On Euler equation and statistical physics (Publications of the Scuola Normale Superiore)

These are a series of lectures given at the 1997 Cattedra Galileiana, at Scuola Normale Superiore. We discuss in these notes two mathematical topics related to Euler equations and turbulence. The general goal is to describe mathematically some coherent structures observed in turbulent flows. The first problem we study concerns 2-dimensional flows: we begin with a system of N points vortices interacting with the natural Coulomb-like force and we consider the associated Gibbs measure. We then show how the measure goes, as N goes to infinity, to a stationary measure which is concentrated on very particular stationary solutions of the two-dimensional Euler equations. We shall study the mathematical properties of these solutions. The second topic we consider is an extension of the previous one to 3-dimensional flows.