Ergodicity for Three-Dimensional Stochastic Navier-Stokes Equations with Markov Switching

TL;DR

This paper investigates the long-term statistical behavior of three-dimensional stochastic Navier-Stokes equations with Markov switching, establishing the existence of ergodic and stationary measures for the system.

Contribution

It introduces a regularization approach and proves the existence of ergodic and stationary measures for the stochastic Navier-Stokes equations with Markov switching.

Findings

Existence of ergodic measure for regularized system.

Existence of stationary measure for original system.

Methodology applicable to similar stochastic PDEs.

Abstract

Asymptotic behavior of the three-dimensional stochastic Navier-Stokes equations with Markov switching in additive noises is studied for incompressible fluid flow in a bounded domain in the three-dimensional space. To study such a system, we introduce a family of regularized equations and investigate the asymptotic behavior of the regularized equations first. The existence an ergodic measure for the regularized system is established via the Krylov-Bogolyubov method. Then the existence of an stationary measure to the original system is obtained by extracting a limit from the ergodic measures of the family of the regularized system.