Transition Threshold for the 2-D Couette Flow in a Finite Channel

In this paper, we study the transition threshold problem for the 2-D Navier–Stokes equations around the Couette flow ( y , 0) at high 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...

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Bibliographic Details
Published in:Archive for rational mechanics and analysis 2020-10, Vol.238 (1), p.125-183
Main Authors: Chen, Qi, Li, Te, Wei, Dongyi, Zhang, Zhifei
Format: Article
Language:English
Subjects:
Boundary layers
Classical Mechanics
Complex Systems
Couette flow
Damping
Estimates
Fluid dynamics
Fluid flow
Fluid- and Aerodynamics
High Reynolds number
Linearization
Mathematical analysis
Mathematical and Computational Physics
Navier-Stokes equations
Physics
Physics and Astronomy
Reynolds number
Spacetime
Theoretical
Two dimensional flow
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Summary:In this paper, we study the transition threshold problem for the 2-D Navier–Stokes equations around the Couette flow ( y , 0) at high 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 a 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 ≦ c R e - 1 2 for some small c independent of Re , then the solution of the 2-D Navier–Stokes equations remains within O ( R e - 1 2 ) of the Couette flow for any time.
ISSN:0003-9527
1432-0673
DOI:10.1007/s00205-020-01538-y