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

• 成果类型: Article
• 署名作者: Bedrossian, Jacob; Germain, Pierre; Masmoudi, Nader
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2017.185.2.4
• 发表日期: 2017
• 页码: 541-608
• 关键词: subcritical transition; diffusion; analyticity; poiseuille; amplitude; EQUATIONS; bounds

摘要

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 parallel to u(in)parallel to H-sigma <= delta Rc(-3/2) for any sigma > 9/2 and some delta = delta(sigma) > 0 depending only on sigma is global in time, remains within O(Re-1/2) of the Couette flow in L-2 for all time, and converges to the class of 2.5 -dimensional streamwise-independent solutions referred to as streaks for times t > Re-1/3. Numerical experiments performed by Reddy et. al. with rough initial data estimated a threshold of ti similar to Re-31/20, which shows very close agreement with our estimate.