REGULARITY OF GLOBAL ATTRACTORS AND EXPONENTIAL ATTRACTORS FOR 2D QUASI-GEOSTROPHIC EQUATIONS WITH FRACTIONAL DISSIPATION

In this paper we investigate the regularity of global attractors and of exponential attractors for two dimensional quasi-geostrophic equations with fractional dissipation in H2 alpha+s(T-2) with alpha > 1/2 and s > 1. We prove the existence of (H2 alpha-+s(T-2), H2 alpha+s(T-2))-global attractor A, that is, A is compact in H2 alpha+s(T-2) and attracts all bounded subsets of H2 alpha-+s(T-2) with respect to the norm of H2 alpha+s(T-2). The asymptotic compactness of solutions in H2 alpha+s(T-2) is established by using commutator estimates for nonlinear terms, the spectral decomposition of solutions and new estimates of higher order derivatives. Furthermore, we show the existence of the exponential attractor in H2 alpha+s(T-2), whose compactness, boundedness of the fractional dimension and exponential attractiveness for the bounded subset of H2 alpha-+s(T-2) are all in the topology of H2 alpha+s(T-2).