Focusing solutions of Vlasov-Poisson equation on a spherical shell

Abstract

We study smooth, spherically-symmetric solutions to the Vlasov-Poisson system and relativistic Vlasov-Poisson system in the plasma physical case. In particular, we construct solutions that initially possess arbitrarily small charge densities and electric fields, but attain arbitrarily large value of these quantities and concentrate on any given spherical shell at some later time.