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The purpose of this paper is twofold. In the first part, we provide a newproof of the global existence of the solutions to the Vlasov-Maxwell systemwith a small initial distribution function. Our approach relies on vector fieldmethods, together with the Glassey-Strauss decomposition of the electromagneticfield, and does not require any support restriction on the initial data orsmallness assumption on the Maxwell field. Contrary to previous works on Vlasovsystems in dimension $3$, we do not modify the linear commutators and avoidthen many technical difficulties.  In the second part of this paper, we prove a modified scattering result forthese solutions. More precisely, we obtain that the electromagnetic field has aradiation field along future null infinity and approaches, for large time, asmooth solution to the vacuum Maxwell equations. As for the Vlasov-Poissonsystem, in constrast, the distribution function converges to a new densityfunction along modifications of the characteristics of the free relativistictransport equation. In order to define these logarithmic corrections, weidentify an effective asymptotic Lorentz force. By considering logarithmicalmodifications of the linear commutators, defined in terms of derivatives of theasymptotic Lorentz force, we finally prove higher order regularity results forthe limit distribution function.

Global existence and modified scattering for the solutions to the Vlasov-Maxwell system with a small distribution function