Skip to main content
SpringerLink
Log in

On the Fokker-Planck-Boltzmann equation

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider the Boltzmann equation perturbed by Fokker-Planck type operator. To overcome the lack of strong a priori estimates and to define a meaningful collision operator, we introduce a notion of renormalized solution which enables us to establish stability results for sequences of solutions and global existence for the Cauchy problem with large data. The proof of stability and existence combines renormalization with an analysis of a defect measure.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bènilan, Ph., Brézis, H., Crandall, M.G.: A semilinear elliptic equation inL 1(R N). Ann. Sc. Norm. Sup. Pisa2, 523–555 (1975)

    Google Scholar 

  2. Cabannes, H.: The discrete Boltzmann equation (Theory and Application). Lecture Notes, University of California, Berkeley 1980

    Google Scholar 

  3. Cercignani, C.: Theory and application of the Boltzmann equation. Edinburgh: Scottish Academic Press 1975

    Google Scholar 

  4. Chapman, S., Cowling, T. G.: The mathematical theory of non-uniform gases. Cambridge: Cambridge University Press 1952

    Google Scholar 

  5. DiPerna, R., Lions, P. L.: On the Cauchy problem for Boltzmann equations: Global existence and weak stability, preprint

  6. DiPerna, R., Lions, P. L.: work in preparation

  7. Grad, H.: Asymptotic theory of the Boltzmann equation II. In: Rarefied Gas Dynamics, Vol. 1. Laurmann J. (ed.). p. 26–59; New York: Academic Press 1963

    Google Scholar 

  8. Hamdache, K.: Sur l'existence globale et comportement asymptotique de quelques solutions de l'equation de Boltzmann. Paris: These d'Etat, Univ. P. et M. Curie 1986

    Google Scholar 

  9. Hamdache, K.: Estimations uniformes des solutions de l'equation de Boltzmann par les methodes de viscosité artificielle et de diffusion de Fokker-Planck. C. R. Acad. Sci. Paris,302, 187–190 (1986)

    Google Scholar 

  10. Hamdache, K.: La methode de viscosité artificielle pour des modeles de la theorie cinétique discréte. To appear in J. Mecan. Theor. Appl.

  11. Hörmander, L.: The analysis of partial differential operators, Vol. I. Berlin Heidelberg, New York: Springer 1983

    Google Scholar 

  12. Illner, R., Global existence results for discrete velocity models of the Boltzmann equation in several dimensions. J. Mecan. Theor. Appl.1, 611–622 (1982)

    Google Scholar 

  13. Loyalka, S. K.: Rarefied gas dynamic problems in environmental sciences. In: Proceedings 15th International Symposium on Rarefied Gas Dynamics, June 1986, Grado Italy. Boffi, V., Cercignani, C.: (eds.) Stuttgart: Teubner 1986

    Google Scholar 

  14. Tartar, L.: Some existence theorems for semilinear hyperbolic systems in one space variable. MRC Report No. 2164, Univ. of Wisconsin-Madison

  15. Truesdell, C., Muncaster, R.: Fundamentals of Maxwell's kinetic theory of a simple monoatomic gas. New York: Academic Press 1980

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Berkeley-California, 94720, Berkeley, CA, USA

    R. J. DiPerna

  2. Place de Lattre de Tassigny, Ceremade, Université Paris-Dauphine, F-75775, Paris Cedex 16, France

    P. L. Lions

Authors
  1. R. J. DiPerna
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. L. Lions
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by A. Jaffe

Rights and permissions

Reprints and permissions

About this article

Cite this article

DiPerna, R.J., Lions, P.L. On the Fokker-Planck-Boltzmann equation. Commun.Math. Phys. 120, 1–23 (1988). https://doi.org/10.1007/BF01223204

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01223204

Keywords

Search

Navigation