Moment theory for the Navier–Stokes equations with a random right side

A theory of the moments of a statistical solution of the Navier–Stokes equations is constructed. The Cauchy problem for an infinite chain of equations which these moments satisfy is written out. Uniqueness of the solution of this Cauchy problem in appropriate function spaces is proved. The solution is not assumed to be positive definite, i.e., it may not be a collection of moments of the statistical solution. The concept of a statistical solution of the Navier-Stokes equations with a random right side is introduced, an equation for the statistical solution is derived, and the connection between this equation and the chain of moment equations is established. The problem of closure of the chain of moment equations is solved in the case of large Reynolds numbers, i.e., a sequence of extremal problems $\mathfrak A^N$ is constructed such that 1) the number of desired functions $M^N=\{M^N_{k,n}\}$ of the extremal problem $\mathfrak A^N$ is finite (and equal to $(N+1)N/2$), and 2) the solution $M^N$ of the problem $\mathfrak A^N$ approximates the solution of the Cauchy problem for the chain of moment equations: $M^N\to M$ as $N\to\infty$. The results are used to solve the problem of closure of the chain of Friedman-Keller moment equations corresponding to the three-dimensional Navier-Stokes system with zero right side, for large Reynolds numbers.