logo

SCIENTIA SINICA Mathematica, Volume 47 , Issue 10 : 1143-1154(2017) https://doi.org/10.1360/N012016-00150

Global existence for the ellipsoidal BGK model with initial large oscillations

More info
  • ReceivedAug 22, 2016
  • AcceptedNov 24, 2016
  • PublishedJan 23, 2017

Abstract

The ellipsoidal BGK model was introduced to fit the correct Prandtl number in the Navier-Stokes approximation of the classical BGK model. In this paper, we establish the global existence of mild solutions to the Cauchy problem on the model for a class of initial data allowed to have large oscillations. The proof is motivated by a recent study of the same topic on the Boltzmann equation.


Funded by

Hong Kong General Research Fund(409913)

Hong Kong General Research Fund(103412)

国家自然科学基金(11401565)


References

[1] Andries P, Le Tallec P, Perlat J P, et al. The Gaussian-BGK model of Boltzmann equation with small Prandtl number. Eur J Mech B Fluids, 2000, 19: 813-830 CrossRef Google Scholar

[2] Bhatnagar P L, Gross E P, Krook M K. A model for collision processes in gases, I: Small amplitude process in charged and neutral one-component systems. Phys Rev, 1954, 94: 511-525 CrossRef Google Scholar

[3] Walender P. On the temperature jump in a rarefied gas. Ark Fysik, 1954, 7: 507-553. Google Scholar

[4] Holway L H. Kinetic theory of shock structure using an ellipsoidal distribution function. In: Rarefied Gas Dynamics, vol. 1. New York: Academic Press, 1966, 193-215. Google Scholar

[5] Andries P, Bourgat J F, Le Tallec P, et al. Numerical comparison between the Boltzmann and ES-BGK models for rarefied gases. Comput Methods Appl Mech Engrg, 2002, 191: 3369-3390 CrossRef Google Scholar

[6] Filbet F, Jin S. An asymptotic preserving scheme for the ES-BGK model of the Boltzmann equation. J Sci Comput, 2011, 46: 204-224 CrossRef Google Scholar

[7] Galli M A, Torczynski R. Investigation of the ellipsoidal-statistical Bhatnagar-Gross-Krook kinetic model applied to gas-phase transport of heat and tangential momentum between parallel walls. Phys Fluids, 2011, 23: 147-154. Google Scholar

[8] Yun S B. Classical solutions for the ellipsoidal BGK model with fixed collision frequency. J Differential Equations, 2015, 259: 6009-6037 CrossRef Google Scholar

[9] Yun S B. Ellipsoidal BGK model near a global Maxwellian. SIAM J Math Anal, 2015, 47: 2324-2354 CrossRef Google Scholar

[10] Guo Y. Bounded solutions for the Boltzmann equation. Quart Appl Math, 2010, 68: 143-148. Google Scholar

[11] Duan R J, Huang F M, Wang Y, et al. Global well-posedness of the Boltzmann equation with large amplitude initial data. ArXiv:1603.06037, 2016. Google Scholar

[12] Perthame B, Pulvirenti M. Weithted $L^\infty$ bounds and uniqueness for the Boltzmann BGK model. Arch Ration Mech Anal, 1993, 125: 289-295 CrossRef Google Scholar

[13] Carrillo J A, Jüngel A, Markowich P A, et al. Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monatsh Math, 2001, 133: 1-82 CrossRef Google Scholar

[14] Gualdani M P, Mischler S, Mouhot C. Factorization for non-symmetric operators and exponential H-theorem. ArXiv:1006.5523, 2010. Google Scholar

Copyright 2020  CHINA SCIENCE PUBLISHING & MEDIA LTD.  中国科技出版传媒股份有限公司  版权所有

京ICP备14028887号-23       京公网安备11010102003388号