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

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 $\mathit{\text{Boussinesq}}\mathit{\text{approximation}}$ 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.