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

TL;DR

This paper establishes a Sobolev regularity threshold for the stability of 3D Couette flow at high Reynolds numbers, showing that perturbations smaller than a certain Re-dependent scale lead to global stability and convergence to streak solutions.

Contribution

The authors derive a precise Re-dependent Sobolev norm threshold for the stability of 3D Couette flow, matching numerical estimates and advancing understanding of flow stability at high Reynolds numbers.

Findings

Initial data with Sobolev norm below Re^{-3/2} remains stable.

Solutions stay close to Couette flow within Re^{-1/2} in L^2 norm.

Flow converges to streak solutions after Re^{1/3} time.

Abstract

We study Sobolev regularity disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number $\textbf{Re}$. 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 which satisfies $\| u_{in} \|_{H^\sigma} \leq \delta\textbf{Re}^{-3/2}$ 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.