On One-dimensional Compressible Navier-Stokes Equations with Degenerate Viscosity and Constant State at Far fields

In this paper, we are concerned with the Cauchy problem for one-dimensional compressible isentropic Navier-Stokes equations with density-dependent viscosity $\mu(\rho)=\rho^\alpha (\alpha>0)$ and pressure $P(\rho)=\rho^{\gamma}\ (\gamma>1)$. We will establish the global existence and asymptotic behavior of weak solutions for any $\alpha>0$ and $\gamma>1$ under the assumption that the density function keeps a constant state at far fields. This enlarges the ranges of $\alpha$ and $\gamma$ and improves the previous results presented by Jiu and Xin. As a result, in the case that $0<\alpha<\frac12$, we obtain the large time behavior of the strong solution obtained by Mellet and Vasseur when the solution has a lower bound (no vacuum).