The mean-field approximation for higher-dimensional Coulomb flows in the scaling-critical L ∞ space

Abstract

In the mean-field scaling regime, a first-order system of particles with binary interactions naturally gives rise to a scalar partial differential equation (PDE), which, depending on the nature of the interaction, corresponds to either the Hamiltonian or gradient flow of the effective energy of the system for a large number of particles. The empirical measure of such systems is a weak solution to this limiting mean-field PDE, and one expects that as the number of particles tends to infinity, it converges along its lifespan in the weak-* sense to a more regular solution of the PDE, provided it does so initially. Much effort has been invested over the years in proving and quantifying this convergence under varying regularity assumptions. When the interaction potential is Coulomb, the mean-field PDE has a scaling invariance which leaves the L ∞ norm unchanged; i.e., L ∞ is a critical function space for the equation. Moreover, the L ∞ norm is either conserved or decreasing, and the equation is globally well-posed in this space, making it a natural choice for studying the dynamics. Building on our previous work (Rosenzweig 2022 Arch. Ration. Mech. Anal. 243 1361–431) for point vortices (i.e. d = 2), we prove quantitative convergence of the empirical measure to the L ∞ solution of the mean-field PDE for short times in dimensions d ⩾ 3. To the best of our knowledge, this is the first such work outside of the 2D case. Our proof is based on a combination of the modulated-energy method of Serfaty (2020 Duke Math. J. 169 2887–935) and a novel mollification argument first introduced by the author in Rosenzweig (2022 Arch. Ration. Mech. Anal. 243 1361–431). Compared to our prior work (Rosenzweig 2022 Arch. Ration. Mech. Anal. 243 1361–431), the new challenge is the non-logarithmic nature of the potential.