Mean field limit of interacting filaments and vector valued non linear PDEs

TL;DR

This paper establishes the mean field limit for families of interacting curves, modeled as 1-currents, converging to a nonlinear PDE, with applications inspired by vortex filaments in turbulence.

Contribution

It introduces a novel framework for analyzing the mean field limit of interacting filaments via 1-currents and links this to a nonlinear PDE, extending vortex filament models.

Findings

Convergence of 1-currents to a mean field current as N approaches infinity.

Interaction with the mean field current in the limit simplifies the dynamics.

Independence property of curves preserved if initially independent.

Abstract

Families of $N$ interacting curves are considered, with long range, mean field type, interaction. A family of curves defines a 1-current, concentrated on the curves, analog of the empirical measure of interacting point particles. This current is proved to converge, as $N$ goes to infinity, to a mean field current, solution of a nonlinear, vector valued, partial differential equation. In the limit, each curve interacts with the mean field current and two different curves have an independence property if they are independent at time zero. This set-up is inspired from vortex filaments in turbulent fluids, although for technical reasons we have to restrict to smooth interaction, instead of the singular Biot-Savart kernel. All these results are based on a careful analysis of a nonlinear flow equation for 1-currents, its relation with the vector valued PDE and the continuous dependence on the initial conditions.