A Lipschitz metric for the Camassa--Holm equation

TL;DR

This paper introduces a Lipschitz metric based on the Wasserstein metric to compare solutions of the Camassa-Holm equation, aiding in analyzing stability and uniqueness despite singularities.

Contribution

It constructs a novel Lipschitz metric for the Camassa-Holm equation solutions, facilitating stability analysis and comparison of initial data.

Findings

The metric effectively compares solutions with singularities.

It provides a framework for stability and uniqueness analysis.

The approach leverages Wasserstein metric properties.

Abstract

We analyze stability of conservative solutions of the Cauchy problem on the line for the Camassa--Holm (CH) equation. Generically, the solutions of the CH equation develop singularities with steep gradients while preserving continuity of the solution itself. In order to obtain uniqueness, one is required to augment the equation itself by a measure that represents the associated energy, and the breakdown of the solution is associated with a complicated interplay where the measure becomes singular. The main result in this paper is the construction of a Lipschitz metric that compares two solutions of the CH equation with the respective initial data. The Lipschitz metric is based on the use of the Wasserstein metric.