1D Vlasov-Poisson equations with electron sheet initial data

Abstract

We construct global weak solutions for both one-component andtwo-component Vlasov-Poisson equations in a single space dimensionwith electron sheet initial data. We give an explicit formula of theweak solution of the one-component Vlasov-Poisson equation providedthe electron sheet remains a graph in the $x$-$v$ plane, and we givesharp conditions on whether the moment of this explicit weaksolution will blow up or not. We introduce new parameters, which wecall 'charge indexes', to construct the global weak solution. Themoment of the weak solution corresponds to a multi-valued solutionto the Euler-Poisson system. Our method guarantees that even ifconcentration in charge develops, it will disappear immediately. Weextend our method to more singular initial data, where charge canconcentrate on points at time $t=0$. Examples show that forone-component Vlasov-Poisson equation our weak solution agrees withthe continuous fission weak solution, which is the zero diffusionlimit of the Fokker-Planck equation. Finally, we propose a novelnumerical method to compute solutions of both one-component andtwo-component Vlasov-Poisson equations and the multi-valued solutionof the one-dimensional Euler-Poisson equation.