Non-uniform continuity on initial data for the two-component b-family system in Besov space

Xing Wu; Cui Li; Jie Cao

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arXiv · 2012.08696

Non-uniform continuity on initial data for the two-component b-family system in Besov space

Xing Wu·Cui Li·Jie Cao

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Abstract

In this paper, we consider the Cauchy problem of a two-component b-family system, which includes the two-component Camassa-Holm system and the two-component Degasperis-Procesi system. It is shown that the solution map of the two-component b-family system is not uniformly continuous on the initial data in Besov spaces $B_{p, r}^{s-1}(\mathbb{R})\times B_{p, r}^s(\mathbb{R})$ with $s>\max\{1+\frac{1}{p}, \frac{3}{2}\}$, $1\leq p, r< \infty$. Our result covers and extends the previous non-uniform continuity in Sobolev spaces $H^{s-1}(\mathbb{R})\times H^s(\mathbb{R})$ for $s>\frac{5}{2}$ (Nonlinear Anal., 2014) to Besov spaces.

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Xing Wu, Cui Li, Jie Cao. 2020-12-16. Non-uniform continuity on initial data for the two-component b-family system in Besov space. https://arxiv.org/abs/2012.08696

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