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

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.