doi: 10.3934/krm.2021033
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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 & Related Models, 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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[1]

Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete & Continuous Dynamical Systems, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361

[2]

Katherine Zhiyuan Zhang. Focusing solutions of the Vlasov-Poisson system. Kinetic & Related Models, 2019, 12 (6) : 1313-1327. doi: 10.3934/krm.2019051

[3]

Blanca Ayuso, José A. Carrillo, Chi-Wang Shu. Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system. Kinetic & Related Models, 2011, 4 (4) : 955-989. doi: 10.3934/krm.2011.4.955

[4]

Silvia Caprino, Guido Cavallaro, Carlo Marchioro. Time evolution of a Vlasov-Poisson plasma with magnetic confinement. Kinetic & Related Models, 2012, 5 (4) : 729-742. doi: 10.3934/krm.2012.5.729

[5]

Jack Schaeffer. Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior. Kinetic & Related Models, 2012, 5 (1) : 129-153. doi: 10.3934/krm.2012.5.129

[6]

Gang Li, Xianwen Zhang. A Vlasov-Poisson plasma of infinite mass with a point charge. Kinetic & Related Models, 2018, 11 (2) : 303-336. doi: 10.3934/krm.2018015

[7]

Gianluca Crippa, Silvia Ligabue, Chiara Saffirio. Lagrangian solutions to the Vlasov-Poisson system with a point charge. Kinetic & Related Models, 2018, 11 (6) : 1277-1299. doi: 10.3934/krm.2018050

[8]

Zili Chen, Xiuting Li, Xianwen Zhang. The two dimensional Vlasov-Poisson system with steady spatial asymptotics. Kinetic & Related Models, 2017, 10 (4) : 977-1009. doi: 10.3934/krm.2017039

[9]

Meixia Xiao, Xianwen Zhang. On global solutions to the Vlasov-Poisson system with radiation damping. Kinetic & Related Models, 2018, 11 (5) : 1183-1209. doi: 10.3934/krm.2018046

[10]

Yulia O. Belyaeva, Björn Gebhard, Alexander L. Skubachevskii. A general way to confined stationary Vlasov-Poisson plasma configurations. Kinetic & Related Models, 2021, 14 (2) : 257-282. doi: 10.3934/krm.2021004

[11]

Jack Schaeffer. On time decay for the spherically symmetric Vlasov-Poisson system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021021

[12]

Lan Luo, Hongjun Yu. Global solutions to the relativistic Vlasov-Poisson-Fokker-Planck system. Kinetic & Related Models, 2016, 9 (2) : 393-405. doi: 10.3934/krm.2016.9.393

[13]

Gerhard Rein, Christopher Straub. On the transport operators arising from linearizing the Vlasov-Poisson or Einstein-Vlasov system about isotropic steady states. Kinetic & Related Models, 2020, 13 (5) : 933-949. doi: 10.3934/krm.2020032

[14]

Silvia Caprino, Guido Cavallaro, Carlo Marchioro. A Vlasov-Poisson plasma with unbounded mass and velocities confined in a cylinder by a magnetic mirror. Kinetic & Related Models, 2016, 9 (4) : 657-686. doi: 10.3934/krm.2016011

[15]

Joackim Bernier, Michel Mehrenberger. Long-time behavior of second order linearized Vlasov-Poisson equations near a homogeneous equilibrium. Kinetic & Related Models, 2020, 13 (1) : 129-168. doi: 10.3934/krm.2020005

[16]

Hyung Ju Hwang, Jaewoo Jung, Juan J. L. Velázquez. On global existence of classical solutions for the Vlasov-Poisson system in convex bounded domains. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 723-737. doi: 10.3934/dcds.2013.33.723

[17]

Francis Filbet, Roland Duclous, Bruno Dubroca. Analysis of a high order finite volume scheme for the 1D Vlasov-Poisson system. Discrete & Continuous Dynamical Systems - S, 2012, 5 (2) : 283-305. doi: 10.3934/dcdss.2012.5.283

[18]

Dongming Wei. 1D Vlasov-Poisson equations with electron sheet initial data. Kinetic & Related Models, 2010, 3 (4) : 729-754. doi: 10.3934/krm.2010.3.729

[19]

Mihaï Bostan. Asymptotic behavior for the Vlasov-Poisson equations with strong uniform magnetic field and general initial conditions. Kinetic & Related Models, 2020, 13 (3) : 531-548. doi: 10.3934/krm.2020018

[20]

Trinh T. Nguyen. Derivative estimates for screened Vlasov-Poisson system around Penrose-stable equilibria. Kinetic & Related Models, 2020, 13 (6) : 1193-1218. doi: 10.3934/krm.2020043

2020 Impact Factor: 1.432

Article outline

[Back to Top]