Skip to main content
SpringerLink
Log in

Formation and Propagation of Discontinuity for Boltzmann Equation in Non-Convex Domains

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

Abstract

The formation and propagation of singularities for the Boltzmann equation in bounded domains has been an important question in numerical studies as well as in theoretical studies. In this paper, we consider the nonlinear Boltzmann solution near Maxwellians under in-flow, diffuse, or bounce-back boundary conditions. We demonstrate that discontinuity is created at the non-convex part of the grazing boundary, and then it propagates only along the forward characteristics inside the domain before it hits on the boundary again.

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. Aoki, K.: Private communications

  2. Arkeryd L., Cercignani C.: A global existence theorem for initial-boundary-value problem for the Boltzmann equation when the boundaries are not isothermal. Arch. Rat. Mech. Anal. 125, 271–287 (1993)

    Article  MathSciNet  Google Scholar 

  3. Arlotti, L., Banasiak, J., Lods, B.: On general transport equations with abstract boundary conditions. The case of divergence free force field. Preprint 2009

  4. Aoki K., Bardos C., Dogbe C., Golse F.: A note on the propagation of boundary induced discontinuities in kinetic theory. Math. Models Methods Appl. Sci. 11(9), 1581–1595 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Aoki K., Takata S., Aikawa H., Golse F.: A rarefied gas flow caused by a discontinuous wall temperature. Phys. Fluids 13(9), 2645–2661 (2001)

    Article  MathSciNet  ADS  Google Scholar 

  6. Alexandre R., Villani C.: On the Boltzmann equation for long-range interactions. Comm. Pure Appl. Math. 55(1), 30–70 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Boudin L., Desvillettes L.: On the singularities of the global small solutions of the full Boltzmann equation. Monatshefte Math. 131, 91–108 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bernis L., Desvillettes L.: Propagation of singularities for classical solutions of the Vlasov-Poisson-Boltzmann equation. Discrete Contin. Dyn. Syst. 24(1), 13–33 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cercignani C.: Propagation phenomena in classical and relativistic rarefied gases. Transport Theory Statist. Phys. 29(3-5), 607–614 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. Cercignani C.: On the initial-boundary value problem for the Boltzmann equation. Arch. Rat. Mech. Anal. 116, 307–315 (1992)

    Article  MathSciNet  Google Scholar 

  11. Cercignani C.: The Boltzmann equation and its applications. Springer, New York (1988)

    Book  MATH  Google Scholar 

  12. Cercignani C., Illner R., Pulvirenti M.: The mathematical theory of dilute gases. Springer, New York (1994)

    MATH  Google Scholar 

  13. 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 

  14. Duan R., Li M.-R., Yang T.: Propagation of singularities in the solutions to the Boltzmann equation near equilibrium. Math. Models Methods Appl. Sci. 18(7), 1093–1114 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Glassey, R.: The Cauchy Problems in Kinetic Theory. Philadelphia: SIAM, 1996

  16. Greenberg, W., van der Mee, C., Protopopescu, V.: Boundary value problems in abstract kinetic theory. Operator Theory: Advances and Applications, 23. Basel: Birkhauser Verlag, 1987

  17. Gressman T., Strain R.: Global Classical Solutions of the Boltzmann Equation without Angular Cut-off. J. Amer. Math. Soc. 24(3), 771–847 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Guiraud, J.-P.: An H theorem for a gas of rigid spheres in a bounded domain, Theories cinetiques classiques et relativistes. Paris: Centre Nat. Recherche Sci., 1975, pp. 29–58

  19. Guo Y.: Singular solutions of the Vlasov-Maxwell system on a half line. Arch. Rat. Mech. Anal. 131(3), 241–304 (1995)

    Article  MATH  Google Scholar 

  20. Guo Y.: Classical solutions to the Boltzmann equation for molecules with an angular cutoff. Arch. Rat. Mech. Anal. 169(4), 305–353 (2003)

    Article  MATH  Google Scholar 

  21. Guo Y.: Decay and Continuity of Boltzmann Equation in Bounded Domains. Arch. Rat. Mech. Anal. 197(3), 713–809 (2010)

    Article  MATH  Google Scholar 

  22. Grad, H.: Asymptotic theory of the Boltzmann equation. II. Rarefied gas dynamics. In: Proceedings of the 3rd international Symposium, (Paris, 1962), Lawmann, J.A. (ed.), New York: Academic Press, 1963, pp. 26–59

  23. Hamdache K.: Initial-boundary value problems for the Boltzmann equation: global existence of weak solutions. Arch. Rat. Mech. Anal. 119(4), 309–353 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hörmander, L.: The analysis of linear partial differential operators. I-IV, Berlin: Springer-Verlag, 2005

  25. Hwang H.-J.: Regularity for the Vlasov-Poisson system in a convex domain. SIAM J. Math. Anal. 36(1), 121–171 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hwang H.-J., Velazquez J.: Global existence for the Vlasov-Poisson system in bounded domains. Arch. Rat. Mech. Anal. 195(3), 763–796 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kim, C.: Boltzmann equation with specular reflection in 2D domains, In preparation.

  28. Lions, P.-L.: Compactness in Boltzmann’s equation via Fourier integral operators and applications. I, II, III. J. Math. Kyoto Univ. 34, no. 2, 391–427, 429–461, 539–584, (1994)

  29. Maslova, N.: Nonlinear evolution equations. Kinetic approach. Rivers Edge, NJ: World Scientific Publishing Co., 1993

  30. Maxwell, J.-C.: On stresses in rarefied gases arising from inequalities of temperature. Phil. Trans. Roy. Soc. London 170, Appendix 231–256 (1879)

    Google Scholar 

  31. Melrose R.B., Sjostrand J.: Singularities of boundary value problems I. Comm. Pure Appl. Math. 31(5), 593–617 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mischler S.: On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system. Commun. Math. Phys. 210(2), 447–466 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. Mouhot C., Villani C.: Regularity theory for the spatially homogeneous Boltzmann equation with cut-off. Arch. Rat. Mech. Anal. 173(2), 169–212 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sone, Y.: Molecular gas dynamics. Theory, techniques, and applications. Modeling and Simulation in Science, Engineering and Technology. Boston, MA: Birkhauser Boston, Inc., 2007

  35. Sone Y., Takata S.: Discontinuity of the velocity distribution function in a rarefied gas around a convex body and the S layer at the bottom of the Knudsen layer. Transport Theor. Stat. Phys. 21, 501–530 (1992)

    Article  ADS  MATH  Google Scholar 

  36. Takata S., Sone Y., Aoki K.: Numerical analysis of a uniform flow of a rarefied gas past a sphere on the basis of the Boltzmann equation for hard-sphere molecules. Phys. Fluids A 5, 716–737 (1993)

    ADS  Google Scholar 

  37. Taylor M.: Reflection of singularities of solutions to systems of differential equations. Comm. Pure Appl. Math. 28(4), 457–478 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ukai, S.: Solutions of the Boltzmann equation. Patterns and waves. Stud. Math. Appl., 18, Amsterdam: North-Holland, 1986, pp. 37–96

  39. Villani, C.: A review of mathematical topics in collisional kinetic theory. Handbook of mathematical fluid dynamics. Vol. I, Amsterdam: North-Holland, 2002, pp. 71–305

  40. Voigt, J.: Functional analytic treatment of the initial boundary value problem for collisionless gases. Habilitationsschrift, Munchen, 1981 (http://www.math.tu-dresden.de/~voigt/vopubl/habilschr/habil80.pdf)

  41. Wennberg B.: Regularity in the Boltzmann Equation and the Radon Transform. Commun. in P.D.E. 19, 2057–2074 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  42. Wennberg B.: The geometry of binary collisions and generalized Radon transforms. Arch. Rat. Mech. Anal. 139(3), 291–302 (1997)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Brown University, Providence, RI, 02917, USA

    Chanwoo Kim

Authors
  1. Chanwoo Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chanwoo Kim.

Additional information

Communicated by H.-T. Yau

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kim, C. Formation and Propagation of Discontinuity for Boltzmann Equation in Non-Convex Domains. Commun. Math. Phys. 308, 641–701 (2011). https://doi.org/10.1007/s00220-011-1355-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-011-1355-1

Keywords

Search

Navigation