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arxiv: 2512.18545 · v2 · submitted 2025-12-21 · 🧮 math.AP · math-ph· math.MP· physics.flu-dyn

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

Gui-Qiang G. Chen , Jiawen Zhang , Shengguo Zhu This is my paper

Pith reviewed 2026-05-16 21:10 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MPphysics.flu-dyn
keywords compressible Navier-Stokesspherical symmetryglobal regularitydegenerate viscositycavitationimplosionadiabatic exponentfar-field vacuum
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The pith

Degenerate density-dependent viscosity ensures global regularity of large spherically symmetric solutions to the compressible Navier-Stokes equations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that solutions to the compressible Navier-Stokes equations with spherically symmetric large initial data stay globally regular when the viscosity coefficients depend degenerately on density. This prevents the formation of singularities like cavitation, where density reaches zero, or implosion in finite time. The result applies in two and three dimensions for broad ranges of the adiabatic exponent and permits vacuum at far field. A reader would care because it addresses a fundamental open problem on whether such flows can develop singularities, showing that degeneracy in viscosity stabilizes the solutions even for large data.

Core claim

For the compressible Navier-Stokes equations with degenerately density-dependent viscosity coefficients, any spherically symmetric initial data with bounded positive density yields a globally regular solution in two and three spatial dimensions that does not develop cavitation or implosion. The result holds for all gamma in (1, infinity) in two dimensions and gamma in (1, 3) in three dimensions, with no restriction on the size of the initial data and allowance for far-field vacuum.

What carries the argument

Degenerately density-dependent viscosity coefficients, which provide the necessary control over density bounds to overcome the coordinate singularity at the origin.

If this is right

  • Global-in-time existence of regular solutions for arbitrary large data.
  • Absence of cavitation and implosion singularities.
  • Applicability to models like the shallow water equations.
  • Validity for all adiabatic exponents gamma in (1, infinity) in 2D and (1, 3) in 3D.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The degeneracy of viscosity may be key to regularity in other symmetric or reduced-dimension fluid problems.
  • This regularity result could guide the development of stable numerical schemes for long-time integration of such equations.
  • Similar techniques might apply to related systems with vacuum boundaries.

Load-bearing premise

The viscosity coefficients are degenerately dependent on density, vanishing appropriately as density goes to zero.

What would settle it

An explicit example or numerical computation of a spherically symmetric initial data leading to density zero or velocity infinity in finite time would disprove the global regularity.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript claims to prove global regularity of spherically symmetric solutions to the degenerate compressible Navier-Stokes equations (with density-dependent viscosity coefficients) for large initial data with bounded positive density. Solutions remain regular for all time in 2D (any γ>1) and 3D (γ∈(1,3)), with no cavitation or implosion, and far-field vacuum is permitted.

Significance. If the estimates close, the result would resolve a longstanding open question on finite-time singularity formation (cavitation/implosion) for large-data spherical flows under degenerate viscosity, extending beyond small-data regimes and allowing vacuum at infinity.

major comments (1)
  1. [A priori estimates (density lower bound via particle-path integration)] The lower bound inf ρ(t,x)>0 is load-bearing for the no-cavitation claim. After reduction to radial coordinates the continuity equation is ∂tρ + ∂r(ρu) + (n-1)r^{-1}ρu =0. Integrating along particle paths gives log ρ(t)=log ρ0 −∫(∂ru+(n-1)r^{-1}u)ds. For n=3 the 2r^{-1}u term is singular at r=0; the BD-type entropy and weighted Sobolev bounds cited in the a priori estimates section do not furnish a uniform bound on sup|u/r| for arbitrarily large negative initial radial velocity near the origin, leaving open the possibility that the integral diverges to +∞ in finite time before dissipation acts.
minor comments (2)
  1. [Abstract and Theorem 1.1] The abstract states γ∈(1,3) in 3D; confirm whether the endpoint γ=3 is included or excluded and state the precise interval in the main theorem.
  2. [Introduction, §1] Explicitly define the precise form of the degenerate viscosity coefficients (μ(ρ)∼ρ, λ(ρ)) and the admissible range of the adiabatic exponent in the introduction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for the detailed and constructive report. The main concern raised pertains to the control of the density lower bound through particle path integration in the presence of the coordinate singularity at the origin. We provide a point-by-point response below and will incorporate clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: [A priori estimates (density lower bound via particle-path integration)] The lower bound inf ρ(t,x)>0 is load-bearing for the no-cavitation claim. After reduction to radial coordinates the continuity equation is ∂tρ + ∂r(ρu) + (n-1)r^{-1}ρu =0. Integrating along particle paths gives log ρ(t)=log ρ0 −∫(∂ru+(n-1)r^{-1}u)ds. For n=3 the 2r^{-1}u term is singular at r=0; the BD-type entropy and weighted Sobolev bounds cited in the a priori estimates section do not furnish a uniform bound on sup|u/r| for arbitrarily large negative initial radial velocity near the origin, leaving open the possibility that the integral diverges to +∞ in finite time before dissipation acts.

    Authors: We thank the referee for this observation. Our a priori estimates, based on the BD entropy and spherical symmetry, provide control over the weighted L^2 norms of u/r and its derivatives. Specifically, the estimates yield a bound on the L^infty norm of u/r through 1D Sobolev embedding in the radial variable with appropriate weights r^{n-1}. This bound is uniform in time and depends on the initial data and the dissipated energy. Consequently, the time integral of sup |u/r| remains finite, preventing the divergence of the log ρ integral in finite time. For the case of large negative initial radial velocity near the origin, the dissipation term in the momentum equation, which includes viscosity terms like div(ρ ∇u), acts to regularize the flow before any potential singularity forms. We will add a new lemma in the revised version to explicitly derive the bound sup |u/r| ≤ C from the existing a priori bounds, addressing this point directly. revision: yes

Circularity Check

0 steps flagged

No circularity; a priori estimates close independently

full rationale

The paper derives global regularity for the degenerate spherically symmetric compressible Navier-Stokes system via BD entropy estimates and weighted Sobolev bounds in radial coordinates. These controls on density lower bounds and velocity gradients are obtained directly from the continuity and momentum equations without reducing to the target statement by definition or by self-citation load-bearing. The handling of the 1/r singularity at the origin is achieved through explicit weighted integrals that remain independent of the final regularity claim. No fitted inputs are renamed as predictions, and no uniqueness theorem is imported from overlapping prior work to force the result. The argument is self-contained against the initial data assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard PDE energy estimates and Sobolev embeddings for the spherically symmetric reduction; no free parameters or new entities are introduced.

axioms (1)
  • standard math Standard a priori energy estimates and Sobolev embeddings hold for the reduced spherically symmetric system
    Invoked to close higher-order estimates after handling the origin singularity.

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Forward citations

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Reference graph

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