Ill-posedness for the higher dimensional Camassa-Holm equations in Besov spaces

TL;DR

This paper demonstrates that the higher dimensional Camassa-Holm equations are ill-posed in certain Besov spaces by constructing initial data leading to discontinuous solutions at the initial time.

Contribution

The paper establishes ill-posedness of the higher dimensional Camassa-Holm equations in specific Besov spaces through explicit initial data construction.

Findings

Solutions are discontinuous at t=0 in the Besov space norm.

Ill-posedness holds for initial data in B^{\sigma}_{p,\infty} with \sigma-2 > ext{max}igrace{1+rac{1}{p}, rac{3}{2}igrace}.

Provides a rigorous mathematical proof of ill-posedness in the specified functional setting.

Abstract

In the paper, by constructing a initial data $u_{0}\in B^{\sigma}_{p,\infty}$ with $\sigma-2>\max\{1+\frac 1 p, \frac 3 2\}$, we prove that the corresponding solution to the higher dimensional Camassa-Holm equations starting from $u_{0}$ is discontinuous at $t=0$ in the norm of $B^{\sigma}_{p,\infty}$, which implies that the ill-posedness for the higher dimensional Camassa-Holm equations in $B^{\sigma}_{p,\infty}$.