Statistical solutions of the Navier-Stokes equations on the phase space of vorticity and the inviscid limits

Using the methods of Foias [Sem. Math. Univ. Padova 48, 219-343 (1972); 49, 9-123 (1973)] and Vishik-Fursikov [Mathematical Problems of Statistical Hydromechanics (Kluwer, Dordrecht, 1988)], we prove the existence and uniqueness of both spatial and space-time statistical solutions of the Navier-Stokes equations on the phase space of vorticity. Here the initial vorticity is in Yudovich space and the initial measure has finite mean enstrophy. We show under further assumptions on the initial vorticity that the statistical solutions of the Navier-Stokes equations converge weakly and the inviscid limits are the corresponding statistical solutions of the Euler equations.