Solutions of the 2D radially symmetric Vlasov-Maxwell system with an initial focusing phase

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. The proof is done by carefully tracking the trajectories of the particles to observe the emergence of a concentrating behavior. It is the first result to discuss this kind of behavior happening in the Vlasov-Maxwell dynamics.