Type: Other
Publication Date: 2013-07-08
Citations: 10
DOI: https://doi.org/10.1090/pspum/087/01434
We construct a global continuous semigroup of weak periodic con- servative solutions to the two-component Camassa{Holm system, ut utxx + u x + 3uux 2uxuxx uuxxx + x = 0 and t + (u )x = 0, for initial data (u; )jt=0 in H 1 L 2. It is necessary to augment the system with an associated energy to identify the conservative solution. We study the stabil- ity of these periodic solutions by constructing a Lipschitz metric. Moreover, it is proved that if the density is bounded away from zero, the solution is smooth. Furthermore, it is shown that given a sequence 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.