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

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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 which satisfies uin 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 Re¹/³. 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.