  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.
