The rate of convergence of approximations for the closure of the Friedman–Keller chain in the case of large Reynolds numbers

The infinite chain of Friedman–Keller equations is studied that describes the evolution of the entire set of moments of a statistical solution of an abstract analogue of the Navier–Stokes system. The problem of closure of this chain is investigated. This problem consists in constructing a sequence of problems $\mathfrak{A}_N=0$ of $N$ unknown functions whose solutions $M^N=(M_1^N,\dots,M_N^N,0,0,\dots)$ approximate the system of moments $M=(M_1,\dots,M_k,\dots)$ as $N\to+\infty$. The case of large Reynolds numbers is considered. Exponential rate of convergence of $~M^N$ to $M$ as $N\to\infty$ is proved.