Equivalence of invariant measures and stationary statistical solutions for theautonomous globally modified Navier-Stokes equations

A new proof of existence of solutions for the three dimensionalsystem of globally modified Navier-Stokes equations introduced in [3] by Caraballo, Kloeden and Real is obtained using a smoother Galerkin scheme. This isthen used to investigate the relationship between invariant measures and statistical solutions of this system in the case of temporally independent forcingterm. Indeed, we are able to prove that a stationary statistical solution is alsoan invariant probability measure under suitable assumptions.