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 Re. Our goal is to estimate how the stability threshold scales in 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 (norm of matrix)u_in(norm of matrix)_Hσ ≤δRe^-3/2 for any σ > 9/2 and some δ = δ(σ) > 0 depending only on σ is global in time, remains within O(Re^-1/2) of the Couette flow in L² for all time, and converges to the class of "2.5-dimensional" streamwise-independent solutions referred to as streaks for times t ≳ Re^1/3. Numerical experiments performed by Reddy et. al. with "rough" initial data estimated a threshold of ~ Re^-31/20, which shows very close agreement with our estimate.