Propagation of Regularity and Long Time Behavior of 3D Massive Relativistic Transport Equation I: Vlasov–Nordström System

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 3 D massive relativistic Vlasov–Nordström system for small initial data without any compact support assumption, which is widely used in the literature for the study of Vlasov equation.