Sur l’effondrement dynamique des étoiles quantiques pseudo-relativistes

Mathieu Lewin[1]

  • [1] CNRS & Université de Cergy-Pontoise (UMR 8088) 95000 Cergy-Pontoise France

Séminaire Laurent Schwartz — EDP et applications (2011-2012)

  • page 1-20
  • ISSN: 2266-0607

Abstract

top
Dans cet exposé, je présente plusieurs modèles quantiques non linéaires permettant de décrire certaines étoiles. Je m’intéresse tout particulièrement à l’effondrement gravitationnel des étoiles trop lourdes, un phénomène modélisé par des solutions qui explosent en temps fini. Je montre l’existence de telles solutions et je décris plusieurs de leurs propriétés au temps d’explosion.

How to cite

top

Lewin, Mathieu. "Sur l’effondrement dynamique des étoiles quantiques pseudo-relativistes." Séminaire Laurent Schwartz — EDP et applications (2011-2012): 1-20. <http://eudml.org/doc/251178>.

@article{Lewin2011-2012,
abstract = {Dans cet exposé, je présente plusieurs modèles quantiques non linéaires permettant de décrire certaines étoiles. Je m’intéresse tout particulièrement à l’effondrement gravitationnel des étoiles trop lourdes, un phénomène modélisé par des solutions qui explosent en temps fini. Je montre l’existence de telles solutions et je décris plusieurs de leurs propriétés au temps d’explosion.},
affiliation = {CNRS & Université de Cergy-Pontoise (UMR 8088) 95000 Cergy-Pontoise France},
author = {Lewin, Mathieu},
journal = {Séminaire Laurent Schwartz — EDP et applications},
language = {fre},
pages = {1-20},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Sur l’effondrement dynamique des étoiles quantiques pseudo-relativistes},
url = {http://eudml.org/doc/251178},
year = {2011-2012},
}

TY - JOUR
AU - Lewin, Mathieu
TI - Sur l’effondrement dynamique des étoiles quantiques pseudo-relativistes
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2011-2012
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
SP - 1
EP - 20
AB - Dans cet exposé, je présente plusieurs modèles quantiques non linéaires permettant de décrire certaines étoiles. Je m’intéresse tout particulièrement à l’effondrement gravitationnel des étoiles trop lourdes, un phénomène modélisé par des solutions qui explosent en temps fini. Je montre l’existence de telles solutions et je décris plusieurs de leurs propriétés au temps d’explosion.
LA - fre
UR - http://eudml.org/doc/251178
ER -

References

top
  1. V. Bach, E. H. Lieb, and J. P. Solovej, Generalized Hartree-Fock theory and the Hubbard model, J. Statist. Phys., 76 (1994), pp. 3–89. Zbl0839.60095MR1297873
  2. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of superconductivity, Phys. Rev., 108 (1957), pp. 1175–1204. Zbl0090.45401MR95694
  3. N. N. Bogoliubov, On a New Method in the Theory of Superconductivity, J. Exp. Theor. Phys., 34 (1958), p. 58. Zbl0090.45501
  4. A.-P. Calderón, Commutators of singular integral operators, Proc. Nat. Acad. Sci. U.S.A., 53 (1965), pp. 1092–1099. Zbl0151.16901MR177312
  5. É. Cancès, C. Le Bris, and Y. Maday, Méthodes mathématiques en chimie quantique. Une introduction, vol. 53 of Collection Mathématiques et Applications, Springer, 2006. Zbl1167.81001MR2426947
  6. T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys., 85 (1982), pp. 549–561. Zbl0513.35007MR677997
  7. J. M. Chadam, The time-dependent Hartree-Fock equations with Coulomb two-body interaction, Commun. Math. Phys., 46 (1976), pp. 99–104. Zbl0322.35043MR411439
  8. J. M. Chadam and R. T. Glassey, Global existence of solutions to the Cauchy problem for time-dependent Hartree equations, J. Math. Phys., 16 (1975), pp. 1122–1130. Zbl0299.35084MR413843
  9. S. Chandrasekhar, The density of white dwarf stars, Philos. Mag., 11 (1931), pp. 592–596. Zbl0001.11103
  10. Idem, The maximum mass of ideal white dwarfs, Astrophys. J., 74 (1931), pp. 81–82. Zbl0002.23502
  11. R. R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), pp. 315–331. Zbl0324.44005MR380244
  12. A. Elgart and B. Schlein, Mean field dynamics of boson stars, Comm. Pure Appl. Math., 60 (2007), pp. 500–545. Zbl1113.81032MR2290709
  13. M. J. Esteban, M. Lewin, and É. Séré, Variational methods in relativistic quantum mechanics, Bull. Amer. Math. Soc. (N.S.), 45 (2008), pp. 535–593. Zbl1288.49016MR2434346
  14. R. L. Frank and E. Lenzmann, On ground states for the L 2 -critical boson star equation, ArXiv e-prints, (2010). 
  15. J. Fröhlich, B. L. G. Jonsson, and E. Lenzmann, Effective dynamics for boson stars, Nonlinearity, 20 (2007), pp. 1031–1075. Zbl1124.35084MR2312382
  16. J. Fröhlich and E. Lenzmann, Blowup for nonlinear wave equations describing boson stars, Comm. Pure Appl. Math., 60 (2007), pp. 1691–1705. Zbl1135.35011MR2349352
  17. Idem, Dynamical collapse of white dwarfs in Hartree- and Hartree-Fock theory, Commun. Math. Phys., 274 (2007), pp. 737–750. Zbl1130.85004MR2328910
  18. C. Hainzl, E. Lenzmann, M. Lewin, and B. Schlein, On blowup for time-dependent generalized Hartree-Fock equations, Ann. Henri Poincaré, 11 (2010), pp. 1023–1052. Zbl1209.85009MR2737490
  19. C. Hainzl and B. Schlein, Stellar collapse in the time dependent Hartree-Fock approximation, Commun. Math. Phys., 287 (2009), pp. 705–717. Zbl1175.85002MR2481756
  20. I. W. Herbst, Spectral theory of the operator ( p 2 + m 2 ) 1 / 2 - Z e 2 / r , Commun. Math. Phys., 53 (1977), pp. 285–294. Zbl0375.35047MR436854
  21. T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, Int. Math. Res. Not., (2005), pp. 2815–2828. Zbl1126.35067MR2180464
  22. T. Kato, Perturbation theory for linear operators, Springer, second ed., 1995. Zbl0836.47009MR1335452
  23. E. Lenzmann, Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10 (2007), pp. 43–64. Zbl1171.35474MR2340532
  24. E. Lenzmann and M. Lewin, Minimizers for the Hartree-Fock-Bogoliubov theory of neutron stars and white dwarfs, Duke Math. J., 152 (2010), pp. 257–315. Zbl1202.49013MR2656090
  25. E. Lenzmann and M. Lewin, On singularity formation for the L 2 -critical Boson star equation. 2011. Zbl1235.35231
  26. E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Commun. Math. Phys., 53 (1977), pp. 185–194. MR452286
  27. E. H. Lieb and W. E. Thirring, Gravitational collapse in quantum mechanics with relativistic kinetic energy, Ann. Physics, 155 (1984), pp. 494–512. MR753345
  28. E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Commun. Math. Phys., 112 (1987), pp. 147–174. Zbl0641.35065MR904142
  29. J.-L. Lions, Espaces intermédiaires entre espaces hilbertiens et applications, Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine (N.S.), 2 (50) (1958), pp. 419–432. Zbl0097.09501MR151829
  30. P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), pp. 109–149. Zbl0704.49004MR778970
  31. Idem, The concentration-compactness principle in the calculus of variations. The locally compact case, Part II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), pp. 223–283. Zbl0704.49004MR778974
  32. Idem, Solutions of Hartree-Fock equations for Coulomb systems, Commun. Math. Phys., 109 (1987), pp. 33–97. Zbl0618.35111MR879032
  33. F. Merle and P. Raphael, Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation, Comm. Math. Phys., 253 (2005), pp. 675–704. Zbl1062.35137MR2116733
  34. F. Merle and Y. Tsutsumi, L 2 concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity, J. Differential Equations, 84 (1990), pp. 205–214. Zbl0722.35047MR1047566
  35. A. Michelangeli and B. Schlein, Dynamical Collapse of Boson Stars, ArXiv e-prints, (2010). Zbl1242.85007MR2909759
  36. H. Nawa, “Mass concentration” phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity. II, Kodai Math. J., 13 (1990), pp. 333–348. Zbl0761.35102MR1078548
  37. M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier analysis, self-adjointness, Academic Press, New York, 1975. Zbl0242.46001MR493420
  38. P. Ring and P. Schuck, The nuclear many-body problem, vol. Texts and Monographs in Physics, Springer Verlag, New York, 1980. MR611683
  39. E. M. Stein, Harmonic analysis : real-variable methods, orthogonality, and oscillatory integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy. Zbl0821.42001MR1232192
  40. W. Thirring, Bosonic black holes, Physics Letters B, 127 (1983), pp. 27 – 29. 
  41. M. I. Weinstein, On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations, Comm. Partial Differential Equations, 11 (1986), pp. 545–565. Zbl0596.35022MR829596

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.