Inventiones mathematicae

, Volume 207, Issue 1, pp 115–290 | Cite as

Regularity of the Boltzmann equation in convex domains

  • Yan Guo
  • Chanwoo Kim
  • Daniela Tonon
  • Ariane Trescases
Article

Abstract

A basic question about regularity of Boltzmann solutions in the presence of physical boundary conditions has been open due to characteristic nature of the boundary as well as the non-local mixing of the collision operator. Consider the Boltzmann equation in a strictly convex domain with the specular, bounce-back and diffuse boundary condition. With the aid of a distance function toward the grazing set, we construct weighted classical \(C^{1}\) solutions away from the grazing set for all boundary conditions. For the diffuse boundary condition, we construct \(W^{1,p}\) solutions for \(1< p<2\) and weighted \( W^{1,p}\) solutions for \(2\le p\le \infty \) as well.

Notes

Acknowledgments

This project was initiated on the study of diffusive reflection boundary condition during the Kinetic Program at ICERM, 2011. We thank referees for their constructive remarks which help greatly improve our paper. Y.Guo’s research is supported in part by a NSF grant #1209437, FRG grant, a Chinese NSF grant #10828103 as well as Beijing International Center for Mathematical Research. He thanks Nader Masmoudi for earlier discussions on the same subject. C. Kim’s research is supported in part by the Herchel Smith fund and a NSF-DMS #1501031 and the University of Wisconsin-Madison Graduate School with funding from the Wisconsin Alumni Research Foundation. He thanks Division of Applied Mathematics, Brown University and KAIST Center for Mathematical Challenges for the kind hospitality and support and also he thanks Clément Mouhot for helpful discussions. A. Trescases thanks the Division of Applied Mathematics, Brown University for its hospitality during her visit during 2011–2012. The authors would like to dedicate this work to the memory of Seiji Ukai.

References

  1. 1.
    Cercignani, C., Illner, R., Pulvirenti, M.: The mathematical theory of dilute gases. In: Applied Mathematical Sciences, vol. 106. Springer, New York (1994)Google Scholar
  2. 2.
    Desvillettes, L., Villani, C.: On the trend to global equilibrium for spatial inhomogeneous kinetic systems: the Boltzmann equation. Invent. Math. 159, 245–316 (2005)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Esposito, R., Guo, Y., Kim, C., Marra, R.: Non-isothermal boundary in the Boltzmann theory and Fourier law. Commun. Math. Phys. 323, 177–239 (2013)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Glassey, R.T.: The Cauchy Problem in Kinetic Theory. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1996)Google Scholar
  5. 5.
    Guiraud, J.P.: An H-Theorem for a Gas of Rigid Spheres in a Bounded Domain, pp. 29–58. CNRS, Paris (1975)Google Scholar
  6. 6.
    Guo, Y.: Regularity of the Vlasov equations in a half space. Indiana. Math. J. 43, 255–320 (1994)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Guo, Y.: Singular solutions of the Vlasov–Maxwell system on a half line. Arch. Rat. Mech. Anal. 131, 241–304 (1995)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Guo, Y.: Classical solutions to the Boltzmann equation for molecules with an angular cutoff. Arch. Rat. Mech. Anal. 169, 305–353 (2003)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Guo, Y.: Decay and continuity of Boltzmann equation in bounded domains. Arch. Rat. Mech. Anal. 197, 713–809 (2010)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Guo, Y., Kim, C., Tonon, D., Trescases, A.: BV-regularity of the Boltzmann equation in non-convex domains. Arch. Rat. Mech. Anal. 220, 1045–1093 (2016)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Hwang, H.-J., Velazquez, J.: Global existence for the Vlasov–Poisson system in bounded domains. Arch. Rat. Mech. Anal. 195, 763–796 (2010)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Kim, C.: Formation and propagation of discontinuity for Boltzmann equation in non-convex domains. Commun. Math. Phys. 308, 641–701 (2011)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Kim, C., Lee, D.: The Boltzmann equation with specular boundary condition in convex domains. arXiv:1604.04342 (submitted)

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Yan Guo
    • 1
  • Chanwoo Kim
    • 2
  • Daniela Tonon
    • 3
  • Ariane Trescases
    • 4
  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA
  2. 2.Department of MathematicsThe University of Wisconsin-MadisonMadisonUSA
  3. 3.CEREMADE (UMR CNRS 7534), Université Paris-Dauphine, PSL Research UniversityParis Cedex 16France
  4. 4.CMLACachanFrance

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