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

ABSTRACT

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.

Keywords:

• Cauchy problem
• Global existence
• Kinetic theory
• Spatial decay
• Vlasov

Mathematics Subject Classification:

• 35L60
• 35Q99
• 82C21
• 82C22
• 82D10