A full compressible Navier–Stokes–Poisson system models the motion of viscous ions under the effect of variable temperature and plays important roles in the study of self-gravitational viscous gaseous stars and in simulations of charged particles in semiconductor devices and plasmas physics. We establish the time-asymptotic nonlinear stability of a planar rarefaction wave to the initial value problem of a three-dimensional full compressible Navier–Stokes–Poisson equation when the initial data are a small perturbation of the planar rarefaction wave. The proof is given by a delicate energy method, which involves highly non-trivial a priori bounds due to the effect of the self-consistent electric field. This appears as the first result on the nonlinear stability of wave patterns to the full compressible Navier–Stokes–Poisson system in multi-dimensions.

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A viscous quantum hydrodynamic system for particle density, current density, energy density, and electrostatic potential, coupled with a Poisson equation, is studied in spatial one dimensional real line. The system is self-consistent in the sense that the electric field, which forms a forcing term in the momentum and energy equations, is determined by the coupled Poisson equation. First, the existence and uniqueness of the stationary solution is proved in an appropriate Sobolev space. Then, exponential stability of the stationary solution is established by constructing an a priori estimate. Since the techniques for classical hydrodynamic equations are not applicable here due to the quantum term, the existence of a local-in-time solution is obtained by showing the existence of local-in-time solutions of a reformulated system via the iteration method.