Lipschitz metric for the Hunter-Saxton equation

TL;DR

This paper introduces a new Lipschitz metric for the Hunter-Saxton equation that ensures solutions' stability over time by providing an exponential bound on their distance.

Contribution

A novel Lipschitz metric is developed for the Hunter-Saxton equation, enabling stability analysis of solutions with explicit exponential bounds.

Findings

The metric guarantees Lipschitz continuity of the solution flow.

Solutions remain stable under the new metric with exponential growth bounds.

The approach enhances understanding of solution stability for the Hunter-Saxton equation.

Abstract

We study stability of solutions of the Cauchy problem for the Hunter-Saxton equation $u_t+uu_x=\frac14(\int_{-\infty}^xu_x^2 dx-\int_{x}^\infty u_x^2 dx)$ with initial data $u_0$. In particular, we derive a new Lipschitz metric $d_\D$ with the property that for two solutions $u$ and $v$ of the equation we have $d_\D(u(t),v(t))\le e^{Ct} d_\D(u_0,v_0)$.