{
  "id": "2010.10009",
  "version": "v1",
  "published": "2020-10-16T18:14:16.000Z",
  "updated": "2020-10-16T18:14:16.000Z",
  "title": "Mean-Field Convergence of Systems of Particles with Coulomb Interactions in Higher Dimensions without Regularity",
  "authors": [
    "Matthew Rosenzweig"
  ],
  "comment": "32 pages. arXiv admin note: text overlap with arXiv:2004.04140",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider first-order conservative systems of particles with binary Coulomb interactions in the mean-field scaling regime in dimensions $d\\geq 3$. We show that if at some time, the associated sequence of empirical measures converges in a suitable sense to a probability measure with bounded density $\\omega^0$ as the number of particles $N\\rightarrow\\infty$, then the sequence converges for short times in the weak-* topology for measures to the unique solution of the mean-field PDE with initial datum $\\omega^0$. This result extends our previous work arXiv:2004.04140 for point vortices (i.e. $d=2$). In contrast to the previous work arXiv:1803.08345, our theorem only requires the limiting measure belong to a scaling-critical function space for the well-posedness of the mean-field PDE, in particular requiring no regularity. Our proof is based on a combination of the modulated-energy method of Serfaty and a novel mollification argument first introduced by the author in arXiv:2004.04140.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-16T18:14:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q35",
      "35Q70"
    ],
    "keywords": [
      "higher dimensions",
      "mean-field convergence",
      "regularity",
      "novel mollification argument first",
      "mean-field pde"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}