Global Lagrangian solutions of the Camassa-Holm equation

TL;DR

This paper constructs global weak conservative solutions for the Camassa-Holm equation on a periodic domain by transforming it into a regularized form using a novel change of variables, ensuring global existence and smooth trajectories.

Contribution

It introduces a new variable transformation that removes singularities, enabling the construction of global solutions and smooth Lagrangian trajectories for the Camassa-Holm equation.

Findings

Established global weak conservative solutions for the CH equation.

Proved global spatial smoothness of Lagrangian trajectories.

Extended geometric methods to the CH equation similar to Hunter-Saxton results.

Abstract

In this paper we construct global weak conservative solutions of the Camassa-Holm equation on the periodic domain. We first express the equation in Lagrangian flow variable $\eta$ and then transform it using a change of variable $\rho=\sqrt{\eta_x}$. The new variable removes the singularity of the CH equation, and we obtain both the global weak conservative solution 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.