Enhanced and unenhanced dampings of the Kolmogorov flow

TL;DR

This paper investigates the decay rates of solutions to the two-dimensional Navier-Stokes equations with Kolmogorov flow, demonstrating conditions for enhanced damping and providing explicit solutions to illustrate metastability phenomena.

Contribution

It introduces a detailed analysis of enhanced and unenhanced damping in Kolmogorov flow, including explicit solutions that clarify metastability and decay behaviors.

Findings

Higher decay rates achieved for solution differences indicating enhanced damping.

Explicit solutions illustrate both enhanced and unenhanced damping scenarios.

Decay rates depend on viscosity and initial conditions.

Abstract

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.