Long-Time Decay to the Global Solution of the 2D Dissipative Quasigeostrophic Equation

Abstract

We study the behavior at infinity in time of any global solutionθ∈C(R+,Ḣ2-2α(R2))of the surface quasigeostrophic equation with subcritical exponent2/3≤α≤1. We prove thatlim⁡t→∞∥θ(t)∥Ḣ2-2α=0. Moreover, we prove also the nonhomogeneous version of the previous result, and we prove that ifθ∈C(R+,Ḣ2-2α(R2))is a global solution, thenlim⁡t→∞∥θ(t)∥H2-2α=0.