Enhanced and unenhanced dampings of the Kolmogorov flow

In the present study, Kolmogorov flow represents the stationary sinusoidal solution $(\sin y,0)$ to a two-dimensional spatially periodic Navier-Stokes system, driven by an external force. This system admits the additional non-stationary solution $(\sin y,0)+e^{-\nu t} (\sin y,0)$, which tends exponentially to the Kolmogorov flow at the minimum decay rate determined by the viscosity $\nu$. Enhanced damping or enhanced dissipation of the problem is obtained by presenting higher decay rate for the difference between a solution and the non-stationary basic solution. Moreover, for the understanding of the metastability problem in an explicit manner, a variety of exact solutions are presented to show enhanced and unenhanced dampings.