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A new approach to the mean-field limit of Vlasov-Fokker-Planck equations

Didier Bresch, Pierre-Emmanuel Jabin, Juan Soler

TL;DR

This work develops a new BBGKY-based framework to rigorously derive the mean-field limit for second-order stochastic particle systems with velocity diffusion, obtaining the Vlasov-Poisson-Fokker-Planck system in dimension $2$ and partial results in dimension $3$. Central to the approach is a novel propagation of weighted $L^q$ bounds for marginals, enabled by diffusion in velocity and an energy-aware weight $e_k$, which circumvents the derivative loss in the hierarchy and accommodates highly singular kernels such as Coulomb/Poisson. The authors prove convergence of the $k$-particle marginals $f_{k,N}$ to the tensorized limit $f^{\\tensor k}$ on a finite time interval $[0,T^*]$, with a quantitative rate in the $L^2$ setting when $K \\in L^2$, and extend the method to first-order systems on bounded domains. These results establish the first rigorous mean-field derivation of Vlasov-Poisson-Fokker-Planck in $d=2$, and offer a versatile tool for handling singular interactions in plasmas and related kinetic models.

Abstract

This article introduces a novel approach to the mean-field limit of stochastic systems of interacting particles, leading to the first ever derivation of the mean-field limit to the Vlasov-Poisson-Fokker-Planck system for plasmas in dimension~$2$ together with a partial result in dimension~$3$. The method is broadly compatible with second order systems that lead to kinetic equations and it relies on novel estimates on the BBGKY hierarchy. By taking advantage of the diffusion in velocity, those estimates bound weighted $L^p$ norms of the marginals or observables of the system, uniformly in the number of particles. This allows to qualitatively derive the mean-field limit for very singular interaction kernels between the particles, including repulsive Poisson interactions, together with quantitative estimates for a general kernel in $L^2$.

A new approach to the mean-field limit of Vlasov-Fokker-Planck equations

TL;DR

This work develops a new BBGKY-based framework to rigorously derive the mean-field limit for second-order stochastic particle systems with velocity diffusion, obtaining the Vlasov-Poisson-Fokker-Planck system in dimension and partial results in dimension . Central to the approach is a novel propagation of weighted bounds for marginals, enabled by diffusion in velocity and an energy-aware weight , which circumvents the derivative loss in the hierarchy and accommodates highly singular kernels such as Coulomb/Poisson. The authors prove convergence of the -particle marginals to the tensorized limit on a finite time interval , with a quantitative rate in the setting when , and extend the method to first-order systems on bounded domains. These results establish the first rigorous mean-field derivation of Vlasov-Poisson-Fokker-Planck in , and offer a versatile tool for handling singular interactions in plasmas and related kinetic models.

Abstract

This article introduces a novel approach to the mean-field limit of stochastic systems of interacting particles, leading to the first ever derivation of the mean-field limit to the Vlasov-Poisson-Fokker-Planck system for plasmas in dimension~ together with a partial result in dimension~. The method is broadly compatible with second order systems that lead to kinetic equations and it relies on novel estimates on the BBGKY hierarchy. By taking advantage of the diffusion in velocity, those estimates bound weighted norms of the marginals or observables of the system, uniformly in the number of particles. This allows to qualitatively derive the mean-field limit for very singular interaction kernels between the particles, including repulsive Poisson interactions, together with quantitative estimates for a general kernel in .
Paper Structure (14 sections, 7 theorems, 106 equations)

This paper contains 14 sections, 7 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. The new result
  4. New stability estimates
  5. The case of first order system
  6. Our notion of entropy solution for the hierarchy; the well-posedness of Eq. \ref{['Liouville']}
  7. The definition
  8. Strong solutions up to the first collision.
  9. Proof of the main results
  10. The BBGKY and Vlasov hierarchies
  11. The key technical lemma
  12. Proof of technical results
  13. Proof of Theorem \ref{['Conv']}
  14. Proof of Theorem \ref{['quantitative']}

Key Result

Theorem 1

Assume that that there exists some constant $\theta>0$ s.t. the potential $\phi$ satisfies and that Let $f$ be the unique smooth solution to the Vlasov equation vlasov with initial data $f^0\in C^\infty(\Pi^d\times {\mathbb{R}}^d)$ such that $\int_{\Pi^d\times {\mathbb{R}}^d} f^0\,e^{\beta\,|v|^2}<\infty$. Consider moreover an entropy solution $f_N$ to Liouville in the sense of subsection Liouvi

Theorems & Definitions (13)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Proposition 1
  • Theorem 3
  • Remark 2
  • Remark 3
  • Lemma 1
  • proof
  • ...and 3 more