The continuous dependence and non-uniform dependence of the rotation Camassa-Holm equation in Besov spaces

TL;DR

This paper proves local well-posedness and continuous dependence for the rotation Camassa-Holm equation in Besov spaces, and shows the solution's dependence on initial data is not uniformly continuous in certain spaces.

Contribution

It establishes well-posedness in broader Besov spaces using new methods and demonstrates non-uniform dependence on initial data.

Findings

Well-posedness in nonhomogeneous Besov spaces with relaxed conditions.

Continuous dependence established via compactness and Lagrangian transformation.

Solution dependence on initial data is not uniformly continuous in supercritical and critical spaces.

Abstract

In this paper, we first establish the local well-posedness and continuous dependence for the rotation Camassa-Holm equation modelling the equatorial water waves with the weak Coriolis effect in nonhomogeneous Besov spaces $B^s_{p,r}$ with $s>1+1/p$ or $s=1+1/p,\ p\in[1,+\infty),\ r=1$ by a new way: the compactness argument and Lagrangian coordinate transformation, which removes the index constraint $s>3/2$ and improves our previous work \cite{guoy1}. Then, we prove the solution is not uniformly continuous dependence on the initial data in both supercritical and critical Besov spaces.