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

TL;DR

This paper develops a Lipschitz metric for conservative solutions of the two-component Camassa--Holm system, enabling stable measurement of solution differences over time within a bounded energy regime.

Contribution

It introduces a Lipschitz metric for the two-component Camassa--Holm system's conservative solutions, ensuring Lipschitz continuity of the solution flow.

Findings

The metric $d_{ ext{D}^M}$ is Lipschitz continuous in time for solutions within a bounded energy ball.

The metric bounds the distance between solutions over time proportionally to their initial distance.

The approach handles the measure-valued energy distribution in the solutions.

Abstract

We construct a Lipschitz metric for conservative solutions of the Cauchy problem on the line for the two-component Camassa--Holm system $u_t-u_{txx}+3uu_x-2u_xu_{xx}-uu_{xxx}+\rho\rho_x=0$, and $\rho_t+(u\rho)_x=0$ with given initial data $(u_0, \rho_0)$. The Lipschitz metric $d_{\D^M}$ has the property that for two solutions $z(t)=(u(t),\rho(t),\mu_t)$ and $\tilde z(t)=(\tilde u(t),\tilde \rho(t),\tilde \mu_t)$ of the system we have $d_{\D^M}(z(t),\tilde z(t))\le C_{M,T} d_{\D^M}(z_0,\tilde z_0)$ for $t\in[0,T]$. Here the measure $\mu_t$ is such that its absolutely continuous part equals the energy $(u^2+u_x^2+\rho^2)(t)dx$, and the solutions are restricted to a ball of radius $M$.