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

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 $0\le \alpha (x)<1$ 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.

59 pages

References in corpus (1)

• Lipschitz metric for the two-component Camassa--Holm system

Cited by in corpus (8)

• The Hunter-Saxton equation with noise
• Uniqueness of conservative solutions for the Hunter-Saxton equation
• Numerical conservative solutions of the Hunter--Saxton equation
• Lipschitz Stability for the Hunter-Saxton Equation
• A Convergent Numerical Algorithm for $\alpha$-Dissipative Solutions of the Hunter-Saxton Equation
• A Lipschitz metric for $\alpha$-dissipative solutions to the Hunter-Saxton equation
• A numerical view on α-dissipative solutions of the Hunter-Saxton equation
• Rate of convergence for numerical $\alpha$-dissipative solutions of the Hunter-Saxton equation