One-dimensional Compressible Navier-stokes-poisson Equation Initial Boundary Value Problem

In this paper, we consider the initial boundary value problem (IBVP) for 1-d compressible Navier-Stokes-Poisson equations with density-dependent viscosities on spatial periodic domainWe investigate the well-posedness for (2) and qualitative behaviers of their solution. The compressible systems are degenerate when vacumm state appears, thus the global existence of solution does not follow directly from theory of the compressible Navier-Stokes equations with constant viscosity. In this paper, we show the global existence of weak solution for general initial data with finite entropy to the IBVP (2). In addition, we prove that there exists a finite number T₀ (T₀＞0), so that when t＞T₀, any vacumm vanishs, and after the vanishing of vacuum, the weak solution gains regularities and becomes to a strong solution, and converges exponentially to the steady state, which is determined by initial data. This extends the previous results of compressible Navier-Stokes equations in [1, 30].