Attractor for the viscous two-component Camassa–Holm equation

This paper is concerned with the attractor for a viscous two-component generalization of the Camassa–Holm equation subject to an external force, where the viscosity term is given by a second order differential operator. The global existence of solution to the viscous two-component Camassa–Holm equation with the periodic boundary condition is studied. We obtain the compact and bounded absorbing set and the existence of the global attractor in H2×H2H2×H2 for the viscous two-component Camassa–Holm equation by uniform prior estimate and many inequalities.