Global Regular Solutions of the Multidimensional Degenerate Compressible Navier-Stokes Equations with Large Initial Data of Spherical Symmetry
Gui-Qiang G. Chen, Jiawen Zhang, Shengguo Zhu
TL;DR
The paper establishes global-in-time regularity for multidimensional degenerate compressible Navier-Stokes equations with density-dependent viscosity under spherical symmetry, showing that large initial data do not develop cavitation or implosion, in contrast to constant-viscosity models. It introduces an enlarged reformulation with φ and ψ and a region-segmentation strategy to obtain uniform density and effective-velocity bounds, enabling global a priori estimates. The results cover 2D for all γ>1 and 3D for γ<3, with both far-field vacuum and strictly positive density cases, and extend naturally to shallow-water type systems in 2D. The techniques—radial weighted estimates, BD entropy, and a careful transport/hyperbolic-parabolic coupling—provide a framework applicable to related nonlinear PDEs with degeneracies at low density.
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.