Long time instability of the Couette flow in low Gevrey spaces

TL;DR

This paper demonstrates that Couette flow becomes unstable under disturbances less smooth than Gevrey class 2, revealing a critical regularity threshold linked to a nonlinear energy cascade mechanism.

Contribution

It establishes the critical regularity for Couette flow instability as the Gevrey class 2, highlighting a nonlinear instability mechanism not previously identified.

Findings

Instability occurs for disturbances less smooth than Gevrey class 2.

Stability and inviscid damping hold for smoother disturbances.

Nonlinear energy cascade drives the instability.

Abstract

We prove the instability of the Couette flow if the disturbances is less smooth than the Gevrey space of class 2. This shows that this is the critical regularity for this problem since it was proved in [5] that stability and inviscid damping hold for disturbances which are smoother than the Gevrey space of class 2. A big novelty is that this critical space is due to an instability mechanism which is completely nonlinear and is due to some energy cascade.