The Lake equation as a supercritical mean-field limit
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2024
- DOI:
- 10.48550/arXiv.2408.14642
- arXiv:
- arXiv:2408.14642
- Bibcode:
- 2024arXiv240814642R
- Keywords:
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- Mathematics - Analysis of PDEs;
- Mathematical Physics;
- Physics - Plasma Physics;
- 35Q35;
- 35Q70;
- 35Q83;
- 35Q82;
- 82C21;
- 82C70;
- 82D10
- E-Print:
- 41 pages