Quantitative mean-field limits for repulsive Coulomb flows at bounded density and Riesz weak--strong stability

Abstract

We establish quantitative mean-field convergence and propagation of chaos for repulsive Coulomb gradient flows at the bounded-density regularity of the limiting equation. The argument couples the dissipative modulated-energy identity with the normalized quadratic transport cost of the full $N$-particle law. The remaining negative mean-square force-error term is used through an exact completion of squares after mollification: the non-Lipschitz remainder is absorbed by this negative term, while a sharp first-order commutator estimate is applied to the mollified Lipschitz field. For the Coulomb equation, the sharp $L^{\infty }$ decay gives the density envelope $m(t)=\Vert {\rho }_0{\Vert }_{L^{\infty }}/(1+t\Vert {\rho }_0{\Vert }_{L^{\infty }})$. A density-adapted transport weight and mollification scale $m(t{)}^{-1/d}$ yield an Osgood comparison. Thus, for every $d\ge 2$ and ${\rho }_0\in {\mathcal{P}}_2({\mathbb{R}}^d)\cap L^{\infty }({\mathbb{R}}^d)$, we obtain quantitative comparison with the global bounded-density Coulomb solution on every prescribed finite interval. For tensorized initial data, the normalized squared Wasserstein distance of the full $N$-particle law, the expected modulated energy, and the time-integrated mean-square force error are bounded by $N^{-2{\gamma }_{T,d}/d}$ for $d\ge 3$ and $((1+\log N)/N{)}^{{\gamma }_{T,2}}$ for $d=2$, where ${\gamma }_{T,d}=(1+T\Vert {\rho }_0{\Vert }_{L^{\infty }}{)}^{-c_d}$. For $d-2<s<d$, we also prove Riesz weak--strong stability for prescribed reference solutions in $L^{\infty }(0,T;B_{\infty ,q}^{s-d+2})$, with Gronwall, Bihari, and Osgood comparisons according to $q$, together with uniqueness in the stated Besov class. Finally, an outlier construction separates modulated-energy convergence and Kac chaos from normalized Wasserstein convergence of the full $N$-particle law.

60 pages