On the asymptotic stability of stratified solutions for the 2D Boussinesq equations with a velocity damping term

TL;DR

This paper proves the asymptotic stability of stratified solutions in the 2D Boussinesq equations with damping, using weighted energy methods and decay estimates in a bounded strip.

Contribution

It establishes the asymptotic stability of stratified solutions for the 2D Boussinesq equations with damping in a bounded domain, a novel result in this setting.

Findings

Proves asymptotic stability for specific perturbations

Uses weighted energy spaces and decay estimates

Employs bootstrap arguments for the proof

Abstract

We consider the 2D Boussinesq equations with a velocity damping term in a strip $\mathbb{T}\times[-1,1]$, with impermeable walls. In this physical scenario, where the \textit{Boussinesq approximation} is accurate when density/temperature variations are small, our main result is the asymptotic stability for a specific type of perturbations of a stratified solution. To prove this result, we use a suitably weighted energy space combined with linear decay, Duhamel's formula and "bootstrap" arguments.