The Mean-Field Limit for a Regularized Vlasov-Maxwell Dynamics

TL;DR

This paper proves the mean-field limit for a regularized relativistic Vlasov-Maxwell system, extending classical results to a more complex electromagnetic setting with energy conservation and stability estimates.

Contribution

It introduces a kinetic formulation and regularization technique for the relativistic Vlasov-Maxwell system, establishing the mean-field limit with energy conservation and stability analysis.

Findings

Established the mean-field limit for the regularized Vlasov-Maxwell system.

Developed a kinetic formulation of Maxwell equations in momentum space.

Proved a stability estimate analogous to Dobrushin's for the system.

Abstract

The present work establishes the mean-field limit of a N-particle system towards a regularized variant of the relativistic Vlasov-Maxwell system, following the work of Braun-Hepp [Comm. in Math. Phys. 56 (1977), 101-113] and Dobrushin [Func. Anal. Appl. 13 (1979), 115-123] for the Vlasov-Poisson system. The main ingredients in the analysis of this system are (a) a kinetic formulation of the Maxwell equations in terms of a distribution of electromagnetic potential in the momentum variable, (b) a regularization procedure for which an analogue of the total energy - i.e. the kinetic energy of the particles plus the energy of the electromagnetic field - is conserved and (c) an analogue of Dobrushin's stability estimate for the Monge-Kantorovich-Rubinstein distance between two solutions of the regularized Vlasov-Poisson dynamics adapted to retarded potentials.