Hilbert Expansion for Coulomb Collisional Kinetic Models

The relativistic Vlasov-Maxwell-Landau (r-VML) system and relativistic Landau equation (r-LAN) are fundamental models describing the dynamics of an electron gas. In this paper, we develop a new weighted energy method and justify the validity of the Hilbert expansion of the r-VML system and r-LAN equation. As the Knudsen number shrinks to zero, we prove the relativistic Euler-Maxwell limit and relativistic Euler limit, respectively. This resolves the long-time open problem of hydrodynamic limits of Landau-type equations.