MeanField Convergence of Systems of Particles with Coulomb Interactions in Higher Dimensions without Regularity
Abstract
We consider firstorder conservative systems of particles with binary Coulomb interactions in the meanfield 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 meanfield 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 scalingcritical function space for the wellposedness of the meanfield PDE, in particular requiring no regularity. Our proof is based on a combination of the modulatedenergy method of Serfaty and a novel mollification argument first introduced by the author in arXiv:2004.04140.
 Publication:

arXiv eprints
 Pub Date:
 October 2020
 DOI:
 10.48550/arXiv.2010.10009
 arXiv:
 arXiv:2010.10009
 Bibcode:
 2020arXiv201010009R
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics;
 35Q35;
 35Q70
 EPrint:
 32 pages. arXiv admin note: text overlap with arXiv:2004.04140