Skip to main content
SpringerLink
Log in

Relative Equilibria in Continuous Stellar Dynamics

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We study a three dimensional continuous model of gravitating matter rotating at constant angular velocity. In the rotating reference frame, by a finite dimensional reduction, we prove the existence of non-radial stationary solutions whose supports are made of an arbitrarily large number of disjoint compact sets, in the low angular velocity and large scale limit. At first order, the solutions behave like point particles, thus making the link with the relative equilibria in N-body dynamics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnol′d V.I.: On conditions for non-linear stability of plane stationary curvilinear flows of an ideal fluid. Dokl. Akad. Nauk SSSR 162, 975–978 (1965)

    MathSciNet  Google Scholar 

  2. Arnol′d V.I.: An a priori estimate in the theory of hydrodynamic stability. Izv. Vysš. Učebn. Zaved. Mat. 1966, 3–5 (1966)

    MATH  Google Scholar 

  3. Batt J., Faltenbacher W., Horst E.: Stationary spherically symmetric models in stellar dynamics. Arch. Rat. Mech. Anal. 93, 159–183 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  4. Batt J., Pfaffelmoser K.: On the radius continuity of the models of polytropic gas spheres which correspond to the positive solutions of the generalized Emden-Fowler equation. Math. Meth. Appl. Sci. 10, 499–516 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  5. Binney J., Tremaine S.: Galactic dynamics. Princeton University Press, Princeton, NJ (1987)

    MATH  Google Scholar 

  6. Dancer E.N., Yan S.: On the superlinear Lazer-McKenna conjecture. J. Diff. Eqs. 210, 317–351 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dancer E.N., Yan S.: On the superlinear Lazer-McKenna conjecture. II. Comm. Part. Diff. Eqs. 30, 1331–1358 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. D’Aprile T., Wei J.: Layered solutions for a semilinear elliptic system in a ball. J. Diff. Eqs. 226, 269–294 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dolbeault J., Fernández J.: Localized minimizers of flat rotating gravitational systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 25, 1043–1071 (2008)

    Article  MATH  Google Scholar 

  10. Dolbeault J., Fernández J., Fernández J.: Stability for the gravitational Vlasov–Poisson system in dimension two. Comm. Part. Diff. Eqs. 31, 1425–1449 (2006)

    Article  MATH  Google Scholar 

  11. Dolbeault J., Markowich P., Oelz D., Schmeiser C.: Non linear diffusions as limit of kinetic equations with relaxation collision kernels. Arch. Rat. Mech. Anal. 186, 133–158 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Elmabsout B.: Sur l’existence de certaines configurations d’équilibre relatif dans le problème des n corps. Cel. Mech. 41, 131–151 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  13. Floer A., Weinstein A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69, 397–408 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  14. Flucher M., Wei J.: Asymptotic shape and location of small cores in elliptic free-boundary problems. Math. Z. 228, 683–703 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gidas B., Ni W.M., Nirenberg L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68, 209–243 (1979)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  16. Guo Y., Rein G.: Existence and stability of Camm type steady states in galactic dynamics. Indiana Univ. Math. J. 48, 1237–1255 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  17. Guo Y., Rein G.: Stable steady states in stellar dynamics. Arch. Rat. Mech. Anal. 147, 225–243 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  18. Guo Y., Rein G.: Isotropic steady states in galactic dynamics. Commun. Math. Phys. 219, 607–629 (2001)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  19. Guo Y., Rein G.: Stable models of elliptical galaxies. Mon. Not. R. Astro. Soc. 344(4), 1296–1306 (2003)

    Article  ADS  Google Scholar 

  20. Guo Y., Rein G.: A non-variational approach to nonlinear stability in stellar dynamics applied to the King model. Commun. Math. Phys. 271, 489–509 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  21. Krieger J., Martel Y., Raphaël P.: Two soliton solutions to the three dimensional gravitational Hartree equation. Comm. Pure Appl. Math. 62, 1501–1550 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  22. Lemou M., Méhats F., Raphaël P.: Orbital stability and singularity formation for Vlasov-Poisson systems . C. R. Math. Acad. Sci. Paris 341, 269–274 (2005)

    MATH  MathSciNet  Google Scholar 

  23. Lemou M., Méhats F., Raphaël P.: The orbital stability of the ground states and the singularity formation for the gravitational Vlasov-Poisson system. Arch. Rat. Mech. Anal. 189, 425–468 (2008)

    Article  MATH  Google Scholar 

  24. Lemou M., Méhats F., Raphaël P.: Stable self-similar blow up dynamics for the three dimensional relativistic gravitational Vlasov-Poisson system. J. Amer. Math. Soc. 21, 1019–1063 (2008)

    Article  MathSciNet  Google Scholar 

  25. Lemou M., Méhats F., Raphaël P.: Structure of the linearized gravitational Vlasov-Poisson system close to a polytropic ground state. SIAM J. Math. Anal. 39, 1711–1739 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. MacMillan W., Bartky W.: Permanent configurations in the problem of four bodies. Trans. Amer. Math. Soc. 34, 838–875 (1932)

    MathSciNet  Google Scholar 

  27. McCann R.J.: Stable rotating binary stars and fluid in a tube. Houston J. Math. 32, 603–631 (2006)

    MATH  MathSciNet  Google Scholar 

  28. Meyer, K., Hall, H.: Introduction to Hamiltonian Dynamical Systems and the N-Body Problem. Vol. 90 of Applied Mathematical Sciences, Berlin-Heidelberg-New York: Springer-Verlag, 1992

  29. Moulton F.R.: The straight line solutions of the problem of n bodies. Ann. of Math. 12(2), 1–17 (1910)

    Article  MathSciNet  Google Scholar 

  30. Palmore J.I.: Classifying relative equilibria. II. Bull. Amer. Math. Soc. 81, 489–491 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  31. Palmore J.I.: Classifying relative equilibria. III. Lett. Math. Phys. 1, 71–73 (1975)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  32. Palmore J.I.: New relative equilibria of the n-body problem. Lett. Math. Phys. 1, 119–123 (1975)

    Article  MathSciNet  ADS  Google Scholar 

  33. Palmore J.I.: Minimally classifying relative equilibria. Lett. Math. Phys. 1, 395–399 (1975)

    Article  MathSciNet  ADS  Google Scholar 

  34. Palmore J.I.: Measure of degenerate relative equilibria. I. Ann. of Math. 104(2), 421–429 (1976)

    Article  MathSciNet  Google Scholar 

  35. Perko L.M., Walter E.L.: Regular polygon solutions of the N-body problem. Proc. Amer. Math. Soc. 94, 301–309 (1985)

    MATH  MathSciNet  Google Scholar 

  36. Rein G.: Flat steady states in stellar dynamics – Existence and stability. Commun. Math. Phys. 205, 229–247 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  37. Rein G.: Reduction and a concentration-compactness principle for energy-Casimir functionals. SIAM J. Math. Anal. 33, 896–912 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  38. Rein G.: Non-linear stability of gaseous stars. Arch. Rat. Mech. Anal. 168, 115–130 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  39. Rein G.: Nonlinear stability of newtonian galaxies and stars from a mathematical perspective. Ann. New York Acad. Sci. 1045, 103–119 (2005)

    Article  ADS  Google Scholar 

  40. Soler J., Soler J.: Orbital stability for polytropic galaxies. Ann. Inst. H. Poincaré Anal. Non Linéaire 23, 781–802 (2006)

    Article  MATH  ADS  Google Scholar 

  41. Schaeffer J.: Steady states in galactic dynamics. Arch. Rat. Mech. Anal. 172, 1–19 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  42. Smale S.: Topology and mechanics II. Invent. Math 11, 45–64 (1970)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  43. Williams W.: Permanent configurations in the problem of five bodies. Trans. Amer. Math. Soc. 44, 562–579 (1938)

    Google Scholar 

  44. Wolansky G.: On nonlinear stability of polytropic galaxies. Ann. Inst. H. Poincaré Anal. Non Linéaire 16, 15–48 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  45. Wolansky G., Ghil M.: An extension of Arnol’d’s second stability theorem for the Euler equations. Phys. D 94, 161–167 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  46. Xie Z., Zhang S.: A simpler proof of regular polygon solutions of the N-body problem. Phys. Lett. A 227, 156–158 (2000)

    Article  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ceremade (UMR CNRS n° 7534), Université Paris-Dauphine, Place de Lattre de Tassigny, 75775, Paris Cédex 16, France

    Juan Campos & Jean Dolbeault

  2. Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

    Manuel del Pino

Authors
  1. Juan Campos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manuel del Pino
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jean Dolbeault
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean Dolbeault.

Additional information

Communicated by H. Spohn

This paper may be reproduced, in its entirety, for non-commercial purposes.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Campos, J., del Pino, M. & Dolbeault, J. Relative Equilibria in Continuous Stellar Dynamics. Commun. Math. Phys. 300, 765–788 (2010). https://doi.org/10.1007/s00220-010-1128-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-010-1128-2

Keywords

Search

Navigation