×

Stellar collapse in the time dependent Hartree-Fock approximation. (English) Zbl 1175.85002

Summary: We prove blow-up in finite time for radially symmetric solutions to the pseudo-relativistic Hartree-Fock equation with negative energy. The non-linear Hartree-Fock equation is commonly used in the physics literature to describe the dynamics of white dwarfs. We extend thereby recent results by J. Fröhlich and E. Lenzmann, who established in [Commun. Pure Appl. Math. 60, No. 11, 1691–1705 (2007; Zbl 1135.35011), Commun. Math. Phys. 274, No. 3, 737–750 (2007; Zbl 1130.85004)] blow-up for solutions to the pseudo-relativistic Hartree equation. As key ingredient for handling the exchange term we use the conservation of the expectation of the square of the angular momentum operator.

MSC:

85-XX Astronomy and astrophysics
35Q75 PDEs in connection with relativity and gravitational theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Chandrasekhar, S.: Phil. Mag. 11, 592 (1931); Astrophys. J. 74, 81 (1931); Rev. Mod. Physics 56, 137 (1984)
[2] Elgart A., Schlein B.: Mean field dynamics of boson stars. Comm. Pure Appl. Math. 60(4), 500–545 (2007) · Zbl 1113.81032 · doi:10.1002/cpa.20134
[3] Fröhlich J., Lenzmann E.: Blowup for nonlinear wave equations describing boson stars. Commun. Pure Appl. Math. 60(11), 1691–1705 (2007) · Zbl 1135.35011 · doi:10.1002/cpa.20186
[4] Fröhlich J., Lenzmann E.: Dynamical collapse of white dwarfs in Hartree- and Hartree-Fock theory. Commun. Math. Phys. 274(3), 737–750 (2007) · Zbl 1130.85004 · doi:10.1007/s00220-007-0290-7
[5] Lieb E.H., Yau H.T.: The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Commun. Math. Phys. 112(1), 147–174 (1987) · Zbl 0641.35065 · doi:10.1007/BF01217684
[6] Lieb E.H., Yau H.T.: A rigorous examination of the Chandrasekhar theory of stellar collapse. Astrophys. J. 323, 140–144 (1987) · doi:10.1086/165813
[7] Lieb E.H., Thirring W.: Gravitational collapse in quantum mechanics with relativistic kinetic energy. Ann. Phys. 155, 494–512 (1984) · doi:10.1016/0003-4916(84)90010-1
[8] Stein E.: Harmonic Analyis. Princeton University Press, Princeton, NJ (1993)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.