Asymptotic behavior of small solutions to the Vlasov--Klein--Gordon system in high dimensions

TL;DR

This paper investigates the long-term behavior of small solutions to the Vlasov--Klein--Gordon system in high-dimensional Minkowski spacetime, using advanced vector field techniques.

Contribution

It introduces a novel approach employing the vector field method and hyperboloidal foliation to analyze asymptotics for the Vlasov--Klein--Gordon system in dimensions n ≥ 4.

Findings

Established asymptotic properties of small solutions in high dimensions

Demonstrated the effectiveness of hyperboloidal foliation in this context

Extended techniques beyond the standard Glassey-Strauss argument

Abstract

We study the asymptotic behavior of small solutions to the Vlasov--Klein--Gordon system in high dimensions. The standard argument of Glassey and Strauss \cite{GS87} for studying small solutions to the Vlasov--Maxwell system does not apply to the Vlasov--Klein--Gordon system due to the massiveness of the Klein--Gordon field. In this paper we use the vector field method and consider solutions in dimensions $ n \geq 4 $ with the hyperboloidal foliation of the Minkowski spacetime to obtain the asymptotic properties for the Vlasov--Klein--Gordon system.