Justification of limit for the Boltzmann equation related to Korteweg theory

Huang, Feimin; Wang, Yi; Wang, Yong; Yang, Tong

doi:10.1090/qam/1440

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.

References

F. V. Atkinson and L. A. Peletier, Similarity solutions of the nonlinear diffusion equation, Arch. Rational Mech. Anal. 54 (1974), 373–392. MR 344559, DOI 10.1007/BF00249197
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, DOI 10.1007/BF01026608
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
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, DOI 10.1142/S0218202591000137
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
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, DOI 10.1007/s10955-005-8087-6
A. V. Bobylev, Generalized Burnett hydrodynamics, J. Stat. Phys. 132 (2008), no. 3, 569–580. MR 2415120, DOI 10.1007/s10955-008-9556-5
Alexander V. Bobylev and Åsa Windfäll, Boltzmann equation and hydrodynamics at the Burnett level, Kinet. Relat. Models 5 (2012), no. 2, 237–260. MR 2911095, DOI 10.3934/krm.2012.5.237
Ludwig Boltzmann, Lectures on gas theory, University of California Press, Berkeley-Los Angeles, Calif., 1964. Translated by Stephen G. Brush. MR 0158708
Russel E. Caflisch, The fluid dynamic limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math. 33 (1980), no. 5, 651–666. MR 586416, DOI 10.1002/cpa.3160330506
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, DOI 10.1007/978-1-4419-8524-8
Sydney Chapman and T. G. Cowling, The mathematical theory of nonuniform gases, 3rd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. An account of the kinetic theory of viscosity, thermal conduction and diffusion in gases; In co-operation with D. Burnett; With a foreword by Carlo Cercignani. MR 1148892
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, DOI 10.1002/cpa.3160420810
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 511671, DOI 10.1016/0362-546X(77)90032-3
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, DOI 10.2307/1971423
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
R. Esposito, Y. Guo, C. Kim, and R. Marra, Stationary solutions to the Boltzmann equation in the hydrodynamic limit. Preprint, arXiv:1502.05324.
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
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, DOI 10.1002/cpa.3011
François Golse, Benoît 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
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, DOI 10.1016/j.matpur.2009.01.013
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.
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<783::AID-ANDP783>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, II, Arch. Ration. Mech. Anal. 158 (2001), no. 3, 173–193, 195–211. MR 1842343, DOI 10.1007/s002050100143
P.-L. Lions and N. Masmoudi, From the Boltzmann equations to the equations of incompressible fluid mechanics. I, II, Arch. Ration. Mech. Anal. 158 (2001), no. 3, 173–193, 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, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2007. Theory, techniques, and applications. 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, DOI 10.14492/hokmj/1470081009
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