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In this paper, we first establish the existence of uniform random attractor for 2D stochastic Navier-Stokes equation in H with deterministic non-autonomous external force being normal in L l o c 2 (R ; V ′) , which is the measurable minimal compact set and uniformly attracts bounded random set in H in the sense of pullback. We also show that uniform random attractor with respect to the deterministic non-autonomous functions belonging to some symbol space coincides with uniform random attractor with respect to the initial time. Then we show that the uniform random attractor for the equation under consideration has regularity property in V when deterministic non-autonomous external force being normal in L l o c 2 (R ; H). [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the asymptotic behavior of solutions of non-autonomous parabolic problems with singular initial data. We first establish the well-posedness of the equation when the initial data belongs to () and (), respectively. When the initial data belongs to , we establish the existence of uniform attractors in for the family of processes with external forces being translation bounded but not translation compact in . When we consider the existence of uniform attractors in , the solution of equation lacks the higher regularity, so we introduce a new type of solution and prove the existence result. For the long time behavior of solutions of the equation in , we only obtain the uniform attracting property in the weak topology. [Copyright &y& Elsevier]