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

TL;DR

This paper constructs a global semigroup of weak periodic conservative solutions for the two-component Camassa-Holm system, analyzing stability, smoothness conditions, and the influence of initial density on solution behavior.

Contribution

It introduces a novel framework for conservative solutions of the two-component Camassa-Holm system, including stability analysis and conditions for smoothness based on initial data.

Findings

Constructed a global continuous semigroup of solutions.

Developed a Lipschitz metric for stability analysis.

Proved smoothness when density is bounded away from zero.

Abstract

We construct a global continuous semigroup of weak periodic conservative solutions to the two-component Camassa-Holm system, $u_t-u_{txx}+\kappa u_x+3uu_x-2u_xu_{xx}-uu_{xxx}+\eta\rho\rho_x=0$ and $\rho_t+(u\rho)_x=0$, for initial data $(u,\rho)|_{t=0}$ 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 $\rho$ is bounded away from zero, the solution is smooth. Furthermore, it is shown that given a sequence $\rho_0^n$ 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.