Star dynamics: Collapse vs. expansion

Hadžić, Mahir

doi:10.1090/qam/1638

Author: Mahir Hadžić
Journal: Quart. Appl. Math. 81 (2023), 329-365
MSC (2020): Primary 35Q31, 35Q75, 37N10
DOI: https://doi.org/10.1090/qam/1638
Published electronically: November 17, 2022
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Abstract: We review a series of recent results on global dynamic properties of radially symmetric self-gravitating compressible Euler flows, which naturally arise in the mathematical description of stars. We focus on the role of scaling invariances and how they interact with nonlinearities to generate imploding finite-time singularities as well as expanding star solutions, arising from smooth initial data. This review paper is based on joint works with Y. Guo, J. Jang, and M. Schrecker.

References

J. Binney and S. Tremaine, Galactic dynamics, 2nd ed., Princeton Series in Astrophysics, vol. 4, Princeton University Press, 2008.
H. Bondi, Spherically symmetrical models in general relativity, Monthly Not. Roy. Astr. Soc. 107 (1947), 410–425. MR 26866, DOI 10.1093/mnras/107.5-6.410
Uwe Brauer and Lavi Karp, Local existence of solutions of self gravitating relativistic perfect fluids, Comm. Math. Phys. 325 (2014), no. 1, 105–141. MR 3182488, DOI 10.1007/s00220-013-1854-3
T. Buckmaster, G. Cao-Labora, and J. Gomez-Serrano, Smooth imploding solutions for 3D compressible fluids, Preprint, arXiv:2208.09445.
Michael P. Brenner and Thomas P. Witelski, On spherically symmetric gravitational collapse, J. Statist. Phys. 93 (1998), no. 3-4, 863–899. MR 1666522, DOI 10.1023/B:JOSS.0000033167.19114.b8
M. E. Cahill and A. H. Taub, Spherically symmetric similarity solutions of the Einstein field equations for a perfect fluid, Comm. Math. Phys. 21 (1971), 1–40. MR 281466, DOI 10.1007/BF01646482
B. J. Carr and A. A. Coley, Self-similarity in general relativity, Classical Quantum Gravity 16 (1999), no. 7, R31–R71. MR 1701065, DOI 10.1088/0264-9381/16/7/201
B. J. Carr, A. A. Coley, M. Goliath, U. S. Nilsson, and C. Uggla, Critical phenomena and a new class of self-similar spherically symmetric perfect-fluid solutions, Phys. Rev. D 61 (2000), 081502, 1–5, (2000).
B. J. Carr and Carsten Gundlach, Spacetime structure of self-similar spherically symmetric perfect fluid solutions, Phys. Rev. D (3) 67 (2003), no. 2, 024035, 13. MR 1971907, DOI 10.1103/PhysRevD.67.024035
S. Chandrasekhar, An introduction to the study of stellar structure, University of Chicago Press, 1939.
S. Chandrasekhar, A general variational principle governing the radial and the non-radial oscillations of gaseous masses, Astrophys. J. 139 (1964), 664–674. MR 162619, DOI 10.1086/147792
S. Chandrasekhar, On stars, their evolution and their stability, Rev. Mod. Phys. 56 (1984), 137.
Yvonne Choquet-Bruhat, General relativity and the Einstein equations, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2009. MR 2473363
G-Q. G Chen, L. He, Y. Wang, and D. Yuan, Global solutions of the compressible Euler-Poisson equations with large initial data of spherical symmetry, Preprint, arXiv:2101.03597, 2021.
M. Choptuik, Universality and scaling in gravitational collapse of a massive scalar field, Phys. Rev. Lett. 70 (1993), 9–12.
Demetrios Christodoulou, Violation of cosmic censorship in the gravitational collapse of a dust cloud, Comm. Math. Phys. 93 (1984), no. 2, 171–195. MR 742192, DOI 10.1007/BF01223743
Demetrios Christodoulou, Examples of naked singularity formation in the gravitational collapse of a scalar field, Ann. of Math. (2) 140 (1994), no. 3, 607–653. MR 1307898, DOI 10.2307/2118619
Demetrios Christodoulou, On the global initial value problem and the issue of singularities, Classical Quantum Gravity 16 (1999), no. 12A, A23–A35. MR 1728432, DOI 10.1088/0264-9381/16/12A/302
Demetrios Christodoulou, The instability of naked singularities in the gravitational collapse of a scalar field, Ann. of Math. (2) 149 (1999), no. 1, 183–217. MR 1680551, DOI 10.2307/121023
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
Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2010. MR 2574377, DOI 10.1007/978-3-642-04048-1
M. C. Dafermos and J. Luk, The interior of dynamical vacuum black holes I: The $C^0$-stability of the Kerr Cauchy horizon, Preprint, arXiv:1710.01722v1, 2017.
Mihalis Dafermos and Alan D. Rendall, An extension principle for the Einstein-Vlasov system in spherical symmetry, Ann. Henri Poincaré 6 (2005), no. 6, 1137–1155. MR 2189379, DOI 10.1007/s00023-005-0235-7
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
F. J. Dyson, Dynamics of a spinning gas cloud, J. Math. Mech. 18 (1968), no. 1, 91–101.
Saskia Kind and Jürgen Ehlers, Initial-boundary value problem for the spherically symmetric Einstein equations for a perfect fluid, Classical Quantum Gravity 10 (1993), no. 10, 2123–2136. MR 1242402, DOI 10.1088/0264-9381/10/10/020
C. R. Evans and J. S. Coleman, Critical phenomena and self-similarity in the gravitational collapse of radiation fluid, Phys. Rev. Lett. 72 (1994), 1782.
Chun-Chieh Fu and Song-Sun Lin, On the critical mass of the collapse of a gaseous star in spherically symmetric and isentropic motion, Japan J. Indust. Appl. Math. 15 (1998), no. 3, 461–469. MR 1651739, DOI 10.1007/BF03167322
P. Goldreich and S. Weber, Homologously collapsing stellar cores, Astrophys. J. 238 (1980), 991–997.
Martin Goliath, Ulf S. Nilsson, and Claes Uggla, Timelike self-similar spherically symmetric perfect-fluid models, Classical Quantum Gravity 15 (1998), no. 9, 2841–2863. Topology of the Universe Conference (Cleveland, OH, 1997). MR 1649680, DOI 10.1088/0264-9381/15/9/028
Magali Grassin, Global smooth solutions to Euler equations for a perfect gas, Indiana Univ. Math. J. 47 (1998), no. 4, 1397–1432. MR 1687130, DOI 10.1512/iumj.1998.47.1608
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
Carsten Gundlach, Critical phenomena in gravitational collapse, Living Rev. Relativ. 2 (1999), 1999-4, 58. MR 1728882, DOI 10.12942/lrr-1999-4
Yan Guo, Mahir Hadžić, and Juhi Jang, Continued gravitational collapse for Newtonian stars, Arch. Ration. Mech. Anal. 239 (2021), no. 1, 431–552. MR 4198723, DOI 10.1007/s00205-020-01580-w
Yan Guo, Mahir Hadžić, and Juhi Jang, Larson-Penston self-similar gravitational collapse, Comm. Math. Phys. 386 (2021), no. 3, 1551–1601. MR 4299130, DOI 10.1007/s00220-021-04175-y
Y. Guo, M. Hadžić, and J. Jang, Naked singularities in the Einstein-Euler system, Preprint, arXiv:2112.10826, 2021.
Y. Guo, M. Hadžić, J. Jang, and M. Schrecker, Gravitational collapse for polytropic gaseous stars: Self-similar solutions, Arch. Rat. Mech. Anal., 10.1007/s00205-022-01827-8.
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 and Juhi Jang, Expanding large global solutions of the equations of compressible fluid mechanics, Invent. Math. 214 (2018), no. 3, 1205–1266. MR 3878730, DOI 10.1007/s00222-018-0821-1
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, 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
T. Harada, Final fate of the spherically symmetric collapse of a perfect fluid, Phys. Rev. D 58 (1998), 104015.
T. Harada, Selfsimilar solutions, critical behavior and convergence to attractor in gravitational collapse, 12th Workshop on General Relativity and Gravitation (JGRG12): Tokyo, Japan, November 25–28, 2002, 2003.
T. Harada, Gravitational collapse and naked singularities, Pramana J. Phys. 63 (2004), 741–753.
T. Harada and M. Maeda, Convergence to a self-similar solution in general relativistic gravitational collapse, Phys. Rev. D 63 (2001), 084022.
B. K. Harrison, K. S. Thorne, M. Wakano, and J. A. Wheeler, Gravitation theory and gravitational collapse, The University of Chicago Press, Chicago and London, 1965.
C. Hunter, The collapse of unstable isothermal spheres, Astrophysical Journal 218 (1977), 834–845.
M. Ifrim and D. Tataru, The compressible Euler equations in a physical vacuum: a comprehensive Eulerian approach, Preprint, arXiv:2007.05668, 2020.
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
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
P. S. Joshi and I. H. Dwivedi, The structure of naked singularity in self-similar gravitational collapse, Comm. Math. Phys. 146 (1992), no. 2, 333–342. MR 1165186, DOI 10.1007/BF02102631
R. B. Larson, Numerical calculations of the dynamics of a collapsing protostar, Mon. Not. R. Astr. Soc. 145 (1969), 271–295.
G. Lemaître, L’Univers en expansion, Ann. Soc. Sci. Bruxelles A 53 (1933), 51.
Elliott H. Lieb and Horng-Tzer Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys. 112 (1987), no. 1, 147–174. MR 904142, DOI 10.1007/BF01217684
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
H. Maeda and T. Harada, Critical phenomena in Newtonian gravity, Phys. Rev. D 64 (2001), 124024.
Tetu Makino, Seiji Ukai, and Shuichi Kawashima, Sur la solution à support compact de l’équations d’Euler compressible, Japan J. Appl. Math. 3 (1986), no. 2, 249–257 (French, with English summary). MR 899222, DOI 10.1007/BF03167100
Tetu Makino and Benoît Perthame, Sur les solutions à symétrie sphérique de l’équation d’Euler-Poisson pour l’évolution d’étoiles gazeuses, Japan J. Appl. Math. 7 (1990), no. 1, 165–170 (French, with English summary). MR 1039243, DOI 10.1007/BF03167897
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
Frank Merle, Pierre Raphaël, Igor Rodnianski, and Jeremie Szeftel, On the implosion of a compressible fluid I: Smooth self-similar inviscid profiles, Ann. of Math. (2) 196 (2022), no. 2, 567–778. MR 4445442, DOI 10.4007/annals.2022.196.2.3
Frank Merle, Pierre Raphaël, Igor Rodnianski, and Jeremie Szeftel, On the implosion of a compressible fluid II: Singularity formation, Ann. of Math. (2) 196 (2022), no. 2, 779–889. MR 4445443, DOI 10.4007/annals.2022.196.2.4
Charles W. Misner and David H. Sharp, Relativistic equations for adiabatic, spherically symmetric gravitational collapse, Phys. Rev. (2) 136 (1964), B571–B576. MR 177783
David W. Neilsen and Matthew W. Choptuik, Critical phenomena in perfect fluids, Classical Quantum Gravity 17 (2000), no. 4, 761–782. MR 1744053, DOI 10.1088/0264-9381/17/4/303
J. R. Oppenheimer and G. M. Volkoff, On massive neutron cores, Physical Review 55 (1939), 374–381.
J. R. Oppenheimer and H. Snyder, On continued gravitational contraction, Phys. Rev. (2) 56 (1939), no. 5, 455–459. MR 3981575, DOI 10.1103/PhysRev.56.455
A. Ori and T. Piran, Naked singularities in self-similar spherical gravitational collapse, Phys. Rev. Lett. 59 (1987), 2137.
Amos Ori and Tsvi Piran, Self-similar spherical gravitational collapse and the cosmic censorship hypothesis, Gen. Relativity Gravitation 20 (1988), no. 1, 7–13. MR 925324, DOI 10.1007/BF00759251
Amos Ori and Tsvi Piran, Naked singularities and other features of self-similar general-relativistic gravitational collapse, Phys. Rev. D (3) 42 (1990), no. 4, 1068–1090. MR 1068847, DOI 10.1103/PhysRevD.42.1068
L. V. Ovsyannikov, A new solution of the equations of hydrodynamics, Dokl. Akad. Nauk SSSR (N.S.) 111 (1956), 47–49 (Russian). MR 0085824
S. Parmeshwar, M. Hadžić, and J. Jang, Global expanding solutions of compressible Euler equations with small initial densities, Quart. Appl. Math. 79 (2021), no. 2, 273–334.
S. Parmeshwar, Global existence for the N body Euler-Poisson system, Arch. Ration. Mech. Anal. 244 (2022), no. 2, 157–208.
R. Penrose, Gravitational collapse: The role of general relativity, Riv. Nuovo Cim. 1 (1969), 252–276.
M. V. Penston, Dynamics of self-gravitating gaseous spheres III, Mon. Not. R. Astr. Soc. 144 (1969), 425–448.
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
A. D. Rendall, The initial value problem for a class of general relativistic fluid bodies, J. Math. Phys. 33 (1992), no. 3, 1047–1053. MR 1149225, DOI 10.1063/1.529766
C. Rickard, M. Hadžić, and J. Jang, Global existence of the nonisentropic compressible Euler equations with vacuum boundary surrounding a variable entropy state, Nonlinearity 34 (2021), no. 1, 33–91.
I. Rodnianski and Y. Shlapentokh-Rothman, Naked singularities for the Einstein vacuum equations: the exterior solution, Preprint, arXiv:1912.08478, 2019.
Olga S. Rozanova, Solutions with linear profile of velocity to the Euler equations in several dimensions, Hyperbolic problems: theory, numerics, applications, Springer, Berlin, 2003, pp. 861–870. MR 2053233
Denis Serre, Solutions classiques globales des équations d’Euler pour un fluide parfait compressible, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 1, 139–153 (French, with English and French summaries). MR 1437182, DOI 10.5802/aif.1563
Steve Shkoller and Thomas C. Sideris, Global existence of near-affine solutions to the compressible Euler equations, Arch. Ration. Mech. Anal. 234 (2019), no. 1, 115–180. MR 3981395, DOI 10.1007/s00205-019-01387-4
Y. Shlapentokh-Rothman, Naked singularities for the Einstein vacuum equations: the interior solution, Preprint, arXiv:2204.09891, 2022.
F. H. Shu, Self-similar collapse of spheres and star formation, Astrophys. J. 214 (1977), 488–497.
Thomas C. Sideris, Global existence and asymptotic behavior of affine motion of 3D ideal fluids surrounded by vacuum, Arch. Ration. Mech. Anal. 225 (2017), no. 1, 141–176. MR 3634025, DOI 10.1007/s00205-017-1106-3
K. P. Stanyukovich, O. Sharshekeev, and V. Ts. Gurovich, Automodel motion of a relativistic gas in general relativity if there is point symmetry, Dokl. Akad. Nauk SSSR 165 (1965), no. 3, 510–513.
Norbert Straumann, General relativity, 2nd ed., Graduate Texts in Physics, Springer, Dordrecht, 2013. MR 2977926, DOI 10.1007/978-94-007-5410-2
K. S. Thorne, The general-relativistic theory of stellar structure and dynamics, Proceedings of the International School of Physics “Enrico Fermi,” Course XXXV, at Varenna, Italy, July 12-24 , ed. L. Gratton, Academic Press, New York, 1966, 166–280.
J. E. Tohline, Hydrodynamic collapse, Fundamentals of Cosmic Physics 8 (1982), 1–82.
R. C. Tolman, Effect of inhomogeneity on cosmological models, Proc. Nat. Acad. Sci. U. S. 20 (1934), 169–176.
R. C. Tolman, Relativitiy, thermodynamics and cosmology, Oxford, 1934.
A. Yahil, Self-similar stellar collapse, Astrophysical Journal 265 (1983), 1047–1055.
Ya. B. Zel’dovich, Hydrodynamical stability of star, Voprosy Kosmogonii 9 (1963), 157–170.
Ya. B. Zel’dovich and I. D. Novikov, Relativistic astrophysics, vol. 1, Chicago University Press, Chicago, 1971.