Decay Rates for Mild Solutions of the Quasi-Geostrophic Equation with Critical Fractional Dissipation in Sobolev-Gevrey Spaces

We study the existence of mild solutions for the Quasi-geostrophic equation with critical fractional dissipation in Sobolev-Gevrey spaces. In order to be more specific, by assuming that the initial data.theta(0) is an element of H (s) (a,0) (R-2) (with a > 0, sigma > 1, s is an element of [0, 1)) is small enough, we prove that there is a unique global in time (mild) solution theta is an element of L(SIC)infinity (R+ ;H-s (a,sigma) (R-2 )) boolean AND L-2 (R+ ;H (s+1/2) (a,sigma) ((R-2)) for this equation. Furthermore, as a consequence, we establish some decay rates for this same solution as time goes to infinity; more precisely, this work also determines the following superior limit: lim sup t(k) (t ->infinity) parallel to theta(t) parallel to (k)(H)(a,sigma) (R2) = 0, for all k >= 0 if s = 0, and for all. > 0 whether s is an element of (0, 1).