  The relativistic Vlasov-Maxwell-Landau (r-VML) system and the relativistic
Landau equation (r-LAN) are fundamental models that describe the dynamics of an
electron gas. In this paper, we introduce a novel weighted energy method and
establish the validity of the Hilbert expansion for the r-VML system and r-LAN
equation. As the Knudsen number shrinks to zero, we rigorously demonstrate the
relativistic Euler-Maxwell limit and relativistic Euler limit, respectively.
This successfully resolves the long-standing open problem regarding the
hydrodynamic limits of Landau-type equations.
