LIMIT BEHAVIOUR OF A DENSE COLLECTION OF VORTEX FILAMENTS

Abstract

The understanding of vortex structures in 3D turbulent fluids is a basic problem. One of the questions is whether some large scale structure can emerge as the macroscopic result of the self-organization of small scale vortex filaments, similarly to the 2D case of point vortices. This paper gives a first step in this direction: a mean field result is proved for a dense collection of vortex filaments. The filaments considered here are described by stochastic processes, including Brownian motion. Under a special rescaling of the energy, a mean field result is proved for a model of 3D vortex filaments described by stochastic processes, including Brownian motion, Brownian bridge, fractional Brownian motion and other semimartingales. Propagation of chaos, variational characterization of the limit Gibbs density h and an equation for h are proved.