Continuity properties of the data-to-solution map and ill-posedness for a two-component Fornberg-Whitham system

TL;DR

This paper investigates the continuity properties of the data-to-solution map for a two-component Fornberg-Whitham system, revealing non-uniform continuity and ill-posedness in certain Sobolev spaces, which impacts the understanding of its solution behavior.

Contribution

It demonstrates the non-uniform continuity of solutions in higher Sobolev spaces and establishes ill-posedness at the critical Sobolev space for the FW system.

Findings

Solutions are not uniformly continuous in $H^{s}$ for $s>3/2$

The FW system is ill-posed in $H^{3/2} imes H^{1/2}$ due to norm inflation

Continuity is Hölder continuous in weaker topologies

Abstract

This work studies a two-component Fornberg-Whitham (FW) system, which can be considered as a model for the propagation of shallow water waves. It's known that its solutions depend continuously on their initial data from the local well-posedness result. In this paper, we further show that such dependence is not uniformly continuous in $H^{s}(R)\times H^{s-1}(R)$ for $s>\frac{3}{2}$, but H\"{o}ler continuous in a weaker topology. Besides, we also establish that the FW system is ill-posed in the critical Sobolev space $H^{\frac{3}{2}}(R)\times H^{\frac{1}{2}}(R)$ by proving the norm inflation.