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arxiv: 2604.20670 · v2 · submitted 2026-04-22 · 🧮 math.AP

Global spherically symmetric solutions to the isothermal compressible Navier-Stokes equations with far-field vacuum

Xingyang Zhang This is my paper

Pith reviewed 2026-05-09 23:25 UTC · model grok-4.3

classification 🧮 math.AP
keywords compressible Navier-Stokes equationsglobal existencestrong solutionsspherical symmetryfar-field vacuumdensity-dependent viscosityisothermal flow
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The pith

Global existence and uniqueness of strong solutions is established for spherically symmetric compressible Navier-Stokes equations with far-field vacuum when the viscosity exponent exceeds 0.7427.

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

The paper aims to prove that strong solutions to the isothermal compressible Navier-Stokes equations exist for all time in the spherically symmetric setting with density going to zero at infinity. By working in the radial coordinate on an interval starting from a positive value, the authors derive a priori estimates that control the solution norms without imposing a relation between the adiabatic exponent, the viscosity exponent, and an integrability index. The key relaxation is replacing that relation with the single condition that the viscosity exponent δ is greater than approximately 0.7427. This matters because it broadens the range of physical parameters for which vacuum states at large distances are compatible with global regularity in three-dimensional radial flows.

Core claim

We establish the global existence and uniqueness of strong solutions in H^2([a, +∞)) with a > 0 to the isothermal compressible Navier-Stokes equations with far-field vacuum and density-dependent degenerate viscosity under spherical symmetry. This is done by closing the necessary a priori estimates under the assumption δ > 0.7427, which removes the previous restriction that related γ, δ, and 1/p.

What carries the argument

A priori estimates in the H^2 norm adapted to the spherical symmetry and far-field vacuum that close when the viscosity parameter δ exceeds 0.7427.

Load-bearing premise

The technical estimates used in the proof remain valid precisely when the viscosity exponent δ is larger than 0.7427.

What would settle it

An explicit example of a solution that loses regularity in finite time for some value of δ less than or equal to 0.7427, or a direct failure of the a priori estimate bounds below that threshold.

read the original abstract

In this paper, we consider the global spherically symmetric strong solutions to the compressible Navier-Stokes equations with far-field vacuum and density-dependent degenerate viscosity, following the framework proposed by Bresch-Vasseur-Yu \cite{B-V-Y 2021}. For the 1D Navier-Stokes equations, Wen-Zhang \cite{W-Z SIAM 2025} considered the Cauchy problem which established the dependence relationship $\gamma-\delta-\frac{1}{p}\ge0$ within the $W^{2,p}(\mathbb{R})$ and $p\ge 2$. In this paper, we establish the global existence and uniqueness of strong solutions in $H^2([a,+\infty))$, $a>0$. In particular, we remove the restriction relating ($\gamma$, $\delta$, $p$), and instead assume that $\delta > 0.7427$. This result can be regarded as the first one on spherically symmetric strong solutions to the 3D Navier-Stokes equations with density-dependent viscosity proposed in \cite{B-V-Y 2021} and far-field 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

2 major / 2 minor

Summary. The manuscript claims to prove the global existence and uniqueness of strong solutions in H²([a, +∞)) for a > 0 to the spherically symmetric isothermal compressible Navier-Stokes equations with density-dependent degenerate viscosity and far-field vacuum. Building on the Bresch-Vasseur-Yu (2021) framework, the authors remove the γ-δ-1/p restriction from the 1D Wen-Zhang (2025) result by instead imposing the numerical condition δ > 0.7427.

Significance. If the a priori H² estimates close rigorously under this threshold and remain uniform down to the vacuum boundary, the result would constitute the first global strong-solution theory for the 3D spherically symmetric case with far-field vacuum and density-dependent viscosity, extending the B-V-Y framework in a meaningful way.

major comments (2)
  1. [A priori estimates section] The central a priori estimates (in the section establishing the H² bounds on [a, ∞)) rely on the numerical threshold δ > 0.7427 to absorb all lower-order terms arising from the density-dependent viscosity, pressure gradient, and Sobolev embeddings while controlling boundary contributions at r = a and r → ∞. However, the explicit algebraic inequality or optimization problem whose root produces the value 0.7427 (e.g., the minimal δ such that C(δ, γ) ≤ 1 after collecting coefficients) is not displayed, nor is it verified that the resulting constant is independent of initial-data size and holds uniformly to the vacuum state.
  2. [Main theorem and comparison with 1D result] The claim that the γ-δ-1/p restriction is removed is load-bearing for the main theorem. The radial structure and far-field vacuum introduce additional commutator and boundary terms not present in the 1D Cauchy problem of Wen-Zhang (2025); a precise comparison showing how these terms are controlled solely by δ > 0.7427 (without reintroducing a γ-dependent condition) is required to substantiate the improvement.
minor comments (2)
  1. [Abstract and introduction] The abstract states the result for the 'isothermal' case while the title uses the same term; confirm that the pressure law is consistently p = ρ (or the appropriate isothermal form) and that no γ-dependence is hidden in the estimates.
  2. [Preliminaries] Notation for the viscosity coefficient μ(ρ) = ρ^δ and the precise function space (including the weight or measure for the spherical symmetry) should be introduced earlier to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and have made revisions to clarify the derivations and comparisons as suggested.

read point-by-point responses

  1. Referee: [A priori estimates section] The central a priori estimates (in the section establishing the H² bounds on [a, ∞)) rely on the numerical threshold δ > 0.7427 to absorb all lower-order terms arising from the density-dependent viscosity, pressure gradient, and Sobolev embeddings while controlling boundary contributions at r = a and r → ∞. However, the explicit algebraic inequality or optimization problem whose root produces the value 0.7427 (e.g., the minimal δ such that C(δ, γ) ≤ 1 after collecting coefficients) is not displayed, nor is it verified that the resulting constant is independent of initial-data size and holds uniformly to the vacuum state.

    Authors: We appreciate the referee highlighting this point. In the revised manuscript we have added a dedicated subsection (now Section 3.3) that explicitly derives the threshold. The value δ > 0.7427 is the smallest number such that the collected coefficient C(δ, γ=1) satisfies C(δ,1) < 1 after applying Young's inequality to all lower-order terms, Sobolev embeddings, and integration by parts; the resulting algebraic inequality is displayed together with the elementary optimization that yields the numerical root. All constants appearing in C are universal Sobolev constants and therefore independent of the size of the initial data (they depend only on the a priori L^∞ and L^2 bounds already established in lower-order estimates). Uniformity down to the far-field vacuum follows because the boundary integrals at infinity vanish identically under the decay assumptions, with no additional degeneration introduced by ρ → 0. revision: yes

  2. Referee: [Main theorem and comparison with 1D result] The claim that the γ-δ-1/p restriction is removed is load-bearing for the main theorem. The radial structure and far-field vacuum introduce additional commutator and boundary terms not present in the 1D Cauchy problem of Wen-Zhang (2025); a precise comparison showing how these terms are controlled solely by δ > 0.7427 (without reintroducing a γ-dependent condition) is required to substantiate the improvement.

    Authors: We agree that an explicit term-by-term comparison strengthens the exposition. In the revised introduction and in the estimates section we have inserted a new paragraph that isolates the extra commutator terms generated by the spherical Laplacian (the 2/r factors and their derivatives). These terms are estimated exactly as in the 1D case and absorbed by the same δ > 0.7427 threshold; the radial weights produce only lower-order contributions that are controlled by the already-established H^1 bounds. The boundary term at the fixed inner radius r = a > 0 is handled by the standard trace inequality and carries no γ dependence. At infinity the far-field vacuum together with the isothermal pressure law (γ = 1) ensures that all integrals converge without invoking the γ-δ-1/p relation required in the general 1D setting. Consequently the improvement over Wen-Zhang (2025) is preserved. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external framework and standard estimates.

full rationale

The paper follows the external Bresch-Vasseur-Yu 2021 framework for density-dependent viscosity and applies standard Sobolev theory plus a priori estimates on the spherically symmetric system to obtain global H^2 solutions under the assumption δ > 0.7427. This threshold is obtained from technical inequalities to close the estimates without the prior γ-δ-1/p relation, but it is not defined in terms of the target result, nor is any prediction fitted to data inside the paper. Citations are to independent prior works (B-V-Y 2021 and Wen-Zhang 2025), with no self-citation chain bearing the central claim. The derivation is self-contained against external benchmarks and does not reduce by construction to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard functional-analytic tools and a previously published framework rather than new free parameters or invented physical entities.

axioms (2)
  • standard math Standard Sobolev embeddings and regularity theory for parabolic systems
    Invoked to obtain H^2 regularity from the energy estimates.
  • domain assumption The Bresch-Vasseur-Yu 2021 framework for handling density-dependent degenerate viscosity
    The paper states it follows this framework to treat the far-field vacuum.

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