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 $μ(ρ)=ρ^α(α>0)$ and pressure $P(ρ)=ρ^γ\ (γ>1)$. We will establish the global existence and asymptotic behavior of weak solutions for any $α>0$ and $γ>1$ under the assumption that the density function keeps a constant state at far fields. This enlarges the ranges of $α$ and $γ$ and improves the previous results presented by Jiu and Xin. As a result, in the case that $0<α<\frac12$, we obtain the large time behavior of the strong solution obtained by Mellet and Vasseur when the solution has a lower bound (no vacuum).

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