A Lipschitz metric for the Hunter-Saxton equation

TL;DR

This paper introduces a Lipschitz metric based on Wasserstein distance to analyze the stability and uniqueness of solutions to the Hunter-Saxton equation, especially near singularities.

Contribution

The paper constructs a novel Lipschitz metric for the Hunter-Saxton equation solutions, facilitating stability analysis and uniqueness proofs.

Findings

Established a Lipschitz metric using Wasserstein distance.

Proved stability of solutions under the new metric.

Provided insights into the behavior of solutions near singularities.

Abstract

We analyze stability of conservative solutions of the Cauchy problem on the line for the (integrated) Hunter-Saxton (HS) equation. Generically, the solutions of the HS 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 HS equation with the respective initial data. The Lipschitz metric is based on the use of the Wasserstein metric.