Global Existence for the Vlasov-Poisson System with Steady Spatial Asymptotics

A collisionless plasma is modelled by the Vlasov-Poisson system in three space dimensions. A fixed background of positive charge—dependant upon only velocity— is assumed. The situation in which mobile negative ions balance the positive charge as x → is considered. Thus, the total positive charge and the total negative charge are both infinite. Smooth solutions with appropriate asymptotic behavior for large x , which were previously shown to exist locally in time, are continued globally. This is done by showing that the charge density decays at least as fast as x −6. This article also establishes decay estimates for the electrostatic field and its derivatives.