Concentrating solutions of the relativistic Vlasov-Maxwell system

TL;DR

This paper constructs smooth, spherically symmetric solutions to the relativistic Vlasov-Maxwell system that can develop arbitrarily large charge densities and electric fields in finite time, even from small initial data.

Contribution

It demonstrates the existence of solutions with arbitrarily large charge concentrations and electric fields, regardless of initial data size or total mass.

Findings

Solutions can concentrate charge density arbitrarily close to the origin.

Electric fields can become arbitrarily large at finite times.

Concentration can occur even with small initial data or large initial mass.

Abstract

We study smooth, global-in-time solutions of the relativistic Vlasov-Maxwell system that possess arbitrarily large charge densities and electric fields. In particular, we construct spherically symmetric solutions that describe a thin shell of equally charged particles concentrating arbitrarily close to the origin and which give rise to charge densities and electric fields as large as one desires at some finite time. We show that these solutions exist even for arbitrarily small initial data or any desired mass. In the latter case, the time at which solutions concentrate can also be made arbitrarily large.