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

Lemou, Mohammed¹ 

¹ CNRS and IRMAR Université de Rennes 1 France

Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exposé no. 18, 17 p.

Résumé

In this work, we prove the nonlinear stability of galaxy models derived from the three dimensional gravitational Vlasov Poisson system, which is a canonical model in astrophysics to describe the dynamics of galactic clusters.

Détail

DOI : 10.5802/slsedp.14

Bibliographie

[1] Aly J.-J., On the lowest energy state of a collisionless self-gravitating system under phase volume constraints. MNRAS 241 (1989), 15. | Zbl | MR

[2] Antonov, A. V., Remarks on the problem of stability in stellar dynamics. Soviet Astr., AJ., 4, 859-867 (1961). | MR

[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. | Zbl

[5] Batt, J.; Faltenbacher, W.; Horst, E., Stationary spherically symmetric models in stellar dynamics, Arch. Rat. Mech. Anal. 93, 159-183 (1986). | Zbl | MR

[6] Binney, J.; Tremaine, S., Galactic Dynamics, Princeton University Press, 1987. | Zbl

[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). | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl

[12] Gardner, C.S., Bound on the energy available from a plasma, Phys. Fluids 6, 1963, 839-840. | MR

[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. | MR

[14] Guo, Y., Variational method for stable polytropic galaxies, Arch. Rat. Mech. Anal. 130 (1999), 163-182. | Zbl | MR

[15] Guo, Y.; Lin, Z., Unstable and stable galaxy models, Comm. Math. Phys. 279 (2008), no. 3, 789–813. | Zbl | MR

[16] Guo, Y.; Rein, G., Stable steady states in stellar dynamics, Arch. Rat. Mech. Anal. 147 (1999), 225–243. | Zbl | MR

[17] Guo, Y.; Rein, G., Isotropic steady states in galactic dynamics, Comm. Math. Phys. 219 (2001), 607–629. | Zbl | MR

[18] Guo, Y., On the generalized Antonov’s stability criterion. Contemp. Math. 263, 85-107 (2000) | Zbl | MR

[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). | Zbl | MR

[20] Hörmander, L, An Introduction to Complex Analysis in Several Variables (3rd Edition ed.), North-Holland, Amsterdam (1990). | Zbl | MR

[21] Hörmander, L., $L2$ estimates and existence theorems for the $\overline{\partial }$ operator, Acta Math. 113 (1965), 89–152. | Zbl

[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. | Zbl | MR

[23] Illner, R.; Neunzert, H., An existence theorem for the unmodified Vlasov equation, Math. Methods Appl. Sci. 1 (1979), no. 4, 530–544. | Zbl | MR

[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. | MR

[25] Kavian, O., Introduction à la théorie des points critiques et applications aux problèmes elliptiques. Mathématiques & Applications (Berlin), 13. Springer-Verlag, Paris, 1993. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[30] Lemou, M.; Méhats, F.; Raphaël, P., A new variational approach to the stability of gravitational systems. Comm. Math. Phys. 302 (2011), 161-224. | MR

[31] Lemou, M.; Méhats, F.; Raphaël, P.,Orbital stability of spherical galactic models. To appear in Invent. Math. | Zbl | MR

[32] Lieb, E. H.; Loss, Analysis. Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001. | Zbl | MR

[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. | Zbl | MR | mathdoc-id

[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. | Zbl | MR | mathdoc-id

[35] Lions, P.-L.; Perthame, B., Propagation of moments and regularity for the $3$-dimensional Vlasov-Poisson system, Invent. Math. 105 (1991), no. 2. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[39] Mouhot, C.; Villani, C. Landau damping, J. Math. Phys. 51 (2010), no. 1, 015204. | Zbl | MR

[40] Mouhot, C.; Villani, C. On Landau damping, to appear in Acta Mathematica. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[43] Sánchez, Ó.; Soler, J., Orbital stability for polytropic galaxies, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), no. 6, 781–802. | Zbl | MR | mathdoc-id

[44] Schaeffer, J., Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Comm. Part. Diff. Eq. 16 (1991), 1313-1335. | Zbl | MR

[45] Schaeffer, J., Steady States in Galactic Dynamics, Arch. Rational, Mech. Anal. 172 (2004), 1–19. | Zbl | MR

[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. | Zbl | MR | mathdoc-id

[48] Wan, Y. H.; Pulvirenti, M., Nonlinear Stability of Circular Vortex Patches, Comm. Math. Phys. 99 (1985), 435–450. | Zbl | MR

[49] Weinstein, M. I., Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), 472–491. | Zbl | MR

[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. | Zbl

[51] Wolansky, G., On nonlinear stability of polytropic galaxies. Ann. Inst. Henri Poincaré, 16, 15-48 (1999). | Zbl | MR | mathdoc-id