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

TL;DR

This paper proves that the solution map for the two-component Camassa-Holm system is not uniformly continuous in certain Sobolev spaces, highlighting limitations in the stability of solutions.

Contribution

It establishes the non-uniform dependence of solutions on initial data for the two-component Camassa-Holm equation in Sobolev spaces.

Findings

Solution map is not uniformly continuous in $H^s(\R) \times H^{s-1}(\R)$ for $s>3/2$

Highlights instability in the solution dependence on initial conditions

Provides insights into the mathematical structure of the two-component Camassa-Holm 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 $H^s(\R)\times H^{s-1}(\R)$ for $s>3/2$.