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

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1. [1] Antonini C., Lower bounds for the $L^2$ minimal periodic blow-up solutions of critical nonlinear Schrödinger equation, Differential Integral Equations15 (6) (2002) 749-768. Zbl1016.35018MR1893845
2. [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. Zbl1170.35528MR2064970
3. [3] Batt J., Faltenbacher W., Horst E., Stationary spherically symmetric models in stellar dynamics, Arch. Rational Mech. Anal.93 (1986) 159-183. Zbl0605.70008MR823117
4. [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. Zbl0533.35029MR695535
5. [5] Binney J., Tremaine S., Galactic Dynamics, Princeton University Press, 1987. Zbl1130.85301
6. [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. Zbl1044.35084MR1978490
7. [7] Chavanis P.-H., Statistical mechanics and thermodynamic limit of self-gravitating fermions in D dimensions, Phys. Rev. E69 (2004) 066126. MR2096500
8. [8] Diperna R.J., Lions P.-L., Global weak solutions of kinetic equations, Rend. Sem. Mat. Univ. Politec. Torino46 (3) (1988) 259-288. Zbl0813.35087MR1101105
9. [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. Zbl0682.35022
10. [10] Fridmann A.M., Polyachenko V.L., Physics of Gravitating Systems, Springer-Verlag, New York, 1984. Zbl0543.70010
11. [11] Glassey R., The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. Zbl0858.76001MR1379589
12. [12] Hmidi T., Keraani S., Blowup theory for the critical nonlinear Schrödinger equations revisited, Int. Math. Res. Not.46 (2005) 2815-2828. Zbl1126.35067MR2180464
13. [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. Zbl0556.35022MR751745
14. [14] Kwong M.K., Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$, Arch. Rational Mech. Anal.105 (3) (1989) 243-266. Zbl0676.35032MR969899
15. [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. I341 (4) (2005) 269-274. Zbl1073.70012MR2164685
16. [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. Zbl1221.35417
17. [17] Lions P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire1 (2) (1984) 109-145. Zbl0541.49009MR778970
18. [18] Lions P.-L., Perthame B., Propagation of moments and regularity for the 3-dimensional Vlasov–Poisson system, Invent. Math.105 (2) (1991). Zbl0741.35061
19. [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. Zbl1033.35102MR1944576
20. [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. Zbl1005.35081MR1888800
21. [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. Zbl0808.35141MR1203233
22. [22] Merle F., Asymptotics for $L^2$ minimal blow-up solutions of critical nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire13 (5) (1996) 553-565. Zbl0862.35013MR1409662
23. [23] Merle F., Raphaël P., On Universality of Blow up Profile for $L^2$ critical nonlinear Schrödinger equation, Invent. Math.156 (2004) 565-672. Zbl1067.35110MR2061329
24. [24] Weinstein M.I., Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys.87 (1983) 567-576. Zbl0527.35023MR691044