Local well-posedness of the fifth-order KdV equations for rough data using short time $X^{s,b}$ structures

Abstract

In this paper, we mainly prove that the following the fifth-order equation arising from the Korteweg-de Vries (KdV) hierarchy \begin{equation*} \begin{cases} \pt u + \px^5 u + c_1\px u\px^2 u + c_2u\px^3 u = 0 \qquad x,t \in \R \\ u(0,x) = u_0(x) \qquad u_0 \in H^s(\R) \end{cases} \end{equation*} is locall well-posed with initial data in $H^s(\R)$ for $s > \frac54$. \\ The method is a short time $X^{s,b}$ space, which is developed by Ionescu-Kenig-Tataru \cite{IKT} in the context of the KP-I equation. In addition, we use a weight on $X^{s,b}$ to reduce the contribution of high-low frequency interaction where the low frequency has large modulation in the proof of energy estimate. As an immediate result from a conservation law, we have the second equation in the KdV hierarchy, $$\partial_t u - \px^5 u -30u^2\px u + 20\px u\px^2 u + 10u\px^3u=0$$ is globally well-posed in $H^2$.\\ Moreover, we introduce the standard \emph{$X^{s,b}$ space} and counter-examples that the nonlinear estimates fails in the usual \emph{$X^{s,b}$ spaces}. We also prove that the KdV equation is locally well-posed with initial data in $H^s(\R)$ for $s > -\frac34$ by using the Picard iteration argument.