On the smallness condition in linear inviscid damping: monotonicity and resonance chains

Abstract

We consider the effects of mixing by smooth bilipschitz shear flows in the linearized Euler equations on $\mathbb{T}_{L}\times\mathbb{R}$. Here, we construct a model which is closely related to a small high frequency perturbation around Couette flow, which exhibits linear inviscid damping for $L$ sufficiently small, but for which damping fails if $L$ is large. In particular, similar to the instabiliity results for convex profiles for a shear flow being bilipschitz is not sufficient for linear inviscid damping to hold. Instead of a 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 mechanism in the nonlinear problem.