Solutions of the 2D Radially Symmetric Vlasov-Maxwell System with an Initial Focusing Phase

TL;DR

This paper constructs radially symmetric solutions to the 2D Vlasov-Maxwell system that start with small initial norms but develop large electric fields and charge densities over time, revealing nonlinear growth phenomena.

Contribution

It demonstrates the existence of solutions with small initial data that evolve to exhibit large electric fields and charge densities, highlighting nonlinear dynamics in the system.

Findings

Solutions with small initial norms can develop large fields over time

Nonlinear growth leads to unbounded electric fields and charge densities

Radially symmetric solutions exhibit significant amplification of initial small data

Abstract

We study radially symmetric solutions to the 2D Vlasov-Maxwell system and construct solutions that initially possess arbitrarily small $C^k$ norms ($k \geq 1$) for the charge densities and the electric fields, but attain arbitrarily large $L^\infty$ norms of them at some later time.