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arxiv: 2605.17121 · v1 · pith:EPCN2BX6new · submitted 2026-05-16 · 🧮 math.AP · math-ph· math.MP· physics.flu-dyn

Global Regular Solutions of the Compressible Navier-Stokes Equations with Nonlinear Density-Dependent Viscosities and Large Initial Data of Spherical Symmetry

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

Pith reviewed 2026-05-20 14:38 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MPphysics.flu-dyn
keywords compressible Navier-Stokesglobal well-posednessdensity-dependent viscosityspherical symmetrylarge initial databarotropic flowvacuum states
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The pith

The compressible Navier-Stokes equations admit global regular solutions with large spherically symmetric initial data when viscosities scale as density to a power between one half and one.

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

This paper proves that the barotropic compressible Navier-Stokes equations possess global regular solutions in two and three dimensions even when the initial data is large and spherically symmetric. The viscosity coefficients are allowed to depend nonlinearly on density through the power law with exponent strictly between one half and one. The initial density stays positive but approaches zero at large distances, so the resulting solutions conserve total mass and momentum. The main technical advance is a uniform upper bound on density that closes the estimates despite degeneracy near vacuum and the coordinate singularity at the origin.

Core claim

For viscosity coefficients depending on density through the power law ρ^δ with δ strictly in (1/2,1), the compressible Navier-Stokes equations for barotropic flow admit global regular solutions with large initial data of spherical symmetry in two and three spatial dimensions. The initial density is positive everywhere but vanishes in the far field, ensuring conservation of total mass and momentum. The analysis obtains a uniform upper bound on density by overcoming the combined difficulties of degeneracy near the far-field vacuum, coordinate singularity at the origin, and nonlinearity of the viscosities.

What carries the argument

A uniform upper bound on the density obtained via a priori estimates that control the combined effects of vacuum degeneracy, origin singularity, and nonlinear viscosity dependence.

Load-bearing premise

The exponent in the viscosity power law must lie strictly between one half and one so that the estimates produce a uniform density upper bound.

What would settle it

An explicit spherically symmetric initial datum with large amplitude for which density blows up in finite time when the exponent equals 0.6 would disprove the global existence result.

read the original abstract

For the physically important case in which the viscosity coefficients depend on the density $\rho$ through a power law (i.e., $\rho^\delta$ with some exponent $\delta \in (\frac{1}{2},1)$), we establish the global well-posedness of regular solutions of the compressible Navier-Stokes equations for barotropic flow with large initial data of spherical symmetry in two and three spatial dimensions. The initial density considered here is positive everywhere but vanishes in the far field, ensuring that the resulting solutions satisfy the conservation laws of total mass and momentum. The most crucial step in our analysis is to obtain a uniform upper bound for the density, which is challenging due to the combined difficulties of degeneracy near the far-field vacuum, coordinate singularity at the origin, and nonlinearity of viscosity coefficients. Furthermore, the methodology developed here can also be applied to the corresponding problem in which the density remains strictly away from the vacuum.

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 paper establishes global well-posedness of regular solutions to the barotropic compressible Navier-Stokes system in 2D and 3D with spherically symmetric large initial data, where the viscosity coefficients are nonlinear functions of density of the form ρ^δ with δ ∈ (1/2,1). The initial density is positive everywhere yet vanishes at infinity, so that total mass and momentum are conserved. The central technical step is the derivation of a time-uniform upper bound on the density that overcomes degeneracy at vacuum, the coordinate singularity at the origin in spherical coordinates, and the nonlinearity of the viscosities.

Significance. If the claimed uniform density bound and subsequent global regularity hold, the result would constitute a meaningful extension of the theory of compressible viscous flows to density-dependent viscosities and large data under spherical symmetry. The approach may also apply to the non-vacuum case mentioned in the abstract. The work addresses a physically relevant regime and provides a concrete range for the exponent δ that closes the estimates.

major comments (1)
  1. [Section 4 (a priori estimates for density bound)] The uniform upper bound on density is load-bearing for the global existence claim. The abstract and introduction acknowledge the origin singularity (terms of the form (n-1)u/r) and degeneracy, but the manuscript must explicitly show that the Gronwall constant in the density-maximum estimate remains independent of the size of the initial momentum; otherwise the bound may depend on initial data and fail to be uniform.
minor comments (2)
  1. [Introduction / Equation (1.1)] Clarify the precise form of both shear and bulk viscosity coefficients (e.g., whether they are identical ρ^δ or have different exponents).
  2. [Abstract / Concluding remarks] The statement that the methodology extends to the strictly positive density case should be accompanied by a brief remark on which estimates simplify when inf ρ > 0.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive major comment. We address the point raised regarding the density bound in Section 4 below and will incorporate clarifications in the revised version.

read point-by-point responses

  1. Referee: [Section 4 (a priori estimates for density bound)] The uniform upper bound on density is load-bearing for the global existence claim. The abstract and introduction acknowledge the origin singularity (terms of the form (n-1)u/r) and degeneracy, but the manuscript must explicitly show that the Gronwall constant in the density-maximum estimate remains independent of the size of the initial momentum; otherwise the bound may depend on initial data and fail to be uniform.

    Authors: We agree that explicit tracking of constants is important for transparency. In the derivation of the time-uniform upper bound on density in Section 4, we first control the velocity field via the energy estimates that incorporate the spherical symmetry terms (n-1)u/r and the nonlinear viscosity ρ^δ. This yields a differential inequality for M(t) := sup ρ(·,t) of the form dM/dt ≤ K M, where the coefficient K is determined by the L^∞ bound on the velocity (obtained from the momentum equation and Sobolev embeddings adapted to spherical coordinates). Applying Gronwall's inequality then produces M(t) ≤ M(0) exp(K t). However, the structure of the estimates (using the conservation of momentum and the decay at infinity) ensures that K is in fact independent of t, giving a time-uniform bound. The constant K does depend on the initial momentum through the initial energy, but since the initial data are fixed (however large), this dependence is admissible and does not prevent the bound from being uniform in time. To address the referee's request, we will revise Section 4 to include an explicit computation of K in terms of the initial momentum and to state clearly that the resulting density bound is independent of t for any fixed initial data. This clarification does not change the main theorem but improves readability. revision: yes

Circularity Check

0 steps flagged

No circularity: standard a priori estimates close independently

full rationale

The paper derives global well-posedness via energy estimates and a uniform density upper bound that is obtained directly from the continuity and momentum equations under the stated power-law viscosity assumption δ ∈ (1/2,1). The spherical symmetry reduction and origin/far-field singularities are controlled by explicit lower-order terms whose domination is shown using the exponent range, without any reduction of the target bound to a fitted parameter, self-definition, or load-bearing self-citation. The argument remains self-contained against the PDE system and initial data assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on obtaining a uniform upper bound for density. This depends on the power-law viscosity range, spherical symmetry reduction, and standard Sobolev-type estimates for the barotropic system; no free parameters or new entities are introduced.

axioms (1)
  • domain assumption Standard assumptions of PDE theory for compressible barotropic fluids, including smoothness of the pressure law and well-posedness of the reduced spherically symmetric system.
    Invoked implicitly to justify the formulation and the applicability of energy methods and maximum principles.

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