Lipschitz metric for the periodic Camassa-Holm equation

TL;DR

This paper introduces a new Lipschitz metric for the periodic Camassa-Holm equation that ensures stability of solutions over time and clarifies its relation to standard function norms.

Contribution

A novel Lipschitz metric is derived for the periodic Camassa-Holm equation, providing stability estimates and connecting it to traditional function space norms.

Findings

The metric guarantees exponential stability of solutions.

The relationship between the new metric and standard norms is established.

Stability estimates are derived for conservative solutions.

Abstract

We study stability of conservative solutions of the Cauchy problem for the periodic Camassa-Holm equation $u_t-u_{xxt}+\kappa u_x+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ 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 $d_\D(u(t),v(t))\le e^{Ct} d_\D(u_0,v_0)$. The relationship between this metric and usual norms in $H^1_{\rm per}$ and $L^\infty_{\rm per}$ is clarified.