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

TL;DR

This paper proves the global existence and uniqueness of spherically symmetric solutions to the degenerate compressible Navier-Stokes equations with large initial data and far field vacuum in three and two dimensions, using a reformulated structure and effective velocity analysis.

Contribution

It introduces a new analytical framework and variable transformation to handle degeneracies and establish global well-posedness for large data with vacuum in spherical symmetry.

Findings

Existence of global spherically symmetric solutions with vacuum

Solutions conserve total mass and have finite energy

Effective velocity satisfies a damped transport equation

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.