Determining modes for the 3D Navier-Stokes equations

TL;DR

This paper introduces a new concept of a determining wavenumber for the 3D Navier-Stokes equations, which remains bounded on the global attractor despite potential blow-ups, linking it to classical turbulence scales.

Contribution

It defines a solution-dependent determining wavenumber for the 3D Navier-Stokes equations with bounded time average on the attractor, connecting it to Kolmogorov's and Grashof's scales.

Findings

The determining wavenumber is finite on the global attractor.

Its time average is uniformly bounded for all solutions.

The bound relates to classical turbulence scales.

Abstract

We introduce a determining wavenumber for the forced 3D Navier-Stokes equations (NSE) defined for each individual solution. Even though this wavenumber blows up if the solution blows up, its time average is uniformly bounded for all solutions on the global attractor. The bound is compared to Kolmogorov's dissipation wavenumber and the Grashof constant.