Boltzmann systems for gas mixtures in 1D torus
Author:
Kung-Chien Wu
Journal:
Quart. Appl. Math. 75 (2017), 415-431
MSC (2010):
Primary 35Q20; Secondary 92C40
DOI:
https://doi.org/10.1090/qam/1460
Published electronically:
December 15, 2016
MathSciNet review:
3636164
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Abstract: We study the 1D Boltzmann equation for a mixture of two gases on a torus with the initial condition of one gas near a vacuum and the other near a Maxwellian equilibrium state. An $L^{\infty }_{x}L^{\infty }_{\xi ,\beta }$ analysis is developed to study this mass diffusion problem, which is based on the Boltzmann equation for the single species hard sphere collision in an earlier work of the author. The decay rate of the solution is algebraic for a small time region and exponential for a large time region. Moreover, the exponential rate depends on the size of the domain.
References
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- Esther S. Daus, Ansgar Jüngel, Clément Mouhot, and Nicola Zamponi, Hypocoercivity for a linearized multispecies Boltzmann system, SIAM J. Math. Anal. 48 (2016), no. 1, 538–568. MR 3457696, DOI https://doi.org/10.1137/15M1017934
- Richard S. Ellis and Mark A. Pinsky, The first and second fluid approximations to the linearized Boltzmann equation, J. Math. Pures Appl. (9) 54 (1975), 125–156. MR 609540
- Seung-Yeal Ha, Se Eun Noh, and Seok Bae Yun, Global existence and stability of mild solutions to the Boltzmann system for gas mixtures, Quart. Appl. Math. 65 (2007), no. 4, 757–779. MR 2370359, DOI https://doi.org/10.1090/S0033-569X-07-01068-6
- Tai-Ping Liu and Shih-Hsien Yu, The Green’s function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math. 57 (2004), no. 12, 1543–1608. MR 2082240, DOI https://doi.org/10.1002/cpa.20011
- Yoshio Sone, Kinetic theory and fluid dynamics, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1919070
- Alexander Sotirov and Shih-Hsien Yu, On the solution of a Boltzmann system for gas mixtures, Arch. Ration. Mech. Anal. 195 (2010), no. 2, 675–700. MR 2592292, DOI https://doi.org/10.1007/s00205-009-0219-8
- Shigeru Takata and Kazuo Aoki, The ghost effect in the continuum limit for a vapor-gas mixture around condensed phases: asymptotic analysis of the Boltzmann equation, Transport Theory Statist. Phys. 30 (2001), no. 2-3, 205–237. The Sixteenth International Conference on Transport Theory, Part I (Atlanta, GA, 1999). MR 1848595, DOI https://doi.org/10.1081/TT-100105368
- Kung-Chien Wu, Pointwise behavior of the linearized Boltzmann equation on a torus, SIAM J. Math. Anal. 46 (2014), no. 1, 639–656. MR 3163241, DOI https://doi.org/10.1137/13090482X
- Kung-Chien Wu, Nonlinear stability of the Boltzmann equation in a periodic box, J. Math. Phys. 56 (2015), no. 8, 081504, 11. MR 3391357, DOI https://doi.org/10.1063/1.4929655
References
- Kazuo Aoki, Claude Bardos, and Shigeru Takata, Knudsen layer for gas mixtures, J. Statist. Phys. 112 (2003), no. 3-4, 629–655. MR 1997264, DOI https://doi.org/10.1023/A%3A1023876025363
- Esther S. Daus, Ansgar Jüngel, Clément Mouhot, and Nicola Zamponi, Hypocoercivity for a linearized multispecies Boltzmann system, SIAM J. Math. Anal. 48 (2016), no. 1, 538–568. MR 3457696, DOI https://doi.org/10.1137/15M1017934
- Richard S. Ellis and Mark A. Pinsky, The first and second fluid approximations to the linearized Boltzmann equation, J. Math. Pures Appl. (9) 54 (1975), 125–156. MR 0609540
- Seung-Yeal Ha, Se Eun Noh, and Seok Bae Yun, Global existence and stability of mild solutions to the Boltzmann system for gas mixtures, Quart. Appl. Math. 65 (2007), no. 4, 757–779. MR 2370359, DOI https://doi.org/10.1090/S0033-569X-07-01068-6
- Tai-Ping Liu and Shih-Hsien Yu, The Green’s function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math. 57 (2004), no. 12, 1543–1608. MR 2082240, DOI https://doi.org/10.1002/cpa.20011
- Yoshio Sone, Kinetic theory and fluid dynamics, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1919070
- Alexander Sotirov and Shih-Hsien Yu, On the solution of a Boltzmann system for gas mixtures, Arch. Ration. Mech. Anal. 195 (2010), no. 2, 675–700. MR 2592292, DOI https://doi.org/10.1007/s00205-009-0219-8
- Shigeru Takata and Kazuo Aoki, The ghost effect in the continuum limit for a vapor-gas mixture around condensed phases: asymptotic analysis of the Boltzmann equation, Transport Theory Statist. Phys. 30 (2001), no. 2-3, 205–237. The Sixteenth International Conference on Transport Theory, Part I (Atlanta, GA, 1999). MR 1848595, DOI https://doi.org/10.1081/TT-100105368
- Kung-Chien Wu, Pointwise behavior of the linearized Boltzmann equation on a torus, SIAM J. Math. Anal. 46 (2014), no. 1, 639–656. MR 3163241, DOI https://doi.org/10.1137/13090482X
- Kung-Chien Wu, Nonlinear stability of the Boltzmann equation in a periodic box, J. Math. Phys. 56 (2015), no. 8, 081504, 11. MR 3391357, DOI https://doi.org/10.1063/1.4929655
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Additional Information
Kung-Chien Wu
Affiliation:
Department of Mathematics, National Cheng Kung University, 701 Tainan, Taiwan – and – National Center for Theoretical Sciences, National Taiwan University, 106, Taipei, Taiwan
MR Author ID:
887455
Email:
kcwu@mail.ncku.edu.tw; kungchienwu@gmail.com
Keywords:
Boltzmann equation,
gas mixtures,
Maxwellian states
Received by editor(s):
June 13, 2016
Received by editor(s) in revised form:
November 2, 2016
Published electronically:
December 15, 2016
Additional Notes:
This work was supported by the Ministry of Science and Technology under the grant 104-2628-M-006-003-MY4 and National Center for Theoretical Sciences. Part of this work was written during a stay at Institute of Mathematics, Academia Sinica; the author thanks Tai-Ping Liu for his kind hospitality.
Article copyright:
© Copyright 2016
Brown University
