The liquid Lane-Emden star is a free boundary problem of compressible Euler-Poisson equation which describes motion of celestial bodies. This model admits a class of static solutions parametrized by its central density. According to Lam [9], when the central density is sufficiently small or the adiabatic constant $\gamma\in [\frac43,2]$, the static solutions are linearly stable. In this article, by constructing periodic solutions to the linearized equation, we prove that even though these solutions are linearly stable, they may not decay in time. Moreover we prove that if the sum of the internal energy and potential energy of this model has an minimizer, then it must be the spherically symmetric solution to the static equation, therefore demonstrating their stability from a variational point of view.
Shuang MIAO
,
Yangyang WANG
. ON STATIC LIQUID LANE-EMDEN STARS[J]. Acta mathematica scientia, Series B, 2025
, 45(6)
: 2579
-2590
.
DOI: 10.1007/s10473-025-0611-y
[1] Chandrasekhar S.An Introduction to the Study of Stellar Structure. Chicago: Dover, 1958
[2] Christodoulou D. The formulation of the two-body problem in general relativity.2015-11-19. https://www.youtube.com/watch?v=FFMSteogq40
[3] Deng Y, Liu T P, Yang T, Yao Z A. Solutions of euler-poisson equations for gaseous stars. Archive for Rational Mechanics and Analysis, 2002, 164: 261-285
[4] Goldreich P, Weber S V. Homologously collapsing stellar cores. Astrophysical Journal, 1980, 238(1): 991-997
[5] Hao Z, Miao S. On nonlinear instability of liquid Lane-Emden stars. Calc Var Partial Differential Equations, 2024, 63(6): Paper 157
[6] Jang J. Nonlinear instability in gravitational euler-poisson systems for. Archive for Rational Mechanics and Analysis, 2008, 188(2): 265-307
[7] Jang J. Nonlinear instability theory of Lane-Emden stars. Communications on Pure and Applied Mathematics, 2014, 67(9): 1418-1465
[8] Jang J, Makino T. Linearized analysis of barotropic perturbations around spherically symmetric gaseous stars governed by the euler-poisson equations. Journal of Mathematical Physics, 2020, 61(5): 051508
[9] Lam K M. Linear stability of liquid Lane-Emden stars. Quarterly of Applied Mathematics, 2024, 82: 639-672
[10] Lieb E H, Loss M. Analysis. Providence, RI: American Mathematical Soc, 2001
[11] Lin S S. Stability of gaseous stars in spherically symmetric motions. SIAM J Math Anal, 1997, 28(3): 539-569
[12] Luo T, Smoller J. Nonlinear dynamical stability of newtonian rotating and non-rotating white dwarfs and rotating super-massive stars. Communications in Mathematical Physics, 2008, 284: 425-457
[13] Luo T, Smoller J. Existence and non-linear stability of rotating star solutions of the compressible Euler-Poisson equations. Arch Ration Mech Anal, 2009, 191(3): 447-496
[14] Makino T. Blowing up solutions of the Euler-Poisson equation for the evolution of gaseous stars. Transport Theory Stat Phys, 1992, 21: 615-624
[15] Miao S, Shahshahani S. On tidal energy in Newtonian two-body motion. Camb J Math, 2019, 7(4): 469-585
[16] Rein G. Non-linear stability of gaseous stars. Archive for Rational Mechanics and Analysis, 2003, 168: 115-130
[17] Shapiro S L, Teukolsky S A. Black Holes, White Dwarfs,Neutron Stars: The Physics of Compact Objects. New York: Wiley, 1983
