Linear stability of liquid Lane-Emden stars
Author:
King Ming Lam
Journal:
Quart. Appl. Math.
MSC (2020):
Primary 35Q85, 35Q35; Secondary 35B35
DOI:
https://doi.org/10.1090/qam/1677
Published electronically:
August 28, 2023
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Abstract: We establish various qualitative properties of liquid Lane-Emden stars in $\mathbb {R}^d$, including bounds for its density profile $\rho$ and radius $R$. Using them we prove that against radial perturbations, the liquid Lane-Emden stars are linearly stable when $\gamma \geq 2(d-1)/d$; linearly stable when $\gamma <2(d-1)/d$ for stars with small relative central density $\rho (0)-\rho (R)$; and linearly unstable when $\gamma <2(d-1)/d$ for stars with large central density. Such dependence on central density is not seen in the gaseous Lane-Emden stars.
References
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- Gerhard Rein, Non-linear stability of gaseous stars, Arch. Ration. Mech. Anal. 168 (2003), no. 2, 115–130. MR 1991989, DOI 10.1007/s00205-003-0260-y
- S. L. Shapiro and S. A. Teukolsky, Black Holes, White Dwarfs and Neutron Stars, John Wiley & Sons, 1983.
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References
- S. Chandrasekhar, An introduction to the study of stellar structure, University of Chicago Press, Chicago, 1939.
- Demetrios Christodoulou, The formation of shocks in 3-dimensional fluids, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2007. MR 2284927, DOI 10.4171/031
- Daniel Coutand and Steve Shkoller, Well-posedness in smooth function spaces for the moving-boundary three-dimensional compressible Euler equations in physical vacuum, Arch. Ration. Mech. Anal. 206 (2012), no. 2, 515–616. MR 2980528, DOI 10.1007/s00205-012-0536-1
- Daniel Coutand, Jason Hole, and Steve Shkoller, Well-posedness of the free-boundary compressible 3-D Euler equations with surface tension and the zero surface tension limit, SIAM J. Math. Anal. 45 (2013), no. 6, 3690–3767. MR 3139610, DOI 10.1137/120888697
- Bernard Dacorogna and Jürgen Moser, On a partial differential equation involving the Jacobian determinant, Ann. Inst. H. Poincaré C Anal. Non Linéaire 7 (1990), no. 1, 1–26 (English, with French summary). MR 1046081, DOI 10.1016/S0294-1449(16)30307-9
- Yinbin Deng, Jianlin Xiang, and Tong Yang, Blowup phenomena of solutions to Euler-Poisson equations, J. Math. Anal. Appl. 286 (2003), no. 1, 295–306. MR 2009638, DOI 10.1016/S0022-247X(03)00487-6
- Daniel Ginsberg, Hans Lindblad, and Chenyun Luo, Local well-posedness for the motion of a compressible, self-gravitating liquid with free surface boundary, Arch. Ration. Mech. Anal. 236 (2020), no. 2, 603–733. MR 4072680, DOI 10.1007/s00205-019-01477-3
- P. Goldreich and S. Weber, Homologously collapsing stellar cores, Astrophys. J. 238 (1980), 991–997.
- Xumin Gu and Zhen Lei, Local well-posedness of the three dimensional compressible Euler-Poisson equations with physical vacuum, J. Math. Pures Appl. (9) 105 (2016), no. 5, 662–723 (English, with English and French summaries). MR 3479188, DOI 10.1016/j.matpur.2015.11.010
- M. Hadžić, J. Jang, and K. M. Lam, Nonradial stability of self-similarly expanding Goldreich-Weber stars, Preprint, arXiv:2212.11420.
- Mahir Hadžić and J. Juhi Jang, A class of global solutions to the Euler-Poisson system, Comm. Math. Phys. 370 (2019), no. 2, 475–505. MR 3994577, DOI 10.1007/s00220-019-03525-1
- Mahir Hadžić and Juhi Jang, Nonlinear stability of expanding star solutions of the radially symmetric mass-critical Euler-Poisson system, Comm. Pure Appl. Math. 71 (2018), no. 5, 827–891. MR 3794516, DOI 10.1002/cpa.21721
- Mahir Hadžić, Zhiwu Lin, and Gerhard Rein, Stability and instability of self-gravitating relativistic matter distributions, Arch. Ration. Mech. Anal. 241 (2021), no. 1, 1–89. MR 4271955, DOI 10.1007/s00205-021-01647-2
- Mahir Hadžić and Zhiwu Lin, Turning point principle for relativistic stars, Comm. Math. Phys. 387 (2021), no. 2, 729–759. MR 4315660, DOI 10.1007/s00220-021-04197-6
- Z. Hao and S. Miao, On nonlinear instability of liquid Lane-Emden stars, Preprint, arXiv:2304.06217.
- J. Mark Heinzle, (In)finiteness of spherically symmetric static perfect fluids, Classical Quantum Gravity 19 (2002), no. 11, 2835–2851. MR 1911313, DOI 10.1088/0264-9381/19/11/307
- M. Ifrim and D. Tataru, The compressible Euler equations in a physical vacuum: a comprehensive Eulerian approach, Preprint, arXiv:2007.05668, 2020.
- Juhi Jang and Nader Masmoudi, Well-posedness of compressible Euler equations in a physical vacuum, Comm. Pure Appl. Math. 68 (2015), no. 1, 61–111. MR 3280249, DOI 10.1002/cpa.21517
- Juhi Jang, Nonlinear instability in gravitational Euler-Poisson systems for $\gamma =\frac 65$, Arch. Ration. Mech. Anal. 188 (2008), no. 2, 265–307. MR 2385743, DOI 10.1007/s00205-007-0086-0
- Juhi Jang, Nonlinear instability theory of Lane-Emden stars, Comm. Pure Appl. Math. 67 (2014), no. 9, 1418–1465. MR 3245100, DOI 10.1002/cpa.21499
- Hans Lindblad, Well posedness for the motion of a compressible liquid with free surface boundary, Comm. Math. Phys. 260 (2005), no. 2, 319–392. MR 2177323, DOI 10.1007/s00220-005-1406-6
- Zhiwu Lin and Chongchun Zeng, Separable Hamiltonian PDEs and turning point principle for stability of gaseous stars, Comm. Pure Appl. Math. 75 (2022), no. 11, 2511–2572. MR 4491877
- Song-Sun Lin, Stability of gaseous stars in spherically symmetric motions, SIAM J. Math. Anal. 28 (1997), no. 3, 539–569. MR 1443608, DOI 10.1137/S0036141095292883
- Tao Luo and Joel Smoller, Existence and non-linear stability of rotating star solutions of the compressible Euler-Poisson equations, Arch. Ration. Mech. Anal. 191 (2009), no. 3, 447–496. MR 2481067, DOI 10.1007/s00205-007-0108-y
- A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, 1984. MR 748308, DOI 10.1007/978-1-4612-1116-7
- Juhi Jang and Tetu Makino, Linearized analysis of barotropic perturbations around spherically symmetric gaseous stars governed by the Euler-Poisson equations, J. Math. Phys. 61 (2020), no. 5, 051508, 42. MR 4097802, DOI 10.1063/1.5088843
- Tetu Makino, Blowing up solutions of the Euler-Poisson equation for the evolution of gaseous stars, Proceedings of the Fourth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics (Kyoto, 1991), 1992, pp. 615–624. MR 1194464, DOI 10.1080/00411459208203801
- Shuang Miao, Sohrab Shahshahani, and Sijue Wu, Well-posedness of free boundary hard phase fluids in Minkowski background and their Newtonian limit, Camb. J. Math. 9 (2021), no. 2, 269–350. MR 4325283, DOI 10.4310/CJM.2021.v9.n2.a1
- T. A. Oliynyk, Dynamical relativistic liquid bodies, Preprint, arXiv:1907.08192, July 2019.
- Gerhard Rein, Non-linear stability of gaseous stars, Arch. Ration. Mech. Anal. 168 (2003), no. 2, 115–130. MR 1991989, DOI 10.1007/s00205-003-0260-y
- S. L. Shapiro and S. A. Teukolsky, Black Holes, White Dwarfs and Neutron Stars, John Wiley & Sons, 1983.
- Thomas C. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys. 101 (1985), no. 4, 475–485. MR 815196
- Yuri Trakhinin, Local existence for the free boundary problem for nonrelativistic and relativistic compressible Euler equations with a vacuum boundary condition, Comm. Pure Appl. Math. 62 (2009), no. 11, 1551–1594. MR 2560044, DOI 10.1002/cpa.20282
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Additional Information
King Ming Lam
Affiliation:
Department of Mathematics, University College London, London WC1E 6XA, United Kingdom
ORCID:
0000-0002-4429-8983
Email:
king.lam.19@ucl.ac.uk, lamkingming@gmail.com
Keywords:
Analysis of partial differential equations,
mathematical physics
Received by editor(s):
June 3, 2023
Received by editor(s) in revised form:
July 3, 2023
Published electronically:
August 28, 2023
Additional Notes:
The author was supported by EPSRC studentship grant EP/R513143/1.
Article copyright:
© Copyright 2023
Brown University
