Communications in Mathematical Sciences

Volume 17 (2019)

Number 2

Concentrating solutions of the relativistic Vlasov–Maxwell system

Pages: 377 – 392



Jonathan Ben-Artzi (School of Mathematics, Cardiff University, Cardiff, United Kingdom)

Simone Calogero (Department of Mathematics, Chalmers Institute of Technology, University of Gothenburg, Sweden)

Stephen Pankavich (Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, Co., U.S.A.)


We study smooth, global-in-time, spherically-symmetric solutions of the relativistic Vlasov–Poisson system that possess arbitrarily large charge densities and electric fields. In particular, we construct 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. As the constructed solutions are spherically-symmetric, they also satisfy the relativistic Vlasov–Maxwell system and thus our results apply to the latter system as well.


kinetic theory, Vlasov–Maxwell, spherical symmetry, charge density, electric field

2010 Mathematics Subject Classification

35L60, 35Q83, 82C22, 82D10

The first author was supported by the UK Engineering and Physical Sciences Research Council’s Early Career Fellowship EP/N020154/1. The final author was supported by US National Science Foundation award DMS-1614586.

Received 28 June 2018

Received revised 6 December 2018

Accepted 6 December 2018

Published 8 July 2019