Dynamics Near the Subcritical Transition of the 3D Couette Flow II: Above Threshold Case

This is the second in a pair of works which study small disturbances to the plane, periodic 3D Couette flow in the incompressible Navier-Stokes equations at high Reynolds number Re. In this work, we show that there is constant 0 < c₀ ≪ 1, independent of Re, such that sufficiently regular disturbances of size ∊ ≲ Re⁻²/^3−δ for any δ > 0 exist at least until t = c_0∊⁻¹ and in general evolve to be O(c₀) due to the lift-up effect. Further, after times t ≳ Re^1/3, the streamwise dependence of the solution is rapidly diminished by a mixing-enhanced dissipation effect and the solution is attracted back to the class of “2.5 dimensional” streamwise-independent solutions (sometimes referred to as “streaks”). The largest of these streaks are expected to eventually undergo a secondary instability at t ≈ ∊⁻¹. Hence, our work strongly suggests, for all (sufficiently regular) initial data, the genericity of the “lift-up effect ⇒ streak growth ⇒ streak breakdown” scenario for turbulent transition of the 3D Couette flow near the threshold of stability forwarded in the applied mathematics and physics literature.