Non-uniform dependence on periodic initial data for the two-component Fornberg-Whitham system in Besov spaces

TL;DR

This paper proves that the solution map for the two-component Fornberg-Whitham system is not uniformly continuous in certain Besov and Sobolev spaces, indicating sensitive dependence on initial data in the periodic setting.

Contribution

It demonstrates non-uniform dependence of solutions on initial data for the system in Besov and Sobolev spaces, extending understanding of its well-posedness properties.

Findings

Non-uniform dependence in Besov spaces for $s> ext{max}igrace 2+rac{1}{p}, rac{5}{2} igrace$

Special case: non-uniform dependence in Sobolev spaces $H^s$ for $s> rac{5}{2}$ when $p=2$, $r=2$

Highlights sensitivity of solutions to initial data in the periodic Fornberg-Whitham system.

Abstract

This paper establishes non-uniform continuity of the data-to-solution map in the periodic case, for the two-component Fornberg-Whitham system in Besov spaces $B^s_{p,r}(\mathbb{T}) \times B^{s-1}_{p,r}(\mathbb{T})$ for $s> \max\{2+\frac{1}{p}, \frac{5}{2}\}$. In particular, when $p=2$ and $r=2$, this proves the non-uniform dependence on initial data for the system in Sobolev spaces $H^s(\mathbb{T})\times H^{s-1}(\mathbb{T})$ for $s> \frac{5}{2}$.