Spatial-decay of solutions to the quasi-geostrophic equation with the critical and the super-critical dissipation

The initial value problem for the two-dimensional dissipative quasi-geostrophic equation derived from geophysical fluid dynamics is studied. The dissipation of this equation is given by the fractional Laplacian. It is known that the half Laplacian is a critical dissipation for the quasi-geostrophic equation. The global existence of solutions upon the suitable condition is also well known, and that solutions of a fractional dissipative equation decay with a polynomial order as the spatial variable tends to infinity. In this paper, far field asymptotics of solutions to the quasi-geostrophic equation are given in the critical and the supercritical cases. Those estimates are derived from the energy methods for the difference between the solution and its asymptotic profile.