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
Full-text PDF Free Access
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Additional Information
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.
References
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- Tai-Ping Liu and Shih-Hsien Yu, Boltzmann equation: micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys. 246 (2004), no. 1, 133–179. MR 2044894, DOI 10.1007/s00220-003-1030-2
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References
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- François Golse, Benoit Perthame, and Catherine Sulem, On a boundary layer problem for the nonlinear Boltzmann equation, Arch. Rational Mech. Anal. 103 (1988), no. 1, 81–96. MR 946970, DOI 10.1007/BF00292921
- François Golse and Laure Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math. 155 (2004), no. 1, 81–161. MR 2025302, DOI 10.1007/s00222-003-0316-5
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- Alexander N. Gorban and Iliya V. Karlin, Structure and approximations of the Chapman-Enskog expansion for the linearized Grad equations, Transport Theory Statist. Phys. 21 (1992), no. 1-2, 101–117. MR 1149364, DOI 10.1080/00411459208203524
- A. N. Gorban and I. V. Karlin, Short wave limit of hydrodynamics: a soluble model, Phys. Rev. Lett. 77 (1996), 282–285.
- Iliya V. Karlin and Alexander N. Gorban, Hydrodynamics from Grad’s equations: what can we learn from exact solutions?, Ann. Phys. 11 (2002), no. 10-11, 783–833. MR 1957348, DOI 10.1002/1521-3889(200211)11:10/11$\langle$783::AID-ANDP783$\rangle$3.0.CO;2-V
- Alexander N. Gorban and Ilya Karlin, Hilbert’s 6th problem: exact and approximate hydrodynamic manifolds for kinetic equations, Bull. Amer. Math. Soc. (N.S.) 51 (2014), no. 2, 187–246. MR 3166040, DOI 10.1090/S0273-0979-2013-01439-3
- Harold Grad, Asymptotic theory of the Boltzmann equation. II, Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos., Palais de l’UNESCO, Paris, 1962) Academic Press, New York, 1963, pp. 26–59. MR 0156656
- Yan Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J. 53 (2004), no. 4, 1081–1094. MR 2095473, DOI 10.1512/iumj.2004.53.2574
- Yan Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Comm. Pure Appl. Math. 59 (2006), no. 5, 626–687. MR 2172804, DOI 10.1002/cpa.20121
- Feimin Huang, Akitaka Matsumura, and Zhouping Xin, Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations, Arch. Ration. Mech. Anal. 179 (2006), no. 1, 55–77. MR 2208289, DOI 10.1007/s00205-005-0380-7
- Feimin Huang, Yi Wang, Yong Wang, and Tong Yang, The limit of the Boltzmann equation to the Euler equations for Riemann problems, SIAM J. Math. Anal. 45 (2013), no. 3, 1741–1811. MR 3066800, DOI 10.1137/120898541
- Feimin Huang, Yi Wang, and Tong Yang, Hydrodynamic limit of the Boltzmann equation with contact discontinuities, Comm. Math. Phys. 295 (2010), no. 2, 293–326. MR 2594329, DOI 10.1007/s00220-009-0966-2
- Feimin Huang, Yi Wang, and Tong Yang, Fluid dynamic limit to the Riemann solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity, Kinet. Relat. Models 3 (2010), no. 4, 685–728. MR 2735911, DOI 10.3934/krm.2010.3.685
- Feimin Huang, Zhouping Xin, and Tong Yang, Contact discontinuity with general perturbations for gas motions, Adv. Math. 219 (2008), no. 4, 1246–1297. MR 2450610, DOI 10.1016/j.aim.2008.06.014
- Feimin Huang and Tong Yang, Stability of contact discontinuity for the Boltzmann equation, J. Differential Equations 229 (2006), no. 2, 698–742. MR 2263572, DOI 10.1016/j.jde.2005.12.004
- Juhi Jang, Vlasov-Maxwell-Boltzmann diffusive limit, Arch. Ration. Mech. Anal. 194 (2009), no. 2, 531–584. MR 2563638, DOI 10.1007/s00205-008-0169-6
- N. Jiang and N. Masmoudi, Boundary layers and incompressible Navier-Stokes-Fourier limit of the Boltzmann equation in bounded domain (I). To appear in Comm. Pure Appl. Math.
- Ning Jiang and Linjie Xiong, Diffusive limit of the Boltzmann equation with fluid initial layer in the periodic domain, SIAM J. Math. Anal. 47 (2015), no. 3, 1747–1777. MR 3343361, DOI 10.1137/130922239
- N. Jiang, C. J. Xu, and H. J. Zhao, Incompressible Navier-Stokes-Fourier limit from the Boltzmann equation: Classical solutions, arXiv:1401.6374
- Yong-Jung Kim, Min-Gi Lee, and Marshall Slemrod, Thermal creep of a rarefied gas on the basis of non-linear Korteweg-theory, Arch. Ration. Mech. Anal. 215 (2015), no. 2, 353–379. MR 3294405, DOI 10.1007/s00205-014-0780-7
- C. David Levermore and Nader Masmoudi, From the Boltzmann equation to an incompressible Navier-Stokes-Fourier system, Arch. Ration. Mech. Anal. 196 (2010), no. 3, 753–809. MR 2644440, DOI 10.1007/s00205-009-0254-5
- P.-L. Lions and N. Masmoudi, From the Boltzmann equations to the equations of incompressible fluid mechanics. I, Arch. Ration. Mech. Anal. 158 (2001), no. 3, 173–193. MR 1842343, DOI 10.1007/s002050100143
- P.-L. Lions and N. Masmoudi, From the Boltzmann equations to the equations of incompressible fluid mechanics. II, Arch. Ration. Mech. Anal. 158 (2001), no. 3, 195–211. MR 1842343, DOI 10.1007/s002050100143
- Shuangqian Liu and Huijiang Zhao, Diffusive expansion for solutions of the Boltzmann equation in the whole space, J. Differential Equations 250 (2011), no. 2, 623–674. MR 2737808, DOI 10.1016/j.jde.2010.07.024
- Tai-Ping Liu, Tong Yang, and Shih-Hsien Yu, Energy method for Boltzmann equation, Phys. D 188 (2004), no. 3-4, 178–192. MR 2043729, DOI 10.1016/j.physd.2003.07.011
- Tai-Ping Liu, Tong Yang, Shih-Hsien Yu, and Hui-Jiang Zhao, Nonlinear stability of rarefaction waves for the Boltzmann equation, Arch. Ration. Mech. Anal. 181 (2006), no. 2, 333–371. MR 2221210, DOI 10.1007/s00205-005-0414-1
- Tai-Ping Liu and Shih-Hsien Yu, Boltzmann equation: micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys. 246 (2004), no. 1, 133–179. MR 2044894, DOI 10.1007/s00220-003-1030-2
- J. C. Maxwell, On the dynamical theory of gases, Phil. Trans. Roy. Soc. London 157 (1866), 49–88.
- Nader Masmoudi, Examples of singular limits in hydrodynamics, Handbook of differential equations: evolutionary equations. Vol. III, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2007, pp. 195–275. MR 2549370, DOI 10.1016/S1874-5717(07)80006-5
- Nader Masmoudi and Laure Saint-Raymond, From the Boltzmann equation to the Stokes-Fourier system in a bounded domain, Comm. Pure Appl. Math. 56 (2003), no. 9, 1263–1293. MR 1980855, DOI 10.1002/cpa.10095
- Laure Saint-Raymond, From the BGK model to the Navier-Stokes equations, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 2, 271–317 (English, with English and French summaries). MR 1980313, DOI 10.1016/S0012-9593(03)00010-7
- M. Slemrod, The problem with Hilbert’s 6th problem, Math. Model. Nat. Phenom. 10 (2015), no. 3, 6–15. MR 3371918, DOI 10.1051/mmnp/201510302
- Yoshio Sone, Molecular gas dynamics, Theory, techniques, and applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2007. MR 2274674, DOI 10.1007/978-0-8176-4573-1
- Seiji Ukai, Solutions of the Boltzmann equation, Patterns and waves, Stud. Math. Appl., vol. 18, North-Holland, Amsterdam, 1986, pp. 37–96. MR 882376, DOI 10.1016/S0168-2024(08)70128-0
- Seiji Ukai and Kiyoshi Asano, The Euler limit and initial layer of the nonlinear Boltzmann equation, Hokkaido Math. J. 12 (1983), no. 3, 311–332. MR 719971
- N. Wolchover, Famous fluid equations are incomplete, Quanta Magazine, July 21, 2015.
- Zhouping Xin and Huihui Zeng, Convergence to rarefaction waves for the nonlinear Boltzmann equation and compressible Navier-Stokes equations, J. Differential Equations 249 (2010), no. 4, 827–871. MR 2652155, DOI 10.1016/j.jde.2010.03.011
- Shih-Hsien Yu, Hydrodynamic limits with shock waves of the Boltzmann equation, Comm. Pure Appl. Math. 58 (2005), no. 3, 409–443. MR 2116619, DOI 10.1002/cpa.20027
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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
MR Author ID:
607798
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
MR Author ID:
1137399
Email:
yongwang@amss.ac.cn
Tong Yang
Affiliation:
Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong
MR Author ID:
303932
Email:
matyang@cityu.edu.hk
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