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On time decay for the spherically symmetric Vlasov-Poisson system
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Linear instability of Vlasov-Maxwell systems revisited-A Hamiltonian approach
The Einstein-Vlasov system in maximal areal coordinates---Local existence and continuation
Department of Mathematics, University of Bayreuth, Germany |
We consider the spherically symmetric, asymptotically flat Einstein-Vlasov system in maximal areal coordinates. The latter coordinates have been used both in analytical and numerical investigations of the Einstein-Vlasov system [
References:
[1] |
H. Andréasson,
Black hole formation from a complete regular past for collisionless matter, Ann. Henri Poincaré, 13 (2012), 1511-1536.
doi: 10.1007/s00023-012-0164-1. |
[2] |
H. Andréasson, M. Kunze and G. Rein,
Existence of axially symmetric static solutions of the Einstein-Vlasov system, Comm. Math. Phys., 308 (2011), 23-47.
doi: 10.1007/s00220-011-1324-8. |
[3] |
H. Andréasson, M. Kunze and G. Rein,
Global existence for the spherically symmetric Einstein-Vlasov system with outgoing matter, Commun. Partial Differential Equations, 33 (2008), 656-668.
doi: 10.1080/03605300701454883. |
[4] |
H. Andréasson, M. Kunze and G. Rein,
Gravitational collapse and the formation of black holes for the spherically symmetric Einstein-Vlasov system, Quart. Appl. Math., 68 (2010), 17-42.
doi: 10.1090/S0033-569X-09-01165-9. |
[5] |
H. Andréasson, M. Kunze and G. Rein,
The formation of black holes in spherically symmetric gravitational collapse, Math. Ann., 350 (2011), 683-705.
doi: 10.1007/s00208-010-0578-3. |
[6] |
H. Andréasson, M. Kunze and G. Rein,
Rotating, stationary, axially symmetric spacetimes with collisionless matter, Comm. Math. Phys., 329 (2014), 787-808.
doi: 10.1007/s00220-014-1904-5. |
[7] |
H. Andréasson and G. Rein,
Formation of trapped surfaces for the spherically symmetric Einstein-Vlasov system, J. Hyperbolic Differ. Equ., 7 (2010), 707-731.
doi: 10.1142/S0219891610002268. |
[8] |
H. Andréasson and G. Rein,
A numerical investigation of the stability of steady states and critical phenomena for the spherically symmetric Einstein–Vlasov system, Classical Quantum Gravity, 23 (2006), 3659-3677.
doi: 10.1088/0264-9381/23/11/001. |
[9] |
J. Batt,
Global symmetric solutions of the initial value problem of stellar dynamics, J. Differential Equations, 25 (1977), 342-364.
doi: 10.1016/0022-0396(77)90049-3. |
[10] |
T. Baumgarte and S. Shapiro, Numerical Relativity: Solving Einstein's Equations on the Computer, Cambridge University Press, 2010.
doi: 10.1017/CBO9781139193344.![]() ![]() |
[11] |
Y. Choquet-Bruhat,
Problème de Cauchy pour le système intégro différentiel d'Einstein-Liouville, Ann. Inst. Fourier (Grenoble), 21 (1971), 181-201.
doi: 10.5802/aif.385. |
[12] |
M. Dafermos,
Spherically symmetric spacetimes with a trapped surface, Classical Quantum Gravity, 22 (2005), 2221-2232.
doi: 10.1088/0264-9381/22/11/019. |
[13] |
M. Dafermos and A. D. Rendall,
An extension principle for the Einstein-Vlasov system in spherical symmetry, Ann. Henri Poincaré, 6 (2005), 1137-1155.
doi: 10.1007/s00023-005-0235-7. |
[14] |
D. Fajman, J. Joudioux and J. Smulevici,
The stability of the Minkowski space for the Einstein-Vlasov system, Anal. PDE, 14 (2021), 425-531.
doi: 10.2140/apde.2021.14.425. |
[15] |
R. T. Glassey and W. A. Strauss,
Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rational Mech. Anal., 92 (1986), 59-90.
doi: 10.1007/BF00250732. |
[16] |
É. Gourgoulhon, $3+1$ Formalism in General Relativity. Bases of Numerical Relativity, Lecture Notes in Physics, 846, Springer, Heidelberg, 2012.
doi: 10.1007/978-3-642-24525-1. |
[17] |
S. Günther, The Einstein-Vlasov System in Maximal Areal Coordinates, Masters thesis, University of Bayreuth, 2019. |
[18] |
S. Günther, J. Körner, T. Lebeda, B. Pötzl, G. Rein, C. Straub and J. Weber, A numerical stability analysis for the Einstein-Vlasov system, Classical Quantum Gravity, 38 (2021), 27pp.
doi: 10.1088/1361-6382/abcbdf. |
[19] |
S. Günther, C. Straub and G. Rein, Collisionless equilibria in general relativity: Stable configurations beyond the first binding energy maximum, Astrophysical J., 918 (2021), 48pp.
doi: 10.3847/1538-4357/ac0eef. |
[20] |
M. Hadžić, Z. Lin and G. Rein,
Stability and instability of self-gravitating relativistic matter distributions, Arch. Ration. Mech. Anal., 241 (2021), 1-89.
doi: 10.1007/s00205-021-01647-2. |
[21] |
M. Hadžić and G. Rein,
On the small redshift limit of steady states of the spherically symmetric Einstein-Vlasov system and their stability, Math. Proc. Cambridge Philos. Soc., 159 (2015), 529-546.
doi: 10.1017/S0305004115000511. |
[22] |
M. Hadžić and G. Rein,
Stability for the spherically symmetric Einstein-Vlasov system–-A coercivity estimate, Math. Proc. Cambridge Philos. Soc., 155 (2013), 529-556.
doi: 10.1017/S030500411300056X. |
[23] |
H. Lindblad and M. Taylor,
Global stability of Minkowski space for the Einstein–Vlasov system in the harmonic gauge, Arch. Ration. Mech. Anal., 235 (2020), 517-633.
doi: 10.1007/s00205-019-01425-1. |
[24] |
E. Malec and N. Ó Murchadha, Optical scalars and singularity avoidance in spherical spacetimes, Phys. Rev. D, 50 (1994), R6033–R6036.
doi: 10.1103/PhysRevD.50.R6033. |
[25] |
T. Ramming and G. Rein,
Spherically symmetric equilibria for self-gravitating kinetic or fluid models in the non-relativistic and relativistic case–-A simple proof for finite extension, SIAM J. Math. Anal., 45 (2013), 900-914.
doi: 10.1137/120896712. |
[26] |
G. Rein, Collisionless kinetic equations from astrophysics–-The Vlasov-Poisson system, in Handbook of Differential Equations: Evolutionary Equations, Handb. Differ. Equ., 3, Elsevier/North-Holland, Amsterdam, 2007,383–476.
doi: 10.1016/S1874-5717(07)80008-9. |
[27] |
G. Rein, The Vlasov-Einstein System with Surface Symmetry, Habilitation thesis, Ludwig-Maximilians-Universität in Munich, 1995. |
[28] |
G. Rein and A. D. Rendall,
Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data, Comm. Math. Phys., 150 (1992), 561-583.
doi: 10.1007/BF02096962. |
[29] |
G. Rein and A. D. Rendall,
Erratum: "Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data", Comm. Math. Phys., 176 (1996), 475-478.
doi: 10.1007/BF02099559. |
[30] |
A. D. Rendall, An introduction to the Vlasov-Einstein system, in Mathematics of Gravitation, Part I, Banach Center Publ., 41, Part I, Polish Acad. Sci. Inst. Math., Warsaw, 1997, 35-68. |
[31] |
A. D. Rendall, Partial Differential Equations in General Relativity, Oxford Graduate Texts in Mathematics, 16, Oxford University Press, Oxford, 2008.
![]() ![]() |
[32] |
M. Taylor, The global nonlinear stability of Minkowski space for the massless Einstein-Vlasov system, Ann. PDE, 3 (2017), 177pp.
doi: 10.1007/s40818-017-0026-8. |
[33] |
R. M. Wald, General Relativity, University of Chicago Press, Chicago, IL, 1984.
doi: 10.7208/chicago/9780226870373.001.0001.![]() ![]() ![]() |
[34] |
J. York, Kinematics and dynamics of general relativity, in Sources of Gravitational Radiation, Cambridge University Press, 1979, 83-126. |
show all references
References:
[1] |
H. Andréasson,
Black hole formation from a complete regular past for collisionless matter, Ann. Henri Poincaré, 13 (2012), 1511-1536.
doi: 10.1007/s00023-012-0164-1. |
[2] |
H. Andréasson, M. Kunze and G. Rein,
Existence of axially symmetric static solutions of the Einstein-Vlasov system, Comm. Math. Phys., 308 (2011), 23-47.
doi: 10.1007/s00220-011-1324-8. |
[3] |
H. Andréasson, M. Kunze and G. Rein,
Global existence for the spherically symmetric Einstein-Vlasov system with outgoing matter, Commun. Partial Differential Equations, 33 (2008), 656-668.
doi: 10.1080/03605300701454883. |
[4] |
H. Andréasson, M. Kunze and G. Rein,
Gravitational collapse and the formation of black holes for the spherically symmetric Einstein-Vlasov system, Quart. Appl. Math., 68 (2010), 17-42.
doi: 10.1090/S0033-569X-09-01165-9. |
[5] |
H. Andréasson, M. Kunze and G. Rein,
The formation of black holes in spherically symmetric gravitational collapse, Math. Ann., 350 (2011), 683-705.
doi: 10.1007/s00208-010-0578-3. |
[6] |
H. Andréasson, M. Kunze and G. Rein,
Rotating, stationary, axially symmetric spacetimes with collisionless matter, Comm. Math. Phys., 329 (2014), 787-808.
doi: 10.1007/s00220-014-1904-5. |
[7] |
H. Andréasson and G. Rein,
Formation of trapped surfaces for the spherically symmetric Einstein-Vlasov system, J. Hyperbolic Differ. Equ., 7 (2010), 707-731.
doi: 10.1142/S0219891610002268. |
[8] |
H. Andréasson and G. Rein,
A numerical investigation of the stability of steady states and critical phenomena for the spherically symmetric Einstein–Vlasov system, Classical Quantum Gravity, 23 (2006), 3659-3677.
doi: 10.1088/0264-9381/23/11/001. |
[9] |
J. Batt,
Global symmetric solutions of the initial value problem of stellar dynamics, J. Differential Equations, 25 (1977), 342-364.
doi: 10.1016/0022-0396(77)90049-3. |
[10] |
T. Baumgarte and S. Shapiro, Numerical Relativity: Solving Einstein's Equations on the Computer, Cambridge University Press, 2010.
doi: 10.1017/CBO9781139193344.![]() ![]() |
[11] |
Y. Choquet-Bruhat,
Problème de Cauchy pour le système intégro différentiel d'Einstein-Liouville, Ann. Inst. Fourier (Grenoble), 21 (1971), 181-201.
doi: 10.5802/aif.385. |
[12] |
M. Dafermos,
Spherically symmetric spacetimes with a trapped surface, Classical Quantum Gravity, 22 (2005), 2221-2232.
doi: 10.1088/0264-9381/22/11/019. |
[13] |
M. Dafermos and A. D. Rendall,
An extension principle for the Einstein-Vlasov system in spherical symmetry, Ann. Henri Poincaré, 6 (2005), 1137-1155.
doi: 10.1007/s00023-005-0235-7. |
[14] |
D. Fajman, J. Joudioux and J. Smulevici,
The stability of the Minkowski space for the Einstein-Vlasov system, Anal. PDE, 14 (2021), 425-531.
doi: 10.2140/apde.2021.14.425. |
[15] |
R. T. Glassey and W. A. Strauss,
Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rational Mech. Anal., 92 (1986), 59-90.
doi: 10.1007/BF00250732. |
[16] |
É. Gourgoulhon, $3+1$ Formalism in General Relativity. Bases of Numerical Relativity, Lecture Notes in Physics, 846, Springer, Heidelberg, 2012.
doi: 10.1007/978-3-642-24525-1. |
[17] |
S. Günther, The Einstein-Vlasov System in Maximal Areal Coordinates, Masters thesis, University of Bayreuth, 2019. |
[18] |
S. Günther, J. Körner, T. Lebeda, B. Pötzl, G. Rein, C. Straub and J. Weber, A numerical stability analysis for the Einstein-Vlasov system, Classical Quantum Gravity, 38 (2021), 27pp.
doi: 10.1088/1361-6382/abcbdf. |
[19] |
S. Günther, C. Straub and G. Rein, Collisionless equilibria in general relativity: Stable configurations beyond the first binding energy maximum, Astrophysical J., 918 (2021), 48pp.
doi: 10.3847/1538-4357/ac0eef. |
[20] |
M. Hadžić, Z. Lin and G. Rein,
Stability and instability of self-gravitating relativistic matter distributions, Arch. Ration. Mech. Anal., 241 (2021), 1-89.
doi: 10.1007/s00205-021-01647-2. |
[21] |
M. Hadžić and G. Rein,
On the small redshift limit of steady states of the spherically symmetric Einstein-Vlasov system and their stability, Math. Proc. Cambridge Philos. Soc., 159 (2015), 529-546.
doi: 10.1017/S0305004115000511. |
[22] |
M. Hadžić and G. Rein,
Stability for the spherically symmetric Einstein-Vlasov system–-A coercivity estimate, Math. Proc. Cambridge Philos. Soc., 155 (2013), 529-556.
doi: 10.1017/S030500411300056X. |
[23] |
H. Lindblad and M. Taylor,
Global stability of Minkowski space for the Einstein–Vlasov system in the harmonic gauge, Arch. Ration. Mech. Anal., 235 (2020), 517-633.
doi: 10.1007/s00205-019-01425-1. |
[24] |
E. Malec and N. Ó Murchadha, Optical scalars and singularity avoidance in spherical spacetimes, Phys. Rev. D, 50 (1994), R6033–R6036.
doi: 10.1103/PhysRevD.50.R6033. |
[25] |
T. Ramming and G. Rein,
Spherically symmetric equilibria for self-gravitating kinetic or fluid models in the non-relativistic and relativistic case–-A simple proof for finite extension, SIAM J. Math. Anal., 45 (2013), 900-914.
doi: 10.1137/120896712. |
[26] |
G. Rein, Collisionless kinetic equations from astrophysics–-The Vlasov-Poisson system, in Handbook of Differential Equations: Evolutionary Equations, Handb. Differ. Equ., 3, Elsevier/North-Holland, Amsterdam, 2007,383–476.
doi: 10.1016/S1874-5717(07)80008-9. |
[27] |
G. Rein, The Vlasov-Einstein System with Surface Symmetry, Habilitation thesis, Ludwig-Maximilians-Universität in Munich, 1995. |
[28] |
G. Rein and A. D. Rendall,
Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data, Comm. Math. Phys., 150 (1992), 561-583.
doi: 10.1007/BF02096962. |
[29] |
G. Rein and A. D. Rendall,
Erratum: "Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data", Comm. Math. Phys., 176 (1996), 475-478.
doi: 10.1007/BF02099559. |
[30] |
A. D. Rendall, An introduction to the Vlasov-Einstein system, in Mathematics of Gravitation, Part I, Banach Center Publ., 41, Part I, Polish Acad. Sci. Inst. Math., Warsaw, 1997, 35-68. |
[31] |
A. D. Rendall, Partial Differential Equations in General Relativity, Oxford Graduate Texts in Mathematics, 16, Oxford University Press, Oxford, 2008.
![]() ![]() |
[32] |
M. Taylor, The global nonlinear stability of Minkowski space for the massless Einstein-Vlasov system, Ann. PDE, 3 (2017), 177pp.
doi: 10.1007/s40818-017-0026-8. |
[33] |
R. M. Wald, General Relativity, University of Chicago Press, Chicago, IL, 1984.
doi: 10.7208/chicago/9780226870373.001.0001.![]() ![]() ![]() |
[34] |
J. York, Kinematics and dynamics of general relativity, in Sources of Gravitational Radiation, Cambridge University Press, 1979, 83-126. |
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