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

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 $B_{p,r}^s(\mathbb{R})$ 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.

The content of arXiv article 2207.08190 "Nowhere-uniform continuity of the solution map of the Camassa-Holm equation in Besov spaces" is included by the content of new updated arXiv article 2112.14104 "Non-uniform dependence on initial data for the Camassa--Holm equation in Besov spaces: Revisited" which is published in Journal of Differential Equations 390 (2024) 426-450