The Cauchy problem for an integrable shallow-water equation

Abstract

We prove that the periodic initial value problem for a completely integrable shallow-water equation is not locally well-posed for initial data in the Sobolev space $H^s(\mathbb{T})$ whenever $s <3/2$. Since on the other hand this problem is locally well-posed in the sense of Hadamard for $s>3/2$ our result suggests that $s=3/2$ is the critical Sobolev index for well-posedness. We also show that the nonperiodic initial value problem is not locally well-posed in $H^s(\mathbb{R})$ for $s <3/2$.