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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

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 & 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.  Google Scholar

[2]

L. Arkeryd, On the Boltzmann equation, Arch. Rational Mech. Anal., 45 (1972), 1-34. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

S. Bochner and K. Chandrasekharan, Fourier Transforms, Princeton University Press, Princeton, 1949.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. DiPerna and P.-L. Lions, On the Fokker-Planck-Boltzmann equation, Comm. Math. Phys., 120 (1988), 1-23.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

R. Laha and V. Rohatgi, Probability Theory, John Wiley & Sons, New York, 1979.  Google Scholar

[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.  Google Scholar

[18]

Y. Morimoto, S. Wang and T. Yang, A new characterization and global regularity of infinite energy solutions to the homogeneous Boltzmann equation, preprint,, , ().   Google Scholar

[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, ().   Google Scholar

[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.  Google Scholar

[21]

I. Schoenberg, Metric spaces and positive definite functions, Trans. Amer. Math. Soc., 44 (1938), 522-536. doi: 10.2307/1989894.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

L. Arkeryd, On the Boltzmann equation, Arch. Rational Mech. Anal., 45 (1972), 1-34. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

S. Bochner and K. Chandrasekharan, Fourier Transforms, Princeton University Press, Princeton, 1949.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. DiPerna and P.-L. Lions, On the Fokker-Planck-Boltzmann equation, Comm. Math. Phys., 120 (1988), 1-23.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

R. Laha and V. Rohatgi, Probability Theory, John Wiley & Sons, New York, 1979.  Google Scholar

[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.  Google Scholar

[18]

Y. Morimoto, S. Wang and T. Yang, A new characterization and global regularity of infinite energy solutions to the homogeneous Boltzmann equation, preprint,, , ().   Google Scholar

[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, ().   Google Scholar

[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.  Google Scholar

[21]

I. Schoenberg, Metric spaces and positive definite functions, Trans. Amer. Math. Soc., 44 (1938), 522-536. doi: 10.2307/1989894.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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