ON STATIC LIQUID LANE-EMDEN STARS

doi:10.1007/s10473-025-0611-y

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (6): 2579-2590.doi: 10.1007/s10473-025-0611-y

ON STATIC LIQUID LANE-EMDEN STARS

Shuang MIAO, Yangyang WANG^*

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

收稿日期:2025-03-12 修回日期:2025-05-28 出版日期:2025-11-25 发布日期:2025-11-14

ON STATIC LIQUID LANE-EMDEN STARS

Shuang MIAO, Yangyang WANG^*

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Received:2025-03-12 Revised:2025-05-28 Online:2025-11-25 Published:2025-11-14
Contact: ^*Yangyang WANG, E-mail: 2019302010102@whu.edu.cn
About author:Shuang MIAO, E-mail: shuang.m@whu.edu.cn
Supported by:
National Key R & D Program of China (2021YFA1001700) and the National Science Foundation of China (12426203, 12221001).

摘要/Abstract

摘要： 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.

关键词: compressible liquid, static solution, uniqueness

Abstract: 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.

Key words: compressible liquid, static solution, uniqueness

中图分类号:

35Q31

Shuang MIAO, Yangyang WANG. ON STATIC LIQUID LANE-EMDEN STARS[J]. 数学物理学报(英文版), 2025, 45(6): 2579-2590.

Shuang MIAO, Yangyang WANG. ON STATIC LIQUID LANE-EMDEN STARS[J]. Acta mathematica scientia,Series B, 2025, 45(6): 2579-2590.

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ON STATIC LIQUID LANE-EMDEN STARS

ON STATIC LIQUID LANE-EMDEN STARS

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