---
author:
- |
    Wengu Chen$^1$, Junfeng Li$^{1,\,2}$ and Changxing Miao$^1$\
    $^1$Institute of Applied Physics and Computational Mathematics\
    P.O.Box 8009, Beijing 100088, China\
    $^2$College of Mathematics, Beijing Normal University,\
    Beijing 100875, China
date: 'E-mail:chenwg@iapcm.ac.cn, junfli@yahoo.com.cn, miao$_-$changxing@iapcm.ac.cn'
title: '**The well-posedness of Cauchy problem for dissipative modified Korteweg de Vries equations [^1]**'
---

[**Abstract.**]{} In this paper we consider some dissipative versions of the modified Korteweg de Vries equation $u_t+u_{xxx}+|D_x|^{\alpha}u+u^2u_x=0$ with $0<\alpha\leq 3$. We prove some well-posedness results on the associated Cauchy problem in the Sobolev spaces $H^s({\Bbb R})$ for $s>1/4-\alpha/4$ on the basis of the $[k;\,Z]-$multiplier norm estimate obtained by Tao in [@Tao] for KdV equation.\
[**2000 Mathematics Subject Classification:**]{} 35Q53, 35A07.

Introduction
============

The $X^{s,\,b}$ spaces, as used by Beals, Bourgain, Kenig-Ponce-Vega, Klainerman-Machedon and others, are fundamental tools to study the low-regularity behavior of nonlinear dispersive equations. It is of particular interest to obtain bilinear or multilinear estimates involving these spaces. By Plancherel’s theorem and duality, these estimates reduce to estimating a weighted convolution integral in terms of the $L^2$ norms of the component functions. In [@Tao], Tao systematically studied the weighted convolution estimates on $L^2$. As a consequence, Tao obtained sharp bilinear estimates for the KdV, wave, and Schrödinger $X^{s,\,b}$ spaces. These estimates are usable for many applications, and Tao in his paper presented some selected applications of these estimates to prove multilinear estimates and local well-posedness results. In this paper, we will get some bilinear estimate on some new type of $X^{s,\,b}$ spaces adapted to the dissipative version of the modified Korteweg de Vries equation and obtain new local well-posedness of associated Cauchy problem. This new type of Bourgain spaces was first introduced by Molinet and Ribaud in [@MR1]. We defer to introduce Tao’s estimates here.

Recently, Molinet and Ribaud consider the Cauchy problem associated with dissipative Korteweg de Vries equations $$\begin{aligned}
\left\{\begin{array}{ll} u_t+u_{xxx}+|D_x|^\alpha u+uu_x=0,\,t\in
{\Bbb
R}_+,\,x\in{\Bbb R},\\
u(0)=\varphi,\end{array}\right.\end{aligned}$$ where $|D_x|^\alpha$ denotes the Fourier multiplier operator with symbol $|\xi|^\alpha$. When $\alpha=1/2$, equation (1) models the evolution of the free surface for shallow water waves damped by viscosity. When $\alpha=2$, equation (1) is the so-called Korteweg de Vries Burgers equation which described the propagation of small amplitude long waves in some nonlinear dispersive media when dissipative effects occur (see[@OS]).

In [@MR1], Molinet and Ribaud proved the global well-posedness of (1) for data in $H^s({\Bbb R}),\,s>-\frac 34$ for all $\alpha>0$. Especially when $\alpha=2$, they proved the global well-posedness of Korteweg de Vries Burgers equation for data in $H^s({\Bbb R}),\,s>
-\frac 34-\frac 1{24}$. The surprising part of this result is that the indice $s=-\frac 34-\frac 1{24}$ is lower than the best known indice $s=- 3/4$ obtained by Kenig, Ponce and Vega in [@KPV1] for the KdV equation and lower than the indice $s=- 1/2$ of the critical Sobolev space for the dissipative Burgers equation $u_t-u_{xx}+uu_x=0$ (see [@B], [@Dix2]). In [@MR2], Molinet and Ribaud improved this result by introducing a new Bourgain type space and working in the space. They showed that Korteweg de Vries Burgers equation is globally well-posed in $H^s({\Bbb R})$ for $s>-1$ and in some sense ill-posed in $H^s({\Bbb
R})$ for $s<-1$.

The main purpose of this paper is to consider the Cauchy problem for the following dissipative versions of the mKdV equation on the real line $$\begin{aligned}
\label{2}
\left\{\begin{array}{ll} u_t+u_{xxx}+|D_x|^{\alpha}u+u^2u_x=0,\,t\in{\Bbb R}_+\\
 u(0)=\varphi \end{array}\right.\end{aligned}$$ with $\alpha\in (0,\,3]$. And we prove local well-posedness results on the associated Cauchy problem in the Sobolev spaces $H^s({\Bbb R})$ for $s>1/4-\alpha/4$ on the basis of the $[k;\,Z]-$multiplier norm estimate obtained by Tao in [@Tao] for KdV equation. We also get the global well-posedness for $1<\alpha\leq 3$ when $s>\frac{1}{4}-\frac{\alpha}{4}$. It is worth pointing out that the method used to obtain the multilinear estimates by Molinet and Ribaud in [@MR2] does not work when $\alpha$ is small enough. If we consider the case $0<\alpha\leq 1$, we can only get that problem (2) is local well-posed for $s>\frac{1}{2}-\frac{\alpha}{2}$ by running the approach of [@MR2]. The basic ideas of processing the multilinear estimates on $X^{s,\,b}$ spaces are using Cauchy-Schwarz inequality and the algebraic smoothing relation in [@MR2]( resonance identity in [@Tao]). In [@Tao], Tao utilizes dyadic decomposition and orthogonality before resorting to Cauchy-Schwarz. The advantages of dyadic decomposition and orthogonality lead to a better estimate when the algebraic smooth relation brings little benefit.

It is clear we get a better local well-posedness for the dissipative version of the mKdV equation than the local well-posedness of mKdV. This explicates that the dissipative term in (\[2\]) plays a key role for the low regularity of the equation. We are interested in finding out the smooth effect of the dissipative term. From the scaling analysis, the critical index of the mKdV equation is $-\frac{1}{2}$, while the critical index of the equation with only dissipative term is $1-\frac{\alpha}{2}$. Thus we conjecture that the mean value $\frac{1}{4}-\frac{\alpha}{4}$ of the two critical indices is the critical index for our dissipative version of the mKdV. For instance, we consider the Korteweg de Vries Burgers equation, $-1$ is exactly the mean value of $-\frac{3}{2}$ the critical scaling index of KdV and $-\frac{1}{2}$ the critical scaling index of Burgers equation. Unfortunately, we can not show the ill-posedness below the index $\frac{1}{4}-\frac{\alpha}{4}$. Recently, the first two authors of this paper [@CL] showed that when $\alpha=2$, (\[2\]) is ill-posedness for $s<-\frac{1}{2}$ in some sense. A similar argument can also get the same ill-posedenss for our problem.

Notations
---------

For a Banach space $X$, we denote by $\|\cdot\|_X$ the norm in $X$. We will use the Sobolev spaces $H^s({\Bbb R})$ and their homogeneous versions $\dot{H}^s({\Bbb R})$ equipped with the norms $\|u\|_{H^s}=\|(1-\Delta)^{s/2}u\|_{L^2}$ and $\|u\|_{\dot{H}^s}=\||D|^{s/2}u\|_{L^2}$. Recall that for $\lambda>0$, $$\begin{aligned}
\|f(\lambda t)\|_{H^s}\leq
(\lambda^{-1/2}+\lambda^{s-1/2})\|f(t)\|_{H^s},\,\|f(\lambda
t)\|_{\dot{H}^s}\sim \lambda^{s-1/2}\|f(t)\|_{\dot{H}^s}.\end{aligned}$$ We also consider the corresponding space-time Sobolev spaces $H^{s,\,b}_{x,\,t}$ endowed with the norm $$\begin{aligned}
\|u\|^2_{H^{s,\,b}}=\int_{{\Bbb
R}^2}<\xi>^{2s}<\tau>^{2b}|\hat{u}(\xi,\,\tau)|^2d\xi d\tau,\end{aligned}$$ where $<\cdot>=(1+|\cdot|^2)^{1/2}$.

Let $U(\cdot)$ be the unitary group in $H^s({\Bbb R}),\,s\in{\Bbb
R}$, which defined the free evolution of the KdV equation, i.e. $$\begin{aligned}
U(t)=\exp\Big(itP(D_x)\Big),\end{aligned}$$ where $P(D_x)$ is the Fourier multiplier with symbol $P(\xi)=\xi^3$. Since the linear symbol of equation (2) is $i(\tau-\xi^3)+|\xi|^\alpha$, by analogy with the spaces introduced by Bourgain in [@Bourgain] for purely dispersive equations and Molinet and Ribaud for KdV-Burgers equation, we define the function space $X^{s,\,b}_\alpha$ endowed with the norm $$\begin{aligned}
\|u\|_{X^{s,\,b}_\alpha}=\|<i(\tau-\xi^3)+|\xi|^\alpha>^b<\xi>^s\hat{u}(\xi,\,\tau)\|_{L^2({\Bbb
R}^2)}.\end{aligned}$$ So that $$\begin{aligned}
\|u\|_{X^{s,\,b}_\alpha}\sim\|<|\tau-\xi^3|+|\xi|^\alpha>^b<\xi>^s\hat{u}(\xi,\,\tau)\|_{L^2({\Bbb
R}^2)}.\end{aligned}$$

Note that since ${\cal F}(U(-t)u)(\xi,\,\tau)={\cal
F}(u)(\xi,\,\tau+\xi^3)$, one can re-express the norm of $X^{s,\,b}_\alpha$ as $$\begin{aligned}
\|u\|_{X^{s,\,b}_\alpha}&=&\|<i\tau+|\xi|^\alpha>^b<\xi>^s\hat{u}(\xi,\,\tau+\xi^3)\|_{L^2({\Bbb
R}^2)}\nonumber\\
&=&\|<i\tau+|\xi|^\alpha>^b<\xi>^s{\cal
F}(U(-t)u)(\xi,\,\tau)\|_{L^2({\Bbb R}^2)}.\end{aligned}$$ For $T\geq 0$, we consider the localized spaces $X^{s,\,b}_{\alpha,\,T}$ endowed with the norm $$\|u\|_{X^{s,\,b}_{\alpha,\,T}}=\inf_{w\in
X^{s,\,b}_\alpha}\{\|w\|_{X^{s,\,b}_\alpha},\,w(t)=u(t)\,{\rm
on}\,[0,\,T]\}.$$ Finally we denote by $W(\cdot)$ the semigroup associated with the free evolution of the equation (2), i.e. $$\forall t\geq 0,\,{\cal
F}_x(W(t)\varphi)(\xi)=\exp[-t|\xi|^\alpha+it\xi^3]\hat{\varphi}(\xi),$$ and we extend $W(\cdot)$ to a linear operator defined on the whole real axis by setting $$\begin{aligned}
\forall t\in{\Bbb R},\,{\cal
F}_x(W(t)\varphi)(\xi)=\exp[-|t||\xi|^\alpha+it\xi^3]\hat{\varphi}(\xi).\end{aligned}$$

Main results
------------

We will mainly work on the integral formulation of (2), i.e. $$\begin{aligned}
u(t)=W(t)\varphi-\frac
13\int_0^tW(t-t')\partial_x(u^3(t'))dt',\,t\geq 0.\end{aligned}$$ Actually, to prove the local existence result, we shall apply a fixed point argument to the following truncated version of (10) $$\begin{aligned}
u(t)=\psi(t)\Big[W(t)\varphi-\frac {\chi_{{\Bbb R}_+}(t)}
3\int_0^tW(t-t')\partial_x(\psi_T^3(t')u^3(t'))dt'\Big],\end{aligned}$$ where $t\in {\Bbb R}$ and, in the sequel of this paper, $\psi$ is a time cut-off function satisfying $$\psi\in C_0^\infty({\Bbb R}),\,{\rm supp}\,\psi\subset [-2,\,2],\,
\psi\equiv 1\,{\rm on}\,[-1,\,1],$$ and $\psi_T(\cdot)=\psi(\cdot/T)$. Indeed, if $u$ solves (11) then $u$ is a solution of (10) on $[0,\,T],\,T<1$.

Let us first state our global well-posedness result on the real line. In this paper we restrict us with $0<\alpha\leq 3$.

Let $\varphi\in H^s({\Bbb R}),\,s>1/4-\alpha/4$. Then there exist some $T>0$ and a unique solution $u$ of (10) in $$\begin{aligned}
Z_T=C([0,\,T],\,H^s)\cap X_{\alpha,\,T}^{s,\,1/2}.\end{aligned}$$ Moreover the map $\varphi\mapsto u$ is smooth from $H^s({\Bbb R})$ to $Z_T$.

If the dissipative term is strong enough, we can also get the global well-posedness.

Let $\varphi\in H^s({\Bbb R}),\,s>1/4-\alpha/4$ and $1<\alpha\leq3$. The existence time of the solution of (10) can be extended to infinity. And $u$ belongs to $C((0,\,+\infty),\,H^\infty({\Bbb
R}))$.

[**Remark**]{}: [*We can not get the global well-posedness for the case $0<\alpha\leq 1$, since that in this case one need a higher order a priori estimate than $L^2$. While in the case $1<\alpha\leq 3$, we need only the a priori estimate in $L^2$. It will be a interesting question to find out a suitable a priori estimate for the dissipative vision of the mKdV and then extend the local well-posedness to the global well-posedness when the dissipative term is very weak.*]{}

This paper is organized as follows. In Section 2 we prove linear estimates in the function space $X^{s,\,1/2}_\alpha$ and in Section 3 we introduce Tao’s $[k;\,Z]-$multiplier norm estimate and derive some trilinear estimate for the nonlinear term $\partial_x(u^3)$ from Tao’s estimate. For Section 4, we consider the local well-posedness while the global well-posedness will be in Section 5.

Linear Estimates
================

In this section we study the linear operator $\psi(\cdot)W(\cdot)$ as well as the linear operator $L$ defined by $$\begin{aligned}
 L:\,f\mapsto\chi_{{\Bbb R}_+}(t)\psi(t)\int_0^tW(t-t')f(t')dt'.
 \end{aligned}$$

Linear estimate for the free term
---------------------------------

Let $s\in {\Bbb R}$. There exists $C>0$ such that $$\begin{aligned}
\|\psi(t)W(t)\varphi\|_{X^{s,\,1/2}_\alpha}\leq
C\|\varphi\|_{H^s},\,\forall\varphi\in H^s({\Bbb R}).\end{aligned}$$

[*Proof*]{}. By definition of $\|\cdot\|_{X^{s,\,1/2}_\alpha}$, $$\begin{aligned}
&&\|\psi(t)W(t)\varphi\|_{X^{s,\,1/2}_\alpha}\nonumber\\
&&=\Big\|<\xi>^s<i(\tau-\xi^3)+|\xi|^\alpha>^{1/2}{\cal
F}_t\Big(\psi(t)e^{-|t||\xi|^\alpha}e^{it\xi^3}\hat{\varphi}
(\xi)\Big)(\tau)\Big\|_{L^2_{\xi,\,\tau}}\nonumber\\
&&=\left\|<\xi>^s\hat{\varphi}(\xi)\Big\|<i\tau+|\xi|^\alpha>^{1/2}
{\cal
F}_t\Big(\psi(t)e^{-|t||\xi|^\alpha}\Big)(\tau)\Big\|_{L^2_\tau}\right\|_{L^2_\xi}\nonumber\\
&&\leq C\left\|<\xi>^s\hat{\varphi}(\xi)\Big\|<\tau>^{1/2} {\cal
F}_t\Big(\psi(t)e^{-|t||\xi|^\alpha}\Big)(\tau)\Big\|_{L^2_\tau}\right\|_{L^2_\xi}\nonumber\\
&&\quad + C\left\|<\xi>^{s+\alpha/2}\hat{\varphi}(\xi)\Big\|{\cal
F}_t\Big(\psi(t)e^{-|t||\xi|^\alpha}\Big)(\tau)\Big\|_{L^2_\tau}\right\|_{L^2_\xi}\nonumber\\
&&\leq C\|\varphi\|_{H^s},\end{aligned}$$ where we used the fact $$\|\psi(t)e^{-|t||\xi|^\alpha}\|_{H^b_t}\leq C<\xi>^{\frac\alpha
2(2b-1)},\,\forall 0\leq b\leq 1,$$ which can be obtained from (19) in [@MR2].

Linear estimates for the forcing term
-------------------------------------

For $w\in {\cal S}({\Bbb R}^2)$ consider $k_\xi$ defined on ${\Bbb R}$ by $$\begin{aligned}
k_\xi(t)=\psi(t)\int_{{\Bbb
R}}\frac{e^{it\tau}-e^{-|t||\xi|^\alpha}}{i\tau+|\xi|^\alpha}\hat{w}(\xi,\,\tau)d\tau.\end{aligned}$$ Then, it holds for all $\xi\in{\Bbb R}$ $$\begin{aligned}
&&\Big\|<i\tau+|\xi|^\alpha>^{1/2}{\cal
F}_t(k_\xi)\Big\|^2_{L^2({\Bbb R})}\nonumber\\
&&\leq C\Big [\Big(\int_{{\Bbb
R}}\frac{|\hat{w}(\xi,\,\tau)|}{<i\tau+|\xi|^\alpha>}d\tau\Big)^2+\Big(\int_{{\Bbb
R}}\frac{|\hat{w}(\xi,\,\tau)|^2}{<i\tau+|\xi|^\alpha>}d\tau\Big)\Big]\end{aligned}$$

By a little modification of Proposition 2 in [@MR2], we can obtain the proof of Lemma 2.

Let $s\in{\Bbb R}$.

a\) There exists $C>0$ such that for all $\upsilon\in {\cal
S}({\Bbb R}^2)$, $$\begin{aligned}
&&\Big\|\chi_{{\Bbb
R}_+}(t)\psi(t)\int_0^tW(t-t')\upsilon(t')dt'\Big\|_{X^{s,\,1/2}_\alpha}\nonumber\\
&&\quad\quad\leq
C\Big[\|\upsilon\|_{X^{s,\,-1/2}_\alpha}+\Big(\int<\xi>^{2s}(\int\frac{|\hat{\upsilon}
(\xi,\,\tau+\xi^3)|}{<i\tau+|\xi|^\alpha>}d\tau)^2d\xi\Big)^{1/2}\Big]\end{aligned}$$

b\) For any $0<\delta<1/2$ there exists $C_\delta>0$ such that for all $\upsilon\in X^{s,\,-1/2+\delta}_\alpha$ $$\begin{aligned}
\Big\|\chi_{{\Bbb
R}_+}(t)\psi(t)\int_0^tW(t-t')\upsilon(t')dt'\Big\|_{X^{s,\,1/2}_\alpha}
\leq C_\delta \|\upsilon\|_{X^{s,\,-1/2+\delta}_\alpha}.\end{aligned}$$

[*Proof*]{}. Assume that $\upsilon\in{\cal S}({\Bbb R}^2)$. Taking the $x-$Fourier transform we get $$\begin{aligned}
&&\chi_{{\Bbb
R}_+}(t)\psi(t)\int_0^tW(t-t')\upsilon(t')dt'\\
&&\quad\quad =U(t)\Big[\chi_{{\Bbb R}_+}(t)\psi(t)\int_{{\Bbb
R}}e^{ix\xi}\int^t_0e^{-|t-t'||\xi|^\alpha}{\cal
F}_x(U(-t')\upsilon(t'))(\xi)dt'd\xi\Big].\end{aligned}$$ Set $\omega(t')=U(-t')\upsilon(t')$. Writing ${\cal
F}_x(\omega)(\xi,\,t)$ with the help of its time Fourier transform and using Fubini’s theorem one infers that $$\begin{aligned}
&&\chi_{{\Bbb
R}_+}(t)\psi(t)\int_0^tW(t-t')\upsilon(t')dt'\nonumber\\
&&\quad\quad =U(t)\Big[\chi_{{\Bbb R}_+}(t)\psi(t)\int_{{\Bbb
R}^2}e^{ix\xi}e^{-t|\xi|^\alpha}\hat{\omega}(\xi,\,\tau)\int^t_0e^{it'\tau}
e^{t'|\xi|^\alpha}dt'd\xi d\tau\Big]\nonumber\\
&&\quad\quad =U(t)\Big[\chi_{{\Bbb R}_+}(t)\psi(t)\int_{{\Bbb
R}^2}e^{ix\xi}\frac{e^{it\tau}-e^{-t|\xi|^\alpha}}{i\tau+|\xi|^\alpha}
\hat{\omega}(\xi,\,\tau)d\xi d\tau\Big].\end{aligned}$$ We set $k_\xi(t)=\psi(t)\int_{\Bbb
R}\frac{e^{it\tau}-e^{-|t||\xi|^\alpha}}{i\tau+|\xi|^\alpha}\hat{\omega}
(\xi,\,\tau)d\tau$. Since $\omega(t)=U(-t)\upsilon(t)\in{\cal
S}({\Bbb R}^2)$, it is clear that for any fixed $\xi\in{\Bbb
R},\,k_\xi$ is continuous on ${\Bbb R}$ and $k_\xi(0)=0$. Then it is not too hard to derive that $\|\chi_{{\Bbb
R}_+}(t)k_\xi(t)\|_{H^b_t}\leq \|k_\xi(t)\|_{H^b_t},\,0\leq b\leq
1$, and $b\neq 1/2$. The case $b=1/2$ follows by Lebesgue dominated convergence theorem. Thus in view of (20) $$\begin{aligned}
&&\Big\|\chi_{{\Bbb
R}_+}(t)\psi(t)\int_0^tW(t-t')\upsilon(t')dt'\Big\|_{X^{s,\,1/2}_\alpha}\\
&&=\Big\|<i\tau+|\xi|^\alpha>^{1/2}<\xi>^s{\cal F}_{x,\,t}
\Big(U(-t)\chi_{{\Bbb
R}_+}(t)\psi(t)\int_0^tW(t-t')\upsilon(t')dt'\Big)\Big\|_{L^2_{\xi,\,\tau}}\\
&&\leq \left\|<\xi>^s\Big\|{\cal F}_{x} \Big(U(-t)\chi_{{\Bbb
R}_+}(t)\psi(t)\int_0^tW(t-t')\upsilon(t')dt'\Big)\Big\|_{H_\tau^{1/2}}\right\|_{L^2_{\xi}}\\
&&\quad +\left\|<\xi>^{s+\alpha/2}\Big\|{\cal F}_{x}
\Big(U(-t)\chi_{{\Bbb
R}_+}(t)\psi(t)\int_0^tW(t-t')\upsilon(t')dt'\Big)\Big\|_{L_\tau^{2}}\right\|_{L^2_{\xi}}\\
&&\leq \Big\|<\xi>^s\|\chi_{{\Bbb
R}_+}(t)k_\xi(t)\|_{H^{1/2}_t}\Big\|_{L^2_\xi}+\Big\|<\xi>^{s+\alpha/2}\|\chi_{{\Bbb
R}_+}(t)k_\xi(t)\|_{L^{2}_t}\Big\|_{L^2_\xi}\\
&&\leq
\Big\|<\xi>^s\|k_\xi(t)\|_{H^{1/2}_t}\Big\|_{L^2_\xi}+\Big\|<\xi>^{s+\alpha/2}\|
k_\xi(t)\|_{L^{2}_t}\Big\|_{L^2_\xi}\\
&&\leq C\left\|<\xi>^s\Big\|<i\tau+|\xi|^\alpha>^{1/2}{\cal
F}_t(k_\xi(t)\Big\|_{L^2_\tau}\right\|_{L^2_\xi}.\end{aligned}$$ Then (18) follows directly from Lemma 2 together with the last estimate.\
To prove (19) we first assume that $\upsilon\in{\cal S}({\Bbb
R}^2)$. Applying Cauchy-Schwartz inequality in $\tau$ on the second term of the right-hand side of (18), one obtains (19) for $\upsilon\in{\cal S}({\Bbb R}^2)$. The result for $\upsilon\in
X^{s,\,-1/2+\delta}_\alpha$ follows by density.

Tao’s $[k;\,Z]-$multiplier norm estimate and its application
============================================================

In this section we introduce Tao’s $[k;\,Z]-$multiplier norm estimate and derive the trilinear estimate needed to obtain the local existence result from Tao’s multiplier norm estimate for KdV equation.

Let $Z$ be any abelian additive group with an invariant measure $d\xi$. For any integer $k\geq 2$, we let $\Gamma_k(Z)$ denote the hyperplane $$\Gamma_k(Z):=\{(\xi_1,\cdots,\,\xi_k)\in
Z^k:\,\xi_1+\cdots+\xi_k=0\}$$ which is endowed with the measure $$\int_{\Gamma_k(Z)}f:=\int_{Z^{k-1}}f(\xi_1,\cdots,\,\xi_{k-1},\,-\xi_1
-\cdots-\xi_{k-1})d\xi_1\cdots d\xi_{k-1}.$$

A $[k;\,Z]-$multiplier is defined to be any function $m:\,\Gamma_k(Z)\rightarrow {\Bbb C}$ which was introduced by Tao in [@Tao]. And the multiplier norm $\|m\|_{[k;\,Z]}$ is defined to be the best constant such that the inequality $$\begin{aligned}
\Big |\int_{\Gamma_k(Z)}m(\xi)\prod_{j=1}^kf_j(\xi_j)\Big |\leq
\|m\|_{[k;\,Z]}\prod_{j=1}^k\|f_j\|_{L^2(Z)},\end{aligned}$$ holds for all test functions $f_j$ on $Z$. Tao systematically studied this kind of weighted convolution estimates on $L^2$ in [@Tao]. To state Tao’s results, we use some notation he used in his paper.

We use $A\lesssim B$ to denote the statement that $A\leq CB$ for some large constant $C$ which may vary from line to line and depend on various parameters, and similarly use $A\ll B$ to denote the statement $A\leq C^{-1}B$. We use $A\sim B$ to denote the statement that $A\lesssim B\lesssim A$.

Any summations over capitalized variables such as $N_j,\,L_j,\,H$ are presumed to be dyadic, i.e., these variables range over numbers of the form $2^k$ for $k\in {\Bbb Z}$. In this paper, we will only consider the $[3;\,Z]$-multiplier. Let $N_1,\,N_2,\,N_3>0$. It will be convenient to define the quantities $N_{max}\geq N_{med}\geq
N_{min}$ to be the maximum, median, and minimum of $N_1,\,N_2,\,N_3$ respectively. Similarly define $L_{max}\geq L_{med}\geq L_{min}$ whenever $L_1,\,L_2,\,L_3>0$. And we also adopt the following summation conventions. Any summation of the form $L_{max}\sim\cdots$ is a sum over the three dyadic variables $L_1,\,L_2,\,L_3\gtrsim 1$, thus for instance $$\sum_{L_{max}\sim H}:=\sum_{L_1,\,L_2,\,L_3\gtrsim 1:\,L_{max}\sim
H}.$$ Similarly, any summation of the form $N_{max}\sim\cdots$ sum over the three dyadic variables $N_1,\,N_2,\,N_3>0$, thus for instance $$\sum_{N_{max}\sim N_{med}\sim
N}:=\sum_{N_1,\,N_2,\,N_3>0:\,N_{max}\sim N_{med}\sim N}.$$ If $\tau,\,\xi$ and $h(\cdot)$ are given, we also adopt the convention that $\lambda$ is short-hand for $$\lambda:=\tau-h(\xi).$$ Similarly we have $$\begin{aligned}
 \lambda_j:=\tau_j-h_j(\xi_j).\end{aligned}$$

In this paper, we do not go further on the general framework of Tao’s weighted convolution estimates. We focus our attention to the $[k;\,Z]-$multiplier norm estimate for KdV equation. During the estimate we need the resonance function $$\begin{aligned}
h(\xi)=\xi_1^3+\xi_2^3+\xi_3^3=-\lambda_1-\lambda_2-\lambda_3,\end{aligned}$$ which measures to what extent the spatial frequencies $\xi_1,\,\xi_2,\,\xi_3$ can resonate with each other.

By dyadic decomposition of the variables $\xi_j,\,\lambda_j$, as well as the function $h(\xi)$, one is led to consider $$\begin{aligned}
\|X_{N_1,\,N_2,\,N_3;\,H;\,L_1,\,L_2,\,L_3}\|_{[3,\,{\Bbb
R}\times{\Bbb R}]},\end{aligned}$$ where $X_{N_1,\,N_2,\,N_3;\,H;\,L_1,\,L_2,\,L_3}$ is the multiplier $$\begin{aligned}
X_{N_1,\,N_2,\,N_3;\,H;\,L_1,\,L_2,\,L_3}(\xi,\,\tau):=\chi_{|h(\xi)|\sim
H}\prod_{j=1}^3\chi_{|\xi_j|\sim N_j}\chi_{|\lambda_j|\sim L_j}.\end{aligned}$$

From the identities $$\xi_1+\xi_2+\xi_3=0$$ and $$\lambda_1+\lambda_2+\lambda_3+h(\xi)=0$$ on the support of the multiplier, we see that $X_{N_1,\,N_2,\,N_3;\,H;\,L_1,\,L_2,\,L_3}$ vanishes unless $$\begin{aligned}
N_{max}\sim N_{med},\end{aligned}$$ and $$\begin{aligned}
L_{max}\sim \max(H,\,L_{med}).\end{aligned}$$ From the resonance identity $$\begin{aligned}
h(\xi)=\xi_1^3+\xi_2^3+\xi_3^3=3\xi_1\xi_2\xi_3\end{aligned}$$ we see that we may assume that $$\begin{aligned}
H\sim N_1N_2N_3,\end{aligned}$$ since the multiplier in (24) vanishes otherwise.

Now we are in the position to state Tao’s $[k;\,Z]-$multiplier norm estimate for KdV equation in non-periodic case.

(see Proposition 6.1 in [@Tao]). Let $H,\,N_1,\,N_2,\,N_3,\,L_1,\,L_2,\,L_3>0$ obey (26), (27), (29).

$\bullet$((++)Coherence) If $N_{max}\sim N_{min}$ and $L_{max}\sim
H$, then we have $$\begin{aligned}
(24)\lesssim L_{min}^{1/2}N_{max}^{-1/4}L_{med}^{1/4}.\end{aligned}$$

$\bullet$((+-)Coherence) If $N_2\sim N_3\gg N_1$ and $H\sim
L_1\gtrsim L_2,\,L_3$, then $$\begin{aligned}
(24)\lesssim
L_{min}^{1/2}N_{max}^{-1}\min(H,\,\frac{N_{max}}{N_{min}}L_{med})^{1/2}.\end{aligned}$$ Similarly for permutations.

$\bullet$ In all other cases, we have $$\begin{aligned}
(24)\lesssim L_{min}^{1/2}N_{max}^{-1}\min(H,\,L_{med})^{1/2}.\end{aligned}$$

With Lemma 4 one can now derive a trilinear estimate involving the new Bourgain spaces $X^{s,\,b}_\alpha$.

Let $s>1/4-\alpha/4$. For all $u_1,\,u_2,\,u_3$ on ${\Bbb
R}\times {\Bbb R}$ and $0<\epsilon\ll 1$, we have $$\begin{aligned}
\|\partial_x(u_1u_2u_3)\|_{X^{s,\,-1/2+\epsilon}_\alpha}\lesssim
\|u_1\|_{X^{s,\,1/2}_\alpha}\|u_2\|_{X^{s,\,1/2}_\alpha}\|u_3\|_{X^{s,\,1/2}_\alpha}.\end{aligned}$$

As seen in [@MR2], Theorem 1 can be reduced to the trilinear estimate Lemma 5 and the linear estimates we obtained in section 2. What we need to do in the following is to prove the trilinear estimate.

[*Proof of Lemma 5.*]{} By duality and Plancherel it suffices to show that $$\begin{aligned}
\left\|\frac{(\xi_1+\xi_2+\xi_3)<\xi_4>^s}{<\tau_4-\xi_4^3+i|\xi_4|^\alpha>^{1/2-\epsilon}
\prod_{j=1}^3<\xi_j>^s<\tau_j-\xi_j^3+i|\xi_j|^\alpha>^{1/2}}\right\|_{[4,\,{\Bbb
R}\times{\Bbb R}]}\lesssim 1.\end{aligned}$$ We estimate $|\xi_1+\xi_2+\xi_3|$ by $<\xi_4>$. We then apply the inequality fractional Leibnitz rule $$<\xi_4>^{s+1}\lesssim <\xi_4>^{1/2}\sum_{j=1}^3<\xi_j>^{s+1/2}$$ where we assume $-1/2<s<1/2$, and symmetry to reduce to $$\begin{aligned}
\left\|\frac{<\xi_1>^{-s}<\xi_3>^{-s}<\xi_2>^{1/2}<\xi_4>^{1/2}}{<\tau_4-\xi_4^3+i|\xi_4|^\alpha>^{1/2-\epsilon}
\prod_{j=1}^3<\tau_j-\xi_j^3+i|\xi_j|^\alpha>^{1/2}}\right\|_{[4,\,{\Bbb
R}\times{\Bbb R}]}\lesssim 1.\end{aligned}$$ We may replace $<\tau_2-\xi_2^3+i|\xi_2|^\alpha>^{1/2}$ by $<\tau_2-\xi_2^3+i|\xi_2|^\alpha>^{1/2-\epsilon}$. By the $TT^*$ identity (see Lemma 3.7, p847 in[@Tao]), the estimate is reduced to the following bilinear estimate.

(Bilinear estimate). Let $s>1/4-\alpha/4$. For all $u,\,v$ on ${\Bbb
R}\times {\Bbb R}$ and $0<\epsilon\ll 1$, we have $$\begin{aligned}
\|uv\|_{L^2({\Bbb R}\times{\Bbb R})}\lesssim
\|u\|_{X^{-1/2,\,1/2-\epsilon}_\alpha({\Bbb R}\times{\Bbb R})
}\|v\|_{X^{s,\,1/2}_\alpha({\Bbb R}\times{\Bbb R})}.\end{aligned}$$

[*Proof of the bilinear estimate*]{}. By Plancherel it suffices to show that $$\begin{aligned}
\left\|\frac{<\xi_1>^{-s}<\xi_2>^{1/2}}{<\tau_1-\xi_1^3+i|\xi_1|^\alpha>^{1/2}
<\tau_2-\xi_2^3+i|\xi_2|^\alpha>^{1/2-\epsilon}}\right\|_{[3,\,{\Bbb
R}\times{\Bbb R}]}\lesssim 1.\end{aligned}$$

There are two approaches to computing these quantities and their generalization. One approach proceeds using the Cauchy-Schwarz inequality, this reducing matters to integrating certain weights on intersections of hypersurfaces $\tau=h(\xi)$; the other utilizes dyadic decomposition and orthogonality before resorting to Cauchy-Schwarz. The advantages of dyadic decomposition are that one can re-use the estimates on dyadic blocks to prove other estimates, and the nature of interactions between different scales of frequency is more apparent.

By dyadic decomposition of the variables $\xi_j,\,\lambda_j,\,h(\xi)$, we may assume that $|\xi_j|\sim
N_j,\,|\lambda_j|\sim L_j,\,|h(\xi)|\sim H$. By the translation invariance of the $[k;\,Z]$-multiplier norm, we can always restrict our estimate on $\lambda_j\gtrsim 1$ and $\max(N_1,N_2,N_3)\gtrsim
1$. The comparison principle and orthogonality (see Schur’s test in [@Tao] p851) reduce our estimate to show that $$\begin{aligned}
&&\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_1,\,L_2,\,L_3\gtrsim
1}\frac{<N_1>^{-s}<N_2>^{1/2}}{\max(L_1,\,<N_1>^\alpha)^{1/2}
\max(L_2,\,<N_2>^\alpha)^{1/2-\epsilon}}\nonumber\\
&&\quad\qquad\quad\quad\quad\qquad\left\|X_{N_1,\,N_2,\,N_3;\,L_{max};\,L_1,\,L_2,\,L_3}\right\|_{[3;\,{\Bbb
R}\times{\Bbb R}]}\lesssim 1\end{aligned}$$ and $$\begin{aligned}
&&\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{max}\sim
L_{med}}\sum_{H\ll L_{max}}
\frac{<N_1>^{-s}<N_2>^{1/2}}{\max(L_1,\,<N_1>^\alpha)^{1/2}
\max(L_2,\,<N_2>^\alpha)^{1/2-\epsilon}}\nonumber\\
&&\quad\qquad\quad\quad\quad\qquad\left\|X_{N_1,\,N_2,\,N_3;\,H;\,L_1,\,L_2,\,L_3}\right\|_{[3;\,{\Bbb
R}\times{\Bbb R}]}\lesssim 1\end{aligned}$$ for all $N\gtrsim 1$. This can be accomplished by Tao’s estimate on dyadic blocks for KdV equation, i.e., Lemma 4 and some concrete summation.

Fix $N\gtrsim 1$. We first prove (37). We may assume (29). By (32) we reduce to $$\begin{aligned}
&&\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{max}\sim L_{med}\gtrsim
N_1N_2N_3}\\
&&\quad\quad\frac{<N_1>^{-s}<N_2>^{1/2}}{\max(L_1,\,<N_1>^\alpha)^{1/2}
\max(L_2,\,<N_2>^\alpha)^{1/2-\epsilon}}
L_{min}^{1/2}N_{min}^{1/2}\lesssim 1.\end{aligned}$$ This high modulation case is easier to handle. We do not need to consider the effect of the dissipative term, i.e., the part with symbol $|\xi|^\alpha$. Just crudely estimating $$<N_1>^{-s}<N_2>^{1/2}\lesssim N^{1/2+\max(0,\,-s)}$$ and $$\max(L_1,\,<N_1>^\alpha)^{1/2}\max(L_2,\,<N_2>^\alpha)^{1/2-\epsilon}
\gtrsim L_{min}^{1/2}L_{med}^{1/2-\epsilon},$$ then performing the $L$ summations, we reduce to $$\begin{aligned}
\sum_{N_{max}\sim N_{med}\sim
N}\frac{N_{min}^\epsilon}{N^{1/2-\max(0,\,-s)-2\epsilon}}\lesssim
1,\end{aligned}$$ which is true if $1/2-\max(0,\,-s)>0$. So, (37) is true if $s>-1/2$.

Now we show the low modulation case (36). We may assume $L_{max}\sim N_1N_2N_3$. We first deal with the contribution where (30) holds. In this case we have $N_1,\,N_2,\,N_3\sim N\gtrsim 1$, so we reduce to $$\begin{aligned}
\sum_{L_{max}\sim
N^3}\frac{N^{-s}N^{1/2}}{\max(L_1,\,N^\alpha)^{1/2}\max(L_2,\,N^\alpha)^{1/2-\epsilon}}
L_{min}^{1/2}N^{-1/4}L_{med}^{1/4}\lesssim 1.\end{aligned}$$ Note that $$\max(L_1,\,N^\alpha)^{1/2}\max(L_2,\,N^\alpha)^{1/2-\epsilon}\gtrsim
\max(L_{min},\,N^\alpha)^{1/2}\max(L_{med},\,N^\alpha)^{1/2-\epsilon}.$$ We may assume that $L_1\leq L_2\leq L_3$. We now need consider three subcases: $L_2\leq N^\alpha,\,L_1\leq N^\alpha\leq
L_2,\,N^\alpha\leq L_1$.

If $L_2\leq N^\alpha$, we reduce to $$\begin{aligned}
\sum_{L_1\leq L_2\leq N^\alpha\lesssim L_3\sim N^3
}\frac{N^{-s}N^{1/4}L_1^{1/2}L_2^{1/4}}{N^{\alpha/2}N^{\alpha(1/2-\epsilon)}}
\lesssim\frac{N^{1/4-s}N^{3\alpha/4}}{N^{\alpha-\alpha\epsilon}}\lesssim
1,\end{aligned}$$ which is true if $1/4-s-\alpha/4<0$, i.e., $s>1/4-\alpha/4$.

If $L_1\leq N^\alpha\leq L_2$, we reduce to $$\begin{aligned}
\sum_{L_1\leq N^\alpha\leq L_2\leq L_3\sim N^3
}\frac{N^{-s}N^{1/4}L_1^{1/2}L_2^{1/4}}{N^{\alpha/2}L_2^{1/2-\epsilon}}
\lesssim\frac{N^{1/4-s}N^{\alpha/2}}{N^{\alpha/2}N^{(1/4-\epsilon)\alpha}}\lesssim
1,\end{aligned}$$ which holds if $s>1/4-\alpha/4$.

If $N^\alpha\leq L_1$, we reduce to $$\begin{aligned}
\sum_{N^\alpha\leq L_1\leq L_2\leq L_3\sim N^3
}\frac{N^{-s}N^{1/4}L_1^{1/2}L_2^{1/4}}{L_1^{1/2}L_2^{1/2-\epsilon}}
\lesssim\frac{N^{1/4-s}}{N^{(1/4-\epsilon)\alpha}}\lesssim 1,\end{aligned}$$ which is true if $s>1/4-\alpha/4$.

Now we deal with the cases where (31) applies. We do not have perfect symmetry and must consider three cases $$\begin{array}{ll}
N\sim N_1\sim N_2\gg N_3;\,& H\sim L_3\gtrsim L_1,\,L_2\\
N\sim N_2\sim N_3\gg N_1;\,& H\sim L_1\gtrsim L_2,\,L_3\\
N\sim N_1\sim N_3\gg N_2;\,& H\sim L_2\gtrsim L_1,\,L_3
\end{array}$$ separately.

In the first case we reduce by (31) to $$\begin{aligned}
&&\sum_{N_3\ll N}\sum_{1\lesssim L_1,\,L_2\lesssim
N^2N_3}\frac{N^{1/2-s}}{\max(L_1,\,N^\alpha)^{1/2}\max(L_2,\,N^\alpha)^{1/2-\epsilon}}\\
&&\quad\qquad\quad\quad\qquad
L_{min}^{1/2}N^{-1}\min(N^2N_3,\,\frac{N}{N_3}L_{med})^{1/2}\lesssim
1.\end{aligned}$$ Performing the $N_3$ summation we reduce to $$\begin{aligned}
\sum_{1\lesssim L_1,\,L_2\lesssim
N^3}\frac{N^{1/2-s}}{\max(L_1,\,N^\alpha)^{1/2}\max(L_2,\,N^\alpha)^{1/2-\epsilon}}
L_{min}^{1/2}N^{-1}N^{3/4}L_{med}^{1/4}\lesssim 1.\end{aligned}$$ which is the same as (38) and so it is true if $s>1/4-\alpha/4$.

We can deal with the second and third cases in a unified way. By asymmetry it suffices to show the worst case. We simplify using the first half of (31) to $$\sum_{N_{min}\ll N}\sum_{1\lesssim L_{min},\,L_{med}\ll
N^2N_{min}}\frac{<N_{min}>^{-s}N^{1/2}}{L_{min}^{1/2}L_{max}^{1/2-\epsilon}}
L_{min}^{1/2}N_{min}^{1/2}\lesssim 1.$$ We may assume $N_{min}\gtrsim N^{-2}$ since the inner sum vanishes otherwise. Performing the $L$ summation we reduce to $$\sum_{N^{-2}\lesssim N_{min}\ll
N}\frac{<N_{min}>^{-s}N^{1/2}N_{min}^{1/2}}{(N^2N_{min})^{1/2-\epsilon}}\lesssim
1$$ which holds if $s>-1/2$.

To finish the proof of (36) it remains to deal with the cases where (32) holds. This reduces to $$\begin{aligned}
&&\sum_{N_{max}\sim N_{med}\sim N}\sum_{L_{max}\sim N_1N_2N_3}\\
&&\quad\quad\frac{<N_1>^{-s}<N_2>^{1/2}}{\max(L_1,\,<N_1>^\alpha)^{1/2}
\max(L_2,\,<N_2>^\alpha)^{1/2-\epsilon}}
L_{min}^{1/2}N^{-1}L_{med}^{1/2}\lesssim 1.\end{aligned}$$ Performing the $L$ summations, we reduce to $$\sum_{N_{max}\sim N_{med}\sim
N}<N_1>^{-s}<N_2>^{1/2}(N_1N_2N_3)^\epsilon N^{-1}\lesssim 1$$ which is true if $s>-1/2$.

Local well-posedness
====================

Let $\varphi$ in $H^s({\Bbb
R}),\,s>\frac{1}{4}-\frac{\alpha}{4}$. We first prove the existence of a solution $u$ of the integral formulation (10) of the equation (2) on some interval $[0,\,T]$ for $T\leq 1$ small enough. Clearly, if $u$ is a solution of the integral equation $u=F(u)$ with $$\begin{aligned}
F(u)=\psi(t)\Big[W(t)\varphi-\chi_{{\Bbb R}_+}(t)
\int_0^tW(t-t')\partial_x(\psi_T^3(t')u^3(t'))dt'\Big],\end{aligned}$$ then $u$ is a solution of (10) on $[0,\,T]$. We are going to solve (39) in the space $$\begin{aligned}
Z=\{u\in
X^{s,\,1/2},\,\|u\|_Z=\|u\|_{X^{s_c^+,\,1/2}}+\nu\|u\|_{X^{s,\,1/2}}<+\infty\},\end{aligned}$$ where $s_c^+\in(\frac{1}{4}-\frac{\alpha}{4},\,s)$ is fixed, and where the constant $\nu$ is defined for all nontrivial $\varphi$ by $$\begin{aligned}
\nu=\frac{\|\varphi\|_{H^{s_c^+}}}{\|\varphi\|_{H^s}}.\end{aligned}$$ To get an contractive factor in our argument, we need the following modified trilinear estimates.

\[mainlema\] Given $s>\frac{1}{4}-\frac{\alpha}{4}$, there exist $C,\,\mu,\,\delta>0$ such that for any triple $(u,\,v,\,w)\in
X^{s,\,1/2}$ with compact support in $[-T,\,T]$ $$\begin{aligned}
\label{triest}
\|\partial_x(uvw)\|_{X^{s,\,-1/2+\delta}}\leq CT^\mu
\|u\|_{X^{s,\,1/2}}\|v\|_{X^{s,\,1/2}}\|w\|_{X^{s,\,1/2}}.\end{aligned}$$

The following lemma is a direct consequence of Lemma \[mainlema\] together with the triangle inequality $$\begin{aligned}
\forall s\geq s_c^+,\,\,<\xi>^s&\leq &
<\xi>^{s_c^+}<\xi_1>^{s-s_c^+}+<\xi>^{s_c^+}<\xi_2>^{s-s_c^+}\nonumber\\&&\quad+
<\xi>^{s_c^+}<\xi-\xi_1-\xi_2>^{s-s_c^+}.\end{aligned}$$

\[uselema\] Given $s_c^+>\frac{1}{4}-\frac{\alpha}{4}$, there exist $C,\,\mu,\,\delta>0$ such that for any $s\geq s_c^+$ and any triple $(u,\,v,\,w)\in X^{s,\,1/2}$ with compact support in $[-T,\,T]$ $$\begin{aligned}
\label{mtrist}
&&\|\partial_x(uvw)\|_{X^{s,\,-1/2+\delta}}\leq CT^\mu
\Big(\|u\|_{X^{s,\,1/2}}\|v\|_{X^{s_c^+,\,1/2}}\|w\|_{X^{s_c^+,\,1/2}}+\|u\|_{X^{s_c^+,\,1/2}}
\nonumber\\
&&\quad\|v\|_{X^{s,\,1/2}}\|w\|_{X^{s_c^+,\,1/2}}
+\|u\|_{X^{s_c^+,\,1/2}}\|v\|_{X^{s_c^+,\,1/2}}\|w\|_{X^{s,\,1/2}}\Big).\end{aligned}$$

The proof of Lemma \[mainlema\] is exactly the same with that of Lemma 5 except changing $f_4(\xi_4,\,\tau_4)$ into $\frac{f_4(\xi_4,\,\tau_4)}{<|\tau_4-\xi_4^3|+|\xi_4|^{\alpha}>^{\delta}}$ and using the following lemma due to Molinet and Ribaud [@MR2].

\[Lemma 4\] Let $f$ with support in $[-T,\,T]$ in time. For any $\delta>0$, there exists $\mu=\mu(\delta)>0$ such that $$\Big\|\frac{\hat{f}(\xi,\,\tau)}{<\tau-\xi^3>^\delta}\Big\|_{L^2_{\xi,\,\tau}}\leq
 C T^\mu\|f\|_{L^2_{t,\,x}}.$$

By Lemmas 7 and 8, there exist $\delta,\,\mu>0$ only depending on $s_c^+$ such that $$\begin{aligned}
&&\|F(u)\|_{X^{s_c^+,\,1/2}}\leq
C\|\varphi\|_{H^{s_c^+}}+CT^\mu\|u\|_{X^{s_c^+,\,1/2}}^3,\\
&&\|F(u)\|_{X^{s,\,1/2}}\leq
C\|\varphi\|_{H^{s}}+CT^\mu\|u\|_{X^{s_c^+,\,1/2}}^2\|u\|_{X^{s,\,1/2}}.\end{aligned}$$ Gathering the above two estimates, one infers that $$\begin{aligned}
\|F(u)\|_Z\leq
C(\|\varphi\|_{H^{s_c^+}}+\nu\|\varphi\|_{H^s})+CT^\mu\|u\|_Z^3.\end{aligned}$$ Next, since $\partial_x(u^3)-\partial_x(v^3)=\partial_x[(u-v)(u^2+uv+v^2)]$, we get in the same way that $$\begin{aligned}
\|F(u)-F(v)\|_{X^{s_c^+,\,1/2}}&\leq &
CT^\mu\|u-v\|_{X^{s_c^+,\,1/2}}
\Big(\|u\|_{X^{s_c^+,\,1/2}}^2\nonumber\\&&\quad
+\|u\|_{X^{s_c^+,\,1/2}}
\|v\|_{X^{s_c^+,\,1/2}}+\|v\|_{X^{s_c^+,\,1/2}}^2\Big),\end{aligned}$$ and $$\begin{aligned}
 \|F(u)-F(v)\|_{X^{s,\,1/2}}&\leq &
CT^\mu\|u-v\|_{X^{s_c^+,\,1/2}}\Big(\|u\|_{X^{s_c^+,\,1/2}}
\|u\|_{X^{s,\,1/2}}\nonumber\\&&+\|u\|_{X^{s_c^+,\,1/2}}
\|v\|_{X^{s,\,1/2}}+\|u\|_{X^{s,\,1/2}}\|v\|_{X^{s_c^+,\,1/2}}\nonumber\\
&&+
\|v\|_{X^{s_c^+,\,1/2}}\|v\|_{X^{s,\,1/2}}\Big)+CT^\mu\|u-v\|_{X^{s,\,1/2}}
\Big(\|u\|_{X^{s_c^+,\,1/2}}^2\nonumber\\&& +\|u\|_{X^{s_c^+,\,1/2}}
\|v\|_{X^{s^+,\,1/2}}+ \|v\|_{X^{s_c^+,\,1/2}}^2\Big)\end{aligned}$$ Combining (48) and (49), we deduce that $$\begin{aligned}
\|F(u)-F(v)\|_Z\leq CT^\mu\|u-v\|_Z\Big(\|u\|_Z+\|v\|_Z\Big)^2.\end{aligned}$$ Setting $T=\Big(128C^3\|\varphi\|_{H^{s_c^+}}^2\Big)^{-1/\mu}$, we deduce that from (47) and (50) that $F$ is strictly contractive on the ball of radius $4C\|\varphi\|_{H^{s_c^+}}$ in $Z$. This proves the existence of a solution $u\in X^{s,\,1/2}$ to the equation (2) on the time interval $[0,\,T]$ with $T=T(\|\varphi\|_{H^{s_c^+}})>0$.

The uniqueness of the solution is exactly the same as in [@MR2]. For brevity, we omit the details.

Global well-posedness for the strong dissipative term.
======================================================

We now turn to the global well-posedness for the case $1<\alpha\leq
3$. In this case, we fix $s_c^+\in(\frac{1}{4}-\frac{\alpha}{4},\,0).$ By an similar argument as the Proposition 2.4 in [@MR2], we can get the following lemma.

\[forglobal\]Let $s\in\Bbb R$ and $\delta>0$. For all $f\in X^{s,-1/2+\delta}$, $$t\longmapsto\int_0^tW(t-t')f(t')dt'\in C(\Bbb R_+,
H^{s+\alpha\delta}).$$ Moreover, if $(f_n)$ is a sequence with $(f_n)$ is a sequence with $f_n\xrightarrow[n\rightarrow\infty]{} 0$ in $X^{s,-1/2+\delta}$, then $$\Big\|\int_0^tW(t-t')f_n(t')dt'\Big\|_{L^\infty(\Bbb
R_+,H^{s+\alpha\delta})}\xrightarrow[n\rightarrow\infty]{}0.$$

It is easily to check that $W(\cdot)\varphi\in
C([0,+\infty);H^s)\cap C((0,+\infty);H^\infty).$ Then it follows from Lemma 5 ,Lemma 10 and the local existence of the solution that $$u\in C([0,T]; H^s)\cap C((0;T];H^{s+\alpha\delta}), \text{ for some } T=T(\|\varphi\|_{H^{s^+_c}}).$$ By induction we have $u\in C((0,T];H^\infty)$. Since that the exist time $T$ of the solution depends only on the norm of date $\|\varphi\|_{H^{s^+_c}},$ to extent the solution into the global sense we need a control on $\|u(t)\|_{H^{s^+_c}}.$ Taking the $L^2$-scalar product of (2) with $u$, we can get that $\|u(t)\|_{L^2}$ is nonincreasing on $(0,T]$. This implies that the solution is global in time.

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[^1]: The work was partly supported by NNSF of China (No.10371080) and China Postdoctoral Science Foundation (No.2005038327).
