One-dimensional Viscous Dependence Free Boundary Problem In The Density Of The Navier-stokes Equations

In this paper,we prove the existence and uniqueness of the weak solution of the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity,i.e.μ(ρ) =ρ~θwithθ∈(0,γ/2],γ> 1.The initial data are a perturbation of a corresponding steady solution and continuously contact with vacuum on the free boundary. The obtained results relax the conditionθ∈(0,γ-1)∩(0,γ/2]in [39] and apply for the one-dimensional Siant-Vcnant model of shallow water .