We adapt the vector field method of Klainerman to the study of
relativistic transport equations. First, we prove robust decay estimates for
velocity averages of solutions to the relativistic massive and massless
transport equations, without any compact support requirements (in
or
) for the
distribution functions. In the second part of this article, we apply our method to the
study of the massive and massless Vlasov–Nordström systems. In the massive case, we
prove global existence and (almost) optimal decay estimates for solutions in dimensions
under
some smallness assumptions. In the massless case, the system decouples
and we prove optimal decay estimates for the solutions in dimensions
for arbitrarily large
data, and in dimension
under some smallness assumptions, exploiting a certain
form of the null condition satisfied by the equations. The
-dimensional
massive case requires an extension of our method and will be treated in future
work.