2020/10/16 by Matthew Rosenzweig, Rosenzweig, Matthew
Mathematics · Physics and Astronomy · #35Q35 #35Q70 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2010.10009
openalex publication_date 2020/10/16 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We consider first-order conservative systems of particles with binary Coulomb\ninteractions in the mean-field scaling regime in dimensions d\≥ 3. We show\nthat if at some time, the associated sequence of empirical measures converges\nin a suitable sense to a probability measure with bounded density \ω0 as\nthe number of particles N\→\∞, then the sequence converges for\nshort times in the weak-* topology for measures to the unique solution of the\nmean-field PDE with initial datum \ω0. This result extends our previous\nwork arXiv:2004.04140 for point vortices (i.e. d=2). In contrast to the\nprevious work arXiv:1803.08345, our theorem only requires the limiting measure\nbelong to a scaling-critical function space for the well-posedness of the\nmean-field PDE, in particular requiring no regularity. Our proof is based on a\ncombination of the modulated-energy method of Serfaty and a novel mollification\nargument first introduced by the author in arXiv:2004.04140.\n