Global Existence and Increased Spatial Decay for the Radial Vlasov‐Poisson System with Steady Spatial Asymptotics

Abstract

A collisionless plasma is modeled by the Vlasov‐Poisson system in three space dimensions. A fixed background of positive charge, which is independent of time and space, is assumed. The situation in which mobile negative ions balance the positive charge as |x|→∞ is considered. Hence the total positive charge, total negative charge, and total energy are infinite. Smooth solutions with appropriate asymptotic behavior for large |x|, which were previously shown to exist locally in time, are continued globally for spherically symmetric data. This is done by showing that the charge density decays at least as fast as |x|−4. Finally, an increased decay rate of |x|−6 is shown in the general case without the assumption of spherical symmetry.