Existence and Lipschitz stability for $\alpha$-dissipative solutions of the two-component Hunter-Saxton system

TL;DR

This paper introduces the concept of $oldsymbol{ ext{ extalpha}- ext{dissipative}}$ solutions for the two-component Hunter-Saxton system and studies their Lipschitz stability using specialized metrics in Lagrangian coordinates.

Contribution

It defines $ ext{ extalpha}- ext{dissipative}$ solutions for the system and analyzes their stability, addressing the non-invariance of the solution space over time.

Findings

Established existence of $ ext{ extalpha}- ext{dissipative}$ solutions.

Proved Lipschitz stability of solutions over time.

Developed a family of metrics in Lagrangian coordinates.

Abstract

We establish the concept of $\alpha$-dissipative solutions for the two-component Hunter-Saxton system under the assumption that either $\alpha(x)=1$ or $0\leq \alpha(x)<1$ for all $x\in \mathbb{R}$. Furthermore, we investigate the Lipschitz stability of solutions with respect to time by introducing a suitable parametrized family of metrics in Lagrangian coordinates. This is necessary due to the fact that the solution space is not invariant with respect to time.