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

Abstract

We consider a stochastic model which describes the motion of a 2D incompressible fluid in a unbounded domain with viscosity and memory effects. This model is different from the classical stochastic Navier-Stokes-Voigt equations due to the absence of the Voigt term $ -\alpha \Delta u_{t}$, and has a much weaker dissipation than the usual Navier-Stokes-Voigt model since only the memory viscoelasticity is present. We are interested in the global well-posedness and long-time behaviors of this model. We first investigate the well-posedness by using the classical Faedo-Galerkin method. Unlike the general method of energy estimate, we then split the solution into two parts and get the low-order and high-order uniform estimates, respectively. Based on the uniform estimates of far-field values of solutions, we further prove the existence and uniqueness of random attractors in unbounded domains with a constructed compact subspace corresponding to memory. Finally, we give the upper semicontinuity of the attractors when stochastic perturbation approaches to zero.