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

By Jacob Bedrossian, Pierre Germain, Nader Masmoudi. 2015

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We study Sobolev regularity disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number $\textbf{Re}$. Our goal is to estimate how the stability threshold scales in $\textbf{Re}$: 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 which satisfies $\| u_{in} \|_{H^\sigma} \leq \delta\textbf{Re}^{-3/2}$ for any $\sigma > 9/2$ and some $\delta = \delta(\sigma) > 0$ depending only on $\sigma$, is global in time, remains within $O(\textbf{Re}^{-1/2})$ 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 $t \gtrsim \textbf{Re}^{1/3}$. Numerical experiments performed by Reddy et. al. with "rough" initial data estimated a threshold of $\sim \textbf{Re}^{-31/20}$, which shows very close agreement with our estimate.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/189292c9bb2ab998d4b92f5aae5f3e8d313a7c65

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In Applied Mathematics

  1. Zhifei Zhang
  2. Weiren Zhao
  3. Dongyi Wei

In Computational Mechanics

  1. Zhifei Zhang
  2. Weiren Zhao
  3. Dongyi Wei

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