Existence, uniqueness and asymptotic stability of invariant measures for the stochastic Allen-Cahn-Navier-Stokes system with singular potential

We study the long-time behaviour of a stochastic Allen-Cahn-Navier-Stokes system modelling the dynamics of binary mixtures of immiscible fluids. The model features two stochastic forcings, one on the velocity in the Navier-Stokes equation and one on the phase variable in the Allen-Cahn equation, and includes the thermodynamically-relevant Flory-Huggins logarithhmic potential. We first show existence of ergodic invariant measures and characterise their support by exploiting ad-hoc regularity estimates and suitable Feller-type and Markov properties. Secondly, we prove that if the noise acting in the Navier-Stokes equation is non-degenerate along a sufficiently large number of low modes, and the Allen-Cahn equation is highly dissipative, then the stochastic flow admits a unique invariant measure and is asymptotically stable with respect to a suitable Wasserstein metric.