Geometric approach on the global conservative solutions of the Camassa–Holm equation

Abstract

In this paper we construct global weak conservative solutions of the Camassa–Holm (CH) equation on the periodic domain. We first express the equation in Lagrangian flow variable η and then transform it using the change of variables ρ=ηx. The new variable removes the singularity of the CH equation, and we obtain both global weak conservative solutions and global spatial smoothness of the Lagrangian trajectories of the CH equation. This work is motivated by J. Lenells who proved similar results for the Hunter–Saxton equation using the geometric interpretation.