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On the spatially homogeneous and isotropic Einstein-Vlasov-Fokker-Planck system with cosmological scalar field
| 1. | Department of Mathematics, Chalmers Institute of Technology, University of Gothenburg, Gothenburg, Sweden |
| 2. | Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, CO, USA |
The Einstein-Vlasov-Fokker-Planck system describes the kinetic diffusion dynamics of self-gravitating particles within the Einstein theory of general relativity. We study the Cauchy problem for spatially homogeneous and isotropic solutions and prove the existence of both global-in-time solutions and solutions that blow-up in finite time depending on the size of certain functions of the initial data. We also derive information on the large-time behavior of global solutions and toward the singularity for solutions which blow-up in finite time. Our results entail the existence of a phase of decelerated expansion followed by a phase of accelerated expansion, in accordance with the physical expectations in cosmology.
References:
| [1] |
J. A. Alcántara, S. Calogero and S. Pankavich,
Spatially homogeneous solutions of the Vlasov-Nordström-Fokker-Planck system, J. Diff. Equations, 257 (2014), 3700-3729.
doi: 10.1016/j.jde.2014.07.006. |
| [2] |
H. Andréasson, The einstein-vlasov system/kinetic theory, Living Rev. Relativity, 5 (2002), 2002-7, 33 pp, URL (cited on June 2016): http://www.livingreviews.org/lrr-2011-4.
doi: 10.12942/lrr-2002-7. |
| [3] |
S. Blaise Tchapnda and N. Noutchegueme,
The surface-symmetric Einstein-Vlasov system with cosmological constant, Math. Proc. Camb. Phil. Soc., 138 (2005), 541-553.
doi: 10.1017/S0305004104008266. |
| [4] |
F. Bouchut,
Existence and uniqueness of a global smooth solution for the Vlasov-Poisson-Fokker-Planck system in three dimensions, J. Funct. Analysis, 111 (1993), 239-258.
doi: 10.1006/jfan.1993.1011. |
| [5] |
S. Calogero, A kinetic theory of diffusion in general relativity with cosmological scalar field, J. Cosm. Astrop. Phys, 11 (2011), 016, 16pp. |
| [6] |
J. A. Carrillo, J. Soler and J. L. Vázquez,
Asymptotic behaviour and self-similarity for the three dimensional Vlasov-Poisson-Fokker-Planck system, J. Funct. Analysis, 141 (1996), 99-132.
doi: 10.1006/jfan.1996.0123. |
| [7] |
D. Fajman, J. Joudioux and J. Smulevici, The Stability of the Minkowski space for the Einstein-Vlasov system, Preprint, arXiv: 1707.06141. |
| [8] |
H. Lee,
Asymptotic behaviour of the Einstein-Vlasov system with a positive cosmological constant, Math. Proc. Camb. Phil. Soc., 137 (2004), 495-509.
doi: 10.1017/S0305004104007960. |
| [9] |
H. Lindblad and M. Taylor, Global stability of Minkowski space for the Einstein-Vlasov system in the harmonic gauge, Preprint, arXiv: 1707.06079. |
| [10] |
R. Maartens and S. D. Maharaj,
Invariant solutions of Liouville's equation in Robertson-Walker space-times, Gen. Rel. Grav., 19 (1987), 1223-1234.
doi: 10.1007/BF00759102. |
| [11] |
K. Ono,
Global existence of regular solutions for the Vlasov-Poisson-Fokker-Planck system, J. Math. Anal. Appl., 263 (2001), 626-636.
doi: 10.1006/jmaa.2001.7640. |
| [12] |
S. Pankavich and N. Michalowksi,
Global classical solutions to the one and one-half dimensional relativistic Vlasov-Maxwell-Fokker-Planck system, Kin. Rel. Models, 8 (2015), 169-199.
doi: 10.3934/krm.2015.8.169. |
| [13] |
S. Pankavich and J. Schaeffer,
Global classical solutions of the "one and one-half dimensional" Vlasov-Maxwell-Fokker-Planck system, Comm. Math. Sci., 14 (2016), 209-232.
doi: 10.4310/CMS.2016.v14.n1.a8. |
| [14] |
S. Weinberg, Cosmology, Oxford University Press, Oxford, 2008. |
show all references
References:
| [1] |
J. A. Alcántara, S. Calogero and S. Pankavich,
Spatially homogeneous solutions of the Vlasov-Nordström-Fokker-Planck system, J. Diff. Equations, 257 (2014), 3700-3729.
doi: 10.1016/j.jde.2014.07.006. |
| [2] |
H. Andréasson, The einstein-vlasov system/kinetic theory, Living Rev. Relativity, 5 (2002), 2002-7, 33 pp, URL (cited on June 2016): http://www.livingreviews.org/lrr-2011-4.
doi: 10.12942/lrr-2002-7. |
| [3] |
S. Blaise Tchapnda and N. Noutchegueme,
The surface-symmetric Einstein-Vlasov system with cosmological constant, Math. Proc. Camb. Phil. Soc., 138 (2005), 541-553.
doi: 10.1017/S0305004104008266. |
| [4] |
F. Bouchut,
Existence and uniqueness of a global smooth solution for the Vlasov-Poisson-Fokker-Planck system in three dimensions, J. Funct. Analysis, 111 (1993), 239-258.
doi: 10.1006/jfan.1993.1011. |
| [5] |
S. Calogero, A kinetic theory of diffusion in general relativity with cosmological scalar field, J. Cosm. Astrop. Phys, 11 (2011), 016, 16pp. |
| [6] |
J. A. Carrillo, J. Soler and J. L. Vázquez,
Asymptotic behaviour and self-similarity for the three dimensional Vlasov-Poisson-Fokker-Planck system, J. Funct. Analysis, 141 (1996), 99-132.
doi: 10.1006/jfan.1996.0123. |
| [7] |
D. Fajman, J. Joudioux and J. Smulevici, The Stability of the Minkowski space for the Einstein-Vlasov system, Preprint, arXiv: 1707.06141. |
| [8] |
H. Lee,
Asymptotic behaviour of the Einstein-Vlasov system with a positive cosmological constant, Math. Proc. Camb. Phil. Soc., 137 (2004), 495-509.
doi: 10.1017/S0305004104007960. |
| [9] |
H. Lindblad and M. Taylor, Global stability of Minkowski space for the Einstein-Vlasov system in the harmonic gauge, Preprint, arXiv: 1707.06079. |
| [10] |
R. Maartens and S. D. Maharaj,
Invariant solutions of Liouville's equation in Robertson-Walker space-times, Gen. Rel. Grav., 19 (1987), 1223-1234.
doi: 10.1007/BF00759102. |
| [11] |
K. Ono,
Global existence of regular solutions for the Vlasov-Poisson-Fokker-Planck system, J. Math. Anal. Appl., 263 (2001), 626-636.
doi: 10.1006/jmaa.2001.7640. |
| [12] |
S. Pankavich and N. Michalowksi,
Global classical solutions to the one and one-half dimensional relativistic Vlasov-Maxwell-Fokker-Planck system, Kin. Rel. Models, 8 (2015), 169-199.
doi: 10.3934/krm.2015.8.169. |
| [13] |
S. Pankavich and J. Schaeffer,
Global classical solutions of the "one and one-half dimensional" Vlasov-Maxwell-Fokker-Planck system, Comm. Math. Sci., 14 (2016), 209-232.
doi: 10.4310/CMS.2016.v14.n1.a8. |
| [14] |
S. Weinberg, Cosmology, Oxford University Press, Oxford, 2008. |
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