Mean field limit for coulomb-type flows

We establish the mean field convergence for systems of points evolving along the gradient flow of their interaction energy when the interaction is the Coulomb potential or a super-Coulombic Riesz potential, for the first time in arbitrary dimension. The proof is based on a modulated energy method using a Coulomb or Riesz distance, assumes that the solutions of the limiting equation are regular enough, and exploits a weak-strong stability property for them. The method can handle the addition of a regular interaction kernel and applies also to conservative and mixed flows. In the Appendix, it is also adapted to prove the mean field convergence of the solutions to Newton’s law with Coulomb or Riesz interaction in the monokinetic case to solutions of an Euler–Poisson type system.