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In this paper, we introduce a new set of vector fields for the relativistictransport equation, which is applicable for general Vlasov-Wave type coupledsystems. By combining the strength of Klainerman vector field method andFourier method, we prove global regularity and scattering for the $3D$ massiverelativistic Vlasov-Nordstr\"om system for small initial data without anycompact support assumption, which is widely used in the literature for thestudy of Vlasov solution.

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