Skip to main content
SpringerLink
Log in

The Initial Value Problem for the Homogeneous Boltzmann Equation

  • Chapter
The Mathematical Theory of Dilute Gases

Part of the book series: Applied Mathematical Sciences ((AMS,volume 106))

  • 1204 Accesses

Abstract

In this chapter we treat the spatially homogeneous Boltzmann equation, i.e., the special case where f does not depend on x. In this case the main difficulty in estimating the collision operator, namely, the pointwise interaction, disappears, and we can develop a rather complete and satisfactory theory. The remaining difficulties are due to large velocities (high energy tails).

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 79.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. L. Arkeryd, “On the Boltzmann equation.” Arch. Rat. Mech. Anal. 45, 1–34 (1972).

    MathSciNet  MATH  Google Scholar 

  2. L. Arkeryd, R. Esposito, and M. Pulvirenti, “The Boltzmann equation for weakly inhomogeneous data,” Commun. Math. Phys. 111, 393–407 (1987).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. T. Carleman, Problèmes Mathématiques dans la Théorie Cinétique des Gaz, Almqvist & Wiksell, Uppsala (1957).

    MATH  Google Scholar 

  4. T. Carelman. “Sur la théorie de l’équation intégro-differentielle de Boltzmann,” Acta Mathematica 60, 91–146 (1933).

    Article  MathSciNet  Google Scholar 

  5. G. DiBlasio, “Differentiability of spatially homogeneous solutions of the Boltzmann equation,” Commun. Math. Phys. 38, 331–340 (1974).

    Article  MathSciNet  ADS  Google Scholar 

  6. T. Elmroth, “On the H-function and convergence towards equilibrium for a space-homogeneous molecular density,” SIAM J. Appl. Math. 44, 150–159 (1984).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. T. Gustafsson, L p - estimates for the nonlinear spatially homogeneous Boltzmann equation, Arch. Rat. Mech. Anal. 92, 23–57 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  8. D. Morgenstern, “Analytical studies related to the Maxwell-Boltzmann equation,” J.Rat. Mech. Anal. 4, 533–555 (1955).

    MathSciNet  MATH  Google Scholar 

  9. A. Ya. Povzner, “The Boltzmann equation in the kinetic theory of gases,” Amer. Math. Soc. Transl. ser. 2, 47, 193–216(1962).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dip. di Matematica, Politecnico di Milano, I-20133, Milano, Italy

    Carlo Cercignani

  2. Dept. of Mathematics, University of Victoria, V8W 3P4, Victoria, B.C., Canada

    Reinhard Illner

  3. Dip. di Matematica, University of Roma, I-00185, Roma, Italy

    Mario Pulvirenti

Authors
  1. Carlo Cercignani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Reinhard Illner
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mario Pulvirenti
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer Science+Business Media New York

About this chapter

Cite this chapter

Cercignani, C., Illner, R., Pulvirenti, M. (1994). The Initial Value Problem for the Homogeneous Boltzmann Equation. In: The Mathematical Theory of Dilute Gases. Applied Mathematical Sciences, vol 106. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8524-8_7

Download citation

  • DOI: https://doi.org/10.1007/978-1-4419-8524-8_7

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-6425-5

  • Online ISBN: 978-1-4419-8524-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation