Limiting absorption principles and linear inviscid damping in the Euler-Boussinesq system in the periodic channel

We consider the long-time behavior of solutions to the two dimensional non-homogeneous Euler equations under the Boussinesq approximation posed on a periodic channel. We study the linearized system near a linearly stratified Couette flow and prove inviscid damping of the perturbed density and velocity field for any positive Richardson number, with optimal rates. Our methods are based on time-decay properties of oscillatory integrals obtained using a limiting absorption principle, and require a careful understanding of the asymptotic expansion of the generalized eigenfunction near the critical layer. As a by-product of our analysis, we provide a precise description of the spectrum of the linearized operator, which, for sufficiently large Richardson number, consists of an essential spectrum (as expected according to classical hydrodynamic problems) as well as discrete neutral eigenvalues (giving rise to oscillatory modes) accumulating towards the endpoints of the essential spectrum.