Concentrations in the one-dimensional Vlasov-Poisson equations: additional regularizations

Abstract

We consider the one component Vlasov-Poisson equations in one dimension. These equations model a collisionless plasma of electrons moving through a uniform background of ions, and serve as a simpler analog of the two-dimensional incompressible Euler equations in vorticity form. We study the behavior of weak solutions to the Vlasov-Poisson equations which arise from non-smooth electron sheet initial data (an electron sheet describes a concentrated beam of electrons). We regularize the Vlasov-Poisson equations with electron sheet initial data in two different ways, smoothing the initial condition and by including collisions modeled by the Fokker-Planck equations. We perform numerical simulations with a finite difference method to examine the solution of the Vlasov-Poisson equations obtained in the limit of vanishing regularization.