Propagation of regularity and long time behavior of $3D$ massive relativistic transport equation I: Vlasov-Nordstr\"om system

TL;DR

This paper establishes global regularity and scattering results for the 3D massive relativistic Vlasov-Nordström system with small initial data, using a novel vector field approach that combines Klainerman and Fourier methods.

Contribution

It introduces a new set of vector fields tailored for the relativistic transport equation, enabling analysis of coupled Vlasov-Wave systems without compact support assumptions.

Findings

Proves global regularity for the 3D Vlasov-Nordström system

Demonstrates scattering behavior for small initial data

Develops a new vector field method for relativistic transport equations

Abstract

In this paper, we introduce a new set of vector fields for the relativistic transport equation, which is applicable for general Vlasov-Wave type coupled systems. By combining the strength of Klainerman vector field method and Fourier method, we prove global regularity and scattering for the $3D$ massive relativistic Vlasov-Nordstr\"om system for small initial data without any compact support assumption, which is widely used in the literature for the study of Vlasov solution.