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Decay of the Boltzmann Equation with the Specular Boundary Condition in Non-convex Cylindrical Domains

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Abstract

The basic question about the existence, uniqueness, and stability of the Boltzmann equation in general non-convex domains with the specular reflection boundary condition has been widely open. In this paper, we consider cylindrical domains whose cross section is generally non-convex analytic bounded planar domain. We establish a global well-posedness and asymptotic stability of the Boltzmann equation with the specular reflection boundary condition. Our method consists of the delicate construction of \({\epsilon}\)-tubular neighborhoods of billiard trajectories which bounce infinitely many times or hit the boundary tangentially at some moment, and sharp estimates of the size of such neighborhoods.

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References

  1. Briant, M.: Instantaneous filling of the vacuum for the full Boltzmann equation in convex domains. Arch. Ration. Mech. Anal. 218(2), 985–1041 (2015). https://doi.org/10.1007/s00205-015-0874-x

    Article  MathSciNet  MATH  Google Scholar 

  2. Briant, M.: Perturbative theory for the Boltzmann equation in bounded domains with different boundary conditions. Kinet. Relat. Models 10(2), 329–371 (2017). https://doi.org/10.3934/krm.2017014

    Article  MathSciNet  MATH  Google Scholar 

  3. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Applied Mathematical Sciences, Vol. 106. Springer, New York, 1994. https://doi.org/10.1007/978-1-4419-8524-8

  4. Chernov, N., Markarian, R.: Chaotic Billiards. Mathematical Surveys and Monographs, Vol. 127. American Mathematical Society, Providence, 2006.https://doi.org/10.1090/surv/127

  5. Devillettes, L., Villani, C.: On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invent. Math. 159(2), 245–316 (2005). https://doi.org/10.1007/s00222-004-0389-9

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Do Carmo, M.: Differential Geometry of Curves and Surfaces. Prentice Hall, Englewood Cliffs (1976)

    MATH  Google Scholar 

  7. 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 (2003). https://doi.org/10.1007/s00220-013-1766-2

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. 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). https://doi.org/10.1007/s40818-017-0037-5

    Article  MathSciNet  Google Scholar 

  9. Guo, Y., Kim, C., Tonon, D., Trescases, A.: Regularity of the Boltzmann equation in convex domains. Invent. Math. 207(1), 115–290 (2017). https://doi.org/10.1007/s00222-016-0670-8

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Guo, Y., Kim, C., Tonon, D., Trescases, A.: BV-regularity of the Boltzmann equation in non-convex domains. Arch. Ration. Mech. Anal. 220(3), 1045–1093 (2016). https://doi.org/10.1007/s00205-015-0948-9

  11. Guo, Y.: Decay and continuity of Boltzmann equation in bounded domains. Arch. Ration. Mech. Anal. 197(3), 713–809 (2010). https://doi.org/10.1007/s00205-009-0285-y

    Article  MathSciNet  MATH  Google Scholar 

  12. Halpern, B.: Strange billiard tables. Tran. Am. Math. Soc. 232, 297–305 (1977). https://doi.org/10.1090/S0002-9947-1977-0451308-7

    Article  MathSciNet  MATH  Google Scholar 

  13. Kim, C.: Formation and propagation of discontinuity for Boltzmann equation in non-convex domains. Commun. Math. Phys. 308(3), 641–701 (2011). https://doi.org/10.1007/s00220-011-1355-1

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Kim, C.: Boltzmann equation with a large potential in a periodic box. Commun. PDE 39(8), 1393–1423 (2014). https://doi.org/10.1080/03605302.2014.903278

    Article  MathSciNet  MATH  Google Scholar 

  15. Kim, C., Yun, S.: The Boltzmann equation near a rotational local maxwellian. SIAM J. Math. Anal. 44(4), 2560–2598 (2012). https://doi.org/10.1137/11084981X

    Article  MathSciNet  MATH  Google Scholar 

  16. Kim, C., Lee, D.: Hölder regularity and decay of the Boltzmann equation in some non-convex domain (in preparation)

  17. Kim, C., Lee, D.: The Boltzmann equation with specular boundary condition in convex domains. Comm. Pure Appl. Math. 71, 411–504 (2018). https://doi.org/10.1002/cpa.21705

    Article  MathSciNet  MATH  Google Scholar 

  18. Shizuta, Y., Asano, K.: Global solutions of the Boltzmann equation in a bounded convex domain. Proc. Jpn. Acad. 53A, 3–5 (1977). https://doi.org/10.3792/pjaa.53.3

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. University of Wisconsin Madison, Madison, WI, USA

    Chanwoo Kim & Donghyun Lee

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  1. Chanwoo Kim
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  2. Donghyun Lee
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Correspondence to Donghyun Lee.

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Communicated by C. Mouhot

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Kim, C., Lee, D. Decay of the Boltzmann Equation with the Specular Boundary Condition in Non-convex Cylindrical Domains. Arch Rational Mech Anal 230, 49–123 (2018). https://doi.org/10.1007/s00205-018-1241-5

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  • DOI: https://doi.org/10.1007/s00205-018-1241-5

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