Stability threshold of two-dimensional Couette flow in Sobolev spaces

Masmoudi, Nader; Zhao, Weiren

doi:10.4171/aihpc/8

Geodesic

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Stability threshold of two-dimensional Couette flow in Sobolev spaces

Masmoudi, Nader ; Zhao, Weiren

Annales de l'I.H.P. Analyse non linéaire, Tome 39 (2022) no. 2, pp. 245-325

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Résumé

We study the stability threshold of the two-dimensional Couette flow in Sobolev spaces at high Reynolds number $Re$ . We prove that if the initial vorticity $Ω_{in}$ satisfies $∥ Ω_{in} {- (- 1) ∥}_{H^{σ}} \leq$ $ε {Re}^{- 1 / 3}$ , then the solution of the two-dimensional Navier–Stokes equation approaches some shear flow which is also close to Couette flow for time $t ≫ {Re}^{1 / 3}$ by a mixing-enhanced dissipation effect, and then converges back to Couette flow when $t \to + \infty$ .

Accepté le : 2021-05-03
Publié le : 2022-02-16

DOI : 10.4171/aihpc/8

Classification : 35B35, 35B40, 35Q30
Keywords: Couette flow, Sobolev Spaces

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     author = {Masmoudi, Nader and Zhao, Weiren},
     title = {Stability threshold of two-dimensional {Couette} flow in {Sobolev} spaces},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {245--325},
     volume = {39},
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     doi = {10.4171/aihpc/8},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/aihpc/8/}
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Masmoudi, Nader; Zhao, Weiren. Stability threshold of two-dimensional Couette flow in Sobolev spaces. Annales de l'I.H.P. Analyse non linéaire, Tome 39 (2022) no. 2, pp. 245-325. doi: 10.4171/aihpc/8

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