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August  2022, 15(4): 605-620. doi: 10.3934/krm.2021033

Kinetic description of stable white dwarfs

1. 

Department of Mathematics, University of Southern California, Los Angeles, CA 90089 USA

2. 

Department of Mathematics, Kyonggi University, Suwon 16227, Republic of Korea

* Corresponding author

Received  June 2021 Published  August 2022 Early access  November 2021

Fund Project: This article is dedicated to the memory of Bob Glassey. JJ is supported in part by the NSF DMS-grant 2009458. JS is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (NRF-2020R1C1C1A01006415)

In this paper, we study fermion ground states of the relativistic Vlasov-Poisson system arising in the semiclassical limit from relativistic quantum theory of white dwarfs. We show that fermion ground states of the three dimensional relativistic Vlasov-Poisson system exist for subcritical mass, the mass density of such fermion ground states satisfies the Chandrasekhar equation for white dwarfs, and that they are orbitally stable as long as solutions exist.

Citation: Juhi Jang, Jinmyoung Seok. Kinetic description of stable white dwarfs. Kinetic and Related Models, 2022, 15 (4) : 605-620. doi: 10.3934/krm.2021033
References:
[1]

S. Chandrasekhar, The maximum mass of ideal white dwarfs, Astrophys. J., 74 (1931), 81.  doi: 10.1086/143324.

[2] S. Chandrasekhar, An Introduction to the Study of Stellar Structure, University of Chicago Press, Chicago, 1939. 
[3]

J. DolbeaultÓ. Sánchez and J. Soler, Asymptotic behaviour for the Vlasov-Poisson system in the stellar-dynamics case, Arch. Ration. Mech. Anal., 171 (2004), 301-327.  doi: 10.1007/s00205-003-0283-4.

[4]

J. Fröhlich and E. Lenzmann, Dynamical collapse of white dwarfs in Hartree-and Hartree-Fock theory, Comm. Math. Phys., 274 (2007), 737-750.  doi: 10.1007/s00220-007-0290-7.

[5]

R. T. Glassey and J. Schaeffer, On symmetric solutions of the relativistic Vlasov–Poisson system, Comm. Math. Phys., 101 (1985), 459-473.  doi: 10.1007/BF01210740.

[6]

F. GolseC. Mouhot and T. Paul, On the mean field and classical limits of quantum mechanics, Commun. Math. Phys., 343 (2016), 165-205.  doi: 10.1007/s00220-015-2485-7.

[7]

Y. Guo, Variational method for stable polytropic galaxies, Arch. Ration. Mech. Anal., 150 (1999), 209-224.  doi: 10.1007/s002050050187.

[8]

Y. Guo and Z. Lin, Unstable and stable galaxy models, Comm. Math. Phys., 279 (2008), 789-813.  doi: 10.1007/s00220-008-0439-z.

[9]

Y. Guo and G. Rein, Isotropic steady states in galactic dynamics, Commun. Math. Phys., 219 (2001), 607-629.  doi: 10.1007/s002200100434.

[10]

Y. Guo and G. Rein, A non-variational approach to nonlinear stability in stellar dynamics applied to the King model, Commun. Math. Phys., 271 (2007), 489-509.  doi: 10.1007/s00220-007-0212-8.

[11]

M. Hadžič and G. Rein, Global existence and nonlinear stability for the Relativistic Vlasov-Poisson system in the gravitational case, Indiana Univ. Math. J., 56 (2007), 2453-2488.  doi: 10.1512/iumj.2007.56.3064.

[12]

J. Jang and J. Seok, On uniformly rotating binary stars and galaxies, preprint.

[13]

N. G. Van Kampen and B. V. Felderhof, Theoretical Methods in Plasma Physics, Amsterdam: North Holland, 1967. doi: 10.1119/1.1974399.

[14]

M. K.-H. Kiessling and A. S. Tahvildar-Zadeh, On the relativistic Vlasov-Poisson system, Indiana Univ. Math. J., 57 (2008), 3177-3207.  doi: 10.1512/iumj.2008.57.3387.

[15]

D. Koester and G. Chanmugam, Physics of white dwarf stars, Rep. Prog. Phys., 53 (1990), 837-915.  doi: 10.1088/0034-4885/53/7/001.

[16]

J. Körner and G. Rein, Strong Lagrangian solutions of the (relativistic) Vlasov-Poisson system for non-smooth, spherically symmetric data, SIAM J. Math. Anal., 53 (2021), 4985-4996.  doi: 10.1137/20M1378910.

[17]

M. LemouF. Méhats and P. Raphaël, On the orbital stability of the ground states and the singularity formation for the gravitational Vlasov-Poisson system, Arch. Rat. Mech. Anal., 189 (2008), 425-468.  doi: 10.1007/s00205-008-0126-4.

[18]

M. LemouF. Méhats and P. Raphaël, Stable ground states for the relativistic gravitational Vlasov-Poisson system, Comm. Partial Differential Equations, 34 (2009), 703-721.  doi: 10.1080/03605300902963369.

[19]

M. LemouF. Méhats and P. Raphaël, Stable self-similar blow-up dynamics for the three dimensional gravitational Vlasov-Poisson system, J. Amer. Math. Soc., 21 (2008), 1019-1063.  doi: 10.1090/S0894-0347-07-00579-6.

[20]

M. LemouF. Méhats and P. Raphaël, Orbital stability of spherical galactic models, Invent. Math., 187 (2012), 145-194.  doi: 10.1007/s00222-011-0332-9.

[21]

E. Lenzmann and M. Lewin, Minimizers for the Hartree-Fock-Bogoliubov theory of neutron stars and white dwarfs, Duke Math. J., 152 (2010), 257-315.  doi: 10.1215/00127094-2010-013.

[22]

E. H. Lieb and M. Loss, \emphAnalysis, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.

[23]

E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Commun. Math. Phys., 112 (1987), 147-174.  doi: 10.1007/BF01217684.

[24]

P.-L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430.  doi: 10.1007/BF01232273.

[25]

T. Luo and J. Smoller, Nonlinear dynamical stability of Newtonian rotating and non-rotating white dwarfs and rotating supermassive stars, Comm. Math. Phys., 284 (2008), 425-457.  doi: 10.1007/s00220-008-0569-3.

[26]

T. Makino, On the existence of positive solutions at infinity for ordinary differential equations of Emden type, Funkcial. Ekvac., 27 (1984), 319-329. 

[27]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differ. Equ., 95 (1992), 281-303.  doi: 10.1016/0022-0396(92)90033-J.

[28]

G. Rein, Collisionless kinetic equations from astrophysics–the Vlasov-Poisson system, Handbook of differential equations: Evolutionary equations. Vol. Ⅲ, (2007), 383–476 doi: 10.1016/S1874-5717(07)80008-9.

show all references

References:
[1]

S. Chandrasekhar, The maximum mass of ideal white dwarfs, Astrophys. J., 74 (1931), 81.  doi: 10.1086/143324.

[2] S. Chandrasekhar, An Introduction to the Study of Stellar Structure, University of Chicago Press, Chicago, 1939. 
[3]

J. DolbeaultÓ. Sánchez and J. Soler, Asymptotic behaviour for the Vlasov-Poisson system in the stellar-dynamics case, Arch. Ration. Mech. Anal., 171 (2004), 301-327.  doi: 10.1007/s00205-003-0283-4.

[4]

J. Fröhlich and E. Lenzmann, Dynamical collapse of white dwarfs in Hartree-and Hartree-Fock theory, Comm. Math. Phys., 274 (2007), 737-750.  doi: 10.1007/s00220-007-0290-7.

[5]

R. T. Glassey and J. Schaeffer, On symmetric solutions of the relativistic Vlasov–Poisson system, Comm. Math. Phys., 101 (1985), 459-473.  doi: 10.1007/BF01210740.

[6]

F. GolseC. Mouhot and T. Paul, On the mean field and classical limits of quantum mechanics, Commun. Math. Phys., 343 (2016), 165-205.  doi: 10.1007/s00220-015-2485-7.

[7]

Y. Guo, Variational method for stable polytropic galaxies, Arch. Ration. Mech. Anal., 150 (1999), 209-224.  doi: 10.1007/s002050050187.

[8]

Y. Guo and Z. Lin, Unstable and stable galaxy models, Comm. Math. Phys., 279 (2008), 789-813.  doi: 10.1007/s00220-008-0439-z.

[9]

Y. Guo and G. Rein, Isotropic steady states in galactic dynamics, Commun. Math. Phys., 219 (2001), 607-629.  doi: 10.1007/s002200100434.

[10]

Y. Guo and G. Rein, A non-variational approach to nonlinear stability in stellar dynamics applied to the King model, Commun. Math. Phys., 271 (2007), 489-509.  doi: 10.1007/s00220-007-0212-8.

[11]

M. Hadžič and G. Rein, Global existence and nonlinear stability for the Relativistic Vlasov-Poisson system in the gravitational case, Indiana Univ. Math. J., 56 (2007), 2453-2488.  doi: 10.1512/iumj.2007.56.3064.

[12]

J. Jang and J. Seok, On uniformly rotating binary stars and galaxies, preprint.

[13]

N. G. Van Kampen and B. V. Felderhof, Theoretical Methods in Plasma Physics, Amsterdam: North Holland, 1967. doi: 10.1119/1.1974399.

[14]

M. K.-H. Kiessling and A. S. Tahvildar-Zadeh, On the relativistic Vlasov-Poisson system, Indiana Univ. Math. J., 57 (2008), 3177-3207.  doi: 10.1512/iumj.2008.57.3387.

[15]

D. Koester and G. Chanmugam, Physics of white dwarf stars, Rep. Prog. Phys., 53 (1990), 837-915.  doi: 10.1088/0034-4885/53/7/001.

[16]

J. Körner and G. Rein, Strong Lagrangian solutions of the (relativistic) Vlasov-Poisson system for non-smooth, spherically symmetric data, SIAM J. Math. Anal., 53 (2021), 4985-4996.  doi: 10.1137/20M1378910.

[17]

M. LemouF. Méhats and P. Raphaël, On the orbital stability of the ground states and the singularity formation for the gravitational Vlasov-Poisson system, Arch. Rat. Mech. Anal., 189 (2008), 425-468.  doi: 10.1007/s00205-008-0126-4.

[18]

M. LemouF. Méhats and P. Raphaël, Stable ground states for the relativistic gravitational Vlasov-Poisson system, Comm. Partial Differential Equations, 34 (2009), 703-721.  doi: 10.1080/03605300902963369.

[19]

M. LemouF. Méhats and P. Raphaël, Stable self-similar blow-up dynamics for the three dimensional gravitational Vlasov-Poisson system, J. Amer. Math. Soc., 21 (2008), 1019-1063.  doi: 10.1090/S0894-0347-07-00579-6.

[20]

M. LemouF. Méhats and P. Raphaël, Orbital stability of spherical galactic models, Invent. Math., 187 (2012), 145-194.  doi: 10.1007/s00222-011-0332-9.

[21]

E. Lenzmann and M. Lewin, Minimizers for the Hartree-Fock-Bogoliubov theory of neutron stars and white dwarfs, Duke Math. J., 152 (2010), 257-315.  doi: 10.1215/00127094-2010-013.

[22]

E. H. Lieb and M. Loss, \emphAnalysis, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.

[23]

E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Commun. Math. Phys., 112 (1987), 147-174.  doi: 10.1007/BF01217684.

[24]

P.-L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430.  doi: 10.1007/BF01232273.

[25]

T. Luo and J. Smoller, Nonlinear dynamical stability of Newtonian rotating and non-rotating white dwarfs and rotating supermassive stars, Comm. Math. Phys., 284 (2008), 425-457.  doi: 10.1007/s00220-008-0569-3.

[26]

T. Makino, On the existence of positive solutions at infinity for ordinary differential equations of Emden type, Funkcial. Ekvac., 27 (1984), 319-329. 

[27]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differ. Equ., 95 (1992), 281-303.  doi: 10.1016/0022-0396(92)90033-J.

[28]

G. Rein, Collisionless kinetic equations from astrophysics–the Vlasov-Poisson system, Handbook of differential equations: Evolutionary equations. Vol. Ⅲ, (2007), 383–476 doi: 10.1016/S1874-5717(07)80008-9.

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