Hydrodynamic limit and Newtonian limit from the relativistic Vlasov-Maxwell-Boltzmann system to the classical Euler-Poisson system

In this paper, around a global smooth irrotational solution to the classical isentropic compressible Euler-Poisson system, we construct classical solutions to the one-species relativistic Vlasov-Maxwell-Boltzmann system on any finite time interval $[0,T]$, and rigorously justify the combined hydrodynamic and Newtonian limits to the Euler-Poisson system. In particular, this yields a rigorous derivation of the compressible Euler-Poisson system, whose Poisson coupling induces an instantaneous electrostatic response and thus no longer preserves a strict finite-speed propagation structure, from a relativistic kinetic model with finite propagation speed. The analysis is based on a Hilbert expansion in $\epsilon$ for the relativistic Vlasov-Maxwell-Boltzmann system, an asymptotic expansion in ${\mathfrak{c}}^{-1}$ for the relativistic Euler-Maxwell system, and estimates that are uniform in $\mathfrak{c}$ and $\epsilon$ for both the expansion coefficients and the remainder terms under the restriction $\mathfrak{c}\epsilon \le 1$. This restriction on $\mathfrak{c}$ is solely for closing the uniform remainder estimates.