Abstract and Applied Analysis

On the Fourier-Transformed Boltzmann Equation with Brownian Motion

Yong-Kum Cho and Eunsil Kim

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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.

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

First available in Project Euclid: 15 June 2015

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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

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  • 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