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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  • [1] F. V. Atkinson and L. A. Peletier, Similarity solutions of the nonlinear diffusion equation, Arch. Rational Mech. Anal. 54 (1974), 373-392. MR 0344559
  • [2] Claude Bardos, François Golse, and David Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations, J. Statist. Phys. 63 (1991), no. 1-2, 323-344. MR 1115587, https://doi.org/10.1007/BF01026608
  • [3] Claude Bardos, C. David Levermore, Seiji Ukai, and Tong Yang, Kinetic equations: fluid dynamical limits and viscous heating, Bull. Inst. Math. Acad. Sin. (N.S.) 3 (2008), no. 1, 1-49. MR 2398020
  • [4] Claude Bardos and Seiji Ukai, The classical incompressible Navier-Stokes limit of the Boltzmann equation, Math. Models Methods Appl. Sci. 1 (1991), no. 2, 235-257. MR 1115292, https://doi.org/10.1142/S0218202591000137
  • [5] A. V. Bobylëv, On the Chapman-Enskog and Grad methods for solving the Boltzmann equation, Dokl. Akad. Nauk SSSR 262 (1982), no. 1, 71-75 (Russian). MR 647994
  • [6] A. V. Bobylev, Instabilities in the Chapman-Enskog expansion and hyperbolic Burnett equations, J. Stat. Phys. 124 (2006), no. 2-4, 371-399. MR 2264613, https://doi.org/10.1007/s10955-005-8087-6
  • [7] A. V. Bobylev, Generalized Burnett hydrodynamics, J. Stat. Phys. 132 (2008), no. 3, 569-580. MR 2415120, https://doi.org/10.1007/s10955-008-9556-5
  • [8] Alexander V. Bobylev and Asa Windfäll, Boltzmann equation and hydrodynamics at the Burnett level, Kinet. Relat. Models 5 (2012), no. 2, 237-260. MR 2911095, https://doi.org/10.3934/krm.2012.5.237
  • [9] Ludwig Boltzmann, Lectures on gas theory, translated by Stephen G. Brush, University of California Press, Berkeley-Los Angeles, Calif., 1964. MR 0158708
  • [10] Russel E. Caflisch, The fluid dynamic limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math. 33 (1980), no. 5, 651-666. MR 586416, https://doi.org/10.1002/cpa.3160330506
  • [11] Carlo Cercignani, Reinhard Illner, and Mario Pulvirenti, The mathematical theory of dilute gases, Applied Mathematical Sciences, vol. 106, Springer-Verlag, New York, 1994. MR 1307620
  • [12] Sydney Chapman and T. G. Cowling, The mathematical theory of nonuniform gases, revised, An account of the kinetic theory of viscosity, thermal conduction and diffusion in gases, 3rd ed., in co-operation with D. Burnett; with a foreword by Carlo Cercignani, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. MR 1148892
  • [13] A. De Masi, R. Esposito, and J. L. Lebowitz, Incompressible Navier-Stokes and Euler limits of the Boltzmann equation, Comm. Pure Appl. Math. 42 (1989), no. 8, 1189-1214. MR 1029125, https://doi.org/10.1002/cpa.3160420810
  • [14] C. J. van Duyn and L. A. Peletier, A class of similarity solutions of the nonlinear diffusion equation, Nonlinear Anal. 1 (1976/77), no. 3, 223-233. MR 0511671, https://doi.org/10.1016/0362-546X(77)90032-3
  • [15] R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. of Math. (2) 130 (1989), no. 2, 321-366. MR 1014927, https://doi.org/10.2307/1971423
  • [16] Raffaele Esposito and Mario Pulvirenti, From particles to fluids, Handbook of mathematical fluid dynamics. Vol. III, North-Holland, Amsterdam, 2004, pp. 1-82. MR 2099033
  • [17]
    R. Esposito, Y. Guo, C. Kim, and R. Marra, Stationary solutions to the Boltzmann equation in the hydrodynamic limit.
    Preprint, arXiv:1502.05324.
  • [18] François Golse, The Boltzmann equation and its hydrodynamic limits, Evolutionary equations. Vol. II, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2005, pp. 159-301. MR 2182829
  • [19] François Golse and C. David Levermore, Stokes-Fourier and acoustic limits for the Boltzmann equation: convergence proofs, Comm. Pure Appl. Math. 55 (2002), no. 3, 336-393. MR 1866367, https://doi.org/10.1002/cpa.3011
  • [20] 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, https://doi.org/10.1007/BF00292921
  • [21] 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, https://doi.org/10.1007/s00222-003-0316-5
  • [22] François Golse and Laure Saint-Raymond, The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials, J. Math. Pures Appl. (9) 91 (2009), no. 5, 508-552 (English, with English and French summaries). MR 2517786, https://doi.org/10.1016/j.matpur.2009.01.013
  • [23] A.N. Gorban, I. V. Karlin, Structure and approximation of the Chapman-Enskog expansion for linearized Grad equations. Soviet Phys. JETP 73 (1991), 637-641.
  • [24] 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, https://doi.org/10.1080/00411459208203524
  • [25] A. N. Gorban and I. V. Karlin, Short wave limit of hydrodynamics: a soluble model, Phys. Rev. Lett. 77 (1996), 282-285.
  • [26] 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, https://doi.org/10.1002/1521-3889(200211)11:10/11$ \langle $783::AID-ANDP783$ \rangle $3.0.CO;2-V
  • [27] 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, https://doi.org/10.1090/S0273-0979-2013-01439-3
  • [28] 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
  • [29] Yan Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J. 53 (2004), no. 4, 1081-1094. MR 2095473, https://doi.org/10.1512/iumj.2004.53.2574
  • [30] Yan Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Comm. Pure Appl. Math. 59 (2006), no. 5, 626-687. MR 2172804, https://doi.org/10.1002/cpa.20121
  • [31] 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, https://doi.org/10.1007/s00205-005-0380-7
  • [32] 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, https://doi.org/10.1137/120898541
  • [33] 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, https://doi.org/10.1007/s00220-009-0966-2
  • [34] 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, https://doi.org/10.3934/krm.2010.3.685
  • [35] 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, https://doi.org/10.1016/j.aim.2008.06.014
  • [36] Feimin Huang and Tong Yang, Stability of contact discontinuity for the Boltzmann equation, J. Differential Equations 229 (2006), no. 2, 698-742. MR 2263572, https://doi.org/10.1016/j.jde.2005.12.004
  • [37] Juhi Jang, Vlasov-Maxwell-Boltzmann diffusive limit, Arch. Ration. Mech. Anal. 194 (2009), no. 2, 531-584. MR 2563638, https://doi.org/10.1007/s00205-008-0169-6
  • [38] 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.
  • [39] 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, https://doi.org/10.1137/130922239
  • [40] N. Jiang, C. J. Xu, and H. J. Zhao, Incompressible Navier-Stokes-Fourier limit from the Boltzmann equation: Classical solutions, arXiv:1401.6374
  • [41] 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, https://doi.org/10.1007/s00205-014-0780-7
  • [42] 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, https://doi.org/10.1007/s00205-009-0254-5
  • [43] 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, https://doi.org/10.1007/s002050100143
  • [44] 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, https://doi.org/10.1007/s002050100143
  • [45] 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, https://doi.org/10.1016/j.jde.2010.07.024
  • [46] 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, https://doi.org/10.1016/j.physd.2003.07.011
  • [47] 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, https://doi.org/10.1007/s00205-005-0414-1
  • [48] 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, https://doi.org/10.1007/s00220-003-1030-2
  • [49] J. C. Maxwell, On the dynamical theory of gases, Phil. Trans. Roy. Soc. London 157 (1866), 49-88.
  • [50] 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, https://doi.org/10.1016/S1874-5717(07)80006-5
  • [51] 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, https://doi.org/10.1002/cpa.10095
  • [52] 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, https://doi.org/10.1016/S0012-9593(03)00010-7
  • [53] M. Slemrod, The problem with Hilbert's 6th problem, Math. Model. Nat. Phenom. 10 (2015), no. 3, 6-15. MR 3371918, https://doi.org/10.1051/mmnp/201510302
  • [54] 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
  • [55] Seiji Ukai, Solutions of the Boltzmann equation, Patterns and waves, Stud. Math. Appl., vol. 18, North-Holland, Amsterdam, 1986, pp. 37-96. MR 882376, https://doi.org/10.1016/S0168-2024(08)70128-0
  • [56] 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
  • [57] N. Wolchover, Famous fluid equations are incomplete, Quanta Magazine, July 21, 2015.
  • [58] 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, https://doi.org/10.1016/j.jde.2010.03.011
  • [59] Shih-Hsien Yu, Hydrodynamic limits with shock waves of the Boltzmann equation, Comm. Pure Appl. Math. 58 (2005), no. 3, 409-443. MR 2116619, https://doi.org/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
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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