The lake equation as a supercritical mean-field limit

We study so-called supercritical mean-field limits of trapped particle systems moving according to Newton’s second law with either Coulomb/super-Coulomb or regular interactions, from which we derive a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="sans-serif">d</mml:mi> </mml:math> -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.