Nonexistence of Hölder 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 Hölder continuity. Precisely speaking, the solution of the Camassa-Holm equation belongs to $\mathcal{C}([0,T];B^s_{p,r})$ but not to $\mathcal{C}^α([0,T];B^s_{p,r})$ with any $α\in(0,1)$. To the best of our knowledge, our work is the first one addressing the issue on the failure of Hölder 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 $B^s_{p,\infty}(\mathbb{R})$.