Existence of solutions and their behavior for the anisotropic quasi-geostrophic equation in Sobolev and Sobolev-Gevrey spaces

This paper establishes that the anisotropic quasi-geostrophic equation, with orders of fractional dissipation alpha and beta, admits a unique mild solution theta in homogeneous Sobolev and Sobolev-Gevrey spaces. (H) oer dot(a,sigma)(s)(R-2) (with a >= 0, sigma >= 1, alpha is an element of(1/2, 3/4], beta is an element of (1/2, 3/4] and max{2 alpha - 1, 2 beta - 1} <= s < 1 - vertical bar alpha - beta vertical bar) provided that the initial data theta(0) is small enough in these spaces (the critical cases alpha = 1/2 and beta = 1/2 are studied as well). Moreover, this work also studies the behavior of this same solution theta through the following decay rate: lim sup(t ->infinity) t(kappa/2) (max{alpha, beta}) parallel to theta(t)parallel to((H) over dota,sigma kappa) = 0, for all kappa >= 0, where a >= 0, sigma >= 1, a is an element of(1/2, 3/4], beta is an element of(1/2, 3/4] and max{2 alpha - 1, 2 beta- 1} <= s <= min{2 - 2 alpha, 2 - 2 beta}). It is important to emphasize that the limit superior above is a consequence of the Gevrey regularity of theta and of the fact that lim(t ->infinity) theta(t)parallel to(L2) = 0, if it is assumed that theta(0) is an element of L-2(R-2). (c) 2023 Elsevier Inc. All rights reserved.