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

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. This allows us, in addition, 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.