A Lipschitz metric for $\alpha$-dissipative solutions to the Hunter-Saxton equation

TL;DR

This paper investigates the Lipschitz stability of $\alpha$-dissipative solutions to the Hunter-Saxton equation, providing a framework to understand how solutions depend continuously on initial data, especially considering energy dissipation.

Contribution

It introduces a Lipschitz metric tailored for $\alpha$-dissipative solutions, extending stability analysis to solutions with spatially varying energy loss.

Findings

Established Lipschitz stability for $\alpha$-dissipative solutions

Developed a generalized characteristic method for stability analysis

Provided insights into energy dissipation effects on solution stability

Abstract

We explore the Lipschitz stability of solutions to the Hunter-Saxton equation with respect to the initial data. In particular, we study the stability of $ \alpha $-dissipative solutions constructed using a generalised method of characteristics approach, where $\alpha$ is a function determining the energy loss at each position in space.