Qualitative properties of stationary measures for three-dimensional Navier–Stokes equations

The paper is devoted to studying the distribution of stationary solutions for 3D Navier–Stokes equations perturbed by a random force. Under a non-degeneracy assumption, we show that the support of such a distribution coincides with the entire phase space, and its finite-dimensional projections are minorised by a measure possessing an almost surely positive smooth density with respect to the Lebesgue measure. Similar assertions are true for weak solutions of the Cauchy problem with a regular initial function. The results of this paper were announced in the short note [A. Shirikyan, Controllability of three-dimensional Navier–Stokes equations and applications, in: Sémin. Équ. Dériv. Partielles, 2005–2006, École Polytech., Palaiseau, 2006].