Asymptotic stability of the $2D$ temperature-dependent tropical climate model with the sharp decay rates

TL;DR

This paper proves the global existence and sharp decay rates of solutions for a 2D temperature-dependent tropical climate model with temperature-dependent diffusion and damping, on the entire plane.

Contribution

It establishes the asymptotic stability and decay rates of solutions for a 2D tropical climate model with temperature-dependent diffusion, considering cases with and without linear damping.

Findings

Global existence of smooth solutions for small initial data.

Sharp decay rates in Sobolev norms for solutions.

Results valid for both damping and non-damping cases.

Abstract

We investigate the asymptotic stability of a tropical climate model posed on $\bR^2$, with temperature-dependent diffusion in the barotropic mode $u$ and linear damping in the first baroclinic mode $v$. We consider two distinct cases for the barotropic component: one with linear damping and one without. For both cases, we prove the small data global existence of smooth solutions. Furthermore, we establish sharp temporal decay estimates for solutions in arbitrary Sobolev norms $H^m (\bR^2)$, $m \ge 0$.