Global existence for the relativistic Vlasov-Poisson system in a two-dimensional bounded domain

TL;DR

This paper proves the global existence of solutions to the relativistic Vlasov-Poisson system in two-dimensional convex bounded domains with specific boundary conditions, using geometric and velocity lemmas.

Contribution

It introduces new geometric techniques and velocity lemmas to establish global solutions for the relativistic Vlasov-Poisson system in bounded domains.

Findings

Global existence of solutions proven for 2D convex domains

Effective boundary analysis using Frenet-Serret formulas

Applicable to both Neumann and Dirichlet boundary conditions

Abstract

In this paper, we prove the global existence of solutions to the relativistic Vlasov-Poisson system for general initial data in convex bounded domains of two space dimensions, assuming the specular reflection boundary conditions for the distribution density. The boundary conditions for the electric potential are considered in two cases: Neumann boundary conditions and homogeneous Dirichlet boundary conditions. The core ideas involve constructing suitable velocity lemmas and applying geometric techniques. In the two-dimensional case, it is crucial to select the arc length as the parameter of the curve and to further combine this with the Frenet-Serret formulas, enabling us to effectively describe the distribution density equation near the boundary and thus establishing a vital connection in the geometric representation.