On the Reynolds time-averaged equations and the long-time behavior of Leray-Hopf weak solutions, with applications to ensemble averages

We consider the three dimensional Navier-Stokes equations and we prove that for Leray-Hopf weak solutions it is possible to characterize (up to sub-sequences) their long-time averages, which satisfy the Reynolds averaged equations. Moreover, we show the validity of the Boussinesq hypothesis, without any additional assumption. Finally, in the last section we consider ensemble averages of solutions and we prove that the fluctuations continue to have a dissipative effect on the mean flow.