Existence and regularity of the global attractor for a weakly damped forced shallow water equation in H1(R)

In this paper we prove the existence of the global attractor for a weakly damped forced shallow water equation in H1(R). Moreover, we show that the global attractor is actually compact in H4(R). The main method is the combination of the energy equation, a suitable splitting of the solutions, and bilinear estimates in Bourgain spaces.