Bi-spatial pullback random attractors of stochastic ϱ-Navier-Stokes equations: Existence, regularity and finite fractal dimension

This paper is concerned with the existence, regularity as well as finite fractal dimension of pullback random attractors of a wide class of non-autonomous stochastic & rhov;-Navier-Stokes equations driven by additive noise. The existence and uniqueness of pullback random attractors of the equations are established in an appropriate & rhov;-weighted L-2-subspace H & rhov;. This attractor is proved to be a bi-spatial attractor that is compact, measurable in another & rhov;-weighted H-0(1)-subspace V & rhov;and attracts all random subsets of H & rhov;under the topology of V & rhov;. The finite fractal dimension of the bi-spatial random attractors is also derived without differentiating the system with respect to time. A spectrum decomposition method is employed to derive the pullback flattening properties (see Kloeden and Langa [Proc. R. Soc. A 463, 163-181(2007)]) of the solutions in V-& rhov; in order to overcome the lack of higher regularity than V & rhov;and the almost sure non-differentiability of the sample paths of the Wiener process. The results of this article are new even when the stochastic & rhov;-Navier-Stokes equation reduces to the standard stochastic Navier-Stokes equation.