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

TL;DR

This paper investigates the behavior of statistical solutions of the incompressible Navier-Stokes equations as viscosity vanishes, establishing conditions under which these solutions converge to solutions of the Euler equations.

Contribution

It introduces a correlation measure framework for statistical solutions and proves their convergence to Euler solutions under weak scaling assumptions.

Findings

Correlation measures are equivalent to Foias-Prodi statistical solutions.

Under weak scaling, Navier-Stokes statistical solutions converge to Euler solutions.

Derived a Karman-Howarth-Monin relation for statistical solutions.

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 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.