Stable self-similar blow up dynamics for the three dimensional relativistic gravitational Vlasov-Poisson system

Journal of the American Mathematical Society

Volumn 21, Issue 4, 2008, Pages 1019-1063

  Lemou, Mohammed ^a   Méhats, Florian ^b   Raphaël, Pierre ^c  

^a UNIVERSITÉ PAUL SABATIER   (France)

^b UNIVERSITÉ DE RENNES 1   (France)

^c CNRS   (France)

Author keywords

Blow up dynamics; Modulation theory; Relativistic gravitational kinetic equations; Virial law; Vlasov Poisson equations

Indexed keywords

EID: 62649174841     PISSN: 08940347     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0894-0347-07-00579-6     Document Type: Article

References (45)

1. Batt, J., Global symmetric solutions of the initial value problem of stellar dynamics, J. Differential Equations 25 (1977), no. 3, 342-364. MR0487082 (58:6749)
2. Batt, J.; Faltenbacher, W.; Horst, E., Stationary spherically symmetric models in stellar dynamics, Arch. Rat. Mech. Anal. 93, 159-183 (1986). MR823117 (87i:85001)
3. Binney, J.; Tremaine, S., Galactic Dynamics, Princeton University Press, 1987.
4. Bouchut, F.; Golse, F.; Pulvirenti, M., Kinetic equations and asymptotic theory, Series in Appl. Math., Gauthiers-Villars, Paris, 2000. MR2065070 (2005d:82102)
5. Bourgain, J.; Wang, W., Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 197-215 (1998). MR1655515 (99m:35219)
6. Choquet-Bruhat, Y.; Noutchegueme, N., Global solutions of the Yang-Mills-Vlasov system with zero rest mass, C. R. Acad. Sci. Paris Ser. I Math. 311 (1990), no. 12, 785-788. MR1082633 (92g:58128)
7. Cote, R., Construction of solutions to the subcritical gKdV equations with a given asymptotic behaviour, J. Funct. Anal. 241 (2006), no. 1, 143-211. MR2264249 (2007h:35279) (Pubitemid 44488085)
8. Dafermos, M., A note on the collapse of small data self-gravitating massless collisionless matter, J. Hyp. Diff. Eq. 3, 589-598. MR2289606
9. Dafermos, M.; Rendall, A., An extension principle for the Einstein-Vlasov system under spherical symmetry, Ann. Henri Poincare 6 (2005), 1137-1155. MR2189379 (2007c:83011) (Pubitemid 41813598)
10. Degond, P., Local existence of solutions of the Vlasov-Maxwell equations and convergence to the Vlasov-Poisson equations for infinite light velocity, Math. Methods Appl. Sci. 8 (1986), no. 4, 533-558. MR870991 (88a:76054) (Pubitemid 17512737)
11. DiPerna, R. J.; Lions, P.-L., Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math. 42 (1989), no. 6, 729-757. MR1003433 (90i:35236)
12. Dolbeault, J., Explosion à temps fini et évanescence en temps grand dans les équations cinétiques de la gravitation, unpublished manuscript.
13. 2 critical non linear Schrödinger equation, to appear in Physica D.
14. Fridmann, A. M.; Polyachenko, V. L., Physics of gravitating systems, Springer-Verlag
15. Frohlich, J.; Lenzman, E., Dynamical Collapse of White Dwarfs in Hartree- and Hartree-Fock Theory, preprint arXiv/math-ph/0608020.
16. Frohlich, J.; Jonsson, L.; Lenzman, E., Boson Stars as Solitary Waves, to appear in Commun. Math. Phys., preprint arXiv/math-ph/0512040.
17. Glassey, R., The Cauchy problem in kinetic theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. MR1379589 (97i:82070)
18. Glassey, R. T.; Schaeffer, J., On symmetric solutions of the relativistic Vlasov-Poisson system, Comm. Math. Phys. 101 (1985), no. 4, 459-473. MR815195 (87c:35138)
19. Glassey, R. T.; Schaeffer, J. W., Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data, Comm. Math. Phys. 119 (1988), no. 3, 353-384. MR969207 (90b:82042)
20. Glassey, R. T.; Schaeffer, J., The relativistic Vlasov-Maxwell system in two space dimensions. I, II, Arch. Rational Mech. Anal. 141 (1998), no. 4, 331-354, 355-374. MR1620506 (99d:82071) (Pubitemid 128062035)
21. Glassey, R. T.; Schaeffer, J., On global symmetric solutions to the relativistic Vlasov-Poisson equation in three space dimensions, Math. Methods Appl. Sci. 24 (2001), no. 3, 143-157. MR1810884 (2001m:82082) (Pubitemid 32227042)
22. Glassey, R. T.; Strauss, W. A., Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rational Mech. Anal. 92 (1986), no. 1, 59-90. MR816621 (87j:82064)
23. Guo, Y.; Rein, G., Isotropic steady states in galactic dynamics, Comm. Math. Phys. 219 (2001), 607-629. MR1838751 (2002g:85001) (Pubitemid 33381621)
24. Hadzic, M; Rein; G., Global existence and nonlinear stability for the relativistic Vlasov- Poisson system in the gravitational case, to appear in Indiana Univ. Math. J.
25. Krieger, J., Schlag, W., Non-generic blow-up solutions for the critical focusing NLS in 1-d, preprint (2005).
26. Lemou, M.; Méhats, F.; Raphaël, P., On the orbital stability of the ground states and the singularity formation for the gravitational Vlasov-Poisson system, to appear in Arch. Rat. Mech. Anal.
27. Lemou, M.; Méhats, F.; Raphaël, P., Structure of the linearized gravitational Vlasov-Poisson system close to a polytropic ground state, to appear in SIAM J. Math. Anal.
28. Lemou, M.; Méhats, F.; Raphaël, P., Uniqueness of the critical mass blow up solution for the four dimensional gravitational Vlasov-Poisson system, to appear in Ann. Inst. H. Poincaé Anal. Non Linéaire.
29. Lieb, E. H.; Yau, H. T., The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys. 112 (1987), no. 1, 147-174. MR904142 (89b:82014)
30. Lions, P.-L.; Perthame, B., Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system. Invent. Math. 105 (1991), no. 2. MR1115549 (92e:35160)
31. Martin-Garcia, J.; Gundlach, C., Self-similar spherically symmetric solutions of the massless Einstein-Vlasov system, Phys. Rev. D (3) 65 (2002), no. 8. MR1899211 (2002m:83010)
32. Merle, F.; Raphaël, P., Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation, Ann. Math. 161 (2005), no. 1, 157-222. MR2150386 (2006k:35277) (Pubitemid 41137799)
33. Merle, F.; Raphaël, P., Sharp upper bound on the blow up rate for critical nonlinear Schrödinger equation, Geom. Funct. Ana 13 (2003), 591-642. MR1995801 (2005j:35207) (Pubitemid 37049166)
34. 2 critical nonlinear Schrödinger equation, Invent. Math. 156, 565-672 (2004). MR2061329 (2006a:35283) (Pubitemid 38739738)
35. Merle, F.; Raphaël, P., On a sharp lower bound on the blow up rate for critical nonlinear Schrödinger equation, J. Amer. Math. Soc. 19 (2006), no. 1, 37-90. MR2169042 (2006j:35223) (Pubitemid 43089183)
36. Merle, F.; Raphaël, P., Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation, Comm. Math. Phys. 253 (2004), no. 3, 675-704. MR2116733 (2006m:35346)
37. Pfaffelmoser, K.; Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Diff. Eq. 95 (1992), 281-303. MR1165424 (93d:35170)
38. 2-critical nonlinear Schrödinger equation in a domain, to appear in Ann. Henri Poincare.
39. Raphaël, P., Stability of the log-log bound for blow up solutions to the critical nonlinear Schrödinger equation, Math. Ann. 331 (2005), 577-609. MR2122541 (2006b:35303) (Pubitemid 40251235)
40. 2 supercritical nonlinear Schrödinger equation, Duke Math. J. 134 no. 2 (2006), 199-258. MR2248831 (Pubitemid 44367791)
41. Rein, G.; Rendall, A. D., Global existence of solutions of the spherically symmetric Vlasov- Einstein system with small initial data. Comm. Math. Phys. 150 (1992), no. 3, 561-583. MR1204320 (94c:83028) (Pubitemid 126035594)
42. Schaeffer, J., Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Comm. Part. Diff. Eq. 16 (1991), 1313-1335. MR1132787 (92g:82113)
43. Schaeffer, J., Steady States in Galactic Dynamics, Arch. Rational, Mech. Anal. 172 (2004), 1-19. MR2048565 (2004m:35263) (Pubitemid 38545913)
44. Van Kampen, N. G.; Felderhof, B. V., Theoretical methods in plasma physics, Amsterdam, North Holland, 1967.
45. Weinstein, M. I., Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983), 567-576. MR691044 (84d:35140)