Hostname: page-component-848d4c4894-cjp7w Total loading time: 0 Render date: 2024-07-04T06:22:46.687Z Has data issue: false hasContentIssue false
  • English
  • Français

Existence and stability of static shells for the Vlasov–Poisson system with a fixed central point mass

Published online by Cambridge University Press:  01 March 2009

ACHIM SCHULZE
ACHIM SCHULZE*
Affiliation:
Mathematisches Institut der Universität Bayreuth D 95440 Bayreuth, Germany. e-mail: achim.schulze@yahoo.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the Vlasov–Poisson system with spherical symmetry and an exterior potential which is induced by a point mass in the center. This system can be used as a simple model for a newtonian galaxy surrounding a black hole. For this system, we establish a global existence result for classical solutions with shell-like initial data, i.e. the support of the density is bounded away from the point mass singularity. We also prove existence and stability of stationary solutions which describe static shells, where we use a variational approach which was established by Y. Guo and G. Rein.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 2 , March 2009 , pp. 489 - 511
Copyright
Copyright © Cambridge Philosophical Society 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]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.Google Scholar
[2]Batt, J.Global symmetric solutions of the initial value problem in stellar dynamics. J. Differential Equations 25 (1977), 342–364.CrossRefGoogle Scholar
[3]Guo, Y.Variational method in polytropic galaxies. Arch. Ration. Mech. Anal. 150 (1999), 209–224.CrossRefGoogle Scholar
[4]Guo, Y.On the generalizes Antonov's stability criterion. Contemp. Math. 263 (2000), 85–107.Google Scholar
[5]Guo, Y. and Rein, G.Existence and stability of camm type steady states in galactic dynamics. Indiana Univ. Math. J. 48 (1999), 1237–1255.Google Scholar
[6]Guo, Y. and Rein, G.Stable steady states in stellar dynamics. Arch. Ration. Mech. Anal. 147 (1999), 225–243.CrossRefGoogle Scholar
[7]Guo, Y. and Rein, G.Isotropic steady states in stellar dynamics. Comm. Math. Phys. 219 (2001), 607–629.CrossRefGoogle Scholar
[8]Horst, E.On the classical solutions of the initial value problem for the unmodified non-linear Vasov equation II. Math. Methods Appl. Sci. 4 (1982), 19–32.CrossRefGoogle Scholar
[9]Lions, P.-L. and Perthame, B.Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system. Invent. Math. 105 (1991), 415–430.CrossRefGoogle Scholar
[10]Pfaffelmoser, K.Global classical solutions of the Vlasov–Poisson system in three dimensions for general initial data. J. Differential Equations 95 (1992), 281–303.CrossRefGoogle Scholar
[11]Rein, G.Static shells for the Vlasov–Poisson and Vlasov–Einstein systems. Indiana Univ. Math. J. 48 (1999), 335–346.CrossRefGoogle Scholar
[12]Rein, G.Reduction and a concentration-compactness principle for energy-Casimir functionals. SIAM J. Math. Anal. 33 (2002), 896–912.CrossRefGoogle Scholar
[13]Rein, G.Stability of spherically symmetric steady states in galactic dynamics against general perturbations. Arch. Ration. Mech. Anal. 161 (2002), 27–42.CrossRefGoogle Scholar
[14]Rein, G.Nonlinear stability of gaseous stars. Arch. Ration. Mech. Anal. 168 (2003), 115–130.CrossRefGoogle Scholar
[15]Rein, G. Collisionless kinetic equations from astrophysics – the Vlasov–Poisson system. In Dafermos, C.M. and Feireisl, E. editors, Handbook of Differential Equations, Evolutionary Equations. Vol. 3 (Elsevier, 2007).Google Scholar
[16]Schaeffer, J.Global existence of smooth solutions to the Vlasov–Poisson system in three dimensions. Comm. Partial Differential Equation 16 (1991), 1313–1335.CrossRefGoogle Scholar
[17]Schulze, A.Existence and stability of static shells for the Vlasov–Poisson system. Analysis 26 (2006), 527–543.Google Scholar
[18]Yosida, K.Functional Analysis (Springer, 1965).Google Scholar