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

Ulrik Skre Fjordholm, Siddhartha Mishra, Franziska Weber

Published 2021-10-10, updated 2022-03-22Version 2

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 Foias-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 Karman-Howarth-Monin relation for statistical solutions and combine it with the weak scaling assumption and a compactness theorem for correlation measures.