Existence of a unique global solution, and its decay at infinity, for the modified supercritical dissipative quasi-geostrophic equation

Our interest in this research is to prove the decay, as time tends to infinity, of a unique global solution for the supercritical case of the modified quasi-gesotrophic equation (MQG) theta(t) + (- Delta)(alpha) theta + u(theta) center dot del theta = 0, with u(theta) = (partial derivative(2)(- Delta)(gamma-2/2) theta,-partial derivative(1)(- Delta)(gamma-2/2) theta), in the non-homogenous Sobolev space H1+gamma-2 alpha(R-2), where alpha is an element of (0, 1/2) and gamma is an element of (1, 2 alpha + 1). To this end, we need consider that the initial data for this equation are small. More precisely, we assume that parallel to theta(0) parallel to (1+gamma-2 alpha)(H) is small enough in order to obtain a unique theta is an element of C([0,infinity); (1+gamma-2 alpha)(R-2)) that solves (MQG) and satisfies the following limit: lim (t -> 8) parallel to theta(t)parallel to (1+gamma-2 alpha)(H) = 0.