Stability of steady-state solutions to Navier-Stokes-Poisson systems

This paper is concerned with a stability problem for compressible Navier-Stokes-Poisson systems. It arises in the modeling of semiconductors with a viscosity term in momentum equations. We prove that smooth solutions exist globally in time near the steady-state solution, and converge to the steady state for large time. In this stability result, we don't give any special assumptions on the given doping profile. The proof is based on the techniques of anti-symmetric matrix and an induction argument on the order of the space derivatives of solutions in energy estimates.