Concentrations in the one-dimensional Vlasov-Poisson equations I: Temporal development and non-unique weak solutions in the single component case

Abstract

Weak solutions of the one-component Vlasov-Poisson equation in a single space dimension are proposed and studied here as a simpler analogue problem for the behavior of weak solutions of the two-dimensional incompressible Euler equations with non-negative vorticity. The physical, structural, and functional analytic analogies between these two problems are developed in detail here. With this background, explicit solutions for electron sheet initial data, the analogue of vortex sheet initial data, are presented, which display the phenomena of singularity formation in finite time as well as the explicit temporal development of charge concentrations. Other rigorous explicit examples with charge concentration are developed where there are non-unique weak solutions with the same initial data. In one of these non-unique weak solutions, an electron sheet completely collapses to a point charge in finite time. The detailed limiting behavior of regularizations such as the diffusive Fokker-Planck equation are developed through a very efficient numerical method which yields extremely high resolution for these simpler analogue problems. A striking consequences of the numerical results reported here is the fact that there is not a selection principle for a unique weak solution in some situations where there are several weak solutions with charge concentration for the same initial data. In particular, two explicit weak solutions with the same initial data are constructed here where it is demonstrated that the zero smoothing limit of time reversible particle methods converges to one of these solutions while the zero diffusion limit of the Fokker-Planck equation converges to the other weak solution.