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

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 $\| u_{in} \|_{H^σ} \leq δ\textbf{Re}^{-3/2}$ for any $σ> 9/2$ and some $δ= δ(σ) > 0$ depending only on $σ$, 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.