An example elucidating the mathematical situation in the statistical non-uniqueness problem of turbulence

Abstract

An instructive example is presented to elucidate the mathematical situation in the non-uniqueness problem of the infinite Friedmann-Keller hierarchy of equations for all multi-point moments within the theory of spatially unbounded Navier-Stokes turbulence. It is shown that the non-uniqueness problem of the Friedmann-Keller hierarchy emerges from the property that the system of equations is defined forward recursively. As a result, this system does not possess a unique general solution, even when the complete infinite system is formally considered. That is, even when imposing a sufficient number of initial conditions to this infinite system, it still does not provide a unique solution. This finding is supported by a Lie-group invariance analysis, in that the imposed example analogous to the Friedmann-Keller hierarchy admits an unclosed Lie algebra which allows for infinitely many functionally different equivalence transformations which all can be made compatible with any specifically chosen initial condition. Hence, if no prior modelling assumptions are made to close the Friedmann-Keller system of equations, the existence of an invariant solution within such a forward recursively defined system is then without value, since it just represents an arbitrary solution among infinitely many other, equally privileged invariant solutions.