Uniqueness of the invariant measure and asymptotic stability for the 2D Navier-Stokes equations with multiplicative noise

We establish the uniqueness and the asymptotic stability of the invariant measure for the two-dimensional Navier-Stokes equations driven by a multiplicative noise which is either bounded or with a sublinear or a linear growth. We work on an "effectively elliptic" setting, that is, we require that the range of the covariance operator contains the unstable directions. We exploit the generalized asymptotic coupling techniques of [12] and [16], used by these authors for the stochastic Navier-Stokes equations with additive noise. Here, we show how these methods are flexible enough to deal with multiplicative noise as well. A crucial role in our argument is played by the Foias-Pro di estimate in expected value, which has a different form (exponential or polynomial decay) according to the growth condition of the multiplicative noise.