Skip to main content
SpringerLink
Log in

From the Boltzmann Description for Mixtures to the Maxwell–Stefan Diffusion Equations

  • Conference paper
  • First Online:
From Particle Systems to Partial Differential Equations (ICPS 2019, ICPS 2018, ICPS 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 352))

  • 725 Accesses

Abstract

This article reviews some recent results on the diffusion limit of the Boltzmann system for gaseous mixtures to the Maxwell–Stefan diffusion equations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. B. Anwasia, The Maxwell-Stefan diffusion limit of a hard-sphere kinetic model for mixtures, From Particle Systems to Partial Differential Equations (Springer, 2021), pp. 15–36

    Google Scholar 

  2. B. Anwasia, M. Bisi, F. Salvarani, A.J. Soares, On the Maxwell-Stefan diffusion limit for a reactive mixture of polyatomic gases in non-isothermal setting. Kineti. Relat. Models 13, 63 (2020)

    Google Scholar 

  3. B. Anwasia, P. Gonçalves, A.J. Soares, On the formal derivation of the reactive Maxwell-Stefan equations from the kinetic theory. EPL (Europhys. Lett.) 129(4), 40005 (2020)

    Article  Google Scholar 

  4. B. Anwasia, P. Gonçalves, A.J. Soares, From the simple reacting sphere kinetic model to the reaction-diffusion system of Maxwell-Stefan type. Commun. Math. Sci. 17(2), 507–538 (2019)

    Google Scholar 

  5. L. Boltzmann, Weitere Studien über das Wärmegleichgewicht unter Gasmoläkulen. Sitzungsberichte der Akademie der Wissenschaften 66, 275–370 (1872)

    MATH  Google Scholar 

  6. A. Bondesan, L. Boudin, B. Grec, A numerical scheme for a kinetic model for mixtures in the diffusive limit using the moment method. Numer. Methods Partial Differ. Equs. 35(3), 1184–1205 (2019)

    Google Scholar 

  7. A. Bondesan, M. Briant, Perturbative Cauchy theory for a flux-incompressible Maxwell-Stefan system in a non-equimolar regime. arXiv:1910.03279

  8. A. Bondesan, M. Briant, Stability of the Maxwell-Stefan system in the diffusion asymptotics of the Boltzmann multi-species equation. arXiv:1910.08357

  9. C. Borgnakke, P.S. Larsen, Statistical collision model for Monte Carlo simulation of polyatomic gas mixture. J. Comput. Phys. 18(4), 405–420 (1975)

    Article  Google Scholar 

  10. D. Bothe, On the Maxwell-Stefan approach to multicomponent diffusion, in Parabolic Problems, Volume 80 of Progress Nonlinear Differential Equations Application (Birkhäuser/Springer Basel AG, Basel, 2011), pp. 81–93

    Google Scholar 

  11. L. Boudin, B. Grec, V. Pavan, The Maxwell-Stefan diffusion limit for a kinetic model of mixtures with general cross sections. Nonlinear Anal. 159, 40–61 (2017)

    Article  MathSciNet  Google Scholar 

  12. L. Boudin, B. Grec, F. Salvarani, A mathematical and numerical analysis of the Maxwell-Stefan diffusion equations. Discret. Contin. Dyn. Syst. Ser. B 17(5), 1427–1440 (2012)

    MathSciNet  MATH  Google Scholar 

  13. L. Boudin, B. Grec, F. Salvarani, The Maxwell-Stefan diffusion limit for a kinetic model of mixtures. Acta Appl. Math. 136, 79–90 (2015)

    Article  MathSciNet  Google Scholar 

  14. J.-F. Bourgat, L. Desvillettes, P. Le Tallec, B. Perthame, Microreversible collisions for polyatomic gases and Boltzmann’s theorem. European J. Mech. B Fluids 13(2), 237–254 (1994)

    MathSciNet  MATH  Google Scholar 

  15. M. Briant, B. Grec, Rigorous derivation of the Fick cross-diffusion system from the multi-species Boltzmann equation in the diffusive scaling. arXiv:2003.07891

  16. L. Desvillettes, R. Monaco, F. Salvarani, A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions. Eur. J. Mech. B Fluids 24(2), 219–236 (2005)

    Article  MathSciNet  Google Scholar 

  17. A. Ern, V. Giovangigli, Multicomponent Transport Algorithms, vol. 24. Lecture Notes in Physics Monographs New Series M (Springer, Berlin, 1994)

    Google Scholar 

  18. V. Giovangigli, Multicomponent Flow Modeling. Modeling and Simulation in Science, Engineering and Technology (Birkhäuser Boston Inc., Boston, MA, 1999)

    Google Scholar 

  19. H. Grad, Principles of the kinetic theory of gases, in Handbuch der Physik (herausgegeben von S. Flügge), Bd. 12, Thermodynamik der Gase (Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958), pp. 205–294

    Google Scholar 

  20. H. Grad, Asymptotic theory of the Boltzmann equation. II, in Rarefied Gas Dynamics (Proceedings 3rd International Symposium, Palais de l’UNESCO, Paris, 1962), Vol. I, pages 26–59. Academic Press, New York, 1963

    Google Scholar 

  21. M. Groppi, G. Spiga, Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas. J. Math. Chem. 26(1–3), 197–219 (1999)

    Article  Google Scholar 

  22. H. Hutridurga, F. Salvarani, Maxwell-Stefan diffusion asymptotics for gas mixtures in non-isothermal setting. Nonlinear Anal. 159, 285–297 (2017)

    Article  MathSciNet  Google Scholar 

  23. H. Hutridurga, F. Salvarani, On the Maxwell-Stefan diffusion limit for a mixture of monatomic gases. Math. Methods Appl. Sci. 40(3), 803–813 (2017)

    Article  MathSciNet  Google Scholar 

  24. M.T. Marron, Simple collision theory of reactive hard spheres. J. Chem. Phys 52(8), 4060–4061 (1970)

    Article  Google Scholar 

  25. J.C. Maxwell, On the dynamical theory of gases. Philos. Trans. R. Soc. 157, 49–88 (1866)

    Google Scholar 

  26. T.F. Morse, Kinetic model equations for a gas mixture. Phys. Fluids 7, 908–918 (1964)

    Article  MathSciNet  Google Scholar 

  27. F. Salvarani, A.J. Soares, On the relaxation of the Maxwell-Stefan system to linear diffusion. Appl. Math. Lett. 85, 15–21 (2018)

    Article  MathSciNet  Google Scholar 

  28. L. Sirovich, Kinetic modeling of gas mixtures. Phys. Fluids 5(8), 908–918 (1962)

    Article  MathSciNet  Google Scholar 

  29. J. Stefan, Über das Gleichgewicht und die Bewegung insbesondere die Diffusion von Gasgemengen. Akad. Wiss. Wien 63, 63–124 (1871)

    Google Scholar 

  30. R. Taylor, R. Krishna, Multicomponent Mass Transfer, vol. 2 (Wiley, New York, 1993)

    Google Scholar 

  31. C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, vol. I (North-Holland, Amsterdam, 2002), pp. 71–305

    Google Scholar 

Download references

Acknowledgements

Work supported by the ANR projects Kimega (ANR-14-ACHN-0030-01) and by the Italian Ministry of Education, University and Research (Dipartimenti di Eccellenza program 2018–2022, Dipartimento di Matematica ’F. Casorati’, Università degli Studi di Pavia). The author thanks the organizers of the series of conferences PSPDE for their generous hospitality and support. He moreover thanks the anonymous referees for their useful comments.

Author information

Authors and Affiliations

  1. Léonard de Vinci Pôle Universitaire, Research Center, Paris La Défense, 92916, France

    Francesco Salvarani

  2. Dipartimento di Matematica, Università degli Studi di Pavia, Pavia, 27100, Italy

    Francesco Salvarani

Authors
  1. Francesco Salvarani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francesco Salvarani .

Editor information

Editors and Affiliations

  1. Laboratoire J.A. Dieudonné, UMR CNRS-UNS, Université Côte d’Azur, Nice Cedex 02, France

    Cédric Bernardin

  2. Centre de Math. Laurent Schwartz (CMLS), École Polytechnique, CMLS, Palaiseau Cedex, France

    François Golse

  3. Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal

    Patrícia Gonçalves

  4. Dipartimento di Matematica e Informatica, Università di Palermo, Palermo, Italy

    Valeria Ricci

  5. Centro de Matemática, Universidade do Minho, Braga, Portugal

    Ana Jacinta Soares

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Salvarani, F. (2021). From the Boltzmann Description for Mixtures to the Maxwell–Stefan Diffusion Equations. In: Bernardin, C., Golse, F., Gonçalves, P., Ricci, V., Soares, A.J. (eds) From Particle Systems to Partial Differential Equations. ICPS ICPS ICPS 2019 2018 2017. Springer Proceedings in Mathematics & Statistics, vol 352. Springer, Cham. https://doi.org/10.1007/978-3-030-69784-6_17

Download citation

Publish with us

Policies and ethics

Search

Navigation