Revisiting Non-Rotating Star Models: Classical Existence and Uniqueness Theory and Scaling Relations

Hangsheng Chen

arxiv: 2602.02631 · v2 · submitted 2026-02-02 · 🧮 math.AP · math-ph· math.MP

Revisiting Non-Rotating Star Models: Classical Existence and Uniqueness Theory and Scaling Relations

Hangsheng Chen This is my paper

Pith reviewed 2026-05-16 08:28 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MP

keywords non-rotating starsEuler-Poisson systemexistence and uniquenessscaling relationspolytropic starsdensity convergence

0 comments

The pith

Non-rotating stellar models governed by the Euler-Poisson system exist and are unique for given total mass under general equations of state, with scaling relations connecting solutions of different masses.

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

The paper establishes existence of solutions to the Euler-Poisson equations for non-rotating stars by extending the variational methods of Auchmuty and Beals, and proves uniqueness by adapting arguments from the quantum-mechanical setting of Lieb and Yau to the classical Newtonian case. It further introduces a scaling transformation that relates any two solutions that differ only in total mass. As the mass parameter approaches zero, the density functions converge to a limit profile while their supports contract or expand at explicit rates determined by the equation of state.

Core claim

For equations of state satisfying standard regularity and monotonicity conditions, the Euler-Poisson system for non-rotating stars admits a unique solution for each prescribed total mass; moreover, a simple scaling maps any such solution to the solution for any other mass, and the corresponding density functions converge with precise support scaling as mass tends to zero.

What carries the argument

The variational formulation of the Euler-Poisson energy functional together with the explicit scaling map that relates solutions of different total masses.

If this is right

Polytropic gaseous stars inherit explicit mass-scaling formulas for radius and central density.
Density profiles converge in L1 and in suitable Sobolev norms as total mass approaches zero.
Support radii of solutions scale like a power of the mass parameter determined by the adiabatic index.
The same scaling and convergence statements apply to any equation of state satisfying the paper's hypotheses.

Where Pith is reading between the lines

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

The scaling relation may simplify numerical continuation methods that track stellar sequences across masses.
Convergence rates as mass vanishes could be used to construct approximate initial data for dynamical simulations of low-mass stars.

Load-bearing premise

The equation of state must be sufficiently regular and monotone so that the energy functional is well-defined and the uniqueness argument from the quantum setting carries over.

What would settle it

Exhibiting two distinct positive density functions that both solve the Euler-Poisson system for the same total mass and the same equation of state would falsify uniqueness.

read the original abstract

This paper presents a systematic study of the properties of non-rotating stellar models governed by the Euler-Poisson system under general equations of state, including the case of polytropic gaseous stars. We revisit and extend existence results by Auchmuty and Beals \cite{AB71}, adapt the uniqueness results from the quantum mechanical framework of Lieb and Yau \cite{LY87} to the classical Newtonian mechanical setting. The results are also synthesized in McCann \cite{McC06} but without proof. The second work we do is applying a scaling method to establish relations between solutions with different total masses. As the mass tends to zero, we analyze convergence properties of the density functions and identify precise rates for the contraction or extension of their supports.

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.

Desk Editor's Note private letter to a colleague

The paper extends existence results and adapts uniqueness for non-rotating Euler-Poisson stars, then adds scaling relations across masses plus convergence rates as mass goes to zero.

read the letter

The core of this work is a clean extension of the Auchmuty-Beals existence theorem for the Euler-Poisson system under general equations of state, together with an adaptation of the Lieb-Yau uniqueness argument from the quantum to the classical Newtonian setting. The new material is the scaling construction that relates solutions of different total masses and the precise rates at which the density support contracts or expands as mass tends to zero. Those scaling and convergence statements are not in the cited literature and constitute the actual increment here. The synthesis with McCann is also useful for readers who want the classical case written out explicitly. The paper is therefore a solid incremental contribution inside mathematical astrophysics and nonlocal elliptic PDEs. It gives rigorous tools that numerical workers on stellar models can cite without having to redo the variational arguments themselves. The main soft spot is the scaling step. The abstract claims the method works for general monotonic equations of state, yet a straightforward rescaling that preserves both Poisson and hydrostatic balance typically requires the pressure law to be homogeneous of a specific degree. That holds for polytropes but fails for generic P(rho) such as rho + rho squared. If the proofs rely on an unstated homogeneity assumption for the scaling relations, the claimed generality is overstated; if they use a different construction that avoids it, that needs to be spelled out clearly. The existence and uniqueness parts look standard and do not raise the same issue. This paper is aimed at specialists in mathematical stellar models and elliptic free-boundary problems. A reader already familiar with the Auchmuty-Beals and Lieb-Yau papers will get the most out of the scaling analysis and the explicit convergence rates. It is worth sending to peer review. The claims are plausible, the citations are appropriate, and the new scaling content is worth checking in detail even if the generality statement needs tightening.

Referee Report

1 major / 3 minor

Summary. The paper revisits and extends existence results for non-rotating stellar models in the Euler-Poisson system under general equations of state (including polytropes), adapting Auchmuty-Beals theorems and Lieb-Yau uniqueness results to the classical Newtonian setting. It further applies a scaling method to relate solutions with different total masses and analyzes convergence of density functions and support sizes as mass tends to zero.

Significance. If the central claims hold, the work synthesizes and extends classical existence/uniqueness theory for stellar models while providing explicit scaling relations and convergence rates as mass vanishes. These could be useful for analyzing limiting behaviors in gaseous star models. The rigorous adaptation of cited theorems (Auchmuty-Beals, Lieb-Yau, McCann) and focus on general EOS (with polytropic examples) are strengths, though the scaling component requires verification against EOS homogeneity assumptions.

major comments (1)

[§3] §3 (Scaling Relations and Mass Dependence): The claimed scaling method to relate solutions with different total masses via a transformation of the form ρ_λ(x) = λ^α ρ(λ^β x) that preserves both the Poisson equation and hydrostatic balance for a fixed P(ρ) requires P to satisfy a homogeneity condition P(λ^γ ρ) = λ^δ P(ρ). This holds for polytropes but fails for generic monotonic EOS (e.g., P(ρ) = ρ + ρ²). The abstract and introduction assert applicability to general EOS, so the mass→0 convergence rates and support contraction claims rest on an unstated assumption not guaranteed by the regularity/monotonicity conditions used for existence/uniqueness. This is load-bearing for the second main contribution.

minor comments (3)

[Abstract] Abstract: The phrasing 'The results are also synthesized in McCann [McC06] but without proof' is ambiguous; clarify whether the present paper supplies the missing proofs or only cites the synthesis.
[§3.1] Notation: The definition of the scaling exponents α, β, γ, δ in the transformation should be stated explicitly with the precise relation to the EOS degree (if any) to avoid reader confusion when comparing to polytropic cases.
[References] References: Ensure the bibliography includes full details for Auchmuty-Beals (1971) and Lieb-Yau (1987) with page numbers for the specific theorems being adapted.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the important distinction between the general existence/uniqueness results and the scaling analysis. We agree that the scaling relations require an additional homogeneity assumption on the equation of state that is not part of the basic monotonicity conditions. We will revise the manuscript to clarify the scope of these claims.

read point-by-point responses

Referee: [§3] §3 (Scaling Relations and Mass Dependence): The claimed scaling method to relate solutions with different total masses via a transformation of the form ρ_λ(x) = λ^α ρ(λ^β x) that preserves both the Poisson equation and hydrostatic balance for a fixed P(ρ) requires P to satisfy a homogeneity condition P(λ^γ ρ) = λ^δ P(ρ). This holds for polytropes but fails for generic monotonic EOS (e.g., P(ρ) = ρ + ρ²). The abstract and introduction assert applicability to general EOS, so the mass→0 convergence rates and support contraction claims rest on an unstated assumption not guaranteed by the regularity/monotonicity conditions used for existence/uniqueness. This is load-bearing for the second main contribution.

Authors: We thank the referee for this precise observation. The scaling transformation ρ_λ(x) = λ^α ρ(λ^β x) preserves the Euler-Poisson structure for a fixed pressure law P only when P satisfies the homogeneity condition P(λ^γ ρ) = λ^δ P(ρ) for exponents γ, δ determined by the choice of α and β. This condition is satisfied by polytropic equations of state but not by arbitrary monotonic functions such as P(ρ) = ρ + ρ². The existence and uniqueness results (adapted from Auchmuty-Beals and Lieb-Yau) hold under the weaker assumptions of monotonicity and suitable regularity on P. However, the scaling relations and the associated mass-to-zero convergence rates are valid specifically under the homogeneity assumption. We will revise the abstract and introduction to explicitly state that the scaling method and convergence analysis apply to homogeneous equations of state (including polytropes), while the existence/uniqueness theory applies more generally. This clarification will ensure the claims are accurately scoped without affecting the validity of the results under the stated conditions. revision: yes

Circularity Check

0 steps flagged

No circularity detected in the derivation chain

full rationale

The paper revisits and extends existence results from Auchmuty and Beals, adapts uniqueness from Lieb and Yau to the Newtonian setting, and applies a scaling method to relate solutions of different total masses. These steps are presented as building on explicitly cited external theorems without any reduction of the central claims to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The scaling relations and convergence rates as mass tends to zero are derived from the applied method rather than being equivalent to the paper's own inputs by construction. The derivation remains self-contained against the external benchmarks of the cited works.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard mathematical assumptions for the equation of state and on the validity of the cited existence and uniqueness theorems; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

domain assumption The pressure-density relation satisfies the conditions required for the variational formulation of Auchmuty-Beals and the adaptation of Lieb-Yau uniqueness to hold.
Implicit in the abstract's reference to general equations of state including polytropes.

pith-pipeline@v0.9.0 · 5423 in / 1362 out tokens · 30393 ms · 2026-05-16T08:28:10.433435+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We revisit and extend existence results by Auchmuty and Beals, adapt the uniqueness results from the quantum mechanical framework of Lieb and Yau to the classical Newtonian mechanical setting... applying a scaling method to establish relations between solutions with different total masses... A=m^{-2/(3γ-4)}, B=m^{(γ-2)/(3γ-4)}
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2.6 (Non-rotating Stars)... σm satisfies A'(σm(x))=[Vσm(x)+λm]+ ... spt σm contained in ball of radius R0(m)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

59 extracted references · 59 canonical work pages

[1]

Alonso-Orán, B

D. Alonso-Orán, B. Kepka, and J. J. L. Velázquez. Rotating solutions to the incompressible euler-poisson equation with external particle, 2023

work page 2023
[2]

Auchmuty

G. Auchmuty. The global branching of rotating stars. Archive for Rational Mechanics and Analysis, 114(2):179–193, Jun 1991

work page 1991
[3]

J. F. G. Auchmuty and R. Beals. Models of rotating stars. The Astrophysical Journal , 165:L79, 04 1971

work page 1971
[4]

J. F. G. Auchmuty and R. Beals. Variational solutions of some nonlinear free boundary problems. Archive for Rational Mechanics and Analysis , 43:255–271, 1971

work page 1971
[5]

Bahouri, J

H. Bahouri, J. Chemin, and R. Danchin. Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften. Springer Berlin Heidelberg, 2011

work page 2011
[6]

J. Batt, W. Faltenbacher, and E. Horst. Stationary spherically symmetric models in stellar dynamics. Archive for Rational Mechanics and Analysis , 93(2):159–183, Jun 1986. 36

work page 1986
[7]

Binney and S

J. Binney and S. Tremaine. Galactic dynamics . 1987

work page 1987
[8]

H. Brezis. Functional analysis, Sobolev spaces and partial differential equations , volume 2. Springer, 2011

work page 2011
[9]

L. A. Caffarelli and A. Friedman. The shape of axisymmetric rotating fluid. Journal of Functional Analysis, 35(1):109–142, 1980

work page 1980
[10]

Campos, M

J. Campos, M. del Pino, and J. Dolbeault. Relative equilibria in continuous stellar dynamics. Communications in Mathematical Physics , 300(3):765–788, Dec 2010

work page 2010
[11]

Chandrasekhar

S. Chandrasekhar. An Introduction to the Study of Stellar Structure . Astrophysical monographs. University of Chicago Press, 1939

work page 1939
[12]

Chandrasekhar

S. Chandrasekhar. Ellipsoidal figures of equilibrium—an historical account. Communications on Pure and Applied Mathematics , 20(2):251–265, 1967

work page 1967
[13]

Chandrasekhar and E

S. Chandrasekhar and E. A. Milne. The Equilibrium of Distorted Polytropes: (I). The Rotational Problem. Monthly Notices of the Royal Astronomical Society , 93(5):390–406, 03 1933

work page 1933
[14]

Chanillo and Y

S. Chanillo and Y. Y. Li. On diameters of uniformly rotating stars. Communications in Mathe- matical Physics , 166(2):417–430, Dec 1994

work page 1994
[15]

H. Chen. Existence for stable rotating star-planet systems, 2026

work page 2026
[16]

H. Chen. Gradient existence and energy finiteness of local minimizers in the Wasserstein l∞ topology for binary-star systems, 2026

work page 2026
[17]

H. Chen, J. J. L. Velázquez, D. Cobb, and R. F.-W.-U. B. B. eines Werks. Existence for stable rotating star-planet systems, 2024

work page 2024
[18]

D. Cobb. Bounded solutions in incompressible hydrodynamics. Journal of Functional Analysis , 286(5):110290, 2024

work page 2024
[19]

Dal Maso

G. Dal Maso. An introduction to Γ-convergence, volume 8 of Progress in Nonlinear Differential Equations and Their Applications . Birkhäuser Boston, MA, 1993

work page 1993
[20]

Deng, T.-P

Y. Deng, T.-P. Liu, T. Yang, and Z.-a. Yao. Solutions of euler-poisson equations¶for gaseous stars. Archive for Rational Mechanics and Analysis , 164(3):261–285, Sep 2002

work page 2002
[21]

L. Evans. Partial Differential Equations . Graduate studies in mathematics. American Mathe- matical Society, 2010

work page 2010
[22]

Y. Guo. Variational method¶for stable polytropic galaxies. Archive for Rational Mechanics and Analysis, 150(3):209–224, Dec 1999. 37

work page 1999
[23]

Guo and Z

Y. Guo and Z. Li. Unstable and stable galaxy models. Communications in Mathematical Physics , 279(3):789–813, May 2008

work page 2008
[24]

Guo and G

Y. Guo and G. Rein. Existence and stability of camm type steady states in galactic dynamics. Indiana University Mathematics Journal , 48(4):1237–1255, 1999

work page 1999
[25]

Guo and G

Y. Guo and G. Rein. Stable steady states in stellar dynamics. Archive for Rational Mechanics and Analysis , 147(3):225–243, Aug 1999

work page 1999
[26]

Guo and G

Y. Guo and G. Rein. Isotropic steady states in galactic dynamics. Communications in Mathe- matical Physics , 219(3):607–629, Jun 2001

work page 2001
[27]

Guo and G

Y. Guo and G. Rein. Stable models of elliptical galaxies. Monthly Notices of the Royal Astro- nomical Society, 344(4):1296–1306, 10 2003

work page 2003
[28]

Guo and G

Y. Guo and G. Rein. A non-variational approach to nonlinear stability in stellar dynamics applied to the king model. Communications in Mathematical Physics , 271(2):489–509, Apr 2007

work page 2007
[29]

Hadžić, G

M. Hadžić, G. Rein, and C. Straub. On the existence of linearly oscillating galaxies. Archive for Rational Mechanics and Analysis , 243(2):611–696, Feb 2022

work page 2022
[30]

U. Heilig. On Lichtenstein’s analysis of rotating newtonian stars. Annales de l’I.H.P. Physique théorique, 60(4):457–487, 1994

work page 1994
[31]

J. Jang. Nonlinear instability theory of lane-emden stars. Communications on Pure and Applied Mathematics, 67(9):1418–1465, 2014

work page 2014
[32]

Jang and T

J. Jang and T. Makino. On slowly rotating axisymmetric solutions of the euler–poisson equations. Archive for Rational Mechanics and Analysis , 225(2):873–900, Aug 2017

work page 2017
[33]

Jang and T

J. Jang and T. Makino. On rotating axisymmetric solutions of the Euler–Poisson equations. Journal of Differential Equations , 266(7):3942–3972, 2019

work page 2019
[34]

Jang and J

J. Jang and J. Seok. On uniformly rotating binary stars and galaxies. Archive for Rational Mechanics and Analysis , 244(2), 2022

work page 2022
[35]

Jardetzky

W. Jardetzky. Theories of Figures of Celestial Bodies . Dover Books on Physics. Dover Publica- tions, 2013

work page 2013
[36]

Lemou, F

M. Lemou, F. Méhats, and P. Raphael. The orbital stability of the ground states and the singularity formation for the gravitational vlasov poisson system. Archive for Rational Mechanics and Analysis , 189(3):425–468, Sep 2008

work page 2008
[37]

Lemou, F

M. Lemou, F. Méhats, and P. Raphaël. A new variational approach to the stability of gravita- tional systems. Communications in Mathematical Physics , 302(1):161–224, Feb 2011. 38

work page 2011
[38]

Lemou, F

M. Lemou, F. Méhats, and P. Raphaël. Orbital stability of spherical galactic models. Inventiones mathematicae, 187(1):145–194, Jan 2012

work page 2012
[39]

Lemou, F

M. Lemou, F. Méhats, and P. Raphael. Orbital stability and singularity formation for vlasov– poisson systems. Comptes Rendus Mathematique , 341(4):269–274, 2005

work page 2005
[40]

Y. Li. On uniformly rotating stars. Archive for Rational Mechanics and Analysis, 115(4):367–393, Dec 1991

work page 1991
[41]

Lichtenstein

L. Lichtenstein. Untersuchungen über die gleichgewichtsfiguren rotierender flüssigkeiten, deren teilchen einander nach dem newtonschen gesetze anziehen. Mathematische Zeitschrift, 36(1):481– 562, Dec 1933

work page 1933
[42]

E. H. Lieb. Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equa- tion. Studies in Applied Mathematics , 57(2):93–105, 1977

work page 1977
[43]

E. H. Lieb and H.-T. Yau. The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Communications in Mathematical Physics , 112(1):147–174, Mar 1987

work page 1987
[44]

Luo and J

T. Luo and J. Smoller. Nonlinear dynamical stability of newtonian rotating and non-rotating white dwarfs and rotating supermassive stars. Communications in Mathematical Physics , 284(2):425–457, Dec 2008

work page 2008
[45]

Luo and J

T. Luo and J. Smoller. Existence and non-linear stability of rotating star solutions of the compressible euler–poisson equations. Archive for Rational Mechanics and Analysis , 191(3):447– 496, Mar 2009

work page 2009
[46]

R. J. McCann. Stable rotating binary stars and fluid in a tube. Houston Journal of Mathematics , 32(2):603–631, 2006

work page 2006
[47]

E. A. Milne. The Equilibrium of a Rotating Star. Monthly Notices of the Royal Astronomical Society, 83(3):118–147, 01 1923

work page 1923
[48]

I. Newton. Philosophiae naturalis principia mathematica . Early English books online. Jussu Societas Regiæ ac typis Josephi Streater, prostant venales apud Sam. Smith, 1687

work page
[49]

Newton, I

I. Newton, I. Cohen, and A. Whitman. The Principia: Mathematical Principles of Natural Phi- losophy. The Principia: Mathematical Principles of Natural Philosophy. University of California Press, 1999

work page 1999
[50]

G. Rein. Non-linear stability of gaseous stars. Archive for Rational Mechanics and Analysis , 168(2):115–130, Jun 2003. 39

work page 2003
[51]

G. Rein. Chapter 5 - collisionless kinetic equations from astrophysics – the vlasov–poisson system. volume 3 of Handbook of Differential Equations: Evolutionary Equations , pages 383–476. North- Holland, 2007

work page 2007
[52]

W. Rudin. Principles of Mathematical Analysis . International series in pure and applied math- ematics. McGraw-Hill, 1976

work page 1976
[53]

A. Schulze. Existence of axially symmetric solutions to the vlasov-poisson system depending on jacobi’s integral. Communications in Mathematical Sciences , 6:711–727, 09 2008

work page 2008
[54]

S. L. Sobolev. On a theorem of functional analysis. Matematicheskii Sbornik , 4:471–497, 1938

work page 1938
[55]

W. A. Strauss and Y. Wu. Steady states of rotating stars and galaxies. SIAM Journal on Mathematical Analysis, 49(6):4865–4914, 2017

work page 2017
[56]

W. A. Strauss and Y. Wu. Rapidly rotating stars. Communications in Mathematical Physics , 368(2):701–721, Jun 2019

work page 2019
[57]

Shell theorem — Wikipedia, the free encyclopedia, 2024

Wikipedia contributors. Shell theorem — Wikipedia, the free encyclopedia, 2024. [Online; accessed 21-April-2024]

work page 2024
[58]

Wolansky

G. Wolansky. On nonlinear stability of polytropic galaxies. Annales de l’Institut Henri Poincaré C, Analyse non linéaire , 16(1):15–48, 1999

work page 1999
[59]

H. v. Zeipel. The Radiative Equilibrium of a Slightly Oblate Rotating Star. Monthly Notices of the Royal Astronomical Society , 84(9):684–702, 07 1924. 40

work page 1924

[1] [1]

Alonso-Orán, B

D. Alonso-Orán, B. Kepka, and J. J. L. Velázquez. Rotating solutions to the incompressible euler-poisson equation with external particle, 2023

work page 2023

[2] [2]

Auchmuty

G. Auchmuty. The global branching of rotating stars. Archive for Rational Mechanics and Analysis, 114(2):179–193, Jun 1991

work page 1991

[3] [3]

J. F. G. Auchmuty and R. Beals. Models of rotating stars. The Astrophysical Journal , 165:L79, 04 1971

work page 1971

[4] [4]

J. F. G. Auchmuty and R. Beals. Variational solutions of some nonlinear free boundary problems. Archive for Rational Mechanics and Analysis , 43:255–271, 1971

work page 1971

[5] [5]

Bahouri, J

H. Bahouri, J. Chemin, and R. Danchin. Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften. Springer Berlin Heidelberg, 2011

work page 2011

[6] [6]

J. Batt, W. Faltenbacher, and E. Horst. Stationary spherically symmetric models in stellar dynamics. Archive for Rational Mechanics and Analysis , 93(2):159–183, Jun 1986. 36

work page 1986

[7] [7]

Binney and S

J. Binney and S. Tremaine. Galactic dynamics . 1987

work page 1987

[8] [8]

H. Brezis. Functional analysis, Sobolev spaces and partial differential equations , volume 2. Springer, 2011

work page 2011

[9] [9]

L. A. Caffarelli and A. Friedman. The shape of axisymmetric rotating fluid. Journal of Functional Analysis, 35(1):109–142, 1980

work page 1980

[10] [10]

Campos, M

J. Campos, M. del Pino, and J. Dolbeault. Relative equilibria in continuous stellar dynamics. Communications in Mathematical Physics , 300(3):765–788, Dec 2010

work page 2010

[11] [11]

Chandrasekhar

S. Chandrasekhar. An Introduction to the Study of Stellar Structure . Astrophysical monographs. University of Chicago Press, 1939

work page 1939

[12] [12]

Chandrasekhar

S. Chandrasekhar. Ellipsoidal figures of equilibrium—an historical account. Communications on Pure and Applied Mathematics , 20(2):251–265, 1967

work page 1967

[13] [13]

Chandrasekhar and E

S. Chandrasekhar and E. A. Milne. The Equilibrium of Distorted Polytropes: (I). The Rotational Problem. Monthly Notices of the Royal Astronomical Society , 93(5):390–406, 03 1933

work page 1933

[14] [14]

Chanillo and Y

S. Chanillo and Y. Y. Li. On diameters of uniformly rotating stars. Communications in Mathe- matical Physics , 166(2):417–430, Dec 1994

work page 1994

[15] [15]

H. Chen. Existence for stable rotating star-planet systems, 2026

work page 2026

[16] [16]

H. Chen. Gradient existence and energy finiteness of local minimizers in the Wasserstein l∞ topology for binary-star systems, 2026

work page 2026

[17] [17]

H. Chen, J. J. L. Velázquez, D. Cobb, and R. F.-W.-U. B. B. eines Werks. Existence for stable rotating star-planet systems, 2024

work page 2024

[18] [18]

D. Cobb. Bounded solutions in incompressible hydrodynamics. Journal of Functional Analysis , 286(5):110290, 2024

work page 2024

[19] [19]

Dal Maso

G. Dal Maso. An introduction to Γ-convergence, volume 8 of Progress in Nonlinear Differential Equations and Their Applications . Birkhäuser Boston, MA, 1993

work page 1993

[20] [20]

Deng, T.-P

Y. Deng, T.-P. Liu, T. Yang, and Z.-a. Yao. Solutions of euler-poisson equations¶for gaseous stars. Archive for Rational Mechanics and Analysis , 164(3):261–285, Sep 2002

work page 2002

[21] [21]

L. Evans. Partial Differential Equations . Graduate studies in mathematics. American Mathe- matical Society, 2010

work page 2010

[22] [22]

Y. Guo. Variational method¶for stable polytropic galaxies. Archive for Rational Mechanics and Analysis, 150(3):209–224, Dec 1999. 37

work page 1999

[23] [23]

Guo and Z

Y. Guo and Z. Li. Unstable and stable galaxy models. Communications in Mathematical Physics , 279(3):789–813, May 2008

work page 2008

[24] [24]

Guo and G

Y. Guo and G. Rein. Existence and stability of camm type steady states in galactic dynamics. Indiana University Mathematics Journal , 48(4):1237–1255, 1999

work page 1999

[25] [25]

Guo and G

Y. Guo and G. Rein. Stable steady states in stellar dynamics. Archive for Rational Mechanics and Analysis , 147(3):225–243, Aug 1999

work page 1999

[26] [26]

Guo and G

Y. Guo and G. Rein. Isotropic steady states in galactic dynamics. Communications in Mathe- matical Physics , 219(3):607–629, Jun 2001

work page 2001

[27] [27]

Guo and G

Y. Guo and G. Rein. Stable models of elliptical galaxies. Monthly Notices of the Royal Astro- nomical Society, 344(4):1296–1306, 10 2003

work page 2003

[28] [28]

Guo and G

Y. Guo and G. Rein. A non-variational approach to nonlinear stability in stellar dynamics applied to the king model. Communications in Mathematical Physics , 271(2):489–509, Apr 2007

work page 2007

[29] [29]

Hadžić, G

M. Hadžić, G. Rein, and C. Straub. On the existence of linearly oscillating galaxies. Archive for Rational Mechanics and Analysis , 243(2):611–696, Feb 2022

work page 2022

[30] [30]

U. Heilig. On Lichtenstein’s analysis of rotating newtonian stars. Annales de l’I.H.P. Physique théorique, 60(4):457–487, 1994

work page 1994

[31] [31]

J. Jang. Nonlinear instability theory of lane-emden stars. Communications on Pure and Applied Mathematics, 67(9):1418–1465, 2014

work page 2014

[32] [32]

Jang and T

J. Jang and T. Makino. On slowly rotating axisymmetric solutions of the euler–poisson equations. Archive for Rational Mechanics and Analysis , 225(2):873–900, Aug 2017

work page 2017

[33] [33]

Jang and T

J. Jang and T. Makino. On rotating axisymmetric solutions of the Euler–Poisson equations. Journal of Differential Equations , 266(7):3942–3972, 2019

work page 2019

[34] [34]

Jang and J

J. Jang and J. Seok. On uniformly rotating binary stars and galaxies. Archive for Rational Mechanics and Analysis , 244(2), 2022

work page 2022

[35] [35]

Jardetzky

W. Jardetzky. Theories of Figures of Celestial Bodies . Dover Books on Physics. Dover Publica- tions, 2013

work page 2013

[36] [36]

Lemou, F

M. Lemou, F. Méhats, and P. Raphael. The orbital stability of the ground states and the singularity formation for the gravitational vlasov poisson system. Archive for Rational Mechanics and Analysis , 189(3):425–468, Sep 2008

work page 2008

[37] [37]

Lemou, F

M. Lemou, F. Méhats, and P. Raphaël. A new variational approach to the stability of gravita- tional systems. Communications in Mathematical Physics , 302(1):161–224, Feb 2011. 38

work page 2011

[38] [38]

Lemou, F

M. Lemou, F. Méhats, and P. Raphaël. Orbital stability of spherical galactic models. Inventiones mathematicae, 187(1):145–194, Jan 2012

work page 2012

[39] [39]

Lemou, F

M. Lemou, F. Méhats, and P. Raphael. Orbital stability and singularity formation for vlasov– poisson systems. Comptes Rendus Mathematique , 341(4):269–274, 2005

work page 2005

[40] [40]

Y. Li. On uniformly rotating stars. Archive for Rational Mechanics and Analysis, 115(4):367–393, Dec 1991

work page 1991

[41] [41]

Lichtenstein

L. Lichtenstein. Untersuchungen über die gleichgewichtsfiguren rotierender flüssigkeiten, deren teilchen einander nach dem newtonschen gesetze anziehen. Mathematische Zeitschrift, 36(1):481– 562, Dec 1933

work page 1933

[42] [42]

E. H. Lieb. Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equa- tion. Studies in Applied Mathematics , 57(2):93–105, 1977

work page 1977

[43] [43]

E. H. Lieb and H.-T. Yau. The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Communications in Mathematical Physics , 112(1):147–174, Mar 1987

work page 1987

[44] [44]

Luo and J

T. Luo and J. Smoller. Nonlinear dynamical stability of newtonian rotating and non-rotating white dwarfs and rotating supermassive stars. Communications in Mathematical Physics , 284(2):425–457, Dec 2008

work page 2008

[45] [45]

Luo and J

T. Luo and J. Smoller. Existence and non-linear stability of rotating star solutions of the compressible euler–poisson equations. Archive for Rational Mechanics and Analysis , 191(3):447– 496, Mar 2009

work page 2009

[46] [46]

R. J. McCann. Stable rotating binary stars and fluid in a tube. Houston Journal of Mathematics , 32(2):603–631, 2006

work page 2006

[47] [47]

E. A. Milne. The Equilibrium of a Rotating Star. Monthly Notices of the Royal Astronomical Society, 83(3):118–147, 01 1923

work page 1923

[48] [48]

I. Newton. Philosophiae naturalis principia mathematica . Early English books online. Jussu Societas Regiæ ac typis Josephi Streater, prostant venales apud Sam. Smith, 1687

work page

[49] [49]

Newton, I

I. Newton, I. Cohen, and A. Whitman. The Principia: Mathematical Principles of Natural Phi- losophy. The Principia: Mathematical Principles of Natural Philosophy. University of California Press, 1999

work page 1999

[50] [50]

G. Rein. Non-linear stability of gaseous stars. Archive for Rational Mechanics and Analysis , 168(2):115–130, Jun 2003. 39

work page 2003

[51] [51]

G. Rein. Chapter 5 - collisionless kinetic equations from astrophysics – the vlasov–poisson system. volume 3 of Handbook of Differential Equations: Evolutionary Equations , pages 383–476. North- Holland, 2007

work page 2007

[52] [52]

W. Rudin. Principles of Mathematical Analysis . International series in pure and applied math- ematics. McGraw-Hill, 1976

work page 1976

[53] [53]

A. Schulze. Existence of axially symmetric solutions to the vlasov-poisson system depending on jacobi’s integral. Communications in Mathematical Sciences , 6:711–727, 09 2008

work page 2008

[54] [54]

S. L. Sobolev. On a theorem of functional analysis. Matematicheskii Sbornik , 4:471–497, 1938

work page 1938

[55] [55]

W. A. Strauss and Y. Wu. Steady states of rotating stars and galaxies. SIAM Journal on Mathematical Analysis, 49(6):4865–4914, 2017

work page 2017

[56] [56]

W. A. Strauss and Y. Wu. Rapidly rotating stars. Communications in Mathematical Physics , 368(2):701–721, Jun 2019

work page 2019

[57] [57]

Shell theorem — Wikipedia, the free encyclopedia, 2024

Wikipedia contributors. Shell theorem — Wikipedia, the free encyclopedia, 2024. [Online; accessed 21-April-2024]

work page 2024

[58] [58]

Wolansky

G. Wolansky. On nonlinear stability of polytropic galaxies. Annales de l’Institut Henri Poincaré C, Analyse non linéaire , 16(1):15–48, 1999

work page 1999

[59] [59]

H. v. Zeipel. The Radiative Equilibrium of a Slightly Oblate Rotating Star. Monthly Notices of the Royal Astronomical Society , 84(9):684–702, 07 1924. 40

work page 1924