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We consider the initial-boundary value problem (IBVP) for the isentropiccompressible Navier-Stokes equations (\textbf{CNS}) in the domain exterior to aball in $\mathbb R^d$ $(d=2\ \text{or} \ 3)$. When viscosity coefficients aregiven as a constant multiple of the mass density $\rho$, based on some analysisof the nonlinear structure of this system, we prove the global existence of theunique spherically symmetric classical solution for (large) initial data withspherical symmetry and far field vacuum in some inhomogeneous Sobolev spaces.Moreover, the solutions we obtained have the conserved total mass and finitetotal energy. $\rho$ keeps positive in the domain considered but decays to zeroin the far field, which is consistent with the facts that the total mass isconserved, and \textbf{CNS} is a model of non-dilute fluids where $\rho$ isbounded away from the vacuum. To prove the existence, on the one hand, weconsider a well-designed reformulated structure by introducing some newvariables, which, actually, can transfer the degeneracies of the time evolutionand the viscosity to the possible singularity of some special source terms. Onthe other hand, it is observed that, for the spherically symmetric flow, theradial projection of the so-called effective velocity $\boldsymbol{v} =U+\nabla\varphi(\rho)$ ($U$ is the velocity of the fluid, and $\varphi(\rho)$ is afunction of $\rho$ defined via the shear viscosity coefficient $\mu(\rho)$:$\varphi'(\rho)=2\mu(\rho)/\rho^2$), verifies a damped transport equation whichprovides the possibility to obtain its upper bound. Then combined with the BDentropy estimates, one can obtain the required uniform a priori estimates ofthe solution. It is worth pointing out that the frame work on thewell-posedness theory established here can be applied to the shallow waterequations.

Global spherically symmetric solutions to degenerate compressible Navier-Stokes equations with large data and far field vacuum