Stabilité orbitale pour le système de Vlasov-Poisson gravitationnel [d'après Lemou-Méhats-Raphaël, Guo, Lin, Rein et al.]

Mouhot, Clément

Séminaire Bourbaki volume 2011/2012 exposés 1043-1058, Astérisque, no. 352 (2013), Exposé no. 1044, 48 p.

MR Zbl

@incollection{AST_2013__352__35_0,
     author = {Mouhot, Cl\'ement},
     title = {Stabilit\'e orbitale pour le syst\`eme de {Vlasov-Poisson} gravitationnel [d'apr\`es {Lemou-M\'ehats-Rapha\"el,} {Guo,} {Lin,} {Rein} et al.]},
     booktitle = {S\'eminaire Bourbaki volume 2011/2012 expos\'es 1043-1058},
     series = {Ast\'erisque},
     note = {talk:1044},
     pages = {35--82},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {352},
     year = {2013},
     mrnumber = {3087342},
     zbl = {1287.70008},
     language = {fr},
     url = {http://archive.numdam.org/item/AST_2013__352__35_0/}
}

TY  - CHAP
AU  - Mouhot, Clément
TI  - Stabilité orbitale pour le système de Vlasov-Poisson gravitationnel [d'après Lemou-Méhats-Raphaël, Guo, Lin, Rein et al.]
BT  - Séminaire Bourbaki volume 2011/2012 exposés 1043-1058
AU  - Collectif
T3  - Astérisque
N1  - talk:1044
PY  - 2013
SP  - 35
EP  - 82
IS  - 352
PB  - Société mathématique de France
UR  - http://archive.numdam.org/item/AST_2013__352__35_0/
LA  - fr
ID  - AST_2013__352__35_0
ER  -

%0 Book Section
%A Mouhot, Clément
%T Stabilité orbitale pour le système de Vlasov-Poisson gravitationnel [d'après Lemou-Méhats-Raphaël, Guo, Lin, Rein et al.]
%B Séminaire Bourbaki volume 2011/2012 exposés 1043-1058
%A Collectif
%S Astérisque
%Z talk:1044
%D 2013
%P 35-82
%N 352
%I Société mathématique de France
%U http://archive.numdam.org/item/AST_2013__352__35_0/
%G fr
%F AST_2013__352__35_0

Mouhot, Clément. Stabilité orbitale pour le système de Vlasov-Poisson gravitationnel [d'après Lemou-Méhats-Raphaël, Guo, Lin, Rein et al.], dans Séminaire Bourbaki volume 2011/2012 exposés 1043-1058, Astérisque, no. 352 (2013), Exposé no. 1044, 48 p. http://archive.numdam.org/item/AST_2013__352__35_0/

Bibliographie
Cité par

[1] R. Abraham & J. E. Marsden - Foundations of mechanics, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, 1978. | MR | Zbl

[2] J. J. Aly - On the lowest energy state of a collisionless self-gravitating system under phase space volume constraints, Monthly Notices Roy. Astronom. Soc. 241 (1989) , n° 1, p. 15-27. | DOI | MR | Zbl

[3] A. V. Antonov - Solution of the problem of stability of a stellar system with the Emden density law and spherical velocity distribution, J. Leningrad Univ. Si. Mekh. Astro. 7 (1962) , p. 135-146.

[4] V. A. Antonov - Remarks on the problems of stability in stellar dynamics, Soviet Astronom. AJ 4 (1960), p. 859-867. | MR

[5] V. I. Arnol'D - On conditions for non-linear stability of plane stationary curvilinear flows of an ideal fluid, Dokl. Akad. Nauk SSSR 162 (1965), p. 975-978. | MR | Zbl

[6] V. I. Arnold - Variational principle for three dimensional steady-state flows of an ideal fluid, J. Appl. Math. Mech. 29 (1965), p. 1002-1008. | DOI | Zbl

[7] V. I. Arnold, On an a priori estimate in the theory of hydrodynamic stability, Am. Math. Soc. Transl. 19 (1969), p. 267-269. | Zbl

[8] A. A. Arsen'Ev - Existence in the large of a weak solution of Vlasov's system of equations, Ž. Vyčisl. Mat. i Mat. Fiz. 15 (1975), p. 136-147. | MR | Zbl

[9] J. Batt, W. Faltenbacher & E. Horst - Stationary spherically symmetric models in stellar dynamics, Arch. Rational Mech. Anal. 93 (1986), n° 2, p. 159-183. | DOI | MR | Zbl

[10] J. Batt, P. J. Morrison & G. Rein - Linear stability of stationary solutions of the Vlasov-Poisson system in three dimensions, Arch. Rational Mech. Anal. 130 (1995), n° 2, p. 163-182. | DOI | MR | Zbl

[11] J. Batt & G. Rein - Global classical solutions of the periodic Vlasov-Poisson system in three dimensions, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), n° 6, p. 411-416. | MR | Zbl

[12] J. Binney & S. Tremaine - Galactic dynamics, Princeton Univ. Press, 1987. | Zbl

[13] W. Braun & K. Hepp - The Vlasov dynamics and its fluctuations in the $1 / N$ limit of interacting classical particles, Comm. Math. Phys. 56 (1977), n° 2, p. 101-113. | DOI | MR | Zbl

[14] S. Calogero, Ó Sánchez & J. Soler - Asymptotic behavior and orbital stability of galactic dynamics in relativistic scalar gravity, Arch. Ration. Mech. Anal. 194 (2009), n° 3, p. 743-773. | DOI | MR | Zbl

[15] T. Cazenave & P.-L. Lions - Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982), n° 4, p. 549-561. | DOI | MR | Zbl

[16] A. Choffrut & V. Šverák - Local structure of the set of steady-state solutions to the $2 D$ incompressible Euler equations, Geom. Fund. Anal. 22 (2012), n° 1, p. 136-201. | DOI | MR | Zbl

[17] R. Diperna & P.-L. Lions - Global weak solutions of kinetic equations, Rend. Sem. Mat. Univ. Politec. Torino 46 (1988), n° 3, p. 259-288. | MR | Zbl

[18] R. Diperna & P.-L. Lions, Solutions globales d'équations du type Vlasov-Poisson, C.R. Acad. Sci. Paris Sér. I Math. 307 (1988), n° 12, p. 655-658. | MR | Zbl

[19] R. Diperna & P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), n° 3, p. 511-547. | DOI | EuDML | MR | Zbl

[20] R. L. Dobrusin - Vlasov equations, Funktsional. Anal, i Prilozhen. 13 (1979), n° 2, p. 48-58. | MR | Zbl

[21] J. Dolbeault, Ó Sánchez & J. Soler - Asymptotic behaviour for the Vlasov-Poisson system in the stellar-dynamics case, Arch. Ration. Mech. Anal. 171 (2004), n° 3, p. 301-327. | DOI | MR | Zbl

[22] J. P. Dorémus & M. R. Feix - Stability of a self gravitating density function of energy and angular momentum, Astron. & Astrophys. 29 (1973), p. 401-407.

[23] J. P. Dorémus, M. R. Feix & G. Baumann - Stability of encounterless spherical stellar systems, Phys. Rev. Letter 26 (1971), p. 725-728. | DOI

[24] A. Eddington - The distribution of stars in globular clusters, Mon. Not. R. Astr. Soc. 76 (1916). | DOI

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

[26] I. Gasser, P. E. Jabin & B. Perthame - Regularity and propagation of moments in some nonlinear Vlasov systems, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), n° 6, p. 1259-1273. | DOI | MR | Zbl

[27] Y. Guo - Stable magnetic equilibria in collisionless plasmas, Comm. Pure Appl. Math. 50 (1997), n° 9, p. 891-933. | DOI | MR | Zbl

[28] Y. Guo, Stable magnetic equilibria in a symmetric collisionless plasma, Comm. Math. Phys. 200 (1999), n° 1, p. 211-247. | DOI | MR | Zbl

[29] Y. Guo, Variational method for stable polytropic galaxies, Arch. Ration. Mech. Anal. 150 (1999), n° 3, p. 209-224. | DOI | MR | Zbl

[30] Y. Guo, On the generalized Antonov stability criterion, in Nonlinear wave equations (Providence, RI, 1998), Contemp. Math., vol. 263, Amer. Math. Soc., 2000, p. 85-107. | DOI | MR | Zbl

[31] Y. Guo & Z. Lin - Unstable and stable galaxy models, Comm. Math. Phys. 279 (2008), n° 3, p. 789-813. | DOI | MR | Zbl

[32] Y. Guo & G. Rein - Stable steady states in stellar dynamics, Arch. Ration. Mech. Anal. 147 (1999), n° 3, p. 225-243. | DOI | MR | Zbl

[33] Y. Guo & G. Rein, Isotropic steady states in galactic dynamics, Comm. Math. Phys. 219 (2001), n° 3, p. 607-629. | DOI | MR | Zbl

[34] Y. Guo & G. Rein, A non-variational approach to nonlinear stability in stellar dynamics applied to the King model, Comm. Math. Phys. 271 (2007), n° 2, p. 489-509. | DOI | MR | Zbl

[35] M. Hadžić - A constraint variational problem arising in stellar dynamics, Quart. Appl. Math. 65 (2007), n° 1, p. 145-153. | DOI | MR | Zbl

[36] M. Hauray & P.-E. Jabin - $N$ -particles approximation of the Vlasov equations with singular potential, Arch. Ration. Mech. Anal. 183 (2007), n° 3, p. 489-524. | DOI | MR | Zbl

[37] M. Hauray & P.-E. Jabin, Particles approximations of Vlasov equations with singular forces : Part 2, prépublication arXiv: 1107.3821. | MR

[38] D. D. Holm, J. E. Marsden, T. Ratiu & A. Weinstein - Nonlinear stability of fluid and plasma equilibria, Phys. Rep. 123 (1985), nos 1-2, p. 116. | MR | Zbl

[39] L. Hörmander - $L^{2}$ estimates and existence theorems for the $\bar{\partial}$ operator, Acta Math. 113 (1965), p. 89-152. | DOI | MR | Zbl

[40] L. Hörmander, An introduction to complex analysis in several variables, third éd., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., 1990. | MR

[41] E. Horst - On the asymptotic growth of the solutions of the Vlasov-Poisson system, Math. Methods Appl. Sci. 16 (1993), n° 2, p. 75-86. | DOI | MR | Zbl

[42] E. Horst & R. Hunze - Weak solutions of the initial value problem for the unmodified nonlinear Vlasov equation, Math. Methods Appl. Sci. 6 (1984), n° 2, p. 262-279. | DOI | MR | Zbl

[43] R. Illner & H. Neunzert - An existence theorem for the unmodified Vlasov equation, Math. Methods Appl. Sci. 1 (1979), n° 4, p. 530-544. | DOI | MR | Zbl

[44] J. H. Jeans - On the theory of star-streaming and the structure of the universe, Mon. Not. R. Astr. Soc. 76 (1915), p. 70-84. | DOI

[45] J. H. Jeans, Problems of cosmogony and stellar dynamics, Cambridge Univ. Press, 1919. | JFM

[46] H. E. Kandrup & J. F. Sygnet - A simple proof of dynamical stability for a class of spherical clusters, Astrophys. J. 298 (1985), n° 1, part 1, p. 27-33. | DOI | MR

[47] I. R. King - The structure of star clusters. III. Some simple dynamical models, Astronomical Journal 71 (1966), p. 64. | DOI

[48] M. D. Kruskal & C. R. Oberman - On the stability of plasma in static equilibrium, Phys. Fluids 1 (1958), p. 275-280. | DOI | MR | Zbl

[49] M. Lemou, F. Méhats & P. Raphael - Orbital stability and singularity formation for Vlasov-Poisson systems, C.R. Math. Acad. Sci. Paris 341 (2005) , n° 4, p. 269-274. | DOI | MR | Zbl

[50] M. Lemou, F. Méhats & P. Raphaël - Uniqueness of the critical mass blow up solution for the four dimensional gravitational Vlasov-Poisson system, Ann. Inst H. Poincaré Anal. Non Linéaire 24 (2007), n° 5, p. 825-833. | DOI | EuDML | MR | Zbl

[51] M. Lemou, F. Méhats & P. Raphael - The orbital stability of the ground states and the singularity formation for the gravitational Vlasov Poisson system, Arch. Ration. Mech. Anal 189 (2008), n° 3, p. 425-468. | DOI | MR | Zbl

[52] M. Lemou, F. Méhats & P. Raphaël - Stable self-similar blow up dynamics for the three dimensional relativistic gravitational Vlasov-Poisson system, J. Amer. Math. Soc. 21 (2008), n° 4, p. 1019-1063. | DOI | MR | Zbl

[53] M. Lemou, F. Méhats & P. Raphaël, Structure of the linearized gravitational Vlasov-Poisson system close to a polytropic ground state, SIAM J. Math. Anal. 39 (2008), n° 6, p. 1711-1739. | DOI | MR | Zbl

[54] M. Lemou, F. Méhats & P. Raphaël, A new variational approach to the stability of gravitational systems, C.R. Math. Acad. Sci. Paris 347 (2009), nos 15-16, p. 979-984. | DOI | MR | Zbl

[55] M. Lemou, F. Méhats & P. Raphaël, Stable ground states for the relativistic gravitational Vlasov-Poisson system, Comm. Partial Differential Equations 34 (2009), nos 7-9, p. 703-721. | DOI | MR | Zbl

[56] M. Lemou, F. Méhats & P. Raphaël, A new variational approach to the stability of gravitational systems, Comm. Math. Phys. 302 (2011), n° 1, p. 161-224. | DOI | MR

[57] M. Lemou, F. Méhats & P. Raphaël, Orbital stability of spherical galactic models, Invent. Math. 187 (2012), n° 1, p. 145-194. | DOI | MR | Zbl

[58] E. Lieb & M. Loss - Analysis, Amer. Math. Soc., 2001. | MR | Zbl

[59] P.-L. Lions - The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), n° 2, p. 109-145. | DOI | EuDML | Numdam | MR | Zbl

[60] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), n° 4, p. 223-283. | DOI | EuDML | Numdam | MR | Zbl

[61] P.-L. Lions & B. Perthame - Propagation of moments and regularity for the $3$ -dimensional Vlasov-Poisson system, Invent. Math. 105 (1991), n° 2, p. 415-430. | DOI | EuDML | MR | Zbl

[62] G. Loeper - Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl. 86 (2006), n° 1, p. 68-79. | DOI | MR | Zbl

[63] D. Lynden-Bell - The stability and vibrations of a gas of stars, Mon. Not. R. Astr. Soc. 124 (1962), p. 279-296. | DOI | Zbl

[64] D. Lynden-Bell, Statistical mechanics of violent relaxation in stellar systems, Mon. Not. R. Astr. Soc. 136 (1967), p. 101-121. | DOI

[65] D. Lynden-Bell, The Hartree-Fock exchange operator and the stability of galaxies, Mon. Not. R. Astr. Soc. 144 (1969), p. 189-217. | DOI

[66] C. Marchioro, E. Miot & M. Pulvirenti - The Cauchy problem for the $3 - D$ Vlasov-Poisson system with point charges, Arch. Ration. Mech. Anal. 201 (2011), n° 1, p. 1-26. | DOI | MR

[67] P. J. Morrison & D. Pfirsch - The free energy of Maxwell-Vlasov equilibria, Phys. Fluids B 2 (1990), n° 6, part 1, p. 1105-1113. | DOI | MR

[68] P. J. Morrison & D. Pfirsch, Dielectric energy versus plasma energy, and Hamiltonian action-angle variables for the Vlasov equation, Phys. Fluids B 4 (1992), n° 10, p. 3038-3057. | DOI | MR

[69] C. Mouhot & C. Villani - On Landau damping, Acta Math. 207 (2011), n° 1, p. 29-201. | DOI | MR | Zbl

[70] C. Pallard - Moment propagation for weak solutions to the Vlasov-Poisson system, Comm. Partial Differential Equations 37 (2012), n° 7, p. 1273-1285. | DOI | MR

[71] J. Perez & J. J. Aly - Stability of spherical stellar systems. I : Analytical results, Mon. Not. R. Astr. Soc. 280 (1996), p. 689-699. | DOI

[72] K. Pfaffelmoser - Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations 95 (1992), n° 2, p. 281-303. | DOI | MR | Zbl

[73] G. Rein - Non-linear stability for the Vlasov-Poisson system-the energy-Casimir method, Math. Methods Appl. Sci. 17 (1994), n° 14, p. 1129-1140. | DOI | MR | Zbl

[74] G. Rein, Stability of spherically symmetric steady states in galactic dynamics against general perturbations, Arch. Ration. Mech. Anal. 161 (2002), n° 1, p. 27-42. | DOI | MR | Zbl

[75] G. Rein & Y. Guo - Stable models of elliptical galaxies, Mon. Not. R. Astr. Soc. 344 (2003), p. 1296-1306. | DOI

[76] R. Robert - Unicité de la solution faible à support compact de l'équation de Vlasov-Poisson, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), n° 8, p. 873-877. | DOI | MR | Zbl

[77] Ó. Sánchez & J. Soler - Orbital stability for polytropic galaxies, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), n° 6, p. 781-802. | DOI | EuDML | Numdam | MR | Zbl

[78] J. Schaeffer - Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Comm. Partial Differential Equations 16 (1991), nos 8-9, p. 1313-1335. | DOI | MR | Zbl

[79] J. Schaeffer, Steady states in galactic dynamics, Arch. Ration. Mech. Anal. 172 (2004), n° 1, p. 1-19. | DOI | MR | Zbl

[80] J. F. Sygnet, G. Des Forets, M. Lachieze-Rey & R. Pellat - Stability of gravitational systems and gravothermal catastrophe in astrophysics, Astrophys. J. 276 (1984), p. 737-745. | DOI

[81] Y. H. Wan - Nonlinear stability of stationary spherically symmetric models in stellar dynamics, Arch. Rational Mech. Anal. 112 (1990), n° 1, p. 83-95. | DOI | MR | Zbl

[82] Y. H. Wan, On nonlinear stability of isotropic models in stellar dynamics, Arch. Ration. Mech. Anal. 147 (1999), n° 3, p. 245-268. | DOI | MR | Zbl

[83] A. Weinstein - The local structure of Poisson manifolds, J. Differential Geom. 18 (1983), n° 3, p. 523-557. | DOI | MR | Zbl

[84] M. I. Weinstein - Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), n° 3, p. 472-491. | DOI | MR | Zbl

[85] M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math. 39 (1986), n° 1, p. 51-67. | DOI | MR | Zbl

[86] H. Wiechen, H. J. Ziegler & K. Schindler - Relaxation of collisionless self gravitating matter : the lowest energy state, Mon. Not. R. Astr. Soc. 232 (1988), p. 623-646. | DOI | Zbl

[87] G. Wolansky - On nonlinear stability of polytropic galaxies, Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999), n° 1, p. 15-48. | DOI | EuDML | Numdam | MR | Zbl

[88] S. Wollman - Global-in-time solutions to the three-dimensional Vlasov-Poisson system, J. Math. Anal. Appl. 176 (1993), n° 1, p. 76-91. | DOI | MR | Zbl

[89] H. Ye, P. J. Morrison & J. D. Crawford - Poisson bracket for the Vlasov equation on a symplectic leaf, Phys. Lett. A 156 (1991), nos 1-2, p. 96-100. | DOI | MR