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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.

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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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