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Due to the strong degeneracy at vacuum, both Euler and Navier-Stokes systems for compressible fluids (in which the viscosity is independent of density) behave singularly [7, 10, 16]. In particular, the classical one-dimensional isentropic Navier-Stokes system picks up unphysical solutions for two gases initially separated by vacuum states [7, 10]. To overcome this difficulty, Liu, Xin and Yang in [10] introduced the modified Navier-Stokes system (1.1) in which the viscosity coefficient depends on the density. It is shown in [10] that at least locally in time, the system (1.1) yields the physically relevant solution. As remarked by Liu, Xin and Yang in [10], the model is also motivated by the physical consideration that in the derivation of the compressible Navier-Stokes equations from the Boltzmann equations, the viscosity is not constant and depends on the temperature. For isentrpoic flow, this dependence is translated into the dependence of the viscosity on the density. For simplicity we consider in this paper

Global Weak Solutions to 1D Compressible Isentropic Navier-Stokes Equations with Density-Dependent Viscosity