\documentclass[reqno]{amsart} \usepackage{amssymb,amsmath}% \usepackage{ifpdf} \ifpdf \usepackage[hyperindex]{hyperref}%,pagebackref \else \expandafter \ifx \csname dvipdfm \endcsname \relax \usepackage[hypertex,hyperindex]{hyperref} \else \usepackage[dvipdfm,hyperindex]{hyperref} \fi \fi \AtBeginDocument{{\noindent\small \sc{Journal of Algebra and Computational Applications}\newline ISSN: 2217-6764, URL: www.ilirias.com/jaca\newline Volume 7 Issue 4(2017), Pages 1-x.} \thanks{\copyright 2017 Ilirias Research Institute, Prishtin\"e, Kosov\"e.} \vspace{9mm}} \begin{document} \title[ Well-posedness for perturbations of the KdV] {Well-posedness for some perturbations of the KdV equation with low regularity data} \author[X. Carvajal, M. Panthee] {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 An\'alise Matem\'atica, Geometria e Sistemas Din\^amicos, Departamento de Matem\'atica, Instituto Superior T\'ecnico, 1049-001 Lisboa, Portugal.\newline Department of Mathematics, Central Department of Mathematics, Tribhuvan University, Kirtipur, Kathmandu, Nepal} \email{mpanthee@math.ist.utl.pt} \thanks{Submitted August 1, 2007. Published January 2, 2008.} \thanks{X. C. was supported by the grant ABC.} \subjclass[2000]{35A07, 35Q53} \keywords{Bourgain spaces; KdV equation; local smoothing effect} \begin{abstract} We study some well-posedness issues of the initial value problem associated with the equation \[ u_t+u_{xxx}+\eta Lu+uu_x=0, \quad x \in \mathbb{R}, \; t\geq 0, \] where $\eta>0$, $\widehat{Lu}(\xi)=-\Phi(\xi)\hat{u}(\xi)$ and $\Phi \in \mathbb{R}$ is bounded above. Using the theory developed by Bourgain and Kenig, Ponce and Vega, we prove that the initial value problem is locally well-posed for given data in Sobolev spaces $H^s(\mathbb{R})$ with regularity below $L^2$. Examples of this model are the Ostrovsky-Stepanyams-Tsimring equation for $\Phi(\xi)=|\xi|-|\xi|^3$, the derivative Korteweg-de Vries-Kuramoto-Sivashinsky equation for $\Phi(\xi)=\xi^2-\xi^4$, and the Korteweg-de Vries-Burguers equation for $\Phi(\xi)=-\xi^2$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem*{remark}{Remark} \section{Introduction} In this paper we consider the initial value problem (IVP) \begin{equation}\label{eq:hs} \begin{gathered} u_t+u_{xxx}+\eta Lu+uu_x=0, \quad x \in \mathbb{R}, \;t\geq 0,\\ u(x,0)=u_0(x), \end{gathered} \end{equation} where $\eta>0$ is a constant and the linear operator $L$ is defined via the Fourier transform by $\widehat{Lu}(\xi)=-\Phi(\xi)\hat{u}(\xi)$. 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 $\Phi(\xi) < C$. Without loss of generality, we suppose that $\Phi(\xi) < 1$. For this, let us perform the following scale change $$ v(x,t) = \frac1{\lambda^2} u\big(\frac x{\lambda} , \frac t{\lambda^3}\big). $$ Then $v$ satisfies the equation \begin{equation}\label{scale.1} \lambda^3 v_t + \lambda^3 v_{xxx} + \eta T v + \lambda^3 vv_x = 0, \end{equation} where $$ \widehat{Tv}(\xi) = \Phi(\lambda \xi) \hat{v}(\xi). $$ If we take $\lambda^3=C$, where $C$ is as earlier, then the Fourier symbol of the new operator $T$ in \eqref{scale.1} is bounded above by $1$. Finally, inverting the scale change, we obtain well-posedness result for the original IVP (\ref{eq:hs}) from that of \eqref{scale.1}. So, throughout this work we consider the IVP (\ref{eq:hs}) with $\Phi(\xi)$ in (\ref{phi}) satisfying $\Phi(\xi) < 1$. Our interest here is to obtain well-posedness results for \eqref{eq:hs} with given data $u_0$ in the Sobolev spaces $H^s(\mathbb{R})$ with regularity below $L^2$. The $L^2$-based Sobolev space $H^s(\mathbb{R})$ is defined by $$ H^s(\mathbb{R}) := \{f\in \mathcal{S}'(\mathbb{R}) : \|f\|_{H^s} < \infty\}, $$ where $$ \|f\|_{H^s}^2 = \int_{\mathbb{R}} (1+|\xi|^2)^s|\hat f(\xi)|^2d\xi, $$ and $\hat f(\xi)$ is the usual Fourier transform given by $$ \hat f(\xi) \equiv \mathcal{F}(f)(\xi) = \frac 1{\sqrt{2\pi}}\int_{\mathbb{R}}e^ {-ix\xi} f(x)\, dx. $$ However, from here onwards, we will neglect the factor $2\pi$ in the definition of the Fourier transform because it does not alter our analysis. Also, we consider the homogeneous Sobolev space $\dot H^s(\mathbb{R})$ defined via the norm $$ \|f\|_{\dot H^s}^2 = \int_{\mathbb{R}} |\xi|^{2s}|\hat f(\xi)|^2d\xi.$$ Before stating the main results of this work, we give some examples that belong to the class considered in (\ref{eq:hs}). The first example of this type is the generalized Ostrovsky-Stepanyams-Tsimring (OST) equation. \begin{equation}\label{eqhs} \begin{gathered} u_t+u_{xxx}-\eta(\mathcal{H}u_x+\mathcal{H}u_{xxx})+u^ku_x=0, \quad x \in \mathbb{R}, \; t\geq 0, \; k\in \mathbb{Z}^+,\\ u(x,0)=u_0(x), \end{gathered} \end{equation} where $\mathcal{H}$ denotes the Hilbert transform: \begin{align*} \mathcal{H}g(x)=\text{ P.V.} \frac{1}{\pi}\int\frac{g(x-\xi)}{\xi}d\xi, \end{align*} $u=u(x,t)$ is a real-valued function and $\eta>0$ is a constant. Equation (\ref{eqhs}) with $k=1$ was derived by Ostrovsky et al. in \cite{O:O} to describe the radiational instability of long waves in a stratified shear flow. Recently, Carvajal and Scialom in \cite{Cv-Sc} considered the IVP (\ref{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 the global well-posedness result for data in $L^2$ when $k=1$. The earlier well-posedness results for (\ref{eqhs}) with $k=1$ can be found in \cite{pa:ba1}, where for given data in $H^s(\mathbb{R})$, local result when $s>1/2$ and global result when $s\geq 1$ have been obtained. Another model that fits in the class (\ref{eq:hs}) is the derivative Korteweg-de Vries-Kuramoto Sivashinsky equation \begin{equation}\label{1eqhs} \begin{gathered} u_t+u_{xxx}+\eta(u_{xx}+u_{xxxx})+uu_x=0, \quad x \in \mathbb{R}, \; t\geq 0,\\ u(x,0)=u_0(x), \end{gathered} \end{equation} where $u=u(x,t)$ is a real-valued function and $\eta>0$ is a constant. This 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 equation (\ref{1eqhs}) is a particular case of Benney-Lin equation \cite{B,TK}; i.e., \begin{equation}\label{2eqhs} \begin{gathered} u_t+u_{xxx}+\eta(u_{xx}+u_{xxxx})+\beta u_{xxxxx}+uu_x=0, \quad x \in \mathbb{R},\; t\geq 0, \\ u(x,0)=u_0(x), \end{gathered} \end{equation} when $\beta=0$. The IVP associated to (\ref{1eqhs}) was studied by Biagioni, Bona, Iorio and Scialom in \cite{BBIS}. They also determined the limiting behavior of solutions as the dissipation tends to zero. Biagioni and Linares proved global well-posedness for the IVP (\ref{2eqhs}) for initial data in $L^2$ in \cite{BL}. Another example is the Korteweg-de Vries-Burgers equation \begin{align}\label{M-R} \begin{gathered} u_t+u_{xxx}-\eta u_{xx}+uu_x=0, \quad x \in \mathbb{R}, \; t\geq 0,\;\eta>0,\\ u(x,0)=u_0(x), \end{gathered} \end{align} Recently, Molinet and Ribaud considered the IVP (\ref{M-R}) in \cite{M-R1} and proved that it is locally well-posed for given data in $H^s$, $s>-1$. The equation (\ref{M-R}) is also known as the parabolic regularization of the KdV equation with $\eta>0$. Some years ago, when the interest was to obtain local results for given data in larger Sobolev spaces, this regularization was used to obtain well-posedness results for $\eta >0$ and then pass the limit $\eta \downarrow 0$. However, this limit is a delicate matter. Now, we state the main results of this work. The first result deals with the local well-posedness for given data in the Sobolev spaces of negative index. \begin{theorem}\label{teorp} The IVP (\ref{eq:hs}) with $\eta>0$ and $\Phi(\xi)$ given by (\ref{phi}) is locally well-posed for any data $u_0 \in H^s(\mathbb{R})$, $s>-3/4$. \end{theorem} To prove this theorem we follow 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 IVP (\ref{eq:hs2}) below). The proof we presented here does not use the Bourgain type space associated to the linear part of the IVP \eqref{eq:hs}; instead it uses the usual Bourgain space associated to the KdV equation. To carry out this scheme, the Proposition \ref{prop3} plays a fundamental role which permits us to use a bilinear estimate for $\partial_x(u^2)$ (see \cite{kpv2:kpv2}), that is a central part of our arguments. The result of the Theorem \ref{teorp} improves the known local well-posedness results for the IVP (\ref{eqhs}) and (\ref{1eqhs}) described above. Note that, the value $s>-3/4$, in the case of the Korteweg-de Vries (KdV) equation, is sharp in the sense that for $s < -3/4$, the IVP associated to the KdV equation is ill-posed. We should mention that, the lack of conserved quantities in the spaces with regularity below $L^2$, prevents us to get global solution using the usual technique. The second result is concerned with the particular case of the IVP (\ref{eq:hs}) for given data in the homogeneous Sobolev space when the Fourier symbol is of the form $\Phi(\xi)=|\xi|^k-|\xi|^{k+2}$, $k\in \mathbb{Z}^+$. \begin{theorem}\label{teorp1} The IVP (\ref{eq:hs}) with $\eta>0$ and $\Phi(\xi)=|\xi|^k-|\xi|^{k+2}$, $k\in \mathbb{Z}^+$, is locally well-posed for any data $u_0 \in \dot{H}^s(\mathbb{R})$, $s>-1/2$. \end{theorem} Although this theorem does not improve the result obtained in Theorem \ref{teorp}, it is interesting on its own because the proof we present here uses different tools, that are simpler than the ones used in the proof of Theorem \ref{teorp}. The main ingredients in the proof are the refined local smoothing effect (see (\ref{x32}) in Corollary \ref{corx2} below), and a Strichartz type estimate (see Proposition \ref{propx0} below). Using these estimates we are able to apply fixed point argument to obtain a local well-posedness result in the homogeneous Sobolev spaces of negative order without the use of Bourgain type spaces. Now we introduce function spaces that will be used for proving Theorem \ref{teorp}. We consider the following IVP associated to the Linear KdV equation \begin{equation}\label{eq:hs0} \begin{gathered} u_t+u_{xxx}=0, \quad x, \; t\in \mathbb{R},\\ u(0)=u_0. \end{gathered} \end{equation} The solution to (\ref{eq:hs0}) is given by $u(x,t)=U(t)u_0(x)$, where the unitary group $U(t)$ is defined as \begin{align}\label{gU} \widehat{U(t)u_0}(\xi)=e^{it\xi^3}\widehat{u_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} \|u\|_{X_{s,b}} \equiv \|U(-t)u\|_{H_{s,b}} &:= \|\langle \tau \rangle^{b} \langle \xi \rangle^{s} \widehat{U(-t)u}(\xi,\tau) \|_{L_{\tau}^2L_{\xi}^2} \\ &=\|\langle \tau-\xi^3 \rangle^{b} \langle \xi \rangle^{s} \widehat u(\xi,\tau) \|_{L_{\tau}^2L_{\xi}^2 {,}} \end{split} \end{equation} where $\widehat u(\xi,\tau)$ is the Fourier transform of $u$ in both space and time variables. The space $X_{s,b}$ is the usual Bourgain space for the KdV equation (see \cite{bou:bou}). Note that (\ref{eq:hs}) is defined only for $t \ge 0$. To use Bourgain's type space, we should be able to write the IVP (\ref{eq:hs}) for all $t \in \mathbb{R}$. For this, we define \begin{equation}\label{eta} \eta (t)\equiv \eta \mathop{\rm sgn}(t)= \begin{cases} \eta & \text{if } t \geq 0 ,\\ -\eta & \text{if } t <0 \end{cases} \end{equation} and write (\ref{eq:hs}) in the form \begin{equation}\label{eq:hs2} \begin{gathered} u_t+u_{xxx}+\eta(t)Lu+uu_x=0, \quad x, \; t \in \mathbb{R} ,\\ u(0)=u_0. \end{gathered} \end{equation} Now we consider the IVP associated to the linear part of (\ref{eq:hs2}), \begin{equation}\label{eq:hs1} \begin{gathered} u_t+u_{xxx}+\eta(t)Lu=0, \quad x, \, t\in \mathbb{R},\\ u(0)=u_0. \end{gathered} \end{equation} The solution to (\ref{eq:hs1}) is given by $u(x,t)=V(t)u_0(x)$ where the semigroup $V(t)$ is defined as \begin{equation}\label{gV} \widehat{V(t)u_0}(\xi)=e^{it\xi^3+\eta |t|\Phi(\xi)}\widehat{u_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 (\ref{gU})). This paper is organized as follows: In Section 2, we prove Theorem \ref{teorp}. In Section 3, we present a refined local smoothing effect when $\Phi(\xi)=|\xi|^k-|\xi|^{k+2}$, $k \in \mathbb{Z}^+$, in (\ref{phi}). In Section 4, we to obtain some Strichartz type estimates. In Section 5, we prove Theorem \ref{teorp1}. \section{Local Well-posedness in $H^s$ for $s>-3/4$} This section is devoted to supply the proof of the Theorem \ref{teorp}. We start by proving some preliminary results. \subsection{Preliminary estimates} \begin{proposition}\label{prop4} Let $s>-3/4$. There exist $b' \in (-\frac{1}{2},0)$ and $\epsilon_s >0$ such that for any $b \in(\frac{1}{2}, b'+1]$ with $1-b+b'\le \epsilon_s$, and $u, v \in X_{s,b}$ \begin{equation*} \|(uv)_x\|_{ X_{s,b'}} \leq c \|u\|_{ X_{s,b}} \,\|v\|_{ X_{s,b}}. \end{equation*} \end{proposition} The proof of the above proposition can be found in \cite{kpv2:kpv2}. 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} Let us define $\psi_{T}(t)\equiv\psi(\frac{t}{T})$ and $\tilde{\psi}_{T}(t)= \mathop{sgn}(t)\psi_{T}(t)$. The following Proposition plays a central role in the proof of our first main result, Theorem \ref{teorp}. This Proposition allows us to work in the usual $X_{s,b}$ space associated to the KdV equation instead of the Bourgain space associated to the IVP (\ref{eq:hs2}). \begin{proposition}\label{prop3} Let $-1/20$ and $|a|>1$, from (\ref{IJ}) and (\ref{IJ1}) we obtain \begin{align*} \|\psi_{T}(t)I_a(t)\|_{H^b} &= \frac{1}{|a|}\|J(at)\|_{H^b} \\ &\le c \frac{\langle a \rangle^b}{|a|^{3/2}}\|J(t)\|_{H^b}\\ &\le \frac{c}{|a|^{(b'+b)/2}}T^{1+b'/2-3b/2}\|f\|_{H^{b'}}. \end{align*} Hence, we arrived at (\ref{eq0}) in this case too. From here onwards, we consider $|a|\leq 1$. Now let $b>1/2$, then we have \begin{align*} I_a(t)&:= \int_0^t e^{|t-t'|a}f(t')dt' =\int_0^t e^{(t-t')\mathop{sgn}(t)a}f(t')dt'\\ &= e^{a|t|}\int_0^te^{-\mathop{sgn}(t)at'} \int_{\mathbb{R}}e^{it'\tau}\hat f(\tau)d\tau dt'\\ &= e^{a|t|}\int_{\mathbb{R}}\hat f(\tau)\int_0^te^{(i\tau-\mathop{sgn}(t)a)t'}dt'd\tau \\ &= e^{a|t|}\int_{\mathbb{R}}\hat f(\tau)\frac{e^{(i\tau-\mathop{sgn}(t)a)t} -1}{i\tau-\mathop{sgn}(t)a}d\tau \\ &= \int_{\mathbb{R}}\hat f(\tau)\frac{e^{i\tau t} -e^{a|t|}}{i\tau-\mathop{sgn}(t)a}d\tau. \end{align*} We have $$ \frac1{\mathop{sgn}(t)a-i\tau}=\mathop{sgn}(t)\frac{a}{a^2+{\tau}^2} +i\frac{\tau}{a^2+{\tau}^2}. $$ If we define $$ p_a(t) = \frac{a}{a^2+t^2}, \quad q_a(t) = \frac{t}{a^2+t^2} $$ and replace $\tau$ by $t'$ we obtain \begin{equation}\label{I1.1} \begin{split} I_a(t)& = \mathop{sgn}(t)\int_{\mathbb{R}}p_a(t') \big[e^{a|t|}-e^{it't}\big]\widehat{f}(t')dt' +ic\int_{\mathbb{R}}q_a(t')\big[e^{a|t|}-e^{it't}\big]\widehat{f}(t')dt'\\ &:=I_{a,1}(t)+I_{a,2}(t). \end{split} \end{equation} \subsection*{Estimate for $I_{a,1}$} We consider two cases. \noindent {\bf Case 1: $|t'|>1/T$.} Let $\widehat{f}(t')\equiv \widehat{f}(t') \chi_{\{|t'|>1/T\}}$. From the definition of $I_{a,1}$ we have \begin{equation}\label{H0} \psi_{T}(t)I_{a,1}(t) = a\mathop{\rm sgn}(t)\psi_T(t) \int_{\mathbb{R}}\frac{\widehat{f}(t')}{a^2+ {t'}^2}\big[e^{a|t|} -e^{itt'}\big]dt' = ah\Big(\frac{t}{T}\Big), \end{equation} where $h(t)=\mathop{sgn}(t)\psi(t) \int_{\mathbb{R}}\{\widehat{f}(t')/(a^2+ {t'}^2)\}\big[e^{aT|t|}-e^{iTtt'}\big]dt'$. We have \begin{align}\label{H} \widehat{h(t)}(\tau)= \int_{\mathbb{R}}\frac{\widehat{f}(t')}{a^2+ {t'}^2}K(a,T,\tau,t')dt', \end{align} where \begin{align*} K(a,T,\tau,t')=\int_{\mathbb{R}}\mathop{sgn}(t)\psi(t)\big[e^{aT|t|} -e^{iTtt'}\big]e^{-it\tau}dt. \end{align*} Integrating by parts, \begin{align*} |K(a,T,\tau,t')|\le c \frac{\langle t'\rangle}{\langle \tau \rangle}, \quad \text{and} \quad |K(a,T,\tau,t')|\le c \frac{\langle t'\rangle}{\langle \tau \rangle^2}+ c\frac{\langle t'\rangle^2}{\langle \tau \rangle^2}\le c \frac{\langle t'\rangle^2}{\langle \tau \rangle^2}. \end{align*} Hence \begin{align*} |K(a,T,\tau,t')|\le c \frac{\langle t'\rangle^{2b}}{\langle \tau \rangle^{2b}}. \end{align*} Therefore, from (\ref{H}) we obtain \begin{align*} |\widehat{h(t)}|(\tau) \le \frac{c}{\langle \tau \rangle^{2b}} \int_{|t'|>1/T}\frac{|\widehat{f}(t')|}{a^2+ {t'}^2}\langle t'\rangle^{2b}dt'\le c \frac{T^{3/2+b'-2b}}{\langle \tau \rangle^{2b}}\|f\|_{H^{b'}.} \end{align*} Now, using (\ref{H0}) we have \begin{align*} \|\psi_{T}(t)I_a(t)\|_{H^b}= |a|\,\|h\Big(\frac{t}{T}\Big)\|_{H^b}\le |a|T^{1/2-b}\|h(t)\|_{H^b} \le cT^{3/2+b'-2b}T^{1/2-b}\|f\|_{H^{b'}.} \end{align*} Hence \begin{align*} \|\psi_{T}(t)I_a(t)\|_{H^b}\le cT^{2+b'-3b}\|f\|_{H^{b'}}\le cT^{1+b'/2-3b/2}\|f\|_{H^{b'}.} \end{align*} \noindent {\bf Case 2: $|t'|\leq 1/T$. } Let $\widehat{f}(t')\equiv \widehat{f}(t') \chi_{\{|t'| \leq 1/T\}}$ and as earlier $\tilde{\psi}_{T}(t)=\mathop{sgn}(t)\psi_{T}(t)$. We have \begin{align*} \mathcal{F}(\psi_{T}(t) I_{a,1}(t))(\tau)&= \int_{\mathbb{R}}e^{-it\tau}\tilde{\psi}_{T}(t)\int_{\mathbb{R}}p_a(t')\big[e^{a|t|}-e^{it't}\big]\hat f(t')dt' dt\\ &= \int_{\mathbb{R}}p_a(t')\hat f(t')\int_{\mathbb{R}}\tilde{\psi}_{T}(t)e^{-it\tau}\big[e^{a|t|}-e^{it't}\big]dtdt'\\ &= \int_{\mathbb{R}}p_a(t')\widehat{f}(t')\{\mathcal{F}\big(\tilde{\psi}_{T}(t)\,e^{a|t|}\big)(\tau)-\mathcal{F}\big(\tilde{\psi}_{T}(t)\big)(\tau-t')\}dt'\\ &= \int_{\mathbb{R}}p_a(t')\widehat{f}(t')\{\mathcal{F}\big(\tilde{\psi}_{T}(t)\,e^{a|t|}\big)(\tau)-\mathcal{F}\big(\tilde{\psi}_{T}(t)\,e^{a|t|}\big)(\tau-t')\}dt'\\ &\quad + \int_{\mathbb{R}}p_a(t')\widehat{f}(t')\{\mathcal{F}\big(\tilde{\psi}_{T}(t)\,e^{a|t|}\big)(\tau-t')-\mathcal{F}\big(\tilde{\psi}_{T}(t)\big)(\tau-t')\}dt'\\ &:=I_{a,11}(\tau)+I_{a,12}(\tau). \end{align*} Since $|p_a(t')|\le 1/ |t'|$, we can estimate the term $I_{a,11}(\tau)$ as in \cite{gtv:gtv}. Therefore we will estimate only the term $I_{a,12}(\tau)$. Let us define $h(t',\tau):= \mathcal{F}\big(\tilde{\psi}_{T}(t)[e^{a|t|}-1]\big)(\tau-t')$, then we have \begin{align}\label{ht} h(t',\tau)=\int_0^a \mathcal{F}\big(|t|\,\tilde{\psi}_{ T}(t)\,e^{s|t|}\big)(\tau-t')ds. \end{align} From (\ref{ineq4}) we have that \begin{align}\label{eq3} |\mathcal{F}\big(|t|\,\tilde{\psi}_{T}(t)\,e^{s|t|}\big)(\tau-t')| \le \frac{c T^{2}}{(1+(|\tau-t'|+|s|)T)^{2}}, \end{align} where $c$ is independent of $s$, $\tau$, $t'$ and $T$. Observe that $0\le s\le a$ if $a\geq 0$ and $a \le s \le 0$ if $a \le 0$. Thus we obtain \begin{align*} |h(t',\tau)|\leq & c T^{2}\int_0^{|a|}\frac{1}{(1+(|\tau-t'|+|s|)T)^{2}} ds\\ &=c T^{2}\frac{|a|}{(1+|\tau-t'|T)(1+|\tau-t'|T+ |a|T)}. \end{align*} As $|Tt'|\le 1$, we have \begin{align*} \frac{1}{1+|\tau-t'|T}\le \frac{2}{1+|\tau|T}. \end{align*} Hence \begin{align*} |h(t',\tau)|\leq c T^{2}\frac{|a|}{(1+|\tau|T)^2}. \end{align*} Using the H\"older's inequality we obtain \begin{equation}\label{I2} \begin{split} |I_{a,12}(\tau)| &=|\int_{\mathbb{R}}p_a(t')\hat f(t')h(t',\tau)dt'|\\ &\le \|f\|_{H^{b'}}\,\Big(\int_{|t'| \le 1/T}\frac{\langle t' \rangle^{-2b'}|h(t',\tau)|^2}{(a^2+{t'}^2)^2}dt'\Big)^{1/2}\\ & \le \frac{c T^{2}}{(1+|\tau|T)^2}\, \|f\|_{H^{b'}}\Big(\int_{|t'| \le 1/T}(1+|t'|^{-2b'})dt'\Big)^{1/2}\\ & \le \frac{c T^{3/2+b'}}{(1+|\tau|T)^2}\, \|f\|_{H^{b'}.} \end{split} \end{equation} Finally, we arrive at \begin{align*} \Big(\int_{\mathbb{R}}(1+|\tau|)^{2b}|I_{a,12}(\tau)|^2 \,d\tau \Big)^{1/2} &\leq c T^{3/2+b'}\,\|f\|_{H^{b'}}\Big(\int_{\mathbb{R}} \frac{1+|\tau|^{2b}}{(1+|\tau|T)^4}d\tau\Big)^{1/2} \\ &\leq c T^{3/2+b'}\,\|f\|_{H^{b'}}\Big(\frac{1}{T^{1/2}} +\frac{1}{T^{b+1/2}}\Big) \\ &\leq c T^{1-b+b'}\,\|f\|_{H^{b'}.} \end{align*} Therefore, in this case we have \[ \|\psi_T I_{a,1}\|_{H^b}\le c T^{1-b+b'}\,\|f\|_{H^{b'}}. \] \subsection*{Estimate for $I_{a,2}$} The estimate for $I_{a,2}$ is similar to that of $I_{a,1}$, exchanging $p_a$ by $q_a$ and $\tilde{\psi}_T(t)$ by $\psi_T(t)$. So, we omit its calculation. \end{proof} In the following remark we present improvement of the estimate obtained in the Lemma \ref{prop2} in some particular cases. Although, this improvement does not help to improve our main result, it will be of interest on its own. \begin{remark} \rm (1) The proof in the case $|t'|\le 1/T$ is valid for all $a <1$. \noindent (2) We know that \[ \widehat{\mathcal{H}g}(\eta)=-i \mathop{\rm sgn}(\eta)\,\widehat{g}(\eta)\quad \text{and}\quad \widehat{q_a}(t)=-i \mathop{\rm sgn}(t)\,e^{a|t|}. \] Thus \begin{align*} I_{a,2}(t) &=-\tilde{\psi_T}(t)\widehat{q_a}(t)\int_{\mathbb{R}}q_a(t')\widehat{f}(t')dt' + \sqrt{2\pi}\tilde{\psi}_T(t)\mathcal{F}^{-1}\mathcal{H}(q_a\widehat{f}\,)(t)\\ &= \tilde{\psi}_T(t)\mathcal{F}^{-1}(q_a)(t)\int_{\mathbb{R}}q_a(t')\widehat{f} (t')dt'+ \sqrt{2\pi}\tilde{\psi}_T(t)\mathcal{F}^{-1}\mathcal{H}(q_a\widehat{f} \,)(t), \end{align*} where $\tilde{\psi_T}(t)=\mathop{sgn}(t)\psi_T(t)$. Consequently, \begin{align*} \widehat{I_{a,2}}(\tau)&=\widehat{\tilde{\psi}_T}\star q_a(\tau) \int_{\mathbb{R}}q_a(t')\widehat{f}(t')dt'+\sqrt{2\pi} \widehat{\tilde{\psi}_T}\star \mathcal{H}(q_a\widehat{f}\,)(\tau)\\ &= \widehat{\tilde{\psi}_T}\star q_a(\tau)\int_{\mathbb{R}}q_a(t') \widehat{f}(t')dt'+\sqrt{2\pi}\mathcal{H}(\widehat{\tilde{\psi}_T}) \star (q_a\widehat{f}\,)(\tau). \end{align*} Similarly, \begin{align*} \widehat{I_{a,1}}(\tau)=\frac{1}{i} \widehat{\psi_T}\star q_a(\tau) \int_{\mathbb{R}}q_a(t')\widehat{f}(t')dt' +\frac{\sqrt{2\pi}}{i}\mathcal{H}(\widehat{\psi_T}) \star (q_a\widehat{f}\,)(\tau). \end{align*} \noindent (3) If $1/2 3/4+b'/2-b> 1+b'/2-3b/2>0$. \noindent (4) If $|a|>1$, then $|q_a(t')| \le c/\langle t'\rangle$, hence \begin{align*} \int_{\mathbb{R}}|q_a(t')\widehat{f}(t')|dt' \le c \int_{\mathbb{R}} \frac{|\widehat{f}(t')|\, \langle t'\rangle^{1-b}}{\langle t'\rangle\, \langle t'\rangle^{1-b}}dt' \le c \|f\|_{H^{b-1}} \, \Big\| \frac{1}{\langle t'\rangle^{b}}\Big\|_{ L^2}, \end{align*} and therefore we obtain a more refined estimate than (\ref{ga}). \noindent(5) In the case $|t'|>1/T$ we can to obtain a better estimate for $I_{a,2}$ because $\psi_T$ is regular (using the inequalities (\ref{ineq1}) and (\ref{ineq2})). In fact, let $\widehat{f}(t')\equiv \widehat{f}(t') \chi_{\{|t'|>1/T\}}$. We have that \[ \| \psi_{T}(t)I_{a,2}(t)\|_{H_t^{b}} \le \|\psi_{T}(t)e^{|t|a}\|_{H_t^{b}} \Big| \int_{\mathbb{R}}q_a(t')\widehat{f}(t')dt'\Big|+ \|\psi_{T}(t)\|_{H_t^{b}} \|\mathcal{F}^{-1}(q_a\widehat{f}\,)(t)\|_{H_t^{b}}. \] Since $|t'|>1/T$ implies $|t'| \simeq \langle t'\rangle$, using the Cauchy-Schwartz inequality we obtain \begin{equation}\label{ga} \begin{split} \int_{\mathbb{R}}|q_a(t')\widehat{f}(t')|dt' &\leq \int_{|t'|>1/T}\frac{|\widehat{f}(t')|}{|t'|}dt'\\ \lesssim&\int_{|t'|>1/T} \frac{|\widehat{f}(t')|\, |\, t'|^{-b'}}{ |\,t'|\, |\,t'|^{-b'}}dt' \\ &\leq \|f\|_{H^{b'}} \, \Big( \int_{|t'|>1/T} \frac{dt'}{|t'|^{2(1+b')}}\Big)^{1/2} \lesssim T^{1/2+b'}\, \|f\|_{H^{b'}.} \end{split} \end{equation} Similarly, \begin{equation} \begin{split} \int_{\mathbb{R}}|q_a(t')\widehat{f}(t')|dt' &\leq c \|f\|_{H^{b'}} \, \Big( \int_{|t'|>1/T} \frac{dt'}{|t'|^{2b'}(a^2+{t'}^2)}\Big)^{1/2}\\ &\leq c \frac{1}{|a|^{1/2+b'}}\,\Big(\int_{\mathbb{R}} \frac{dt'}{|t'|^{2b'}(1+{t'}^2)}\Big)^{1/2} \|f\|_{H^{b'}} \\ &\leq c \frac{1}{|a|^{1/2+b'}}\, \|f\|_{H^{b'}.} \label{ga1} \end{split} \end{equation} Hence from (\ref{ga}) and (\ref{ga1}) we obtain \begin{align}\label{ga2} \int_{\mathbb{R}}|q_a(t')\widehat{f}(t')|dt' \le \,c \Big(\frac{T}{|a |}\Big)^{1/4+b'/2} \, \|f\|_{H^{b'}\text{,}} \end{align} and \begin{align*} \|\mathcal{F}^{-1}(q_a\widehat{f}\,)(t)\|_{ H_t^{b}}^2 \lesssim & \int_{|t'|>1/T}\frac{|\widehat{f}(t')|^2}{\langle t'\rangle^{2(1-b)}}dt' \\ \lesssim & \int_{|t'|>1/T}\frac{|\widehat{f}(t')|^2}{\langle t'\rangle^{-2b'}|t'|^{2(1-b+b')}}dt'\\ &\leq T^{2(1-b+b')}\|f\|_{H^{b'}}^2. \end{align*} So from inequality (\ref{ineq1}) we have \begin{align*} \|\psi_{T}(t)\|_{H_t^{b}}\|\mathcal{F}^{-1}(q_a\widehat{f}\,)(t)\|_{ H_t^{b}} \le c T^{1/2-b}\,T^{(1-b+b')}\|f\|_{H^{b'}}\le c T^{3/2-2b+b'}\|f\|_{H^{b'}.} \end{align*} On the other hand, if $a<-1$ from inequalities (\ref{ineq2}) and (\ref{ga2}) we obtain \begin{align*} &\|\psi_{T}(t)\,e^{a|t|}\|_{H_t^{b}}\int_{\mathbb{R}}|q_a(t') \widehat{f}(t')|dt' \\ &\leq c \Big(\frac{T}{|a|}\Big)^{1/2}\Big(1+ \frac{1}{T^{b}}\Big)(1+|a|^b)\Big(\frac{T}{|a|}\Big)^{1/4+b'/2}\|f\|_{H^{b'}}\\ &\leq c \Big(\frac{T}{|a|}\Big)^{3/4+b'/2}\Big(1+ \frac{1}{T^{b}}\Big)(1+|a|^b)\|f\|_{H^{b'}}\\ &\leq c T^{3/4+b'/2-b}\Big(1+\frac{1}{|a|^{3/4+b'/2-b}}\Big)\|f\|_{H^{b'}}\\ &\leq c T^{3/4+b'/2-b}\|f\|_{H^{b'}}. \end{align*} Now if $|a|<1$ using the inequality (\ref{ineq2}) we obtain: \begin{align*} \|\psi_{T}(t)\,e^{a|t|}\|_{H_t^{b}}\int_{\mathbb{R}}|q_a(t')\widehat{f}(t')|dt' &\leq c T^{1/2+b'}\|f\|_{H^{b'}}T^{1/2}\Big(1+ \frac{1}{|T|^b}\Big)\\ &\leq c T^{1/2+b'}\,T^{1/2-b}\|f\|_{H^{b'}}\\ &\leq c T^{1-b+b'}\|f\|_{H^{b'}}\\ &\leq c T^{3/4+b'/2-b}\|f\|_{H^{b'}}. \end{align*} Therefore, in this case \begin{align*} \| \psi_{T}(t)I_a(t)\|_{H_t^{b}}\le c T^{3/4+b'/2-b}\, \|f\|_{H^{b'}}, \end{align*} where $c$ is a constant independent of $a$, $f$ and $T$. \end{remark} Now we prove Proposition \ref{prop3} which plays a crucial role in the proof of the first main result of this work. \begin{proof}[Proof of Proposition \ref{prop3}] To prove (\ref{eq1}), we use the estimate (\ref{ineq2}) with $T=1$ and $a= \Phi(\xi)<1$, to obtain \begin{align*} \|\psi(t)\,e^{\eta \Phi(\xi)|t|}\|_{H_t^{b}}^2 \le c(\eta), \end{align*} where $c(\eta)$ is a constant. Therefore, \begin{align*} &\|\psi_{T}(t)V(t)u_{0}\|_{ X_{s,b}} \le c(\eta)\Big( \int_{\mathbb{R}}\langle \xi\rangle^{2s}|\widehat{u_0}(\xi)|^2 d\xi \Big)^{1/2} \le c(\eta)\|u_0\|_{H^s.} \end{align*} Now, we move to prove (\ref{eq2}). From definition \eqref{xsb-norm} of the $X_{s, b}$ norm, we have \begin{equation}\label{x2.2} \begin{split} \|\psi_T(t)\int_0^tV(t&-t') (uu_x)(t') dt'\|_{X_{s, b}} \\ &= \|U(-t)\psi_T(t)\int_0^tV(t-t') (uu_x)(t') dt'\|_{H_{s, b}}\\ &= \|\langle \tau\rangle^b\langle\xi\rangle^s \mathcal{F}_{\xi\tau}{\big[\psi_T(t)\int_0^tU(-t')\tilde U(t-t') (uu_x)(t') dt'\big]}\|_{L_{\tau}^2L_{\xi}^2}\\ &= \|\langle\xi\rangle^s\|\psi_T(t)\int_0^t e^{-it'\xi^3} e^{|t-t'| \Phi(\xi)}\widehat{uu_x}(t', \xi) dt'\|_{H_t^b}\|_{L_{\xi}^2}. \end{split} \end{equation} If we fix the variable $\xi$ and suppose $f_\xi(t')=e^{-it' \xi^3} \widehat{uu_x}(t', \xi)$ the estimate \eqref{eq2} follows from \eqref{x2.2} using (\ref{eq0}). \end{proof} \subsection{Proof of the Theorem \ref{teorp}} \begin{proof} As discussed in the introduction, we will use Bourgain's space associated to the KdV group to prove well-posedness of (\ref{eq:hs}); therefore we need to consider that (\ref{eq:hs2}) is defined for all $t$. Now consider (\ref{eq:hs2}) in its equivalent integral form \begin{equation}\label{int1} u(t)=V(t)u_{0}- \int_{0}^{t}V(t-t')(uu_x)(t')dt', \end{equation} where $V(t)$ is the semigroup associated with the linear part given by (\ref{gV}). Note that, if for all $t\in \mathbb{R}$, $u(t)$ satisfies \[ u(t)=\psi(t)V(t)u_{0}- \psi_{T}(t) \int_{0}^{t}V(t-t')(uu_x)(t')dt', \] then $u(t)$ satisfies (\ref{int1}) in $[-T,T]$. We define an application \[ \Psi(u)(t)= \psi(t)\,V(t)u_0-\psi_{T}(t) \int_0^t V(t-t')(uu_x)(t')dt'. \] Let $s>-3/4$, and $u_0\in H^s$. Let $b$ and $b'$ be two numbers given by Proposition \ref{prop4}, such that $\theta\equiv \min\{1+b'/2-3b/2,\, 3/4+s/3-b\}>0$. For $M>0$, let us define a ball \begin{align*} X_{s,b}^M= \{u\in X_{s,b} : \|u\|_{X_{s,b}}\leq M \}. \end{align*} We will prove that there exists $M$ such that the application $\Psi$ maps $X_{s,b}^M$ into $X_{s,b}^M$ and is a contraction. Let $u\in X_{s,b}^M$. Then using Propositions \ref{prop4}, \ref{prop3} and the definition of $X_{s,b}^M$ we obtain \[ \|\Psi(u)\|_{X_{s,b}}\leq c\|u_0\|_s+c T^\theta \|(uu_x)\|_{X_{s,b'}} \leq \frac{M}{4}+ cT^\theta M^2\leq \frac{M}{2}, \] where we have chosen $M=4c\|u_0\|_{H^s}$ and $00$, $u_0 \in L^q$, $0\le s < (k+3-p)/p+1/p_1$ and $p\ge 2$, $p_1 \ge 2$, then \begin{align}\label{x5} \|D_x^s V_k(t)u_0\|_{L_T^p L_x^{p_1}} \leq \frac{c(\eta)}{(k+3-p(s+1)+p/p_1)^{1/p}}\,(T^{1/p}e^{2\eta T} +T^{\epsilon})\, \|u_0\|_{L^q}, \end{align} where $\epsilon=\epsilon(p,k,s,p_1)=(k+3-p(s+1))/(k+2)+p/((k+1)p_1)$ and $1/p+1/q=1$. \end{theorem} \begin{corollary}\label{teox20} Let $u_0 \in L^q$, $T >0$, $2\le p< k+3$, and $0\le s < (k+3-p)/p$, then \begin{equation}\label{x50} \|D_x^s V_k(t)u_0\|_{L_T^p L_x^\infty} \leq \frac{c(\eta)}{(k+3-p(s+1))^{1/p}}\,(T^{1/p}e^{2\eta T}+T^{\epsilon_0})\, \|u_0\|_{L^q}, \end{equation} where $\epsilon_0=\epsilon(p,k,s)=(k+3-p(s+1))/(k+2)$ and $1/p+1/q=1$. \end{corollary} In particular, the case when $p=2$ is interesting, which is stated as follows. \begin{corollary}\label{corx2} {\bf (1)} If $u_0 \in L^{2}$, $p_1 \ge 2$, $0\le s< 1+ (k-1)/2+1/p_1$, $0< T < 1$ and $\gamma= \min\{1/2, \epsilon(2,k,s,p_1)\}$ then \begin{equation}\label{x33} \|D_x^s V_k(t)u_0\|_{L_T^2 L_x^{p_1}} \leq \frac{c(\eta)\,T^\gamma}{(1+(k-1)/2+1/p_1-s)^{1/2}} \|u_0\|_{L^2}. \end{equation} \noindent{\bf (2)} If $u_0 \in \dot{H}^{s}$, $-k/22)} \equiv J_1+J_2. \end{equation} In $J_1$ by (\ref{tg1}), Minkowski and H\"older's inequalities and Proposition (\ref{Px6}) we obtain \begin{equation} \begin{aligned} J_1 \leq & c \int_{\mathbb{R}}\varphi_{T}(t) \|e^{\eta t \Phi(\xi)}\|_{L^{r_1}(|\xi|\le 2)}\|\mathcal{F}^{-1}(g(\cdot,t))(\xi) \|_{L^{p_1}(|\xi|\le 2)}dt \\ \le& c e^{2\eta T}\|\varphi_{T}\|_{L^p} \,\|g\|_{L_t^q L_x^{q_1}} \le c e^{2\eta T}T^{1/p} \,\|g\|_{L_t^q L_x^{q_1}}, \end{aligned} \label{j1} \end{equation} where $c$ is a constant, $1/q=1/r_1+1/p_1$ and $1/p_1+1/q_1=1$. Similarly for $J_2$ we have \begin{align*} J_2 \leq \int_{\mathbb{R}}\varphi_{T}(t) \|\,\xi^s\, e^{\eta t \Phi(\xi)}\|_{L^{r_1}(|\xi|> 2)}\|g(\cdot,t)\|_{L_x^{q_1}}dt . \end{align*} For $t>0$, we have \begin{align*} \| \,|\xi|^se^{-\eta t|\xi|^{k+2}/2}\|_{L^{r_1}(|\xi|>2)}\, \le \,\frac{c(\eta)}{t^{s/(k+2)+1/((k+2)r_1)}}. \end{align*} Therefore, for $0 \le s<(k+3-p)/p+1/p_1$ we obtain \begin{align}\label{j2} J_2 \le c(\eta)\Big\|\frac{\varphi_{T}(t)}{t^{s/(k+2)+1/((k+2)r_1)}}\Big\|_{L^p} \,\|g\|_{L_t^q L_x^{q_1}} \le c(\eta)\, T^{\epsilon}\,\|g\|_{L_t^q L_x^{q_1}}, \end{align} where $\epsilon=(k+3-p(s+1))/(k+2)+p/((k+1)p_1)$. From (\ref{tg2}), (\ref{j1}) and (\ref{j2}) we obtain \[ \|Lg\|_{L^q} \le \frac{c(\eta)}{(k+3-p(s+1)+p/p_1)^{1/p}}\, (T^{1/p}e^{2\eta T}+T^{\epsilon})\,\|g\|_{L_t^q L_x^{q_1}}. \] \end{proof} \section{Some Strichartz type estimates} \begin{proposition}\label{propx0} Let $2\le p$, $k \ge 1$, \,$c_{p,k}=\frac{p-2}{2p(k+2)}$, $00\,. \] Then \begin{align*} \|V_k(t)u_0\|_{L_T^r L_x^p}\le c(\eta,sqr_0)T^{\{\frac{1}{r}+\frac{s}{(k+2)}-c_{p,k}\}}\|u_0\|_{{H}^{s}{,}} \end{align*} where $1/q+1/p=1$, $r_0=2/(2-q)$. \end{proposition} \begin{proof} Let $\Phi(\xi)= |\xi|^k-|\xi|^{k+2}$. By (\ref{x6}) we have \begin{align*} \|V_k(t)u_0\|_{L_T^r L_x^p} &\leq c \,\|\widehat{V_k(t)u_0}\|_{L_T^r L_\xi^q}\\ &\leq c \, \|e^{\eta t\Phi(\xi) }\widehat{u_0}\|_{L_T^r L_\xi^q(|\xi| \le 2)} +c \|e^{\eta t \Phi(\xi)}\widehat{u_0}\|_{L_T^r L_\xi^q(|\xi| > 2)}\\ &\equiv J_1+J_2. \end{align*} In $J_1$, using H\"older's inequality we have \begin{align*} J_1 \le c3^{-s}e^{\eta T}T^{1/r}. \end{align*} To estimate $J_2$, by H\"older's inequality we obtain \begin{align*} &\int_{|\xi| > 2}e^{q\eta t\Phi(\xi) }|\widehat{u_0}(\xi)|^q d\xi\\ &\le \int_{|\xi| > 2}e^{-q\eta t|\xi|^{k+2}/2}(1+|\xi|)^{-sq}(1+|\xi|)^{sq}| \widehat{u_0}(\xi)|^q d\xi \\ &\le \|e^{-q\eta t|\xi|^{(k+2)}/2}(1+|\xi|)^{-sq}\|_{L^{r_{0}}(|\xi|>2)} \|u_0\|_{H^{s}}^{q} \\ &\le c(\eta)\Big[\frac{1}{t^{1/((k+2)r_{0})}}+\frac{1}{t^{1/((k+2)r_{0}) -sq/(k+2)}}\Big(\int_{\mathbb{R}}e^{-|y|^{(k+2)}}|y|^{-sqr_0}dy\Big)^{1/r_0} \Big]\|u_0\|_{H^{s}}^{q}, \end{align*} where $r_0=2/(2-q)$. Therefore \begin{align*} J_2 \le c(\eta,sqr_0)\, \big\| \frac{1}{t^{1/((k+2)r_{0}q)-s/(k+2)}}\big\|_{L_T^r}\|u_0\|_{H^{s}} \le c(\eta, sqr_0)\,T^{\frac{1}{r}+\frac{s}{(k+2)}-c_{p,k}}\|u_0\|_{H^{s}}, \end{align*} where $c_{p,k}=\frac{p-2}{2p(k+2)}$. \end{proof} \begin{corollary}\label{propx3} Let $0