Research article Special Issues

On the particle approximation to stationary solutions of the Boltzmann equation

  • Received: 26 January 2019 Accepted: 24 July 2019 Published: 14 August 2019
  • We discuss the problem of the approximation of the solutions of the stationary Boltzmann equation, driven by diffuse boundary conditions at varying temperature, by means of stochastic particle systems. In particular we extend a previous results, by substituting the hypothesis of a cutoff on small and large velocities with the presence of an external field.

    Citation: Mario Pulvirenti. On the particle approximation to stationary solutions of the Boltzmann equation[J]. Mathematics in Engineering, 2019, 1(4): 699-714. doi: 10.3934/mine.2019.4.699

    Related Papers:

  • We discuss the problem of the approximation of the solutions of the stationary Boltzmann equation, driven by diffuse boundary conditions at varying temperature, by means of stochastic particle systems. In particular we extend a previous results, by substituting the hypothesis of a cutoff on small and large velocities with the presence of an external field.


    加载中


    [1] Aoki K, Golse F (2011) On the speed of approach to equilibrium for a collisionless gas. Kinet Relat Mod 4: 87–107. doi: 10.3934/krm.2011.4.87
    [2] Bird GA (1994) Molecular Gas Dynamics and the Direct Simulation of Gas Flows, 2Eds., Oxford University Press.
    [3] Bodineau T, Gallagher I, Saint-Raymond L (2016) The Brownian motion as the limit of a deterministic system of hard-spheres. Invent Math 203: 493–553. doi: 10.1007/s00222-015-0593-9
    [4] Bodineau T, Gallagher I, Saint-Raymond L (2017) From hard sphere dynamics to the Stokes-Fourier equations: An L2 analysis of the Boltzmann-Grad limit. arXiv:1511.03057.
    [5] Bodineau T, Gallagher I, Saint-Raymond L, et al. One-sided convergence in the Boltzmann-Grad limit. Ann Fac Sci Toulouse Math (to appear).
    [6] Caprino S, De Masi A, Presutti E, et al. (1991) A derivation of the Broadwell equation. Commun Math Phys 135: 443–465. doi: 10.1007/BF02104115
    [7] Caprino S, Pulvirenti M (1996) The Boltzmann-Grad limit for a one-dimensional Boltzmann equation in a stationary state. Commun Math Phys 177: 63–81. doi: 10.1007/BF02102430
    [8] Cercignani C, Illner R, Pulvirenti M (1994) The Mathematical Theory of Dilute Gases, New York: Springer-Verlag.
    [9] Caprino S, Pulvirenti M, Wagner W (1998) A particle systems approximating stationary solutions to the Boltzmann equation. SIAM J Math Anal 4: 913–934.
    [10] Cercignani C (1983) The Grad limit for a system of soft spheres. Commun Pure Appl Math 36: 479–494. doi: 10.1002/cpa.3160360406
    [11] Esposito1 R, Guo Y, Kim C, et al. (2013) Non-isothermal boundary in the Boltzmann theory and Fourier law. Commun Math Phys 323: 177–239. doi: 10.1007/s00220-013-1766-2
    [12] Gallagher I, Saint-Raymond L, Texier B (2014) From Newton to Boltzmann: Hard Spheres and Short-Range Potentials, Series of Zurich Lectures in Advanced Mathematics, Vol 18, European Mathematical Society, and erratum to Chapter 5.
    [13] Goldstain S, Lebowitz J, Presutti E (1979) Mechanical system with stochastic boundaries. Colloquia Mathemtica Societatis Janos Bolyai 27. Random Fields, Eszergom (Hungary).
    [14] Kuo HW, Liu TP, Tsai LC (2014) Equilibrating effects of boundary and collision in rarefied gases. Commun Math Phys 328: 421–480. doi: 10.1007/s00220-014-2042-9
    [15] Illner R, Pulvirenti M (1986) Global validity of the Boltzmann equation for a two-dimensional rare gas in the vacuum. Commun Math Phys 105: 189–203. doi: 10.1007/BF01211098
    [16] Illner R, Pulvirenti M (1989) Global validity of the Boltzmann equation for a two- and three-dimensional rare gas in vacuum: Erratum and improved result. Commun Math Phys 121: 143–146. doi: 10.1007/BF01218628
    [17] Kac M (1956) Foundations of kinetic theory, In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley and Los Angeles.
    [18] Kac M (1959) Probability and Related Topics in Physical Sciences, London-New York: Interscience Publishers.
    [19] Lods B, Mokhtar-Kharroubi M, Rudnicki R (2018) Invariant density and time asymptotic for collisionless kinetic equations with partially diffuse boundary operators. arXiv:1812.05397v1 [math.AP].
    [20] Lachowicz M , Pulvirenti M (1990) A stochastic system of particles modelling the Euler equation. Arch Ration Mech Anal 109: 81–93. doi: 10.1007/BF00377981
    [21] Lanford OE (1975) Time evolution of large classical systems, In: Moser, J. Editor, Dynamical Systems, Theory and Applications, Berlin: Springer-Verlag, 1–111.
    [22] Paul T, Pulvirenti M, Simonella S (2019) On the size of chaos in the mean field dynamics. Arch Rat Mech Anal 231: 285–317. doi: 10.1007/s00205-018-1280-y
    [23] Pulvirenti M, Saffirio C, Simonella S (2014) On the validity of the Boltzmann equation for short-range potentials. Rev Math Phys 26: Article ID 1450001.
    [24] Pulvirenti M, Simonella S (2017) The Boltzmann-Grad limit of a hard sphere system: Analysis of the correlation error. Invent math 207: 1135–1237. doi: 10.1007/s00222-016-0682-4
    [25] Pulvirenti M, Wagner W, Zavelani Rossi MB (1994) Convergence of particle schemes for the Boltzmann equation. Eur J Mech B/Fluids 13: 339–351.
    [26] Rjasanow S, Wagner W (2005) Stochastic Numerics for the Boltzmann Equation, Springer Series in Computational Mathematics, Vol 37, Springer, Berlin, Heidelberg.
    [27] Villani C (2002) A review of mathematical topics in collisional kinetic theory, In: Hand-book of mathematical fluid dynamics, Elsevier Science, Vol I, 71–305.
  • Reader Comments
  • © 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2903) PDF downloads(491) Cited by(0)

Article outline

Figures and Tables

Figures(2)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog