In this paper, we obtain the optimal instability threshold of the Couette flow for Navier–Stokes equations with small viscosity $\nu >0$, when the perturbations are in the critical spaces $H_x^1{L}_y^2$. More precisely, we introduce a new dynamical approach to prove the instability for some perturbation of size ${\nu }^{\frac{1}{2}-{\delta }_0}$ with any small ${\delta }_0>0$, which implies that ${\nu }^{\frac{1}{2}}$ is the sharp stability threshold. In our method, we prove a transient exponential growth without referring to eigenvalue or pseudo-spectrum. As an application, for the linearized Euler equations around shear flows that are near the Couette flow, we provide a new tool to prove the existence of growing modes for the corresponding Rayleigh operator and give a precise location of the eigenvalues.

中文翻译：

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研究库埃特流附近不稳定性的动力学方法

在本文中，我们获得了小粘度Navier-Stokes方程的Couette流的最佳不稳定阈值$\nu >0$，当扰动位于临界空间时$H_X^1{L}_y^2$。更准确地说，我们引入了一种新的动力学方法来证明某些尺寸扰动的不稳定性${\nu }^{\frac{1}{2}-{\delta }_0}$与任何小${\delta }_0>0$，这意味着${\nu }^{\frac{1}{2}}$是锐稳定性阈值。在我们的方法中，我们在不参考特征值或伪谱的情况下证明了瞬态指数增长。作为一种应用，对于靠近库埃特流的剪切流周围的线性化欧拉方程，我们提供了一种新工具来证明相应瑞利算子的增长模式的存在，并给出特征值的精确位置。