Non linear stability of spherical gravitational systems described by the Vlasov-Poisson equation

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1. Aly J.-J., On the lowest energy state of a collisionless self-gravitating system under phase volume constraints. MNRAS 241 (1989), 15. Zbl0682.70009MR1021616
2. Antonov, A. V., Remarks on the problem of stability in stellar dynamics. Soviet Astr., AJ.,4, 859-867 (1961). MR131633
3. Antonov, A. V., Solution of the problem of stability of a stellar system with the Emden density law and spherical velocity distribution. J. Leningrad Univ. Se. Mekh. Astro. 7, 135-146 (1962). 
4. Arsen’ev, A. A., Global existence of a weak solution of Vlasov’s system of equations, U.S.S.R. Computational Math. and Math. Phys. 15 (1975), 131–141. Zbl0302.35009
5. Batt, J.; Faltenbacher, W.; Horst, E., Stationary spherically symmetric models in stellar dynamics, Arch. Rat. Mech. Anal. 93, 159-183 (1986). Zbl0605.70008MR823117
6. Binney, J.; Tremaine, S., Galactic Dynamics, Princeton University Press, 1987. Zbl1136.85001
7. Diperna, R. J.; Lions, P.-L., Global weak solutions of kinetic equations, Rend. Sem. Mat. Univ. Politec. Torino 46 (1988), no. 3, 259–288 (1990). Zbl0813.35087MR1101105
8. Diperna, R. J.; Lions, P.-L., Solutions globales d’équations du type Vlasov-Poisson, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 12, 655–658. Zbl0682.35022MR967806
9. Dolbeault, J.; Sánchez, Ó.; Soler, J., Asymptotic behaviour for the Vlasov-Poisson system in the stellar-dynamics case, Arch. Ration. Mech. Anal. 171 (2004), no. 3, 301–327. Zbl1057.70009MR2038342
10. Doremus, J. P.; Baumann, G.; Feix, M. R., Stability of a Self Gravitating System with Phase Space Density Function of Energy and Angular Momentum, Astronomy and Astrophysics 29 (1973), 401. 
11. Fridmann, A. M.; Polyachenko, V. L., Physics of gravitating systems, Springer-Verlag, 1984. Zbl0543.70010
12. Gardner, C.S., Bound on the energy available from a plasma, Phys. Fluids 6, 1963, 839-840. MR152330
13. Gillon, D.; Cantus, M.; Doremus, J. P.; Baumann, G., Stability of self-gravitating spherical systems in which phase space density is a function of energy and angular momentum, for spherical perturbations, Astronomy and Astrophysics 50 (1976), no. 3, 467–470. MR456303
14. Guo, Y., Variational method for stable polytropic galaxies, Arch. Rat. Mech. Anal. 130 (1999), 163-182. Zbl0953.70011MR1738118
15. Guo, Y.; Lin, Z., Unstable and stable galaxy models, Comm. Math. Phys. 279 (2008), no. 3, 789–813. Zbl1140.85304MR2386728
16. Guo, Y.; Rein, G., Stable steady states in stellar dynamics, Arch. Rat. Mech. Anal. 147 (1999), 225–243. Zbl0935.70011MR1709211
17. Guo, Y.; Rein, G., Isotropic steady states in galactic dynamics, Comm. Math. Phys. 219 (2001), 607–629. Zbl0974.35093MR1838751
18. Guo, Y., On the generalized Antonov’s stability criterion. Contemp. Math.263, 85-107 (2000) Zbl0974.35092MR1777637
19. Guo, Y.; Rein, G., A non-variational approach to nonlinear Stability in stellar dynamics applied to the King model, Comm. Math. Phys., 271, 489-509 (2007). Zbl1130.85002MR2287914
20. Hörmander, L, An Introduction to Complex Analysis in Several Variables (3rd Edition ed.), North-Holland, Amsterdam (1990). Zbl0685.32001MR1045639
21. Hörmander, L., $L2$ estimates and existence theorems for the $\overline{\partial }$ operator, Acta Math. 113 (1965), 89–152. Zbl0158.11002
22. Horst, E.; Hunze, R., Weak solutions of the initial value problem for the unmodified nonlinear Vlasov equation, Math. Methods Appl. Sci. 6 (1984), no. 2, 262–279. Zbl0556.35022MR751745
23. Illner, R.; Neunzert, H., An existence theorem for the unmodified Vlasov equation, Math. Methods Appl. Sci. 1 (1979), no. 4, 530–544. Zbl0415.35076MR548686
24. Kandrup, H. E.; Sygnet, J. F., A simple proof of dynamical stability for a class of spherical clusters. Astrophys. J. 298 (1985), no. 1, part 1, 27–33. MR807234
25. Kavian, O., Introduction à la théorie des points critiques et applications aux problèmes elliptiques. Mathématiques & Applications (Berlin), 13. Springer-Verlag, Paris, 1993. Zbl0797.58005MR1276944
26. Lemou, M.; Méhats, F.; Raphaël, P., Orbital stability and singularity formation for Vlasov-Poisson systems. C. R. Math. Acad. Sci. Paris 341 (2005), no. 4, 269–274. Zbl1073.70012MR2164685
27. Lemou, M.; Méhats, F.; Raphaël, P., On the orbital stability of the ground states and the singularity formation for the gravitational Vlasov-Poisson system, Arch. Rat. Mech. Anal. 189 (2008), no. 3, 425–468. Zbl1221.35417MR2424993
28. Lemou, M.; Méhats, F,; Raphaël, P., Stable ground states for the relativistic gravitational Vlasov-Poisson system, Comm. Partial Diff. Eq. 34 (2009), no. 7, 703–721. Zbl1179.35054MR2560298
29. Lemou, M.; Méhats, F.; Raphaël, P., A new variational approach to the stability of gravitational systems. C. R. Math. Acad. Sci. Paris 347 (2009), no. 4, 979–984. Zbl1170.85305MR2542905
30. Lemou, M.; Méhats, F.; Raphaël, P., A new variational approach to the stability of gravitational systems. Comm. Math. Phys. 302 (2011), 161-224. Zbl05859339MR2770012
31. Lemou, M.; Méhats, F.; Raphaël, P.,Orbital stability of spherical galactic models. To appear in Invent. Math. Zbl1232.35170MR2874937
32. Lieb, E. H.; Loss, Analysis. Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001. Zbl0966.26002MR1817225
33. Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109–145. Zbl0541.49009MR778970
34. Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223–283. Zbl0704.49004MR778974
35. Lions, P.-L.; Perthame, B., Propagation of moments and regularity for the $3$-dimensional Vlasov-Poisson system, Invent. Math. 105 (1991), no. 2. Zbl0741.35061MR1115549
36. Lynden-Bell, D., The Hartree-Fock exchange operator and the stability of galaxies, Mon. Not. R. Astr. Soc. 144, 1969, 189–217. 
37. Marchioro, C.; Pulvirenti, M., Some considerations on the nonlinear stability of stationary planar Euler flows, Comm. Math. Phys. 100 (1985), no. 3, 343–354. Zbl0625.76060MR802550
38. Marchioro, C.; Pulvirenti, M., A note on the nonlinear stability of a spatially symmetric Vlasov-Poisson flow, Math. Methods Appl. Sci. 8 (1986), no. 2, 284Ð288. Zbl0609.35008MR845931
39. Mouhot, C.; Villani, C. Landau damping, J. Math. Phys. 51 (2010), no. 1, 015204. Zbl1247.82081MR2605837
40. Mouhot, C.; Villani, C. On Landau damping, to appear in Acta Mathematica. Zbl1239.82017MR2827908
41. Mossino, J., Inégalités isopérimétriques et applications en physique. (French) [Isoperimetric inequalities and applications to physics] Travaux en Cours. [Works in Progress] Hermann, Paris, 1984. Zbl0537.35002MR733257
42. Pfaffelmoser, K., Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Diff. Eq. 95 (1992), 281-303. Zbl0810.35089MR1165424
43. Sánchez, Ó.; Soler, J., Orbital stability for polytropic galaxies, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), no. 6, 781–802. Zbl1110.35011MR2271693
44. Schaeffer, J., Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Comm. Part. Diff. Eq. 16 (1991), 1313-1335. Zbl0746.35050MR1132787
45. Schaeffer, J., Steady States in Galactic Dynamics, Arch. Rational, Mech. Anal. 172 (2004), 1–19. Zbl1061.85001MR2048565
46. Sygnet, J.-F.; Des Forets, G.; Lachieze-Rey, M.; Pellat, R., Stability of gravitational systems and gravothermal catastrophe in astrophysics, Astrophys. J. 276 (1984), no. 2, 737–745. 
47. Talenti, G., Elliptic equations and rearrangements. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3 (1976), no. 4, 697–718. Zbl0341.35031MR601601
48. Wan, Y. H.; Pulvirenti, M., Nonlinear Stability of Circular Vortex Patches, Comm. Math. Phys. 99 (1985), 435–450. Zbl0584.76062MR795112
49. Weinstein, M. I., Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), 472–491. Zbl0583.35028MR783974
50. Wiechen, H., Ziegler, H.J., Schindler, K. Relaxation of collisionless self gravitating matter: the lowest energy state, Mon. Mot. R. ast. Soc (1988) 223, 623-646. Zbl0667.70018
51. Wolansky, G., On nonlinear stability of polytropic galaxies. Ann. Inst. Henri Poincaré, 16, 15-48 (1999). Zbl0927.70019MR1668556