  In this paper, the Cauchy problem for the one-dimensional (1-D) isentropic
compressible Navier-Stokes equations (\textbf{CNS}) is considered. When the
viscosity $\mu(\rho)$ depends on the density $\rho$ in a sublinear power law ($
\rho^\delta$ with $0<\delta\leq 1$), based on an elaborate analysis of the
intrinsic singular structure of this degenerate system, we prove the
global-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 all
point $x\in \mathbb{R}$ but decays to zero in the far field, which is
consistent 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 below
away from zero. The key to the proof is the introduction of a well-designed
reformulated structure by introducing some new variables and initial
compatibility conditions, which, actually, can transfer the degeneracies of the
time evolution and the viscosity to the possible singularity of some special
source terms. Then, combined with the BD entropy estimates and transport
properties of the so-called effective velocity $v=u+\varphi(\rho)_x$ ($u$ is
the velocity of the fluid, and $\varphi(\rho)$ is a function of $\rho$ defined
by $\varphi'(\rho)=\mu(\rho)/\rho^2$), one can obtain the required uniform a
priori estimates of corresponding solutions.
