Non-uniform dependence on initial data for the Camassa--Holm equation in Besov spaces: Revisited

TL;DR

This paper investigates the non-uniform dependence of the solution map of the Camassa--Holm equation in Besov spaces, showing it is not uniformly continuous and strengthening previous results, with implications for related equations.

Contribution

It improves and extends previous work by proving the solution map is nowhere uniformly continuous in certain Besov spaces for the Camassa--Holm equation and related models.

Findings

Solution map is not uniformly continuous in specified Besov spaces.

Strengthened results showing the map is nowhere uniformly continuous.

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

Abstract

In the paper, we revisit the uniform continuity properties of the data-to-solution map of the Camassa--Holm equation on the real-line case. We show that the data-to-solution map of the Camassa--Holm equation is not uniformly continuous on the initial data in Besov spaces $B_{p, r}^s(\mathbb{R})$ with $s>\frac{1}{2}$ and $1\leq p, r< \infty$, which improves the previous works [Himonas et al., Asian J. Math., 11 (2007)], [Li et al., J. Differ. Equ., 269 (2020)] and [Li et al., J. Math. Fluid Mech., 23 (2021)]. Furthermore, we present a strengthening of our previous work in [Li et al., J. Differ. Equ., 269 (2020)] and prove that the data-to-solution map for the Camassa--Holm equation is nowhere uniformly continuous in $B^s_{p,r}(\mathbb{R})$ 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.