Properties of time-dependent statistical solutions of the three-dimensional Navier-Stokes equations

TL;DR

This paper introduces a new intermediate type of statistical solution for the 3D Navier-Stokes equations, bridging existing concepts and enhancing analytical properties for turbulence modeling.

Contribution

It proposes and analyzes a novel statistical solution type that combines features of previous approaches, improving mathematical rigor in turbulence studies.

Findings

Defined a new statistical solution type for Navier-Stokes equations.

Proved analytical properties of the new solution.

Bridged gap between existing statistical solution concepts.

Abstract

This work is devoted to the concept of statistical solution of the Navier-Stokes equations, proposed as a rigorous mathematical object to address the fundamental concept of ensemble average used in the study of the conventional theory of fully developed turbulence. Two types of statistical solutions have been proposed in the 1970's, one by Foias and Prodi and the other one by Vishik and Fursikov. In this article, a new, intermediate type of statistical solution is introduced and studied. This solution is a particular type of a statistical solution in the sense of Foias and Prodi which is constructed in a way akin to the definition given by Vishik and Fursikov, in such a way that it possesses a number of useful analytical properties.