On Euler equations and statistical physics. (English) Zbl 1024.35085

Cattedra Galileiana. 1. Pisa: Scuola Normale Superiore, Classe di Scienze, vi, 74 p. (1998).
This book contains a series of lectures given by the author at the Scuola Normale Superiore di Pisa (Italy). It deals with two mathematical topics related to the Euler equations and turbulence.
The contents are as follows. Chapter I is devoted to the presentation of the problems. Chapter II contains a brief description of facts concerning the Euler equations \(\partial_tu+u\cdot\nabla u+\nabla p=f\), \(\nabla\cdot u=0\); here \(u\) is the velocity vector field, \(p\) the pressure and \(f\) the external force. Chapter III deals with two-dimensional flows. The chapter begins with a system of \(N\) point vortices interacting with the natural Coulomb-like force and the associated Gibbs measure.
In Chapter IV the limit of the measure as \(N\to\infty\) is investigated. It is shown that the measure tends to a stationary measure which is concentrated on very particular solutions of the two-dimensional Euler equations. These stationary solutions are characterized by a variational principle which yields a classical semilinear elliptic equation, the so-called mean field equation. Chapter V is devoted to the study of the mean field equation. Chapter VI deals with an extension of the preceding theory to three-dimensional flows in the case of thin, slowly-varying, nearly parallel interacting vortex filaments.


35Q30 Navier-Stokes equations
76M35 Stochastic analysis applied to problems in fluid mechanics
82D15 Statistical mechanics of liquids