Asymptotic stability threshold of the 2D Couette flow in a finite channel
Abstract
In this paper we study the asymptotic stability threshold of the Couette flow for the 2D Navier–Stokes equations in a finite channel \Omega={\mathbb{T}}\times[-1,1] with Navier-slip boundary condition. It was proved that if the initial velocity v_{0} satisfies \|v_{0}-(y, 0)\|_{H^{4}}\le c\nu^{1/3} for some small c independent of the viscosity coefficient \nu , then the solution of the 2D Navier–Stokes equations rapidly converges to some shear flow close to Couette flow for t\gg \nu^{-1/3} . Moreover, we prove the optimal enhanced dissipation and inviscid damping estimates. To this end, we develop a new approach that does not rely on the construction of the Fourier multiplier. Therefore, our approach opens a way toward the asymptotic stability threshold problem for other laminar flows in a domain with a physical boundary.
Fields of Study & Subjects
Navier-Stokes equation solutions
Stability and Controllability of Differential Equations
Advanced Mathematical Modeling in Engineering
Publication details
- Journal / source
- Annales de l Institut Henri Poincaré C Analyse Non Linéaire
- Publisher
- European Mathematical Society - EMS - Publishing House GmbH
- Volume
- 43
- Issue
- 3
- Pages
- 501–540
- Publication date
- 27 February 2026