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arXiv · 2408.14642

The Lake equation as a supercritical mean-field limit

Matthew Rosenzweig·Sylvia Serfaty
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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.

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Matthew Rosenzweig, Sylvia Serfaty. 2024-08-26. The Lake equation as a supercritical mean-field limit. https://arxiv.org/abs/2408.14642

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