TY - JOUR

T1 - On the stability threshold for the 3D Couette flow in Sobolev regularity

AU - Bedrossian, Jacob

AU - Germain, Pierre

AU - Masmoudi, Nader

N1 - Publisher Copyright:
© 2017 Department of Mathematics, Princeton University.

PY - 2017

Y1 - 2017

N2 - We study Sobolev regularity disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number Re. Our goal is to estimate how the stability threshold scales in Re: the largest the initial perturbation can be while still resulting in a solution that does not transition away from Couette flow. In this work we prove that initial data that satisfies (norm of matrix)uin(norm of matrix)Hσ ≤δRe-3/2 for any σ > 9/2 and some δ = δ(σ) > 0 depending only on σ is global in time, remains within O(Re-1/2) of the Couette flow in L2 for all time, and converges to the class of "2.5-dimensional" streamwise-independent solutions referred to as streaks for times t ≳ Re1/3. Numerical experiments performed by Reddy et. al. with "rough" initial data estimated a threshold of ~ Re-31/20, which shows very close agreement with our estimate.

AB - We study Sobolev regularity disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number Re. Our goal is to estimate how the stability threshold scales in Re: the largest the initial perturbation can be while still resulting in a solution that does not transition away from Couette flow. In this work we prove that initial data that satisfies (norm of matrix)uin(norm of matrix)Hσ ≤δRe-3/2 for any σ > 9/2 and some δ = δ(σ) > 0 depending only on σ is global in time, remains within O(Re-1/2) of the Couette flow in L2 for all time, and converges to the class of "2.5-dimensional" streamwise-independent solutions referred to as streaks for times t ≳ Re1/3. Numerical experiments performed by Reddy et. al. with "rough" initial data estimated a threshold of ~ Re-31/20, which shows very close agreement with our estimate.

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U2 - 10.4007/annals.2017.185.2.4

DO - 10.4007/annals.2017.185.2.4

M3 - Article

AN - SCOPUS:85014681696

VL - 185

SP - 541

EP - 608

JO - Annals of Mathematics

JF - Annals of Mathematics

SN - 0003-486X

IS - 2

ER -