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

Abstract: We consider the initial-boundary value problem (IBVP) for the isentropic compressible Navier-Stokes equations (\textbf{CNS}) in the domain exterior to a ball in $\mathbb R^d$ $(d=2\ \text{or} \ 3)$. When viscosity coefficients are given as a constant multiple of the mass density $\rho$, based on some analysis of the nonlinear structure of this system, we prove the global existence of the unique spherically symmetric classical solution for (large) initial data with spherical symmetry and far field vacuum in some inhomogeneous Sobolev spaces. Moreover, the solutions we obtained have the conserved total mass and finite total energy. $\rho$ keeps positive in the domain considered but decays to zero in the far field, which is consistent with the facts that the total mass is conserved, and \textbf{CNS} is a model of non-dilute fluids where $\rho$ is bounded away from the vacuum. To prove the existence, on the one hand, we consider a well-designed reformulated structure by introducing some new variables, which, actually, can transfer the degeneracies of the time evolution and the viscosity to the possible singularity of some special source terms. On the other hand, it is observed that, for the spherically symmetric flow, the radial projection of the so-called effective velocity $\boldsymbol{v} =U+\nabla \varphi(\rho)$ ($U$ is the velocity of the fluid, and $\varphi(\rho)$ is a function of $\rho$ defined via the shear viscosity coefficient $\mu(\rho)$: $\varphi'(\rho)=2\mu(\rho)/\rho^2$), verifies a damped transport equation which provides the possibility to obtain its upper bound. Then combined with the BD entropy estimates, one can obtain the required uniform a priori estimates of the solution. It is worth pointing out that the frame work on the well-posedness theory established here can be applied to the shallow water equations.