Stability threshold of the 2D Couette flow in Sobolev spaces

Abstract

We study the stability threshold of the 2D Couette flow in Sobolev spaces at high Reynolds number $Re$. We prove that if the initial vorticity ${\Omega }_{in}$ satisfies $\Vert {\Omega }_{in}-(-1){\Vert }_{H^{\sigma }}\le \epsilon R{e}^{-1/3}$, then the solution of the 2D Navier-Stokes equation approaches to some shear flow which is also close to Couette flow for time $t\gg R{e}^{1/3}$ by a mixing-enhanced dissipation effect and then converges back to Couette flow when $t\to +\infty$.

References in corpus (7)

• Transition threshold for the 3D Couette flow in a finite channel
• Enhanced dissipation in the Navier-Stokes equations near the Poiseuille flow
• Large time behavior and asymptotic stability of the two-dimensional Euler and linearized Euler equations
• Linear inviscid damping in Gevrey spaces
• Metastability of Kolmogorov flows and inviscid damping of shear flows
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• Stability for two-dimensional plane Couette flow to the incompressible Navier-Stokes equations with Navier boundary conditions

Cited by in corpus (6)

• A dynamical approach to the study of instability near Couette flow
• Uniform linear inviscid damping and enhanced dissipation near monotonic shear flows in high Reynolds number regime (I): the whole space case
• Metastability for the dissipative quasi-geostrophic equation and the non-local enhancement
• Stability threshold for 2D shear flows of the Boussinesq system near Couette
• Inviscid damping of monotone shear flows for 2D inhomogeneous Euler equation with non-constant density in a finite channel
• Normalized solutions for Sobolev critical Schrödinger-Bopp-Podolsky systems