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

TL;DR

This paper investigates the long-term decay of solutions to anisotropic quasi-geostrophic equations, showing they tend to zero in Sobolev and Lebesgue spaces as time approaches infinity.

Contribution

It establishes the decay to zero over time of global solutions in Sobolev and Lebesgue spaces for anisotropic quasi-geostrophic equations.

Findings

Solutions decay to zero in $L^p$ spaces as time goes to infinity.

The Sobolev norm of solutions also tends to zero over time.

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 $\lim\limits_{t\rightarrow +\infty}\|\theta(t)\|_{H^s}=0$.