\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 142, 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 2010 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2010/142\hfil Well-posedness and ill-posedness] {Remark on well-posedness and ill-posedness for the KdV equation} \author[Takamori Kato\hfil EJDE-2010/142\hfilneg] {Takamori Kato} \address{Takamori Kato \newline Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan} \email{d08003r@math.nagoya-u.ac.jp} \thanks{Submitted August 19, 2010. Published October 8, 2010.} \subjclass[2000]{35Q55} \keywords{KdV equation; well-posedness; ill-posedness; Cauchy problem; \hfill\break\indent Fourier restriction norm; low regularity} \begin{abstract} We consider the Cauchy problem for the KdV equation with low regularity initial data given in the space $H^{s,a}(\mathbb{R})$, which is defined by the norm $\| \varphi \|_{H^{s,a}}=\| \langle \xi \rangle^{s-a} |\xi|^a \widehat{\varphi} \|_{L_{\xi}^2}.$ We obtain the local well-posedness in $H^{s,a}$ with $s \geq \max\{-3/4,-a-3/2\}$, $-3/2< a \leq 0$ and $(s,a) \neq (-3/4,-3/4)$. The proof is based on Kishimoto's work \cite{Ki09} which proved the sharp well-posedness in the Sobolev space $H^{-3/4}(\mathbb{R})$. Moreover we prove ill-posedness when $s< \max\{-3/4,-a-3/2\}$, $a\leq -3/2$ or $a >0$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \section{Introduction} We consider the Cauchy problem of the Korteweg-de Vries equation as follows; $$\label{KdV} \begin{gathered} \partial_t u + \partial_{x}^3 u- 3 \partial_x (u)^2 =0, \quad (t,x) \in [0,T] \times \mathbb{R}, \\ u(0, x)=u_0(x), \quad x \in \mathbb{R}. \end{gathered}$$ Here the given data $u_0$ and an unknown function $u$ are real-valued. We consider \eqref{KdV} with initial data given in the space $H^{s,a}(\mathbb{R})$, which is defined by the norm \begin{align*} \| \varphi \|_{H^{s,a}}:=\| \langle \xi \rangle^{s-a} |\xi|^a \widehat{\varphi} (\xi) \|_{L_{\xi}^2}, \end{align*} where $\langle \xi \rangle:= (1+ | \xi|^2)^{1/2}$ and $\widehat{u}$ is the Fourier transform of $u$. The KdV equation was originally derived by Korteweg and de Vries \cite{KV} as a model for the propagation of shallow water waves along a canal. This equation is completely integrable in the sense that there are Lax formulations, which have an infinite number of conservation laws as follows; $\int u^2 dx,\quad \int (\partial_x u)^2+2 u^3 dx,\quad \int (\partial_x^2 u)^2+ 5 \partial_x(\partial_x u)^2 +\frac{5}{2} u^4 dx, \quad \text{etc.}$ Our main aim is to prove the local well-posedness (LWP for short) for \eqref{KdV} with low regularity initial data given in $H^{s,a}(\mathbb{R})$. The main tool is the Fourier restriction norm method introduced by Bourgain \cite{Bo}. We recall some known results of LWP for \eqref{KdV} with initial data given in the Sobolev space $H^s(\mathbb{R})$. The viscosity method was applied to establish LWP for \eqref{KdV} with $s>3/2$, (see \cite{BS75}). Kenig, Ponce and Vega \cite{KPV93} proved LWP for $s>3/4$ by the iterative approach exploiting the local smoothing effect for the Airy operator $e^{-t \partial_x^3}$. Bourgain \cite{Bo} established the Fourier restriction norm method and showed LWP for $s \geq 0$ by this method, which was improved to $s>-3/4$ by Kenig, Ponce and Vega \cite{KPV96}. In \cite{KPV01}, they also proved that the data-to-solution map fails to be uniformly continuous as a map from $H^s$ to $C([0,T];H^s)$ for $s <-3/4$, (see also \cite{CCT}). Kishimoto \cite{Ki09} showed LWP and the global well-posedness for \eqref{KdV} at the critical regularity $s=-3/4$, (see also \cite{Gu}). In \cite{Tz}, Tzvetkov proved the flow map $\dot{H}^s \ni u_0 \mapsto u(t) \in \dot{H}^s$ cannot be $C^2$ for $s<-3/4$. Under the following assumptions we obtain the following well-posedness result which is generalization of \cite{Ki09}. $$\label{co_op} s \geq \max \bigl\{ -\frac{3}{4}, -a-\frac{3}{2}\bigr\},\quad -\frac{3}{2}< a \leq 0, \quad (s,a) \neq (-\frac{3}{4},-\frac{3}{4})\,.$$ \begin{theorem} \label{thm_well} Let $s,a$ satisfy \eqref{co_op}. Then \eqref{KdV} is locally well-posed in $H^{s,a}$. \end{theorem} We put $s_a=-a-3/2$ and $B_r(\mathcal{X}):= \{ u \in \mathcal{X}; \| u \|_{\mathcal{X}} \leq r \}$ for a Banach space $\mathcal{X}$. We obtain ill-posedness for \eqref{KdV} in the following sense when $s< \max\{-3/4,-a-3/2 \}$, $a \leq -3/2$ or $a >0$. \begin{theorem} \label{thm_ill} \begin{itemize} \item[(i)] Let $r>1$ and $-3/2< a <-3/4$. Then, from Proposition~\ref{prop_well} below, there exist $T>0$ and the flow map for \eqref{KdV} $B_r(H^{s_a,a}) \ni u_0 \mapsto u(t) \in H^{s_a,a}$ for any $t \in (0,T]$. The flow map is discontinuous on $B_r(H^{s_a,a})$ (with $H^{s,a}$ topology) to $H^{s_a,a}$ (with $H^{s,a}$ topology) for any $s0$ such that the flow map for \eqref{KdV}, $u_0 \mapsto u(t)$, is $C^2$ as a map from $B_r(H^{s,a})$ to $H^{s,a}$ for $t \in (0,T]$. \item[(iii)] Let $s <-3/4$ and $a \in \mathbb{R}$. Then there is no $T>0$ such that the flow map for \eqref{KdV}, $u_0 \mapsto u(t)$, is $C^3$ as a map from $B_r(H^{s,a})$ to $H^{s,a}$ for any $t \in (0,T]$. \end{itemize} \end{theorem} We consider \eqref{KdV} with initial data given in the homogeneous Sobolev space $\dot{H}^s(\mathbb{R})$. Noting $\dot{H}^{s}(\mathbb{R})=H^{s,a}(\mathbb{R})$ if $s=a$, we immediately obtain the following results. \begin{corollary} \label{dot_well} Let $-3/4 < s \leq 0$. Then \eqref{KdV} is well-posed in $\dot{H}^{s}$. \end{corollary} \begin{corollary} \label{dot_ill} \begin{itemize} \item[(i)] Let $r>1$, $s_s-s-3/2$ and $-3/20$ and the flow map for \eqref{KdV} $B_r(H^{s_s,s}) \ni u_0 \mapsto u(t ) \in H^{s_s,s}$ for any $t \in (0,T]$. The flow map is discontinuous on $B_r(H^{s_s,s})$ (with $\dot{H}^{s}$ topology) to $H^{s_s,s}$ (with $\dot{H}^{s}$ topology). \item[(ii)] Let $s>0$ or $s \leq -3/2$. Then there is no $T>0$ such that the flow map for \eqref{KdV}, $u_0 \mapsto u(t)$, is $C^2$ as a map from $B_r(\dot{H}^s)$ to $\dot{H}^s$ for $t \in (0, T]$. \end{itemize} \end{corollary} \noindent\textbf{Remark.} We do not know whether LWP for \eqref{KdV} holds or not in $H^{-3/4,-3/4}$. In the present paper, we only prove LWP when $s \geq \max\{-3/4, -a-3/2\}$, -3/20, \end{align*} then we have $$\label{es_dy_1-2} \| |\xi|^{1/2}~f*g \|_{L_{\tau,\xi}^{2}} \lesssim K^{-1/2} \| f \|_{\hat{X}_{(2,1)}^{0,1/2}} \| g \|_{\hat{X}_{(2,1)}^{0,1/2}}.$$ \end{lemma} \begin{lemma} \label{lem_dy_2} Assume thatf$is supported on$A_{j}$and$g $is an arbitrary test function for$j \geq 0$. If a non-empty set$\Omega \subset \mathbb{R}^2satisfies \begin{align*} K:=\inf \{|\xi+\xi_1|;\exists \tau,\tau_1 \text{ s.t. } (\tau,\xi) \in \Omega,~(\tau_1,\xi_1) \in \operatorname{supp}f \}>0, \end{align*} then $$\label{es_dy_2-2} \| f* g \|_{L_{\xi, \tau}^2(\Omega \cap B_k)} \lesssim 2^{k/2}~K^{-1/2}~ \| f \|_{\hat{X}_{(2,1)}^{0,1/2} } \| |\xi|^{-1/2} g \|_{L_{\tau,\xi}^2}.$$ \end{lemma} For the proof of these lemmas, refer the reader to \cite[Lemmas 3.2 and 3.3]{Ki09}. Here we putU(t):=\exp (-t \partial_x^3)$and a smooth cut-off function$\varphi (t)$satisfying$\varphi (t)= 1$for$|t|<1$and$\varphi (t)=0$for$|t|>2 $. For a Banach space$\mathcal{X}$,$\| \cdot \|_{\mathcal{X}}$denotes$\| u \|_{\mathcal{X}}=\| \widehat{u} \|_{\hat{\mathcal{X}}} $. We mention the linear estimates below. \begin{proposition} \label{prop_linear1} Let$s,a \in \mathbb{R}$and$u(t)=\varphi (t) U(t) u_0$. Then the following estimate holds. $\| u\|_{Z^{s, a}} +\| u \|_{L_t^{\infty}( \mathbb{R};H_{x}^{s,a})} \lesssim \| u_0 \|_{H^{s,a}}.$ \end{proposition} \begin{proposition}\label{prop_linear2} Let$s,a \in \mathbb{R}and $u(t)=\varphi (t) \int_{0}^{t} U(t-s) F(s) ds .$ Then $\| u \|_{Z^{s, a}}+ \| u \|_{L_t^{\infty} (\mathbb{R}; H_x^{s,a})} \lesssim \|\mathcal{F}_{\tau,\xi}^{-1} \langle \tau-\xi^3 \rangle^{-1} \widehat{F} \|_{Z^{s,a}}+ \| \langle \xi \rangle^{s-a} ~|\xi|^{a}~ \langle \tau-\xi^3 \rangle^{-1} \widehat{F} \|_{L_{\xi}^2 L_{\tau}^1}.$ \end{proposition} The proofs of these two propositions are given in \cite{GTV}. \section{Proof of the bilinear estimates} In this section, we give the proof of the bilinear estimates \eqref{BE-1} and \eqref{BE-2}. We use the following notation for simplicity, \begin{align*} A_{0. \item[(iii)] $j,j_2 \geq 20$, $|j-j_1| \leq 10$, $0< j_2 < j+11$ and $C(j,j_1,j_2) \sim 2^{-\delta j_2}+2^{-\delta (j-j_2) }$ for some $\delta>0$. \item[(iv)] $j_1,j_2 \geq 20$, $j=0$ and $C(j,j_1,j_2) \sim 1$. \item[(v)] $j,j_1 \geq 20$, $j_2=0$ and $C(j,j_1,j_2) \sim 1$. \end{itemize} \end{proposition} We remark that the cases (iii), (v) are also true with $j_1$ and $j_2$ exchanged because of symmetry. Using this proposition and $\| f \|_{\hat{Z}^{s,a}}^2 \sim \sum_{j} \|f \|_{\hat{Z}^{s,a}(A_j)}^2$, we obtain \eqref{BE-1} and \eqref{BE-2} in the same manner as the proof inc \cite[Theorem 2.2]{KiS}. \begin{proof} We only prove \eqref{BE-X}--\eqref{BE-Y} in the case $s \geq \max\{-3/4,-a-3/2 \}$, $-3/20$ is a sufficiently small number. First, we prove \eqref{BE-X}. (I) Estimate for (i). In this case, we can assume $j,j_1,j_2 \leq 30$. From \eqref{imb}, the left hand side of \eqref{BE-X} is bounded by $C\| |\xi|^{a+1}~\langle \tau-\xi^3 \rangle^{-1/2+\varepsilon} f*g \|_{L_{\xi, \tau}^2 }$. We use the H\"{o}lder inequality and the Young inequality to obtain \begin{align*} \| |\xi|^{a+1} ~\langle \tau-\xi^3 \rangle^{-1/2+\varepsilon} f*g \|_{L_{\xi,\tau}^2} & \lesssim \| f*g \|_{L_{\xi}^{\infty} L_{\tau}^4 } \\ & \lesssim \| f \|_{L_{\xi}^2 L_{\tau}^{8/5}}\| g \|_{L_{\xi}^2 L_{\tau}^{8/5}} \lesssim \| f \|_{\hat{X}^{s,a,1/4}} \| g \|_{\hat{X}^{s,a,1/4}}. \end{align*} (II) Estimate for (iv). We prove $$\label{es_hhl} \| \langle \tau-\xi^3 \rangle^{-1}~\xi~f*g \|_{\hat{X}_L^a (A_0)} \lesssim \| f \|_{\hat{X}_{(2,1)}^{s,1/2}} \| g \|_{\hat{X}_{(2,1)}^{s,1/2}}.$$ (IIa) We consider the estimate \eqref{es_hhl} in the case $|\xi| \leq 2^{-2j_1}$. In this case, the left hand side of \eqref{es_hhl} is bounded by $C \| |\xi|^{a+1} \langle \tau \rangle^{-1/2+\varepsilon} f*g \|_{L_{\tau,\xi}^2}$ from \eqref{imb}. We use H\"{o}lder's inequality and Young's inequality to have \begin{align*} \| |\xi|^{a+1} \langle \tau \rangle^{-1/2+\varepsilon} f*g \|_{L_{\tau,\xi}^2} & \lesssim 2^{-2sj_1} \| |\xi|^{a+1} \|_{L_{\xi}^2 (|\xi| \leq 2^{-2j_1})} \| (\langle \xi \rangle^s f) * (\langle \xi \rangle^s g) \|_{L_{\xi}^{\infty} L_{\tau}^4} \\ & \lesssim 2^{-2(s+a+3/2) j_1} \| \langle \xi \rangle^s f \|_{L_{\tau,\xi}^2} \| \langle \xi \rangle^s g \|_{L_{\xi}^2 L_{\tau}^{4/3}} \\ & \lesssim 2^{-2(s+a+3/2)j_1} \| f \|_{\hat{X}_{(2,1)}^{s,1/2}} \| g \|_{\hat{X}_{(2,1)}^{s,1/2}}. \end{align*} We prove only the case $2^{-2 j_1} \leq |\xi| \leq 1$ below. (IIb) In the case $2^{k_{\rm max}}=2^{k_2}$, we have $2^{k_2} \gtrsim |\xi| 2^{j_1}$. Since $|\xi|^{a+1} \leq |\xi|^{-s-1/2}$ and $2^{-k_2/2} \lesssim 2^{-k/4} (|\xi| 2^{j_1})^{-1/4}$, we use \eqref{es_dy_2-2} with $K_2 \sim 2^{j_1}$ to have \begin{align*} (\text{L.H.S.}) & \lesssim 2^{-2sj_1} \sum_{k \geq 0} 2^{-k/2} \| |\xi|^{a+1}~ (\langle \xi \rangle^s f) * (\langle \xi \rangle^s g) \|_{L_{\tau,\xi}^2 (B_k)}\\ & \lesssim 2^{j_1} \sum_{k \geq 0} 2^{-k/2} \| \langle |\xi| 2^{2j_1} \rangle^{-s-1/2}(\langle \xi \rangle^s f) *(\langle \xi \rangle^s g) \|_{L_{\tau,\xi}^2 (B_k)} \\ & \lesssim 2^{j_1} \sum_{k \geq 0} 2^{-3k/4} \| \langle |\xi| 2^{2j_1} \rangle^{-(s+3/4)} (\langle \xi \rangle^s f) *(\langle \xi \rangle^s \langle \tau-\xi^3 \rangle^{1/2} g) \|_{L_{\tau,\xi}^2 (B_k)} \\ & \lesssim \sum_{k \geq 0} 2^{-k/4} \|f \|_{\hat{X}_{(2,1)}^{s,1/2}} \| g \|_{\hat{X}_{(2,1)}^{s,1/2}}. \end{align*} In the same manner as above, we obtain the desired estimate in the case $2^{k_{\rm max}}=2^{k_1}$. (IIc) We consider the estimate \eqref{es_hhl} in the case $2^{k_{\rm max}}=2^{k}$. If $2^{k_{\rm max}} \gg |\xi| 2^{2j_1}$, then we have $2^{k_{\rm max}} \sim 2^{k_1}$ or $2^{k_{\rm max}} \sim 2^{k_2}$. Thus we only consider the case $2^{k_{\rm max}} \sim |\xi| 2^{2 j_1}$. (IIc-1) In the case $-3/40$. Thus we only consider the case (i) and (iv). (IV) Estimate of (i). In this case, the left hand side of \eqref{BE-Y} is bounded by $C \| |\xi|^{a+1} \langle \tau-\xi^3 \rangle^{-1/2+\varepsilon} f*g \|_{L_{\tau,\xi}^2}$. In the same manner as (I), we have the desired estimate. (V) Estimate of (vi). We prove \begin{align} \label{Y-vi} \| |\xi|^{a+1} \langle \tau \rangle^{-1} f*g \|_{L_{\xi}^2 L_{\tau}^1 (A_0)} \lesssim \| f \|_{\hat{X}_{(2,1)}^{s,1/2}} \| g \|_{\hat{X}_{(2,1)}^{s,1/2}}. \end{align} We easily obtain \eqref{Y-vi} in the case $|\xi| \leq 2^{-2 j_1}$. Therefore we only consider the case $2^{-2 j_1} \leq |\xi| \leq 1$ below. (Va) We consider the estimate \eqref{Y-vi} in the case $2^{k_{\rm max}}=2^{k_1}$ or $2^{k_2}$. Note that the left hand side of \eqref{Y-vi} is bounded by $C \sum_{k \geq 0} 2^{-k/2} \||\xi|^{a+1} f*g \|_{L_{\tau,\xi}^2(B_k)}$ from \eqref{es-L-1}. In the same manner as (IIb), we obtain \eqref{Y-vi} in the case $2^{k_{\rm max}}=2^{k_1}$ or $2^{k_2}$. (Vb) We consider the estimate \eqref{Y-vi} in the case $2^{k_{\rm max}}=2^{k}$. From $|\xi|^{a+1} \leq | \xi |^{-s-1/2}$, we have $|\xi|^{a+1} \langle \tau \rangle^{-1} \lesssim |\xi|^{-3/4} 2^{2sj_1-j_1/2}$. We use the H\"{o}lder inequality and the Young inequality to have \begin{align*} \text{(L.H.S.)} & \lesssim 2^{ - j_1/2} \| |\xi|^{-3/4} (\langle \xi \rangle^s f)* (\langle \xi \rangle^s g) \|_{L_{\xi}^{2} L_{\tau}^1} \\ & \lesssim 2^{-j_1/2} \| |\xi|^{-3/4} \|_{L_{\xi}^2 (|\xi| \geq 2^{-2j_1})} \| \langle \xi \rangle^s f \|_{L_{\xi}^2 L_{\tau}^1} \| \langle \xi \rangle^s g \|_{L_{\xi}^2 L_{\tau}^1} \\ & \lesssim \| f \|_{\hat{X}_{(2,1)}^{s,1/2}} \| g \|_{\hat{X}_{(2,1)}^{s,1/2}}. \end{align*} \end{proof} \section{Proof of the main results} In this section, we give the proofs of Theorem~\ref{thm_well} and \ref{thm_ill}. Here $Z_{T}^{s,a}$ is defined by the norm \begin{align*} \|u \|_{Z_T^{s,a}}:=\inf \bigl\{ \| v \|_{Z^{s,a}};u(t)=v(t)\text{ on } t \in [0,T] \bigr\}. \end{align*} We obtain the following main result. \begin{proposition} \label{prop_well} Let $s,a$ satisfy \eqref{co_op} and $r>1$. (Existence) For any $u_0 \in B_r(H^{s,a})$, there exist $T \sim r^{-6/(3+2 \min\{s,a\})}$ and $u \in C([0,T];H^{s,a}) \cap Z^{s,a}_T$ satisfying the following integral form for \eqref{KdV}; \begin{align} \label{integral-1} u(t)= U(t) u_0 +3 \int_{0}^t U(t-s) \partial_x (u(s))^2 ds. \end{align} Moreover the data-to-solution map $B_r(H^{s,a}) \ni u_0 \mapsto u \in C([0,T]; H^{s,a}) \cap Z^{s,a}_{T}$ is Lipschitz continuous. (Uniqueness) Assume that $u,v \in C([0,T];H^{s,a}) \cap Z_{T}^{s,a}$ satisfy \eqref{integral-1}. Then, $u=v$ on $t \in [0,T]$. \end{proposition} \begin{proof} We first prove the existence of the solution to \eqref{integral-1}. The KdV equation is scale invariant with respect to the transform $u(t,x) \mapsto u_{\lambda}(t,x) :=\lambda^{-2} u(\lambda^{-3}t, \lambda^{-1}x), \quad \lambda \geq 1.$ A simple calculation shows \begin{align*} \|u_{\lambda}(0,\cdot) \|_{H^{s,a}} \leq \lambda^{-3/2-\min \{s,a \}} \| u_0 \|_{H^{s,a}}. \end{align*} Therefore, we can assume that initial data is small enough. From this, we use Propositions~\ref{BE-ES-1},~\ref{prop_linear1} and~\ref{prop_linear2} to prove the existence of the solution by Banach's fixed point argument. For the details, see the proof in \cite[Proposition 4.1]{KT}. We next prove the uniqueness of solutions by the argument in \cite{MT}. We define the space ${W}^{s,a}$ by the norm $\|u \|_{W^{s,a}}:= \| u \|_{Z^{s, a}} + \|u \|_{L^{\infty}(\mathbb{R};H^{s,a})}.$ In the same manner as the proof in \cite[Theorem 2.5]{MT}, we obtain, for $1/2 \leq b <1$, $$\label{uni-lem} w \in X_{(1,1),T_{\lambda}}^{s,a,b} ,\quad w(0,x)=0 \quad \Rightarrow \quad \lim_{\delta \to +0} \| w|_{[0,\delta]} \|_{X_{(1,1),\delta}^{s,a,b} }=0,$$ where $T_{\lambda}:= \lambda^{3} T$, $\lambda \geq 1$ and the space $X_{(1,1)}^{s,a,b}$ defined by $\| u \|_{X_{(1,1)}^{s,a,b}} := \bigl\| \bigl\{ \| \langle \xi \rangle^{s-a} |\xi|^a \langle \tau-\xi^3 \rangle^{b} \widehat{u} \|_{L_{\tau,\xi}^2(A_j \cap B_k)} \bigr\}_{j ,k \geq 0} \bigr\|_{l_{j,k}^{1}}.$ Let $u \in W^{s,a}$ satisfy $u(0,x)=0$ and $\varepsilon$ is an arbitrary positive number. Since $W^{s,a}$ contains $\mathcal{Z}$ densely, we can choose $v \in \mathcal{Z}$ satisfying $\| v-u \|_{W^{s,a}} < \varepsilon$. From the definition, we have \begin{align*} \|v(0) \|_{H^{s,a}} = \| v(0)-u(0) \|_{H^{s,a}} \lesssim \| u-v \|_{W^{s,a}} < \varepsilon. \end{align*} Note that $\sup_{t \in \mathbb{R}} \| u \|_{H^{s,a}} \lesssim \| u \|_{W^{s,a}} \lesssim \| u \|_{X^{s,a,b}},$ for $1/2 < b <1$. By the above argument, we have \begin{align*} \| u \|_{W_{T}^{s,a}} \lesssim & \| u-v \|_{W_{T}^{s,a}}+ \| v-U(t)v(0) \|_{W_{T}^{s,a}}+ \| U(t) v(0) \|_{X_T^{s,a,b}} \\ \lesssim & \varepsilon + \| v-U(t)v(0) \|_{W_{T}^{s,a}}+ \| v(0) \|_{H^{s,a}} \\ \lesssim & \varepsilon+\|v-U(t)v(0) \|_{W_{T}^{s,a}}. \end{align*} Since the second term tends to $0$ as $T \to 0$ from \eqref{uni-lem}, we have \begin{align} \label{uni_lem-2} \lim_{T \to 0} \| u \|_{W_T^{s,a}}=0. \end{align} By combining Propositions \ref{BE-ES-1}, \ref{prop_linear1}, \ref{prop_linear2} and \eqref{uni_lem-2}, we have uniqueness. For the details, see \cite{Ki09}. \end{proof} Next, we prove Theorem \ref{thm_ill} (i)--(iii). We first consider Theorem \ref{thm_ill} (i). In \cite{BT}, Bejenaru and Tao, for the quadratic Schr\"{o}dinger equation with nonlinear term $u^2$, proved the discontinuity of the data-to-solution map for any $s<-1$. We essentially follow their argument to obtain the following proposition. \begin{proposition} \label{prop_ill} Let $s 0$ and a sequence of initial data $\{ \phi_{N,\delta} \}_{N=1}^{\infty} \in H^{\infty}$ satisfying the following three conditions for any $t \in (0,T]$, \begin{itemize} \item[(1)] $\| \phi_{N, \delta} \|_{H^{s_a,a}} \sim \delta$, \item[(2)] $\| \phi_{N, \delta} \|_{H^{s,a}} \to 0$ as $N \to \infty$, \item[(3)] $\| u_{N,\delta} (t) \|_{H^{s,a}} \gtrsim \delta^2$, \\ where $u_{N, \delta}(t)$ is the solution to \eqref{KdV} obtained in Proposition~\ref{prop_well} with the initial data $\phi_{N,\delta}$. \end{itemize} \end{proposition} \begin{proof} Let $N \gg 1$. We put the initial data $\phi_{N,\delta}$ as follows; $\phi_{N, \delta}(x)= \delta N^{a+5/2} \cos(N x) \int_{-\gamma}^{\gamma} e^{i \xi x} d\xi,$ where $\gamma:=N^{-2}$. By a simple calculation, we have $$\label{initial-Fo} \widehat{\phi}_{N, \delta}(\xi) \sim \delta N^{a+5/2} \chi_{B^{+}}(\xi) +\delta N^{a+5/2} \chi_{B^{-}}(\xi),$$ where $B^{\pm}:=[\pm N-\gamma,~ \pm N+\gamma].$ Therefore, $$\label{initial-norm} \| \phi_{N,\delta} \|_{H^{s,a}} \sim \delta N^{s+a+3/2}, \quad \| U(t) \phi_{N, \delta} \|_{H^{s,a}} =\| \phi_{N,\delta} \|_{H^{s,a}} \sim \delta N^{s+a+3/2 } .$$ Since $\| \phi_{N,\delta} \|_{H^{s_a}} \sim \delta$, we have $T=T(\delta)>0$ and the solution $u_{N, \delta}$ to \eqref{KdV} with the initial data $\phi_{N,\delta}$ by Proposition~\ref{prop_well}. Let $t \in (0, T]$. A quadratic term $A_2$ of the Taylor expansion is defined by $A_2(u_0)(t) := 3 \int_{0}^{t} U(t-s) \partial_x(U(s) u_0)^2 ds.$ A simple calculation shows that $$\label{qua-Fo} \widehat{A}_2(u_0) (t)= \exp (i \xi^3 t) \int \frac{1-\exp(-iq(\xi,\xi_1) t)}{q(\xi,\xi_1)} \widehat{u}_0 (\xi_1) \widehat{u}_0(\xi-\xi_1) d\xi_1,$$ where $q(\xi,\xi_1):=3\xi \xi_1 (\xi-\xi_1)$. By similar argument to the proof in \cite[Theorem 1.2]{KT}, we obtain $$\label{qua-norm} \| A_2(u_0) (t) \|_{H^{s,a}} \gtrsim \delta^2.$$ Now we put $v_{N,\delta}(t):= u_{ N,\delta}(t) -U(t)\phi_{N,\delta}- A_2(\phi_{N ,\delta})(t)$. Since the data-to-solution map is Lipschitz continuous for $s=s_a$, we obtain $$\label{er-norm} \| v_{N, \delta} (t) \|_{H^{s_a,a}} \lesssim \delta^3,$$ by using Propositions \ref{BE-ES-1}, \ref{prop_linear1} and \ref{prop_linear2}. From \eqref{initial-norm}, \eqref{qua-norm} and \eqref{er-norm}, we obtain $\| u_{ N,\delta}(t) \|_{H^{s,a}} \geq \| A_2 (\phi_{N ,\delta})(t) \|_{H^{s,a}}- \|v_{N,\delta}(t) \|_{H^{s,a} } - \| U(t) \phi_{N, \delta} \|_{H^{s,a}} \gtrsim \delta^2,$ for all $N \gg 1$. Since $\| \phi_{N,\delta} \|_{H^{s,a}} \to 0$ as $N \to \infty$, this shows the discontinuity of the flow map. We next prove Theorem~\ref{thm_ill} (ii). We only prove that the following estimate fails. $$\label{qua_br} \|A_2 (u_0) (t) \|_{H^{s,a}} \lesssim \| u_0 \|_{H^{s,a}}^2,$$ for $|t|$ bounded by the general argument. For details, see \cite{Ho}. Let $N \gg 1$. We put a smooth initial data as follows; $\phi_N(x):= N^{-s+1} \cos(Nx) \int_{-\gamma}^{\gamma} e^{i \xi x} d\xi +N^{2a+1} \cos(N^{-2}x ) \int_{-\gamma/2}^{\gamma/2} e^{i \xi x} d\xi.$ A straightforward computation shows that $$\label{initial-3} \widehat{\phi}_N (\xi) \sim N^{-s+1} (\chi_{B^{+}}(\xi) + \chi_{B^{-}}(\xi)) +N^{2a+1} \chi_{C} (\xi),$$ where $C:= [\gamma/2, 3\gamma/2]$. Clearly, $\| \phi_N \|_{H^{s,a}} \sim 1$. Substituting \eqref{initial-3} into \eqref{qua-Fo}, we have \begin{align*} |\widehat{A}_2 (\phi_N)(t)|& \lesssim N^{-2s} |\xi|~ \chi_{[-\gamma/2,\gamma/2]}(\xi) + N^{-s+a} |\xi|~\chi_{ [\pm N, \pm N+ \gamma] }(\xi) \\ &\quad + \text{(remainder terms)}. \end{align*} Therefore, $$\label{qu_norm-3} \| A_{2}(\phi_N) (t) \|_{H^{s,a}} \gtrsim N^{-2s} \Bigl( \int_{-\gamma/2}^{\gamma/2} |\xi|^{2a+2} d\xi \Bigr)^{1/2} +N^{-s+2a} \Bigl( \int_{N}^{N +\gamma} \langle \xi \rangle^{2s+2} \Bigr)^{1/2}.$$ If $a \leq -3/2$, the first term of the right hand side of \eqref{qu_norm-3} diverges. When we assume $a \geq -3/2$, the right hand side of \eqref{qu_norm-3} is greater than $C (N^{-2(s+a+3/2)}+N^{2a} )$. In the case $0< a$ or $s< -a-3/2$, we have $\| A_2(\phi_{N}) (t) \|_{H^{s,a}} \to \infty$ as $N \to \infty$, which shows the claim since $\| \phi_N \|_{H^{s,a}} \sim 1$. Finally, we consider Theorem \ref{thm_ill} (iii). Similar to the proof of Theorem~\ref{thm_ill} (ii), we only prove that the following estimate fails for $|t|$ bounded. $$\label{cub_br} \| A_3 (u_0) (t) \|_{H^{s,a}} \lesssim \| u_0 \|_{H^{s,a}}^3,$$ where $A_3$ is the cubic term of the Taylor expansion. We put the sequence of initial data $\{ \psi \}_{N=1}^{\infty} \in H^{\infty}$ as follows; \begin{align*} \psi_{N}(x)= N^{-s+1/4}~\cos(Nx) \int_{-N^{-1/2}}^{N^{1/2}} e^{i \xi x} d\xi. \end{align*} Similar to this data is used in \cite{Bo97}. In the same manner as the argument in \cite{Bo97}, we prove \eqref{cub_br} fails. \end{proof} \section{Appendix} We mention the typical counterexamples of \eqref{BE-3} in the case \eqref{co_cr1}. \begin{example}[high-high-low interaction] \label{exa1} \rm We define the rectangles $P_1, P_2$ as follows; \begin{gather*} P_1:= \bigl\{ (\tau,\xi) \in \mathbb{R}^2~ ;~ |\xi-N | \leq N^{-1/2}, ~~ |\tau-( 3N^2 \xi -2 N^3) | \leq 1/2 \bigr\}, \\ P_2:= \bigl\{ (\tau,\xi) \in \mathbb{R}^2;(-\tau,-\xi) \in A_1 \bigr\}. \end{gather*} Here we put $$\label{rec-1} f(\tau,\xi):= \chi_{P_1}(\tau,\xi), \quad g(\tau,\xi):=\chi_{P_2}(\tau,\xi).$$ Then $$\label{int-1} f*g(\tau,\xi) \gtrsim N^{-1/2}~\chi_{R_1} (\tau,\xi),$$ where $R_1:= \bigl\{ (\tau,\xi) \in \mathbb{R}^2; \xi \in [ 1/2 N^{-1/2},3/4 N^{-1/2} ], ~~ |\tau- 3N^2 \xi | \leq 1/2 \bigr\}.$ Inserting \eqref{rec-1} and \eqref{int-1} into \eqref{BE-3}, the necessary condition for \eqref{BE-3} is $b \leq 4s/3+a/3+3/2$. If \eqref{BE-3} for $s=-3/4$, $b \leq a/3+1/2$. \end{example} \begin{example}[high-low-high interaction] \label{exa2} \rm We define the rectangle $Q:=\bigl\{ (\tau,\xi) \in \mathbb{R}^2 ;|\xi-2N^{-1/2}| \leq N^{-1/2},~~ |\tau-3N^2 \xi | \leq 1/2 \bigr\}.$ Here we put $$\label{rec-2} f(\tau,\xi) = \chi_{P_1}(\tau,\xi), \quad g(\tau,\xi)=\chi_{Q}(\tau,\xi).$$ Then \begin{align} \label{int-2} f*g(\tau,\xi) \gtrsim N^{-1/2}~\chi_{R_2}(\tau,\xi), \end{align} where $R_2:= \bigl\{ (\tau,\xi) \in \mathbb{R}^2; |\xi-N| \leq N^{-1/2}/2 , ~ |\tau- (3N^2 \xi- 2 N^3 ) | \leq 1/2 \bigr\}.$ Substituting \eqref{rec-2} and \eqref{int-2} into \eqref{BE-3}, the necessary condition for \eqref{BE-3} is $b \geq a/3+1/2$. \end{example} \begin{example}[high-high-high interaction] \label{exa3} \rm We put $$\label{rec-3} f(\tau,\xi)= \chi_{P_1}(\tau,\xi), \quad g(\tau,\xi)= \chi_{P_1}(\tau,\xi).$$ Then \begin{align} \label{int-3} f*g(\tau,\xi) \gtrsim N^{-1/2}~\chi_{R_3} (\tau,\xi), \end{align} where $R_3:= \bigl\{ (\tau,\xi) \in \mathbb{R}^2;| \xi -2N | \leq N^{-1/2}/2 , ~ | \tau- (3N^2 \xi -4 N^3) | \leq 1/2 \bigr\}.$ Inserting \eqref{rec-3} and \eqref{int-3} into \eqref{BE-3}, the necessary condition for \eqref{BE-3} is $b \leq 1/2$ for $s=-3/4$. On the other hand, we put $$\label{rec-4} f(\tau,\xi)=\chi_{R_3}(\tau,\xi), \quad g(\tau,\xi)=\chi_{P_2}(\tau,\xi).$$ Then \begin{align} \label{int-4} f*g(\tau,\xi) \gtrsim N^{-1/2}~\chi_{R_4}(\tau,\xi), \end{align} where $R_4:= \bigl\{ (\tau,\xi) \in \mathbb{R}^2; |\xi-N | \leq N^{-1/2}/4, ~ |\tau- (3N^2 \xi-2N^3 ) | \leq 1/2 \bigr\}.$ Substituting \eqref{rec-4} and \eqref{int-4} into \eqref{BE-3}, the necessary condition for \eqref{BE-3} is $b \geq 1/2$ for $s=-3/4$. \end{example} \subsection*{Acknowledgements} The author would like to thank his adviser Kotaro Tsugawa for many helpful conversation and encouragement. \begin{thebibliography}{00} \bibitem{BT} J. Bejenaru and T. Tao; \emph{Sharp well-posedness and ill-posedness results for a quadratic nonlinear Schr\"{o}dinger equation}, J. Funct. Anal. \textbf{233} (2006), 228--259. \bibitem{BS75} J. L. Bona and R. Smith; \emph{The initial value problem for the Korteweg-de Vries equation}, Philos. Trans. Roy. Soc. 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