Explicit solutions and linear inviscid damping in the Euler-Boussinesq equation near a stratified Couette flow in the periodic strip

TL;DR

This paper derives explicit solutions for the linearized Boussinesq equations near stratified Couette flow and proves optimal inviscid damping rates for density and velocity perturbations across all positive Richardson numbers.

Contribution

It provides explicit solutions and establishes inviscid damping with optimal rates for the linearized system near stratified Couette flow, using limiting absorption and oscillatory integral techniques.

Findings

Explicit solutions obtained via limiting absorption principle.

Inviscid damping proven for all positive Richardson numbers.

Optimal decay rates for density and velocity perturbations.

Abstract

This short note provides explicit solutions to the linearized Boussinesq equations around the stably stratified Couette flow posed on $\mathbb{T}\times\mathbb{R}$. We consider the long-time behavior of such solutions and prove inviscid damping of the perturbed density and velocity field for any positive Richardson number, with optimal rates. The explicit solution is obtained through the limiting absorption principle whereas the inviscid damping is proved using oscillatory integral methods.