Skip to main content
SpringerLink
Log in

Hydrodynamic Limit of the Boltzmann Equation with Contact Discontinuities

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

Abstract

The hydrodynamic limit for the Boltzmann equation is studied in the case when the limit system, that is, the system of Euler equations contains contact discontinuities. When suitable initial data is chosen to avoid the initial layer, we prove that there exist a family of solutions to the Boltzmann equation globally in time for small Knudsen number. And this family of solutions converge to the local Maxwellian defined by the contact discontinuity of the Euler equations uniformly away from the discontinuity as the Knudsen number ε tends to zero. The proof is obtained by an appropriately chosen scaling and the energy method through the micro-macro decomposition.

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. Asona F., Ukai S.: The Euler limit and the initial layer of the nonlinear Boltzmann equation. Hokkaido Math. J. 12, 303–324 (1983)

    Google Scholar 

  2. Atkinson F.V., Peletier L.A.: Similarity solutions of the nonlinear diffusion equation. Arch. Rat. Mech. Anal. 54, 373–392 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bardos, C., Golse, F., Levermore, D.: Fluid dynamic limits of kinetic equations, I. Formal derivations. J. Stat. Phys. 63, 323–344 (1991); II. Convergence proofs for the Boltzmann equation. Comm. Pure Appl. Math. 46, 667–753 (1993)

    Google Scholar 

  4. Boltzmann, L.: (translated by Stephen G. Brush),: Lectures on Gas Theory. New York: Dover Publications, Inc., 1964

  5. Caflish R.E.: The fluid dynamical limit of the nonlinear Boltzmann equation. Comm. Pure Appl. Math. 33, 491–508 (1980)

    Google Scholar 

  6. Cercignani C., Illner R., Pulvirenti M.: The Mathematical Theory of Dilute Gases. Springer-Verlag, Berlin (1994)

    MATH  Google Scholar 

  7. Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-Uniform Gases. Cambridge: Cambridge University Press, 3rd edition, 1990

  8. Diperna R.J., Lions P.L.: On the Cauchy problem for Boltzmann equation: global existence and weak stability. Ann. Math. 130, 321–366 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  9. Duyn C.T., Peletier L.A.: A class of similarity solution of the nonlinear diffusion equation. Nonlinear Anal., T.M.A. 1, 223–233 (1977)

    Article  MATH  Google Scholar 

  10. Esposito, R., Pulvirenti, M.: From Particle to Fluids. Handbook of mathematical fluid dynamics, in press

  11. Goodman J.: Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Rat. Mech. Anal. 95(4), 325–344 (1986)

    Article  MATH  Google Scholar 

  12. Goodman J., Xin Z.: Viscous limits for piecewise smooth solutions to systems of conservation laws. Arch. Rat. Mech. Anal. 121(3), 235–265 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Grad, H.: Asymptotic theory of the Boltzmann equation II, In: Rarefied Gas Dynamics, J. A. Laurmann, ed., Vol. 1, New York: Academic Press, 1963, pp. 26–59

  14. Guo Y.: The Boltzmann equation in the whole space. Indiana Univ. Math. J. 53(4), 1081–1094 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Huang F.M., Matsumura A., Shi X.: On the stability of contact discontinuity for compressible Navier-Stokes equations with free boundary. Osaka J. Math. 41(1), 193–210 (2004)

    MathSciNet  MATH  Google Scholar 

  16. Huang F.M., Matsumura A., Xin Z.P.: Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations. Arch. Rat. Mech. Anal. 179(1), 55–77 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Huang F.M., Wang Y.: Large time behavior of the Boltzmann equation with specular reflective boundary conditions. J. Diff. Eqs. 242(2), 399–429 (2007)

    Article  Google Scholar 

  18. Huang F.M., Xin Z.P., Yang T.: Contact discontinuities with general perturbation for gas motion. Adv. Math. 219(4), 1246–1297 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Huang F.M., Zhao H.J.: On the global stability of contact discontinuity for compressible Navier-Stokes equations. Rend. Sem. Mat. Univ. Padova 109, 283–305 (2003)

    MathSciNet  MATH  Google Scholar 

  20. Kawashima S., Matsumura A.: Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion. Commun. Math. Phys. 101(1), 97–127 (1985)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. Lachowicz M.: On the initial layer and existence theorem for the nonlinear Boltzmann equation. Math. Methods Appl. Sci. 9(3), 342–366 (1987)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Liu T.P.: Nonlinear stability of shock waves for viscous conservation laws. Mem. Amer. Math. Soc. 56(329), 1–108 (1985)

    Google Scholar 

  23. Liu T.P., Xin Z.P.: Pointwise decay to contact discontinuities for systems of viscous conservation laws. Asian J. Math. 1, 34–84 (1997)

    MathSciNet  MATH  Google Scholar 

  24. Liu T.P., Yang T., Yu S.H.: Energy method for the Boltzmann equation. Physica D 188(3-4), 178–192 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. Liu T.P., Yang T., Yu S.H., Zhao H.J.: Nonlinear stability of rarefaction waves for the Boltzmann equation. Arch. Rat. Mech. Anal. 181(2), 333–371 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu T.P., Yu S.H.: Boltzmann equation: Micro-macro decompositions and positivity of shock profiles. Commun. Math. Phys. 246(1), 133–179 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. Matsumura A., Nishihara K.: On the stability of traveling wave solutions of a one-dimensional model system for compressible viscous gas. Japan J. Appl. Math. 2(1), 17–25 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  28. Maxwell, J.C.: The Scientific Papers of James Clerk Maxwell. Cambridge University Press, 1890: (a) On the dynamical theory of gases, Vol. II, p. 26. (b) On stresses in rarefied gases arising from inequalities of temperature, Vol. II, p. 681

  29. Nishida T.: Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation. Commun. Math. Phys. 61, 119–148 (1978)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. Smoller J.: Shock Waves and Reaction-diffusion Equations. Springer, New York (1994)

    MATH  Google Scholar 

  31. Ukai S.: On the existence of global solutions of mixed problem for non-linear Boltzmann equation. Proc. Japan Acad. 50, 179–184 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ukai S.: Les solutions globales de l′équation de Boltzmann dans l′espace tout entier et dans le demi-espace. C.R. Acad. Sci. Paris 282, 317–320 (1976)

    MathSciNet  MATH  Google Scholar 

  33. Xin, Z.P.: On nonlinear stability of contact discontinuities. In: Hyperbolic Problems: Theory, Numerics, Applications (Stony Brook, NY, 1994), River Edge, NJ: World Sci. Publishing, River Edge, NJ, 1996, pp. 249–257

  34. Xin Z.P.: Zero dissipation limit to rarefaction waves for the one-dimentional Navier-Stokes equations of compressible isentropic gases. Commun. Pure Appl. Math XLVI, 621–665 (1993)

    Article  Google Scholar 

  35. Yu S.H.: Hydrodynamic limits with shock waves of the Boltzmann equations. Commun. Pure Appl. Math. 58(3), 409–443 (2005)

    Article  MATH  Google Scholar 

  36. Yang T., Zhao H.J.: A half space problem to the Boltzmann equaiton. Commun. Math. Phys. 268(3), 569–605 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics, AMSS, Academia Sinica, China

    Feimin Huang & Yi Wang

  2. Department of Mathematics, City University of HongKong, Kowloon, HongKong

    Tong Yang

Authors
  1. Feimin Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yi Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tong Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Feimin Huang.

Additional information

Communicated by H. Spohn

Rights and permissions

Reprints and permissions

About this article

Cite this article

Huang, F., Wang, Y. & Yang, T. Hydrodynamic Limit of the Boltzmann Equation with Contact Discontinuities. Commun. Math. Phys. 295, 293–326 (2010). https://doi.org/10.1007/s00220-009-0966-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-009-0966-2

Keywords

Search

Navigation