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 solution $\theta \in \mathcal{C}({\mathbb{R}}^{+},{\dot{H}}^{2-2\alpha }({\mathbb{R}}^2\left)\right)$ of the surface quasigeostrophic equation with subcritical exponent $2/3\le \alpha \le 1$. We prove that ${\lim }_{t\to \infty }\parallel \theta \left(t\right){\parallel }_{{\dot{H}}^{2-2\alpha }}=0$. Moreover, we prove also the nonhomogeneous version of the previous result, and we prove that if $\theta \in \mathcal{C}({\mathbb{R}}^{+},{\dot{H}}^{2-2\alpha }({\mathbb{R}}^2\left)\right)$ is a global solution, then ${\lim }_{t\to \infty }\parallel \theta \left(t\right){\parallel }_{H^{2-2\alpha }}=0$.