Shock profiles for the Navier--Stokes--Fourier--Poisson system under the Boltzmann relation

Abstract

The one-dimensional compressible Navier--Stokes--Fourier--Poisson system under the Boltzmann relation is studied for the existence of small-amplitude viscous shock profiles connecting the Rankine--Hugoniot states of the quasineutral Euler system with effective pressure $P_{\mathrm{eff}}=\rho (\theta +1)$. This profile is unique up to translation, of Lax type, and orbitally stable in time without a zero-mass assumption on the perturbation. The proof is based on a center-manifold reduction of the traveling-wave ODE and on the method of $a$-contraction with shifts, together with a relative-entropy functional augmented by electric energy and a sharp estimate of the viscous flux associated with $P_{\mathrm{eff}}$.