Development of Implosions of Solutions to the Three-Dimensional Degenerate Compressible Navier-Stokes Equations

TL;DR

This paper investigates finite-time implosion phenomena in solutions to the three-dimensional degenerate compressible Navier-Stokes equations with nonlinear viscosity, identifying conditions under which smooth solutions develop singularities due to insufficient viscous damping.

Contribution

It introduces a threshold for the power-law density dependence of viscosity, demonstrating finite-time implosion for initial data below this threshold, advancing understanding of singularity formation in degenerate Navier-Stokes equations.

Findings

Finite-time implosion occurs for certain initial data with nonlinear viscosity.

A threshold exponent determines the transition between global regularity and finite-time blow-up.

Decay estimates for velocity gradients help control the singular growth of density.

Abstract

A fundamental open problem in the theory of the multidimensional compressible Navier-Stokes equations is whether smooth solutions can develop singularities in finite time. For constant viscosity coefficients, recent remarkable results show that there exist smooth initial data for which the corresponding smooth solutions of the barotropic flow undergo finite-time implosion at the origin, with the density blowing up to infinity. In contrast, when the viscosity coefficients depend linearly on the density (as in the shallow water case), it has been established that, for general large spherically symmetric initial data, the solutions remain globally regular. These results indicate that the qualitative behavior of multidimensional solutions is sensitive to the structure of the viscosity coefficients. In this paper, we investigate the case of nonlinear viscosity coefficients with power-law density dependence. We identify a threshold value, depending on the adiabatic exponent, such that, for any power below this threshold, there exists a class of smooth initial data with strictly positive density for which the corresponding smooth solutions implode in finite time at the origin. The key issue is to show that, in this regime, the degenerate viscous terms are not sufficiently strong to suppress the convective mechanism driving the implosion. Establishing this rigorously is highly nontrivial due to the degenerate structure of the Navier-Stokes equations. To overcome this difficulty, we first derive a pointwise estimate for the density and then obtain spatial decay estimates for the velocity gradient via carefully constructed weighted high-order energy estimates and interpolation inequalities. The resulting decay rate is sufficiently rapid to compensate for the singular growth of the density, leading to uniform-in-time control of the viscous terms and ultimately to the formation of implosion.