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 $\text{Re}$. Our goal is to estimate how the stability threshold scales in $\text{Re}$: 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 $\Vert u_{in}{\Vert }_{H^{\sigma }}\le \delta {\text{Re}}^{-3/2}$ for any $\sigma >9/2$ and some $\delta =\delta (\sigma )>0$ depending only on $\sigma$, is global in time, remains within $O({\text{Re}}^{-1/2})$ 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 $t\gtrsim {\text{Re}}^{1/3}$. Numerical experiments performed by Reddy et. al. with "rough" initial data estimated a threshold of $\sim {\text{Re}}^{-31/20}$, which shows very close agreement with our estimate.

arXiv comment53 pages