Comptes Rendus
Mathematical Problems in Mechanics
Orbital stability and singularity formation for Vlasov–Poisson systems
[Stabilité orbitale et formation de singularité pour des systèmes de Vlasov–Poisson]
Comptes Rendus. Mathématique, Volume 341 (2005) no. 4, pp. 269-274.

Nous considérons le système de Vlasov–Poisson gravitationnel en dimensions N=3 et N=4 et replaçons l'étude de la stabilité non linéaire des états stationnaires dans le cadre des techniques de concentration compacité. En dimension N=3 où le problème est sous-critique, nous démontrons la stabilité orbitale dans l'espace d'énergie des polytropes qui sont des solutions stationnaires de type ground state, ce qui améliore pour cette classe les résultats déjà publiés. En dimension N=4 où le problème est L1 critique, les polytropes sont obtenus dans la lignée de Weinstein [M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys. 87 (1983) 567–576] comme minimiseurs d'une inégalité de type Gagliardo–Nirenberg. Un fait remarquable est maintenant l'existence d'une symétrie conforme qui nous permet d'écrire des solutions explosives explicites de masse critique. Ceci constitue à notre connaissance le premier résultat de description d'une formation de singularité dans le cadre des équations cinétiques de type Vlasov. Un résultat général de concentration de masse est enfin obtenu pour les solutions explosives.

We study the gravitational Vlasov–Poisson system in dimension N=3 and N=4 and consider the problem of nonlinear stability of steady states solutions within the framework of concentration compactness techniques. In dimension N=3 where the problem is subcritical, we prove the orbital stability in the energy space of the polytropes which are ground state type stationary solutions, which improves the already published results for this class. In dimension N=4 where the problem is L1 critical, polytropic steady states are obtained following Weinstein [M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys. 87 (1983) 567–576] by minimizing a suitable Gagliardo Nirenberg type inequality. Now a striking feature is the existence of a pseudo-conformal symmetry which allows us to derive explicit critical mass finite time blow up solutions. This is to our knowledge the first result of description of a singularity formation in a Vlasov setting. A general mass concentration phenomenon is eventually obtained for finite time blow up solutions.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2005.06.018

Mohammed Lemou 1 ; Florian Méhats 1 ; Pierre Raphael 2

1 MIP (UMR CNRS 5640), université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse cedex, France
2 Département de mathématiques, CNRS et université Paris Sud, bâtiment 425, 91405 Orsay, France
@article{CRMATH_2005__341_4_269_0,
     author = {Mohammed Lemou and Florian M\'ehats and Pierre Raphael},
     title = {Orbital stability and singularity formation for {Vlasov{\textendash}Poisson} systems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {269--274},
     publisher = {Elsevier},
     volume = {341},
     number = {4},
     year = {2005},
     doi = {10.1016/j.crma.2005.06.018},
     language = {en},
}
TY  - JOUR
AU  - Mohammed Lemou
AU  - Florian Méhats
AU  - Pierre Raphael
TI  - Orbital stability and singularity formation for Vlasov–Poisson systems
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 269
EP  - 274
VL  - 341
IS  - 4
PB  - Elsevier
DO  - 10.1016/j.crma.2005.06.018
LA  - en
ID  - CRMATH_2005__341_4_269_0
ER  - 
%0 Journal Article
%A Mohammed Lemou
%A Florian Méhats
%A Pierre Raphael
%T Orbital stability and singularity formation for Vlasov–Poisson systems
%J Comptes Rendus. Mathématique
%D 2005
%P 269-274
%V 341
%N 4
%I Elsevier
%R 10.1016/j.crma.2005.06.018
%G en
%F CRMATH_2005__341_4_269_0
Mohammed Lemou; Florian Méhats; Pierre Raphael. Orbital stability and singularity formation for Vlasov–Poisson systems. Comptes Rendus. Mathématique, Volume 341 (2005) no. 4, pp. 269-274. doi : 10.1016/j.crma.2005.06.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.06.018/

[1] F. Bouchut; F. Golse; M. Pulvirenti Kinetic Equations and Asymptotic Theory, Ser. Appl. Math., Gauthiers-Villars, Paris, 2000

[2] T. Cazenave; P.-L. Lions Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys., Volume 85 (1982) no. 4, pp. 549-561

[3] P.-H. Chavanis Statistical mechanics and thermodynamic limit of self-gravitating fermions in D dimensions, Phys. Rev. E, Volume 69 (2004), p. 066126

[4] J. Dolbeault; Ó. Sánchez; J. Soler Asymptotic behaviour for the Vlasov–Poisson system in the stellar-dynamics case, Arch. Rational Mech. Anal., Volume 171 (2004) no. 3, pp. 301-327

[5] R. Glassey The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996

[6] Y. Guo Variational method for stable polytropic galaxies, Arch. Rational Mech. Anal., Volume 130 (1995), pp. 163-182

[7] Y. Guo; G. Rein Isotropic steady states in galactic dynamics, Commun. Math. Phys., Volume 219 (2001), pp. 607-629

[8] E. Horst; R. Hunze Weak solutions of the initial value problem for the unmodified non-linear Vlasov equation, Math. Methods Appl. Sci., Volume 6 (1984), pp. 262-279

[9] M. Lemou, F. Méhats, P. Raphael, On the orbital stability of the ground states and the singularity formation for the gravitational Vlasov Poisson system, submitted for publication

[10] P.-L. Lions The concentration-compactness principle in the calculus of variations. The locally compact case, I. Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 1 (1984) no. 2, pp. 109-145

[11] F. Merle; P. Raphaël Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation, Ann. of Math., Volume 161 (2005) no. 1, pp. 157-222

[12] F. Merle; P. Raphaël Profiles and quantization of the blow up mass for critical non linear Schrödinger equation, Commun. Math. Phys., Volume 253 (2004) no. 3, pp. 675-704

[13] F. Merle; Y. Tsutsumi L2 concentration of blow up solutions for the nonlinear Schrödinger equation with critical power nonlinearity, J. Differential Equations, Volume 84 (1990), pp. 205-214

[14] Ó. Sánchez, J. Soler, Orbital stability for polytropic galaxies, Preprint

[15] M.I. Weinstein Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., Volume 87 (1983), pp. 567-576

  • Juhi Jang; Jinmyoung Seok On uniformly rotating binary stars and galaxies, Archive for Rational Mechanics and Analysis, Volume 244 (2022) no. 2, pp. 443-499 | DOI:10.1007/s00205-022-01766-4 | Zbl:1514.85001
  • Mohammed Lemou Non linear stability of spherical gravitational systems described by the Vlasov-Poisson equation, Séminaire Laurent Schwartz. EDP et Applications, Volume 2011-2012 (2013), p. ex | DOI:10.5802/slsedp.14 | Zbl:1319.35266
  • Mohammed Lemou; Florian Méhats; Pierre Raphaël Orbital stability of spherical galactic models, Inventiones Mathematicae, Volume 187 (2012) no. 1, pp. 145-194 | DOI:10.1007/s00222-011-0332-9 | Zbl:1232.35170
  • Mohammed Lemou; Florian Méhats; Pierre Raphaël A new variational approach to the stability of gravitational systems, Communications in Mathematical Physics, Volume 302 (2011) no. 1, pp. 161-224 | DOI:10.1007/s00220-010-1182-9 | Zbl:1339.35320
  • Juan Campos; Manuel del Pino; Jean Dolbeault Relative equilibria in continuous stellar dynamics, Communications in Mathematical Physics, Volume 300 (2010) no. 3, pp. 765-788 | DOI:10.1007/s00220-010-1128-2 | Zbl:1203.35280
  • Nikolaos Bournaveas; Vincent Calvez Critical mass phenomenon for a chemotaxis kinetic model with spherically symmetric initial data, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, Volume 26 (2009) no. 5, pp. 1871-1895 | DOI:10.1016/j.anihpc.2009.02.001 | Zbl:1171.92003
  • Mohammed Lemou; Florian Méhats; Pierre Raphaël Stable Ground States for the Relativistic Gravitational Vlasov–Poisson System, Communications in Partial Differential Equations, Volume 34 (2009) no. 7, p. 703 | DOI:10.1080/03605300902963369
  • Jean Dolbeault; Javier Fernández Localized minimizers of flat rotating gravitational systems, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, Volume 25 (2008) no. 6, pp. 1043-1071 | DOI:10.1016/j.anihpc.2007.01.001 | Zbl:1157.35112
  • Mohammed Lemou; Florian Méhats; Pierre Raphael The orbital stability of the ground states and the singularity formation for the gravitational Vlasov-Poisson system, Archive for Rational Mechanics and Analysis, Volume 189 (2008) no. 3, pp. 425-468 | DOI:10.1007/s00205-008-0126-4 | Zbl:1221.35417
  • Mohammed Lemou; Florian Méhats; Pierre Raphaël Uniqueness of the critical mass blow up solution for the four-dimensional gravitational Vlasov-Poisson system, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, Volume 24 (2007) no. 5, pp. 825-833 | DOI:10.1016/j.anihpc.2006.07.003 | Zbl:1138.35090
  • Yan Guo; Gerhard Rein A non-variational approach to nonlinear stability in stellar dynamics applied to the King model, Communications in Mathematical Physics, Volume 271 (2007) no. 2, pp. 489-509 | DOI:10.1007/s00220-007-0212-8 | Zbl:1130.85002

Cité par 11 documents. Sources : Crossref, zbMATH

Commentaires - Politique


Il n'y a aucun commentaire pour cet article. Soyez le premier à écrire un commentaire !


Publier un nouveau commentaire:

Publier une nouvelle réponse: