Global Regular Solutions of the Degenerate Compressible Navier-Stokes Equations with Large Initial Data of Spherical Symmetry

TL;DR

This paper proves that large spherically symmetric solutions to the degenerate compressible Navier-Stokes equations remain globally regular, avoiding singularities like cavitation or implosion, even with vacuum at infinity.

Contribution

It establishes global regularity results for large initial data with density-dependent viscosity, overcoming coordinate singularities at the origin.

Findings

Solutions stay regular for all time in 2D and 3D.

No cavitation or implosion occurs for the considered initial data.

Results apply to a wide range of adiabatic exponents without size restrictions.

Abstract

A fundamental open problem in the theory of the compressible Navier-Stokes equations is whether regular spherically symmetric flows can develop singularities, such as cavitation or implosion, in finite time. A formidable challenge lies in how the well-known coordinate singularity at the origin can be overcome to control the lower or upper bound of the density. In this paper, when the viscosity coefficients are degenerately density-dependent (as in the shallow water equations), we prove that, for general large spherically symmetric initial data with bounded positive density, solutions remain globally regular and cannot undergo cavitation or implosion in two and three spatial dimensions. Moreover, the far-field vacuum is allowed for the data under consideration here. Our results hold for all adiabatic exponents $\gamma\in(1,\infty)$ in two dimensions, and for physical adiabatic exponents $\gamma\in (1, 3)$ in three dimensions, without any restriction on the size of the initial data.