The global attractor of the viscous damped forced Ostrovsky equation

The existence of the global attractor of the viscous damped forced Ostrovsky equation in L̃2(R) is proved. An asymptotic smoothing effect of the equation is also shown, namely, that for the forces in L̃2(R), the global attractor in L̃2(R) is actually a compact set in H̃3(R). The energy equation method is used in conjunction with a suitable splitting of the solutions; the regularization properties of the equation in the context of Bourgain spaces are extensively exploited.