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

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) × H s−1 (R) for s > 3 2 , but Höler continuous in a weaker topology. Besides, we also establish that the FW system is ill-posed in the critical Sobolev space H 3 2 (R) × H 1 2 (R) by proving the norm inflation.