\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 $$\begin{gathered} \partial_t u+ia\,\partial_{x}^{2}u+b \partial_{x}^{3}u+ic |u|^{2}u +d |u|^{2}\partial_{x}u+e u^{2}\partial_{x}\bar{u}=0 \quad \text{in } \mathbb{R}^{2}, \\ u(x,0)=\varphi(x) \quad {for }\quad x\in \mathbb{R}, \end{gathered} \label{e1.1}$$ 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 $$\begin{gathered} \partial_t v+b \partial_{x}^{3}v+i(c-\frac{(d-e)a}{3b}) |v|^{2}v +d |v|^{2}\partial_{x}v+e v^{2}\partial_{x}\bar{v}=0 \quad\text{in } \mathbb{R}^{2}, \\ v(x,0)=e^{i\frac{ax}{3b}}\varphi(x) \quad\text{for } x\in \mathbb{R}. \end{gathered} \label{e1.2}$$ 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 $\frac{6}{7}>1$, 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 $$\delta\sim \|I\varphi\|_{H^{1}(\mathbb{R})}^{-\theta},\quad \theta=12+ \label{e1.3}$$ and the solution satisfies the estimate $$\|Iu||_{X_{1,\frac{1}{2}+}^{\delta}}\leq C\|I\varphi\|_{H^{1}(\mathbb{R})}. \label{e1.4}$$ \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{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{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{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} 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 $$\|u\|_{L_{x}^{2}L_t^{2}({\mathbb{R}}\times[0, \delta])}\leq \delta^{\frac{1}{2}}\|u\|_{L_t^{\infty}([0, \delta],L_{x}^{2})} \leq C\delta^{\frac{1}{2}}\|u\|_{X_{0,\frac{1}{2}+}^{\delta}}. \label{e2.19}$$ Interpolating \eqref{e2.19} with \eqref{e2.16} we get $$\|u\|_{L_{x}^{4}L_t^{4}({\mathbb{R}}\times[0, \delta])}\leq C\delta^{\frac{1}{6}}\|u\|_{X_{0,\frac{1}{2}+}^{\delta}}. \label{e2.20}$$ From \cite{g1} we have the following bilinear estimate: $$\|D_{x}^{\frac{1}{2}}I_{-}^{\frac{1}{2}}(u_{1},u_{2})\|_{L_{xt}^{2}}\leq C\|u_{1}\|_{X_{0,\frac{1}{2}+}}\|u_{2}\|_{X_{0,\frac{1}{2}+}}, \label{e2.21}$$ 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 $$\|u_{1}u_{2}\|_{L_{xt}^{2}} \leq \frac{C}{N}\|u_{1}\|_{X_{0,\frac{1}{2}+}}\|u_{2}\|_{X_{0,\frac{1}{2}+}}. \label{e2.22}$$ \end{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 $$|J_{1}|\leq C(N^{-1+}\delta^{\frac{2}{3}}+N^{-2+}) \|Iu\|_{X_{1,\frac{1}{2}+}^{\delta}}^{4}. \label{e2.25}$$ 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 $X_{s,\frac{1}{2}+}^{T}$. \begin{thebibliography}{00} \bibitem{b1} J. 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