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

TL;DR

This paper proves the global existence of smooth solutions to the Vlasov-Poisson system with specific steady asymptotic conditions, demonstrating decay of charge density and electrostatic fields in a plasma with infinite total charge.

Contribution

It extends local solutions to global ones by establishing decay estimates for charge density and electrostatic fields under steady spatial asymptotics.

Findings

Charge density decays at least as fast as x^{-6}

Global existence of smooth solutions is established

Spatial decay estimates for electrostatic field and derivatives are proven

Abstract

A collisionless plasma is modeled 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 tends to infinity 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 spatial decay estimates for the electrostatic field and its derivatives.