Construction of a turbulent measure for the set of Navier-Stokes equations

The previously reported statistical solution (Foias, 1972) for a measure, from which an average overall solution of the Navier-Stokes equations in the theory of turbulence is derived, is generalized. The extended formalism of the statistical solution permits the averaging of functionals which are nonlocal in time from the solution of the Navier-Stokes equations for a given set of initial data and outside forces. The pure measure satisfies an expression which coincides in essence with a generalization of Hopf's equation for the characteristic functional proposed by Leuvis and Kraichenan (1962).