Dynamics of Partial Differential Equations

Volume 7 (2010)

Number 2

Free Turbulence on R³ and T³

Pages: 107 – 160

DOI: http://dx.doi.org/10.4310/DPDE.2010.v7.n2.a1

Author

François Vigneron (Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées, UMR 8050 du CNRS, Créteil, France)

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 (including the author) 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 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.

Keywords

turbulence, Navier-Stokes equations, Kolmogorov spectrum

2010 Mathematics Subject Classification

Primary 76-xx. Secondary 35-xx.

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