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

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. For the barotropic Navier-Stokes system with constant viscosity coefficients, recent striking results have shown that such implosions do indeed occur. In this paper, we show that the situation is fundamentally different 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. Our results hold for all adiabatic exponents $γ\in (1,\infty)$ in two dimensions, and for physical adiabatic exponents $γ\in (1, 3)$ in three dimensions, without any restriction on the size of the initial data. To achieve these results, we make carefully designed weighted radial estimates via a region segmentation method, which is the key for obtaining uniform control over both the density and the effective velocity. The methodology developed here should also be useful for solving other related nonlinear partial differential equations involving similar difficulties.