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

Abstract

We construct a Lipschitz metric for conservative solutions of the Cauchy problem on the line for the two-component Camassa--Holm system $u_t-u_{txx}+3u{u}_x-2u_x{u}_{xx}-u{u}_{xxx}+\rho {\rho }_x=0$, 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 $M$.

References in corpus (1)

• The geometry of the two-component Camassa-Holm and Degasperis-Procesi equations

Cited by in corpus (1)

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