论文标题
有限通道中3D COUETTE流的过渡阈值
Transition threshold for the 3D Couette flow in a finite channel
论文作者
论文摘要
在本文中，我们研究了在有限通道$ \ Mathbb {T} \ times [-1,1] \ times \ times \ times \ mathbb {t} $中，在High Reynolds Number Number $ {RE} $上研究3D平面Couette Flow $（y，0,0）$的非线性稳定性。众所周知，对于任何雷诺的数字，平面轴向流量稳定。但是，对于高雷诺数的小而有限的扰动，它可能变得不稳定，并过渡到湍流。这是所谓的Sommerfeld悖论。该悖论的一个解决方案是研究过渡阈值问题，这与干扰会导致流动不稳定以及对雷诺数数量的依赖的依赖。这项工作表明，如果初始速度$ v_0 $满足$ \ | v_0-（y，0,0）\ | _ {h^2} \ le c_0 {re}^{re}^{ - 1} $对于某些$ c_0> 0> 0 $ re $，则独立于$ re $，那么3D Navier-Stokes earge and Tresmition in Tirm and tirs and do cout cout cout cout in tirm and do time and do cout cout countion countion in Tirm and do cout cout cout countion cou_0> 0 $ l^\ infty $ sense，并迅速收敛到$ t \ gg re^{\ frac 13} $的条纹解决方案，这是由于混合增强的耗散效果。该结果证实了Treefethen等人提出的过渡阈值猜想（Science，261（1993），578-584）。为此，我们开发了解决方案估计方法，以建立围绕流量$（v（t，y，z），0,0）$的完整线性化navier-stokes系统的时空估计，其中$ v（t，y，z）$是一个小的扰动（但独立于$ re $ $ $ $）。
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 $\|v_0-(y,0,0)\|_{H^2}\le c_0{Re}^{-1}$ 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 $t\gg Re^{\frac 13}$ 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 $y$.
