Abstract.
By minimizing the energy for the Vlasov-Poisson system under a constraint, Guo and Rein have constructed a large class of isotropic, spherically symmetric steady states. They have shown that an isolated minimizer is automatically dynamically stable under general (i.e., not necessarily symmetric) perturbations. The main result of this work is to remove the assumption that the minimizer must be isolated, so minimizers are stable even if they are not isolated. It is also shown that the Lagrange multipliers associated with all minimizers have the same value. Finally, an example where two distinct minimizers exist is studied numerically.
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References
Aly, J.J.: On the lowest energy state of a collisionless selfgravitating system under phase space volume constraints. Monthly Notices Royal Astronomical Soc. 241, 15–27 (1989)
Binney, J., Tremaine, S.: Galactic Dynamics. Princeton University Press, 1987
Braasch, P., Rein, G., Vukadinović, J.: Nonlinear stability of stationary plasmas - an extension of the energy-Casimir method. SIAM J. Applied Math. 59, 831–844 (1999)
Fridman, A.M., Polyachenko, V.L.: Physics of Gravitating Systems I. Springer-Verlag, New York, 1984
Guo, Y.: Variational method in polytropic galaxies. Arch. Rational Mech. Anal. 150, 209–224 (1999)
Guo, Y.: On the generalized Antonov’s stability criterion. Contem. Math. 263, 85–107 (2000)
Guo, Y., Rein, G.: Stable steady states in stellar dynamics. Arch. Rational Mech. Anal. 147, 225–243 (1999)
Guo, Y., Rein, G.: Existence and stability of Camm type steady states in galactic dynamics. Indiana Univ. Math. J. 48, 1237–1255 (1999)
Guo, Y., Rein, G.: Isotropic steady states in galactic dynamics. Commun. Math. Phys. 219, 607–629 (2001)
Guo, Y., Rein, G.: Stable models of elliptical galaxies. Preprint
Guo, Y., Strauss, W.: Nonlinear instability of double-humped equilibria. Ann. Inst. Henri Poincaré 12, 339–352 (1995)
Guo, Y., Strauss, W.: Instability of periodic BGK equilibria. Comm. Pure Appl. Math. 48, 861–894 (1995)
Horst, E.: On the asymptotic growth of the solutions of the Vlasov-Poisson system. Math. Meth. Appl. Sci. 16, 75–85 (1993)
Lions, P.L., Perthame, B.: Propagation of moments and regularity for the three dimensional Vlasov-Poisson system. Invent. Math. 105, 415–430 (1991)
Pfaffelmoser, K.: Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data. J. Diff. Eqns. 95, 281–303 (1992)
Rein, G.: Nonlinear stability for the Vlasov-Poisson system - the energy - Casimir method. Math. Meth. Appl. Sci. 17, 1129–1140 (1994)
Rein, G.: Flat steady states in stellar dynamics - existence and stability. Commun. Math. Phys. 205, 229–247 (1999)
Rein, G.: Stability of spherically symmetric steady states in galactic dynamics against general perturbations. Arch. Rational Mech. Anal. 161, 27–42 (2002)
Schaeffer, J.: Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions. Commun. Part. Diff. Eqns. 16, 1313–1335 (1991)
Wan, Y.-H.: On nonlinear stability of isotropic models in stellar dynamics. Arch. Rational Mech. Anal. 147, 245–268 (1999)
Wolansky, G.: On nonlinear stability of polytropic galaxies. Ann. Inst. Henri Poincaré 16, 15–48 (1999)
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P.-L. Lions
Dedicated to David Schaeffer on the occasion of his 60th birthday.
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Schaeffer, J. Steady States in Galactic Dynamics. Arch. Rational Mech. Anal. 172, 1–19 (2004). https://doi.org/10.1007/s00205-004-0308-7
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DOI: https://doi.org/10.1007/s00205-004-0308-7