Dynamics of stochastic three dimensional Navier–Stokes–Voigt equations on unbounded domains

The aim of this paper is to study the long time behavior of the following stochastic 3D Navier–Stokes–Voigt equation u t − ν Δ u − α 2 Δ u t + ( u ⋅ ∇ ) u + ∇ p = g ( x ) + ε h d ω d t in an arbitrary (bounded or unbounded) domain satisfying the Poincaré inequality. By famous J. Ballʹs energy equation method, we obtain a unique random attractor A ε for the random dynamical system generated by the equation. Moreover, we prove that the random attractor A ε tends to the global attractor A 0 of the deterministic equation in the sense of Hausdorff semi-distance as ε → 0 .