Abstract and Applied Analysis

On the Fourier-Transformed Boltzmann Equation with Brownian Motion

Yong-Kum Cho and Eunsil Kim

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We establish a global existence theorem, and uniqueness and stability of solutions of the Cauchy problem for the Fourier-transformed Fokker-Planck-Boltzmann equation with singular Maxwellian kernel, which may be viewed as a kinetic model for the stochastic time-evolution of characteristic functions governed by Brownian motion and collision dynamics.

Article information

Source
Abstr. Appl. Anal., Volume 2015, Special Issue (2015), Article ID 318618, 9 pages.

Dates
First available in Project Euclid: 15 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1434398646

Digital Object Identifier
doi:10.1155/2015/318618

Mathematical Reviews number (MathSciNet)
MR3339665

Zentralblatt MATH identifier
1356.35153

Citation

Cho, Yong-Kum; Kim, Eunsil. On the Fourier-Transformed Boltzmann Equation with Brownian Motion. Abstr. Appl. Anal. 2015, Special Issue (2015), Article ID 318618, 9 pages. doi:10.1155/2015/318618. https://projecteuclid.org/euclid.aaa/1434398646


Export citation

References

  • C. Villani, “A review of mathematical topics in collisional kinetic theory,” in Handbook of Mathematical Fluid Dynamics, vol. 1, pp. 71–305, North-Holland, Amsterdam, The Netherlands, 2002.
  • K. Hamdache, “Estimations uniformes des solutions de l'equation de Boltzmann par les methodes de viscosité artificielle et de diffusion de Fokker-Planck,” Comptes Rendus de l'Académie des Sciences, vol. 302, no. 5, pp. 187–190, 1986.
  • R. J. DiPerna and P.-L. Lions, “On the Fokker-Planck-Boltzmann equation,” Communications in Mathematical Physics, vol. 120, no. 1, pp. 1–23, 1988.
  • H.-L. Li and A. Matsumura, “Behaviour of the Fokker-Planck-Boltzmann equation near a Maxwellian,” Archive for Rational Mechanics and Analysis, vol. 189, no. 1, pp. 1–44, 2008.
  • L. Xiong, T. Wang, and L. Wang, “Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation,” Kinetic and Related Models, vol. 7, no. 1, pp. 169–194, 2014.
  • M.-Y. Zhong and H.-L. Li, “Long time behavior of the Fokker-Planck-Boltzmann equation with soft potential,” Quarterly of Applied Mathematics, vol. 70, no. 4, pp. 721–742, 2012.
  • H.-L. Li, “Diffusive property of the Fokker-Planck-Boltzmann equation,” Bulletin of the Institute of Mathematics, vol. 2, no. 4, pp. 921–933, 2007.
  • T. Goudon, “On Boltzmann equations and Fokker-Planck asymptotics: influence of grazing collisions,” Journal of Statistical Physics, vol. 89, no. 3-4, pp. 751–776, 1997.
  • L. Arkeryd, “On the Boltzmann equation. I. Existence,” Archive for Rational Mechanics and Analysis, vol. 45, pp. 1–16, 1972.
  • L. Arkeryd, “Intermolecular forces of infinite range and the Boltzmann equation,” Archive for Rational Mechanics and Analysis, vol. 77, no. 1, pp. 11–21, 1981.
  • C. Villani, “On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations,” Archive for Rational Mechanics and Analysis, vol. 143, no. 3, pp. 273–307, 1998.
  • A. V. Bobylev, “Fourier transform method in the theory of the Boltzmann equation for Maxwell molecules,” Doklady Akademii Nauk SSSR, vol. 225, pp. 1041–1044, 1975.
  • R. M. Blumenthal and R. K. Getoor, “Some theorems on stable processes,” Transactions of the American Mathematical Society, vol. 95, pp. 263–273, 1960.
  • M. Bisi, J. A. Carrillo, and G. Toscani, “Contractive metrics for a Boltzmann equation for granular gases: diffusive equilibria,” Journal of Statistical Physics, vol. 118, no. 1-2, pp. 301–331, 2005.
  • A. Pulvirenti and G. Toscani, “The theory of the nonlinear Boltzmann equation for Maxwell molecules in Fourier representation,” Annali di Matematica Pura ed Applicata, vol. 171, pp. 181–204, 1996.
  • G. Toscani and C. Villani, “Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas,” Journal of Statistical Physics, vol. 94, no. 3-4, pp. 619–637, 1999.
  • J. A. Carrillo and G. Toscani, “Contractive probability metrics and asymptotic behavior of dissipative kinetic equations,” Rivista di Matematica della Università di Parma, vol. 6, pp. 75–198, 2007.
  • A. V. Bobylev and C. Cercignani, “Self-similar solutions of the Boltzmann equation and their applications,” Journal of Statistical Physics, vol. 106, no. 5-6, pp. 1039–1071, 2002.
  • M. Cannone and G. Karch, “Infinite energy solutions to the homogeneous Boltzmann equation,” Communications on Pure and Applied Mathematics, vol. 63, no. 6, pp. 747–778, 2010.
  • Y. Morimoto, “A remark on Cannone-Karch solutions to the homogeneous Boltzmann equation for Maxwellian molecules,” Kinetic and Related Models, vol. 5, no. 3, pp. 551–561, 2012.
  • B. Petersen, Introduction to the Fourier Transform & Pseudo-Differential Operators, Pitman, 1983.
  • Y.-K. Cho, “On the Boltzmann equation with the symmetric stable Lévi processčommentComment on ref. [9?]: Please update the information of this reference, if possible.,” To appear in Kinetic and Related Models.
  • E. Wild, “On Boltzmann's equation in the kinetic theory of gases,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 47, pp. 602–609, 1951. \endinput