Focusing solutions of the Vlasov-Poisson system

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 \begin{document}$ C^k $\end{document} norms ( \begin{document}$ k \geq 1 $\end{document} ) for the charge densities and the electric fields, but attain arbitrarily large \begin{document}$ L^\infty $\end{document} norms of them at some later time.