Finite energy scattering for the Lorentz-Maxwell equation

TL;DR

This paper establishes a scattering theory for the Lorentz-Maxwell equations with finite-energy data, showing particle speed convergence and electromagnetic field decomposition into a soliton plus free Maxwell solution.

Contribution

It provides the first scattering result for finite-energy data in a field-particle system, specifically for the Lorentz-Maxwell equations with small charge-to-mass ratio.

Findings

Particle speed converges to a limit as time approaches infinity.

Electromagnetic field decomposes into a soliton and a free Maxwell solution.

First scattering result for finite-energy data in a field-particle equation.

Abstract

In the case where the charge of the particle is small compared to its mass, we describe the asymptotics of the Lorentz-Maxwell equation for any finite-energy data. As time goes to infinity, we prove that the speed of the particle converges to a certain limit, whereas the electromagnetic field can be decomposed into a soliton plus a free solution of the Maxwell equation. It is the first instance of a scattering result for general finite energy data in a field-particle equation.