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

References in corpus (5)

• On an integrable two-component Camassa-Holm shallow water system
• Ill-posedness of the Camassa-Holm and related equations in the critical space
• A 2 - Component Generalization of the Degasperis - Procesi Equation
• Nonlinear wave interactions for the Benjamin-Ono equation
• The well-posedness, blow-up and travelling waves for a two-component Fornberg-Whitham system