The Lake equation as a supercritical mean-field limit

TL;DR

This paper derives a generalized Lake equation as a supercritical mean-field limit of particle systems with Coulomb interactions, establishing its universality and mathematical foundation through advanced analytical methods.

Contribution

It introduces a new supercritical mean-field limit leading to a generalized Lake equation, revealing its universal dependence on particle density.

Findings

Derivation of a $ ext{d}$-dimensional Lake equation from particle systems.

Establishment of the Lake equation's universality in this scaling limit.

Application of a modulated-energy method and obstacle problem regularity theory.

Abstract

We study so-called supercritical mean-field limits of systems of trapped particles moving according to Newton's second law with either Coulomb/super-Coulomb or regular interactions, from which we derive a $\mathsf{d}$-dimensional generalization of the Lake equation, which coincides with the incompressible Euler equation in the simplest setting, for monokinetic data. This supercritical mean-field limit may also be interpreted as a combined mean-field and quasineutral limit, and our assumptions on the rates of these respective limits are shown to be optimal. Our work provides a mathematical basis for the universality of the Lake equation in this scaling limit -- a new observation -- in the sense that the dependence on the interaction and confinement is only through the limiting spatial density of the particles. Our proof is based on a modulated-energy method and takes advantage of regularity theory for the obstacle problem for the fractional Laplacian.