Annales Henri Poincaré

, Volume 11, Issue 6, pp 1023–1052 | Cite as

On Blowup for Time-Dependent Generalized Hartree–Fock Equations

  • Christian Hainzl
  • Enno Lenzmann
  • Mathieu Lewin
  • Benjamin Schlein
Article

Abstract

We prove finite-time blowup for spherically symmetric and negative energy solutions of Hartree–Fock and Hartree–Fock–Bogoliubov-type equations, which describe the evolution of attractive fermionic systems (e.g. white dwarfs). Our main results are twofold: first, we extend the recent blowup result of Hainzl and Schlein (Comm. Math. Phys. 287:705–714, 2009) to Hartree–Fock equations with infinite rank solutions and a general class of Newtonian type interactions. Second, we show the existence of finite-time blowup for spherically symmetric solutions of a Hartree–Fock–Bogoliubov model, where an angular momentum cutoff is introduced. We also explain the key difficulties encountered in the full Hartree–Fock–Bogoliubov theory.

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References

  1. 1.
    Bach, V., Fröhlich, J., Jonsson L.: Bogolubov–Hartree–Fock mean field theory for neutron stars and other systems with attractive interactions. J. Math. Phys. 50, 102102, 22 (2009)Google Scholar
  2. 2.
    Bach V., Lieb E.H., Solovej J.P.: Generalized Hartree–Fock theory and the Hubbard model. J. Statist. Phys. 76, 3–89 (1994)MATHCrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Bender M., Heenen P.-H., Reinhard P.-G.: Self-consistent mean-field models for nuclear structure. Rev. Mod. Phys. 75, 121–180 (2003)CrossRefADSGoogle Scholar
  4. 4.
    Bove A., Da Prato G., Fano G.: On the Hartree–Fock time-dependent problem. Commun. Math. Phys. 49, 25–33 (1976)CrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Chadam J.M.: The time-dependent Hartree–Fock equations with Coulomb two-body interaction. Commun. Math. Phys. 46, 99–104 (1976)MATHCrossRefMathSciNetADSGoogle Scholar
  6. 6.
    Chadam J.M., Glassey R.T.: Global existence of solutions to the Cauchy problem for time-dependent Hartree equations. J. Math. Phys. 16, 1122–1130 (1975)MATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Chandrasekhar S.: The maximum mass of ideal white dwarfs. Astrophys. J. 74, 81–82 (1931)MATHCrossRefADSGoogle Scholar
  8. 8.
    Dean D.J., Hjorth-Jensen M.: Pairing in nuclear systems: from neutron stars to finite nuclei. Rev. Mod. Phys. 75, 607–656 (2003)CrossRefADSGoogle Scholar
  9. 9.
    Fröhlich J., Lenzmann E.: Blowup for nonlinear wave equations describing boson stars. Commun. Pure Appl. Math. 60, 1691–1705 (2007)MATHCrossRefGoogle Scholar
  10. 10.
    Fröhlich J., Lenzmann E.: Dynamical collapse of white dwarfs in Hartree- and Hartree–Fock theory. Commun. Math. Phys. 274, 737–750 (2007)MATHCrossRefGoogle Scholar
  11. 11.
    Hainzl C., Lewin M., Sparber C.: Existence of global-in-time solutions to a generalized Dirac-Fock type evolution equation. Lett. Math. Phys. 72, 99–113 (2005)MATHCrossRefMathSciNetADSGoogle Scholar
  12. 12.
    Hainzl C., Schlein B.: Stellar collapse in the time dependent Hartree–Fock approximation. Commun. Math. Phys. 287, 705–717 (2009)MATHCrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Lenzmann E.: Well-posedness for semi-relativistic Hartree equations of critical type. Math. Phys. Anal. Geom. 10, 43–64 (2007)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Lenzmann E., Lewin M.: Minimizers for the Hartree–Fock–Bogoliubov theory of neutron stars and white dwarfs. Duke Math. J. 152, 257–315 (2010)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Lieb, E.H., Loss, M.: Analysis, Graduate Studies in Mathematics, vol. 14, 2nd edn. American Mathematical Society, Providence (2001)Google Scholar
  16. 16.
    Lieb E.H., Thirring W.E.: Gravitational collapse in quantum mechanics with relativistic kinetic energy. Ann. Phys. 155, 494–512 (1984)CrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Lieb E.H., Yau H.-T.: The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Commun. Math. Phys. 112, 147–174 (1987)MATHCrossRefMathSciNetADSGoogle Scholar
  18. 18.
    Reed M., Simon B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York (1978)MATHGoogle Scholar
  19. 19.
    Ring P., Schuck P.: The Nuclear Many-body Problem. Texts and Monographs in Physics. Springer, New York (1980)Google Scholar
  20. 20.
    Simon, B.: Trace ideals and their applications. London Mathematical Society Lecture Note Series, vol. 35. Cambridge University Press, Cambridge (1979)Google Scholar
  21. 21.
    Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series, vol. 43. Princeton University Press, Princeton (1993) (With the assistance of Timothy S. Murphy)Google Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Christian Hainzl
    • 1
  • Enno Lenzmann
    • 2
  • Mathieu Lewin
    • 3
  • Benjamin Schlein
    • 4
  1. 1.Department of MathematicsUABBirminghamUSA
  2. 2.Department of Mathematical SciencesUniversity of CopenhagenCopenhagen ØDenmark
  3. 3.CNRS & Laboratoire de Mathématiques, (CNRS UMR 8088)Université de Cergy-PontoiseCergy-PontoiseFrance
  4. 4.Institute for Applied MathematicsUniversity of BonnBonnGermany

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