Linear Inviscid Damping for Monotone Shear Flows

In this article, we prove linear stability, scattering and inviscid damping with optimal decay rates for the linearized 2D Euler equations around a large class of strictly monotone shear flows, $(U(y),0)$, in a periodic channel under Sobolev perturbations. Here, we consider the settings of both an infinite periodic channel of period $L$, ${\mathbb{T}}_L\times \mathbb{R}$, as well as a finite periodic channel, ${\mathbb{T}}_L\times [0,1]$, with impermeable walls. The latter setting is shown to not only be technically more challenging, but to exhibit qualitatively different behavior due to boundary effects.