This paper is devoted to the study of long term behavior of the twodimensional random Navier-Stokes equations driven by colored noise defined in bounded and unbounded domains. We prove the existence and uniqueness of pullback random attractors for the equations with Lipschitz diffusion terms. In the case of additive noise, we show the upper semi-continuity of these attractors when the correlation time of the colored noise approaches zero. When the equations are defined on unbounded domains, we establish the pullback asymptotic compactness of the solutions by Ball's idea of energy equations in order to overcome the difficulty introduced by the noncompactness of Sobolev embeddings.Given τ ∈ R and ω ∈ Ω, consider the following non-autonomous two-dimensional random Navier-Stokes equations driven by colored noise:along with homogeneous Dirichlet boundary condition, where ν is a positive constant, f is a given function defined on R × O, G is a nonlinear function satisfying certain conditions, and ζ δ (θ t ω) is the colored noise with correlation time δ > 0.The Navier-Stokes equations are used to model the flow of an incompressible viscous fluid of constant density enclosed in a region O with rigid boundary ∂O, where u(t, x) ∈ R 2 and p(t, x) ∈ R are, respectively, the velocity and the pressure of the fluid at point x ∈ O and time t > τ , ν > 0 is the kinematic viscosity of the fluid and f = f (t, x) ∈ R 2 is the time dependent external force. The existence of global attractors for the Navier-Stokes equations has been extensively studied in the literature, see, e.g., [5,7,12,13,19,34,46,48,51] for the deterministic case, and [11,22,24,56] for the stochastic case. More work on random attractors can be found in [8,9,14,15,16,17,20,21,22,24,25,26,27,29,37,42,47,53,54] for the autonomous stochastic equations; and in [18,23,30,31,55,57,58] for the nonautonomous stochastic systems. In this paper, we will investigate pullback random attractors of the non-autonomous random system (2) driven by colored noise.The random system (2) can be considered as an approximation of the following Stratonovich stochastic equations: ∂u ∂t − ν∆u + (u · ∇)u = f (t, x) − ∇p + G(t, x, u) • dW dt , x ∈ Q, t > τ, div u = 0, x ∈ Q, t > τ,

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We consider the long-time behavior for stochastic 2D nematic liquid crystals flows with the velocity field perturbed by an additive noise. The presence of the noises destroys the basic balance law of the nematic liquid crystals flows, so we can not follow the standard argument to obtain uniform a priori estimates for the stochastic flow even in the weak solution space under non-periodic boundary conditions. To overcome the difficulty we use a new technique some kind of logarithmic energy estimates to obtain the uniform estimates which improve the previous result for the orientation field that grows exponentially w.r.t.time t. Considering the existence of random attractor, the common method is to derive uniform a priori estimates in functional space which is more regular than the solution space. We can follow the common method to prove the existence of random attractor in the weak solution space. However, if we consider the existence of random attractor in the strong solution space, it is very difficult and very complicated for such highly nonlinear stochastic system with no basic balance law and non-periodic boundary conditions. Here, we use a compactness arguments of the stochastic flow and regularity of the solutions to the stochastic model to obtain the existence of the random attractor in the strong solution space, which implies the support of the invariant measure lies in a more regular space. As far as we know, it is the first article to attack the long-time behavior of stochastic nematic liquid crystals.

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