Transition threshold for the 3D Couette flow in a finite channel

Abstract

In this paper, we study nonlinear stability of the 3D plane Couette flow $(y,0,0)$ at high Reynolds number $Re$ in a finite channel $\mathbb{T}\times [-1,1]\times \mathbb{T}$. It is well known that the plane Couette flow is linearly stable for any Reynolds number. However, it could become nonlinearly unstable and transition to turbulence for small but finite perturbations at high Reynolds number. This is so-called Sommerfeld paradox. One resolution of this paradox is to study the transition threshold problem, which is concerned with how much disturbance will lead to the instability of the flow and the dependence of disturbance on the Reynolds number. This work shows that if the initial velocity $v_0$ satisfies $\Vert v_0-(y,0,0){\Vert }_{H^2}\le c_0{Re}^{-1}$ for some $c_0>0$ independent of $Re$, then the solution of the 3D Navier-Stokes equations is global in time and does not transition away from the Couette flow in the $L^{\infty }$ sense, and rapidly converges to a streak solution for $t\gg R{e}^{\frac{1}{3}}$ due to the mixing-enhanced dissipation effect. This result confirms the transition threshold conjecture proposed by Trefethen et al.(Science, 261(1993), 578-584). To this end, we develop the resolvent estimate method to establish the space-time estimates for the full linearized Navier-Stokes system around the flow $(V(t,y,z),0,0)$, where $V(t,y,z)$ is a small perturbation(but independent of $Re$) of the Couette flow $y$.

arXiv comment147 pages

References in corpus (11)

• Enhanced dissipation and inviscid damping in the inviscid limit of the Navier-Stokes equations near the 2D Couette flow
• Transition threshold for the 3D Couette flow in a finite channel
• Inviscid damping near the Couette flow in a channel
• The Sobolev stability threshold for 2D shear flows near Couette
• Diffusion and mixing in fluid flow via the resolvent estimate
• Metastability of Kolmogorov flows and inviscid damping of shear flows
• Transition threshold for the 3D Couette flow in Sobolev space
• Pseudospectral and spectral bounds for the Oseen vortices operator
• Linear inviscid damping and vorticity depletion for shear flows
• Stability of the Couette flow at high Reynolds number in 2D and 3D
• Resolvent estimates for 2 dimensional perturbations of plane Couette Flow

Cited by in corpus (17)

• Transition threshold for the 3D Couette flow in a finite channel
• Enhanced dissipation in the Navier-Stokes equations near the Poiseuille flow
• Stability threshold of the 2D Couette flow in Sobolev spaces
• A dynamical approach to the study of instability near Couette flow
• Nonlinear inviscid damping near monotonic shear flows
• 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
• Stirring Speeds Up Chemical Reaction
• Inviscid damping of monotone shear flows for 2D inhomogeneous Euler equation with non-constant density in a finite channel
• Inviscid damping and enhanced dissipation of the boundary layer for 2D Navier-Stokes linearized around Couette flow in a channel
• Enhanced dissipation, hypoellipticity for passive scalar equations with fractional dissipation
• On the $L^{\infty }$ stability of Prandtl expansions in Gevrey class
• Enhanced dissipation for the 2D Couette flow in critical space
• Linear stability of the Couette flow in the 3D isentropic compressible Navier-Stokes equations
• Stability of Couette flow for 2D Boussinesq system in a uniform magnetic field
• Linear stability of the Couette flow for the non-isentropic compressible fluid
• Stability of the Couette flow under the 2D steady Navier-Stokes flow