The Lake equation as a supercritical mean-field limit
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910578025431040 |
|---|---|
| author | Rosenzweig, Matthew Serfaty, Sylvia |
| author_facet | Rosenzweig, Matthew Serfaty, Sylvia |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14642 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Lake equation as a supercritical mean-field limit Rosenzweig, Matthew Serfaty, Sylvia Analysis of PDEs Mathematical Physics Plasma Physics 35Q35, 35Q70, 35Q83, 35Q82, 82C21, 82C70, 82D10 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. |
| title | The Lake equation as a supercritical mean-field limit |
| topic | Analysis of PDEs Mathematical Physics Plasma Physics 35Q35, 35Q70, 35Q83, 35Q82, 82C21, 82C70, 82D10 |
| url | https://arxiv.org/abs/2408.14642 |