Local well-posedness for Kawahara equation

Abstract

We consider the Cauchy problem for the Kawahara equation, which is a fifth-order KdV equation. This paper establishes the local well-posedness with initial data given in the Sobolev space $H^s(\mathbb{R})$. Previously, Chen, Li, Miao, and Wu (2009) proved the local well-posedness for $s>-7/4$, which has been improved to $s \geq -7/4$ by Chen and Guo. We improve this result to $s \geq -2$. The main idea is to modify the Bourgain space. Similar arguments are used by Bejenaru and Tao (2006). Moreover, we prove ill-posedness for $s<-2$ by using the argument by Bejenaru and Tao (2006).