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

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 $B^{\sigma}_{p,\infty}$.