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, ut − utxx + κux + 3uux − 2uxuxx − uuxxx + ηρρx = 0 and ρt + (uρ)x = 0, for initial data (u, ρ)|t=0 in H1 per ×Lper. 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 ρ0 of initial values for the densities that tend to zero, then the associated solutions un 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.