A Note on a Generalized Two Component Camassa-Holm System in Sobolev Spaces

TL;DR

This paper investigates a generalized two-component Camassa-Holm system, demonstrating that the data-to-solution map is not uniformly continuous in Sobolev spaces, highlighting the system's sensitive dependence on initial data.

Contribution

It establishes the nonuniform dependence of solutions on initial data for the generalized system, extending understanding of its well-posedness in Sobolev spaces.

Findings

Data-to-solution map is not uniformly continuous.

Sharpness of continuity results established.

Approximate solutions used to prove nonuniform dependence.

Abstract

In this paper, we consider a generalized two component Camassa-Holm system. Based on local well-posedness results and lifespan estimates, we establish sharpness of continuity on the data-to-solution map by showing that it is not uniformly continuous from product Sobolev spaces $H^s \times H^{s}$ to $C([0,T]; H^s \times H^{s})$. The proof of nonuniform dependence is based upon approximate solutions.