Global Regular Solutions of the Compressible Navier-Stokes Equations with Nonlinear Density-Dependent Viscosities and Large Initial Data of Spherical Symmetry

TL;DR

This paper proves the global existence of regular solutions to the compressible Navier-Stokes equations with density-dependent viscosities and large spherical initial data, addressing vacuum and singularity challenges.

Contribution

It establishes the first global well-posedness results for these equations with nonlinear density-dependent viscosities and large initial data of spherical symmetry.

Findings

Proved global regular solutions exist under specified conditions.

Obtained uniform upper bounds for the density despite vacuum and singularities.

Method applicable to cases with strictly positive density.

Abstract

For the physically important case in which the viscosity coefficients depend on the density $\rho$ through a power law (i.e., $\rho^\delta$ with some exponent $\delta \in (\frac{1}{2},1)$), we establish the global well-posedness of regular solutions of the compressible Navier-Stokes equations for barotropic flow with large initial data of spherical symmetry in two and three spatial dimensions. The initial density considered here is positive everywhere but vanishes in the far field, ensuring that the resulting solutions satisfy the conservation laws of total mass and momentum. The most crucial step in our analysis is to obtain a uniform upper bound for the density, which is challenging due to the combined difficulties of degeneracy near the far-field vacuum, coordinate singularity at the origin, and nonlinearity of viscosity coefficients. Furthermore, the methodology developed here can also be applied to the corresponding problem in which the density remains strictly away from the vacuum.