Lipschitz Stability for the Hunter-Saxton Equation

TL;DR

This paper investigates the Lipschitz stability over time for solutions to the Hunter-Saxton equation with a focus on $eta$-dissipative solutions, establishing stability results in both Lagrangian and Eulerian frameworks.

Contribution

The paper introduces metrics in Lagrangian and Eulerian coordinates and proves Lipschitz stability for $eta$-dissipative solutions of the Hunter-Saxton equation.

Findings

Lipschitz stability established in Lagrangian coordinates

Lipschitz stability established in Eulerian coordinates

Metrics defined for stability analysis

Abstract

We study the Lipschitz stability in time for $\alpha$-dissipative solutions to the Hunter-Saxton equation, where $\alpha \in [0,1]$ is a constant. We define metrics in both Lagrangian and Eulerian coordinates, and establish Lipschitz stability for those metrics.