Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Justification of limit for the Boltzmann equation related to Korteweg theory


Authors: Feimin Huang, Yi Wang, Yong Wang and Tong Yang
Journal: Quart. Appl. Math. 74 (2016), 719-764
MSC (2010): Primary 35Q35, 35B65, 76N10
DOI: https://doi.org/10.1090/qam/1440
Published electronically: June 17, 2016
MathSciNet review: 3539030
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Abstract: Under the diffusion scaling and a scaling assumption on the microscopic component, a non-classical fluid dynamic system was derived by Bardos et al. (2008) that is related to the system of ghost effect derived by Sone (2007) in a different setting. By constructing a non-trivial solution to the limiting system that is closely related to the Korteweg theory, we prove that there exists a sequence of smooth solutions of the Boltzmann equation that converge to the limiting solution when the Knudsen number vanishes. This provides the first rigorous nonlinear derivation of Korteweg theory from the Boltzmann equation and re-emphasizes the importance of Korteweg theory for the problem of thermal creep flow.


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Additional Information

Feimin Huang
Affiliation: Institute of Applied Mathematics, AMSS, CAS, Beijing 100190, People’s Republic of China — and — Beijing Center of Mathematics and Information Sciences, Beijing 100048, People’s Republic of China
Email: fhuang@amt.ac.cn

Yi Wang
Affiliation: Institute of Applied Mathematics, AMSS, CAS, Beijing 100190, People’s Republic of China — and — Beijing Center of Mathematics and Information Sciences, Beijing 100048, People’s Republic of China
Email: wangyi@amss.ac.cn

Yong Wang
Affiliation: Institute of Applied Mathematics, AMSS, CAS, Beijing 100190, People’s Republic of China
Email: yongwang@amss.ac.cn

Tong Yang
Affiliation: Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong
Email: matyang@cityu.edu.hk

DOI: https://doi.org/10.1090/qam/1440
Keywords: Boltzmann equation, Knudsen number, diffusive scaling, diffusion wave
Received by editor(s): February 29, 2016
Published electronically: June 17, 2016
Article copyright: © Copyright 2016 Brown University


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