Sharp well-posedness and ill-posedness of the Cauchy problem for the higher-order KdV

Abstract

In this paper, we investigate the Cauchy problem for the higher-order KdV-type equation \begin{eqnarray*} u_{t}+(-1)^{j+1}\partial_{x}^{2j+1}u + \frac{1}{2}\partial_{x}(u^{2}) = 0,j\in N^{+},x\in\mathbf{T}= [0,2πλ) \end{eqnarray*} with low regularity data and $\lambda \ge 1$. Firstly, we show that the Cauchy problem for the periodic higher-order KdV equation is locally well-posed in $H^s(\mathbf{T})$ with $s\ge -j+\frac{1}{2},j\ge 2.$ By using some new Strichartz estimate and some new function spaces, we also show that the Cauchy problem for the periodic higher-order KdV equation is ill-posed in $H^s(\mathbf{T})$ with $s<-j+\frac{1}{2},j\ge 2$ in the sense that the solution map is $C^3.$ The result of this paper improves the result of \cite{H} with $j\ge 2$.