pith. sign in

arxiv: 2604.18306 · v3 · submitted 2026-04-20 · 🧮 math.AP

Global Well-Posedness of Classical Solutions to the Multi-Dimensional Degenerate Compressible Navier-Stokes Equations with Large Spherically Symmetric Initial Data

Qinghao Lei This is my paper

Pith reviewed 2026-05-10 03:53 UTC · model grok-4.3

classification 🧮 math.AP
keywords Navier-Stokes equationsdegenerate viscosityglobal existencespherically symmetricclassical solutionscompressible fluidsbarotropic flow
0
0 comments X

The pith

Spherically symmetric classical solutions exist globally for arbitrarily large initial data in the degenerate compressible Navier-Stokes equations under specified conditions on the viscosity exponent.

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

This paper establishes the global existence and uniqueness of spherically symmetric classical solutions to the barotropic compressible Navier-Stokes equations with degenerate viscosity in two and three dimensions. The viscosity is taken as power laws μ(ρ)=ρ^α and λ(ρ)=(α-1)ρ^α that obey the BD entropy relation, with initial data that are spherically symmetric and have positive far-field density. For N=2 the conditions are α greater than about 0.54 and any γ>1, while for N=3 α greater than about 0.68 and γ less than 6α-3. The solutions are shown to avoid vacuum states at all finite times. A sympathetic reader would care because this shows that symmetry can control the behavior of these singular equations even when data are large, preventing finite-time blowup.

Core claim

For the multi-dimensional barotropic compressible Navier-Stokes equations with degenerate viscosity coefficients μ(ρ)=ρ^α and λ(ρ)=(α−1)ρ^α satisfying the BD entropy relation, and for arbitrarily large spherically symmetric initial data with non-vacuum far-field density, there exist global-in-time unique spherically symmetric classical solutions when α ∈ (0.54369,1) and γ ∈ (1,∞) for N=2, and when α ∈ (0.67661,1) and γ ∈ (1,6α−3) for N=3 in both bounded domains and the whole space. In the two-dimensional case, the restriction on α can be relaxed to α ∈ (9−6√2,1) under additional weighted integrability conditions on the initial data. The solution remains free of vacuum for all positive times.

What carries the argument

The specific choice of viscosity coefficients that satisfy the BD entropy relation, which provides additional control on the density through an entropy estimate, together with the assumption of spherical symmetry that reduces the problem to a one-dimensional radial system.

If this is right

  • The density stays positive everywhere for all time if it starts positive at infinity.
  • Global classical solutions are unique within the class of spherically symmetric functions.
  • The result applies equally to the whole space and to bounded domains in three dimensions.
  • For two-dimensional flows, the admissible range for the viscosity exponent α can be extended downward if the data satisfy extra decay conditions at infinity.

Where Pith is reading between the lines

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

  • If the symmetry assumption is dropped, solutions might form vacuum or singularities at lower thresholds for α.
  • These techniques could be adapted to study other degenerate parabolic systems arising in fluid mechanics or materials science.
  • Computational verification of the critical α values could be done by simulating the radial equations with large data.

Load-bearing premise

The initial data must be exactly spherically symmetric and the far-field density must be positive and constant.

What would settle it

Constructing or numerically finding a spherically symmetric initial datum with α below the stated lower bound that leads to either vacuum formation or a singularity in finite time would falsify the global existence claim.

read the original abstract

This paper is concerned with the global existence and uniqueness of classical solutions to the barotropic compressible Navier-Stokes equations with degenerate viscosity coefficients in three-dimensional bounded domains or in the whole space $\mathbb{R}^N$ $(N=2,3)$ with non-vacuum far-field density. Specifically, we assume that the shear viscosity coefficient $\mu(\rho)=\rho^\alpha$ and the bulk viscosity coefficient $\lambda(\rho)=(\alpha-1)\rho^\alpha$, which satisfy the BD entropy relation. For arbitrarily large spherically symmetric initial data, we establish the global existence and uniqueness of spherically symmetric classical solutions under the following conditions: for $N=2$, $\alpha \in (0.54369,1)$ and $\gamma \in (1,\infty)$; for $N=3$ (both bounded domains and the whole space), $\alpha \in (0.67661,1)$ and $\gamma \in (1,6\alpha-3)$. In the two-dimensional case $\mathbb{R}^2$, the restriction on $\alpha$ can be further relaxed to $\alpha \in (9-6\sqrt{2},1)$ provided that the initial data satisfy additional weighted integrability conditions. Moreover, we show that the solution will not exhibit vacuum in any finite time provided that no vacuum is present initially.

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 establishes global existence and uniqueness of spherically symmetric classical solutions to the barotropic compressible Navier-Stokes equations with degenerate viscosities μ(ρ)=ρ^α and λ(ρ)=(α−1)ρ^α (satisfying the BD entropy relation) in N=2,3, for arbitrarily large spherically symmetric initial data with non-vacuum far-field density. The result holds in bounded domains or R^N under the conditions: N=2 with α∈(0.54369,1) and γ>1; N=3 with α∈(0.67661,1) and γ∈(1,6α−3). A relaxed lower bound α>9−6√2 is given in 2D under extra weighted integrability. The solutions remain non-vacuum for all finite time if initially so.

Significance. If the a priori estimates close, the result extends the theory of degenerate compressible Navier-Stokes to large data by exploiting spherical symmetry to reduce to effectively one-dimensional radial problems and combining BD entropy with weighted spherical norms r^{N−1}. This yields explicit (numerical) ranges for the degeneracy exponent α that permit control of density away from zero and infinity together with velocity gradients, which is a meaningful advance over small-data results in the degenerate setting.

major comments (2)
  1. [a priori estimates section (leading to the conditions on α and γ)] The lower bounds α>0.54369 (N=2) and α>0.67661 (N=3) are load-bearing for the global existence claim, as they are the minimal values allowing absorption of the convective and pressure terms into the BD-entropy dissipation ∫ρ^{2α−1}|∇ρ|^2 after integration against the spherical weights. In the a priori estimate section deriving the higher-order energy inequality (the step that produces the numerical thresholds), the precise interpolation constants and test-function choices used to close the Gronwall inequality should be stated explicitly so that the decimals can be independently verified; a small shift in those constants would alter the admissible interval for γ near the upper limit 6α−3.
  2. [energy estimates for N=3 (the step closing the bound on γ)] For N=3 the upper restriction γ<6α−3 must be compatible with the lower bound on α; when α↓0.67661 the admissible γ-interval collapses to a very small neighborhood of 1. The manuscript should verify in the energy estimates that the constants remain uniform as γ approaches this upper threshold from below, or identify any additional restrictions that appear in that limit.
minor comments (2)
  1. The abstract states the ranges with six-decimal precision; if these arise from solving a quadratic or cubic inequality, it would be clearer to record the exact algebraic condition on α before giving the numerical root.
  2. The title refers to “multi-dimensional” while the results are stated only for N=2,3; a brief clarification in the introduction would avoid any ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below.

read point-by-point responses

  1. Referee: The lower bounds α>0.54369 (N=2) and α>0.67661 (N=3) are load-bearing. In the a priori estimate section deriving the higher-order energy inequality, the precise interpolation constants and test-function choices used to close the Gronwall inequality should be stated explicitly so that the decimals can be independently verified.

    Authors: We agree that explicit constants will improve verifiability. In the revised manuscript we will state the precise interpolation inequalities (including the specific forms of Young's inequality and the weighted Sobolev embeddings adapted to spherical symmetry) and the test functions employed in the higher-order estimates. The thresholds 0.54369 and 0.67661 are the minimal values obtained by solving the resulting algebraic conditions for absorption into the BD-entropy dissipation; we will include the exact expressions leading to these bounds. revision: yes

  2. Referee: For N=3 the upper restriction γ<6α−3 must be compatible with the lower bound on α; when α↓0.67661 the admissible γ-interval collapses to a very small neighborhood of 1. The manuscript should verify that the constants remain uniform as γ approaches this upper threshold from below, or identify any additional restrictions.

    Authors: In the energy estimates the absorption constants depend on the gap 6α−3−γ but remain finite for any fixed γ strictly below the threshold. As α approaches its lower bound from above, the admissible γ-interval shrinks toward (1,1+), yet for each fixed pair (α,γ) satisfying the stated conditions the estimates close with finite constants. We will add a remark clarifying that no further restrictions appear and that uniformity holds on any compact subinterval of the admissible range. revision: partial

Circularity Check

0 steps flagged

No circularity: standard a priori estimates close under explicit exponent conditions

full rationale

The derivation proceeds by reducing the N-dimensional spherically symmetric system to a 1D radial problem, applying the given BD entropy structure (μ=ρ^α, λ=(α-1)ρ^α) to obtain weighted energy inequalities, and closing higher-order estimates via integration by parts and Gronwall-type arguments on the radial domain. The numerical thresholds on α (0.54369 for N=2, 0.67661 for N=3) are the explicit lower bounds required for the dissipation terms to dominate the convective and pressure contributions for the stated γ intervals; they are obtained by solving the resulting algebraic inequalities on the exponents rather than by fitting to data or by self-referential definition. No step reduces the claimed existence result to its own inputs by construction, and the proof relies on standard PDE techniques without load-bearing self-citations or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard PDE analysis tools and the BD entropy relation for the chosen viscosity; no free parameters are fitted to data, and no new physical entities are introduced.

axioms (2)
  • standard math Standard Sobolev embeddings, interpolation inequalities, and basic energy estimates for parabolic systems hold in the spherically symmetric setting.
    Invoked to close the a priori estimates that prevent blow-up and vacuum formation.
  • domain assumption The BD entropy relation is satisfied exactly by the power-law viscosities μ(ρ)=ρ^α and λ(ρ)=(α−1)ρ^α.
    This algebraic identity is used to obtain the key dissipation structure for the degenerate system.

pith-pipeline@v0.9.0 · 5546 in / 1537 out tokens · 71475 ms · 2026-05-10T03:53:07.322039+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

52 extracted references · 52 canonical work pages · 1 internal anchor

  1. [1]

    R. A. Adams and J. J. F. Fournier, Sobolev spaces, Elsevier/Academic Press, Amsterdam, 2003

    work page 2003

  2. [2]

    Bresch and B

    D. Bresch and B. Desjardins, Sur un mod` ele de Saint-Venant visqueux et sa limite quasi-g´ eostrophique, C. R. Math. Acad. Sci. Paris335(2002), no. 12, 1079–1084

    work page 2002

  3. [3]

    Bresch and B

    D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys.238(2003), no. 1-2, 211–223

    work page 2003

  4. [4]

    Bresch, B

    D. Bresch, B. Desjardins and C.-K. Lin, On some compressible fluid models: Ko- rteweg, lubrication, and shallow water systems, Comm. Partial Differential Equa- tions28(2003), no. 3-4, 843–868

    work page 2003

  5. [5]

    Bresch, A

    D. Bresch, A. F. Vasseur and C. Yu, Global existence of entropy-weak solutions to the compressible Navier-Stokes equations with non-linear density dependent viscosities, J. Eur. Math. Soc. (JEMS)24(2022), no. 5, 1791–1837. 44

    work page 2022

  6. [6]

    G. C. Cai and J. Li, Existence and exponential growth of global classical solutions to the compressible Navier-Stokes equations with slip boundary conditions in 3D bounded domains, Indiana Univ. Math. J.72(2023), no. 6, 2491–2546

    work page 2023

  7. [7]

    Y. Cao, H. Li and S. Zhu, Global spherically symmetric solutions to degenerate compressible Navier-Stokes equations with large data and far field vacuum, Calc. Var. Partial Differential Equations63(2024), no. 9, Paper No. 230, 46 pp

    work page 2024

  8. [8]

    Chapman and T

    S. Chapman and T. G. Cowling, The mathematical theory of non-uniform gases: An account of the kinetic theory of viscosity, thermal conduction, and diffusion in gases, Cambridge Univ. Press, New York, 1960

    work page 1960

  9. [9]

    G.-Q. G. Chen, J. Zhang and S. Zhu, Global Regular Solutions of the Multidimen- sional Degenerate Compressible Navier-Stokes Equations with Large Initial Data of Spherical Symmetry, arXiv:2512.18545

    work page internal anchor Pith review Pith/arXiv arXiv

  10. [10]

    Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl. (9)83(2004), no. 2, 243–275

    work page 2004

  11. [11]

    Cho and H

    Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equa- tions with nonnegative initial densities, Manuscripta Math.120(2006), no. 1, 91–129

    work page 2006

  12. [12]

    H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isen- tropic compressible fluids, J. Differential Equations190(2003), no. 2, 504–523

    work page 2003

  13. [13]

    L. C. Evans, Partial differential equations, second edition, Graduate Studies in Mathematics, 19, Amer. Math. Soc., Providence, RI, 2010

    work page 2010

  14. [14]

    X. Fan, J. X. Li and J. Li, Global existence of strong and weak solutions to 2D compressible Navier-Stokes system in bounded domains with large data and vacuum, Arch. Ration. Mech. Anal.245(2022), no. 1, 239–278

    work page 2022

  15. [15]

    Feireisl, A

    E. Feireisl, A. Novotn´ y and H. Petzeltov´ a, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech.3(2001), no. 4, 358–392

    work page 2001

  16. [16]

    Z. H. Guo, Q. Jiu and Z. Xin, Spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients, SIAM J. Math. Anal.39(2008), no. 5, 1402–1427

    work page 2008

  17. [17]

    Z. H. Guo, H. L. Li and Z. Xin, Lagrange structure and dynamics for solutions to the spherically symmetric compressible Navier-Stokes equations, Comm. Math. Phys.309(2012), no. 2, 371–412

    work page 2012

  18. [18]

    Hoff, Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data, Trans

    D. Hoff, Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data, Trans. Amer. Math. Soc.303(1987), no. 1, 169–181

    work page 1987

  19. [19]

    Hoff, Global solutions of the Navier-Stokes equations for multidimensional com- pressible flow with discontinuous initial data, J

    D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional com- pressible flow with discontinuous initial data, J. Differential Equations120(1995), no. 1, 215–254. 45

    work page 1995

  20. [20]

    Hoff, Strong convergence to global solutions for multidimensional flows of com- pressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch

    D. Hoff, Strong convergence to global solutions for multidimensional flows of com- pressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Rational Mech. Anal.132(1995), no. 1, 1–14

    work page 1995

  21. [21]

    Hoff, Compressible flow in a half-space with Navier boundary conditions, J

    D. Hoff, Compressible flow in a half-space with Navier boundary conditions, J. Math. Fluid Mech.7(2005), no. 3, 315–338

    work page 2005

  22. [22]

    Huang and J

    X.-D. Huang and J. Li, Existence and blowup behavior of global strong solutions to the two-dimensional barotrpic compressible Navier-Stokes system with vacuum and large initial data, J. Math. Pures Appl. (9)106(2016), no. 1, 123–154

    work page 2016

  23. [23]

    Huang and J

    X.-D. Huang and J. Li, Global well-posedness of classical solutions to the Cauchy problem of two-dimensional barotropic compressible Navier-Stokes system with vacuum and large initial data, SIAM J. Math. Anal.54(2022), no. 3, 3192–3214

    work page 2022

  24. [24]

    Huang, J

    X.-D. Huang, J. Li and Z. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math.65(2012), no. 4, 549–585

    work page 2012

  25. [25]

    Huang, W

    X.-D. Huang, W. Meng and X. Zhang, On global classical and weak solutions with arbitrary large initial data to the multi-dimensional viscous Saint-Venant system and compressible Navier-Stokes equations subject to the BD entropy condition under spherical symmetry, arXiv:2512.15029

    work page arXiv

  26. [26]

    Huang et al., Global large strong solutions to the radially symmetric compress- ible Navier-Stokes equations in 2D solid balls, J

    X. Huang et al., Global large strong solutions to the radially symmetric compress- ible Navier-Stokes equations in 2D solid balls, J. Differential Equations396(2024), 393–429

    work page 2024

  27. [27]

    Jiang and P

    S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Comm. Math. Phys.215(2001), no. 3, 559– 581

    work page 2001

  28. [28]

    Q. Jiu, Y. Wang and Z. Xin, Global well-posedness of 2D compressible Navier- Stokes equations with large data and vacuum, J. Math. Fluid Mech.16(2014), no. 3, 483–521

    work page 2014

  29. [29]

    Q. Jiu, Y. Wang and Z. Xin, Global classical solution to two-dimensional com- pressible Navier-Stokes equations with large data inR 2, Phys. D376/377(2018), 180–194

    work page 2018

  30. [30]

    Y. I. Kanel’, A model system of equations for the one-dimensional motion of a gas, Differencial’ nye Uravnenija4(1968), 721–734

    work page 1968

  31. [31]

    Kawashima and T

    S. Kawashima and T. Nishida, Global solutions to the initial value problem for the equations of one-dimensional motion of viscous polytropic gases, J. Math. Kyoto Univ.21(1981), no. 4, 825–837

    work page 1981

  32. [32]

    A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, Prikl. Mat. Meh.41(1977), no. 2J. Appl. Math. Mech.41(1977), no. 2

    work page 1977

  33. [33]

    Y. Li, R. H. Pan and S. Zhu, On classical solutions for viscous polytropic fluids with degenerate viscosities and vacuum, Arch. Ration. Mech. Anal.234(2019), no. 3, 1281–1334. 46

    work page 2019

  34. [34]

    J. Li, Z. Liang, On local classical solutions to the Cauchy problem of the two- dimensional barotropic compressible Navier-Stokes equations with vacuum, J. Math. Pures Appl. (9)102(2014), no. 4, 640–671

    work page 2014

  35. [35]

    Global Existence of Weak Solutions to the Barotropic Compressible Navier-Stokes Flows with Degenerate Viscosities

    J. Li and Z. Xin, Global Existence of Weak Solutions to the Barotropic Compress- ible Navier-Stokes Flows with Degenerate Viscosities, arXiv:1504.06826

    work page Pith review arXiv

  36. [36]

    Li and Z

    J. Li and Z. Xin, Global well-posedness and large time asymptotic behavior of classical solutions to the compressible Navier-Stokes equations with vacuum, Ann. PDE5(2019), no. 1, Paper No. 7, 37 pp

    work page 2019

  37. [37]

    Lions, Mathematical Topics in Fluid Mechanics

    P.L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1: Incompressible Mod- els, Oxford University Press, New York, 1996

    work page 1996

  38. [38]

    Lions, Mathematical Topics in Fluid Mechanics

    P.L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2: Compressible Models, Oxford University Press, New York, 1998

    work page 1998

  39. [39]

    Matsumura, T

    A. Matsumura, T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ.20(1) (1980), 67–104

    work page 1980

  40. [40]

    Mellet and A

    A. Mellet and A. F. Vasseur, On the barotropic compressible Navier-Stokes equa- tions, Comm. Partial Differential Equations32(2007), no. 1-3, 431–452

    work page 2007

  41. [41]

    Nash, Le probl` eme de Cauchy pour les ´ equations diff´ erentielles d’un fluide g´ en´ eral, Bull

    J. Nash, Le probl` eme de Cauchy pour les ´ equations diff´ erentielles d’un fluide g´ en´ eral, Bull. Soc. Math. France90(1962), 487–497 (French)

    work page 1962

  42. [42]

    Nirenberg, On elliptic partial differential equations, Ann

    L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3)13(1959), 115–162

    work page 1959

  43. [43]

    Novotn´ y and I

    A. Novotn´ y and I. Straˇ skraba, Introduction to the mathematical theory of com- pressible flow, Oxford Lecture Series in Mathematics and its Applications, 27, Oxford Univ. Press, Oxford, 2004

    work page 2004

  44. [44]

    Salvi and I

    R. Salvi and I. Straˇ skraba, Global existence for viscous compressible fluids and their behavior ast→ ∞, J. Fac. Sci. Univ. Tokyo Sect. IA Math.40(1993), no. 1, 17–51

    work page 1993

  45. [45]

    Serre, Solutions faibles globales des ´ equations de Navier-Stokes pour un fluide compressible, C

    D. Serre, Solutions faibles globales des ´ equations de Navier-Stokes pour un fluide compressible, C. R. Acad. Sci. Paris S´ er. I Math.303(1986), no. 13, 639–642

    work page 1986

  46. [46]

    Serre, Sur l’´ equation monodimensionnelle d’un fluide visqueux, compressible et conducteur de chaleur, C

    D. Serre, Sur l’´ equation monodimensionnelle d’un fluide visqueux, compressible et conducteur de chaleur, C. R. Acad. Sci. Paris S´ er. I Math.303(1986), no. 14, 703–706

    work page 1986

  47. [47]

    Serrin, On the uniqueness of compressible fluid motions, Arch

    J. Serrin, On the uniqueness of compressible fluid motions, Arch. Rational Mech. Anal.3(1959), 271–288 (1959)

    work page 1959

  48. [48]

    T.-P. Liu, Z. Xin and T. Yang, Vacuum states for compressible flow, Discrete Contin. Dynam. Systems4(1998), no. 1, 1–32

    work page 1998

  49. [49]

    V. A. Vaigant and A. V. Kazhikhov, On existence of global solutions to the two- dimensional Navier–Stokes equations for a compressible viscous fluid, Sib. Math. J. 36 (6) (1995) 1283–1316. 47

    work page 1995

  50. [50]

    A. F. Vasseur and C. Yu, Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations, Invent. Math.206(2016), no. 3, 935–974

    work page 2016

  51. [51]

    Zhang and J

    P. Zhang and J. N. Zhao, The existence of local solutions for the compressible Navier-Stokes equations with the density-dependent viscosities, Commun. Math. Sci.12(2014), no. 7, 1277–1302

    work page 2014

  52. [52]

    Zhang, Spherically symmetric strong solution of compressible flow with large data and density-dependent viscosities, J

    X. Zhang, Spherically symmetric strong solution of compressible flow with large data and density-dependent viscosities, J. Math. Anal. Appl.549(2025), no. 2, Paper No. 129488, 29 pp. 48

    work page 2025