Hydrodynamic Limit of the Incompressible Navier-Stokes-Fourier-Maxwell System with Ohm's Law from the Vlasov-Maxwell-Boltzmann System: Hilbert Expansion Approach

We prove a global-in-time limit from the two-species Vlasov-Maxwell-Boltzmann system with a cutoff hard potential collision kernel to the two-fluid incompressible Navier-Stokes-Fourier-Maxwell system with Ohm's law. Besides the techniques developed for the classical solutions to the Vlasov-Maxwell-Boltzmann equations in the past years, such as the nonlinear energy method and micro-macro decomposition are employed, in this paper, key roles are played by the decay properties of both the electric field and the wave equation with linear damping of the divergence free magnetic field. This is a companion paper of Jiang and Luo (Ann PDE, 8(1), Paper No. 4, 126, 2022) in which Hilbert expansion was not employed but only the hard sphere case was considered.