On the Boltzmann equation with the symmetric stable Lévy process

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March 2015, 8(1): 53-77. doi: 10.3934/krm.2015.8.53

On the Boltzmann equation with the symmetric stable Lévy process

Yong-Kum Cho ^1,

1.	Department of Mathematics, College of Natural Sciences, Chung-Ang University, 84 Heukseok-Ro, Dongjak-Gu, Seoul 156-756

Received June 2014 Revised October 2014 Published December 2014

Abstract
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As for the spatially homogeneous Boltzmann equation of Maxwellian molecules with the fractional Fokker-Planck diffusion term, we consider the Cauchy problem for its Fourier-transformed version, which can be viewed as a kinetic model for the stochastic time-evolution of characteristic functions associated with the symmetric stable Lévy process and the Maxwellian collision dynamics. Under a non-cutoff assumption on the kernel, we establish a global existence theorem with maximum growth estimate, uniqueness and stability of solutions.

Keywords: positive definite., characteristic function, equicontinuity, Lévy process, Fourier transform, cutoff, collision, Boltzmann equation.

Mathematics Subject Classification: 35Q82, 47G20, 76P05, 82B4.

Citation: Yong-Kum Cho. On the Boltzmann equation with the symmetric stable Lévy process. Kinetic and Related Models, 2015, 8 (1) : 53-77. doi: 10.3934/krm.2015.8.53

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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