Abstract
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.
Original language | English (US) |
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Pages (from-to) | 541-608 |
Number of pages | 68 |
Journal | Annals of Mathematics |
Volume | 185 |
Issue number | 2 |
DOIs | |
State | Published - 2017 |
ASJC Scopus subject areas
- Mathematics (miscellaneous)