From the Boltzmann equation for gas mixture to the two-fluid incompressible hydrodynamic system

Zhendong Fang; Kunlun Qi

doi:10.3934/krm.2025017

Article Contents

2026, Volume 19: 119-159. Doi: 10.3934/krm.2025017

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From the Boltzmann equation for gas mixture to the two-fluid incompressible hydrodynamic system

Zhendong Fang^{1, 2,} and
Kunlun Qi^{3, 4, ,}

1.
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
2.
School of Mathematics, South China University of Technology, Guangzhou, 510641, China
3.
Simons Laufer Mathematical Sciences Institute, Berkeley, CA 94720, USA
4.
Department of Computational Mathematics, Science and Engineering and Department of Mathematics, Michigan State University, East Lansing, MI 48823, USA

^*Corresponding author: Kunlun Qi
^*Corresponding author: Kunlun Qi

Received: June 2025

Revised: August 2025

Early access: September 2, 2025

Published online: September 2, 2025

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Abstract

In this paper, we study the hydrodynamic limit transition from the Boltzmann equation for gas mixtures to the two-fluid macroscopic system. Employing a meticulous dimensionless analysis, we derive several novel hydrodynamic models via the moments method. For a certain class of scaled Boltzmann equations governing gas mixtures of two species, we rigorously establish the two-fluid incompressible Navier-Stokes-Fourier system as the hydrodynamic limit. This validation is achieved through the Hilbert expansion around the global Maxwellian and refined energy estimates based on the Macro-Micro decomposition.

Keywords:

Hydrodynamic limit,
Boltzmann equation,
Gas mixture,
Incompressible Navier-Stokes-Fourier equation,
Dimensionless analysis.

Mathematics Subject Classification: Primary: 35B25; 35Q30; 35Q20.

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References

[1]	R. Alexandre and C. Villani, On the Boltzmann equation for long-range interactions, Communications on Pure and Applied Mathematics, 55 (2002), 30-70. doi: 10.1002/cpa.10012.
[2]	K. Aoki, C. Bardos and S. Takata, Knudsen layer for gas mixtures, Journal of Statistical Physics, 112 (2003), 629-655. doi: 10.1023/A:1023876025363.
[3]	D. Arsénio, From Boltzmann's equation to the incompressible Navier-Stokes-Fourier system with long-range interactions, Archive for Rational Mechanics and Analysis, 206 (2012), 367-488. doi: 10.1007/s00205-012-0557-9.
[4]	D. Arsénio and L. Saint-Raymond, From the Vlasov-Maxwell-Boltzmann System to Incompressible Viscous Electro-Magneto-Hydrodynamics. Vol. 1, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2019. doi: 10.4171/193.
[5]	C. Bardos, F. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation, Communications on Pure and Applied Mathematics, 46 (1993), 667-753. doi: 10.1002/cpa.3160460503.
[6]	C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations, Journal of Statistical Physics, 63 (1991), 323-344. doi: 10.1007/BF01026608.
[7]	C. Bardos and S. Ukai, The classical incompressible Navier-Stokes limit of the Boltzmann equation, Mathematical Models and Methods in Applied Sciences, 1 (1991), 235-257. doi: 10.1142/S0218202591000137.
[8]	C. Bardos and X. Yang, The classification of well-posed kinetic boundary layer for hard sphere gas mixtures, Communications in Partial Differential Equations, 37 (2012), 1286-1314. doi: 10.1080/03605302.2011.624149.
[9]	S. Bastea, R. Esposito, J. L. Lebowitz and R. Marra, Binary fluids with long range segregating interaction. I. Derivation of kinetic and hydrodynamic equations, Journal of Statistical Physics, 101 (2000), 1087-1136. doi: 10.1023/A:1026481706240.
[10]	C. Bianca and C. Dogbe, On the Boltzmann gas mixture equation: Linking the kinetic and fluid regimes, Communications in Nonlinear Science and Numerical Simulation, 29 (2015), 240-256. doi: 10.1016/j.cnsns.2015.05.015.
[11]	A. Bondesan and M. Briant, Stability of the Maxwell-Stefan system in the diffusion asymptotics of the Boltzmann multi-species equation, Communications in Mathematical Physics, 382 (2021), 381-440. doi: 10.1007/s00220-021-03976-5.
[12]	M. Briant, From the Boltzmann equation to the incompressible Navier–Stokes equations on the torus: a quantitative error estimate, Journal of Differential Equations, 259 (2015), 6072-6141, URL https://www.sciencedirect.com/science/article/pii/S0022039615003812.
[13]	M. Briant and E. S. Daus, The Boltzmann equation for a multi-species mixture close to global equilibrium, Archive for Rational Mechanics and Analysis, 222 (2016), 1367-1443. doi: 10.1007/s00205-016-1023-x.
[14]	M. Briant, S. Merino-Aceituno and C. Mouhot, From Boltzmann to incompressible Navier-Stokes in {S}obolev spaces with polynomial weight, Anal. Appl. (Singap.), 17 (2019), 85-116. doi: 10.1142/S021953051850015X.
[15]	R. E. Caflisch, The fluid dynamic limit of the nonlinear Boltzmann equation, Communications on Pure and Applied Mathematics, 33 (1980), 651-666. doi: 10.1002/cpa.3160330506.
[16]	C. Cercignani, The Boltzmann Equation and Its Applications, vol. 67 of Applied Mathematical Sciences, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.
[17]	A. De Masi, R. Esposito and J. L. Lebowitz, Incompressible Navier-Stokes and Euler limits of the Boltzmann equation, Communications on Pure and Applied Mathematics, 42 (1989), 1189-1214. doi: 10.1002/cpa.3160420810.
[18]	L. Desvillettes and F. Glose, A remark concerning the Chapman-Enskog asymptotics, in Advances in Kinetic Theory and Computing, Ser. Adv. Math. Appl. Sci., World Sci. Publ., River Edge, NJ, 22 (1994), 191-203. doi: 10.1142/9789814354165_0007.
[19]	R. J. DiPerna and P.-L. Lions, On the cauchy problem for Boltzmann equations: Global existence and weak stability, Annals of Mathematics, 130 (1989), 321-366. doi: 10.2307/1971423.
[20]	C. Dogbe, Fluid dynamic limits for gas mixture. I. Formal derivations, Mathematical Models and Methods in Applied Sciences, 18 (2008), 1633-1672. doi: 10.1142/S0218202508003157.
[21]	R. Duan, W.-X. Li and L. Liu, Gevrey regularity of mild solutions to the non-cutoff Boltzmann equation, Advances in Mathematics, 395 (2022), Paper No. 108159, 75 pp. URL https://www.sciencedirect.com/science/article/pii/S0001870821005983. doi: 10.1016/j.aim.2021.108159.
[22]	R. Duan, S. Liu, S. Sakamoto and R. M. Strain, Global mild solutions of the Landau and non-cutoff Boltzmann equations, Communications on Pure and Applied Mathematics, 74 (2021), 932-1020. doi: 10.1002/cpa.21920.
[23]	Z. Fang, Formal derivations from Boltzmann equation to three stationary equations, 2024. arXiv: 2403.11058.
[24]	Z. Fang, K. Qi and H. Wen, The small Deborah number limit for the fluid-particle flows: Incompressible case, Mathematical Models and Methods in Applied Sciences, 34 (2024), 2265-2304. doi: 10.1142/S0218202524500489.
[25]	G. Francois and L. Saint-Raymond, Hydrodynamic limits for the Boltzmann equation, Rivista di Matematica della Universita di Parma, 4** (2005), 1-144.
[26]	I. Gallagher and I. Tristani, On the convergence of smooth solutions from Boltzmann to Navier-Stokes, Annales Henri Lebesgue, 3 (2020), 561-614. doi: 10.5802/ahl.40.
[27]	P. Gervais, On the convergence from Boltzmann to Navier-Stokes-Fourier for general initial data, SIAM Journal on Mathematical Analysis, 55 (2023), 805-848. doi: 10.1137/22M1471687.
[28]	F. Golse and L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Inventiones Mathematicae, 155 (2004), 81-161. doi: 10.1007/s00222-003-0316-5.
[29]	F. Golse and L. Saint-Raymond, The incompressible Navier–Stokes limit of the Boltzmann equation for hard cutoff potentials, Journal de Mathématiques Pures et Appliquées, 91 (2009), 508-552, URL https://www.sciencedirect.com/science/article/pii/S0021782409000154. doi: 10.1016/j.matpur.2009.01.013.
[30]	P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, Journal of the American Mathematical Society, 24 (2011), 771-847. doi: 10.1090/S0894-0347-2011-00697-8.
[31]	Y. Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Archive for Rational Mechanics and Analysis, 169 (2003), 305-353.
[32]	Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Inventiones mathematicae, 153 (2003), 593-630. doi: 10.1007/s00222-003-0301-z.
[33]	Y. Guo, The Boltzmann equation in the whole space, Indiana University Mathematics Journal, 53 (2004), 1081-1094.
[34]	Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Communications on Pure and Applied Mathematics, 59 (2006), 626-687. doi: 10.1002/cpa.20121.
[35]	Y. Guo, Decay and continuity of the Boltzmann equation in bounded domains, Archive for Rational Mechanics and Analysis, 197 (2010), 713-809. doi: 10.1007/s00205-009-0285-y.
[36]	Y. Guo, F. Huang and Y. Wang, Hilbert expansion of the Boltzmann equation with specular boundary condition in half-space, Archive for Rational Mechanics and Analysis, 241 (2021), 231-309. doi: 10.1007/s00205-021-01651-6.
[37]	Y. Guo, J. Jang and N. Jiang, Local Hilbert expansion for the Boltzmann equation, Kinetic and Related Models, 2 (2009), 205-214. doi: 10.3934/krm.2009.2.205.
[38]	Y. Guo, J. Jang and N. Jiang, Acoustic limit for the Boltzmann equation in optimal scaling, Communications on Pure and Applied Mathematics, 63 (2010), 337-361. doi: 10.1002/cpa.20308.
[39]	D. Hilbert, Begründung der kinetischen gastheorie (german), Mathematische Annalen, 72 (1912), 562-577. doi: 10.1007/BF01456676.
[40]	C. Imbert and L. Silvestre, The weak Harnack inequality for the Boltzmann equation without cut-off, Journal of the European Mathematical Society (JEMS), 22 (2020), 507-592. doi: 10.4171/jems/928.
[41]	C. Imbert and L. E. Silvestre, Global regularity estimates for the Boltzmann equation without cut-off, Journal of the American Mathematical Society, 35 (2022), 625-703. doi: 10.1090/jams/986.
[42]	J. Jang, Vlasov-Maxwell-Boltzmann diffusive limit, Archive for Rational Mechanics and Analysis, 194 (2009), 531-584. doi: 10.1007/s00205-008-0169-6.
[43]	J. Jang and N. Jiang, Acoustic limit of the Boltzmann equation: classical solutions, Discrete and Continuous Dynamical Systems. Series A, 25 (2009), 869-882.
[44]	J. Jang and C. Kim, Incompressible Euler limit from Boltzmann equation with diffuse boundary condition for analytic data, Annals of PDE, 7 (2021), Paper No. 22,103 pp.
[45]	N. Jiang, C. D. Levermore and N. Masmoudi, Remarks on the acoustic limit for the Boltzmann equation, Communications in Partial Differential Equations, 35 (2010), 1590-1609. doi: 10.1080/03605302.2010.496096.
[46]	N. Jiang and Y.-L. Luo, From Vlasov-Maxwell-Boltzmann system to two-fluid incompressible Navier-Stokes-Fourier-Maxwell system with Ohm's law: convergence for classical solutions, Annals of PDE, 8 (2022), Paper No. 4,126pp. doi: 10.1007/s40818-022-00117-6.
[47]	N. Jiang, Y.-L. Luo and T.-F. Zhang, Hydrodynamic limit of the incompressible Navier-Stokes-Fourier-Maxwell system with Ohm's law from the Vlasov-Maxwell-Boltzmann system: Hilbert expansion approach, Archive for Rational Mechanics and Analysis, 247 (2023), Paper No. 55, 85pp. doi: 10.1007/s00205-023-01888-3.
[48]	N. Jiang and N. Masmoudi, Boundary layers and incompressible Navier-Stokes-Fourier limit of the Boltzmann equation in bounded domain I, Communications on Pure and Applied Mathematics, 70 (2017), 90-171. doi: 10.1002/cpa.21631.
[49]	N. Jiang and L. Xiong, Diffusive limit of the Boltzmann equation with fluid initial layer in the periodic domain, SIAM Journal on Mathematical Analysis, 47 (2015), 1747-1777. doi: 10.1137/130922239.
[50]	N. Jiang, C.-J. Xu and H. Zhao, Incompressible Navier-Stokes-Fourier limit from the Boltzmann equation: Classical solutions, Indiana University Mathematics Journal, 67 (2018), 1817-1855. doi: 10.1512/iumj.2018.67.5940.
[51]	C. Kim and D. Lee, The Boltzmann equation with specular boundary condition in convex domains, Communications on Pure and Applied Mathematics, 71 (2018), 411-504. doi: 10.1002/cpa.21705.
[52]	L.-B. He and J.-C. Jang, On the Cauchy problem for the cutoff Boltzmann equation with small initial data, Journal of Statistical Physics, 190 (2023), Paper No. 52, 25 pp. doi: 10.1007/s10955-023-03065-y.
[53]	T.-P. Liu, T. Yang and S.-H. Yu, Energy method for Boltzmann equation, Physica D. Nonlinear Phenomena, 188 (2004), 178-192. doi: 10.1016/j.physd.2003.07.011.
[54]	N. Masmoudi and L. Saint-Raymond, From the Boltzmann equation to the Stokes-Fourier system in a bounded domain, Communications on Pure and Applied Mathematics, 56 (2003), 1263-1293. doi: 10.1002/cpa.10095.
[55]	S. Mischler, Kinetic equations with Maxwell boundary conditions, Annales scientifiques de l'École Normale SupÉrieure, 43 (2010), 719-760. doi: 10.24033/asens.2132.
[56]	T. Nishida, Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation, Communications in Mathematical Physics, 61 (1978), 119-148. doi: 10.1007/BF01609490.
[57]	K. Qi, On the measure valued solution to the inelastic Boltzmann equation with soft potentials, Journal of Statistical Physics, 183 (2021), Paper No. 27, 31 pp.
[58]	K. Qi, Measure valued solution to the spatially homogeneous Boltzmann equation with inelastic long-range interactions, Journal of Mathematical Physics, 63 (2022), 021503, 22pp. doi: 10.1063/5.0062859.
[59]	M. Rachid, Incompressible Navier-Stokes-Fourier limit from the Landau equation, Kinetic and Related Models, 14 (2021), 599-638. doi: 10.3934/krm.2021017.
[60]	A. Radjesvarane, M. Yoshinori, U. Seiji, C. J. Xu and Y. Tong, The Boltzmann equation without angular cutoff in the whole space: I, global existence for soft potential, Journal of Functional Analysis, 262 (2012), 915-1010, URL https://www.sciencedirect.com/science/article/pii/S0022123611003752.
[61]	L. Saint-Raymond, Hydrodynamic Limits of the Boltzmann Equation, Lecture Notes in Math., 1971, Springer-Verlag, Berlin, 2009.
[62]	A. Sotirov and S.-H. Yu, On the solution of a Boltzmann system for gas mixtures, Archive for Rational Mechanics and Analysis, 195 (2010), 675-700. doi: 10.1007/s00205-009-0219-8.
[63]	S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proceedings of the Japan Academy, 50 (1974), 179-184. doi: 10.3792/pja/1195519027.
[64]	C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, Vol. I, North-Holland, Amsterdam, 2002, 71-305. doi: 10.1016/S1874-5792(02)80004-0.
[65]	Y. Wang, Decay of the two-species Vlasov-Poisson-Boltzmann system, Journal of Differential Equations, 254 (2013), 2304-2340. doi: 10.1016/j.jde.2012.12.007.
[66]	T. Wu and X. Yang, Hydrodynamic limit of Boltzmann equations for gas mixture, Journal of Differential Equations, 377 (2023), 418-468. doi: 10.1016/j.jde.2023.09.009.