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) no. 5, pp. 825-833. 
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     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},
     pages = {825--833},
     publisher = {Elsevier},
     volume = {24},
     number = {5},
     year = {2007},
     doi = {10.1016/j.anihpc.2006.07.003},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2006.07.003/}
}
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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) no. 5, pp. 825-833. doi : 10.1016/j.anihpc.2006.07.003. http://www.numdam.org/articles/10.1016/j.anihpc.2006.07.003/

[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 | Zbl

[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

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

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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

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

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

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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