Dynamics Near the Subcritical Transition of the 3D Couette Flow I: Below Threshold Case - Bedrossian, Jacob; Germain, Pierre; Masmoudi, Nader (Memoirs of the American Mathematical Society, 2020)
The authors study small disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number Re. They prove that for sufficiently regular initial data of size $ε ≤ c_0Re^-1$ 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 → ∞ $. For times $t \gtrsim Re^1/3$, 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.