A mathematical theory of isolated systems in relativistic plasma physics

TL;DR

This paper investigates the existence, properties, and conservation laws of isolated solutions to the relativistic Vlasov-Maxwell system with initial data on a backward hyperboloid, including results on mass-energy conservation and global solutions in one dimension.

Contribution

It introduces a framework for defining isolated solutions with specific boundary conditions and proves mass-energy conservation and global existence in one dimension.

Findings

Mass-energy is conserved on backward hyperboloids.

Mass-energy on forward hyperboloids is non-increasing.

Global classical solutions exist in one dimension.

Abstract

The existence and the properties of isolated solutions to the relativistic Vlasov-Maxwell system with initial data on the backward hyperboloid $t=-\sqrt{1+|x|^2}$ are investigated. Isolated solutions of Vlasov-Maxwell can be defined by the condition that the particle density is compactly supported on the initial hyperboloid and by imposing the absence of incoming radiation on the electromagnetic field. Various consequences of the mass-energy conservation laws are derived by assuming the existence of smooth isolated solutions which match the inital data. In particular, it is shown that the mass-energy of isolated solutions on the backward hyperboloids and on the surfaces of constant proper time are preserved and equal, while the mass-energy on the forward hyperboloids is non-increasing and uniformly bounded by the mass-energy on the initial hyperboloid. Moreover the global existence and uniqueness of classical solutions in the future of the initial surface is established for the one dimensional version of the system.