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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[19]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[19]

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

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