October  2019, 12(5): 945-967. doi: 10.3934/krm.2019036

Macroscopic regularity for the relativistic Boltzmann equation with initial singularities

1. 

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

2. 

College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029, China

* Corresponding author: Weiyuan Zou*

Received  January 2018 Revised  November 2018 Published  July 2019

Fund Project: The second author is supported by the Fundamental Research Funds for the Central Universities ZY1937.

In this paper, it is proved that the macroscopic parts of the relativistic Boltzmann equation will be continuous, even though the macroscopic components are discontinuity initially. The Lorentz transformation plays an important role to prove the continuity of nonlinear term.

Citation: Yan Yong, Weiyuan Zou. Macroscopic regularity for the relativistic Boltzmann equation with initial singularities. Kinetic and Related Models, 2019, 12 (5) : 945-967. doi: 10.3934/krm.2019036
References:
[1]

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

[2]

L. Boudin and L. Desvillettes, On the singularities of the global small solution soft the full Boltzmann equation, Monatsch. Math., 131 (2000), 91-108.  doi: 10.1007/s006050070015.

[3]

R. J. Duan, M. R. Li and T. Yang, Propagation of singularities in the solutions to the Boltzmann equation near equilibrium, Math. Models Methods Appl. Sci., 18 (2008), 1093-1114. doi: 10.1142/S0218202508002966.

[4]

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.

[5]

M. Dudyński and M. L. Ekiel-Jeżewska, Causality of the linearized relativistic Boltzmann equation, Phys. Rev. Lett., 55 (1985), 2831-2834.  doi: 10.1103/PhysRevLett.55.2831.

[6]

M. Dudyński and M. L. Ekiel-Jeżewska, Errata: Causality of the linearized relativistic Boltzmann equation, Investigación Oper. 6 (1985), 2228.

[7]

M. Dudyński and M. L. Ekiel-Jeżewska, On the linearized relativistic Boltzmann equation. Ⅰ. Existence of solutions, Commun. Math. Phys., 115 (1988), 607-629.  doi: 10.1007/BF01224130.

[8]

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

[9]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics, Philadelphia, 1996. doi: 10.1137/1.9781611971477.

[10]

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

[11]

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.

[12]

R. T. Glassey, Global solutions to the Cauchy problem for the relativistic Boltzmann equation with near-vacuum data, Commun. Math. Phys., 264 (2006), 705-724.  doi: 10.1007/s00220-006-1522-y.

[13]

F. GolseP. L. LionsB. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal., 76 (1988), 110-125.  doi: 10.1016/0022-1236(88)90051-1.

[14]

L. Hsiao and H. Yu, Asymptotic stability of the relativistic Maxwellian, Math. Meth. Appl. Sci., 29 (2006), 1481-1499.  doi: 10.1002/mma.736.

[15]

F. M. Huang and Y. Wang, Macroscopic regularity for the Boltzmann equation, Acta Math. Sci. Ser. B Engl. Ed., 38 (2018), 1549-1566.  doi: 10.1016/S0252-9602(18)30831-2.

[16]

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

[17]

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.

[18]

R. M. Strain, An Energy Method in Collisional Kinetic Theory, Ph.D. dissertation, Division of Applied Mathematics, Brown University, May 2005.

[19]

Y. Wang, Global well-posedness of the relativistic Boltzmann equation, SIAM J. Math. Anal., 50 (2018), 5637-5694.  doi: 10.1137/17M112600X.

show all references

References:
[1]

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

[2]

L. Boudin and L. Desvillettes, On the singularities of the global small solution soft the full Boltzmann equation, Monatsch. Math., 131 (2000), 91-108.  doi: 10.1007/s006050070015.

[3]

R. J. Duan, M. R. Li and T. Yang, Propagation of singularities in the solutions to the Boltzmann equation near equilibrium, Math. Models Methods Appl. Sci., 18 (2008), 1093-1114. doi: 10.1142/S0218202508002966.

[4]

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.

[5]

M. Dudyński and M. L. Ekiel-Jeżewska, Causality of the linearized relativistic Boltzmann equation, Phys. Rev. Lett., 55 (1985), 2831-2834.  doi: 10.1103/PhysRevLett.55.2831.

[6]

M. Dudyński and M. L. Ekiel-Jeżewska, Errata: Causality of the linearized relativistic Boltzmann equation, Investigación Oper. 6 (1985), 2228.

[7]

M. Dudyński and M. L. Ekiel-Jeżewska, On the linearized relativistic Boltzmann equation. Ⅰ. Existence of solutions, Commun. Math. Phys., 115 (1988), 607-629.  doi: 10.1007/BF01224130.

[8]

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

[9]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics, Philadelphia, 1996. doi: 10.1137/1.9781611971477.

[10]

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

[11]

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.

[12]

R. T. Glassey, Global solutions to the Cauchy problem for the relativistic Boltzmann equation with near-vacuum data, Commun. Math. Phys., 264 (2006), 705-724.  doi: 10.1007/s00220-006-1522-y.

[13]

F. GolseP. L. LionsB. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal., 76 (1988), 110-125.  doi: 10.1016/0022-1236(88)90051-1.

[14]

L. Hsiao and H. Yu, Asymptotic stability of the relativistic Maxwellian, Math. Meth. Appl. Sci., 29 (2006), 1481-1499.  doi: 10.1002/mma.736.

[15]

F. M. Huang and Y. Wang, Macroscopic regularity for the Boltzmann equation, Acta Math. Sci. Ser. B Engl. Ed., 38 (2018), 1549-1566.  doi: 10.1016/S0252-9602(18)30831-2.

[16]

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

[17]

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.

[18]

R. M. Strain, An Energy Method in Collisional Kinetic Theory, Ph.D. dissertation, Division of Applied Mathematics, Brown University, May 2005.

[19]

Y. Wang, Global well-posedness of the relativistic Boltzmann equation, SIAM J. Math. Anal., 50 (2018), 5637-5694.  doi: 10.1137/17M112600X.

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