Linear Inviscid Damping for Monotone Shear Flows

TL;DR

This paper establishes linear stability, scattering, and inviscid damping with optimal decay rates for 2D Euler equations around monotone shear flows in periodic channels, accounting for boundary effects.

Contribution

It proves the first optimal decay results for linearized 2D Euler around monotone shear flows in bounded channels, including boundary effects.

Findings

Proves linear stability and inviscid damping for monotone shear flows.

Demonstrates different behaviors in finite versus infinite channels due to boundaries.

Establishes optimal decay rates for perturbations in Sobolev spaces.

Abstract

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.