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We consider a stochastic model which describes the motion of a 2Dincompressible fluid in a unbounded domain with viscosity and memory effects.This model is different from the classical stochastic Navier-Stokes-Voigtequations due to the absence of the Voigt term $ -\alpha \Delta u_{t}$, and hasa much weaker dissipation than the usual Navier-Stokes-Voigt model since onlythe memory viscoelasticity is present. We are interested in the globalwell-posedness and long-time behaviors of this model. We first investigate thewell-posedness by using the classical Faedo-Galerkin method. Unlike the generalmethod of energy estimate, we then split the solution into two parts and getthe low-order and high-order uniform estimates, respectively. Based on theuniform estimates of far-field values of solutions, we further prove theexistence and uniqueness of random attractors in unbounded domains with aconstructed compact subspace corresponding to memory. Finally, we give theupper semicontinuity of the attractors when stochastic perturbation approachesto zero.

Asymptotic behavior for 2D stochastic Navier-Stokes equations with memory in unbounded domains