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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 40, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/40\hfil Well-posedness of KdV type equations]
{Well-posedness of KdV type equations}

\author[X. Carvajal, M. Panthee \hfil EJDE-2012/40\hfilneg]
{Xavier Carvajal, Mahendra Panthee} % in alphabetical order

\address{Xavier Carvajal \newline
Instituto de Matem\'atica - UFRJ
Av. Hor\'acio Macedo, Centro de Tecnologia
 Cidade Universit\'aria, Ilha do Fund\~ao,
 Caixa Postal 68530
21941-972 Rio de Janeiro,  RJ, Brasil}
\email{carvajal@im.ufrj.br}

\address{Mahendra Panthee \newline
Centro de  Matem\'atica, Universidade do Minho,
4710-057, Braga, Portugal}
\email{mpanthee@math.uminho.pt}

\thanks{Submitted September 9, 2011. Published March 14, 2012.}
\subjclass[2000]{35A07, 35Q53}
\keywords{Initial value problem; well-posedness; Bourgain spaces, KdV equation}

\begin{abstract}
 In this work, we study the initial value problems associated to some linear
 perturbations of KdV equations. Our focus is in the well-posedness issues
 for initial data given in the $L^2$-based Sobolev spaces.
 We develop a method that allows us to treat the problem in the Bourgain's space
 associated to the KdV equation. With this method, we can use the multilinear
 estimates developed in the KdV context, thereby getting analogous well-posedness
 results for linearly perturbed equations.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

In this article, we consider the initial value problem (IVP)
\begin{equation}\label{eq:hs}
 \begin{gathered}
  v_t+v_{xxx}+\eta Lv+(v^{k+1})_x=0, \quad x \in \mathbb{R}, \; t\geq 0,\; k \in \mathbb{Z}^+,\\
     v(x,0)=v_0(x),
 \end{gathered}
\end{equation}
and
\begin{equation}\label{eq:hs-1}
 \begin{gathered}
  u_t+u_{xxx}+\eta Lu+(u_x)^{k+1}=0, \quad x \in \mathbb{R}, \,t\geq 0,\; k\in \mathbb{Z}^+,\\
  u(x,0)=u_0(x),
 \end{gathered}
\end{equation}
where $\eta>0$ is a constant; $u=u(x, t)$, $v=v(x,t)$ are real valued functions and the 
linear operator $L$ is defined via the Fourier transform by 
$\widehat{Lf}(\xi)=-\Phi(\xi)\hat{f}(\xi)$.

The Fourier symbol
\begin{align}\label{phi}
\Phi(\xi)=\sum_{j=0}^{n}\sum_{i=0}^{2m}c_{i,j}\xi^i |\xi|^j, \quad c_{i,j} \in \mathbb{R},
\; c_{2m,n}=-1,
\end{align}
is a real valued function which is bounded above; i.e., there is a constant $C$ such 
that $\Phi(\xi) < C$.

We observe that,  if $u$ is a solution of \eqref{eq:hs-1} then $v=u_x$ is a solution 
of \eqref{eq:hs} with initial data $v_0 = (u_0)_x$. That is why \eqref{eq:hs} is called  
the derivative equation of \eqref{eq:hs-1}.

In this work, we are interested in investigating the  well-posedness results 
to the IVPs \eqref{eq:hs-1} and \eqref{eq:hs} for given data in the low
regularity  Sobolev spaces $H^s(\mathbb{R})$. Recall that, for $s\in \mathbb{R}$, 
the $L^2$-based Sobolev spaces $H^s(\mathbb{R})$ are defined by
$$
H^s(\mathbb{R}) := \{f\in \mathcal{S}'(\mathbb{R}) : \|f\|_{H^s} < \infty\},
$$
where 
$$
\|f\|_{H^s} := \Big(\int_{\mathbb{R}} (1+|\xi|^2)^s|\hat f(\xi)|^2d\xi\Big)^{1/2},
$$
and $\hat f(\xi)$ is the usual Fourier transform given by
$$
\hat f(\xi) \equiv \mathcal{F}(f)(\xi) 
:= \frac 1{\sqrt{2\pi}}\int_{\mathbb{R}}e^ {-ix\xi} f(x)\, dx.
$$
The factor $\frac1{\sqrt{2\pi}}$ in the definition of the Fourier transform  
does not alter our analysis, so we will omit  it.

The notion of well-posedness we use is the standard one. We say that an IVP 
for given data in a Banach space $X$ is locally well-posed, if there exists 
a certain time interval $[-T, T]$ and a unique solution depending continuously
upon the initial data and the solution satisfies the persistence property; 
i.e., the solution describes a continuous curve in $X$ in the time interval  $[-T, T]$.
If the above properties are true for any time interval, we say that the IVP is globally 
well-posed.

Before stating the main results of this work, we present some particular examples 
that belong to the class considered in \eqref{eq:hs} and \eqref{eq:hs-1} and 
discuss the known well-posedness results about them.

The first examples belonging to the classes  \eqref{eq:hs} and \eqref{eq:hs-1} are
\begin{equation}\label{eqhs}
   \begin{gathered}
     v_t+v_{xxx}-\eta(\mathcal{H}v_x+\mathcal{H}v_{xxx})+(v^{k+1})_x=0, \quad 
 x \in \mathbb{R}, \; t\geq     0, \; k\in \mathbb{Z}^+,\\
     v(x,0)=v_0(x),
   \end{gathered}
\end{equation}
and
\begin{equation}\label{eqhs-1}
   \begin{gathered}
     u_t+u_{xxx}-\eta(\mathcal{H}u_x+\mathcal{H}u_{xxx})+(u_x)^{k+1}=0, \quad 
x \in \mathbb{R}, \; t\geq     0, \; k\in \mathbb{Z}^+,\\
     u(x,0)=u_0(x),
   \end{gathered}
\end{equation}
respectively, where $\mathcal{H}$ denotes the Hilbert transform
\begin{align*}
\mathcal{H}g(x)=\operatorname{P.V.} \frac{1}{\pi}\int\frac{g(x-\xi)}{\xi}d\xi;
\end{align*}
$u=u(x,t)$, $v=v(x,t)$ are real-valued functions and $\eta>0$ is a constant.

The equation in \eqref{eqhs} with $k=1$ was derived by Ostrovsky et al
\cite{O:O} to describe the radiational  instability of long
waves in a stratified shear flow.  Recently,  Carvajal and Scialom \cite{Cv-Sc} 
considered the IVP \eqref{eqhs} and proved the
local well-posedness results for given data  in $H^s$, $s \geq 0$ when
$k=1,2,3$. They also obtained an \emph{a priori} estimate for given data in $L^2(\mathbb{R})$
there by proving  global well-posedness result. The earlier well-posedness results 
for \eqref{eqhs} with $k=1$  can be found in \cite{pa:ba1}, where for given 
data in $H^s(\mathbb{R})$, local well-posedness when $s>1/2$ and global well-posedness when
$s\geq 1$ have been proved. In \cite{pa:ba1}, IVP \eqref{eqhs-1}, when $k=1$, 
is also considered to prove global well-posedness for given data in $H^s(\mathbb{R})$, 
$s\geq1$.

Another two  models that fit in the classes \eqref{eq:hs-1} and \eqref{eq:hs} respectively 
are the Korteweg-de Vries-Kuramoto Sivashinsky (KdV-KS) equation
\begin{equation}\label{1eqhs-1}
   \begin{gathered}
     u_t+u_{xxx}+\eta(u_{xx}+u_{xxxx})+(u_x)^2=0, \quad x \in \mathbb{R}, \; t\geq 0,\\
     u(x,0)=u_0(x),
   \end{gathered}
\end{equation}
 and its derivative equation
\begin{equation}\label{1eqhs}
   \begin{gathered}
     v_t+v_{xxx}+\eta(v_{xx}+v_{xxxx})+vv_x=0, \quad x \in \mathbb{R}, \;t\geq 0,\\
     v(x,0)=v_0(x),
   \end{gathered}
\end{equation}
where $u=u(x,t)$, $v=v(x,t)$ are real-valued functions and $\eta>0$ is a constant.

The KdV-KS equation arises as a model for long waves in a viscous fluid flowing down 
an inclined plane and also describes drift waves in a plasma (see \cite{CKTR, TK}). 
The KdV-KS equation is very interesting in the sense that it combines the dispersive 
characteristics of the Korteweg-de Vries equation and dissipative characteristics of 
the Kuramoto-Sivashinsky equation. Also, it is worth noticing that 
 \eqref{1eqhs} is a particular case of the Benney-Lin equation \cite{B,TK}; i.e.,
\begin{equation}\label{2eqhs}
   \begin{gathered}
    v_t+v_{xxx}+\eta(v_{xx}+v_{xxxx})+\beta v_{xxxxx}+vv_x=0, \quad
 x \in \mathbb{R},\;t\geq 0,\\
  v(x,0)=v_0(x),
 \end{gathered}
\end{equation}
when $\beta=0$.

The IVPs  \eqref{1eqhs-1} and \eqref{1eqhs} were studied by Biagioni, Bona, 
Iorio and Scialom \cite{BBIS}. The authors in \cite{BBIS} proved that the 
IVPs \eqref{1eqhs-1} and \eqref{1eqhs} are  locally well-posed for given data 
in $H^s$, $s\geq 1$ with $\eta >0$. They also constructed appropriate \emph{a priori}
estimates and used them to prove global well-posedness too.
The limiting behavior of solutions as the dissipation tends to zero
(i.e., $\eta\to 0$) has also been studied in \cite{BBIS}. The IVP \eqref{2eqhs} associated 
to the Benney-Lin equation is also widely studied in the literature \cite{B, BL, TK}.
Regarding well-posedness issues for the IVP \eqref{2eqhs} the work of Biagioni 
and Linares \cite{BL} is worth mentioning, where they proved global well-posedness  
for given data in $L^2(\mathbb{R})$.

Now, we state the main results of this work. The first result deals with the
local well-posedness results for the IVP \eqref{eq:hs}, while the second result
deals the same for the IVP \eqref{eq:hs-1}, with low regularity data.

\begin{theorem}\label{teorp-1}
 Let $\eta>0$ be fixed and $\Phi(\xi)$ be as given by \eqref{phi}, then  
the IVP \eqref{eq:hs} is locally well-posed for any data $v_0 \in H^s(\mathbb{R})$, 
in the following cases:
\begin{gather*}
k=1, \quad s>-3/4,\\
k=2, \quad s>1/4,\\
k=3, \quad s>-1/6,\\
k=4, \quad s>0.
\end{gather*}
\end{theorem}

\begin{theorem}\label{teorp}
Let $\eta>0$ be fixed and $\Phi(\xi)$ be as given by \eqref{phi}, then  
the IVP \eqref{eq:hs-1}  is locally well-posed for any data $u_0 \in H^s(\mathbb{R})$, 
in the following cases:
\begin{gather*}
k=1, \quad s>1/4,\\
k=2, \quad s>5/4,\\
k=3, \quad s>5/6,\\
k=4, \quad s>1.
\end{gather*}
\end{theorem}

The first main result, Theorem \ref{teorp-1}, deals with the quite general Fourier 
symbol and generalized nonlinearity. As discussed above, some particular cases are 
studied in the recent literature. In particular, the result of 
Theorem \ref{teorp-1} improves the  local well-posedness result for \eqref{eqhs} 
with $k=3$ obtained  in \cite{Cv-Sc}.  It is worth noticing that, when $\eta =0$ 
and $k=2$, the IVP \eqref{eq:hs} turns out the modified KdV equation. 
We know that for the modified KdV equation local well-posedness holds for data
in $H^s$, $s\geq 1/4$ and we have ill-posedness for $s<1/4$. However, for $k=2$,
 $\Phi(\xi) = |\xi|-|\xi|^3$ and $\eta >0$  it has been proved  in \cite{Cv-Sc} 
that the local well-posedness holds for $s\geq 0$. Therefore, it would really 
be interesting to study the limiting behavior when $\eta \to 0$.
As noted in \cite{Cv-Sc}, it is still an open problem.

At this point, we would like to note that the first main result for $k=1$ is 
just the reproduction of our earlier result in \cite{XC-MP}. Although the result
presented in \cite[Theorem 1.1]{XC-MP} is correct, in the due course of time, 
we found a misleading argument employed in the proof. More precisely, 
the estimate \cite[(2.5)]{XC-MP} was not as it should have been. 
In this work, this flaw is corrected (see Lemma \ref{xav5}, below). 
This correction leads us to develop  the contraction mapping scheme in the 
space $X_{s-p(b-\frac12), b}$.

The second main result, Theorem \ref{teorp},  in particular, improves the 
local well-posedness results for \eqref{eqhs-1} with $k=1$ obtained in  \cite{pa:ba1} 
and for \eqref{1eqhs-1} obtained in \cite{BBIS}.

To prove the main results we follow the techniques used in \cite{XC-MP}. 
The main idea is to use  the theory developed by Bourgain
\cite{bou:bou} and Kenig, Ponce and Vega \cite{kpv2:kpv2}. The
main ingredients in the proof  are estimates
in the integral equation associated to an extended IVP that is defined for all 
$t\in \mathbb{R}$ (see IVPs \eqref{eq:hs2-1} and \eqref{eq:hs2} below).
The main idea is to use the usual Bourgain space associated to  the KdV equation 
instead of that associated to the linear part of the IVPs \eqref{eq:hs} and 
\eqref{eq:hs-1}. To carry out this scheme, the Proposition \ref{prop3} plays a 
fundamental role which permits us to use  a multilinear estimates for 
$\partial_x(u^2)$, $\partial_x(u^3)$ $\partial_x(u^4)$ and $\partial_x(u^5)$  
proved respectively  in
\cite{kpv2:kpv2,Tao,Axel,MSYG}.

As noted earlier, the IVPs \eqref{eq:hs-1} and \eqref{eq:hs} are globally well-posed 
for given data in $H^s(\mathbb{R})$, $s\geq 1$. As the models under consideration do not
have conserved quantities, the global well-posedness have been proved by constructing 
appropriate \emph{a priori} estimates. 
However, for given data in $H^s(\mathbb{R})$, $s< 1$
no \emph{a priori} estimates are available. 
Also, the lack of conserved quantities
prevent us to use the recently introduced $I$-method \cite{CKSTT, CKSTT-2}, 
to obtain global solution for the low regularity data.

Now we introduce  function spaces that will  be used to prove
the main results.  We consider the following IVP associated
to the linear KdV equation
\begin{equation}\label{eq:hs0}
   \begin{gathered}
     w_t+w_{xxx}=0, \quad x, \;t\in \mathbb{R},\\
     w(0)=w_0.
   \end{gathered}
\end{equation}
The solution to \eqref{eq:hs0} is given by
$w(x,t)=[U(t)w_0](x)$, where the unitary group $U(t)$ is defined as
\begin{align}\label{gU}
\widehat{U(t)w_0}(\xi)=e^{it\xi^3}\widehat{w_0}(\xi).
\end{align}
For $s,b\in \mathbb{R}$, we define the space $X_{s,b}$ as  the
completion of the Schwartz space $S(\mathbb{R}^2)$ with respect to
the norm
\begin{equation}\label{xsb-norm}
 \begin{split}
  \|w\|_{X_{s,b}} \equiv  \|U(-t)w\|_{H_{s,b}}
&:= \|\langle \tau \rangle^{b}
  \langle \xi \rangle^{s}  \widehat{U(-t)w}(\xi,\tau) \|_{L_{\tau}^2L_{\xi}^2} \\
 &= \|\langle \tau-\xi^3 \rangle^{b}
  \langle \xi \rangle^{s}  \widehat w(\xi,\tau) \|_{L_{\tau}^2L_{\xi}^2 {\textstyle,}}
 \end{split}
\end{equation}
where $\widehat w(\xi,\tau)$ is the Fourier transform of $w$ in both space 
and time variables, and $\langle\cdot\rangle = (1+|\cdot|^2)^{1/2}$. 
The space $X_{s,b}$ is the usual Bourgain space for the KdV equation 
(see \cite{bou:bou}) and using the Sobolev embedding theorem one has that 
$X_{s,b}\subset C(\mathbb{R}; H^s(\mathbb{R}))$, whenever $b>1/2$.

 Note that, the IVPs \eqref{eq:hs-1} and \eqref{eq:hs} are defined only for $t \ge 0$. 
To use Bourgain's type space, we should be able to write these IVPs for all 
$t \in \mathbb{R}$.
For this, we define
\begin{equation}\label{eta}
\eta (t)\equiv \eta \operatorname{sgn}(t)= \begin{cases}
 \eta & \text{if } t \geq 0 ,\\
 -\eta & \text{if } t <0
\end{cases}
\end{equation}
and write the  IVPs \eqref{eq:hs} and \eqref{eq:hs-1} in the following forms
\begin{equation}\label{eq:hs2}
   \begin{gathered}
     v_t+v_{xxx}+\eta(t)Lv+(v^{k+1})_x=0, \quad x,  t \in \mathbb{R},\; k\in \mathbb{Z}^+,\\
     v(0)=v_0,
   \end{gathered}
\end{equation}
and
\begin{equation}\label{eq:hs2-1}
   \begin{gathered}
     u_t+u_{xxx}+\eta(t)Lu+(u_x)^{k+1}=0, \quad x,  t \in \mathbb{R}, \; k\in \mathbb{Z}^+,\\
     u(0)=u_0,
   \end{gathered}
\end{equation}
respectively. From here onwards we consider the IVPs \eqref{eq:hs2} and \eqref{eq:hs2-1} 
 instead of  \eqref{eq:hs} and \eqref{eq:hs-1} respectively.

Now we consider the IVP associated to the linear parts of  \eqref{eq:hs2-1} 
and \eqref{eq:hs2},
\begin{equation}\label{eq:hs1}
   \begin{gathered}
     w_t+w_{xxx}+\eta(t)Lw=0, \quad x, \; t\in \mathbb{R},\\
     w(0)=w_0.
   \end{gathered}
\end{equation}
The solution to \eqref{eq:hs1} is given by $w(x,t)=V(t)w_0(x)$
where the  semigroup $V(t)$ is defined as
\begin{equation}\label{gV}
\widehat{V(t)w_0}(\xi)=e^{it\xi^3+\eta
|t|\Phi(\xi)}\widehat{w_0}(\xi).
\end{equation}

Observe that, defining  $\widetilde{U}(t)$ by
$\widehat{\widetilde{U}(t)u_0}(\xi)=e^{\eta
|t|\Phi(\xi)}\widehat{u_0}(\xi)$, the semigroup $V(t)$ can be written as
$V(t)=U(t)\widetilde{U}(t)$ where $U(t)$ is the unitary group
associated to the KdV equation (see \eqref{gU}).

This paper is organized as follows: In Section \ref{sec-2}, we prove 
some preliminary estimates and in Section \ref{sec-3} we prove the main results.

\section{Preliminary estimates}\label{sec-2}

This section is devoted to obtain some preliminary estimates that are essential 
in the proof of the main results.
Before going to details, we consider a cut-off function 
$\psi  \in C^{\infty}(\mathbb{R})$, such that $0\leq \psi(t) \leq 1$,
\begin{equation}\label{psi}
 \psi(t)= \begin{cases}
 1 & \text{if } |t|  \leq 1,\\
 0 & \text{if } |t| \ge 2.
\end{cases}
\end{equation}
Also, we define $\psi_{T}(t)\equiv\psi(\frac{t}{T})$.

Let $p=2m+n$, observe that the Fourier symbol given in  \eqref{phi} can be written as
\begin{equation} \label{phi1}
\begin{split}
\Phi(\xi) &= -|\xi|^{p}+ \sum_{\substack{0 \le i \le 2m, \,0 \le j\le n, \\
(i,j) \neq (2m,n)}}c_{i,j}\xi^i |\xi|^j, \quad c_{i,j} \in \mathbb{R},  \\
&=-|\xi|^{p}+\Phi_1(\xi),
\end{split}
\end{equation}
where the degree of $\Phi_1$ is less than $p$. In what follows, we present some
elementary  lemmas.

\begin{lemma}\label{xav4}
There exists $M>0$ such that for all $|\xi| \geq M$, one has that
\begin{equation}\label{xavi9}
\Phi(\xi)=-|\xi|^{p}+\Phi_1(\xi) <-1.
\end{equation}
\end{lemma}

\begin{proof}
The inequality \eqref{xavi9} is a direct consequence of
\begin{equation*}
\lim_{|\xi| \to \infty} \frac{\Phi_1(\xi) +1}{ |\xi|^p}=0.
\end{equation*}
\end{proof}

\begin{lemma}\label{xav7}
The Fourier symbol $\Phi(\xi)$ satisfies the estimate
\begin{equation}
\langle\Phi(\xi)\rangle \le c \langle \xi \rangle^{p}.
\end{equation}
\end{lemma}

\begin{proof}
It is not difficult see that
\begin{align*}
\langle\Phi(\xi)\rangle  &\le  \langle|\xi|^p\rangle + \langle\Phi_1(\xi)\rangle \\
&\leq \langle\xi\rangle^p + \sum_{\substack{0 \le i \le 2m, \,0 \le j\le n, \\ 
 (i,j) \neq (2m,n)}}|c_{i,j}|\langle\xi^i |\xi|^j \rangle \\
&\leq \langle\xi\rangle^p + \sum_{\substack{0 \le i \le 2m, \,0 \le j\le n, \\ 
 (i,j) \neq (2m,n)}}|c_{i,j}|\langle\xi \rangle^{i +j}\\
&\leq \langle\xi\rangle^p \Big(1 + \sum_{\substack{0 \le i \le 2m, \,0 \le j\le n, \\
 (i,j) \neq (2m,n)}}|c_{i,j}|\Big).
\end{align*}
\end{proof}

\begin{lemma}\label{xav5}
Let $0<T \leq 1$, $1/2 \le b \le 1$ and $a \le B$. Then we have
\begin{equation}\label{xavi1}
\|\Psi_{T}(\cdot)e^{a|\cdot|}\|_{H_t^{b}}\leq c e^{2 B}
\big( T^{\frac12-b}+ |a|^{b-1/2}\big).
\end{equation}
\end{lemma}

\begin{proof}
Let $h(t)=\Psi(t)\,e^{a|t|T}$, so that $h_T(t)=\Psi_T(t)\,e^{a|t|}$.
 A straight forward calculation yields
\begin{equation}
\|\Psi_T(\cdot)e^{a|\cdot|}\|_{H_t^{b}}
= \|h_T\|_{H_t^{b}}\leq c\,T^{1/2}\|{h}_{L^2}+c\,T^{1/2-b}\|D_t^{b}h\|_{L^2}.
\label{chave4}
\end{equation}
We know that
\begin{equation}
\|h\|_{L^2}^2 =\int_{-2}^{2}|\Psi(t)|^2\,e^{2 a |t|T}\,dt
 \leq 4\,e^{4B T}\|\Psi\|_{L^{\infty}}^2. \label{novouno}
\end{equation}
To bound the term $\|D_t^{b}h\|_{L^2}$, we  explore $\widehat{h}(\tau)$
by integrating by parts two times, and obtain
\begin{align*}
\widehat{h}(\tau)
&=\int_0^{+\infty}\Psi(t)e^{aTt}e^{-it\tau}\,dt
 + \int_{-\infty}^0\Psi(t)e^{-aTt}e^{-it\tau}\,dt   \\
&=\frac{-1}{aT-i\tau}
 \Big(1+\int_0^{+\infty}\frac{d\Psi}{dt}(t)\,e^{t(aT-i\tau)}\,dt\Big)\\
&\quad -\frac{1}{aT+i\tau}\Big(1-\int_{-\infty}^0\frac{d\Psi}{dt}(t)
 e^{-t(aT+i\tau)}\,dt\Big)  \\
&=\frac{-2aT}{(aT)^2+\tau^2}+\frac{1}{(aT-i\tau)^2}\int_0^{+\infty}
 \frac{d^2\Psi}{dt^2}(t)\,e^{t(aT-i\tau)}\,dt\\
&\quad +\frac{1}{(aT+i\tau)^2}\int_{-\infty}^0\frac{d^2\Psi}{dt^2}(t)
 e^{-t(aT+i\tau)}\,dt .  %\label{novodos}
\end{align*}
From this we have that
\begin{align}
|\widehat{h}(\tau)|&\leq \frac{2|a|T}{(aT)^2+\tau^2}
+\frac{4\,e^{2B T}\|\frac{d^2\Psi}{dt^2}\|_{L^{\infty}}}{(aT)^2+\tau^2}, \label{novotres} \\
|\widehat{h}(\tau)|&\leq 4\,e^{2B T}\|\Psi\|_{L^{\infty}} \le ce^{2B}.\label{novoquatro}
\end{align}
From \eqref{novotres} and \eqref{novoquatro}, we obtain
\begin{equation}
|\widehat{h}(\tau)| \leq \frac{2|a|T +ce^{2B} }{1+(aT)^2+\tau^2}. \label{novocinco}
\end{equation}
Multiplying \eqref{novocinco} by $|\tau|^{b}$, taking square and integrating on
$\mathbb{R}$, we obtain
\begin{equation}
\begin{aligned}
\|D_t^{b}h\_{L^2}^2
&=\| |\tau|^{b}\widehat{h}(\tau)\|_{L^2}^2\\
&\le ca^2T^2\int_{\mathbb{R}}\frac{|\tau|^{2b}}{(1+a^2T^2+\tau^2)^2}\,d\tau
 + ce^{4B}\int_{\mathbb{R}}\frac{|\tau|^{2b}}{(1+a^2T^2+\tau^2)^2}\,d\tau  \\
&\le ca^2T^2\int_{\mathbb{R}}\frac{|\tau|^{2b}}{(a^2T^2+\tau^2)^2}\,d\tau
 + ce^{4B} \int_{\mathbb{R}}\frac{|\tau|^{2b}}{(1+\tau^2)^2}\,d\tau  \\
&\le c |aT|^{2b-1}+ce^{4B} \\
& \leq c\,e^{4B}\langle  aT\rangle^{2b-1},
\end{aligned} \label{novoseis}
\end{equation}
where in the second inequality we used $\tau=|a|T x$. Thus
\begin{equation}\label{xavi3}
\|\Psi_{T}(\cdot)e^{a|\cdot|}\|_{H_t^{b}}
\leq c e^{2 B} \left(T^{1/2} +T^{1/2-b}+|a |^{b-1/2}\right).
\end{equation}
Since $T\leq 1$, we conclude \eqref{xavi1} from \eqref{chave4}, \eqref{novouno},
\eqref{novoseis} and \eqref{xavi3}.
\end{proof}


\begin{remark} \rm
Considering $T=1$, the estimate \eqref{xavi1} yields
\begin{equation}\label{xavi1.11}
\|\Psi_{T}(\cdot)e^{a|\cdot|}\|_{H_t^{b}}\leq c e^{2 B}\langle a \rangle^{b-1/2}.
\end{equation}
\end{remark}

I what follows we present some results from the earlier works \cite{xavirica}
and \cite{XC-MP}. Before providing the exact announcement we gather some 
elementary estimates.

\begin{proposition}
For any functions $\varphi$, $g$  such that $\varphi g\in H^1$  and 
$\operatorname{supp}{ \varphi} \subset  [-L,L]$ we have
\begin{equation}\label{eq:5}
\|\varphi g\|_{L^2} \le C L \|\frac{d}{dt}(\varphi g)|\|_{L^2},
\end{equation}
where $C$ is independent of $g, \varphi, L$.
\end{proposition}

\begin{proof}
We have
$$
\|\varphi g\|_{L^2}^2=\int_{-L}^{L} |g(x)\,\varphi (x)|^2 dx 
\le 2\,L  \|\varphi\, g \|_{L^{\infty}}^2.
$$
Now, using the known inequality $ \|u\|_{L^{\infty}}^2\le c\|u\|_{L^{2}}\|u'\|_{L^{2}}$,
we obtain
$$
\|g \varphi \|_{L^2}^2 \le C L\|g\,\varphi \|_{L^{2}}\|\frac{d}{dt}(g\,\varphi) \|_{L^{2}},
$$
thereby getting the required estimate.
\end{proof}

\begin{lemma} \label{lem-rv1} 
The following estimate holds 
\begin{equation}\label{rv.1}
\|\Psi_T g\|_{H^1} \le C \|\Psi_{2T} g\|_{H^1}.
\end{equation}
\end{lemma}

\begin{proof}
We have
$$
\|\Psi_T g\|_{H^1}  \sim \|\Psi_T g\|_{L^2}+  \|\frac{d}{dt}(\Psi_T g)\|_{L^2}.
$$
It is obvious that $\|\Psi_T g\|_{L^2} \le \|\Psi_{2T} g\|_{L^2}$. 
Thus to get the desired estimate \eqref{rv.1} it is sufficient to prove that
\begin{equation}\label{eq:1}
\|\frac{d}{dt}(\Psi_T g)\|_{L^2} \le C  \|\frac{d}{dt}(\Psi_{2T} g)\|_{L^2}.
\end{equation}

To prove \eqref{eq:1}, observe that in the support of $\Psi_T$ one has $g=g \Psi_{2T}$.
On the other hand
\begin{equation} \label{eq:2}
\|\frac{d}{dt}(\Psi_T g)\|_{L^2}
= \|\frac{d}{dt}(\Psi_T)g+ \Psi_T\frac{d}{dt}(g)\|_{L^2}
\le \|\frac{d}{dt}(\Psi_T)g\|_{L^2}+\|\Psi_T\frac{d}{dt}(g)\|_{L^2}.
\end{equation}
From the observation above ($g=g \Psi_{2T}$ in the support of $\Psi_T$) we obtain
\begin{equation}\label{eq:3}
\|\Psi_T\frac{d}{dt}(g)\|_{L^2} =\|\Psi_T\frac{d}{dt}(g \Psi_{2T} )\|_{L^2}
 \le \|\frac{d}{dt}(g \Psi_{2T} )\|_{L^2}.
\end{equation}
We have
\begin{equation} \label{eq:4}
\begin{aligned}
\|\Psi'(\frac{t}{T})g\|_{L^2}^2
&=\int_{\mathbb{R}}|\Psi'(\frac{t}{T})|^2 |g(t)\Psi_{2T} |^2 dt  \\
&\le \|g\Psi_{2T}\|_{L^{\infty}}^2 \int_{\mathbb{R}}|\Psi'(\frac{t}{T})|^2 dt \\
&=T \|g\Psi_{2T}\|_{L^{\infty}}^2\int_{\mathbb{R}}|\Psi'(\tau)|^2 d\tau  \\
&\le C_{\Psi'} T \|g\Psi_{2T}\|_{L^{\infty}}^2.
\end{aligned}
\end{equation}
Now, using the known inequality $ \|u\|_{L^{\infty}}^2\le c\|u\|_{L^{2}}\|u'\|_{L^{2}}$;
from \eqref{eq:4} and \eqref{eq:5}  it follows that
\[ %\label{eq:5}
\|\Psi'(\frac{t}{T})g\|_{L^2}^2
\le C_{\Psi'} T \|g\Psi_{2T}\|_{L^{2}}\|\frac{d}{dt}(g\Psi_{2T})\|_{L^{2}}
\le C_{\Psi'} T^2 \|\frac{d}{dt}(g\Psi_{2T})\|_{L^{2}}^2.
\]
This completes the proof.
\end{proof}


\begin{proposition}\label{prop300}
Let $0\leq b\leq 1$, $B_1 \le B_2 \le 0$. Then
\begin{equation}
\|\Psi_T(t)\,\int_0^te^{B_1|t-x|}\,f(x)\,dx\|_{H^b}
\leq C\,(1+T)\,\|\Psi_{2T}(t)\int_0^te^{B_2\,|t-x|}\,f(x)\,dx\|_{H^b},\label{chave7}
\end{equation}
where $C=C_{\Psi}= C\max \big\{\|\Psi\|_{L^{\infty}},
\|\frac{d\Psi}{dt}\|_{L^{\infty}}\big\}$ is a constant independent
 of $B_1$, $B_2$ and $f$.
\end{proposition}

\begin{proof}
The proof of this result follows by using estimate \eqref{rv.1} 
from Lemma \ref{lem-rv1}. For details we refer to \cite{xavirica}.
\end{proof}

\begin{lemma}\label{lema2.4}
Let $-1/2<b'\le 0$, $1/2<b \le b'/3+2/3$, $T \in (0,1]$, $|a|< B$. Then
\begin{equation}\label{eq2.14}
\|\psi_T(t) \int_0^t e^{a|t-t'|}f(t') dt'\|_{H_t^b} \le c_{B, \psi} T^{1+b'/2-3b/2 }
\|f\|_{H^{b'}},
\end{equation}
where $c_{B, \psi}$ is a constant independent of $a$, $f$ and $T$.
\end{lemma}

A detailed proof of the above  lemma  has been presented in \cite{XC-MP}, so omit it.
We start with following Proposition  that plays a central role in  the proof of
the main results of this work. The result of this Proposition allows
us to work in the usual  $X_{s,b}$ space associated to the KdV group $U(t)$ 
defined by \eqref{gU} instead of the Bourgain space associated to the group $V(t)$ 
defined by \eqref{gV}.


\begin{proposition}\label{prop3}
Let $b>1/2$ and $-1/2<b' \le 0$, $T \in (0,1]$. Then 
\begin{equation}\label{eq1}
  \|\psi(t)V(t)u_{0}\|_{X_{s,b}} \le  c
  \|u_{0}\|_{s+p(b-1/2)}.
\end{equation}
If $1/2 < b \le b'/3+2/3$, $s\in \mathbb{R}$ then
\begin{equation}\label{eq2}
  \| \psi_{T}(t)\int_{0}^{t}V(t-t')F(t')dt'\|_{ X_{s,b}}
  \leq c\; T^{1+b'/2-3b/2}
  \|F\|_{ X_{s,b'}},
\end{equation}
where $c$ is a constant.
\end{proposition}

\begin{proof}
To prove \eqref{eq1}, we have
\begin{align*}
\|\psi(t)V(t)u_{0}\|_{X_{s,b}} =\|\langle \xi \rangle^s \widehat{u_0}(\xi)
 \|\psi(t)e^{\Phi(\xi) \eta|t|} \|_{H_t^b}\|_{L_{\xi}^2}.
\end{align*}
Using Lemma \ref{xav7} and Lemma \ref{xav5}, we obtain
\begin{align*}
\|\psi(t)V(t)u_{0}\|_{X_{s,b}}  
&\leq \|\langle \xi \rangle^s \widehat{u_0}(\xi) 
 \|\psi(t)e^{\Phi(\xi) \eta|t|}\|_{H_t^b}\|_{L_{\xi}^2}\\
&\leq c \|\langle \xi \rangle^s \widehat{u_0}(\xi) \langle\Phi(\xi) \rangle^{b-\frac12}\|_{L_{\xi}^2}\\
&\leq c \|\langle \xi \rangle^s \widehat{u_0}(\xi) 
\langle\xi \rangle^{p(b-\frac12)}\|_{L_{\xi}^2}.
\end{align*}
This proves \eqref{eq1}.

Now, to prove \eqref{eq2}, let $M$ be as in Lemma \ref{xav4}. 
From the definition of Bourgain's space, we have
\begin{align*}
&\| \psi_{T}(t)\int_{0}^{t}V(t-t')F(t')dt'\|_{X_{s,b}}\\
&=\|\langle \xi \rangle^s  \|\psi_T(t) \int_0^t e^{-it' \xi^3}e^{\Phi(\xi) \eta|t-t'|}
 \widehat{F(t')}(\xi) dt'\|_{H_t^b}\|_{L_{\xi}^2} \\
&\le  \|\langle \xi \rangle^s  \|\psi_T(t) \int_0^t e^{-it' \xi^3}e^{\Phi(\xi) \eta|t-t'|} 
 \widehat{F(t')}(\xi) dt'\|_{H_t^b}\|_{L_{\xi}^2(|\xi|< M)}  \\
&\quad + \|\langle \xi \rangle^s  \|\psi_T(t) \int_0^t e^{-it' \xi^3}e^{\Phi(\xi) 
 \eta|t-t'|} \widehat{F(t')}(\xi) dt'\|_{H_t^b}\|_{L_{\xi}^2(|\xi|\geq M)} \\
&=:  I_1+I_2.
\end{align*}
To estimate $I_1$,  note that for $|\xi|<M$, one has
$$
|\Phi(\xi)| \le\sum_{j=0}^{n}\sum_{i=0}^{2m}|c_{i,j}|\,
 |\xi|^i |\xi|^j \le \sum_{j=0}^{n}\sum_{i=0}^{2m}|c_{i,j}|\, M^{i+j}=:c_M.
$$
Therefore,  using Lemma \ref{lema2.4}, we obtain
\begin{align*}
I_1 &\le c_{M, \psi}T^{1+b'/2-3b/2}\|\langle \xi \rangle^s  
 \| e^{-it \xi^3} \widehat{F(t)}\|_{H_t^{b'}}\|_{L_{\xi}^2(|\xi|\le M)}\\
&\le  c_{M, \psi}T^{1+b'/2-3b/2}\|F\|_{X_{s,b'}}.
\end{align*}
To estimate $I_2$,  we observe that  for $|\xi|\geq M$, one can write the
Fourier symbol as $\Phi(\xi)=(\Phi(\xi)+1)-1$, where from Lemma \ref{xav4},
 $\Phi(\xi)+1 <0$. Now, using Proposition \ref{prop300} and Lemma \ref{lema2.4}, we obtain
\begin{align*}
I_2 &\le   a(1+T)\|\langle \xi \rangle^s  \|\psi_{2T}(t) 
 \int_0^t e^{-it' \xi^3}e^{-\eta|t-t'|} 
 \widehat{F(t')}(\xi) dt'\|_{H_t^b}\|_{L_{\xi}^2(|\xi|\geq M)} \\
&\le  c_{ \psi}T^{1+b'/2-3b/2}\|F\|_{X_{s,b'}}.
\end{align*}
\end{proof}

In what follows we record the familiar multilinear estimate in the Bourgain's space 
associated to the KdV group.

\begin{proposition}\label{prop4}
Let $k=1,2,3,4$, and $s > a_k$. There exist $\gamma  \in  (\frac12, 1)$ and $r(s)>0$
such that if $b$ and $b'$ are two numbers satisfying $\frac12<b \le b'+1 < \gamma$  and
$b'+\frac12 \le r(s)$, then for $u \in X_{s,b}$ the following estimate holds
\begin{equation}\label{bil-1}
 \|(u^{k+1})_x\|_{X_{s,b'}} \leq c\,\|u\|_{X_{s,b}}^{k+1},
\end{equation}
where
\begin{equation}\label{eq-b22}
a_1 = -\frac34,\quad a_2 = \frac14,\quad a_3 =-\frac16,\quad a_4=0.
\end{equation}
\end{proposition}

For the proof of the above proposition, we refer to  
 \cite{kpv2:kpv2,JC}, \cite{Tao}, \cite{Axel}, \cite{MSYG} respectively 
for $k=1$, $k=2$, $k=3$ and $k=4$.

Before providing another multilinear estimate to prove Theorem \ref{teorp}, 
we introduce some new notation from \cite{Tao}, and auxiliary  results.

For any Abelian additive group $Z$ with an invariant measure $d\xi$, we use 
$\Gamma_k(Z)$ to denote the hyperplane
$$
\Gamma_k(Z):=\{(\xi_1, \dots , \xi_k) \in Z^k:  \xi_1+ \dots + \xi_k=0\},
\quad k\geq 2,
$$
 endowed with the measure
$$
\int_{\Gamma_k(Z)}f:=\int_{Z^{k-1}}f(\xi_1,\dots , \xi_{k-1}, -\xi_1
-\dots -\xi_{k-1})d \xi_1 \dots d\xi_{k-1}.
$$
We define a $[k;Z]$-multiplier to be any function 
$m: \Gamma_k(Z) \to \mathbb{C}$ and also define $\|m\|_{[k;Z]}$ 
to be the best constant such that the inequality
$$
|\int_{\Gamma_k(Z)}m(\xi)\prod_{j=1}^{k}f_j(\xi_j)| 
\le \|m\|_{[k;Z]}\prod_{j=1}^{k}\|f_j\|_{L^2(Z)},
$$
holds for all test functions $f_j$ on $Z$. Note that, in our case the Abelian
group $Z$ will be Euclidean space $\mathbb{R}^{n+1}$ with Lebesgue measure.

In what follows, we state in the form of Lemmas, some properties satisfied 
by the $[k;Z]$-multiplier,   whose proof can be found in \cite{Tao}.

\begin{lemma}[Comparison principle]
If $m$ and $M$ are $[k;Z]$ multipliers such that $|m(\xi)| \le M(\xi)$ for all 
$\xi \in \Gamma_k(Z)$, then $\|m\|_{[k;Z]}\le \|M\|_{[k;Z]}$ and
$$
\|m(\xi)\prod_{j=1}^k a_j(\xi_j)\|_{[k;Z]} \le \|m\|_{[k;Z]}\prod_{j=1}^k \|a_j\|_{\infty},
$$
where $a_1, \dots, a_k$ are functions from $Z$ to $\mathbb{R}$.
\end{lemma}

\begin{lemma}
For any $[k;Z]$-multiplier $m:Z^k \to \mathbb{R}$, the following properties hold:
\begin{enumerate}
\item $TT^{*}$ identity:
$$
\|m(\xi_1,\dots,\xi_k) \overline{m(-\xi_{k+1},\dots,
 -\xi_{2k})}\|_{[2k;Z]}=\|m(\xi_1,\dots,\xi_k) \|_{[k+1;Z]}^2.
$$
\item Translation invariance:  
$$
\|m(\xi)\|_{[k;Z]}=\|m(\xi+\xi_0)\|_{[k;Z]},
$$ 
for any $\xi_0 \in \Gamma_k(Z)$.

\item Averaging: 
$$
\|m \ast \mu\|_{[k;Z]} \leq \|m\|_{[k;Z]}\|\mu\|_{L^1({\Gamma_k(Z)})},
$$ 
for any finite measure $\mu$ on $\Gamma_k(Z)$.
\end{enumerate}
\end{lemma}

The following proposition is crucial in proving multilinear estimates that are 
essential in the proof of the second main result of this work.

\begin{proposition}\label{prop1}
Let $ k =2,3,4,5$. Under the hypothesis of the Proposition \ref{prop4}, we have
\begin{equation}\label{eq:prop}
\|\prod_{j=1}^{k}u_j\|_{X_{s,b'}} \leq c\prod_{j=1}^{k}\|u_j\|_{X_{s,b}},  \quad s> s_k,
\end{equation}
where $s_2=-3/4$, $s_3=1/4$, $s_4=-1/6$, $s_5=0$.
\end{proposition}


\begin{proof}
To prove the estimate  \eqref{eq:prop}, we will use the techniques developed by 
Tao in \cite{Tao} on $[k,Z]$ multipliers.
Consider $u_j \in X_{s,b}$ for $j=1,\dots k$, $u_{k+1} \in X_{-s,-b'}$ and 
use properties of the  Fourier transform, to obtain
\begin{align*}
&\int_{\mathbb R^2} (\prod_{j=1}^{k}u_j)(\xi, \tau) 
 \overline{u_{k+1}}(\xi, \tau) d\xi d\tau \\
&= \int_{\mathbb R^2}  \int_{\mathbb R^{(k-1)\times(k-1)}}\widehat{u_1}(\xi_1,\tau_1)
 \widehat{u_2}(\xi_2,\tau_2)\dots\\
&\quad \widehat{u_{k}}(\xi-\sum_{j=1}^{k-1}\xi_j,\tau-\sum_{j=1}^{k-1}\tau_j) 
 \widehat{u_{k+1}}(-\xi,-\tau)d\xi_1d\tau_1 \dots d\xi_{k-1} d\tau_{k-1}d\xi d\tau\\
&=:\int_{\stackrel{\xi_1+\xi_2+\dots+\xi_{k+1}=0}{\tau_1+\tau_2+\dots
 +\tau_{k+1}=0}}\prod_{j=1}^{k+1}\widehat{u_j}(\xi_j,\tau_j)d\xi_1d\tau_1\dots d\xi_4 
d\tau_4,
\end{align*}
Therefore, using duality proving \eqref{eq:prop} is equivalent to proving
\begin{align*}
&\int_{\stackrel{\xi_1+\xi_2+\dots+\xi_{k+1}=0}{\tau_1+\tau_2+\dots
+\tau_{k+1}=0}} \prod_{j=1}^{k+1}\widehat{u_j}(\xi_j,\tau_j)d\xi_1d
\tau_1\dots d\xi_{k+1} d\tau_{k+1} \\
&\lesssim \prod_{j=1}^{k}\|u_j\|_{X_{s,b}}
\|u_{k+1}\|_{X_{-s,-b'}}.
\end{align*}
Let
\begin{gather*}
\langle \xi_j \rangle^s \langle \tau_j- \xi_j^3 \rangle^b\widehat{u_j}(\xi, \tau)
 =\widehat{f_j}(\xi, \tau), \quad j=1,\dots,k, \\
\langle \xi_{k+1}\rangle^{-s} \langle \tau_{k+1}- \xi_{k+1}^3 
 \rangle^{-b'}\widehat{u_{k+1}}(\xi, \tau) =\widehat{f_{k+1}}(\xi, \tau).
\end{gather*}
Now with these considerations, proving \eqref{eq:prop} is equivalent to proving
\begin{equation} \label{Taoxav}
\begin{split}
&\int_{\stackrel{\xi_1+\xi_2+\dots+\xi_{k+1}=0}{\tau_1+\tau_2+\dots+\tau_{k+1}=0}}
m((\xi_1,\tau_1),\dots (\xi_{k+1}, \tau_{k+1}))\\
&\times \prod_{j=1}^{k+1}
\widehat{f_j}(\xi_j,\tau_j)d\xi_1d\tau_1\dots d\xi_{k+1} d\tau_{k+1} \\
&\lesssim  \prod_{j=1}^{k+1}\|f_j\|_{L_{x,t}^{2}},
\end{split}
\end{equation}
where
\begin{equation}\label{Taoxav1}
m((\xi_1,\tau_1),\dots (\xi_{k+1},\tau_{k+1}))
=\frac{ \langle\xi_{k+1} \rangle^s}{\prod_{j=1}^{k} \langle\xi_j
\rangle^s \prod_{j=1}^{k+1}\langle\tau_j-\xi_j^3\rangle^{b_j}},
\end{equation}
and  $b_1=\dots =b_k=b$, $b_{k+1}=-b'$.
So, we need to prove that the $[k+1,\mathbb{R}^2]$-multiplier estimate is finite; i.e.,
$\|m\|_{[k+1;\mathbb{R}^2]}< \infty$.

We know from Proposition \ref{prop4} that the  $[k+1,\mathbb{R}^2]$-multiplier estimate
\begin{equation}\label{Taoxav12}
\tilde{m}((\xi_1,\tau_1),\dots (\xi_{k+1},\tau_{k+1}))
=\frac{|\xi_{k+1}| \langle\xi_{k+1} \rangle^s}{\prod_{j=1}^{k} \langle\xi_j 
\rangle^s \prod_{j=1}^{k+1}\langle\tau_j-\xi_j^3\rangle^{b_j}},
\end{equation}
where  $b_1=\dots =b_k=b$, $b_{k+1}=-b'$;  is finite.

Observe that we may restrict the multiplier \eqref{Taoxav1} to the region
$|\xi_{k+1} | \ge 1$,  (since the general case then follows by an averaging over 
unit time scales). The $|\xi_j| \le  1$ behavior of $m$ is usually identical 
to its $|\xi_j| \sim 1$ behavior, see Section 4 on $X_{s,b}$ spaces in
 \cite[page 17]{Tao}.

In the high frequencies, we have
$m \le \tilde{m}$,
and the Comparison principle implies that $\|m\|_{[k+1;\mathbb{R}^2]}< \infty$ as required.
\end{proof}

\begin{remark} \rm
We note that the multilinear estimates without derivative hold in the $X_{s,b}$ 
spaces with low regularity than that with derivative. For example, in the case 
 $k=3$  the inequality \eqref{eq:prop} holds true for $s>-1/4$, see \cite{XC},
 and with derivative holds for $s\geq 1/4$, see  \eqref{bil-1} in
Proposition \ref{prop4} above.
\end{remark}


The following Lemma  is an immediate consequence of Propositions \ref{prop4} and
 \ref{prop1} and will be used in the proof of Theorem \ref{teorp}.

\begin{lemma}\label{Cor-1}
 Let $ k =1,2,3,4$. Under the hypothesis of  Proposition \ref{prop4}, we have
\begin{equation}\label{eq.cor}
\|(u_x)^{k+1}\|_{X_{s,b'}} \leq c\|u\|_{X_{s,b}}^{k+1},
\end{equation}
whenever,
\begin{equation}\label{conk2}
\begin{gathered}
k=1, \quad s>1/4,\\
k=2, \quad s>5/4,\\
k=3, \quad s>5/6,\\
k=4, \quad s>1.
\end{gathered}
\end{equation}
\end{lemma}

\begin{proof}
Let $k=1, 2, 3, 4$, and consider $s$ satisfying \eqref{conk2}. 
As $\langle\xi\rangle^s=\langle\xi\rangle^{s-1}\langle\xi\rangle$, we have
\begin{equation}\label{eq:333}
\| (u_x)^{k+1}\|_{X_s, b'} \leq\|D_x(u_x)^{k+1}\|_{X_{s-1, b'}}
+\| (u_x)^{k+1}\|_{X_{s-1}, b'}.
\end{equation}
For the  first term we have
\begin{equation}\label{rv-c1}
\|D_x(u_x)^{k+1}\|_{X_{s-1,b'}}
\leq c\,\|u_x\|_{X_{s-1,b}}^{k+1} \leq c\|u\|_{X_{s,b}}^{k+1},
\end{equation}
where in the first inequality the bilinear estimate \eqref{bil-1} has been used.

To estimate the second term in \eqref{eq:333}, we use \eqref{eq:prop} to obtain
\begin{align}
\| (u_x)^{k+1}\|_{X_{s-1}, b'} \le c\| u_x\|_{X_{s-1}, b}^{k+1}\leq c\|u\|_{X_{s,b}}^{k+1},
\end{align}
which completes  the proof of \eqref{eq.cor}.
\end{proof}

\section{Proof of main results}\label{sec-3}


\begin{proof}[Proof of Theorem \ref{teorp-1}]
As discussed in the introduction, we will use Bourgain's space associated to
 the KdV group to prove well-posedness for the IVP \eqref{eq:hs}, therefore
 we need to consider the IVP \eqref{eq:hs2} that is defined for all $t$. 
Now consider the IVP \eqref{eq:hs2} in its equivalent integral form
\begin{equation} \label{int1}
v(t)=V(t)v_{0}- \int_{0}^{t}V(t-t')(v^{k+1})_x(t')dt',
\end{equation}
where
$V(t)$ is the semigroup associated with the linear part given by \eqref{gV}.

Note that, if for all $t\in \mathbb{R}$, $v(t)$ satisfies
\[
v(t)=\psi(t)V(t)v_{0}- \psi_{T}(t)\int_{0}^{t}V(t-t')(v^{k+1})_x(t')dt',
\]
with $T\in (0, 1]$, then $v(t)$ satisfies \eqref{int1} in $[-T,T]$.
We define an application
\[
  \Psi(v)(t)= \psi(t)\,V(t)v_0-\psi_{T}(t)\,\int_0^t V(t-t')(v^{k+1})_x(t')dt'.
\]

Assume $k\in\{1,2,3,4\}$ and $s>a_k$, where $a_k$ is given by \eqref{eq-b22}. 
Let $v_0\in H^s$ and let us define $b:=1/2 + \epsilon$, $b':=-1/2+4 \epsilon$,
with $0< \epsilon \ll 1$ satisfying
\begin{equation}\label{epsilon-1}
0<\epsilon< \min\big\{ \frac{s-a_k}p, \frac14\big(\gamma-\frac12\big), \frac{r(s)}4\big\},
\end{equation}
 where $\gamma$ and $r(s)$ are as in Proposition \ref{prop4}.
 With these choices of $b$ and $b'$ it is easy to verify that all the conditions 
of  Propositions \ref{prop3} and \ref{prop4}, and Lemma \ref{lema2.4} are satisfied. 
For $M>0$, let us define a ball
\[
  X_{s-p(b-\frac12),b}^M= \{f\in X_{s-p(b-\frac12),b} :
  \|f\|_{X_{s-p(b-\frac12),b}}\leq M \}.
\]
We will prove that there exists $M$ such that the application  $\Psi$ maps 
 $X_{s-p(b-\frac12),b}^M $ into
$X_{s-p(b-\frac12),b}^M$ and is a contraction. Let $v\in X_{s-p(b-\frac12),b}^M$.
By using Proposition \ref{prop3}, we obtain
\begin{equation}\label{eq3.5}
  \|\Psi(v)\|_{X_{s-p(b-\frac12),b}}
  \leq  c\|v_0\|_{H^s}+c\,T^\alpha \|(v^{k+1})_x\|_{X_{s-p(b-\frac12),b'}},
\end{equation}
 where $\alpha:=1+\frac{b'}2-\frac{3b}2 =\frac{\epsilon}2>0$.
The use of Proposition \ref{prop4} in \eqref{eq3.5} yields
 \begin{equation}\label{eq3.6}
  \|\Psi(v)\|_{X_{s-p(b-\frac12),b}}\leq   c\|v_0\|_{H^s}+c\,T^\alpha \|v\|_{X_{s-p(b-\frac12),b}}^{k+1},
\end{equation}
whenever
\begin{equation}\label{eq3.7}
\begin{gathered}
s-p(b-\frac12)>-3/4, \quad \text{for }  k =1,\\
s-p(b-\frac12)>1/4, \quad \text{for }  k =2,\\
s-p(b-\frac12)>-1/6, \quad \text{for }  k =3,\\
s-p(b-\frac12)>0, \quad \text{for } k =4,
\end{gathered}
\end{equation}
holds, which is true because of the choice of $b$ and arbitrarily small $\epsilon$ 
satisfying \eqref{epsilon-1}.
 Now, using the definition of $X_{s-p(b-\frac12),b}^M$, one obtains
 \begin{equation}\label{eq3.8}
  \|\Psi(v)\|_{X_{s-p(b-\frac12),b}}\leq \frac{M}{4}+ cT^\alpha M^{k+1}\leq  \frac{M}{2},
\end{equation}
where we have chosen $M=4c\|v_0\|_{H^s}$ and $cT^\alpha M^k=1/4$.
Therefore, from \eqref{eq3.8} we see that the application $\Psi$ maps 
$X_{s-p(b-\frac12),b}$ into itself. A similar argument proves that $\Psi$ is a contraction. 
 Hence $\Psi$ has a fixed point $v$ which is a  solution of the IVP \eqref{eq:hs} 
such that $u \in C([-T,T], H^{s-p(b-\frac12)})$.

 Since $\epsilon>0$ is arbitrarily small satisfying \eqref{epsilon-1} 
and $b=\frac12+\epsilon$, this concludes the proof of the theorem.
 \end{proof}

\begin{proof}[Proof of Theorem \ref{teorp}]
This proof is analogous to that of  Theorem \ref{teorp-1}. 
The only difference is that, in this case, we use  Lemma \ref{Cor-1} instead 
of Proposition \ref{prop4}.
\end{proof}

\section{A priori estimate: global solutions}\label{global}

In this section we find an \emph{a priori} estimate that leads to conclude global
 well-posedness of the IVPs \eqref{eq:hs} and \eqref{eq:hs-1}.

\begin{lemma}\label{a-priori}
Let $v_0\in H^3(\mathbb{R})$ and $v \in  C([0, T ],H^3(\mathbb{R}))$ be the solution of \eqref{eq:hs} with
initial data $v(x, 0) = v_0$. Then the following a priori estimate
\begin{equation}\label{eq4.1}
\|v(t)\|_{L^2} \leq C \|v_0\|_{L^2}e^{C\eta T},
\end{equation}
holds.
\end{lemma}

\begin{proof}
We multiply  \eqref{eq:hs} by $v$ and integrate by parts to obtain
\begin{equation}\label{eq4.2}
\frac12\frac{d}{dt}\int v^2(x)dx +\eta\int v(x)Lv(x)dx=0.
\end{equation}
Now using our assumption on the Fourier symbol $\Phi$ of $L$ 
from \eqref{phi}, Plancherel's identity we obtain from \eqref{eq4.2} that
\begin{equation}\label{eq4.3}
\frac12\frac{d}{dt}\|v(t)\|_{L^2}^2 
= \eta\int \widehat{v}(\xi)\Phi(\xi)\bar{\widehat{v}}(\xi)d\xi 
\leq C\eta\int\int\widehat{v}(\xi)\bar{\widehat{v}}(\xi)d\xi = C\eta\|v(t)\|_{L^2}^2.
\end{equation}
Now, integrating \eqref{eq4.3} in $[0, t]$ for $t\in [0, T]$, and applying Gronwall's 
inequality, we obtain the required an \emph{a priori} estimate \eqref{eq4.1}.
\end{proof}

\begin{remark} \rm
As in Lemma \ref{a-priori}, differentiating equation \eqref{eq:hs-1} with respect 
to $x$, multiplying the resulting equation by $u_x$ and the integrating by parts 
and using Plancherel's identity and Gronwall's inequality, we obtain the following 
an  \emph{a priori} estimate
\begin{equation}\label{eq4.4}
\|\partial_xu(t)\|_{L^2} \leq C \|\partial_xu_0\|_{L^2}e^{C\eta T}.
\end{equation}
\end{remark}

Now, with the \emph{a priori} estimates \eqref{eq4.1} and \eqref{eq4.4} at hand,
one can prove the following global results for the IVPs \eqref{eq:hs} 
and \eqref{eq:hs-1} for some particular values of $k$.

\begin{theorem}\label{global-1}
Let $k = 1, 3$, and $v_0\in H^s(\mathbb{R})$, $s\geq 0$, then the local solution
of \eqref{eq:hs} obtained in Theorem \ref{teorp-1} can be extended globally in time.
\end{theorem}

\begin{theorem}\label{global-2}
Let $k = 1, 3$, and $u_0\in H^s(\mathbb{R})$, $s\geq 1$, then the local solution
of \eqref{eq:hs-1} obtained in Theorem \ref{teorp} can be extended globally in time.
\end{theorem}

\subsection*{Acknowledgments} 
X. Carvajal was supported by grants E-26/111.564/2008
and E-26/ 110.560/2010 from  FAPERJ Brazil, grant 303849/2008-8
from the National Council of Technological and 
Scientific Development (CNPq) Brazil.

M. Panthee was supported  by grant Est-C/MAT/UI0013/2011
from FEDER Funds through ``Programa
Operacional Factores de Competitividade - COMPETE'',
and by grant PTDC/MAT/109844/2009 from Portuguese Funds through FCT - 
``Funda\c c\~ao para a Ci\^encia e a Tecnologia''.
Part of this research was done while M. Panthee was visiting the Institute
of Mathematics, Federal University of Rio de Janeiro, Brazil.
 He wishes to thank for the support received during his visit.

The authors are thankful to the anonymous referee for his or her numerous 
corrections and remarks on early versions of this work.

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\end{document}