Ill-posedness for the Camassa-Holm equation in $B_{p,1}^1\cap C^{0,1}$

Abstract

In this paper, we study the Cauchy problem for the Camassa-Holm equation on the real line. By presenting a new construction of initial data, we show that the solution map in the smaller space $B_{p,1}^1\cap C^{0,1}$ with $p\in (2,\infty ]$ is discontinuous at origin. More precisely, $u_0\in B_{p,1}^1\cap C^{0,1}$ can guarantee that the Camassa-Holm equation has a unique local solution in $W^{1,p}\cap C^{0,1}$, however, this solution is instable and can have an inflation in $B_{p,1}^1\cap C^{0,1}$ for certain initial data.

arXiv admin note: text overlap with arXiv:2104.05973, arXiv:2112.14104