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

TL;DR

This paper demonstrates the ill-posedness of the Camassa-Holm equation in a specific Besov and Lipschitz space, showing that the solution map is discontinuous at the origin due to initial data construction.

Contribution

It introduces a new initial data construction to prove the discontinuity of the solution map in the space $B_{p,1}^{1}\cap C^{0,1}$ for the Camassa-Holm equation.

Findings

Solution map discontinuous at origin in the specified space

Existence of initial data leading to solution inflation

Ill-posedness in the considered function space

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.