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

About this Title

Jacob Bedrossian, Pierre Germain and Nader Masmoudi

Publication: Memoirs of the American Mathematical Society
Publication Year: 2022; Volume 279, Number 1377
ISBNs: 978-1-4704-7225-2 (print); 978-1-4704-7231-3 (online)
DOI: https://doi.org/10.1090/memo/1377
Published electronically: July 28, 2022

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Table of Contents

Chapters

  • 1. Introduction
  • 2. Outline of the proof
  • 3. Regularization and continuation
  • 4. Multiplier and paraproduct tools
  • 5. High norm estimate on $Q^2$
  • 6. High norm estimate on $Q^3$
  • 7. High norm estimate on $Q^1_0$
  • 8. High norm estimate on $Q^1_{\neq }$
  • 9. Coordinate system controls
  • 10. Enhanced dissipation estimates
  • 11. Sobolev estimates
  • A. Fourier analysis conventions, elementary inequalities, and Gevrey spaces
  • B. Some details regarding the coordinate transform
  • C. Definition and analysis of the norms
  • D. Elliptic estimates

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 Re. In this work, we show that there is constant $0 < c_0 \ll 1$, independent of $\mathbf {Re}$, such that sufficiently regular disturbances of size $\epsilon \lesssim \mathbf {Re}^{-2/3-\delta }$ for any $\delta > 0$ exist at least until $t = c_0\epsilon ^{-1}$ and in general evolve to be $O(c_0)$ due to the lift-up effect. Further, after times $t \gtrsim \mathbf {Re}^{1/3}$, 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 $t \approx \epsilon ^{-1}$. 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.

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