One-dimensional Compressible Navier-stokes Equations With Initial Boundary Value Problems

In recent years, the compressible 1D Navier-Stokes systems with density-dependent viscosities are studied by many authors. The typical model of these equations is the viscous Saint-Venant system describing the motion of shallow water. These compressible systems are degenerate when vacuum states appears. Hence, for general initial data, the global existence of weak solution can't be derived directly from the renormalization method for compressible Navier-Stokes equations with constant viscosity. In this paper we consider initial-boundary-value problems for the 1D compressible Navier-Stokes systems with dispersion (surface tension),We show that, the entropy weak solutions to (1) for general large initial data exist globally in time. Furthermore, it can be shown that, there exists a finite time T₀ > 0, so that the density is uniformly positive from below when t > T₀. And, the weak solution gets enough regularity and become a strong one. Moreover, the solution tends to the non-vacuum equilibrium state at an exponential time rate.