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.
Categories: math.AP physics.flu-dyn