Steady Euler-Poisson systems: A differential/integral equation formulation with general constitutive relations

The Cauchy problem and the initial-boundary value problem for the Euler-Poisson system have been extensively investigated, together with a study of scaled and unscaled asymptotic limits. The pressure-density relationships employed have included both the adiabatic (isentropic) relation as well as the ideal gas law (isothermal). The study most closely connected to this one is that of Nishibata and Suzuki [S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors, Osaka J. Math. 44 (2007) 639-665], where a power law was employed in the context of the subsonic case, covering both the isothermal and adiabatic cases. These authors characterize the steady solution as an asymptotic limit. In this paper, we consider only the steady case, in much greater generality, and with more transparent arguments, than heretofore. We are able to identify both subsonic and supersonic regimes, and correlate them to one-sided boundary values of the momentum and concentration. We employ the novelty of a differential/integral equation formulation.