On the Smallness Condition in Linear Inviscid Damping: Monotonicity and Resonance Chains

TL;DR

This paper investigates the conditions under which linear inviscid damping occurs for shear flows, revealing that bilipschitz shear profiles can exhibit damping or failure depending on domain size, due to a cascade of resonances.

Contribution

It introduces a new resonance cascade mechanism explaining damping failure in shear flows, extending understanding beyond eigenvalue-based analyses.

Findings

Damping occurs for small domain sizes but fails for large sizes.

A new resonance cascade mechanism is identified as the cause of damping failure.

Damping behavior depends critically on the shear profile and domain size.

Abstract

We consider the linearized Euler equations around a smooth, bilipschitz shear profile $U(y)$ on $\mathbb{T}_L \times \mathbb{R}$. We construct an explicit flow which exhibits linear inviscid damping for $L$ sufficiently small, but for which damping fails if $L$ is large. In particular, similar to the instability results for convex profiles for a shear flow being bilipschitz is not sufficient for linear inviscid damping to hold. Instead of an eigenvalue-based argument the underlying mechanism here is shown to be based on a new cascade of resonances moving to higher and higher frequencies in $y$, which is distinct from the echo chain mechanism in the nonlinear problem.