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 H2α+s (T 2 ) with α > 1 2 and s > 1. We prove the exis tence of (H2α−+s (T 2 ), H2α+s (T 2 ))-global attractor A, that is, A is compact in H2α+s (T 2 ) and attracts all bounded subsets of H2α−+s (T 2 ) with respect to the norm of H2α+s (T 2 ). The asymptotic compactness of solutions in H2α+s (T 2 ) is established by using commutator estimates for nonlinear terms, the spectral decomposition of solutions and new estimates of higher order derivatives. Fur thermore, we show the existence of the exponential attractor in H2α+s (T 2 ), whose compactness, boundedness of the fractional dimension and exponential attractiveness for the bounded subset of H2α−+s (T 2 ) are all in the topology of H2α+s (T 2 ).