Stability threshold of two-dimensional Couette flow in Sobolev spaces

Nader Masmoudi; Weiren Zhao

doi:10.4171/AIHPC/8

Stability threshold of two-dimensional Couette flow in Sobolev spaces

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Abstract

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 Ωin satisfies kΩin -.-1/kHσ ≤ "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 Re1=3 by a mixing-enhanced dissipation effect, and then converges back to Couette flow when t C1.

Original language	English (US)
Pages (from-to)	245-325
Number of pages	81
Journal	Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume	39
Issue number	2
DOIs	https://doi.org/10.4171/AIHPC/8
State	Published - 2022

Keywords

Couette flow
Sobolev spaces

ASJC Scopus subject areas

Analysis
Mathematical Physics
Applied Mathematics

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title = "Stability threshold of two-dimensional Couette flow in Sobolev spaces",

abstract = "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 Ωin satisfies kΩin -.-1/kHσ ≤ {"}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 Re1=3 by a mixing-enhanced dissipation effect, and then converges back to Couette flow when t C1.",

keywords = "Couette flow, Sobolev spaces",

author = "Nader Masmoudi and Weiren Zhao",

note = "Funding Information: Acknowledgments. The work of N.M. is supported by NSF grant DMS-1716466 and by Tamkeen under the NYU Abu Dhabi Research Institute grant of the center SITE. Publisher Copyright: {\textcopyright} 2022 Elsevier Masson SAS. All rights reserved.",

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AU - Masmoudi, Nader

AU - Zhao, Weiren

N1 - Funding Information: Acknowledgments. The work of N.M. is supported by NSF grant DMS-1716466 and by Tamkeen under the NYU Abu Dhabi Research Institute grant of the center SITE. Publisher Copyright: © 2022 Elsevier Masson SAS. All rights reserved.

PY - 2022

Y1 - 2022

N2 - 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 Ωin satisfies kΩin -.-1/kHσ ≤ "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 Re1=3 by a mixing-enhanced dissipation effect, and then converges back to Couette flow when t C1.

AB - 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 Ωin satisfies kΩin -.-1/kHσ ≤ "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 Re1=3 by a mixing-enhanced dissipation effect, and then converges back to Couette flow when t C1.

KW - Couette flow

KW - Sobolev spaces

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JF - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

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ER -

Stability threshold of two-dimensional Couette flow in Sobolev spaces

Abstract

Keywords

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