Dynamics near the subcritical transition of the 3D Couette flow II: Above threshold case
Jacob Bedrossian, Pierre Germain, Nader Masmoudi
TL;DR
The paper advances the mathematical understanding of the 3D Navier–Stokes dynamics near plane Couette flow in the subcritical regime by establishing that, above a sharp threshold ε ≲ Re^{-2/3-δ}, small disturbances persist up to time t ~ c0 ε^{-1} and evolve toward the class of streaks due to lift-up, with rapid damping of nonzero streamwise modes by mixing-enhanced dissipation. It introduces a refined Gevrey-based energy framework and a near-Lagrangian coordinate transform to control the growth from lift-up while capturing the secondary instability pathway leading to streak breakdown. A detailed bootstrap argument bounds high-norm quantities for the velocity components, the coordinate system, and the auxiliary g,C variables, balancing nonlinear interactions via a toy-model-informed norm design and paraproduct decomposition. The results support the conventional narrative lift-up → streak growth → streak breakdown as a generic route to transition near the stability threshold, extending previous subcritical analyses to the above-threshold regime and clarifying the role of regularity and enhanced dissipation in sustaining transient large-amplitude streaks. Applicability to high-Re turbulence near threshold is reinforced by the explicit ε–ν balance and the explicit time scales τ_ED ∼ ν^{-1/3} and τ_NL ∼ ε^{-1/2} that govern the nonlinear-to-dissipative competition.
Abstract
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 $\textbf{Re}$. In this work, we show that there is constant $0 < c_0 \ll 1$, independent of $\textbf{Re}$, such that sufficiently regular disturbances of size $ε\lesssim \textbf{Re}^{-2/3-δ}$ for any $δ> 0$ exist at least until $t = c_0ε^{-1}$ and in general evolve to be $O(c_0)$ due to the lift-up effect. Further, after times $t \gtrsim \textbf{Re}^{1/3}$, 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 $t \approx ε^{-1}$. Hence, our work strongly suggests, for all (sufficiently regular) initial data, the genericity of the "lift-up effect $\Rightarrow$ streak growth $\Rightarrow$ streak breakdown" scenario for turbulent transition of the 3D Couette flow near the threshold of stability forwarded in the applied mathematics and physics literature.