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An instructive example is presented to elucidate the mathematical situationin the non-uniqueness problem of the infinite Friedmann-Keller hierarchy ofequations for all multi-point moments within the theory of spatially unboundedNavier-Stokes turbulence. It is shown that the non-uniqueness problem of theFriedmann-Keller hierarchy emerges from the property that the system ofequations is defined forward recursively. As a result, this system does notpossess a unique general solution, even when the complete infinite system isformally considered. That is, even when imposing a sufficient number of initialconditions to this infinite system, it still does not provide a uniquesolution. This finding is supported by a Lie-group invariance analysis, in thatthe imposed example analogous to the Friedmann-Keller hierarchy admits anunclosed Lie algebra which allows for infinitely many functionally differentequivalence transformations which all can be made compatible with anyspecifically chosen initial condition. Hence, if no prior modelling assumptionsare made to close the Friedmann-Keller system of equations, the existence of aninvariant solution within such a forward recursively defined system is thenwithout value, since it just represents an arbitrary solution among infinitelymany other, equally privileged invariant solutions.

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