Lipschitz metric for the Hunter-Saxton equation

Abstract

We study stability of solutions of the Cauchy problem for the Hunter-Saxton equation $u_t+u{u}_x=\frac{1}{4}({\int }_{-\infty }^x{u}_x^{2}dx-{\int }_x^{\infty }{u}_x^{2}dx)$ 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 $d_\D(u(t),v(t))\le e^{Ct} d_\D(u_0,v_0)$.