Universality of attractors at weak dissipation and particles distribution in turbulence

Abstract

We study stationary solutions to the continuity equation for weakly compressible flows. These describe non-equilibrium steady states of weakly dissipative dynamical systems. Compressibility is a singular perturbation that changes the steady state density from a constant "microcanonical" distribution into a singular multifractal measure supported on the "strange attractor". We introduce a representation of the latter and show that the space-averaged properties are described universally by a log-normal distribution determined by a single structure function. The spectrum of fractal dimensions is derived. Application to the problem of distribution of particles in turbulence gives testable predictions for real turbulence and stresses the role of pressure fluctuations.