  The Vlasov-Maxwell-Landau (VML) system and the Vlasov-Maxwell-Boltzmann (VMB)
system are fundamental models in dilute collisional plasmas. In this paper, we
are concerned with the hydrodynamic limits of both the VML and the non-cutoff
VMB systems in the entire space. Our primary objective is to rigorously prove
that, within the framework of Hilbert expansion, the unique classical solution
of the VML or non-cutoff VMB system converges globally over time to the smooth
global solution of the Euler-Maxwell system as the Knudsen number approaches
zero.
  The core of our analysis hinges on deriving novel interplay energy estimates
for the solutions of these two systems, concerning both a local Maxwellian and
a global Maxwellian, respectively. Our findings address a problem in the
hydrodynamic limit for Landau-type equations and non-cutoff Boltzmann-type
equations with a magnetic field. Furthermore, the approach developed in this
paper can be seamlessly extended to assess the validity of the Hilbert
expansion for other types of kinetic equations.
