Free Turbulence on R^3 and T^3

TL;DR

This paper introduces a mathematical framework for analyzing turbulence in Newtonian fluids on R^3 and T^3, bridging physics and mathematics by studying spectral properties of turbulence through PDE solutions.

Contribution

It provides a deterministic dictionary for spectral analysis of turbulence, validating classical physics statements and exploring turbulence's compatibility with Navier-Stokes solution smoothness.

Findings

Validated classical turbulence statements mathematically.

Established a deterministic spectral analysis framework.

Explored turbulence's relation to Navier-Stokes smoothness.

Abstract

The hydrodynamics of Newtonian fluids has been the subject of a tremendous amount of work over the past eighty years, both in physics and mathematics. Sadly, however, a mutual feeling of incomprehension has often hindered scientific contacts. This article provides a dictionary that allows mathematicians to define and study the spectral properties of Kolmogorov-Obukov turbulence in a simple deterministic manner. In other words, this approach fits turbulence into the mathematical framework of studying the qualitative properties of solutions of PDEs, independently from any a-priori model of the structure of the flow. To check that this new approach is correct, this article proves some of the classical statements that can be found in physics textbooks. This is followed by an investigation of the compatibility between turbulence and the smoothness of solutions of Navier-Stokes in 3D, which was the initial motivation of this study.