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Nonlinear stability of stationary spherically symmetric models in stellar dynamics. (English) Zbl 0727.70015
The author considers a system of particles in the three dimensional space without any collision among them. The system can be described by distribution functions f(x,v,t) satisfying the Vlasov equation and the Poisson law. Then $$f_ 0$$, for which $$\partial f/\partial t=0$$, is nonlinearly stable under some conditions, and is spherically symmetric if $$f(x,v)=f(Sx,Sv)$$ with any rotation S.
The author proves that under an appropriate condition of the state $$f_ 0$$, $$f_ 0$$ is nonlinearly stable subject to general perturbations and that under a regularity condition, $$f_ 0$$ is nonlinearly stable subject to spherically symmetric perturbations.
Reviewer: Y.Kozai (Tokyo)

##### MSC:
 70F15 Celestial mechanics 85A05 Galactic and stellar dynamics
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##### References:
 [1] Antonov, V. A., Remarks on the problems of stability in stellar dynamics, Soviet Astronomy, AJ. 4 (1961), 859-867. [2] Antonov, V. A., Solution of the Problem of Stability of a Stellar System with the Emden Density law and Spherical Velocity Distribution, J. Leningrad Univ. Ser. Mat., Mekh. Astro. 7, 135-46 (1962). [3] Arnol’d, V. I., Conditions for nonlinear stability of the stationary plane curvilinear flows of an ideal fluid, Soviet Math. Dokl. 6, 773-777 (1965). · Zbl 0141.43901 [4] Batt, J., Global symmetric solutions of the initial value problem of stellar dynamics, J. Diff. Equations 25, 342-364 (1977). · Zbl 0366.35020 · doi:10.1016/0022-0396(77)90049-3 [5] Batt, J., ?The nonlinear Vlasov-Poisson system of partial differential equations in stellar dynamics?, Publications de l’.U.R.E. Math. Pures et Appl. 5, 1-30 (1983). [6] Batt, J., The present state of the investigation of the Vlasov-Poisson system and of the Vlasov-Maxwell system in stellar dynamics and plasma physics, preprint, Ludwig-Maximilians-Universität München, 1988. [7] Doremus, J. P., Feix, M. R., & Baumann, G., Stability of Encounterless Spherical Stellar Systems. Phys. Rev. Letters 26, 725-728 (1971). · doi:10.1103/PhysRevLett.26.725 [8] Fridman, A. M. & Polyachenko, V. L., Physics of Gravitating Systems I, Equilibrium and Stability, Springer-Verlag, 1984. · Zbl 0543.70010 [9] Gillon, D., Doremus, J. P., & Baumann, G., Stability of self-gravitating systems with phase space density a function of energy and angular momentum for aspherical modes, Astron. & Astrophys. 48, 467-474 (1976). [10] Holm, D., Marsden, J., Ratiu, J., & Weinstein, A., Nonlinear stability of fluid and plasma equilibria, Physics Reports 123, 1-116 (1985). · Zbl 0717.76051 · doi:10.1016/0370-1573(85)90028-6 [11] Hénon, M., Numerical experiments on the stability of stellar systems, Astron. and Astroph. 24, 229-238 (1973). [12] Lewis, D., Marsden, J. E., & Ratiu, T., Stability and bifurcation of a rotating planar liquid drop, J. Math. Phys. 28, (1987). · Zbl 0651.76021 [13] Marsden, J. E., A group theoretic approach to the equations of plasma physics, Can. Math. Bull. 25, 129-142 (1982). · Zbl 0492.58015 · doi:10.4153/CMB-1982-019-9 [14] Sygnet, J. F., Forets, G. D., Lachieze-Rey, M., & Pellat, R., Stability of gravitational systems and gravothermal catastrophe in astrophysics, The Astrophysical Journal 276, 737-745 (1984). · doi:10.1086/161659 [15] Wan, Y. H., The stability of rotating vortex patches, Comm. in Math. Physics 107, 1-20 (1986). · Zbl 0624.76055 · doi:10.1007/BF01206950 [16] Wan, Y. H., Variational principles for Hill’s spherical vortex and nearly spherical vortices, Trans, of Amer. Math. Soc., 1988. · Zbl 0661.76017 [17] Wan, Y. H., & Pulvirenti, M., Nonlinear stability of circular vortex patches, Comm. in Math. Physics 99, 435-450 (1985). · Zbl 0584.76062 · doi:10.1007/BF01240356
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