Skip to main content
SpringerLink
Log in

From the Boltzmann Equation to an Incompressible Navier–Stokes–Fourier System

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We establish a Navier–Stokes–Fourier limit for solutions of the Boltzmann equation considered over any periodic spatial domain of dimension two or more. We do this for a broad class of collision kernels that relaxes the Grad small deflection cutoff condition for hard potentials and includes for the first time the case of soft potentials. Appropriately scaled families of DiPerna–Lions renormalized solutions are shown to have fluctuations that are compact. Every limit point is governed by a weak solution of a Navier–Stokes–Fourier system for all time.

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Bardos C., Golse F., Levermore D.: Sur les limites asymptotiques de la théorie cinétique conduisant à la dynamique des fluides incompressibles. C.R. Acad. Sci. Paris Sr. I Math. 309, 727–732 (1989)

    MATH  MathSciNet  Google Scholar 

  2. Bardos C., Golse F., Levermore D.: Fluid dynamic limits of kinetic equations I: formal derivations. J. Stat. Phys. 63, 323–344 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  3. Bardos C., Golse F., Levermore C.D.: Fluid dynamic limits of kinetic equations II: convergence proofs for the Boltzmann equation. Commun. Pure Appl. Math. 46, 667–753 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bardos C., Golse F., Levermore C.D.: Acoustic and Stokes limits for the Boltzmann equation. C.R. Acad. Sci. Paris 327, 323–328 (1999)

    MathSciNet  ADS  Google Scholar 

  5. Bardos C., Golse F., Levermore C.D.: The Acoustic limit for the Boltzmann equation. Archive Rat. Mech. Anal. 153, 177–204 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cercignani C.: The Boltzmann Equation and its Applications. Springer, New York (1988)

    MATH  Google Scholar 

  7. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, Vol. 106. Springer, New York, (1994)

  8. Constantin, P., Foias, C.: Navier–Stokes Equations. Chicago Lectures in Mathematics. The University of Chicago Press, Chicago, 1988

  9. DiPerna R.J., Lions P.-L.: On the Cauchy problem for the Boltzmann equation: global existence and weak stability results. Ann. Math. 130, 321–366 (1990)

    Article  MathSciNet  Google Scholar 

  10. Glassey, R.: The Cauchy Problem in Kinetic Theory. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1996

  11. Golse, F.: From kinetic to macroscopic models. Kinetic Equations and Asymptotic Theory, Vol. 4 (Eds. Perthame B. and Desvillettes L.) Series in Applied Mathematics. Gauthier-Villars, Paris, 41–126, 2000

  12. Golse F., Levermore C.D.: Stokes–Fourier and acoustic limits for the Boltsmann equation: convergence proofs. Commun. Pure Appl. Math. 55, 336–393 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Golse F., Lions P.-L., Perthame B., Sentis R.: Regularity of the moments of the solution of a transport equation. J. Funct. Anal. 76, 110–125 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  14. Golse F., Poupaud F.: Un résultat de compacité pour l’équation de Boltzmann avec potentiel mou. Application au problème de demi-espace. C.R. Acad. Sci. Paris Sr. I Math. 303, 583–586 (1986)

    MATH  MathSciNet  Google Scholar 

  15. Golse F., Saint-Raymond L.: The Navier–Stokes limit for the Boltzmann equation. C.R. Acad. Sci. Paris Sr. I Math. 333, 897–902 (2001)

    MATH  MathSciNet  ADS  Google Scholar 

  16. Golse F., Saint-Raymond L.: Velocity averaging in L 1 for the transport equation. C.R. Acad. Sci. Paris Sr. I Math. 334, 557–562 (2002)

    MATH  MathSciNet  Google Scholar 

  17. Golse F., Saint-Raymond L.: The Navier–Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155, 81–161 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Golse, F., Saint-Raymond, L.: The Incompressible Navier–Stokes Limit of the Boltzmann Equation for Hard Cutoff Potentials, preprint 2008

  19. Grad, H.: Principles of the kinetic theory of gases. Handbuch der Physik, Vol. 12 (Ed. Flügge S.) Springer, Berlin, 205–294 (1958)

  20. Hilbert, D.: Begründung der kinetischen Gastheorie. Math. Annalen 72, 562–577 (1912); English: Foundations of the kinetic theory of gases. Kinetic Theory, Vol. 3 (Ed. Brush G.) Pergamon Press, Oxford, 89–101, 1972

  21. Jiang, N., Levermore, C.D., Masmoudi, N.: Remarks on the acoustic limits for the Boltzmann equation. Commun. P.D.E., submitted 2009

  22. Jiang, N., Masmoudi, N.: From the Boltzmann Equation to the Navier–Stokes–Fourier System in a Bounded Domain, preprint 2009

  23. Leray J.: Sur le mouvement d’un fluide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)

    Article  MATH  MathSciNet  Google Scholar 

  24. Levermore C.D.: Entropic convergence and the linearized limit for the Boltzmann equation. Commun. P.D.E. 18, 1231–1248 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  25. Levermore, C.D., Sun, W.: Compactness of the gain parts of the linearized Boltzmann operator with weakly cutoff kernels. Commun. P.D.E., submitted 2009

  26. Lions, P.-L.: Compactness in Boltzmann’s equation via Fourier integral operators and applications, I, II, & III. J. Math. Kyoto Univ. 34, 391–427, 429–461, 539–584 (1994)

    Google Scholar 

  27. Lions P.-L., Masmoudi N.: Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77, 585–627 (1998)

    MATH  MathSciNet  Google Scholar 

  28. Lions P.-L., Masmoudi N.: Une approche locale de la limite incompressible. C.R. Acad. Sci. Paris Sr. I Math. 329, 387–392 (1999)

    MATH  MathSciNet  ADS  Google Scholar 

  29. Lions P.-L., Masmoudi N.: From the Boltzmann equations to the equations of incompressible fluid mechanics, I. Archive Rat. Mech. Anal. 158, 173–193 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  30. Lions P.-L., Masmoudi N.: From the Boltzmann equations to the equations of incompressible fluid mechanics II. Archive Rat. Mech. Anal. 158, 195–211 (2001)

    Article  MathSciNet  Google Scholar 

  31. Masmoudi, N.: Examples of singular limits in hydrodynamis. Evolutionary Equations, Handb. Differ. Equ., Vol. 3. Elsevier/North-Holland, Amsterdam, 195–276, 2007

  32. Masmoudi N., Saint-Raymond L.: From the Boltzmann equation to the Stokes–Fourier system in a bounded domain. Commun. Pure Appl. Math 56, 1263–1293 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  33. Saint-Raymond L.: Du modèl BGK de l’équation de Boltzmann aux équations d’ Euler des fluides incompressibles. Bull. Sci. Math. 126, 493–506 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  34. Saint-Raymond L.: From the Boltzmann BGK equation to the Navier–Stokes system. Ann. Sci. École Norm. Sup. 36, 271–317 (2003)

    MATH  MathSciNet  Google Scholar 

  35. Saint-Raymond L.: Convergence of solutions to the Boltzmann equation in the incompressible Euler limit. Archive Rat. Mech. Anal. 166, 47–80 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  36. Sun, W.: Fredholm Alternatives for Linearized Boltzmann Collision Operators with Weakly Cutoff Kernels, (preprint 2005). Chapter I of Mathematical Problems Arising when Connecting Kinetic to Fluid Regimes. Ph.D. Dissertation, University of Maryland, 2008

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Institute for Physical Science and Technology (IPST), University of Maryland, College Park, MD, 20742-4015, USA

    C. David Levermore

  2. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012, USA

    Nader Masmoudi

Authors
  1. C. David Levermore
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nader Masmoudi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nader Masmoudi.

Additional information

Communicated by Y. Brenier

Rights and permissions

Reprints and permissions

About this article

Cite this article

David Levermore, C., Masmoudi, N. From the Boltzmann Equation to an Incompressible Navier–Stokes–Fourier System. Arch Rational Mech Anal 196, 753–809 (2010). https://doi.org/10.1007/s00205-009-0254-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-009-0254-5

Keywords

Search

Navigation