Uniqueness of the critical mass blow up solution for the four dimensional gravitational Vlasov-Poisson system

Lemou, Mohammed; Méhats, Florian; Raphaël, Pierre

Lemou, Mohammed ; Méhats, Florian ; Raphaël, Pierre

Annales de l'I.H.P. Analyse non linéaire, Tome 24 (2007), p. 825-833 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Publié le : 2007-01-01

Zbl pre05228824

DOI : https://doi.org/10.1016/j.anihpc.2006.07.003

@article{AIHPC_2007__24_5_825_0,
     author = {Lemou, Mohammed and M\'ehats, Florian and Rapha\"el, Pierre},
     title = {Uniqueness of the critical mass blow up solution for the four dimensional gravitational Vlasov-Poisson system},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {24},
     year = {2007},
     pages = {825-833},
     doi = {10.1016/j.anihpc.2006.07.003},
     zbl = {pre05228824},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2007__24_5_825_0}
}

Lemou, Mohammed; Méhats, Florian; Raphaël, Pierre. Uniqueness of the critical mass blow up solution for the four dimensional gravitational Vlasov-Poisson system. Annales de l'I.H.P. Analyse non linéaire, Tome 24 (2007) pp. 825-833. doi : 10.1016/j.anihpc.2006.07.003. http://gdmltest.u-ga.fr/item/AIHPC_2007__24_5_825_0/

Bibliographie

[1] Antonini C., Lower bounds for the $L^{2}$ minimal periodic blow-up solutions of critical nonlinear Schrödinger equation, Differential Integral Equations 15 (6) (2002) 749-768. | MR 1893845 | Zbl 1016.35018

[2] Banica V., Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain, Ann. Sc. Norm. Super. Pisa Cl. Sci. 3 (1) (2004) 139-170. | Numdam | MR 2064970 | Zbl pre02217257

[3] Batt J., Faltenbacher W., Horst E., Stationary spherically symmetric models in stellar dynamics, Arch. Rational Mech. Anal. 93 (1986) 159-183. | MR 823117 | Zbl 0605.70008

[4] Berestycki H., Lions P.-L., Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (4) (1983) 313-345. | MR 695535 | Zbl 0533.35029

[5] Binney J., Tremaine S., Galactic Dynamics, Princeton University Press, 1987. | Zbl 1130.85301

[6] Burq N., Gérard P., Tzvetkov N., Two singular dynamics of the nonlinear Schrödinger equation on a plane domain, Geom. Funct. Anal. 13 (1) (2003) 1-19. | MR 1978490 | Zbl 1044.35084

[7] Chavanis P.-H., Statistical mechanics and thermodynamic limit of self-gravitating fermions in D dimensions, Phys. Rev. E 69 (2004) 066126. | MR 2096500

[8] Diperna R.J., Lions P.-L., Global weak solutions of kinetic equations, Rend. Sem. Mat. Univ. Politec. Torino 46 (3) (1988) 259-288. | MR 1101105 | Zbl 0813.35087

[9] Diperna R.J., Lions P.-L., Solutions globales d'équations du type Vlasov-Poisson, C. R. Acad. Sci. Paris Sér. I Math. 307 (12) (1988) 655-658. | Zbl 0682.35022

[10] Fridmann A.M., Polyachenko V.L., Physics of Gravitating Systems, Springer-Verlag, New York, 1984. | Zbl 0543.70010

[11] Glassey R., The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. | MR 1379589 | Zbl 0858.76001

[12] Hmidi T., Keraani S., Blowup theory for the critical nonlinear Schrödinger equations revisited, Int. Math. Res. Not. 46 (2005) 2815-2828. | MR 2180464 | Zbl 1126.35067

[13] Horst E., Hunze R., Weak solutions of the initial value problem for the unmodified nonlinear Vlasov equation, Math. Methods Appl. Sci. 6 (2) (1984) 262-279. | MR 751745 | Zbl 0556.35022

[14] Kwong M.K., Uniqueness of positive solutions of $Δ u - u + u^{p} = 0$ in $R^{n}$ , Arch. Rational Mech. Anal. 105 (3) (1989) 243-266. | MR 969899 | Zbl 0676.35032

[15] Lemou M., Méhats F., Raphael P., On the orbital stability of the ground states and the singularity formation for the gravitational Vlasov Poisson system, C. R. Math. Acad. Sci. Paris, Ser. I 341 (4) (2005) 269-274. | MR 2164685 | Zbl 1073.70012

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

[17] 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 (2) (1984) 109-145. | Numdam | MR 778970 | Zbl 0541.49009

[18] Lions P.-L., Perthame B., Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math. 105 (2) (1991). | Zbl 0741.35061

[19] Martel Y., Merle F., Nonexistence of blow-up solution with minimal $L^{2}$ -mass for the critical gKdV equation, Duke Math. J. 115 (2) (2002) 385-408. | MR 1944576 | Zbl 1033.35102

[20] Martel Y., Merle F., Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation, Ann. of Math. (2) 155 (1) (2002) 235-280. | MR 1888800 | Zbl 1005.35081

[21] Merle F., Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Math. J. 69 (2) (1993) 427-454. | MR 1203233 | Zbl 0808.35141

[22] Merle F., Asymptotics for $L^{2}$ minimal blow-up solutions of critical nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (5) (1996) 553-565. | Numdam | MR 1409662 | Zbl 0862.35013

[23] Merle F., Raphaël P., On Universality of Blow up Profile for $L^{2}$ critical nonlinear Schrödinger equation, Invent. Math. 156 (2004) 565-672. | MR 2061329 | Zbl 1067.35110

[24] Weinstein M.I., Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983) 567-576. | MR 691044 | Zbl 0527.35023