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

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.