A constraint variational problem arising in stellar dynamics
Author:
Mahir Hadzic
Journal:
Quart. Appl. Math. 65 (2007), 145-153
MSC (2000):
Primary 49K20
DOI:
https://doi.org/10.1090/S0033-569X-06-01032-9
Published electronically:
December 14, 2006
MathSciNet review:
2313153
Full-text PDF Free Access
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Abstract: We use the compactness result of A. Burchard and Y. Guo to analyze the reduced ‘energy’ functional arising naturally in the stability analysis of steady states of the Vlasov-Poisson system (cf. Sánchez and Soler, to appear, and Hadžić, 2005). We consider the associated variational problem and present a new proof that puts it in the general framework for tackling the variational problems of this type, given by Y. Guo and G. Rein (cf. Rein, 2005 and Rein, 2002).
References
- Almut Burchard and Yan Guo, Compactness via symmetrization, J. Funct. Anal. 214 (2004), no. 1, 40–73. MR 2079885, DOI https://doi.org/10.1016/j.jfa.2004.04.005
- Hadžić, M.: Compactness and stability of some systems of nonlinear PDE-s in galactic dynamics. Diploma thesis, University of Vienna, 2005.
- G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR 0046395
- Elliott H. Lieb and Michael Loss, Analysis, 2nd ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. MR 1817225
- Rein, G.: Nonlinear Stability of Newtonian Galaxies and Stars from a Mathematical Perspective, Nonlinear Dynamics in Astronomy and Physics. Annals of the New York Academy of Sciences, 1045, 103–119, 2005.
- Gerhard Rein, Reduction and a concentration-compactness principle for energy-Casimir functionals, SIAM J. Math. Anal. 33 (2001), no. 4, 896–912. MR 1884728, DOI https://doi.org/10.1137/P0036141001389275
- Sánchez, Ó., J. Soler: Orbital stability for polytropic galaxies. Annales de l’Institut Henri Poincare, “Analyse non lineaire ". To appear.
References
- Burchard, A., Y. Guo: Compactness via Symmetrization. J. Functional Analysis, 214, 40–73, 2004. MR 2079885 (2005g:46061)
- Hadžić, M.: Compactness and stability of some systems of nonlinear PDE-s in galactic dynamics. Diploma thesis, University of Vienna, 2005.
- Hardy, G., J. E. Littlewood, G. Polya: Inequalities. Cambridge Mathematical Press, Cambridge 1952, 2nd edition. MR 0046395 (13:727e)
- Lieb, E. H., M. Loss: Analysis. American Math. Soc., Providence, 2001. MR 1817225 (2001i:00001)
- Rein, G.: Nonlinear Stability of Newtonian Galaxies and Stars from a Mathematical Perspective, Nonlinear Dynamics in Astronomy and Physics. Annals of the New York Academy of Sciences, 1045, 103–119, 2005.
- Rein, G.: Reduction and a concentration-compactness principle for energy-Casimir functionals. SIAM J. on Math. Anal., 33, 896–912, 2002. MR 1884728 (2003a:35006)
- Sánchez, Ó., J. Soler: Orbital stability for polytropic galaxies. Annales de l’Institut Henri Poincare, “Analyse non lineaire ". To appear.
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Additional Information
Mahir Hadzic
Affiliation:
Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
Email:
hadzic@dam.brown.edu
Received by editor(s):
April 20, 2006
Published electronically:
December 14, 2006
Additional Notes:
This paper is a part of the author’s Diploma thesis. I would like to thank Gerhard Rein for introducing me to the problem and for many stimulating discussions.
Article copyright:
© Copyright 2006
Brown University
The copyright for this article reverts to public domain 28 years after publication.
