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June  2015, 8(2): 381-394. doi: 10.3934/krm.2015.8.381

Galactic dynamics in MOND---Existence of equilibria with finite mass and compact support

Gerhard Rein 1,

1. 

Fakultät für Mathematik, Physik und Informatik, Universität Bayreuth, D-95440 Bayreuth, Germany

Received  September 2014 Revised  November 2014 Published  March 2015

We consider a self-gravitating collisionless gas where the gravitational interaction is modeled according to MOND (modified Newtonian dynamics). For the resulting modified Vlasov-Poisson system we establish the existence of spherically symmetric equilibria with compact support and finite mass. In the standard situation where gravity is modeled by Newton's law the latter properties only hold under suitable restrictions on the prescribed microscopic equation of state. Under the MOND regime no such restrictions are needed.
Keywords: Modified Newtonian dynamics, steady states, finite mass., Vlasov equation, galactic dynamics.
Mathematics Subject Classification: Primary: 35Q83, 35Q75; Secondary: 85A1.
Citation: Gerhard Rein. Galactic dynamics in MOND---Existence of equilibria with finite mass and compact support. Kinetic & Related Models, 2015, 8 (2) : 381-394. doi: 10.3934/krm.2015.8.381
References:
[1]

J. Batt, W. Faltenbacher and E. Horst, Stationary spherically symmetric models in stellar dynamics, Arch. Rational Mech. Anal., 93 (1986), 159-183. doi: 10.1007/BF00279958.  Google Scholar

[2]

J. Binney and S. Tremaine, Galactic Dynamics, Princeton University Press, Princeton, 1987. doi: 10.1063/1.2811635.  Google Scholar

[3]

B. Famaey and S. McGaugh, Modified Newtonian dynamics (MOND): Observational phenomenology and relativistic extensions, Living Rev. Relativity, 15 (2012), p10. doi: 10.12942/lrr-2012-10.  Google Scholar

[4]

Y. Guo and G. Rein, Stable steady states in stellar dynamics, Arch. Rational Mech. Anal., 147 (1999), 225-243. doi: 10.1007/s002050050150.  Google Scholar

[5]

Y. Guo and G. Rein, A non-variational approach to nonlinear stability in stellar dynamics applied to the King model, Commun. Math. Phys., 271 (2007), 489-509. doi: 10.1007/s00220-007-0212-8.  Google Scholar

[6]

M. Lemou, F. Méhats and P. Raphaël, Orbital stability of spherical galactic models, Invent. math., 187 (2012), 145-194. doi: 10.1007/s00222-011-0332-9.  Google Scholar

[7]

M. Milgrom, Light and dark in the universe, preprint,, , ().   Google Scholar

[8]

M. Milgrom, The MOND paradigm, preprint,, , ().   Google Scholar

[9]

M. Milgrom, Quasi-linear formulation of MOND, Mon. Not. R. Astron. Soc., 403 (2010), 886-895. doi: 10.1111/j.1365-2966.2009.16184.x.  Google Scholar

[10]

M. Núñez, On the gravitational potential of modified Newtonian dynamics, J. Math. Phys., 54 (2013), 082502, 8pp. doi: 10.1063/1.4817858.  Google Scholar

[11]

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.  Google Scholar

[12]

G. Rein, Collisionless kinetic equations from astrophysics-The Vlasov-Poisson system, in Handbook of Differential Equations, Evolutionary Equations, Vol. 3 (eds. C. M. Dafermos and E. Feireisl), Elsevier, 2007, 383-476. doi: 10.1016/S1874-5717(07)80008-9.  Google Scholar

[13]

G. Rein and A. Rendall, Compact support of spherically symmetric equilibria in non-relativistic and relativistic galactic dynamics, Math. Proc. Camb.\Phil. Soc., 128 (2000), 363-380. doi: 10.1017/S0305004199004193.  Google Scholar

[14]

J. Schaeffer, A class of counterexamples to Jeans' Theorem for the Vlasov-Einstein system, Commun. Math. Phys., 204 (1999), 313-327. doi: 10.1007/s002200050647.  Google Scholar

show all references

References:
[1]

J. Batt, W. Faltenbacher and E. Horst, Stationary spherically symmetric models in stellar dynamics, Arch. Rational Mech. Anal., 93 (1986), 159-183. doi: 10.1007/BF00279958.  Google Scholar

[2]

J. Binney and S. Tremaine, Galactic Dynamics, Princeton University Press, Princeton, 1987. doi: 10.1063/1.2811635.  Google Scholar

[3]

B. Famaey and S. McGaugh, Modified Newtonian dynamics (MOND): Observational phenomenology and relativistic extensions, Living Rev. Relativity, 15 (2012), p10. doi: 10.12942/lrr-2012-10.  Google Scholar

[4]

Y. Guo and G. Rein, Stable steady states in stellar dynamics, Arch. Rational Mech. Anal., 147 (1999), 225-243. doi: 10.1007/s002050050150.  Google Scholar

[5]

Y. Guo and G. Rein, A non-variational approach to nonlinear stability in stellar dynamics applied to the King model, Commun. Math. Phys., 271 (2007), 489-509. doi: 10.1007/s00220-007-0212-8.  Google Scholar

[6]

M. Lemou, F. Méhats and P. Raphaël, Orbital stability of spherical galactic models, Invent. math., 187 (2012), 145-194. doi: 10.1007/s00222-011-0332-9.  Google Scholar

[7]

M. Milgrom, Light and dark in the universe, preprint,, , ().   Google Scholar

[8]

M. Milgrom, The MOND paradigm, preprint,, , ().   Google Scholar

[9]

M. Milgrom, Quasi-linear formulation of MOND, Mon. Not. R. Astron. Soc., 403 (2010), 886-895. doi: 10.1111/j.1365-2966.2009.16184.x.  Google Scholar

[10]

M. Núñez, On the gravitational potential of modified Newtonian dynamics, J. Math. Phys., 54 (2013), 082502, 8pp. doi: 10.1063/1.4817858.  Google Scholar

[11]

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.  Google Scholar

[12]

G. Rein, Collisionless kinetic equations from astrophysics-The Vlasov-Poisson system, in Handbook of Differential Equations, Evolutionary Equations, Vol. 3 (eds. C. M. Dafermos and E. Feireisl), Elsevier, 2007, 383-476. doi: 10.1016/S1874-5717(07)80008-9.  Google Scholar

[13]

G. Rein and A. Rendall, Compact support of spherically symmetric equilibria in non-relativistic and relativistic galactic dynamics, Math. Proc. Camb.\Phil. Soc., 128 (2000), 363-380. doi: 10.1017/S0305004199004193.  Google Scholar

[14]

J. Schaeffer, A class of counterexamples to Jeans' Theorem for the Vlasov-Einstein system, Commun. Math. Phys., 204 (1999), 313-327. doi: 10.1007/s002200050647.  Google Scholar

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