Regularity and asymptotic behavior of 1D compressible Navier–Stokes–Poisson equations with free boundary

In this paper, firstly, we consider the regularity of solutions in H i ( [ 0 , 1 ] ) ( i = 2 , 4 ) to the 1D Navier–Stokes–Poisson equations with density-dependent viscosity and the initial density that is connected to vacuum with discontinuities, and the viscosity coefficient is proportional to ρ θ with 0 < θ < 1 . Furthermore, we get the asymptotic behavior of the solutions when the viscosity coefficient is a constant. This is a continuation of [S.J. Ding, H.Y. Wen, L. Yao, C.J. Zhu, Global solutions to one-dimensional compressible Navier–Stokes–Poisson equations with density-dependent viscosity, J. Math. Phys. 50 (2009) 023101], where the existence and uniqueness of global weak solutions in H 1 ( [ 0 , 1 ] ) for both cases: μ ( ρ ) = ρ θ , 0 < θ < 1 and μ = constant have been established.