Long time dynamics of 2D quasi-geostrophic equations with damping in LpLp

The long time behavior of solutions of 2D quasi-geostrophic equations with damping on the whole space is studied. The existence of the global attractor in LpLp is proved. New difficulties come from nonlocal terms and the Banach space LpLp. We overcome them by using commutator estimates and positivity inequalities associated with fractional Laplacian.