Arbitrarily large solutions of the Vlasov-Poisson system

TL;DR

This paper constructs specific solutions to the Vlasov-Poisson system demonstrating that they can have arbitrarily large charge densities and electric fields, even starting from small initial conditions or with prescribed mass.

Contribution

It introduces two classes of spherically-symmetric solutions that can develop arbitrarily large densities and fields over time, highlighting new dynamic behaviors of the system.

Findings

Solutions can reach arbitrarily large densities and fields.

Small initial data can evolve into large values.

Solutions can be tailored to reach large values at prescribed times.

Abstract

We study smooth, global-in-time solutions of the Vlasov-Poisson system in the plasma physical case that possess arbitrarily large charge densities and electric fields. In particular, we construct two classes of solutions with this property. The first class are spherically-symmetric solutions that initially possess arbitrarily small density and field values, but attain arbitrarily large values of these quantities at some later time. Additionally, we construct a second class of spherically-symmetric solutions that possess any desired mass and attain arbitrarily large density and field values at any later prescribed time.