Global Hilbert Expansion for the Relativistic Vlasov–Maxwell–Boltzmann System

Consider the relativistic Vlasov–Maxwell–Boltzmann system describing the dynamics of an electron gas in the presence of a fixed ion background. Thanks to recent works Germain and Masmoudi (Ann Sci Éc Norm Supér 47(3):469–503, 2014), Guo et al. (J Math Phys 55(12):123102, 2014) and Deng et al. (Arch Ration Mech Anal 225(2):771–871, 2017), we establish the global-in-time validity of its Hilbert expansion and derive the limiting relativistic Euler–Maxwell system as the mean free path goes to zero. Our method is based on the L^2-L^∞ framework and the Glassey–Strauss Representation of the electromagnetic field, with auxiliary H^1 estimates and W^1,∞ estimates to control the characteristic curves and corresponding L^∞ norm.