August  2019, 12(4): 749-764. doi: 10.3934/krm.2019029

Stationary solutions to the boundary value problem for the relativistic BGK model in a slab

Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea

* Corresponding author: Seok-Bae Yun

Received  April 2018 Revised  February 2019 Published  May 2019

Fund Project: Seok-Bae Yun is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03935955)

In this paper, we are concerned with the boundary value problem in a slab for the stationary relativistic BGK model of Marle type, which is a relaxation model of the relativistic Boltzmann equation. In the case of fixed inflow boundary conditions, we establish the existence of unique stationary solutions.

Citation: Byung-Hoon Hwang, Seok-Bae Yun. Stationary solutions to the boundary value problem for the relativistic BGK model in a slab. Kinetic & Related Models, 2019, 12 (4) : 749-764. doi: 10.3934/krm.2019029
References:
[1]

J. L. Anderson and A. C. Payne, The relativistic Burnett equations and sound propagation, Physica A, 85 (1976), 261-286. doi: 10.1016/0378-4371(76)90050-9.

[2]

J. L. Anderson and H. R. Witting, A relativistic relaxational time model for the Boltzmann equation, Physica, 74 (1974), 466-488.

[3]

J. H. Bang and S.-B. Yun, Stationary solution for the ellipsoidal BGK model in slab, J. Differential Equations, 261 (2016), 5803-5828. doi: 10.1016/j.jde.2016.08.022.

[4]

A. BellouquidJ. CalvoJ. Nieto and J. Soler, On the relativistic BGK-Boltzmann model: Asymptotics and hydrodynamics, J. Stat. Phys, 149 (2012), 284-316. doi: 10.1007/s10955-012-0600-0.

[5]

A. BellouquidJ. Nieto and L. Urrutia, Global existence and asymptotic stability near equilibrium for the relativistic BGK model, Nonlinear Anal, 114 (2015), 87-104. doi: 10.1016/j.na.2014.10.020.

[6]

K. Bichteler, On the Cauchy problem of the relativistic Boltzmann equation, Commun. Math. Phys, 4 (1967), 352-364. doi: 10.1007/BF01653649.

[7]

S. Calogero, The Newtonian limit of the relativistic Boltzmann equation, J. Math. Phys, 45 (2004), 4042-4052. doi: 10.1063/1.1793328.

[8]

C. Cercignani and G. M. Kremer, The Relativistic Boltzmann Equation: Theory and Applications, Progress in Mathematical Physics, 22. Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8165-4.

[9]

C. Cercignani and G. M. Kremer, Moment closure of the relativistic Anderson and Witting model equation, Physica A, 290 (2001), 192-202. doi: 10.1016/S0378-4371(00)00403-9.

[10]

Y. ChenY. Kuang and H. Tang, Second-order accurate genuine BGK schemes for the ultra-relativistic flow simulations, J. Comput. Phys, 349 (2017), 300-327. doi: 10.1016/j.jcp.2017.08.022.

[11]

M. Dudyński, On the linearized relativistic Boltzmann equation. Ⅱ. Existence of hydro-dynamics, J. Stat. Phys, 57 (1989), 199-245. doi: 10.1007/BF01023641.

[12]

M. Dudyński and M. L. Ekiel-Jeżewska, On the linearized relativistic Boltzmann equation. I. Existence of solutions, Commun. Math. Phys, 115 (1985), 607-629. doi: 10.1007/BF01224130.

[13]

M. Dudyński and M. L. Ekiel-Jeżewska, Global existence proof for relativistic Boltzmann equation, J. Stat. Phys, 66 (1992), 991-1001. doi: 10.1007/BF01055712.

[14]

W. FlorkowskiR. Ryblewski and M. Strickland, Anisotropic hydrodynamics for rapidly expanding systems, Nuclear Physics A, 916 (2013), 249-259. doi: 10.1016/j.nuclphysa.2013.08.004.

[15]

R. T. Glassey and W. A. Strauss, Asymptotic stability of the relativistic Maxwellian, Publ. Res. Inst. Math. Sci, 29 (1993), 301-347. doi: 10.2977/prims/1195167275.

[16]

R. T. Glassey and W. A. Strauss, Asymptotic stability of the relativistic Maxwellian via fourteen moments, Trans. Th. Stat. Phys, 24 (1995), 657-678. doi: 10.1080/00411459508206020.

[17]

Y. Guo and R. M. Strain, Momentum regularity and stability of the relativistic Vlasov-Maxwell-Boltzmann system, Comm. Math. Phys, 310 (2012), 649-673. doi: 10.1007/s00220-012-1417-z.

[18]

R. Hakim and L. Mornas, Collective effects on transport coefficients of relativistic nuclear matter, Phys. Rev. C, 47 (1993), 2846-2860. doi: 10.1103/PhysRevC.47.2846.

[19]

R. HakimL. MornasP. Peter and H. D. Sivak, Relaxation time approximation for relativistic dense matter, Phys. Rev. D, 46 (1992), 4603-4629. doi: 10.1103/PhysRevD.46.4603.

[20]

Z. Jiang, On the relativistic Boltzmann equation, Acta Math. Sci, 18 (1998), 348-360. doi: 10.1016/S0252-9602(17)30724-5.

[21]

Z. Jiang, On the Cauchy problem for the relativistic Boltzmann equation in a periodic box: global existence, Transport Theory Statist. Phys, 28 (1999), 617-628. doi: 10.1080/00411459908214520.

[22]

F. Jüttner, Das Maxwellsche Gesetz der Geschwindigkeitsverteilung in der Relativtheorie, Ann. Phys, 339 (1911), 856-882.

[23]

H. Lee and A. Rendall, The spatially homogeneous relativistic Boltzmann equation with a hard potential, Comm. Partial Differential Equations, 38 (2013), 2238-2262. doi: 10.1080/03605302.2013.827709.

[24]

A. Lichnerowicz and R. Marrot, Propriétés statistiques des ensembles de particules en relativité restreinte, C. R. Acad. Sci. Paris, 210 (1940), 759-761.

[25]

G. M. Kremer, Diffusion of relativistic gas mixtures in gravitational fields, Physica A, 393 (2014), 76-85. doi: 10.1016/j.physa.2013.09.019.

[26]

G. M. Kremer, Relativistic gas in a Schwarzschild metric, J. Stat. Mech, 2013 (2013), P04016, 13pp. doi: 10.1088/1742-5468/2013/04/p04016.

[27]

A. Majorana, Relativistic relaxation models for a simple gas, J. Math. Phys, 31 (1990), 2042-2046. doi: 10.1063/1.528655.

[28]

C. Marle, Sur l'établissement des equations de l'hydrodynamique des fluides relativistes dissipatifs, I. L'equation de Boltzmann relativiste, Ann. Inst. Henri Poincaré, 10 (1969), 67-126.

[29]

C. Marle, Sur l'établissement des equations de l'hydrodynamique des fluides relativistes dissipatifs. II. Méthodes de résolution approchée de l'equation de Boltzmann relativiste, Ann. Inst. Henri Poincaré, 10 (1969), 127-194.

[30]

C. Marle, Modele cinétique pour l'établissement des lois de la conduction de la chaleur et de la viscosité en théorie de la relativité, C. R. Acad. Sci. Paris, 260 (1965), 6300-6302.

[31]

M. Mendoza, B. M. Boghosian, H. J. Herrmann and S. Succi, Derivation of the lattice Boltzmann model for relativistic hydrodynamics, Phys. Rev. D, 82 (2010), 105008. doi: 10.1103/PhysRevD.82.105008.

[32]

J. Speck and R. M. Strain, Hilbert expansion from the Boltzmann equation to relativistic fluids, Commun. Math. Phys, 304 (2011), 229-280. doi: 10.1007/s00220-011-1207-z.

[33]

R. M. Strain, Asymptotic stability of the relativistic Boltzmann equation for the soft potentials, Commun. Math. Phys, 300 (2010), 529-597. doi: 10.1007/s00220-010-1129-1.

[34]

R. M. Strain, Global Newtonian limit for the relativistic Boltzmann equation near vacuum, SIAM J. Math. Anal, 42 (2010), 1568-1601. doi: 10.1137/090762695.

[35]

R. M. Strain and S.-B. Yun, Spatially homogenous Boltzmann equation for relativistic particles, SIAM J. Math. Anal, 46 (2014), 917-938. doi: 10.1137/130923531.

[36]

R. M. Strain and K. Zhu, Large-time decay of the soft potential relativistic Boltzmann equation in $\mathbb{R}^3_x$, Kinet. Relat. Models, 5 (2012), 383-415. doi: 10.3934/krm.2012.5.383.

[37]

H. Struchtrup, Projected moments in relativistic kinetic theory, Physica A, 253 (1998), 555-593. doi: 10.1016/S0378-4371(98)00037-5.

[38]

M. Takamoto and S. -I. Inutsuka, The relativistic kinetic dispersion relation: Comparison of the relativistic Bhatnagar-Gross-Krook model and Grad's 14-moment expansion, Physica A, 389 (2010), 4580-4603. doi: 10.1016/j.physa.2010.06.021.

[39]

R. D. Tenreiro and R. Hakim, Transport properties of the relativistic degenerate electron gas in a strong magnetic field: Covariant relaxation-time model, Phys. Rev. D, 15 (1977), 1435-1447. doi: 10.1103/PhysRevD.15.1435.

show all references

References:
[1]

J. L. Anderson and A. C. Payne, The relativistic Burnett equations and sound propagation, Physica A, 85 (1976), 261-286. doi: 10.1016/0378-4371(76)90050-9.

[2]

J. L. Anderson and H. R. Witting, A relativistic relaxational time model for the Boltzmann equation, Physica, 74 (1974), 466-488.

[3]

J. H. Bang and S.-B. Yun, Stationary solution for the ellipsoidal BGK model in slab, J. Differential Equations, 261 (2016), 5803-5828. doi: 10.1016/j.jde.2016.08.022.

[4]

A. BellouquidJ. CalvoJ. Nieto and J. Soler, On the relativistic BGK-Boltzmann model: Asymptotics and hydrodynamics, J. Stat. Phys, 149 (2012), 284-316. doi: 10.1007/s10955-012-0600-0.

[5]

A. BellouquidJ. Nieto and L. Urrutia, Global existence and asymptotic stability near equilibrium for the relativistic BGK model, Nonlinear Anal, 114 (2015), 87-104. doi: 10.1016/j.na.2014.10.020.

[6]

K. Bichteler, On the Cauchy problem of the relativistic Boltzmann equation, Commun. Math. Phys, 4 (1967), 352-364. doi: 10.1007/BF01653649.

[7]

S. Calogero, The Newtonian limit of the relativistic Boltzmann equation, J. Math. Phys, 45 (2004), 4042-4052. doi: 10.1063/1.1793328.

[8]

C. Cercignani and G. M. Kremer, The Relativistic Boltzmann Equation: Theory and Applications, Progress in Mathematical Physics, 22. Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8165-4.

[9]

C. Cercignani and G. M. Kremer, Moment closure of the relativistic Anderson and Witting model equation, Physica A, 290 (2001), 192-202. doi: 10.1016/S0378-4371(00)00403-9.

[10]

Y. ChenY. Kuang and H. Tang, Second-order accurate genuine BGK schemes for the ultra-relativistic flow simulations, J. Comput. Phys, 349 (2017), 300-327. doi: 10.1016/j.jcp.2017.08.022.

[11]

M. Dudyński, On the linearized relativistic Boltzmann equation. Ⅱ. Existence of hydro-dynamics, J. Stat. Phys, 57 (1989), 199-245. doi: 10.1007/BF01023641.

[12]

M. Dudyński and M. L. Ekiel-Jeżewska, On the linearized relativistic Boltzmann equation. I. Existence of solutions, Commun. Math. Phys, 115 (1985), 607-629. doi: 10.1007/BF01224130.

[13]

M. Dudyński and M. L. Ekiel-Jeżewska, Global existence proof for relativistic Boltzmann equation, J. Stat. Phys, 66 (1992), 991-1001. doi: 10.1007/BF01055712.

[14]

W. FlorkowskiR. Ryblewski and M. Strickland, Anisotropic hydrodynamics for rapidly expanding systems, Nuclear Physics A, 916 (2013), 249-259. doi: 10.1016/j.nuclphysa.2013.08.004.

[15]

R. T. Glassey and W. A. Strauss, Asymptotic stability of the relativistic Maxwellian, Publ. Res. Inst. Math. Sci, 29 (1993), 301-347. doi: 10.2977/prims/1195167275.

[16]

R. T. Glassey and W. A. Strauss, Asymptotic stability of the relativistic Maxwellian via fourteen moments, Trans. Th. Stat. Phys, 24 (1995), 657-678. doi: 10.1080/00411459508206020.

[17]

Y. Guo and R. M. Strain, Momentum regularity and stability of the relativistic Vlasov-Maxwell-Boltzmann system, Comm. Math. Phys, 310 (2012), 649-673. doi: 10.1007/s00220-012-1417-z.

[18]

R. Hakim and L. Mornas, Collective effects on transport coefficients of relativistic nuclear matter, Phys. Rev. C, 47 (1993), 2846-2860. doi: 10.1103/PhysRevC.47.2846.

[19]

R. HakimL. MornasP. Peter and H. D. Sivak, Relaxation time approximation for relativistic dense matter, Phys. Rev. D, 46 (1992), 4603-4629. doi: 10.1103/PhysRevD.46.4603.

[20]

Z. Jiang, On the relativistic Boltzmann equation, Acta Math. Sci, 18 (1998), 348-360. doi: 10.1016/S0252-9602(17)30724-5.

[21]

Z. Jiang, On the Cauchy problem for the relativistic Boltzmann equation in a periodic box: global existence, Transport Theory Statist. Phys, 28 (1999), 617-628. doi: 10.1080/00411459908214520.

[22]

F. Jüttner, Das Maxwellsche Gesetz der Geschwindigkeitsverteilung in der Relativtheorie, Ann. Phys, 339 (1911), 856-882.

[23]

H. Lee and A. Rendall, The spatially homogeneous relativistic Boltzmann equation with a hard potential, Comm. Partial Differential Equations, 38 (2013), 2238-2262. doi: 10.1080/03605302.2013.827709.

[24]

A. Lichnerowicz and R. Marrot, Propriétés statistiques des ensembles de particules en relativité restreinte, C. R. Acad. Sci. Paris, 210 (1940), 759-761.

[25]

G. M. Kremer, Diffusion of relativistic gas mixtures in gravitational fields, Physica A, 393 (2014), 76-85. doi: 10.1016/j.physa.2013.09.019.

[26]

G. M. Kremer, Relativistic gas in a Schwarzschild metric, J. Stat. Mech, 2013 (2013), P04016, 13pp. doi: 10.1088/1742-5468/2013/04/p04016.

[27]

A. Majorana, Relativistic relaxation models for a simple gas, J. Math. Phys, 31 (1990), 2042-2046. doi: 10.1063/1.528655.

[28]

C. Marle, Sur l'établissement des equations de l'hydrodynamique des fluides relativistes dissipatifs, I. L'equation de Boltzmann relativiste, Ann. Inst. Henri Poincaré, 10 (1969), 67-126.

[29]

C. Marle, Sur l'établissement des equations de l'hydrodynamique des fluides relativistes dissipatifs. II. Méthodes de résolution approchée de l'equation de Boltzmann relativiste, Ann. Inst. Henri Poincaré, 10 (1969), 127-194.

[30]

C. Marle, Modele cinétique pour l'établissement des lois de la conduction de la chaleur et de la viscosité en théorie de la relativité, C. R. Acad. Sci. Paris, 260 (1965), 6300-6302.

[31]

M. Mendoza, B. M. Boghosian, H. J. Herrmann and S. Succi, Derivation of the lattice Boltzmann model for relativistic hydrodynamics, Phys. Rev. D, 82 (2010), 105008. doi: 10.1103/PhysRevD.82.105008.

[32]

J. Speck and R. M. Strain, Hilbert expansion from the Boltzmann equation to relativistic fluids, Commun. Math. Phys, 304 (2011), 229-280. doi: 10.1007/s00220-011-1207-z.

[33]

R. M. Strain, Asymptotic stability of the relativistic Boltzmann equation for the soft potentials, Commun. Math. Phys, 300 (2010), 529-597. doi: 10.1007/s00220-010-1129-1.

[34]

R. M. Strain, Global Newtonian limit for the relativistic Boltzmann equation near vacuum, SIAM J. Math. Anal, 42 (2010), 1568-1601. doi: 10.1137/090762695.

[35]

R. M. Strain and S.-B. Yun, Spatially homogenous Boltzmann equation for relativistic particles, SIAM J. Math. Anal, 46 (2014), 917-938. doi: 10.1137/130923531.

[36]

R. M. Strain and K. Zhu, Large-time decay of the soft potential relativistic Boltzmann equation in $\mathbb{R}^3_x$, Kinet. Relat. Models, 5 (2012), 383-415. doi: 10.3934/krm.2012.5.383.

[37]

H. Struchtrup, Projected moments in relativistic kinetic theory, Physica A, 253 (1998), 555-593. doi: 10.1016/S0378-4371(98)00037-5.

[38]

M. Takamoto and S. -I. Inutsuka, The relativistic kinetic dispersion relation: Comparison of the relativistic Bhatnagar-Gross-Krook model and Grad's 14-moment expansion, Physica A, 389 (2010), 4580-4603. doi: 10.1016/j.physa.2010.06.021.

[39]

R. D. Tenreiro and R. Hakim, Transport properties of the relativistic degenerate electron gas in a strong magnetic field: Covariant relaxation-time model, Phys. Rev. D, 15 (1977), 1435-1447. doi: 10.1103/PhysRevD.15.1435.

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