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October  2018, 11(5): 1063-1083. doi: 10.3934/krm.2018041

On the spatially homogeneous and isotropic Einstein-Vlasov-Fokker-Planck system with cosmological scalar field

Simone Calogero 1, and  Stephen Pankavich 2,,

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

* Corresponding author: S. Pankavich

Received  December 2016 Revised  September 2017 Published  May 2018

Fund Project: The second author is supported by the US National Science Foundation under awards DMS-1211667 and DMS-1614586.

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.

Keywords: Einstein-Vlasov-Fokker-Planck, blow-up, global existence, asymptotic behavior, cosmological expansion.
Mathematics Subject Classification: Primary: 35Q83, 83F05; Secondary: 35Q76, 83C05.
Citation: Simone Calogero, Stephen Pankavich. On the spatially homogeneous and isotropic Einstein-Vlasov-Fokker-Planck system with cosmological scalar field. Kinetic & Related Models, 2018, 11 (5) : 1063-1083. doi: 10.3934/krm.2018041
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

S. Calogero, A kinetic theory of diffusion in general relativity with cosmological scalar field, J. Cosm. Astrop. Phys, 11 (2011), 016, 16pp.  Google Scholar

[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.  Google Scholar

[7]

D. Fajman, J. Joudioux and J. Smulevici, The Stability of the Minkowski space for the Einstein-Vlasov system, Preprint, arXiv: 1707.06141. Google Scholar

[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.  Google Scholar

[9]

H. Lindblad and M. Taylor, Global stability of Minkowski space for the Einstein-Vlasov system in the harmonic gauge, Preprint, arXiv: 1707.06079. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

S. Weinberg, Cosmology, Oxford University Press, Oxford, 2008.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

S. Calogero, A kinetic theory of diffusion in general relativity with cosmological scalar field, J. Cosm. Astrop. Phys, 11 (2011), 016, 16pp.  Google Scholar

[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.  Google Scholar

[7]

D. Fajman, J. Joudioux and J. Smulevici, The Stability of the Minkowski space for the Einstein-Vlasov system, Preprint, arXiv: 1707.06141. Google Scholar

[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.  Google Scholar

[9]

H. Lindblad and M. Taylor, Global stability of Minkowski space for the Einstein-Vlasov system in the harmonic gauge, Preprint, arXiv: 1707.06079. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

S. Weinberg, Cosmology, Oxford University Press, Oxford, 2008.  Google Scholar

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