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On the Boltzmann equation with the symmetric stable Lévy process
1. | Department of Mathematics, College of Natural Sciences, Chung-Ang University, 84 Heukseok-Ro, Dongjak-Gu, Seoul 156-756 |
References:
[1] |
R. Alexandre and M. El Safadi, Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations, I. Non-cutoff case and Maxwellian molecules, Math. Models Methods Appl. Sci., 15 (2005), 907-920.
doi: 10.1142/S0218202505000613. |
[2] |
L. Arkeryd, On the Boltzmann equation, Arch. Rational Mech. Anal., 45 (1972), 1-34. |
[3] |
M. Bisi, J. A. Carrillo and G. Toscani, Contractive metrics for a Boltzmann equation for granular gases: Diffusive equilibria, J. Stat. Phys., 118 (2005), 301-331.
doi: 10.1007/s10955-004-8785-5. |
[4] |
R. Blumenthal and R. Getoor, Some theorems on stable processes, Trans. Amer. Math. Soc., 95 (1960), 263-273.
doi: 10.1090/S0002-9947-1960-0119247-6. |
[5] |
A. V. Bobylev, Fourier transform method in the theory of the Boltzmann equation for Maxwell molecules, Dokl. Akad. Nauk SSSR, 225 (1975), 1041-1044. |
[6] |
A. V. Bobylev and C. Cercignani, Self-silimiar solutions of the Boltzmann equation and their applications, J. Stat. Phys., 106 (2002), 1039-1071.
doi: 10.1023/A:1014037804043. |
[7] |
S. Bochner and K. Chandrasekharan, Fourier Transforms, Princeton University Press, Princeton, 1949. |
[8] |
M. Cannone and G. Karch, Infinite energy solutions to the homogeneous Boltzmann equation, Comm. Pure Appl. Math., 63 (2010), 747-778.
doi: 10.1002/cpa.20298. |
[9] |
M. Cannone and G. Karch, On self-similar solutions to the homogeneous Boltzmann equation, Kinetic and Related Models, 6 (2013), 801-808.
doi: 10.3934/krm.2013.6.801. |
[10] |
J. A. Carrillo and G. Toscani, Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Univ. Parma, 6 (2007), 75-198. |
[11] |
Y.-K. Cho, A quadratic Fourier representation of the Boltzmann collision operator with an application to the stability problem, Kinetic and Related Models, 5 (2012), 441-458.
doi: 10.3934/krm.2012.5.441. |
[12] |
R. DiPerna and P.-L. Lions, On the Fokker-Planck-Boltzmann equation, Comm. Math. Phys., 120 (1988), 1-23. |
[13] |
I. Gamba, V. Panferov and C. Villani, On the Boltzmann equation for diffusively excited granular media, Comm. Math. Phys., 246 (2004), 503-541.
doi: 10.1007/s00220-004-1051-5. |
[14] |
T. Goudon, On Boltzmann equations and Fokker-Planck asymptotics: Influence of grazing collisions, J. Stat. Phys., 89 (1997), 752-776.
doi: 10.1007/BF02765543. |
[15] |
K. Hamdache, Estimations uniformes des solutions de l'equation de Boltzmann par les methodes de viscosité artificielle et de diffusion de Fokker-Planck, A. R. Acad. Sci. Paris, 302 (1986), 187-190. |
[16] |
R. Laha and V. Rohatgi, Probability Theory, John Wiley & Sons, New York, 1979. |
[17] |
Y. Morimoto, A remark on Cannone-Karch solutions to the homogeneous Boltzmann equation for Maxwellian molecules, Kinetic and Related Models, 5 (2012), 551-561.
doi: 10.3934/krm.2012.5.551. |
[18] |
Y. Morimoto, S. Wang and T. Yang, A new characterization and global regularity of infinite energy solutions to the homogeneous Boltzmann equation, preprint, arXiv:1407.7701. |
[19] |
Y. Morimoto and T. Yang, Smoothing effect of the homogeneous Boltzmann equation with measure valued initial datum, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press. |
[20] |
A. Pulvirenti and G. Toscani, The theory of the nonlinear Boltzmann equation for Maxwell molecules in Fourier representation, Ann. Mat. Pura Appl., 171 (1996), 181-204.
doi: 10.1007/BF01759387. |
[21] |
I. Schoenberg, Metric spaces and positive definite functions, Trans. Amer. Math. Soc., 44 (1938), 522-536.
doi: 10.2307/1989894. |
[22] |
G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas, J. Stat. Phys., 94 (1999), 619-637.
doi: 10.1023/A:1004589506756. |
[23] |
C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal., 143 (1998), 273-307.
doi: 10.1007/s002050050106. |
[24] |
C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics,, North-Holland, Amsterdam, I (2002), 71-305.
doi: 10.1016/S1874-5792(02)80004-0. |
show all references
References:
[1] |
R. Alexandre and M. El Safadi, Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations, I. Non-cutoff case and Maxwellian molecules, Math. Models Methods Appl. Sci., 15 (2005), 907-920.
doi: 10.1142/S0218202505000613. |
[2] |
L. Arkeryd, On the Boltzmann equation, Arch. Rational Mech. Anal., 45 (1972), 1-34. |
[3] |
M. Bisi, J. A. Carrillo and G. Toscani, Contractive metrics for a Boltzmann equation for granular gases: Diffusive equilibria, J. Stat. Phys., 118 (2005), 301-331.
doi: 10.1007/s10955-004-8785-5. |
[4] |
R. Blumenthal and R. Getoor, Some theorems on stable processes, Trans. Amer. Math. Soc., 95 (1960), 263-273.
doi: 10.1090/S0002-9947-1960-0119247-6. |
[5] |
A. V. Bobylev, Fourier transform method in the theory of the Boltzmann equation for Maxwell molecules, Dokl. Akad. Nauk SSSR, 225 (1975), 1041-1044. |
[6] |
A. V. Bobylev and C. Cercignani, Self-silimiar solutions of the Boltzmann equation and their applications, J. Stat. Phys., 106 (2002), 1039-1071.
doi: 10.1023/A:1014037804043. |
[7] |
S. Bochner and K. Chandrasekharan, Fourier Transforms, Princeton University Press, Princeton, 1949. |
[8] |
M. Cannone and G. Karch, Infinite energy solutions to the homogeneous Boltzmann equation, Comm. Pure Appl. Math., 63 (2010), 747-778.
doi: 10.1002/cpa.20298. |
[9] |
M. Cannone and G. Karch, On self-similar solutions to the homogeneous Boltzmann equation, Kinetic and Related Models, 6 (2013), 801-808.
doi: 10.3934/krm.2013.6.801. |
[10] |
J. A. Carrillo and G. Toscani, Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Univ. Parma, 6 (2007), 75-198. |
[11] |
Y.-K. Cho, A quadratic Fourier representation of the Boltzmann collision operator with an application to the stability problem, Kinetic and Related Models, 5 (2012), 441-458.
doi: 10.3934/krm.2012.5.441. |
[12] |
R. DiPerna and P.-L. Lions, On the Fokker-Planck-Boltzmann equation, Comm. Math. Phys., 120 (1988), 1-23. |
[13] |
I. Gamba, V. Panferov and C. Villani, On the Boltzmann equation for diffusively excited granular media, Comm. Math. Phys., 246 (2004), 503-541.
doi: 10.1007/s00220-004-1051-5. |
[14] |
T. Goudon, On Boltzmann equations and Fokker-Planck asymptotics: Influence of grazing collisions, J. Stat. Phys., 89 (1997), 752-776.
doi: 10.1007/BF02765543. |
[15] |
K. Hamdache, Estimations uniformes des solutions de l'equation de Boltzmann par les methodes de viscosité artificielle et de diffusion de Fokker-Planck, A. R. Acad. Sci. Paris, 302 (1986), 187-190. |
[16] |
R. Laha and V. Rohatgi, Probability Theory, John Wiley & Sons, New York, 1979. |
[17] |
Y. Morimoto, A remark on Cannone-Karch solutions to the homogeneous Boltzmann equation for Maxwellian molecules, Kinetic and Related Models, 5 (2012), 551-561.
doi: 10.3934/krm.2012.5.551. |
[18] |
Y. Morimoto, S. Wang and T. Yang, A new characterization and global regularity of infinite energy solutions to the homogeneous Boltzmann equation, preprint, arXiv:1407.7701. |
[19] |
Y. Morimoto and T. Yang, Smoothing effect of the homogeneous Boltzmann equation with measure valued initial datum, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press. |
[20] |
A. Pulvirenti and G. Toscani, The theory of the nonlinear Boltzmann equation for Maxwell molecules in Fourier representation, Ann. Mat. Pura Appl., 171 (1996), 181-204.
doi: 10.1007/BF01759387. |
[21] |
I. Schoenberg, Metric spaces and positive definite functions, Trans. Amer. Math. Soc., 44 (1938), 522-536.
doi: 10.2307/1989894. |
[22] |
G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas, J. Stat. Phys., 94 (1999), 619-637.
doi: 10.1023/A:1004589506756. |
[23] |
C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal., 143 (1998), 273-307.
doi: 10.1007/s002050050106. |
[24] |
C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics,, North-Holland, Amsterdam, I (2002), 71-305.
doi: 10.1016/S1874-5792(02)80004-0. |
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