# Focusing Solutions of the Vlasov-Poisson System

## TL;DR

This paper constructs smooth, spherically symmetric solutions to the Vlasov-Poisson systems that start with small norms but evolve to have large L^1 norms, revealing complex dynamics in plasma physics models.

## Contribution

It demonstrates the existence of solutions with initially small norms that grow large over time in the Vlasov-Poisson systems, highlighting nonlinear growth phenomena.

## Key findings

- Solutions with small initial norms can develop large L^1 norms over time
- The results apply to both classical and relativistic Vlasov-Poisson systems
- The solutions are smooth and spherically symmetric

## Abstract

We study smooth, spherically-symmetric solutions to the Vlasov-Poisson system and relativistic Vlasov-Poisson system in the plasma physical case. We construct solutions that initially possess arbitrarily small C^k norms for the charge densities and the electric fields, but attain arbitrarily large L^1 norms of them at some later time.