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

Jacob Bedrossian, Pierre Germain, Nader Masmoudi

Research output: Contribution to journalArticlepeer-review


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 = c0∊−1 and in general evolve to be O(c0) due to the lift-up effect. Further, after times t ≳ Re1/3, 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.

Original languageEnglish (US)
Pages (from-to)1-147
Number of pages147
JournalMemoirs of the American Mathematical Society
Issue number1377
StatePublished - Sep 2022

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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