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 $$v_0$$ v 0 satisfies $$\Vert v_0-(y, 0)\Vert _{H^2}\leqq cRe^{-\frac{1}{2}}$$ ‖ 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 $$O(Re^{-\frac{1}{2}})$$ O ( R e - 1 2 ) of the Couette flow for any time.

中文翻译：

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有限通道中二维库埃特流的过渡阈值

在本文中，我们研究了有限通道中高雷诺数 Re 下 Couette 流 (y, 0) 周围的二维 Navier-Stokes 方程的过渡阈值问题。我们开发了一种系统方法来建立线性化算子的分解估计和线性化 Navier-Stokes 方程的时空估计。特别是，三种重要的影响——增强耗散、无粘阻尼和边界层——以尖锐的形式整合到时空估计中。作为应用，我们证明如果初速度 $$v_0$$ v 0 满足 $$\Vert v_0-(y, 0)\Vert _{H^2}\leqq cRe^{-\frac{1}{ 2}}$$ ‖ v 0 - ( y , 0 ) ‖ H 2 ≦ c Re - 1 2 对于一些独立于 Re 的小 c ，那么二维 Navier-Stokes 方程的解仍然在 $$O( Re^{-\frac{1}{2}})$$ O ( Re - 1 2 ) 的 Couette 流。