### Abstract

We consider the 2D Navier–Stokes equation on T× R, with initial datum that is ε-close in H^{N} to a shear flow (U(y), 0), where ‖U(y)-y‖HN+4≪1 and N> 1. We prove that if ε≪ ν^{1 / 2}, where ν denotes the inverse Reynolds number, then the solution of the Navier–Stokes equation remains ε-close in H^{1} 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 ν^{1 / 2} for 2D shear flows close to the Couette flow.

Original language | English (US) |
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Pages (from-to) | 2051-2075 |

Number of pages | 25 |

Journal | Journal of Nonlinear Science |

Volume | 28 |

Issue number | 6 |

DOIs | |

State | Published - Dec 1 2018 |

### All Science Journal Classification (ASJC) codes

- Modeling and Simulation
- Engineering(all)
- Applied Mathematics

### Keywords

- Enhanced dissipation
- Inviscid damping
- Stability of shear flows

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## Cite this

Bedrossian, J., Vicol, V., & Wang, F. (2018). The Sobolev Stability Threshold for 2D Shear Flows Near Couette.

*Journal of Nonlinear Science*,*28*(6), 2051-2075. https://doi.org/10.1007/s00332-016-9330-9