Echo Chains as a Linear Mechanism: Norm Inflation, Modified Exponents and Asymptotics

TL;DR

This paper investigates linear instability mechanisms in the Euler equations near Couette flow, revealing norm inflation in Gevrey spaces, and refines the understanding of resonance cancellations affecting high-frequency behavior.

Contribution

It demonstrates that echo chains cause norm inflation as a linear instability and provides a more precise analysis of resonance cancellations, removing previous constraints.

Findings

Norm inflation occurs in Gevrey spaces over time.

Resonance cancellations lead to a modified high-frequency exponent.

Solutions can diverge in Gevrey regularity while converging in Sobolev space.

Abstract

In this article we show that the Euler equations, when linearized around a low frequency perturbation to Couette flow, exhibit norm inflation in Gevrey-type spaces as time tends to infinity. Thus, echo chains are shown to be a (secondary) linear instability mechanism. Furthermore, we develop a more precise analysis of cancellations in the resonance mechanism, which yields a modified exponent in the high frequency regime. In addition it allows us to remove a logarithmic constraint on the perturbations present in prior works by Bedrossian, Deng and Masmoudi and to construct solutions which are initially in a Gevrey class for which the velocity asymptotically converges in Sobolev regularity but diverges in Gevrey regularity.