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Stability for the spherically symmetric Einstein–Vlasov system–a coercivity estimate

Published online by Cambridge University Press:  10 September 2013

MAHIR HADŽIĆ  and
GERHARD REIN
MAHIR HADŽIĆ
Affiliation:
Massachusetts Institute of Technology, Department of Mathematics, Cambridge, MA 02139-4307, U.S.A. e-mail: hadzic@math.mit.edu
GERHARD REIN
Affiliation:
Mathematisches Institut der Universität BayreuthD-95440 Bayreuth, Germany. e-mail: gerhard.rein@uni-bayreuth.de
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Abstract

The stability of static solutions of the spherically symmetric, asymptotically flat Einstein–Vlasov system is studied using a Hamiltonian approach based on energy-Casimir functionals. The main results are a coercivity estimate for the quadratic part of the expansion of the natural energy-Casimir functional about an isotropic steady state, and the linear stability of such steady states. The coercivity bound shows in a quantified way that this quadratic part is positive definite on a class of linearly dynamically accessible perturbations, provided the particle distribution of the steady state is a strictly decreasing function of the particle energy and provided the steady state is not too relativistic. In contrast to the stability theory for isotropic steady states of the gravitational Vlasov-Poisson system the monotonicity of the particle distribution alone does not determine the stability character of the state, a fact which was observed by Ze'ldovitch et al. in the 1960's. The result in an essential way exploits the non-linear structure of the Einstein equations satisfied by the steady state and is not just a perturbation result of the analogous coercivity bounds for the Newtonian case. It should be an essential step in a fully non-linear stability analysis for the Einstein–Vlasov system.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 3 , November 2013 , pp. 529 - 556
Copyright
Copyright © Cambridge Philosophical Society 2013 

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References

REFERENCES

[1]Andréasson, H.On static shells and the Buchdahl inequality for the spherically symmetric Einstein–Vlasov system. Commun. Math. Phys. 274 (2007), 409–425.CrossRefGoogle Scholar
[2]Andréasson, H.The Einstein–Vlasov system/kinetic theory. Living Rev. Relativity 14, 4. URL (cited on 16.1.2012) (2011).CrossRefGoogle ScholarPubMed
[3]Andréasson, H., Kunze, M. and Rein, G.The formation of black holes in spherically symmetric gravitational collapse. Math. Ann. 350 (2011), 683–705.CrossRefGoogle Scholar
[4]Andréasson, H., Kunze, M., and Rein, G.Existence of axially symmetric static solutions of the Einstein–Vlasov system. Commun. Math. Phys. 308 (2011), 23–47.CrossRefGoogle Scholar
[5]Andréasson, H. and Rein, G.A numerical investigation of the stability of steady states and critical phenomena for the spherically symmetric Einstein–Vlasov system. Class. Quantum Grav. 23 (2006), 3659–3677.CrossRefGoogle Scholar
[6]Batt, J., Morrison, P. and Rein, G.Linear stability of stationary solutions of the Vlasov–Poisson system in three dimensions. Arch. Rational Mech. Anal. 130 (1995), 163–182.CrossRefGoogle Scholar
[7]Guo, Y.Variational method in polytropic galaxies. Arch. Rational Mech. Anal. 150 (1999), 209–224.CrossRefGoogle Scholar
[8]Guo, Y. and Lin, Z.Unstable and stable galaxy models. Commun. Math. Phys. 279 (2008), 789–813.CrossRefGoogle Scholar
[9]Guo, Y. and Rein, G.Isotropic steady states in galactic dynamics. Commun. Math. Phys. 219 (2001), 607–629.CrossRefGoogle Scholar
[10]Guo, Y. and Rein, G.A non-variational approach to nonlinear stability in stellar dynamics applied to the King model. Commun. Math. Phys. 271 (2007), 489–509.CrossRefGoogle Scholar
[11]Hadžić, M. and Rein, G.Global existence and nonlinear stability for the relativistic Vlasov-Poisson system in the gravitational case. Indiana Univ. Math. J. 56 (2007), 2453–2488.CrossRefGoogle Scholar
[12]Ipser, J. and Thorne, K.Relativistic, spherically symmetric star clusters I. Stability theory for radial perturbations. Astrophys. J. 154 (1968), 251–270.CrossRefGoogle Scholar
[13]Kandrup, H. and Morrison, P.Hamiltonian structure of the Vlasov-Einstein system and the problem of stability for spherical relativistic star clusters. Ann. Physics 225 (1993), 114–166.CrossRefGoogle Scholar
[14]Kandrup, H. and Sygnet, J. F.A simple proof of dynamical stability for a class of spherical clusters. Astrophys. J. 298 (1985), 27–33.CrossRefGoogle Scholar
[15]Lemou, M., Mehats, F., and Raphaël, P.Stable ground states for the relativistic gravitational Vlasov–Poisson system. Commun. Partial Differential Eqns. 34 (2009), 703–721.CrossRefGoogle Scholar
[16]Lemou, M., Mehats, F., and Raphaël, P.A new variational approach to the stability of gravitational systems. Commun. Math. Phys. 302 (2011), 161–224.CrossRefGoogle Scholar
[17]Lemou, M., Mehats, F. and Raphaël, P.Orbital stability of spherical systems. Inventiones Math. 187 (2012), 145–194.CrossRefGoogle Scholar
[18]Mouhot, C. Stabilité orbitale pour le système de Vlasov–Poisson gravitationnel, d'après Lemou–Méhats–Raphaël, Guo, Lin, Rein et al. Séminaire Nicolas Bourbaki, Nov. 2011. arXiv:1201.2275 (2012).Google Scholar
[19]Rein, G.Static solutions of the spherically symmetric Vlasov–Einstein system. Math. Proc. Camb. Phil. Soc. 115 (1994), 559–570.CrossRefGoogle Scholar
[20]Rein, G.The Vlasov–Einstein System with Surface Symmetry. (Habilitationsschrift, München 1995).Google Scholar
[21]Rein, G.Static shells for the Vlasov–Poisson and Vlasov–Einstein systems. Indiana Univ. Math. J. 48 (1999), 335–346.CrossRefGoogle Scholar
[22]Rein, G.Collisionless kinetic equations from astrophysics—The Vlasov–Poisson system. In Handbook of Differential Equations, Evolutionary Equations Vol. 3 (2007), edited by Dafermos, C. M. and Feireisl, E., Elsevier.Google Scholar
[23]Rein, G. and Rendall, A.Global existence of solutions of the spherically symmetric Vlasov–Einstein system with small initial data. Commun. Math. Phys. 150 (1992), 561–583. Erratum: Commun. Math. Phys. 176 (1996), 475–478.CrossRefGoogle Scholar
[24]Rein, G. and Rendall, A.The Newtomian limit of the spherically symmetric Vlasov–Einstein system. Commun. Math. Phys. 150 (1992), 585–591.CrossRefGoogle Scholar
[25]Rein, G. and Rendall, A.Smooth static solutions of the spherically symmetric Vlasov–Einstein system. Ann. Inst. H. Poincaré, Physique Théorique 59 (1993), 383–397.Google Scholar
[26]Rein, G. and Rendall, A.Compact support of spherically symmetric equilibria in non-relativistic and relativistic galactic dynamics. Math. Proc. Camb. Phil. Soc. 128 (2000), 363–380.CrossRefGoogle Scholar
[27]Wang, Z., Guo, Y., Lin, Z. and Zhang, P. Unstable galaxy models. Preprint, http://arxiv.org/abs/1303.2988Google Scholar
[28]Wolansky, G.Static solutions of the Vlasov–Einstein system. Arch. Rational Mech. Anal. 156 (2001), 205–230.CrossRefGoogle Scholar
[29]Zel'dovich, Ya. B. and Novikov, I. D.Relativistic Astrophysics, Vol 1 (Chicago University Press 1971).Google Scholar
[30]Zel'dovich, Ya. B. and Podurets, M. A.The evolution of a system of gravitationally interacting point masses. Soviet Astronomy–AJ 9 (1965), 742–749. Translated from Astronomicheskii Zhurnal 42.Google Scholar