Compressible Navier–Stokes equations with vacuum state in the case of general pressure law

In this paper, we consider the one‐dimensional compressible isentropic Navier–Stokes equations with a general ‘pressure law’ and the density‐dependent viscosity coefficient when the density connects to vacuum continuously. Precisely, the viscosity coefficient µ is proportional to ρθ and 0<θ<1, where ρ is the density. And the pressure P = P(ρ) is a general ‘pressure law’. The global existence and the uniqueness of weak solutions is proved, and a decay result for the pressure as t→ + ∞ is given. It is also proved that no vacuum states and no concentration states develop, and the free boundary do not expand to infinite.