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In this paper, the Cauchy problem for the one-dimensional (1-D) isentropiccompressible Navier-Stokes equations (\textbf{CNS}) is considered. When theviscosity $\mu(\rho)$ depends on the density $\rho$ in a sublinear power law ($\rho^\delta$ with $0<\delta\leq 1$), based on an elaborate analysis of theintrinsic singular structure of this degenerate system, we prove theglobal-in-time well-posedness of regular solutions with conserved total mass,momentum, and finite total energy in some inhomogeneous Sobolev spaces.Moreover, the solutions we obtained satisfy that $\rho$ keeps positive for allpoint $x\in \mathbb{R}$ but decays to zero in the far field, which isconsistent with the facts that the total mass of the whole space is conserved,and \textbf{CNS} is a model of non-dilute fluids where $\rho$ is bounded belowaway from zero. The key to the proof is the introduction of a well-designedreformulated structure by introducing some new variables and initialcompatibility conditions, which, actually, can transfer the degeneracies of thetime evolution and the viscosity to the possible singularity of some specialsource terms. Then, combined with the BD entropy estimates and transportproperties of the so-called effective velocity $v=u+\varphi(\rho)_x$ ($u$ isthe velocity of the fluid, and $\varphi(\rho)$ is a function of $\rho$ definedby $\varphi'(\rho)=\mu(\rho)/\rho^2$), one can obtain the required uniform apriori estimates of corresponding solutions.

Global regular solutions for 1-D degenerate compressible Navier-Stokes equations with large data and far field vacuum