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

TL;DR

This paper proves global existence and enhanced spatial decay rates for solutions to the radial Vlasov-Poisson system with steady asymptotics, extending previous local results to global solutions.

Contribution

It establishes global existence of solutions with specific decay rates, including an improved decay rate of x^{-6} in the non-spherical case.

Findings

Charge density decays at least as fast as x^{-4} for symmetric solutions.

In the general case, decay rate improves to x^{-6}.

Solutions exist globally in time under the given asymptotic conditions.

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 tends to infinity is considered. Hence the total positive charge, total negative charge, and total energy are all 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.