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The Boltzmann Equation with Time-periodic Boundary Temperature

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Abstract

This paper is concerned with the boundary-value problem on the Boltzmann equation in bounded domains with diffuse-reflection boundary where the boundary temperature is time-periodic. We establish the existence of time-periodic solutions with the same period for both hard and soft potentials, provided that the time-periodic boundary temperature is sufficiently close to a stationary one which has small variations around a positive constant. The dynamical stability of time-periodic profiles is also proved under small perturbations, and this in turn yields the non-negativity of the profile. For the proof, we develop new estimates in the time-periodic setting.

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References

  1. Aoki, K., Kosuge, S., Fujiwara, S., Goudon, T. Unsteady motion of a slightly rarefied gas caused by a plate oscillating in its normal direction. Phys. Rev. Fluids, 2: 013402 (2017)

    Article  Google Scholar 

  2. Duan, R.J., Huang, F.M., Wang, Y., Yang, T. Global well–posedness of the Boltzmann equation with large amplitude initial data. Arch. Ration. Mech. Anal., 225(1): 375–424 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Duan, R.J., Huang, F.M., Wang, Y., Zhang, Z. Effects of soft interaction and non–isothermal boundary upon long–time dynamics of rarefied gas. Preprint, arXiv: 1807.05700, 2018

    Google Scholar 

  4. Duan, R.J., Liu, S.Q. Time–periodic solutions of the Vlasov–Poisson–Fokker–Planck system. Acta Math. Sci. Ser. B (Engl. Ed.), 35(4): 876–886 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Duan, R.J., Ukai, S., Yang, T., Zhao, H.J. Optimal decay estimates on the linearized Boltzmann equation with time dependent force and their application. Comm. Math. Phys., 277(1): 189–236 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Duan, R.J., Zhang, Z. Solutions to a moving boundary problem on the Boltzmann equation. Preprint, arXiv: 1812.04797, 2018

    Google Scholar 

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

    Article  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): Art. 1, 119 pp (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. Esposito, R., Guo, Y., Marra, R. Hydrodynamic limit of a kinetic gas flow past an obstacle. Preprint, arXiv:1702.05656v1

  10. Glassey, R.T. The Cauchy Problem in Kinetic Theory. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1996

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Guo, Y. Decay and continuity of the Boltzmann equation in bounded domains. Arch. Ration. Mech. Anal., 197(3): 713–809 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Guo, Y., Strain, R. Exponential decay for soft potentials near Maxwellian. Arch. Ration. Mech. Anal., 187(2): 287–339 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kim, C. Formation and propagation of discontinuity for Boltzmann equation in non–convex domains. Comm. Math. Phys., 308(3): 641–701 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Liu, S.Q., Yang, X.F. The initial boundary value problem for the Boltzmann equation with soft potential. Arch. Ratio. Mech. Anal., 223(1): 463–541 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ukai, S. Time–periodic solutions of the Boltzmann equation. Discrete Contin. Dyn. Syst., 14(3): 579–596 (2006)

    MathSciNet  MATH  Google Scholar 

  18. Tsuji, T., Aoki, K. Moving boundary problems for a rarefied gas: Spacially one–dimensional case. J. Comput. Phys., 250: 574–600 (2017)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, China

    Renjun Duan

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Yong Wang

  3. University of Chinese Academy of Sciences, Beijing, 100190, China

    Yong Wang

  4. Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, China

    Zhu Zhang

Authors
  1. Renjun Duan
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  2. Yong Wang
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  3. Zhu Zhang
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Corresponding author

Correspondence to Renjun Duan.

Additional information

This paper is dedicated to Professor Philippe G. Ciarlet on the occasion of his 80th birthday.

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Duan, R., Wang, Y. & Zhang, Z. The Boltzmann Equation with Time-periodic Boundary Temperature. Acta Math. Appl. Sin. Engl. Ser. 35, 174–208 (2019). https://doi.org/10.1007/s10255-019-0803-0

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  • DOI: https://doi.org/10.1007/s10255-019-0803-0

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