We consider the 3D Navier--Stokes systems with randomly rapidly oscillating right--hand sides. Under the assumption that the random functions are ergodic and statistically homogeneous in space variables or in time variables we prove that the trajectory attractors of these systems tend to the trajectory attractors of homogenized 3D Navier--Stokes systems whose right--hand sides are the average of the corresponding terms of the original systems. We do not assume that the Cauchy problem for the considered 3D Navier--Stokes systems is uniquely solvable.
In 2006, Gorodetski proved that central fibres of perturbed skew products are Hölder continuous with respect to the base point. In this paper, we give an explicit estimate of this Hölder exponent. Moreover, we extend Gorodetski's result from the case when the fibre maps are close to the identity to a much wider class of maps that satisfy the so-called modified dominated splitting condition. In many cases (for example, in the case of skew products over the solenoid or over linear Anosov diffeomorphisms of the torus), the Hölder exponent is close to 1. This allows one to overcome the so-called Fubini nightmare, in some sense. Namely, we prove that the union of central fibres that are strongly atypical from the point of view of ergodic theory, has Lebesgue measure zero despite the lack of absolute continuity of the holonomy map for the central foliation. This result is based on a new kind of ergodic theorem, which we call special. To prove our main result, we revisit the theory of Hirsch, Pugh and Shub, and estimate the contraction constant of the graph transform map. © 2012 IOP Publishing Ltd & London Mathematical Society.