Vacuum States On Compressible Navier-Stokes Equations With General Density-Dependent Viscosity And General Pressure Law

In this paper, we study the compressible Navier-Stokes equations which can be written, in Euler coordinates, aswhere the pressure P(p) and the viscosityμ(ρ) are general functions of the densityρ. When the initial density connects to vacuum state with a jump, we prove the global existence and the uniqueness of weak solutions by using the line method. For this, some new prior estimates are obtained to take care of the general viscosity coefficientμ(ρ) instead ofρ~θ.