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 connects to vacuum state with a jump in density. Precisely, the viscosity coefficient μ is proportional to ρϑ and 0<ϑ½, where ρ is the density, and the pressure P = P(ρ) is a general pressure. The global existence and the uniqueness of weak solution are proved.