Initial boundary value problems for the two-component shallow water systems

Abstract

In this paper we study initial boundary value problems of three types of two-component shallow water systems on the half line subject to homogeneous Dirichlet boundary conditions. We first prove local well-possedness of the two-component Camassa–Holm system, the modified two-component Camassa–Holm system, and the two-component Degasperis–Procesi system in the Besov spaces. Then, we are able to specify certain conditions on the initial data which on the one hand guarantee global existence and on the other hand produce solutions with a finite lifespan. Moreover, in the case of finite time singularities we are able to describe the precise blow-up scenario for breaking waves. Finally we investigate global weak solutions for the two-component Camassa–Holm system and the modified two-component Camassa–Holm system on the half line, respectively. Our approach is based on sharp extension results for functions on the half line and several symmetry preserving properties of the systems under discussion.