In this paper, the Cauchy problem for the three-dimensional (3-D) isentropic compressible Navier-Stokes equations is considered. When viscosity coefficients are given as a constant multiple of the density's power (${\rho }^{\delta }$ with $0<\delta <1$), based on some analysis of the nonlinear structure of this system, we identify a class of initial data admitting a local regular solution with far field vacuum and finite energy in some inhomogeneous Sobolev spaces by introducing some new variables and initial compatibility conditions, which solves an open problem of degenerate viscous flow partially mentioned by Bresh-Desjardins-Metivier [3], Jiu-Wang-Xin [11] and so on. Moreover, in contrast to the classical theory in the case of the constant viscosity, we show that one cannot obtain any global regular solution whose $L^{\infty }$ norm of u decays to zero as time t goes to infinity.

在本文中，考虑了三维 (3-D) 等熵可压缩 Navier-Stokes 方程的柯西问题。当粘度系数以密度幂的常数倍数给出时（${\rho }^{\delta }$ 和 $0<\delta <1$)，基于对该系统非线性结构的一些分析，我们通过引入一些新的变量和初始相容条件，确定了一类在一些非齐次Sobolev空间中允许具有远场真空和有限能量的局部正则解的初始数据，从而求解Bresh-Desjardins-Metivier [3]、Jiu-Wang-Xin [11] 等部分提到的退化粘性流的开放问题。此外，在恒定粘度的情况下，与经典理论相反，我们表明人们无法获得任何全局正则解，其${升}^{\infty }$随着时间t趋于无穷大，u 的范数衰减为零。