American Institute of Mathematical Sciences

Advanced
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.  doi: 10.1142/S0218202505000613.  Google Scholar

[2]

L. Arkeryd, On the Boltzmann equation,, Arch. Rational Mech. Anal., 45 (1972), 1.   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.  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.  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.   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.  doi: 10.1023/A:1014037804043.  Google Scholar

[7]

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

[8]

M. Cannone and G. Karch, Infinite energy solutions to the homogeneous Boltzmann equation,, Comm. Pure Appl. Math., 63 (2010), 747.  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.  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.   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.  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.   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.  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.  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.   Google Scholar

[16]

R. Laha and V. Rohatgi, Probability Theory,, John Wiley & Sons, (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.  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.  doi: 10.1007/BF01759387.  Google Scholar

[21]

I. Schoenberg, Metric spaces and positive definite functions,, Trans. Amer. Math. Soc., 44 (1938), 522.  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.  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.  doi: 10.1007/s002050050106.  Google Scholar

[24]

C. Villani, A review of mathematical topics in collisional kinetic theory,, in Handbook of Mathematical Fluid Dynamics, I (2002), 71.  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.  doi: 10.1142/S0218202505000613.  Google Scholar

[2]

L. Arkeryd, On the Boltzmann equation,, Arch. Rational Mech. Anal., 45 (1972), 1.   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.  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.  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.   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.  doi: 10.1023/A:1014037804043.  Google Scholar

[7]

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

[8]

M. Cannone and G. Karch, Infinite energy solutions to the homogeneous Boltzmann equation,, Comm. Pure Appl. Math., 63 (2010), 747.  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.  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.   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.  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.   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.  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.  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.   Google Scholar

[16]

R. Laha and V. Rohatgi, Probability Theory,, John Wiley & Sons, (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.  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.  doi: 10.1007/BF01759387.  Google Scholar

[21]

I. Schoenberg, Metric spaces and positive definite functions,, Trans. Amer. Math. Soc., 44 (1938), 522.  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.  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.  doi: 10.1007/s002050050106.  Google Scholar

[24]

C. Villani, A review of mathematical topics in collisional kinetic theory,, in Handbook of Mathematical Fluid Dynamics, I (2002), 71.  doi: 10.1016/S1874-5792(02)80004-0.  Google Scholar

[1]

Yongxia Zhao, Rongming Wang, Chuancun Yin. Optimal dividends and capital injections for a spectrally positive Lévy process. Journal of Industrial & Management Optimization, 2017, 13 (1) : 1-21. doi: 10.3934/jimo.2016001
[2]

Hongjun Gao, Fei Liang. On the stochastic beam equation driven by a Non-Gaussian Lévy process. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1027-1045. doi: 10.3934/dcdsb.2014.19.1027
[3]

Alexander Alekseenko, Jeffrey Limbacher. Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in $ \mathcal{O}(N^2) $ operations using the discrete fourier transform. Kinetic & Related Models, 2019, 12 (4) : 703-726. doi: 10.3934/krm.2019027
[4]

Yong-Kum Cho. A quadratic Fourier representation of the Boltzmann collision operator with an application to the stability problem. Kinetic & Related Models, 2012, 5 (3) : 441-458. doi: 10.3934/krm.2012.5.441
[5]

Léo Glangetas, Hao-Guang Li, Chao-Jiang Xu. Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation. Kinetic & Related Models, 2016, 9 (2) : 299-371. doi: 10.3934/krm.2016.9.299
[6]

Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Uniqueness of solutions for the non-cutoff Boltzmann equation with soft potential. Kinetic & Related Models, 2011, 4 (4) : 919-934. doi: 10.3934/krm.2011.4.919
[7]

Zhaohui Huo, Yoshinori Morimoto, Seiji Ukai, Tong Yang. Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff. Kinetic & Related Models, 2008, 1 (3) : 453-489. doi: 10.3934/krm.2008.1.453
[8]

Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 187-212. doi: 10.3934/dcds.2009.24.187
[9]

Lingbing He, Yulong Zhou. High order approximation for the Boltzmann equation without angular cutoff. Kinetic & Related Models, 2018, 11 (3) : 547-596. doi: 10.3934/krm.2018024
[10]

Ralf Kirsch, Sergej Rjasanow. The uniformly heated inelastic Boltzmann equation in Fourier space. Kinetic & Related Models, 2010, 3 (3) : 445-456. doi: 10.3934/krm.2010.3.445
[11]

Thomas Chen, Ryan Denlinger, Nataša Pavlović. Moments and regularity for a Boltzmann equation via Wigner transform. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 4979-5015. doi: 10.3934/dcds.2019204
[12]

Yong-Kum Cho. On the homogeneous Boltzmann equation with soft-potential collision kernels. Kinetic & Related Models, 2015, 8 (2) : 309-333. doi: 10.3934/krm.2015.8.309
[13]

Wen Chen, Song Wang. A finite difference method for pricing European and American options under a geometric Lévy process. Journal of Industrial & Management Optimization, 2015, 11 (1) : 241-264. doi: 10.3934/jimo.2015.11.241
[14]

Yong-Kum Cho, Hera Yun. On the gain of regularity for the positive part of Boltzmann collision operator associated with soft-potentials. Kinetic & Related Models, 2012, 5 (4) : 769-786. doi: 10.3934/krm.2012.5.769
[15]

Jean-Marie Barbaroux, Dirk Hundertmark, Tobias Ried, Semjon Vugalter. Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction. Kinetic & Related Models, 2017, 10 (4) : 901-924. doi: 10.3934/krm.2017036
[16]

Nicolas Fournier. A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff. Kinetic & Related Models, 2019, 12 (3) : 483-505. doi: 10.3934/krm.2019020
[17]

Kevin Zumbrun. L∞ resolvent bounds for steady Boltzmann's Equation. Kinetic & Related Models, 2017, 10 (4) : 1255-1257. doi: 10.3934/krm.2017048
[18]

Jiangyan Peng, Dingcheng Wang. Asymptotics for ruin probabilities of a non-standard renewal risk model with dependence structures and exponential Lévy process investment returns. Journal of Industrial & Management Optimization, 2017, 13 (1) : 155-185. doi: 10.3934/jimo.2016010
[19]

Esther S. Daus, Shi Jin, Liu Liu. Spectral convergence of the stochastic galerkin approximation to the boltzmann equation with multiple scales and large random perturbation in the collision kernel. Kinetic & Related Models, 2019, 12 (4) : 909-922. doi: 10.3934/krm.2019034
[20]

Sihem Guerarra. Positive and negative definite submatrices in an Hermitian least rank solution of the matrix equation AXA*=B. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 15-22. doi: 10.3934/naco.2019002

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences