  We study small disturbances to the periodic, plane Couette flow in the 3D
incompressible Navier-Stokes equations at high Reynolds number $\textbf{Re}$.
We prove that for sufficiently regular initial data of size $\epsilon \leq
c_0\textbf{Re}^{-1}$ for some universal $c_0 > 0$, the solution is global,
remains within $O(c_0)$ of the Couette flow in $L^2$, and returns to the
Couette flow as $t \rightarrow \infty$. For times $t \gtrsim
\textbf{Re}^{1/3}$, the streamwise dependence is damped by a mixing-enhanced
dissipation effect and the solution is rapidly attracted to the class of "2.5
dimensional" streamwise-independent solutions referred to as streaks. Our
analysis contains perturbations that experience a transient growth of kinetic
energy from $O(\textbf{Re}^{-1})$ to $O(c_0)$ due to the algebraic linear
instability known as the lift-up effect. Furthermore, solutions can exhibit a
direct cascade of energy to small scales. The behavior is very different from
the 2D Couette flow, in which stability is independent of $\textbf{Re}$,
enstrophy experiences a direct cascade, and inviscid damping is dominant
(resulting in a kind of inverse energy cascade). In 3D, inviscid damping will
play a role on one component of the velocity, but the primary stability
mechanism is the mixing-enhanced dissipation. Central to the proof is a
detailed analysis of the interplay between the stabilizing effects of the
mixing and enhanced dissipation and the destabilizing effects of the lift-up
effect, vortex stretching, and weakly nonlinear instabilities connected to the
non-normal nature of the linearization.
