Periodic conservative solutions for the two-component Camassa-Holm system

Abstract

We construct a global continuous semigroup of weak periodic conservative solutions to the two-component Camassa-Holm system, $u_t-u_{txx}+κu_x+3uu_x-2u_xu_{xx}-uu_{xxx}+ηρρ_x=0$ and $ρ_t+(uρ)_x=0$, for initial data $(u,ρ)|_{t=0}$ in $H^1_{\rm per}\times L^2_{\rm per}$. It is necessary to augment the system with an associated energy to identify the conservative solution. We study the stability of these periodic solutions by constructing a Lipschitz metric. Moreover, it is proved that if the density $ρ$ is bounded away from zero, the solution is smooth. Furthermore, it is shown that given a sequence $ρ_0^n$ of initial values for the densities that tend to zero, then the associated solutions $u^n$ will approach the global conservative weak solution of the Camassa-Holm equation. Finally it is established how the characteristics govern the smoothness of the solution.

To appear in Spectral Analysis, Differential Equations and Mathematical Physics, Proc. Symp. Pure Math., Amer. Math. Soc