Non-uniform dependence for the two-component Camassa-Holm shallow water system

Abstract

In this paper, we consider the solution map of the initial value problem to the two-component Camassa-Holm equation on the line. We prove that the solution map of this problem is not uniformly continuous in Sobolev spaces $H^s(\mathbb{R})\times H^{s-1}(\mathbb{R})$ for $s>3/2$.

15 pages