American Institute of Mathematical Sciences

Advanced
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.  doi: 10.1007/BF00279958.  Google Scholar

[2]

J. Binney and S. Tremaine, Galactic Dynamics,, Princeton University Press, (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).  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.  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.  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.  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.  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).  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.  doi: 10.1137/120896712.  Google Scholar

[12]

G. Rein, Collisionless kinetic equations from astrophysics-The Vlasov-Poisson system,, in Handbook of Differential Equations, (2007), 383.  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.  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.  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.  doi: 10.1007/BF00279958.  Google Scholar

[2]

J. Binney and S. Tremaine, Galactic Dynamics,, Princeton University Press, (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).  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.  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.  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.  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.  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).  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.  doi: 10.1137/120896712.  Google Scholar

[12]

G. Rein, Collisionless kinetic equations from astrophysics-The Vlasov-Poisson system,, in Handbook of Differential Equations, (2007), 383.  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.  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.  doi: 10.1007/s002200050647.  Google Scholar

[1]

Simone Calogero, Juan Calvo, Óscar Sánchez, Juan Soler. Dispersive behavior in galactic dynamics. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 1-16. doi: 10.3934/dcdsb.2010.14.1
[2]

Elbaz I. Abouelmagd, Juan L. G. Guirao, Aatef Hobiny, Faris Alzahrani. Dynamics of a tethered satellite with variable mass. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1035-1045. doi: 10.3934/dcdss.2015.8.1035
[3]

Yuncheng You. Asymptotical dynamics of the modified Schnackenberg equations. Conference Publications, 2009, 2009 (Special) : 857-868. doi: 10.3934/proc.2009.2009.857
[4]

Carmen Cortázar, Manuel Elgueta, Jorge García-Melián, Salomé Martínez. Finite mass solutions for a nonlocal inhomogeneous dispersal equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1409-1419. doi: 10.3934/dcds.2015.35.1409
[5]

Alexander Kemarsky, Frédéric Paulin, Uri Shapira. Escape of mass in homogeneous dynamics in positive characteristic. Journal of Modern Dynamics, 2017, 11: 369-407. doi: 10.3934/jmd.2017015
[6]

Xiaoyu Zeng, Yimin Zhang. Asymptotic behaviors of ground states for a modified Gross-Pitaevskii equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5263-5273. doi: 10.3934/dcds.2019214
[7]

Dmitri Finkelshtein, Yuri Kondratiev, Yuri Kozitsky. Glauber dynamics in continuum: A constructive approach to evolution of states. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1431-1450. doi: 10.3934/dcds.2013.33.1431
[8]

Monika Joanna Piotrowska, Joanna Górecka, Urszula Foryś. The role of optimism and pessimism in the dynamics of emotional states. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 401-423. doi: 10.3934/dcdsb.2018028
[9]

Sylvain Sorin, Cheng Wan. Finite composite games: Equilibria and dynamics. Journal of Dynamics & Games, 2016, 3 (1) : 101-120. doi: 10.3934/jdg.2016005
[10]

Arno Berger, Doan Thai Son, Stefan Siegmund. Nonautonomous finite-time dynamics. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 463-492. doi: 10.3934/dcdsb.2008.9.463
[11]

Roy Malka, Vered Rom-Kedar. Bacteria--phagocyte dynamics, axiomatic modelling and mass-action kinetics. Mathematical Biosciences & Engineering, 2011, 8 (2) : 475-502. doi: 10.3934/mbe.2011.8.475
[12]

Gianluca Frasca-Caccia, Peter E. Hydon. Locally conservative finite difference schemes for the modified KdV equation. Journal of Computational Dynamics, 2019, 6 (2) : 307-323. doi: 10.3934/jcd.2019015
[13]

Katherine A. Newhall, Gregor Kovačič, Ildar Gabitov. Polarization dynamics in a resonant optical medium with initial coherence between degenerate states. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020189
[14]

Samir K. Bhowmik, Dugald B. Duncan, Michael Grinfeld, Gabriel J. Lord. Finite to infinite steady state solutions, bifurcations of an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 57-71. doi: 10.3934/dcdsb.2011.16.57
[15]

Jin Li, Jianhua Huang. Dynamics of a 2D Stochastic non-Newtonian fluid driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2483-2508. doi: 10.3934/dcdsb.2012.17.2483
[16]

Soohyun Bae. Weighted $L^\infty$ stability of positive steady states of a semilinear heat equation in $\R^n$. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 823-837. doi: 10.3934/dcds.2010.26.823
[17]

Hisashi Okamoto, Takashi Sakajo, Marcus Wunsch. Steady-states and traveling-wave solutions of the generalized Constantin--Lax--Majda equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3155-3170. doi: 10.3934/dcds.2014.34.3155
[18]

Viktor I. Gerasimenko, Igor V. Gapyak. Hard sphere dynamics and the Enskog equation. Kinetic & Related Models, 2012, 5 (3) : 459-484. doi: 10.3934/krm.2012.5.459
[19]

Proscovia Namayanja. Chaotic dynamics in a transport equation on a network. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3415-3426. doi: 10.3934/dcdsb.2018283
[20]

Maxime Herda, Luis Miguel Rodrigues. Anisotropic Boltzmann-Gibbs dynamics of strongly magnetized Vlasov-Fokker-Planck equations. Kinetic & Related Models, 2019, 12 (3) : 593-636. doi: 10.3934/krm.2019024

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences