On the vanishing viscosity limit of statistical solutions of the incompressible Navier-Stokes equations

Abstract

We study statistical solutions of the incompressible Navier--Stokes equation and their vanishing viscosity limit. We show that a formulation using correlation measures, which are probability measures accounting for spatial correlations, and moment equations is equivalent to statistical solutions in the Foiac{s}--Prodi sense. Under the assumption of weak scaling, a weaker version of Kolmogorov's self-similarity at small scales hypothesis that allows for intermittency corrections, we show that the limit is a statistical solution of the incompressible Euler equations. To pass to the limit, we derive a K'arm'an--Howarth--Monin relation for statistical solutions and combine it with the weak scaling assumption and a compactness theorem for correlation measures.