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

Abstract

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 wellpossedness of the two-component Camassa-Holm system, the modified two-component Camassa-Holm system, and the two-component DegasperisProcesi 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.

Analyticity of the Cauchy Problem for a Three-Component Generalization of Camassa–Holm Equation

“…Recently, the well-posedness, the scattering problem, and some qualitative properties for the CH Equation (4) were studied in Refs. [12][13][14][15][16][17][18][19] and the references therein.…”

Lipschitz metric for the modified two-component Camassa–Holm system

“…The Cauchy problem and initial boundary value problem for system (1.1) have been investigated in many works, cf. [17,18,19,22,25,26,27]. However, in the present paper, we reformulate the considered system to a semilinear system of ODEs by means of a transformation between Eulerian and Lagrangian coordinates, which is distinct from those in [18,19,22].…”

On the modified multi-component Camassa-Holm system in higher dimensions

“…x ) −1 ρ x = 0, m = u − u xx , ρ t + (uρ) x = 0, which was firstly proposed in [26] and proved that it allows singular solutions in both variables m and ρ, not just the fluid momentum. The Cauchy problem and initial boundary value problem for (M2CH) have been investigated in many works, see the discussions in [21,22,30,33,34].…”