Compressible navier-stokes equations with density-dependent viscosity, vacuum and gravitational force in the case of general pressure

This is a continuation of the article (Comm. Partial Differential Equations 26 (2001) 965). In this article, the authors consider the one-dimensional compressible isentropic Navier-Stokes equations with gravitational force, fixed boundary condition, a general pressure and the density-dependent viscosity coefficient when the viscous gas con-nects to vacuum state with a jump in density. Precisely, the viscosity coefficient u is proportional to pθ and 0 < θ < 1/2, where p is the density, and the pressure P =P(p) is a general pressure. The global existence and the uniqueness of weak solution are proved.