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

Abstract

The aim of this paper is to study the long time behavior of the following stochastic 3D Navier–Stokes–Voigt equationut−νΔu−α2Δut+(u⋅∇)u+∇p=g(x)+εhdωdt in an arbitrary (bounded or unbounded) domain satisfying the Poincaré inequality. By famous J. Ball's energy equation method, we obtain a unique random attractor Aε for the random dynamical system generated by the equation. Moreover, we prove that the random attractor Aε tends to the global attractor A0 of the deterministic equation in the sense of Hausdorff semi-distance as ε→0.