Kolmogorov's dissipation number and the number of degrees of freedom for the 3D Navier-Stokes equations

TL;DR

This paper establishes a mathematical link between Kolmogorov's turbulence theory and the degrees of freedom in 3D Navier-Stokes flows by proving the existence of a time-dependent determining wavenumber bounded by Kolmogorov's dissipation number.

Contribution

It proves the existence of a time-dependent determining wavenumber for 3D Navier-Stokes equations, bounded by Kolmogorov's dissipation number, extending prior 2D results.

Findings

Existence of a time-dependent determining wavenumber for 3D flows.

Bounded time average of the wavenumber by Kolmogorov's dissipation number.

Applicable to solutions on the global attractor with moderate intermittency.

Abstract

Kolmogorov's theory of turbulence predicts that only wavenumbers below some critical value, called Kolmogorov's dissipation number, are essential to describe the evolution of a three-dimensional fluid flow. A determining wavenumber, first introduced by Foias and Prodi for the 2D Navier-Stokes equations, is a mathematical analog of Kolmogorov's number. The purpose of this paper is to prove the existence of a time-dependent determining wavenumber for the 3D Navier-Stokes equations whose time average is bounded by Kolmogorov's dissipation wavenumber for all solutions on the global attractor whose intermittency is not extreme.