The Sobolev Stability Threshold for 2D Shear Flows Near Couette

Jacob Bedrossian, Vlad Vicol, Fei Wang

Research output: Contribution to journalArticlepeer-review


We consider the 2D Navier–Stokes equation on T× R, with initial datum that is ε-close in HN to a shear flow (U(y), 0), where ‖U(y)-y‖HN+4≪1 and N> 1. We prove that if ε≪ ν1 / 2, where ν denotes the inverse Reynolds number, then the solution of the Navier–Stokes equation remains ε-close in H1 to (etν∂yyU(y),0) for all t> 0. Moreover, the solution converges to a decaying shear flow for times t≫ ν- 1 / 3 by a mixing-enhanced dissipation effect, and experiences a transient growth of gradients. In particular, this shows that the stability threshold in finite regularity scales no worse than ν1 / 2 for 2D shear flows close to the Couette flow.

Original languageEnglish (US)
Pages (from-to)2051-2075
Number of pages25
JournalJournal of Nonlinear Science
Issue number6
StatePublished - Dec 1 2018


  • Enhanced dissipation
  • Inviscid damping
  • Stability of shear flows

ASJC Scopus subject areas

  • Modeling and Simulation
  • General Engineering
  • Applied Mathematics


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