Mean-Field Limits for Some Riesz Interaction Gradient Flows

This paper is concerned with the mean-field limit for the gradient flow evolution of particle systems with pairwise Riesz interactions, as the number of particles tends to infinity. Based on the method introduced by Serfaty [Mean-Field Limits of the Gross--Pitaevskii and Parabolic Ginzburg--Landau equations, preprint, arXiv:1507.03821, 2015] in the context of the Ginzburg--Landau vortices, using regularity and stability properties of the limiting equation, we prove a mean-field limit result in dimensions $1$ and $2$ in cases for which this problem was still open.