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We consider first-order conservative systems of particles with binary Coulombinteractions in the mean-field scaling regime in dimensions $d\geq 3$. We showthat if at some time, the associated sequence of empirical measures convergesin a suitable sense to a probability measure with bounded density $\omega^0$ asthe number of particles $N\rightarrow\infty$, then the sequence converges forshort times in the weak-* topology for measures to the unique solution of themean-field PDE with initial datum $\omega^0$. This result extends our previouswork arXiv:2004.04140 for point vortices (i.e. $d=2$). In contrast to theprevious work arXiv:1803.08345, our theorem only requires the limiting measurebelong to a scaling-critical function space for the well-posedness of themean-field PDE, in particular requiring no regularity. Our proof is based on acombination of the modulated-energy method of Serfaty and a novel mollificationargument first introduced by the author in arXiv:2004.04140.

Mean-Field Convergence of Systems of Particles with Coulomb Interactions in Higher Dimensions without Regularity