Mean field limit of interacting filaments for 3D Euler equations

TL;DR

This paper demonstrates that smooth solutions to the 3D Euler equations can be derived as a mean field limit of interacting vortex filaments modeled by 1-currents, extending previous work with mollified relations.

Contribution

It establishes a rigorous connection between vortex filament dynamics and the 3D Euler equations using the framework of 1-currents, advancing the mathematical understanding of fluid flow modeling.

Findings

3D Euler solutions obtained as mean field limits of vortex filaments

Use of 1-currents to describe filament interactions

Extension of previous mollified PDE results to true Euler equations

Abstract

The 3D Euler equations, precisely local smooth solutions of class $H^s$ with $s>5/2$, are obtained as a mean field limit of finite families of interacting curves, the so called vortex filaments, described by means of the concept of $1$-currents. This work is a continuation of the a previous work, where a preliminary result in this direction was obtained, with the true Euler equations replaced by a vector valued non linear PDE with a mollified Biot-Savart relation.