The Vlasov-Poisson system, which models the statistical behavior of many-particle systems, is known to have global-in-time classical solutions (in three dimensions). However, the underlying particle systems (of attractive or repulsive particles) may have singularities appearing in finite time. For instance, attractive particles (stars) can collapse to a single point in finite time. It is therefore interesting to ask how close to a singularity can the Vlasov-Poisson system get?
This has recently been done by Rein & Taegert in the attractive case, however the repulsive case remained open. The main result states that for any constants there exists initial data with density whose norm is initially bounded by but that at some later time is greater than . The main tool is obtaining a priori estimates for particle trajectories and choosing initial data carefully. This data is chosen to be supported on a spherical shell about the origin, with initial velocities pointing inwards.