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In this paper, we prove the linear inviscid damping and voticity depletionphenomena for the linearized Euler equations around the Kolmogorov flow. Theseresults confirm Bouchet and Morita's predictions based on numerical analysis.By using the wave operator method introduced by Li, Wei and Zhang, we solveBeck and Wayne's conjecture on the optimal enhanced dissipation rate for the2-D linearized Navier-Stokes equations around the bar state called Kolmogorovflow. The same dissipation rate is proved for the Navier-Stokes equations ifthe initial velocity is included in a basin of attraction of the Kolmogorovflow with the size of $\nu^{\frac 23+}$, here $\nu$ is the viscositycoefficient.

Linear inviscid damping and enhanced dissipation for the Kolmogorov flow