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 ec Norm Super 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-infinity framework and the Glassey-Strauss Representation of the electromagnetic field, with auxiliary H-1 estimates and W-1,W-infinity estimates to control the characteristic curves and corresponding L-infinity norm.