  In this paper, we study the hydrodynamic limits of both the Landau equation
and the Vlasov-Maxwell-Landau system in the whole space. Our main purpose is
two-fold: the first one is to give a rigorous derivation of the compressible
Euler equations from the Landau equation via the Hilbert expansion; while the
second one is to prove, still in the setting of Hilbert expansion, that the
unique classical solution of the Vlasov-Maxwell-Landau system converges, which
is shown to be globally in time, to the resulting global smooth solution of the
Euler-Maxwell system, as the Knudsen number goes to zero. The main ingredient
of our analysis is to derive some novel interplay energy estimates on the
solutions of the Landau equation and the Vlasov-Maxwell-Landau system which are
small perturbations of both a local Maxwellian and a global Maxwellian,
respectively. Our result solves an open problem in the hydrodynamic limit for
the Landau-type equations with Coulomb potential and the approach developed in
this paper can seamlessly be used to deal with the problem on the validity of
the Hilbert expansion for other types of kinetic equations.
