Dynamics Near the Subcritical Transition of the 3D Couette Flow I: Below Threshold Case by Pierre Germain, Nader Masmoudi, Jacob Bedrossian (Paperback, 2020)

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Our analysis contains perturbations that experience a transient growth of kinetic energy from O(Re-1) to O(c0) due to the algebraic linear instability known as the lift-up effect. Furthermore, solutions can exhibit a direct cascade of energy to small scales.

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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 $\epsilon \leq c_0\mathbf {Re}^-1$ 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 ightarrow \infty $. For times $t \gtrsim \mathbf {Re}^1/3$, 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.