In this paper, we prove the linear inviscid damping and voticity depletion phenomena for the linearized Euler equations around the Kolmogorov flow. These results confirm Bouchet and Morita's predictions based on numerical analysis. By using the wave operator method introduced by Li, Wei and Zhang, we solve Beck and Wayne's conjecture on the optimal enhanced dissipation rate for the 2-D linearized Navier-Stokes equations around the bar state called Kolmogorov flow. The same dissipation rate is proved for the Navier-Stokes equations if the initial velocity is included in a basin of attraction of the Kolmogorov flow with the size of $\nu^{\frac 23+}$, here $\nu$ is the viscosity coefficient.

中文翻译：

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Kolmogorov 流的线性无粘阻尼和增强耗散

在本文中，我们证明了围绕 Kolmogorov 流的线性化欧拉方程的线性无粘阻尼和气浮损耗现象。这些结果证实了 Bouchet 和 Morita 基于数值分析的预测。通过使用 Li、Wei 和 Zhang 引入的波算子方法，我们解决了贝克和韦恩关于称为 Kolmogorov 流的棒状态的二维线性化 Navier-Stokes 方程的最优增强耗散率的猜想。如果初始速度包含在大小为 $\nu^{\frac 23+}$ 的 Kolmogorov 流的吸引力盆地中，则 Navier-Stokes 方程证明了相同的耗散率，这里 $\nu$ 是粘度系数。