Space-Time Statistical Solutions of the Navier-Stokes Equations for Arbitrary Reynolds Numbers

Abstract

We construct a space-time statistical solution for the Navier-Stokes system in a bounded domain and, also, for an operator generalization of this system. At t = 0 we define the initial measure μ, i.e. the measure on the space of initial values, and we suppose that the mean energy of initial values with respect to the measure μ is finite. From any such measure μ a statistical solution P is obtained whose restriction at t=0 coincides with μ. When the uniqueness theorem for individual solutions occurs the uniqueness theorem for statistical solutions is proved.