Nonexistence of H\"{o}lder continuous solution for the Camassa-Holm equation in Besov spaces

Abstract

In the paper, we show that the continuity of the solution can not be improved to the H\"{o}lder continuity. Precisely speaking, the solution of the Camassa-Holm equation belongs to $\mathcal{C}([0,T];B^s_{p,r})$ but not to $\mathcal{C}^\alpha([0,T];B^s_{p,r})$ with any $\alpha\in(0,1)$. To the best of our knowledge, our work is the first one addressing the issue on the failure of H\"{o}lder continuous in time of solution to the classical Camassa-Holm equation. As a by-product, we establish the ill-posedness for the Camassa-Holm equation in $B^s_{p,\infty}(\mathbb{R})$ with $s>\max\big\{1+1/p, 3/2\big\}$ with $p\in[1,\infty]$ by proving the solution map to the Camassa-Holm equation starting from $u_0$ is discontinuous at $t = 0$ in $B^s_{p,\infty}(\mathbb{R})$.