DiPerna-Lions flow for relativistic particles in an electromagnetic field

TL;DR

This paper proves the existence and uniqueness of a DiPerna-Lions flow for relativistic particles influenced by electromagnetic fields, even with singular initial conditions and force fields in $L^2$, advancing mathematical understanding of such systems.

Contribution

It establishes the DiPerna-Lions flow for relativistic particles under Lorentz force with singular initial conditions and electromagnetic fields solving Maxwell's equations.

Findings

Existence and uniqueness of the flow proven.

Analysis of cancellations over trajectories with $L^2$ force fields.

Application to particles in electromagnetic fields with singular initial data.

Abstract

We show the existence and uniqueness of a DiPerna-Lions flow for relativistic particles subject to a Lorentz force in an electromagnetic field. The electric and magnetic fields solves the linear Maxwell system in the void but for singular initial conditions. As the corresponding force field is only in $L^2$, we have to perform a careful analysis of the cancellations over a trajectory.