Skip to main content
SpringerLink
Log in
Book cover

Free Boundary Problems pp 107–116Cite as

Birkhäuser

Nonlinear Diffusion Models for Self-gravitating Particles

  • Conference paper
  • First Online:

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 154))

Abstract

This paper deals with parabolic-elliptic systems of drift-diffusion type modelling gravitational interaction of particles. The main feature is presence of a nonlinear diffusion describing physically relevant density-pressure relations. We study the existence of solutions of the evolution problem, and recall results on the existence of steady states, and the blow up of solutions in cases when drift prevails the diffusion.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. P. Biler, J. Dolbeault, M.J. Esteban, P.A. Markowich, T. Nadzieja, Steady states for Streater’s energy-transport models of self-gravitating particles. “Transport in Transition Regimes”, N. Ben Abdallah, A. Arnold, P. Degond et al., eds., Springer IMA Volumes in Mathematics and Its Applications 135, New York: Springer (2004), 37–56.

    Chapter  Google Scholar 

  2. P. Biler, A. Krzywicki, T. Nadzieja, Self-interaction of Brownian particles coupled with thermodynamic processes. Rep. Math. Phys. 42 (1998), 359–372.

    Article  MathSciNet  Google Scholar 

  3. P. Biler, Ph. Laurençot, T. Nadzieja, On an evolution system describing self-gravitating Fermi-Dirac particles. Adv. Diff. Eq. 9 (2004), 563–586.

    MathSciNet  MATH  Google Scholar 

  4. P. Biler, T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles, I. Colloq. Math. 66 (1994), 319–334.

    Article  MathSciNet  Google Scholar 

  5. P. Biler, T. Nadzieja, Structure of steady states for Streater’s energy-transport models of gravitating particles. Topol. Methods Nonlinear Anal. 19 (2002), 283–301.

    Article  MathSciNet  Google Scholar 

  6. P. Biler, T. Nadzieja, Global and exploding solutions in a model of self-gravitating systems. Rep. Math. Phys. 52 (2003), 205–225.

    Article  MathSciNet  Google Scholar 

  7. P. Biler, T. Nadzieja, R. Stańczy, Nonisothermal systems of self-interacting Fermi-Dirac particles. “Nonlocal Elliptic and Parabolic Problems”, P. Biler, G. Karch, T. Nadzieja, eds., Banach Center Publications 66, Warsaw: Polish Acad. Sci. (2004), 61–78.

    Chapter  Google Scholar 

  8. P. Biler, R. Stańczy, Parabolic-elliptic systems with general density-pressure relations. “Variational Problems and Related Topics”, RIMS, Kyoto, M. Misawa, T. Suzuki, eds., Sūrikaisekikenkyūsho Kōkyūroku 1405 (2004), 31–53.

    Google Scholar 

  9. P.-H. Chavanis, Generalized kinetic equations and effective thermodynamics. “Nonlocal Elliptic and Parabolic Problems”, P. Biler, G. Karch, T. Nadzieja, eds., Banach Center Publications 66, Warsaw: Polish Acad. Sci. (2004), 79–101.

    Chapter  Google Scholar 

  10. C.J. van Duijn, I.A. Guerra, M.A. Peletier, Global existence conditions for a non-local problem arising in statistical mechanics. Adv. Diff. Eq. 9 (2004), 133–158.

    MATH  Google Scholar 

  11. I.A. Guerra, T. Nadzieja, Convergence to stationary solutions in a model of self-gravitating systems. Colloq. Math. 98 (2003), 42–47.

    Article  MathSciNet  Google Scholar 

  12. D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, II. Jahresber. Deutsch. Math.-Verein. 105 (2003), 103–165, and 106 (2004), 51–69.

    MathSciNet  MATH  Google Scholar 

  13. A. Raczyński, On a nonlocal elliptic problem. Appl. Math. (Warsaw) 26 (1999), 107–119.

    MathSciNet  MATH  Google Scholar 

  14. A. Raczyński, Steady states for polytropic equation of self-gravitating gas. Math. Meth. Appl. Sci. 28 (2005), 1881–1896.

    Article  MathSciNet  Google Scholar 

  15. R. Stańczy, Steady states for a system describing self-gravitating Fermi-Dirac particles. Diff. Int. Eq. 18 (2005), 567–582.

    MathSciNet  MATH  Google Scholar 

  16. R. Stańczy, Self-attracting Fermi-Dirac particles in canonical and microcanonical setting. Math. Meth. Appl. Sci. 28 (2005), 975–990.

    Article  MathSciNet  Google Scholar 

  17. Y. Sugiyama, Global existence in the supercritical cases and finite time blow-up in the subcritical cases to degenerate Keller-Segel systems. 1–21, preprint.

    Google Scholar 

  18. G. Wolansky, On steady distributions of self-attracting clusters under friction and fluctuations, Arch. Rational Mech. Anal. 119 (1992), 355–391.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, PL-50-384, Wrocław, Poland

    Piotr Biler & Robert Stańczy

Authors
  1. Piotr Biler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Robert Stańczy
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Departamento de Matemática Faculdade de Ciências e Tecnologia, Universidade de Coimbra, Apartado 3008, 3001-454, Coimbra, Portugal

    Isabel Narra Figueiredo

  2. Universidade de Lisboa / CMAF, Av. Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal

    José Francisco Rodrigues

  3. Departamento de Matemática, Universidade do Minho, Campus de Gualtar, 4710-057, Braga, Portugal

    Lisa Santos

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Birkhäuser Verlag Basel/Switzerland

About this paper

Cite this paper

Biler, P., Stańczy, R. (2006). Nonlinear Diffusion Models for Self-gravitating Particles. In: Figueiredo, I.N., Rodrigues, J.F., Santos, L. (eds) Free Boundary Problems. International Series of Numerical Mathematics, vol 154. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-7719-9_11

Download citation

Publish with us

Policies and ethics

Search

Navigation