Hausdorff Sub-norm Spaces and Continuity of Random Attractors for Bi-stochastic g-Navier–Stokes Equations with Respect to Tempered Forces.

We study the continuity of pullback random attractors A λ , where the parameter belongs to a complete metric space Λ . By a Hausdorff sub-norm space, we mean the collection of all nonempty compact subsets of the state space, equipped by the Hausdorff metric as well as a sub-norm. Under weaker conditions, we prove that the binary map (λ , s) → A λ (s , θ s ω) is continuous at all points of Λ ∗ × R with respect to the Hausdorff metric, where Λ ∗ is residual and dense in Λ , and that the binary map is continuous under the sub-norm if and only if each fibre of attractors is a point. The proofs of these results are based on the discussion of continuous operators on the Hausdorff sub-norm space as well as the theory of Baire category. For the g-Navier–Stokes equation driven by random density, stochastic noise and time-dependent forces, we establish the residual continuity and full upper semi-continuity of pullback random attractors on the Fréchet space formed from all backward tempered forces. [ABSTRACT FROM AUTHOR]