Skip to main content
SpringerLink
Log in

The Boltzmann Equation in Bounded Domains

  • Chapter
  • First Online:
Nonlinear Partial Differential Equations

Part of the book series: Advanced Courses in Mathematics - CRM Barcelona ((ACMBIRK))

  • 2322 Accesses

Abstract

Boundary effects play a crucial role in the dynamics of gases governed by the Boltzmann equation

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

Access this chapter

Log in via an institution
eBook
USD 24.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

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.

Bibliography

  1. Arkeryd, L., On the strong L1 trend to equilibrium for the Boltzmann equation, Studies in Appl. Math. 87 (1992), 283–288.

    MathSciNet  MATH  Google Scholar 

  2. Arkeryd, L., Cercignani, C., A global existence theorem for the initial boundary value problem for the Boltzmann equation when the boundaries are not isothermal, Arch. Rational Mech. Anal. 125 (1993), 271–288.

    Article  MathSciNet  Google Scholar 

  3. Arkeryd, L., Esposito, R., Marra, R., and Nouri, A., Stability of the laminar solution of the Boltzmann equation for the Benard problem, Bull. Inst. Math. Academia Sinica 3 (2008), 51–97.

    MathSciNet  MATH  Google Scholar 

  4. Arkeryd, L., Esposito, R., Pulvirenti, M., The Boltzmann equation for weakly inhomogeneous data, Commun. Math. Phys. 111 (1987), no. 3, 393–407.

    Article  MathSciNet  MATH  Google Scholar 

  5. Alexandre, R., Desvillettes, L., Villani, C., and Wennberg, B., Entropy dissipation and long-range interactions, Arch. Rational. Mech. Anal. 152 (2000), no. 4, 327–355.

    Article  MathSciNet  MATH  Google Scholar 

  6. Arkeryd, L., Heintz, A., On the solvability and asymptotics of the Boltzmann equation in irregular domains, Comm. Partial Differential Equations 22 (1997), no. 11–12, 2129–2152.

    Article  MathSciNet  MATH  Google Scholar 

  7. Beals, R., Protopopescu, V., Abstract time-depedendent transport equations, J. Math. Anal. Appl. 212 (1987), 370–405.

    Article  MathSciNet  Google Scholar 

  8. Cercignani, C., The Boltzmann Equation and Its Applications, Springer- Verlag, 1988.

    Google Scholar 

  9. Cercignani, C., Equilibrium states and the trend to equlibrium in a gas according to the Boltzmann equation, Rend. Mat. Appl. 10 (1990), 77–95.

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Cannoe, R., Cercignani, C., A trace theorem in kinetic theory, Appl. Math. Letters 4 (1991), 63–67.

    Article  Google Scholar 

  12. Cercignani, C., Illner, R., and Pulvirenti, M., The Mathematical Theory of Dilute Gases, Springer-Verlag, 1994.

    Google Scholar 

  13. Chernov, N., Markarian, R., Chaotic Billiards, AMS, 2006.

    Google Scholar 

  14. Deimling, K., Nonlinear Functional Analysis, Springer-Verlag, 1988.

    Google Scholar 

  15. Desvillettes, L., Convergence to equilibrium in large time for Boltzmann and BGK equations, Arch. Rational Mech. Anal. 110 (1990), 73–91.

    Article  MathSciNet  MATH  Google Scholar 

  16. Desvillettes, L., Villani, C., On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Invent. Math. 159 (2005), no. 2, 245–316.

    Article  MathSciNet  MATH  Google Scholar 

  17. Diperna, R., Lions, P.-L., On the Cauchy problem for the Boltzmann equation, Ann. of Math. 130 (1989), 321–366.

    Article  MathSciNet  MATH  Google Scholar 

  18. Diperna, R., Lions, P.-L., Global weak solution of Vlasov–Maxwell systems, Comm. Pure Appl. Math. 42 (1989), 729–757.

    Article  MathSciNet  MATH  Google Scholar 

  19. Esposito, L., Guo, Y., Marra, R., Phase transition in a Vlasov–Boltzmann system, Commun. Math. Phys. 296 (2010), 1–33.

    Article  MathSciNet  MATH  Google Scholar 

  20. Guo, Y., The Vlasov–Poisson–Boltzmann system near Maxwellians, Comm. Pure Appl. Math. 55 (2002) no. 9, 1104–1135.

    Article  MathSciNet  MATH  Google Scholar 

  21. Guo, Y., The Vlasov–Maxwell–Boltzmann system near Maxwellians, Invent. Math. 153 (2003), no. 3, 593–630.

    Article  MathSciNet  MATH  Google Scholar 

  22. Guo, Y., Singular solutions of the Vlasov–Maxwell system on a half line, Arch. Rational Mech. Anal. 131 (1995), no. 3, 241–304.

    Article  MathSciNet  MATH  Google Scholar 

  23. Guo, Y., Regularity for the Vlasov equations in a half-space, Indiana Univ. Math. J. 43 (1994), no. 1, 255–320.

    Article  MathSciNet  MATH  Google Scholar 

  24. Guo, Y., Jang, J., and Jiang, N., Acoustic limit of the Boltzmann equation in optimal scaling, Comm. Pure Appl. Math. 63 (2010), no. 3, 337–361.

    MathSciNet  MATH  Google Scholar 

  25. Glassey, R., The Cauchy Problems in Kinetic Theory, SIAM, 1996.

    Google Scholar 

  26. Glassey, R., Strauss, W. A., Asymptotic stability of the relativistic Maxwellian, Publ. Res. Inst. Math. Sci. 29 (1993), no. 2, 301–347.

    Article  MathSciNet  MATH  Google Scholar 

  27. Guo, Y., Strauss, W. A., Instability of periodic BGK equilibria, Comm. Pure Appl. Math. 48 (1995), no. 8, 861–894.

    Article  MathSciNet  MATH  Google Scholar 

  28. Grad, H., Principles of the kinetic theory of gases, Handbuch der Physik XII (1958), 205–294.

    Google Scholar 

  29. Grad, H., Asymptotic theory of the Boltzmann equation, II. Rarefied gas dynamics, 3rd Symposium, Paris, 1962, 26–59.

    Google Scholar 

  30. Guiraud, J. P., An H-theorem for a gas of rigid spheres in a bounded domain. Th´eories cin´etique classique et relativiste (1975), G. Pichon, ed., CNRS, Paris, 29–58.

    Google Scholar 

  31. Hamdache, K., Initial boundary value problems for the Boltzmann equation. Global existence of weak solutions, Arch. Rational Mech. Anal. 119 (1992), 309–353.

    Google Scholar 

  32. Hwang, H.-J., Regularity for the Vlasov–Poisson system in a convex domain, SIAM J. Math. Anal. 36 (2004), no. 1, 121–171.

    Article  MathSciNet  MATH  Google Scholar 

  33. Hwang, H.-J., Vel´azquez, J., Global existence for the Vlasov–Poisson system in bounded domains, Arch. Rational Mech. Anal. 195 (2010), no. 3, 763–796.

    Google Scholar 

  34. Liu, T.-P.; Yu, S.-H., Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation, Comm. Pure Appl. Math. 60 (2007), no. 3, 295–356.

    Article  MathSciNet  MATH  Google Scholar 

  35. Liu, T.-P., Yu, S.-H., Green’s function of the Boltzmann equation, 3-D waves, Bull. Inst. Math. Acad. Sin. (N.S.) 1 (2006), no. 1, 1–78.

    Google Scholar 

  36. Liu, T.-P., Yu, S.-H., The Green’s function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math. 57 (2004), no. 12, 1543–1608.

    Article  MathSciNet  MATH  Google Scholar 

  37. Maslova, N. B., Nonlinear Evolution Equations, Kinetic Approach, World Scientific Publishing, Singapore, 1993.

    Book  MATH  Google Scholar 

  38. Mischler, S., On the initial boundary value problem for the Vlasov–Poisson–

    Google Scholar 

  39. Boltzmann system, Commun. Math. Phys. 210 (2000) 447–466.

    Google Scholar 

  40. Masmoudi, N., Saint-Raymond, L., From the Boltzmann equation to the Stokes–Fourier system in a bounded domain, Comm. Pure Appl. Math. 56 (2003), no. 9, 1263–1293.

    Article  MathSciNet  MATH  Google Scholar 

  41. Shizuta, Y., On the classical solutions of the Boltzmann equation, Comm. Pure Appl. Math. 36 (1983), 705–754.

    Article  MathSciNet  MATH  Google Scholar 

  42. Shizuta, Y., Asano, K., Global solutions of the Boltzmann equation in a bounded convex domain, Proc. Japan Acad. 53A (1977), 3–5.

    MathSciNet  Google Scholar 

  43. Strain, R., Guo, Y., Exponential decay for soft potentials near Maxwellians, Arch. Rational Mech. Anal. 187 (2008), no. 2, 287–339.

    Article  MathSciNet  MATH  Google Scholar 

  44. Tabachnikov, S., Billiards, SMF, 1995.

    Google Scholar 

  45. Ukai, S., Solutions of the Boltzmann equation. Pattern and Waves – Qualitative Analysis of Nonlinear Differential Equations, Studies of Mathematics and its Applications vol. 18, Kinokuniya-North-Holland, Tokyo, 1986, 37–96.

    Article  MathSciNet  Google Scholar 

  46. Ukai, S., private communications.

    Google Scholar 

  47. Ukai, S., Asano, K., On the initial boundary value problem of the linearized Boltzmann equation in an exterior domain, Proc. Japan Acad. 56 (1980), 12–17.

    Article  MathSciNet  MATH  Google Scholar 

  48. Villani, C., Hypocoercivity, Memoirs of the AMS, vol. 202, no. 950, 2009.

    Google Scholar 

  49. Vidav, I., Spectra of perturbed semigroups with applications to transport theory, J. Math. Anal. Appl. 30 (1970), 264–279.

    Article  MathSciNet  MATH  Google Scholar 

  50. Yang, T., Zhao, H.-J., A half-space problem for the Boltzmann equation with specular reflection boundary condition, Commun. Math. Phys. 255 (2005), no. 3, 683–726.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    Yan Guo

Authors
  1. Yan Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Basel AG

About this chapter

Cite this chapter

Guo, Y. (2012). The Boltzmann Equation in Bounded Domains. In: Nonlinear Partial Differential Equations. Advanced Courses in Mathematics - CRM Barcelona. Springer, Basel. https://doi.org/10.1007/978-3-0348-0191-1_3

Download citation

Publish with us

Policies and ethics

Search

Navigation