Chapter 5 Collisionless Kinetic Equations from Astrophysics - The Vlasov-Poisson System

Handbook of Differential Equations: Evolutionary Equations

Volumn 3, Issue , 2007, Pages 383-476

  Rein, Gerhard ^a  

^a UNIVERSITY OF BAYREUTH   (Germany)

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[No Author keywords available]

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EID: 36049049331     PISSN: 18745717     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S1874-5717(07)80008-9     Document Type: Chapter

References (116)

1. Aly J.J. On the lowest energy state of a collisionless selfgravitating system under phase space volume constraints. Mon. Not. R. Astron. Soc. 241 (1989) 15-27
2. Aly J.J. Existence of a minimum energy state for a constrained collisionless gravitational system (2005) Preprint
3. Andréasson H. The Einstein-Vlasov system/kinetic theory. Living Reviews in Relativity (2005), Max Planck Institute for Gravitational Physics. http://www.livingreviews.org/lrr-2005-2 available at
4. Arsen′ev A.A. Global existence of a weak solution of Vlasov′s system of equations. Comput. Math. Math. Phys. 15 (1975) 131-141
5. Bardos C., and Degond P. Global existence for the Vlasov-Poisson system in 3 space variables with small initial data. Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985) 101-118
6. Batt J. Ein Existenzbeweis für die Vlasov-Gleichung der Stellardynamik bei gemittelter Dichte. Arch. Ration. Mech. Anal. 13 (1963) 296-308
7. Batt J. Global symmetric solutions of the initial value problem in stellar dynamics. J. Differential Equations 25 (1977) 342-364
8. Batt J., Berestycki H., Degond P., and Perthame B. Some families of solutions of the Vlasov-Poisson system. Arch. Ration. Mech. Anal. 104 (1988) 79-103
9. Batt J., Faltenbacher W., and Horst E. Stationary spherically symmetric models in stellar dynamics. Arch. Ration. Mech. Anal. 93 (1986) 159-183
10. Batt J., Kunze M., and Rein G. On the asymptotic behaviour of a one-dimensional, monocharged plasma. Adv. Differential Equations 3 (1998) 271-292
11. Batt J., Morrison P., and Rein G. Linear stability of stationary solutions of the Vlasov-Poisson system in three dimensions. Arch. Ration. Mech. Anal. 130 (1995) 163-182
12. Batt J., and Rein G. Global classical solutions of the periodic Vlasov-Poisson system in three dimensions. C. R. Acad. Sci. Paris 313 (1991) 411-416
13. Binney J., and Tremaine S. Galactic Dynamics (1987), Princeton Univ. Press, Princeton, NJ
14. Bouchut F. Existence and uniqueness of a global smooth solution for the Vlasov-Poisson-Fokker-Planck system in three dimensions. J. Funct. Anal. 111 (1993) 239-258
15. Bouchut F., Golse F., and Pallard C. Classical solutions and the Glassey-Strauss theorem for the 3d Vlasov-Maxwell system. Arch. Ration. Mech. Anal. 170 (2003) 1-15
16. Burchard A., and Guo Y. Compactness via symmetrization. J. Funct. Anal. 214 (2004) 40-73
17. Calogero S. Global classical solutions to the 3D Nordström-Vlasov system (2005). http://arXiv:math-ph/0507030 Preprint; available at
18. Cercignani C., Illner R., and Pulvirenti M. The Mathematical Theory of Dilute Gases (1994), Springer-Verlag, New York
19. Deng Y., Liu T.-P., Yang T., and Yao Z.-A. Solutions of Euler-Poisson equations for gaseous stars. Arch. Ration. Mech. Anal. 164 (2002) 261-285
20. Dietz C., and Sandor V. The hydrodynamical limit of the Vlasov-Poisson system. Transport Theory Statist. Phys. 28 (1999) 499-520
21. DiPerna R.J., and Lions P.-L. Global weak solutions of Vlasov-Maxwell systems. Comm. Pure Appl. Math. XLII (1989) 729-757
22. DiPerna R.J., and Lions P.-L. On the Cauchy problem for Boltzmann equations: Global existence and weak stability. Ann. of Math. 130 (1989) 321-366
23. Dolbeault J., and Rein G. Time-dependent rescalings and Lyapunov functionals for the Vlasov-Poisson and Euler-Poisson systems, and for related models of kinetic equations, fluid dynamics and quantum physics. Math. Models Methods Appl. Sci. 11 (2001) 407-432
24. Dolbeault J., Sanchez O., and Soler J. Asymptotic behaviour for the Vlasov-Poisson system in the stellar-dynamics case. Arch. Ration. Mech. Anal. 171 (2004) 301-327
25. Ehlers J., and Rienstra W. The locally isotropic solutions of the Liouville and Poisson equations. Astrophys. J. 155 (1969) 105-116
26. Fridman A.M., and Polyachenko V.L. Physics of Gravitating Systems I (1984), Springer-Verlag, New York
27. Gasser I., Jabin P.-E., and Perthame B. Regularity and propagation of moments in some nonlinear Vlasov systems. Proc. Roy. Soc. Edinburgh Sect. A 130 (2000) 1259-1273
28. Glassey R.T. The Cauchy Problem in Kinetic Theory (1996), SIAM, Philadelphia, PA
29. Glassey R.T., and Schaeffer J. On symmetric solutions of the relativistic Vlasov-Poisson system. Comm. Math. Phys. 101 (1985) 459-473
30. Glassey R.T., and Schaeffer J. Global existence of the relativistic Vlasov-Maxwell system with nearly neutral initial data. Comm. Math. Phys. 119 (1988) 353-384
31. Glassey R.T., and Schaeffer J. On the "one and one-half dimensional" relativistic Vlasov-Maxwell system. Math. Methods Appl. Sci. 13 (1990) 169-179
32. Glassey R.T., and Schaeffer J. The "two and one-half dimensional" relativistic Vlasov-Maxwell system. Comm. Math. Phys. 185 (1997) 257-284
33. Glassey R.T., and Schaeffer J. The relativistic Vlasov-Maxwell system in two space dimensions, I and II. Arch. Ration. Mech. Anal. 141 (1998) 331-354
34. R.T. G., and Schaeffer J. The relativistic Vlasov-Maxwell system in two space dimensions, I and II. Arch. Ration. Mech. Anal. 141 (1998) 355-374
35. Glassey R.T., and Schaeffer J. On global symmetric solutions of the relativistic Vlasov-Poisson equation in three space dimensions. Math. Methods Appl. Sci. 24 (2001) 143-157
36. Glassey R.T., and Strauss W.A. Remarks on collisionless plasmas. Contemp. Math. 28 (1984) 269-279
37. Glassey R.T., and Strauss W.A. Singularity formation in a collisionless plasma could occur only at high velocities. Arch. Ration. Mech. Anal. 92 (1986) 59-90
38. Glassey R.T., and Strauss W.A. Absence of shocks in an initially dilute collisionless plasma. Comm. Math. Phys. 113 (1987) 191-208
39. Guo Y. Global weak solutions to the Vlasov-Maxwell system with boundary conditions. Comm. Math. Phys. 154 (1993) 245-263
40. Guo Y. Regularity for the Vlasov equations in a half space. Indiana Univ. Math. J. 43 (1994) 255-320
41. Guo Y. Singular solutions of the Vlasov-Maxwell system on a half line. Arch. Ration. Mech. Anal. 131 (1995) 241-304
42. Guo Y. Variational method in polytropic galaxies. Arch. Ration. Mech. Anal. 150 (1999) 209-224
43. Guo Y. On the generalized Antonov′s stability criterion. Contemp. Math. 263 (2000) 85-107
44. Guo Y. The Vlasov-Poisson-Boltzmann system near vacuum. Comm. Math.Phys. 218 (2001) 293-313
45. Guo Y. The Vlasov-Maxwell-Boltzmann system near Maxwellians. Invent. Math. 153 (2003) 593-630
46. Guo Y., and Grotta Ragazzo C. On steady states in a collisionless plasma. Comm. Pure Appl. Math. XVIX (1996) 1145-1174
47. Guo Y., and Rein G. Stable steady states in stellar dynamics. Arch. Ration. Mech. Anal. 147 (1999) 225-243
48. Guo Y., and Rein G. Existence and stability of Camm type steady states in galactic dynamics. Indiana Univ. Math. J. 48 (1999) 1237-1255
49. Guo Y., and Rein G. Isotropic steady states in galactic dynamics. Comm. Math. Phys. 219 (2001) 607-629
50. Guo Y., and Rein G. Stable models of elliptical galaxies. Mon. Not. R. Astron. Soc. 344 (2003) 1396-1406
51. Guo Y., and Rein G. A non-variational approach to nonlinear stability in stellar dynamics applied to the King model (2006). http://arXiv:math-ph/0602058 Preprint; available at
52. Guo Y., and Strauss W.A. Nonlinear instability of double-humped equilibria. Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995) 339-352
53. Guo Y., and Strauss W.A. Unstable BGK solitary waves and collisionless shocks. Comm. Math. Phys. 195 (1998) 267-293
54. Hadžić M. Compactness and stability of some systems of nonlinear PDE-s in galactic dynamics. Diploma thesis (2005), University of Vienna
55. M. Hauray and P.-E. Jabin, N-particle approximation of the Vlasov equations with singular potential, Arch. Ration. Mech. Anal., to appear.
56. Heinzle M., Rendall A.D., and Uggla C. Theory of Newtonian self-gravitating stationary spherically symmetric systems (2004). http://arXiv:math-ph/0408045 Preprint; available at
57. Holm D.D., Marsden J.E., Ratiu T., and Weinstein A. Nonlinear stability of fluid and plasma equilibria. Phys. Reports 123 (1985) 1-116 1/2
58. Horst E. On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation I. Math. Methods Appl. Sci. 3 (1981) 229-248
59. Horst E. On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation II. Math. Methods Appl. Sci. 4 (1982) 19-32
60. Horst E. Symmetric plasmas and their decay. Comm. Math.Phys. 126 (1990) 613-633
61. Horst E. On the asymptotic growth of the solutions of the Vlasov-Poisson system. Math. Methods Appl. Sci. 16 (1993) 75-85
62. Horst E., and Hunze R. Weak solutions of the initial value problem for the unmodified non-linear Vlasov equation. Math. Methods Appl. Sci. 6 (1984) 262-279
63. Hwang H.J. Regularity for the Vlasov-Poisson system in a convex domain. SIAM J. Math. Anal. 36 (2004) 121-171
64. Illner R., and Rein G. Time decay of the solutions of the Vlasov-Poisson system in the plasma physical case. Math. Methods Appl. Sci. 19 (1996) 1409-1413
65. Jang J. Nonlinear instability in gravitational Euler-Poisson system for γ 6/5 (2005) Preprint
66. Jeans J. On the theory of star-streaming and the structure of the universe. Mon. Not. R. Astron. Soc. 76 (1915) 70-84
67. Jeans J. On the theory of star-streaming and the structure of the universe. Mon. Not. R. Astron. Soc. 76 (1916) 552-567
68. Klainerman S., and Staffilani G. A new approach to study the Vlasov-Maxwell system. Commun. Pure Appl. Anal. 1 (2002) 103-125
69. Kruse K.-O., and Rein G. A stability result for the relativistic Vlasov-Maxwell system. Arch. Ration. Mech. Anal. 121 (1992) 187-203
70. Kunze M., and Rendall A.D. The Vlasov-Poisson system with radiation damping. Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (2001) 857-886
71. Kurth R. Das Anfangswertproblem der Stellardynamik. Z. Astrophys. 30 (1952) 213-229
72. Kurth R. A global particular solution to the initial-value problem of stellar dynamics. Quart. Appl. Math., 36 (1978) 325-329
73. Landau L.D., and Lifschitz E.M. Lehrbuch der Theoretischen Physik, Bd. X. Physikalische Kinetik (1983), Akademie-Verlag, Berlin
74. Lemou M., Méhats F., and Raphael P. On the orbital stability of the ground states and the singularity formation for the gravitational Vlasov-Poisson system (2005) Preprint
75. Liboff R.L. Kinetic Theory: Classical, Quantum, and Relativistic Descriptions (2003), Springer-Verlag, New York
76. Lieb E.H., and Loss M. Analysis. Amer. Math. Soc. (1996), Providence, RI
77. Lin Z. Instability of periodic BGK waves. Math. Res. Lett. 8 (2001) 521-534
78. Lin Z. Nonlinear instability of periodic BGK waves for Vlasov-Poisson system. Comm. Pure Appl. Math. LVIII (2005) 505-528
79. Z. Lin W.A. and Strauss, Linear stability and instability of relativistic Vlasov-Maxwell systems, Comm. Pure Appl. Math., to appear.
80. Z. Lin W.A. and Strauss, Nonlinear stability and instability of relativistic Vlasov-Maxwell systems, Comm. Pure Appl. Math., to appear.
81. Lions P.-L. The concentration-compactness principle in the calculus of variations. The locally compact case. Part 1. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984) 109-145
82. Lions P.-L., and Perthame B. Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system. Invent. Math. 105 (1991) 415-430
83. Loeper G. Uniqueness of the solution to the Vlasov-Poisson system with bounded density (2005). http://arXiv:math.AP/0504140 Preprint; available at
84. Majda A., Majda G., and Zheng Y. Concentrations in the one-dimensional Vlasov-Poisson equations, I: Temporal development and non-unique weak solutions in the single component case. Phys. D 74 (1994) 268-300
85. Majda A., and Zheng Y. Existence of global weak solutions to one-component Vlasov-Poisson and Fokker-Planck-Poisson systems in one space dimension with measures as initial data. Comm. Pure Appl. Math. XLVII (1994) 1365-1401
86. Mawhin J., and Willem M. Critical Point Theory and Hamiltonian Systems (1989), Springer-Verlag, New York
87. Neunzert H. An Introduction to the Nonlinear Boltzmann-Vlasov Equation. Kinetic Theories and the Boltzmann Equation, Lecture Notes in Math., Vol. 1048 (1984), Springer-Verlag, Berlin
88. Pallard C. On the boundedness of the momentum support of solutions to the relativistic Vlasov-Maxwell system. Indiana Univ. Math. J., 54 (2005) 1395-1410
89. Perthame B. Time decay, propagation of low moments and dispersive effects for kinetic equations. Comm. Partial Differential Equations 21 (1996) 659-686
90. Perthame B. Mathematical tools for kinetic equations. Bull. Amer. Math. Soc. 41 (2004) 205-244
91. Pfaffelmoser K. Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data. J. Differential Equations 95 (1992) 281-303
92. Rein G. Generic global solutions of the relativistic Vlasov-Maxwell system of plasma physics. Comm. Math. Phys. 135 (1990) 41-78
93. Rein G. Existence of stationary, collisionless plasmas in bounded domains. Math. Methods Appl. Sci. 15 (1992) 365-374
94. Rein G. Nonlinear stability for the Vlasov-Poisson system - the energy-Casimir method. Math. Methods Appl. Sci. 17 (1994) 1129-1140
95. Rein G. Growth estimates for the solutions of the Vlasov-Poisson system in the plasma physics case. Math. Nachr. 191 (1998) 269-278
96. Rein G. Flat steady states in stellar dynamics - existence and stability. Comm. Math. Phys. 205 (1999) 229-247
97. Rein G. Stationary and static stellar dynamic models with axial symmetry. Nonlinear Anal. 41 (2000) 313-344
98. Rein G. Stability of spherically symmetric steady states in galactic dynamics against general perturbations. Arch. Ration. Mech. Anal. 161 (2002) 27-42
99. Rein G. Reduction and a concentration-compactness principle for energy-Casimir functionals. SIAM J. Math. Anal. 33 (2002) 896-912
100. Rein G. Nonlinear stability of gaseous stars. Arch. Ration. Mech. Anal. 168 (2003) 115-130
101. Rein G. Global weak solutions to the relativistic Vlasov-Maxwell system revisited. Commun. Math. Sci. 2 (2004) 145-158
102. Rein G., and Rendall A.D. Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data. Comm. Math. Phys. 150 (1992) 561-583
103. Rein G., and Rendall A.D. Global existence of classical solutions to the Vlasov-Poisson system in a three-dimensional, cosmological setting. Arch. Ration. Mech. Anal. 126 (1994) 183-201
104. Rein G., and Rendall A.D. Compact support of spherically symmetric equilibria in non-relativistic and relativistic galactic dynamics. Math. Proc. Cambridge Philos. Soc. 128 (2000) 363-380
105. Robert R. Unicité de la solution faible á support compact de l′équation de Vlasov-Poisson. C.R. Acad. Sci. Paris 324 (1997) 873-877
106. O. Sánchez and J. Soler, Orbital stability for polytropic galaxies, Ann. Inst. H. Poincaré Anal. Non Linéaire, to appear.
107. Schaeffer J. The good, the bad, and the ugly (1991) Private communication
108. Schaeffer J. Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions. Comm. Partial Differential Equations 16 (1991) 1313-1335
109. Schaeffer J. Steady states in galactic dynamics. Arch. Ration. Mech. Anal. 172 (2004) 1-19
110. E. S. Singular Integrals and Differentiability Properties of Functions (1970), Princeton Univ. Press, Princeton, NJ
111. Vlasov A.A. Zh. Eksper. Teor. Fiz. 8 (1938) 291
112. Vlasov A.A. Vlasov equation and plasma dispersion relation. J. Phys. (U.S.S.R.) 9 (1945) 25
113. Weckler J. On the initial-boundary-value problem for the Vlasov-Poisson system: Existence of weak solutions and stability. Arch. Ration. Mech. Anal. 130 (1995) 145-161
114. Wolansky G. On nonlinear stability of polytropic galaxies. Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999) 15-48
115. Wolansky G. Static solutions of the Vlasov-Einstein system. Arch. Ration. Mech. Anal. 156 (2001) 205-230
116. Zhidkov P. On global solutions for the Vlasov-Poisson system. Electron. J. Differ. Equ. 58 (2004) 1-11