\\
&\quad - 4k_{2}d  \mathop{\rm Re}\int_{0}^{\delta}\!\!\int_{{\mathbb{R}}}
|Iu|^{2}I\bar{u}\Big(I(|u|^{2}\partial_{x}u)-|Iu|^{2}\partial_{x}Iu\Big)
\,dx\,dt\\
&\quad - 4k_{2}e  \mathop{\rm Re}\int_{0}^{\delta}\!\!\int_{{\mathbb{R}}}
|Iu|^{2}I\bar{u}\Big(I(u^{2}\partial_{x}\bar{u})-(Iu)^{2}\partial_{x}I
\bar{u}\Big)\,dx\,dt\\
&\quad + 4k_{2}c  \mathop{\rm Im}\int_{0}^{\delta}\!\!\int_{{\mathbb{R}}}
|Iu|^{2}I\bar{u}\Big(I(|u|^{2}u)-|Iu|^{2}Iu\Big)\,dx\,dt.\\
\end{aligned}
\label{e2.7}
\end{equation}
\end{lemma}

\begin{proof} From \eqref{e1.1}, we have
\begin{gather*}
\partial_t Iu=-b \partial_{x}^{3}Iu-ic I(|u|^{2}u)-d I(|u|^{2}
\partial_{x}u)-e I(u^{2}\partial_{x}\bar{u}),
\\
\partial_{t}\partial_{x} Iu=-b \partial_{x}^{4}Iu-ic
\partial_{x}I(|u|^{2}u)
-d \partial_{x}I(|u|^{2}\partial_{x}u)-e \partial_{x}I(u^{2}
\partial_{x}\bar{u}),
\\
\partial_{t}\partial_{x} I\bar{u}=-b \partial_{x}^{4}I\bar{u}+ic
\partial_{x}I(|u|^{2}\bar{u})
-d \partial_{x}I(|u|^{2}\partial_{x}\bar{u})-e \partial_{x}I(\bar{u}^{2}
\partial_{x}u).
\end{gather*}
 From the above equalities and using integration by part we obtain
\begin{align*}
\frac{d}{dt} E(Iu)
&= 2k_{1} \mathop{\rm Re}\int \partial_{x}I\bar{u}\partial_{t}\partial_{x}Iu\\
&\quad +4k_{2} \mathop{\rm Re}\int
|Iu|^{2}I\bar{u}\partial_{t}Iu+k_{3} \mathop{\rm Im}\Big(\int
\partial_{x}I\bar{u}\partial_{t}Iu
+\int Iu\partial_{t}\partial_{x}I\bar{u}\Big)\\
&= 2k_{1}d \mathop{\rm Re}\int
\partial_{x}^{2}I\bar{u}I(|u|^{2}\partial_{x}u)+2k_{1}e \mathop{\rm Re}\int
\partial_{x}^{2}IuI(u^{2}\partial_{x}\bar{u})\\
&\quad -4k_{2}b \mathop{\rm Re}\int|Iu|^{2}I\bar{u}\partial_{x}^{3}Iu
  - 2k_{1}c  \mathop{\rm Im}\int
\partial_{x}^{2}I\bar{u}I(|u|^{2}u)\\
&\quad -2k_{3}d \mathop{\rm Im}\int
\partial_{x}I\bar{u}I(|u|^{2}\partial_{x}u)
-2k_{3}e \mathop{\rm Im}\int
\partial_{x}I\bar{u}I(u^{2}\partial_{x}\bar{u})\\
&\quad - 2k_{3}c \mathop{\rm Re}\int
\partial_{x}I\bar{u}I(|u|^{2}u)+4k_{2}c  \mathop{\rm Im}\int
|Iu|^{2}I\bar{u}I(|u|^{2}u)\\
&\quad - 4k_{2}d  \mathop{\rm Re}\int
|Iu|^{2}I\bar{u}I(|u|^{2}\partial_{x}u)-4k_{2}e  \mathop{\rm Re}\int
|Iu|^{2}I\bar{u}I(|u|^{2}\partial_{x}\bar{u}).
\end{align*} % \label{e2.8}
We note that
$$
\mathop{\rm Re}\int
\partial_{x}^{2}I\bar{u}(Iu)^{2}\partial_{x}I\bar{u}=\mathop{\rm Re}\int
\partial_{x}^{2}I\bar{u}|Iu|^{2}\partial_{x}Iu
$$
and
$$
\mathop{\rm Re}\int |Iu|^{2}I\bar{u}\partial_{x}^{3}Iu=
-\mathop{\rm Re}\int
\partial_{x}^{2}I\bar{u}(Iu)^{2}\partial_{x}I\bar{u}-2
\mathop{\rm Re}\int
\partial_{x}^{2}I\bar{u}|Iu|^{2}\partial_{x}Iu.
$$
Hence
\begin{equation}
\begin{aligned}
&2k_{1}d \mathop{\rm Re}\int \partial_{x}^{2}I\bar{u}I(|u|^{2}
\partial_{x}u)+2k_{1}e \mathop{\rm Re}\int
\partial_{x}^{2}I\bar{u}I(u^{2}\partial_{x}\bar{u})\\
& -4k_{2}b\mathop{\rm Re}\int|Iu|^{2}I\bar{u}\partial_{x}^{3}Iu
\\
&= 2k_{1}d \mathop{\rm Re}\int
\partial_{x}^{2}I\bar{u}\Big(I(|u|^{2}\partial_{x}u)-|Iu|^{2}
\partial_{x}Iu\Big)\\
&\quad + 2k_{1}e \mathop{\rm Re}\int
\partial_{x}^{2}I\bar{u}\Big(I(u^{2}\partial_{x}\bar{u})
-(Iu)^{2}\partial_{x}I\bar{u}\Big).
\end{aligned} \label{e2.9}
\end{equation}
Integration by part yields
$$
\mathop{\rm Im}\int \partial_{x}^{2}I\bar{u}|Iu|^{2}Iu
=-\mathop{\rm Im}\int (\partial_{x}I\bar{u})^{2}(Iu)^{2}.
$$
It follows from the above equality that
\begin{align*}
&-2k_{1}c  \mathop{\rm Im}\int
\partial_{x}^{2}I\bar{u}I(|u|^{2}u)-2k_{3}d \mathop{\rm Im}\int
\partial_{x}I\bar{u}I(|u|^{2}\partial_{x}u)
-2k_{3}e \mathop{\rm Im}\int
\partial_{x}I\bar{u}I(u^{2}\partial_{x}\bar{u})\\
&=-2k_{1}c  \mathop{\rm Im}\int
\partial_{x}^{2}I\bar{u}\Big(I(|u|^{2}u)-|Iu|^{2}Iu\Big)\\
&\quad -2k_{3}d \mathop{\rm Im}\int
\partial_{x}I\bar{u}\Big(I(|u|^{2}\partial_{x}u)-|Iu|^{2}\partial_{x}Iu\Big)\\
&\quad -2k_{3}e \mathop{\rm Im}\int
\partial_{x}I\bar{u}\Big(I(u^{2}\partial_{x}\bar{u})
-(Iu)^{2}\partial_{x}I\bar{u}\Big).
\end{align*} %\label{e2.10}
Observe that
\begin{gather}
\mathop{\rm Re}\int \partial_{x}I\bar{u}|Iu|^{2}Iu=0, \label{e2.11}\\
\mathop{\rm Im}\int |Iu|^{2}I\bar{u}|Iu|^{2}Iu=0, \label{e2.12} \\
\mathop{\rm Re}\int |Iu|^{2}I\bar{u}|Iu|^{2}\partial_{x}Iu=0, \label{e2.13}\\
\mathop{\rm Re}\int |Iu|^{2}I\bar{u}(Iu)^{2}\partial_{x}I\bar{u}=0.
\label{e2.14}
\end{gather}
Hence by \eqref{e2.9}-\eqref{e2.14}, we have
\begin{align*}
\frac{d}{dt}  E(Iu)
&= 2k_{1}d  \mathop{\rm Re}\int_{{\mathbb{R}}}\partial_{x}^{2}
 I\bar{u}\Big(I(|u|^{2}\partial_{x}u)-|Iu|^{2}\partial_{x}Iu\Big)dx\\
&\quad +2k_{1}e  \mathop{\rm Re}\int_{{\mathbb{R}}}\partial_{x}^{2}
 I\bar{u}\Big(I(u^{2}\partial_{x}\bar{u})-(Iu)^{2}\partial_{x}I\bar{u}\Big)dx\\
&\quad -2k_{1}c  \mathop{\rm Im}\int_{{\mathbb{R}}}\partial_{x}^{2}
 I\bar{u}\Big(I(|u|^{2}u)-|Iu|^{2}Iu\Big)dx \\
&\quad -2k_{3}e  \mathop{\rm Im}\int_{{\mathbb{R}}}\partial_{x}
 I\bar{u}\Big(I(u^{2}\partial_{x}\bar{u})-(Iu)^{2}\partial_{x}I\bar{u}\Big)dx\\
&\quad -2k_{3}d  \mathop{\rm Im}\int_{{\mathbb{R}}}\partial_{x}
 I\bar{u}\Big(I(|u|^{2}\partial_{x}u)-|Iu|^{2}\partial_{x}Iu\Big)dx\\
&\quad -2k_{3}c  \mathop{\rm Re}\int_{{\mathbb{R}}}\partial_{x}I\bar{u}
 \Big(I(|u|^{2}u)-|Iu|^{2}Iu\Big)dx\\
&\quad -4k_{2}d  \mathop{\rm Re}\int_{{\mathbb{R}}}|Iu|^{2}I\bar{u}\Big(I(|u|^{2}
  \partial_{x}u)-|Iu|^{2}\partial_{x}Iu\Big)dx \\
&\quad -4k_{2}e  \mathop{\rm Re}\int_{{\mathbb{R}}}|Iu|^{2}I\bar{u}\Big(I(u^{2}\partial_{x}\bar{u})-(Iu)^{2}\partial_{x}I\bar{u}\Big)dx\\
&\quad +4k_{2}c  \mathop{\rm Im}\int_{{\mathbb{R}}}|Iu|^{2}I\bar{u}
  \Big(I(|u|^{2}u)-|Iu|^{2}Iu\Big)dx.
\end{align*} %\label{e2.15}
Integrating both sides of the above expression, over the interval $[0,\delta]$,
we obtain \eqref{e2.7}.
\end{proof}

Next we apply Lemma \ref{lem2.1} to deduce an exact estimate on the increment
of the modified energy $E(Iu)$ in terms of the norm
$\|Iu\|_{X_{1,\frac{1}{2}+}^{\delta}}$. Before stating the result,
we give a few simple preliminary estimates.

The following embedding inequality is established in \cite{k1}:
\begin{gather}
\|u\|_{L_{xt}^{8}}\leq C\|u\|_{X_{0,\frac{1}{2}+}}, \label{e2.16}\\
\|u\|_{L_{t}^{\infty}L_{x}^{2}}\leq C\|u\|_{X_{0,\frac{1}{2}+}},
\label{e2.17}\\
\|D_{x}^{\frac{1}{6}}u\|_{L_{xt}^{6}}\leq
C\|u\|_{X_{0,\frac{1}{2}+}}. \label{e2.18}
\end{gather}
By H\"{o}lder inequality and \eqref{e2.17}, we have
\begin{equation}
\|u\|_{L_{x}^{2}L_t^{2}({\mathbb{R}}\times[0, \delta])}\leq
\delta^{\frac{1}{2}}\|u\|_{L_t^{\infty}([0, \delta],L_{x}^{2})} \leq
C\delta^{\frac{1}{2}}\|u\|_{X_{0,\frac{1}{2}+}^{\delta}}.
\label{e2.19}
\end{equation}
Interpolating \eqref{e2.19} with \eqref{e2.16} we get
\begin{equation}
\|u\|_{L_{x}^{4}L_t^{4}({\mathbb{R}}\times[0, \delta])}\leq
C\delta^{\frac{1}{6}}\|u\|_{X_{0,\frac{1}{2}+}^{\delta}}.
\label{e2.20}
\end{equation}
 From \cite{g1} we have the following bilinear estimate:
\begin{equation}
\|D_{x}^{\frac{1}{2}}I_{-}^{\frac{1}{2}}(u_{1},u_{2})\|_{L_{xt}^{2}}\leq
C\|u_{1}\|_{X_{0,\frac{1}{2}+}}\|u_{2}\|_{X_{0,\frac{1}{2}+}},
\label{e2.21}
\end{equation}
where
\[
\big({I_{-}^{\alpha}}(u_{1}u_{2})\big) \hat{} (\xi,\tau)=
\int_{\xi=\xi_{1}+\xi_{2},\,\tau=\tau_{1}+\tau_{2}}
|\xi_{1}-\xi_{2}|^{\alpha}\tilde{u_{1}}(\xi_{1},\tau_{1})
\tilde{u_{2}}(\xi_{2},\tau_{2})d\xi_{1}d\tau_{1}.
\]
 Also, we need the following refined Strichartz estimate.

\begin{lemma} \label{lm2.2}
Let $u_{1}$, $u_{2}$ be such that
$\mathop{\rm supp} u_{1}\subset\{|\xi|\sim N\}$ and
$\mathop{\rm supp} u_{2}\subset\{|\xi|\ll N\}$, then
\begin{equation}
\|u_{1}u_{2}\|_{L_{xt}^{2}}
\leq \frac{C}{N}\|u_{1}\|_{X_{0,\frac{1}{2}+}}\|u_{2}\|_{X_{0,\frac{1}{2}+}}.
\label{e2.22}
\end{equation}
\end{lemma}
It is not difficult to prove the above, using the same argument as
the one of \cite[Lemma 7.1]{c2}, so we omit it.

\begin{lemma} \label{lem2.3}
If $u$ is the solution of \eqref{e1.1} on $[0,\delta]$ in the sense
of Theorem \ref{thm1.1}, then
\begin{align*}
&|E(Iu(\delta))- E(I\varphi)|\\
&\leq C(N^{-1+}\delta^{\frac{2}{3}}+N^{-2+})
\|Iu\|_{X_{1,\frac{1}{2}+}^{\delta}}^{4}
+C(N^{-\frac{5}{2}+}\delta^{\frac{1}{2}}+N^{-3+})
\|Iu\|_{X_{1,\frac{1}{2}+}^{\delta}}^{6}.
\end{align*} % \label{e2.23}}
\end{lemma}

\begin{proof} We denote the nine terms on the right-hand side of
\eqref{e2.7} in their appearing order by $J_{1}$, $J_{2}$, $\dots$,
$J_{9}$, respectively. In the sequel we only consider $J_{1}$ and
$J_{7}$ because the other terms can be readily controlled by the
bound of $J_{1}$ and $J_{7}$.
\smallskip

\noindent\textbf{Estimate of  $J_{1}$.}
It suffices to prove that for
any  $(u_1,u_2,u_3,u_4)\in C([0,\delta],S({\mathbb{R}}))^4$ such
that the frequency support of each $u_j$ is located in a dyadic band
$|\xi|\sim N_{j}$ (i.e., $C_1 N_j\leq |\xi|\leq C_2 N_j$) for some
positive numbers $N_j$ ($j=1,2,3,4$), there holds
\begin{align*}
I_{1}&:=\int_{0}^{\delta}\Big(\int_{\ast}
\Big|\frac{m(\xi_{1}+\xi_{2}+\xi_{3})-m(\xi_{1})m(\xi_{2})m(\xi_{3})}
{m(\xi_{1})m(\xi_{2})m(\xi_{3})}\Big|\\
&\quad\times |\xi_{3}|\xi_{4}^{2}
|\hat{u_{1}}(\xi_{1},t)\hat{u_{2}}(\xi_{2},t)
\hat{u_{3}}(\xi_{3},t)\hat{u_{4}}(\xi_{4},t)|\Big)dt
\\
&\leq C(N^{-1+}\delta^{\frac{2}{3}}+N^{-2+})N_{\rm max}^{0-}
\prod_{j=1}^{4}\|u_{j}\|_{X_{1,\frac{1}{2}+}^{\delta}}.
\end{align*} %\label{e2.24}
where $N_{\rm max}=\max\{N_1,N_2,N_3,N_4\}$ and $\ast$ denotes
integration on the set $\sum _{j=1}^{4}\xi_j=0$. Indeed, once this
estimate is proved, then the Littlewood-Paley decomposition
immediately implies that
\begin{equation}
  |J_{1}|\leq C(N^{-1+}\delta^{\frac{2}{3}}+N^{-2+})
\|Iu\|_{X_{1,\frac{1}{2}+}^{\delta}}^{4}. \label{e2.25}
\end{equation}
First. All the frequencies are equivalent, namely,
$|\xi_{1}|\sim |\xi_{2}|\sim |\xi_{3}|\sim |\xi_{4}|\geq CN$.
Using H\"{o}lder inequality and \eqref{e2.20} we see that
\begin{align*}
I_{1} &\leq
C(\frac{N_{1}}{N})^{3(1-s)}N_{3}N_{4}^{2}
\prod_{j=1}^{4}\|u_{j}\|_{L_{x}^{4}L_t^{4}({\mathbb{R}}\times[0, \delta])}\\
&\leq C(\frac{N_{1}}{N})^{3(1-s)}N_{3}N_{4}^{2}(N_{1}N_{2}N_{3}
 N_{4})^{-1}\delta^{\frac{2}{3}}
\prod_{j=1}^{4}\|u_{j}\|_{X_{1,\frac{1}{2}+}^{\delta}}\\
&\leq C\delta^{\frac{2}{3}}N^{-1+}N_{\rm max}^{0-}
 \prod_{j=1}^{4}\|u_{j}\|_{X_{1,\frac{1}{2}+}^{\delta}}.
\end{align*}

Second. Three of the frequencies are equivalent.
We shall deal with the most difficult case
$|\xi_{1}|\sim |\xi_{3}|\sim |\xi_{4}|\geq CN$ and
$|\xi_{2}|\ll |\xi_{1}|,|\xi_{3}|,|\xi_{4}|$.
The other two cases  $|\xi_{1}|\sim |\xi_{2}|\sim |\xi_{3}|\geq CN$ and
$|\xi_{1}|\sim |\xi_{2}|\sim |\xi_{4}|\geq CN$ can be solved easily
 by the same argument as the case $1^\circ$ and the difficult case,
 respectively. Since
$\xi_{1}+\xi_{2}+\xi_{3}+\xi_{4}=0$, the largest two of the
frequencies must have different sign. We may assume that they are
$\xi_{1}$ and $\xi_{4}$ for the other cases can be considered
similarly. Thus we have
$$
N_{4}\sim |\xi_{1}-\xi_{4}|\sim N_{3}\sim |\xi_{3}+\xi_{2}|\sim
|\xi_{1}+\xi_{4}|.
$$
Utilizing \eqref{e2.21} and \eqref{e2.22}, we obtain
\begin{align*}
I_{1}
&\leq C(\frac{N_{1}}{N})^{2(1-s)}\langle(\frac{N_{2}}{N})^{1-s}\rangle
N_{3}N_{4}\|D_{x}^{\frac{1}{2}}I_{-}^{\frac{1}{2}}(u_{1},u_{4})\|_{L_{xt}^{2}}
\|u_{2}u_{3}\|_{L_{xt}^{2}}\\
&\leq
C(\frac{N_{1}}{N})^{2(1-s)}\langle(\frac{N_{2}}{N})^{1-s}\rangle
N_{3}N_{4}N_{1}^{-1}\langle N_{2}\rangle^{-1}N_{3}^{-2}N_{4}^{-1}
\prod_{j=1}^{4}\|u_{j}\|_{X_{1,\frac{1}{2}+}^{\delta}}\\
&\leq
CN^{-2+}N_{\rm max}^{0-}\prod_{j=1}^{4}\|u_{j}\|_{X_{1,\frac{1}{2}+}^{\delta}}.
\end{align*}

Third. Exact two of the frequencies are equivalent.
We only consider the most difficult case $|\xi_{1}|\sim
|\xi_{4}|\geq CN$ and $|\xi_{2}|, |\xi_{3}|\ll |\xi_{1}|,
|\xi_{4}|$.

$3.1^\circ$\ \ $|\xi_{2}|,|\xi_{3}|\leq N$
Applying the mean value theorem and \eqref{e2.22} yield
\begin{align*}
I_{1}
&\leq C\frac{N_{2}+N_{3}}{N_{1}} N_{3}N_{4}^{2}
\|u_{1}u_{2}\|_{L_{xt}^{2}}\|u_{3}u_{4}\|_{L_{xt}^{2}}\\
&\leq C\frac{N_{2}+N_{3}}{N_{1}}
N_{3}N_{4}^{2}N_{1}^{-2}N_{4}^{-2}\langle N_{2}\rangle^{-1}\langle
N_{3}\rangle^{-1}
\prod_{j=1}^{4}\|u_{j}\|_{X_{1,\frac{1}{2}+}^{\delta}}\\
&\leq
CN^{-2+}N_{\rm max}^{0-}\prod_{j=1}^{4}\|u_{j}\|_{X_{1,\frac{1}{2}+}^{\delta}}.
\end{align*}
$3.2^\circ$\ \ $|\xi_{2}|\geq N$ ( $|\xi_{3}|\geq N$ can be
considered with the same argument).

By \eqref{e2.22} we obtain
\begin{align*}
I_{1}
&\leq C(\frac{N_{2}}{N})^{1-s}\langle
(\frac{N_{3}}{N})^{1-s}\rangle N_{3}N_{4}^{2}
\|u_{1}u_{2}\|_{L_{xt}^{2}}\|u_{3}u_{4}\|_{L_{xt}^{2}}\\
&\leq C(\frac{N_{2}}{N})^{1-s}\langle (\frac{N_{3}}{N})^{1-s}\rangle
N_{3}N_{4}^{2}N_{1}^{-2}N_{4}^{-2}\langle N_{2}\rangle^{-1}\langle
N_{3}\rangle^{-1}
\prod_{j=1}^{4}\|u_{j}\|_{X_{1,\frac{1}{2}+}^{\delta}}\\
&\leq
CN^{-2+}N_{\rm max}^{0-}\prod_{j=1}^{4}\|u_{j}\|_{X_{1,\frac{1}{2}+}^{\delta}}.
\end{align*}

\noindent\textbf{Estimate of $J_{7}$.}
 Similarly as before we only need
to prove that for any triple $(u_1,u_2,\dots,u_6)$ similar as before
there holds
\begin{align*}
I_{7}&:=\int_{0}^{\delta}\Big(\int_{\ast}\Big|
\frac{m(\xi_{4}+\xi_{5}+\xi_{6})-m(\xi_{4})m(\xi_{5})m(\xi_{6})}
{m(\xi_{4})m(\xi_{5})m(\xi_{6})}\Big||\xi_{6}|\prod_{j=1}^{6}
|\hat{u_{j}}(\xi_{j},t)|
dt
\\
&\leq C(N^{-\frac{5}{2}+}\delta^{\frac{1}{2}}+N^{-3+})N_{\rm max}^{0-}
\prod_{j=1}^{6}\|u_{j}\|_{X_{1,\frac{1}{2}+}^{\delta}}.
\end{align*} %\label{e2.26}}
where $N_{\rm max}=\max\{N_1,N_2,\dots,N_6\}$ and $\ast$ denotes
integration on the set $\sum _{j=1}^{6}\xi_j=0$.

First. At least three of $\xi_{i}$'s satisfy $|\xi_{i}|\geq CN$.
Let the largest three of $|\xi_{i}|$ be $N_{1}^{\ast}$,
$N_{2}^{\ast}$ and $N_{3}^{\ast}$. Then by H\"{o}lder inequality
$L_{x,t}^{6}-L_{x,t}^{6}-L_{x,t}^{6}-L_{x,t}^{2}-L_{x,t}^{\infty}-L_{x,t}^{\infty}$,
\eqref{e2.17}, \eqref{e2.18} and Sobolev embedding we have
\begin{align*}
I_{7} &\leq C(\frac{N_{1}^{\ast}}{N})^{1-s}(\frac{N_{2}^{\ast}}{N})^{1-s}
(\frac{N_{3}^{\ast}}{N})^{1-s}
N_{1}^{\ast}{N_{1}^{\ast}}^{-\frac{7}{6}}{N_{2}^{\ast}}^{-\frac{7}{6}}{N_{3}^{\ast}}^{-\frac{7}{6}}
\delta^{\frac{1}{2}}\prod_{j=1}^{6}\|u_{j}\|_{X_{1,\frac{1}{2}+}^{\delta}}\\
&\leq
C{N_{1}^{\ast}}^{\frac{5}{6}-s}{N_{2}^{\ast}}^{-\frac{1}{6}-s}{N_{3}^{\ast}}^{-\frac{1}{6}-s}
\delta^{\frac{1}{2}}\prod_{j=1}^{6}\|u_{j}\|_{X_{1,\frac{1}{2}+}^{\delta}}\\
&\leq
CN^{-\frac{5}{2}+}\delta^{\frac{1}{2}}
N_{\rm max}^{0-}\prod_{j=1}^{6}\|u_{j}\|_{X_{1,\frac{1}{2}+}^{\delta}}.
\end{align*}

Second. Exactly two of $|\xi_{i}|\geq CN$ and the others $\ll
N$.
For example, $|\xi_{4}|,|\xi_{6}|\geq CN$. Then , using Sobolev
embedding, \eqref{e2.17} and \eqref{e2.22}, we get
\begin{align*}
I_{7}
&\leq C(\frac{N_{4}}{N})^{1-s}(\frac{N_{6}}{N})^{1-s} N_{6}
\|u_{1}u_{4}\|_{L_{xt}^{2}}\|u_{2}u_{6}\|_{L_{xt}^{2}}
\|u_{3}\|_{L_{xt}^{\infty}}\|u_{5}\|_{L_{xt}^{\infty}}\\
&\leq C(\frac{N_{4}}{N})^{1-s}(\frac{N_{6}}{N})^{1-s}
N_{6}N_{4}^{-2}N_{6}^{-2}
\prod_{j=1}^{6}\|u_{j}\|_{X_{1,\frac{1}{2}+}^{\delta}}\\
&\leq
CN^{-3+}N_{\rm max}^{0-}\prod_{j=1}^{6}\|u_{j}\|_{X_{1,\frac{1}{2}+}^{\delta}}.
\end{align*}
\end{proof}

\section{Proof of Theorem \ref{thm1.1}}

For completeness, we give the proof of Theorem \ref{thm1.1} in this section
(see also \cite{p1,w1}). For any fixed $T>0$, we want to construct the
solution of the time initial value \eqref{e1.1} on the interval $[0, T]$.

Since $\|I\varphi\|_{H^{1}(\mathbb{R})}^{2}\leq CN^{2(1-s)}$, it follows
from \eqref{e2.6} that
$$
|E(I\varphi)|\leq C'N^{2(1-s)}\leq 2C'N^{2(1-s)},
$$
which, by \eqref{e2.5}, implies
$\|I\varphi\|_{H^{1}(\mathbb{R})}^{2}\leq \hat{C}N^{2(1-s)}$
with $\hat{C}=\hat{C}(2C')$.
Applying Theorem \ref{thm1.1} we know that the solution $u$ exists on
$[0,\delta]$ with
\begin{gather*}
  \delta\geq C''\|I\varphi\|_{H^{1}(\mathbb{R})}^{-\theta}
  \geq C''\big(\hat{C}N^{(1-s)}\big)^{-\theta}
  =C_0 N^{-(1-s)\theta},\\
\|Iu(t)\|_{X_{1,\frac{1}{2}+}^{\delta}}\leq
C\|I\varphi\|_{H^{1}(\mathbb{R})}\leq\hat{C}CN^{1-s}\quad \mbox{for }
0\leq t\leq\delta.
\end{gather*}
By Lemma \ref{lem2.3}, we have
$$
|E(Iu(\delta))- E(I\varphi)|\leq
C'''[(N^{-1+}\delta^{\frac{2}{3}}+N^{-2+})
N^{4(1-s)}+(N^{-\frac{5}{2}+}\delta^{\frac{1}{2}}+N^{-3+})
N^{6(1-s)}],
$$
where $C'''$ depends only on $\hat{C}C$. As long as
$$
  C'''[(N^{-1+}\delta^{\frac{2}{3}}+N^{-2+})
N^{4(1-s)}+(N^{-\frac{5}{2}+}\delta^{\frac{1}{2}}+N^{-3+})
N^{6(1-s)}]\leq C'N^{2(1-s)},
$$
we have
$$
  |E(Iu(\delta))|\leq 2C'N^{2(1-s)}.
$$
It follows, by considering $\delta$ as the initial time, using $I
u(\delta)$ as the initial value, and applying Theorem \ref{thm1.1}, that the
problem \eqref{e1.1} has a solution on ${\mathbb{R}}\times [\delta, 2\delta]$.
In this way we succeed to extend the solution of \eqref{e1.1} to the time
interval $[0,2\delta]$.

The above argument can be repeated for $K$ steps as long as the
following condition on $K$ is satisfied:
$$
 C'''[(N^{-1+}\delta^{\frac{2}{3}}+N^{-2+})
N^{4(1-s)}+(N^{-\frac{5}{2}+}\delta^{\frac{1}{2}}+N^{-3+})
N^{6(1-s)}]K\leq C'N^{2(1-s)}.
$$
In order to extend the solution to the time interval $[0,T]$, we
must have $K\delta\geq T$, or $K\geq T\delta^{-1}$. Since the
minimum of all such $K$ satisfies $K\sim T\delta^{-1}$, to arrive at
this goal we only need to have
$$
  CC'''[(N^{-1+}\delta^{\frac{2}{3}}+N^{-2+})
N^{4(1-s)}+(N^{-\frac{5}{2}+}\delta^{\frac{1}{2}}+N^{-3+})
N^{6(1-s)}]T\delta^{-1}\leq C'N^{2(1-s)}.
$$
Since $\delta\geq C_0 N^{-(1-s)\theta}$, this can be fulfilled if we
can choose a sufficiently large number $N$ so that
\begin{align*}
&CC'''C_0^{-1}[(N^{-1+}N^{\frac{(1-s)\theta}{3}}+N^{-2+}N^{(1-s)
 \theta})N^{4(1-s)}\\
&+(N^{-\frac{5}{2}+}N^{\frac{(1-s)\theta}{2}}
 +N^{-3+}N^{(1-s)\theta})N^{6(1-s)}]T\\
&\leq C'N^{2(1-s)}.
\end{align*}
Though direct computation, we know that the above condition is
satisfied if $s>6/7$.
Hence, the solution exists on ${\mathbb{R}}\times [0,T]$ for any $T>0$,
and it belongs to and is unique in $X_{s,\frac{1}{2}+}^{T}$.

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\end{document}