On the stability threshold for the 3D Couette flow in Sobolev regularity

Abstract

We study Sobolev regularity disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number \textbfRe. Our goal is to estimate how the stability threshold scales in $\textbf{Re}$: the largest the initial perturbation can be while still resulting in a solution that does not transition away from Couette flow. In this work we prove that initial data that satisfies $\|u_{\rm in}\|_{H^\sigma} \leq \delta\textbf{Re}^{-3/2}$ for any $\sigma > 9/2$ and some $\delta = \delta(\sigma) > 0$ depending only on $\sigma$ is global in time, remains within $O(\textbf{Re}^{-1/2})$ of the Couette flow in $L^2$ for all time, and converges to the class of “2.5-dimensional” streamwise-independent solutions referred to as streaks for times $t \gtrsim \textbf{Re}^{1/3}$. Numerical experiments performed by Reddy et. al. with “rough” initial data estimated a threshold of $\sim \textbf{Re}^{-31/20}$, which shows very close agreement with our estimate.