Skip to main content
SpringerLink
Log in

A Revisit of the Velocity Averaging Lemma: On the Regularity of Stationary Boltzmann Equation in a Bounded Convex Domain

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

In the present work, we adopt the idea of velocity averaging lemma to establish regularity for stationary linearized Boltzmann equations in a bounded convex domain. Considering the incoming data, with four iterations, we establish regularity in fractional Sobolev space in space variable up to order \(1^-\).

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Data Availability

We do not use any data.

References

  1. Bourgain, J., Brezis, H., Mironescu, P.: Another look at Sobolev spaces. Optimal control and partial differential equations, pp. 439–455. IOS, Amsterdam (2001)

  2. Caflisch, R.E.: The Boltzmann equation with a soft potential. I. Linear, spatially-homogeneous. Commun. Math. Phys. 74(1), 71–95 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  3. Cessenat, M.: Théorèmes de trace Lp pour des espaces de fonctions de la neutronique. Comptes rendus des séances de l’Académie des sciences. Série 1, Mathématique 299(16), 831–834 (1984)

    MATH  Google Scholar 

  4. Cessenat, M.: Théorèmes de trace pour des espaces de fonctions de la neutronique. Comptes rendus des séances de l’Académie des sciences. Série 1, Mathématique 300(3), 89–92 (1985)

    MATH  Google Scholar 

  5. Chen, I.-K., Liu, T.-P., Takata, S.: Boundary singularity for thermal transpiration problem of the linearized Boltzmann equation. Arch. Ration. Mech. Anal. 212(2), 575–595 (2014)

    Article  MathSciNet  Google Scholar 

  6. Chen, I.-K., Hsia, C.-H.: Singularity of macroscopic variables near boundary for gases with cutoff hard potential. SIAM J. Math. Anal. 47(6), 4332–4349 (2015)

    Article  MathSciNet  Google Scholar 

  7. Chen, I.-K.: Regularity of stationary solutions to the linearized Boltzmann equations. Regularity of stationary solutions to the linearized Boltzmann equations. SIAM J. Math. Anal. 50(1), 138–161 (2018)

    Article  MathSciNet  Google Scholar 

  8. Chen, I.-K., Hsia, C.-H., Kawagoe, D.: Regularity for diffuse reflection boundary problem to the stationary linearized Boltzmann equation in a convex domain. Annales de l’Institut Henri Poincaré C Analyse non linéaire 36(3), 745–782 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  9. Chen, H., Kim, C.: Regularity of stationary Boltzmann equation in convex domains. Arch. Ration. Mech. Anal. 244(3), 1099–1222 (2022)

    Article  MathSciNet  Google Scholar 

  10. Chen, H.: Regularity of Boltzmann equation with Cercignani–Lampis boundary in convex domain. (2021). arXiv:2105.10052

  11. Chuang, P.-H.: Velocity averaging lemmas and their application to Boltzmann equation. Mater’s thesis (2019). https://doi.org/10.6342/NTU201901751

  12. Desvillettes, L.: About the use of the Fourier transform for the Boltzmann equation. Summer School on Methods and Models of Kinetic Theory (M &MKT 2002). Riv. Mat. Univ. Parma (7)2*, 1–99 (2003)

  13. DeVore, R., Petrova, G.: The averaging lemma. J. Am. Math. Soc. 14(2), 279–296 (2001)

    Article  MathSciNet  Google Scholar 

  14. DiPerna, R., Lions, P.-L., Meyer, Y.: Lp regularity of velocity averages. Ann. Inst. H. Poincaré C Anal. Non Linéaire 8(3–4), 271–287 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  15. Di Nezzaa, E., Palatucciab, G., Valdinociac, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math 136(5), 521–573 (2012)

    Article  MathSciNet  Google Scholar 

  16. DiPerna, R., Lions, P.-L.: On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math. 130(2), 321–366 (1989)

    Article  MathSciNet  Google Scholar 

  17. do Carmo, M.P.: Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition. Dover Publications, Inc., Mineola, NY (2016)

    Google Scholar 

  18. Esposito, R., Guo, Y., Kim, C., Marra, R.: Non-isothermal boundary in the Boltzmann theory and Fourier law. Commun. Math. Phys. 323(1), 177–239 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  19. Esposito, R., Guo, Y., Kim, C., Marra, R.: Stationary solutions to the Boltzmann equation in the hydrodynamic limit. Ann. PDE 4(1), 1–119 (2018)

    Article  MathSciNet  Google Scholar 

  20. Golse, F., Perthame, B., Sentis, R.: Un rsultat de compacité pour les équations de transport. C. R. Acad. Sci. Paris I Math. 301(7), 341–344 (1985)

    MathSciNet  MATH  Google Scholar 

  21. Golse, F., Lions, P.-L., Perthame, B., Sentis, R.: Regularity of the moments of the solution of a transport equation. J. Funct. Anal. 76(1), 110–125 (1988)

    Article  MathSciNet  Google Scholar 

  22. Golse, F., Poupaud, F.: Stationary solutions of the linearized Boltzmann equation in a half-space. Math. Methods Appl. Sci. 11(4), 483–502 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  23. Grad, H.: Asymptotic theory of the Boltzmann equation. II. In: 1963 Rarefied Gas Dynamics. Proc. 3rd Internat. Sympos., Palais de l’UNESCO, Paris, Vol. I, 26–59 (1962)

  24. Guiraud, J.-P.: Probleme aux limites intérieur pour l’équation de Boltzmann linéaire. J. Mécanique 9, 443–490 (1970)

    MathSciNet  MATH  Google Scholar 

  25. Guo, Y., Kim, C., Tonon, D., Trescases, A.: BV-regularity of the Boltzmann equation in non-convex domains. Arch. Rat. Mech. Anal. 220(3), 1045–1093 (2016)

    Article  MathSciNet  Google Scholar 

  26. Guo, Y., Kim, C., Tonon, D., Trescases, A.: Regularity of the Boltzmann equation in convex domains. Invent. Math. 207(1), 115–290 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  27. Jabin, P.-E., Perthame, B.: Regularity in kinetic formulations via averaging lemmas. A tribute to J. L. Lions. ESAIM Control Optim. Calc. Var. 8, 761–774 (2002)

    Article  MathSciNet  Google Scholar 

  28. Kawagoe, D..: Regularity of solutions to the stationary transport equation with the incoming boundary data. unpublished manuscript (2018)

  29. Kuo, H.-W., Liu, T.-P., Noh, S.E.: Mixture Lemma, Bulletin. Inst. Math. Acad. Sin. (N.S.) 5(1), 1–10 (2010)

    MathSciNet  MATH  Google Scholar 

  30. Lin, Y.-C., Lyu, M.-J., Wang, H., Wu, K.-C.: Space-time behavior of the solution to the Boltzmann equation with soft potentials. J. Differ. Equ. 322, 180–236 (2022)

    Article  ADS  MathSciNet  Google Scholar 

  31. Liu, T.-P., Yu, S.-H.: The Green’s function and large-time behavior of solutions for the one-dimensional Boltzmann equation. Commun. Pure Appl. Math. 57(12), 1543–1608 (2004)

    Article  MathSciNet  Google Scholar 

  32. Wu, K.-C.: Pointwise behavior of the linearized Boltzmann equation on a torus. SIAM J. Math. Anal. 46, 639–656 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The first author is supported in part by MOST grant 108-2628-M-002 -006 -MY4 and 106-2115-M-002 -011 -MY2. The third author is supported by NCTS and MOST grant 104-2628-M-002-007-MY3.

Author information

Authors and Affiliations

  1. Institute of Applied Mathematical Sciences, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., Taipei, 10617, Taiwan

    I.-Kun Chen & Chun-Hsiung Hsia

  2. Department of Mathematics, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., Taipei, 10617, Taiwan

    Ping-Han Chuang & Jhe-Kuan Su

Authors
  1. I.-Kun Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ping-Han Chuang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chun-Hsiung Hsia
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jhe-Kuan Su
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jhe-Kuan Su.

Ethics declarations

Conflict of interest

We do not have any conflict of interest.

Additional information

Communicated by Hal Tasaki.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, IK., Chuang, PH., Hsia, CH. et al. A Revisit of the Velocity Averaging Lemma: On the Regularity of Stationary Boltzmann Equation in a Bounded Convex Domain. J Stat Phys 189, 17 (2022). https://doi.org/10.1007/s10955-022-02977-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10955-022-02977-5

Keywords

Search

Navigation