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Transition threshold for the 2-D Couette flow in a finite channel

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arxiv 1808.08736 v1 pith:AHKNS4S2 submitted 2018-08-27 math.AP

Qi Chen , Te Li , Dongyi Wei , Zhifei Zhang This is my paper · ORCID
classification math.AP
keywords couetteequationsestimatesflownavier-stokeschannelfinitefrac
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abstract

In this paper, we study the transition threshold problem for the 2-D Navier-Stokes equations around the Couette flow $(y,0)$ at large Reynolds number $Re$ in a finite channel. We develop a systematic method to establish the resolvent estimates of the linearized operator and space-time estimates of the linearized Navier-Stokes equations. In particular, three kinds of important effects: enhanced dissipation, inviscid damping and boundary layer, are integrated into the space-time estimates in a sharp form. As an application, we prove that if the initial velocity $v_0$ satisfies $\|v_0-(y, 0)\|_{H^2}\le cRe^{-\frac 12}$ for some small $c$ independent of $Re$, then the solution of the 2-D Navier-Stokes equations remains within $O(Re^{-\frac 12})$ of the Couette flow for any time.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Stability threshold of the 2D Couette flow in Sobolev spaces

    math.AP 2019-08 conditional novelty 8.0 of 10

    For 2D Navier-Stokes near Couette flow, H^σ vorticity perturbations of size ≤ ε Re^{-1/3} are globally stable with inviscid damping and enhanced dissipation.

  2. Enhanced dissipation for the 2D Couette flow in critical space

    math.AP 2019-08 conditional novelty 7.0 of 10

    For 2D Navier-Stokes near Couette, ν^{1/2}-small perturbations in H^log_x L^2_y undergo enhanced dissipation at rate ν^{1/3} and inviscid damping, with L^2 initial data sufficient for any threshold exponent above 1/2.

  3. On the stability of laminar flows between plates

    math-ph 2019-08 conditional novelty 7.0 of 10

    Monotone shear flows that are either nearly linear or have no inflection point are linearly stable in the large Reynolds limit, in a periodic channel with no-slip or traction boundary conditions.

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