Camassa-Holm equation on shallow water wave equation

On the Camassa-Holm equation

This talk is focused on recent progress of studies for the Camassa-Holm equation. First, we will give a brief review on the derivations, well-posedness for the strong solution, blow-up phenomenon and existence of the weak solutions. Then, infinite propagation speed for the Camassa-Holm equation will be proved in the following sense: the corresponding solution u(x, t) with compactly supported in...

Wave-breaking for the Weakly Dissipative Modified Camassa-Holm Equation

The equation (1) arises from an intrinsic (arc-length preserving) invariant planar curve flow in Euclidean geometry and it can be regarded as a Euclidean-invariant version of the Camassa-Holm equation in [1]. It has the form of a modified Camassa-Holm equation with cubic nonlinearity. By Fuchssteiner [2] and Olver and Rosenau[3], it can be derived as a new integrable system by applying the gene...

A Singular Limit Problem for Conservation Laws Related to the Camassa-holm Shallow Water Equation

We consider a shallow water equation of Camassa-Holm type, containing nonlinear dispersive effects as well as fourth order dissipative effects. We prove that as the diffusion and dispersion parameters tend to zero, with a condition on the relative balance between these two parameters, smooth solutions of the shallow water equation converge to discontinuous solutions of a scalar conservation law...

Decomposition Method for Camassa-holm Equation

Attractors for the Viscous Camassa-holm Equation

Abstract. We consider the viscous Camassa-Holm equation subject to an external force, where the viscosity term is given by second order differential operator in divergence form. We show that under some mild assumptions on the viscosity term, one has global wellposedness both in the periodic case and the case of the whole line. In the periodic case, we show the existence of global attractors in ...