Nowhere-uniform continuity of the solution map of the Camassa-Holm equation in Besov spaces

TL;DR

This paper proves that the solution map for the Camassa-Holm equation is nowhere uniformly continuous in certain Besov spaces, extending previous results and applying to related equations like the Degasperis-Procesi.

Contribution

It strengthens prior work by establishing nowhere uniform continuity of the solution map in Besov spaces for the Camassa-Holm and related equations.

Findings

Solution map is nowhere uniformly continuous in specified Besov spaces.

Method applies to the b-family of equations including Camassa-Holm.

Extends previous results on the regularity of the solution map.

Abstract

In the paper, we gave a strengthening of our previous work in [32] (J. Differ. Equ. 269 (2020)) and proved that the data-to-solution map for the Camassa-Holm equation is nowhere uniformly continuous in $B^s_{p,r}(\R)$ with $s>\max\{1+1/{p},3/2\}$ and $(p,r)\in [1,\infty]\times[1,\infty)$. The method applies also to the b-family of equations which contain the Camassa-Holm and Degasperis-Procesi equations.