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

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