Lie symmetries of the lundgren−monin−novikov hierarchy

Abstract

In this work we consider the statistical approach to turbulence represented by the Lundgren−Monin−Novikov (LMN) hierarchy of equations for the probability density functions (PDFs). After a review of the properties that the PDFs have to satisfy, we first show the basic Galilean invariance of the LMN equations; then we discuss the extended Galilean one and finally we present a transformation of the PDFs and examine the conditions which have to be satisfied so that this transformation represents a symmetry of the LMN hierarchy corresponding in the Multi−Point Correlation (MPC) approach to one of the so called statistical symmetries found using the Lie symmetry machinery in [6] for the infinite hierarchy of equations satisfied by the correlation functions from which some decay exponents of turbulent scaling law could be worked out.