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Enhanced and unenhanced dampings of the Kolmogorov flow

Zhi-Min Chen

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 $(\sin y,0)+e^{-νt} (\sin y,0)$, which tends exponentially to the Kolmogorov flow at the minimum decay rate determined by the viscosity $ν$. 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.

Enhanced and unenhanced dampings of the Kolmogorov flow

Abstract

In the present study, Kolmogorov flow represents the stationary sinusoidal solution to a two-dimensional spatially periodic Navier-Stokes system, driven by an external force. This system admits the additional non-stationary solution , which tends exponentially to the Kolmogorov flow at the minimum decay rate determined by the viscosity . 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.

Paper Structure

This paper contains 7 sections, 5 theorems, 100 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['th1']}
  3. Preliminary estimates
  4. Proof of Theorem \ref{['th1']}
  5. Enhanced and unenhanced dampings of exact solutions
  6. Exact solutions of (\ref{['NS']}) with $a=0,\, 1$
  7. More exact solutions of (\ref{['NS']}) with $a=0$

Key Result

Theorem 1.1

For any $\alpha>1$, $\tau>0$, $\delta >0$ and $a\in \{0, \,1\}$, then the following assertions hold true. (i) Any solution $\omega$ to the linear equation (n10) initiated from $\omega(0) \in L_2( \mathbb{T}_\alpha)$ with $P_{\ne0}\omega(0) =\omega(0)$ satisfies provided that $\nu>0$ is sufficiently small. (ii)For any solution $\omega$ to the nonlinear equation (NS) presented in the perturbed for

Figures (2)

  • Figure 1: Vorticity contours for exact solution (\ref{['s3']}) when $\nu=0.01$.
  • Figure 2: Vorticity contours of solutions (\ref{['s4']}) in (a)-(b), (\ref{['s20']}) in (c) and (\ref{['s21']}) in (d) for $\nu=0.01$.

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 3.1