Ergodicity for three-dimensional stochastic Navier-Stokes equations with Markovian switching

Asymptotic behavior of the three-dimensional stochastic Navier-Stokes equations with Markovian switching for additive noises is studied for an incompressible fluid flow in a bounded domain. The objective of this article is to prove the existence and identification of a stationary measure for such stochastic systems. To achieve this goal, a family of regularized equations is introduced in order to investigate the asymptotic behavior of their solutions. The existence of an ergodic measure for the regularized system is established by the Krylov-Bogolyubov method. The existence and identification of a stationary measure corresponding to the original system are obtained as a sequential limit point of the family of the ergodic measures that correspond to the regularized systems. The main result of this work is that under mild integrability conditions on the initial data and external force, there exists a stationary measure for the solution of the three-dimensional stochastic Navier-Stokes equations with Markovian switching.