Long-time behavior of global solutions of anisotropic quasi-geostrophic equations in Sobolev space

Abstract

We study the behavior at infinity in time of the global solution of the anisotropic quasi-geostrophic equation $\theta \in C_b({\mathbb{R}}^{+},H^s({\mathbb{R}}^2))$. We prove that this solution decays to zero as time goes to infinity in $L^p({\mathbb{R}}^2)$, $p\in [2,+\infty ],$ moreover, we prove also that $\underset{t\to +\infty }{\lim }\Vert \theta (t){\Vert }_{H^s}=0$.