Bi-spatial pullback random attractors of stochastic <i>ϱ</i>-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 ϱ-Navier-Stokes equations driven by additive noise. The existence and uniqueness of pullback random attractors of the equations are established in an appropriate ϱ-weighted L2-subspace Hϱ. This attractor is proved to be a bi-spatial attractor that is compact, measurable in another ϱ-weighted H01-subspace Vϱand attracts all random subsets of Hϱunder the topology of Vϱ. 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 Vϱin order to overcome the lack of higher regularity than Vϱand the almost sure non-differentiability of the sample paths of the Wiener process. The results of this article are new even when the stochastic ϱ-Navier-Stokes equation reduces to the standard stochastic Navier-Stokes equation.