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We show that a smooth, small enough Cauchy datum launches a unique classicalsolution of the relativistic Vlasov-Darwin (RVD) system globally in time. Asimilar result is claimed in Comm. Math. Sci. 6, 749-764 (2008) following thework in Int. Mat. Res. Not. 57191, 1-31 (2006). Our proof does not requireestimates derived from the conservation of the total energy, nor thosepreviously given on the transverse component of the electric field. Theseestimates are crucial in the references cited above. Instead, we exploit theformulation of the RVD system in terms of the generalized space and momentumvariables. By doing so, we produce a simple a-priori estimate on the transversecomponent of the electric field. We widen the functional space required for theCauchy datum to extend the solution globally in time, and we improve decayestimates given in Comm. Math. Sci. 6, 749-764 (2008) on the electromagneticfield and its space derivatives. Our method extends the constructive proofpresented in "Collisionless kinetic equations from astrophysics: TheVlasov-Poisson system. Handbook of Differential Equations: EvolutionaryEquations, Vol.3, Elsevier, 2007" to solve the Cauchy problem for theVlasov-Poisson system with a small initial datum.

Global Classical Solutions of the Relativistic Vlasov-Darwin System with Small Cauchy Data: the Generalized Variables Approach