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 ) (y,0,0) at high Reynolds number R e {Re} in a finite channel T × [ − 1 , 1 ] × T \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 v_0 satisfies ‖ v 0 − ( y , 0 , 0 ) ‖ H 2 ≤ c 0 R e − 1 \|v_0-(y,0,0)\|_{H^2}\le c_0{Re}^{-1} for some c 0 > 0 c_0>0 independent of R e Re , then the solution of the 3D Navier-Stokes equations is global in time and does not transit away from the Couette flow in the L ∞ L^\infty sense, and rapidly converges to a streak solution for t ≫ R e 1 3 t\gg Re^{\frac 13} due to the mixing-enhanced dissipation effect. This result confirms the threshold result obtained by Chapman via an asymptotic analysis(JFM 2002). The most key ingredient of the proof is the resolvent estimates for the full linearized 3D Navier-Stokes system around the flow ( V ( y , z ) , 0 , 0 ) (V(y,z), 0,0) , where V ( y , z ) V(y,z) is a small perturbation(but independent of R e Re ) of y y .