Exponential mixing properties of the stochastic tamed 3D Navier–Stokes equation with degenerate noise

Abstract

The current paper is devoted to the stochastic tamed 3D Naiver-Stokes equation with degenerate noise with degenerate random forcing. We prove that the associated Markov semigroup possesses an exponentially attracting invariant measure by showing that the corresponding dynamical system possesses a strong type of Lyapunov structure along with a special type of gradient inequality and is of a relatively weak form of irreducibility.