Long-Time Behavior of Global Solutions of Anisotropic Quasi-Geostrophic Equations in Sobolev Space

In this study, we investigate the long-time behavior of global solution anisotropic quasi-geostrophic equation, denoted by $$\theta $$ , where belongs to space $$C_b({\mathbb {R}}^+,H^s({\mathbb {R}}^2))$$ . Our results demonstrate that decays zero as time approaches infinity in $$L^p({\mathbb {R}}^2)$$ norm, with $$p\ge 2$$ Additionally, establish limit $$\Vert \theta (t)\Vert _{H^s}$$ t tends infinity.