[Stabilité orbitale et formation de singularité pour des systèmes de Vlasov–Poisson]
Nous considérons le système de Vlasov–Poisson gravitationnel en dimensions et et replaçons l'étude de la stabilité non linéaire des états stationnaires dans le cadre des techniques de concentration compacité. En dimension 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 où le problème est 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 and and consider the problem of nonlinear stability of steady states solutions within the framework of concentration compactness techniques. In dimension 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 where the problem is 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.
Accepté le :
Publié le :
@article{CRMATH_2005__341_4_269_0,
author = {Lemou, Mohammed and M\'ehats, Florian and Raphael, Pierre},
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},
url = {http://archive.numdam.org/articles/10.1016/j.crma.2005.06.018/}
}
TY - JOUR AU - Lemou, Mohammed AU - Méhats, Florian AU - Raphael, Pierre 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 UR - http://archive.numdam.org/articles/10.1016/j.crma.2005.06.018/ DO - 10.1016/j.crma.2005.06.018 LA - en ID - CRMATH_2005__341_4_269_0 ER -
%0 Journal Article %A Lemou, Mohammed %A Méhats, Florian %A Raphael, Pierre %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 %U http://archive.numdam.org/articles/10.1016/j.crma.2005.06.018/ %R 10.1016/j.crma.2005.06.018 %G en %F CRMATH_2005__341_4_269_0
Lemou, Mohammed; Méhats, Florian; Raphael, Pierre. Orbital stability and singularity formation for Vlasov–Poisson systems. Comptes Rendus. Mathématique, Tome 341 (2005) no. 4, pp. 269-274. doi : 10.1016/j.crma.2005.06.018. http://archive.numdam.org/articles/10.1016/j.crma.2005.06.018/
[1] Kinetic Equations and Asymptotic Theory, Ser. Appl. Math., Gauthiers-Villars, Paris, 2000
[2] Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys., Volume 85 (1982) no. 4, pp. 549-561
[3] Statistical mechanics and thermodynamic limit of self-gravitating fermions in D dimensions, Phys. Rev. E, Volume 69 (2004), p. 066126
[4] 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] The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996
[6] Variational method for stable polytropic galaxies, Arch. Rational Mech. Anal., Volume 130 (1995), pp. 163-182
[7] Isotropic steady states in galactic dynamics, Commun. Math. Phys., Volume 219 (2001), pp. 607-629
[8] 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] 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] 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] 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] 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] Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., Volume 87 (1983), pp. 567-576
Cité par Sources :
