Global Weak Solution To Compressible Navier-Stokes Equations With Density-Dependent Viscosity, Vacuum And Gravitational Force

In this paper, we study the one-dimensional isentropic Navier-Stokes equations with gravitational forceThe viscosity coefficientμis proportional toρ~θand 0 <θ< 1, whereρis the density, and the pressure P(ρ) = Aρ~γis a the pressure, y(τ) is the free boundary. When the density connects to vacuum discontinuously, we get the global existence and the uniqueness of weak solution by constructing an approximate solution sequence through mollifying the initial data.