The purpose of this paper is two-fold. In the first part, we provide a new proof of the
global existence of the solutions to the Vlasov–Maxwell system with a small initial
distribution function. Our approach relies on vector field methods, together with the
Glassey–Strauss decomposition of the electromagnetic field, and does not require
any support restriction on the initial data or smallness assumption on the
Maxwell field. Contrary to previous works on Vlasov systems in dimension
,
we do not modify the linear commutators and avoid then many technical
difficulties.
In the second part of this paper, we prove a modified scattering result for these
solutions. More precisely, we obtain that the electromagnetic field has a
radiation field along future null infinity and approaches, for large time, a smooth
solution to the vacuum Maxwell equations. As for the Vlasov–Poisson system,
in contrast, the distribution function converges to a new density function
along
modifications of the characteristics of the free relativistic transport
equation. In order to define these logarithmic corrections, we identify
an effective asymptotic Lorentz force. By considering logarithmical
modifications of the linear commutators, defined in terms of derivatives of the
asymptotic Lorentz force, we finally prove higher-order regularity results for
.
