Properties of stationary statistical solutions of the three-dimensional Navier-Stokes equations

TL;DR

This paper investigates stationary statistical solutions of the 3D Navier-Stokes equations, focusing on their properties, support, and relation to the weak global attractor, with implications for understanding turbulent flows.

Contribution

It introduces a modified definition of stationary statistical solutions and analyzes their support, regularity, and recurrence properties within the 3D Navier-Stokes framework.

Findings

Stationary solutions are supported on the weak global attractor.

They are carried by a more regular subset containing Leray-Hopf solutions.

Recurrence results for these measures are established.

Abstract

The stationary version of a modified definition of statistical solution for the three-dimensional incompressible Navier-Stokes equations introduced in a previous work is investigated. Particular types of such stationary statistical solutions and their analytical properties are addressed. Results on the support and carriers of these stationary statistical solutions are also given, showing in particular that they are supported on the weak global attractor and are carried by a more regular part of the weak global attractor containing Leray-Hopf weak solutions which are locally strong solutions. Two recurrence-type results related to these measures are also proved.