We consider the viscous Camassa-Holm equation subject to an external force,
where the viscosity term is
given by second order differential operator in divergence form. We show that under some mild assumptions on the viscosity term, one has global well-posedness both in
the periodic case and the case of the whole line. In the periodic case,
we show the existence of global attractors in the energy space $H^1$,
provided the external force is in the class $L^2(I)$. Moreover, we establish
an asymptotic smoothing effect, which states that the elements of the attractor are
in fact in the smoother Besov space B2 2, ∞$(I)$.
Identical results (after adding an appropriate linear damping term)
are obtained in the case of the whole line.
Mathematics Subject Classification: Primary: 35Q35, 35Q58, 37K40; Secondary: 35B65, 76B15.