Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Commutators, mean-field, and supercritical mean-field limits for Coulomb/Riesz gases

Matthew Rosenzweig

Abstract

This paper is intended as a companion to the author's talk "Commutator estimates and mean-field limits for Coulomb/Riesz gases" at the 2025 Journées équations aux dérivées partielles in Aussois. The goal is to provide a concise, accessible account of sharp commutator estimates recently obtained for modulated energies associated to Coulomb/Riesz interactions and how these estimates lead to optimal results for mean-field and supercritical mean-field limits of Coulomb/Riesz gas dynamics via the modulated-energy method. The exposition centers on the works arXiv:2408.14642, arXiv:2407.15650 with Serfaty and arXiv:2511.13461, arXiv:2601.02326 with Hess-Childs and Serfaty.

Commutators, mean-field, and supercritical mean-field limits for Coulomb/Riesz gases

Abstract

This paper is intended as a companion to the author's talk "Commutator estimates and mean-field limits for Coulomb/Riesz gases" at the 2025 Journées équations aux dérivées partielles in Aussois. The goal is to provide a concise, accessible account of sharp commutator estimates recently obtained for modulated energies associated to Coulomb/Riesz interactions and how these estimates lead to optimal results for mean-field and supercritical mean-field limits of Coulomb/Riesz gas dynamics via the modulated-energy method. The exposition centers on the works arXiv:2408.14642, arXiv:2407.15650 with Serfaty and arXiv:2511.13461, arXiv:2601.02326 with Hess-Childs and Serfaty.
Paper Structure (21 sections, 7 theorems, 76 equations)

This paper contains 21 sections, 7 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Commutator estimates
  3. Coulomb/super-Coulomb case
  4. sub-Coulomb case
  5. Localized and higher-order estimates
  6. Transport regularity and defective estimates
  7. Mean-field limits
  8. Background and motivation
  9. Main results
  10. Modulated energy and commutators
  11. Optimal regularity assumptions
  12. Positive temperature
  13. Supercritical mean-field limits
  14. Background and motivation
  15. The combined mean-field and quasineutral limit
  16. ...and 6 more sections

Key Result

Theorem 2.1

Let $-2<\mathsf{s}<\mathsf{d}$. Let $\mu \in L^1\cap L^p$ for $\frac{\mathsf{d}}{\mathsf{d}-\mathsf{s}}<p\le \infty$ with $\int_{{\mathbb{R}}^\mathsf{d}}d\mu = 1$, and associated to $\mu$, define the length scales When $-2<\mathsf{s}\le 0$, assume that $\int_{({\mathbb{R}}^\mathsf{d})^2}|{\mathsf{g}}|(x-y)d|\mu|^{\otimes 2}<\infty$ to ensure that the modulated energy is finite. Let $v$ be a Lipsc

Theorems & Definitions (14)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • Theorem 2.5: Global estimates
  • Theorem 2.6: Localized estimates
  • Remark 2.7
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • ...and 4 more