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August  2022, 15(4): 681-719. doi: 10.3934/krm.2021040

The Einstein-Vlasov system in maximal areal coordinates---Local existence and continuation

Sebastian Günther and  Gerhard Rein ,

Department of Mathematics, University of Bayreuth, Germany

* Corresponding author: Gerhard Rein

Dedicated to the memory of Prof. Robert T. Glassey

Received  May 2021 Published  August 2022 Early access  November 2021

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 [3,8,18,19], but neither a local existence theorem nor a suitable continuation criterion has so far been established for the corresponding nonlinear system of PDEs. We close this gap. Although the analysis follows lines similar to the corresponding result in Schwarzschild coordinates, essential new difficulties arise from to the much more complicated form which the field equations take, while at the same time it becomes easier to control the necessary, highest order derivatives of the solution. The latter observation may be useful in subsequent investigations.

Keywords: Einstein-Vlasov system, maximal areal coordinates, local existence, continuation of solutions.
Mathematics Subject Classification: Primary: 35Q85, 35Q76; Secondary: 85A05.
Citation: Sebastian Günther, Gerhard Rein. The Einstein-Vlasov system in maximal areal coordinates---Local existence and continuation. Kinetic and Related Models, 2022, 15 (4) : 681-719. doi: 10.3934/krm.2021040
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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