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

Abstract

This paper establishes that the anisotropic quasi-geostrophic equation, with orders of fractional dissipation α and β, admits a unique mild solution θ in homogeneous Sobolev and Sobolev-Gevrey spaces H˙a,σs(R2) (with a≥0, σ≥1, α∈(12,34], β∈(12,34] and max⁡{2α−1,2β−1}≤s<1−|α−β|) provided that the initial data θ0 is small enough in these spaces (the critical cases α=12 and β=12 are studied as well). Moreover, this work also studies the behavior of this same solution θ through the following decay rate:limsupt→∞tκ2max⁡{α,β}‖θ(t)‖H˙a,σκ=0, for all κ≥0, where a≥0, σ≥1, α∈(12,34], β∈(12,34] and max⁡{2α−1,2β−1}≤s≤min⁡{2−2α,2−2β}). It is important to emphasize that the limit superior above is a consequence of the Gevrey regularity of θ and of the fact thatlimt→∞⁡‖θ(t)‖L2=0, if it is assumed that θ0∈L2(R2).