Ergodicity for the Two-Dimensional Stochastic Navier-Stokes Equations with Shear-Thickening Viscosity

The two-dimensional stochastic Navier-Stokes equations with a shear-thickening viscosity driven by multiplicative noise are studied. After the problem is formulated as an abstract stochastic evolution equation on suitable function spaces, the existence and uniqueness of strong solutions are established through local monotonicity methods. Moment estimates are then obtained and used to prove the existence of an ergodic invariant probability measure via the Krylov-Bogolyubov technique. The uniqueness of the ergodic invariant measure is subsequently established by constructing a generalized coupling through a controlled auxiliary equation. Finally, the results are extended to other types of viscosities as well as to different stochastic hydrodynamical models.