Rigorous derivation of the compressible Navier–Stokes equations from the two-fluid Navier–Stokes–Maxwell equations
Authors:
Yi Peng and Huaqiao Wang
Journal:
Quart. Appl. Math.
MSC (2020):
Primary 35Q35, 35Q30, 35A09, 35B40
DOI:
https://doi.org/10.1090/qam/1665
Published electronically:
June 21, 2023
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Abstract: In this paper, we rigorously derive the compressible one-fluid Navier–Stokes equations from the scaled compressible two-fluid Navier–Stokes–Maxwell equations under the assumption that the initial data are well prepared. We justify the singular limit by proving the uniform decay of the error system, which is obtained by using the elaborate energy estimates.
References
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- Yi Peng, Huaqiao Wang, and Qiuju Xu, Derivation of the Hall-MHD equations from the Navier-Stokes-Maxwell equations, J. Nonlinear Sci. 32 (2022), no. 6, Paper No. 90, 27. MR 4491095, DOI 10.1007/s00332-022-09850-5
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- A. I. Vol’pert and S. I. Hujeav, On the Cauchy problem for composite systems of nonlinear differential equations, Mat. Sbornik 16 (1972), no. 4, 517–544.
- Linjie Xiong, Incompressible limit of isentropic Navier-Stokes equations with Navier-slip boundary, Kinet. Relat. Models 11 (2018), no. 3, 469–490. MR 3810835, DOI 10.3934/krm.2018021
- JianWei Yang and Shu Wang, Convergence of compressible Navier-Stokes-Maxwell equations to incompressible Navier-Stokes equations, Sci. China Math. 57 (2014), no. 10, 2153–2162. MR 3247540, DOI 10.1007/s11425-014-4792-4
- Jianwei Yang and Shu Wang, Convergence of the Euler-Maxwell two-fluid system to compressible Euler equations, J. Math. Anal. Appl. 417 (2014), no. 2, 889–903. MR 3194520, DOI 10.1016/j.jmaa.2014.02.035
- Jianwei Yang and Shu Wang, Non-relativistic limit of two-fluid Euler-Maxwell equations arising from plasma physics, ZAMM Z. Angew. Math. Mech. 89 (2009), no. 12, 981–994. MR 2590892, DOI 10.1002/zamm.200900267
- S. Zheng, Nonlinear parabolic equations and hyperbolic-parabolic coupled systems, CRC Press, 1995.
References
- Christophe Besse, Pierre Degond, Fabrice Deluzet, Jean Claudel, Gérard Gallice, and Christian Tessieras, A model hierarchy for ionospheric plasma modeling, Math. Models Methods Appl. Sci. 14 (2004), no. 3, 393–415. MR 2047577, DOI 10.1142/S0218202504003283
- Gui-Qiang Chen, Joseph W. Jerome, and Dehua Wang, Compressible Euler-Maxwell equations, Proceedings of the Fifth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics (Maui, HI, 1998), 2000, pp. 311–331. MR 1770435, DOI 10.1080/00411450008205877
- David Gérard-Varet, Daniel Han-Kwan, and Frédéric Rousset, Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries, Indiana Univ. Math. J. 62 (2013), no. 2, 359–402. MR 3158514, DOI 10.1512/iumj.2013.62.4900
- David Gérard-Varet, Daniel Han-Kwan, and Frédéric Rousset, Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries II, J. Éc. polytech. Math. 1 (2014), 343–386 (English, with English and French summaries). MR 3322792, DOI 10.5802/jep.13
- Song Jiang and Fucai Li, Rigorous derivation of the compressible magnetohydrodynamic equations from the electromagnetic fluid system, Nonlinearity 25 (2012), no. 6, 1735–1752. MR 2929600, DOI 10.1088/0951-7715/25/6/1735
- Song Jiang, QiangChang Ju, HaiLiang Li, and Yong Li, Quasi-neutral limit of the full bipolar Euler-Poisson system, Sci. China Math. 53 (2010), no. 12, 3099–3114. MR 2746309, DOI 10.1007/s11425-010-4114-4
- Qiangchang Ju and Yong Li, Quasineutral limit of the two-fluid Euler-Poisson system in a bounded domain of $\mathbb {R}^3$, J. Math. Anal. Appl. 469 (2019), no. 1, 169–187. MR 3857516, DOI 10.1016/j.jmaa.2018.09.010
- Qiangchang Ju, Hailiang Li, Yong Li, and Song Jiang, Quasi-neutral limit of the two-fluid Euler-Poisson system, Commun. Pure Appl. Anal. 9 (2010), no. 6, 1577–1590. MR 2684049, DOI 10.3934/cpaa.2010.9.1577
- Tosio Kato, Nonstationary flows of viscous and ideal fluids in ${\mathbf { R}}^{3}$, J. Functional Analysis 9 (1972), 296–305. MR 0481652, DOI 10.1016/0022-1236(72)90003-1
- S. Kawashima, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Kyoto University, 1984.
- Sergiu Klainerman and Andrew Majda, Compressible and incompressible fluids, Comm. Pure Appl. Math. 35 (1982), no. 5, 629–651. MR 668409, DOI 10.1002/cpa.3160350503
- Sergiu Klainerman and Andrew Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math. 34 (1981), no. 4, 481–524. MR 615627, DOI 10.1002/cpa.3160340405
- Yeping Li, The asymptotic behavior and the quasineutral limit for the bipolar Euler-Poisson system with boundary effects and a vacuum, Chinese Ann. Math. Ser. B 34 (2013), no. 4, 529–540. MR 3072246, DOI 10.1007/s11401-013-0782-z
- Yachun Li, Yue-Jun Peng, and Ya-Guang Wang, From two-fluid Euler-Poisson equations to one-fluid Euler equations, Asymptot. Anal. 85 (2013), no. 3-4, 125–148. MR 3156622
- Yachun Li, Yue-Jun Peng, and Shuai Xi, The combined non-relativistic and quasi-neutral limit of two-fluid Euler–Maxwell equations, Z. Angew. Math. Phys. 66 (2015), no. 6, 3249–3265. MR 3428463, DOI 10.1007/s00033-015-0569-z
- F. J. McGrath, Nonstationary plane flow of viscous and ideal fluids, Arch. Rational Mech. Anal. 27 (1967), 329–348. MR 221818, DOI 10.1007/BF00251436
- Yue-Jun Peng and Shu Wang, Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters, Discrete Contin. Dyn. Syst. 23 (2009), no. 1-2, 415–433. MR 2449086, DOI 10.3934/dcds.2009.23.415
- Yuejun Peng and Shu Wang, Convergence of compressible Euler-Maxwell equations to compressible Euler-Poisson equations, Chinese Ann. Math. Ser. B 28 (2007), no. 5, 583–602. MR 2358943, DOI 10.1007/s11401-005-0556-3
- Yue-Jun Peng and Shu Wang, Convergence of compressible Euler-Maxwell equations to incompressible Euler equations, Comm. Partial Differential Equations 33 (2008), no. 1-3, 349–376. MR 2398233, DOI 10.1080/03605300701318989
- Yue-Jun Peng and Shu Wang, Rigorous derivation of incompressible e-MHD equations from compressible Euler-Maxwell equations, SIAM J. Math. Anal. 40 (2008), no. 2, 540–565. MR 2438776, DOI 10.1137/070686056
- Yue-Jun Peng and Ya-Guang Wang, Boundary layers and quasi-neutral limit in steady state Euler-Poisson equations for potential flows, Nonlinearity 17 (2004), no. 3, 835–849. MR 2057130, DOI 10.1088/0951-7715/17/3/006
- Yue-Jun Peng, Shu Wang, and Qilong Gu, Relaxation limit and global existence of smooth solutions of compressible Euler-Maxwell equations, SIAM J. Math. Anal. 43 (2011), no. 2, 944–970. MR 2801184, DOI 10.1137/100786927
- Yi Peng, Huaqiao Wang, and Qiuju Xu, Derivation of the Hall-MHD equations from the Navier-Stokes-Maxwell equations, J. Nonlinear Sci. 32 (2022), no. 6, Paper No. 90, 27. MR 4491095, DOI 10.1007/s00332-022-09850-5
- Marie Hélène Vignal, A boundary layer problem for an asymptotic preserving scheme in the quasi-neutral limit for the Euler-Poisson system, SIAM J. Appl. Math. 70 (2010), no. 6, 1761–1787. MR 2596500, DOI 10.1137/070703272
- A. I. Vol’pert and S. I. Hujeav, On the Cauchy problem for composite systems of nonlinear differential equations, Mat. Sbornik 16 (1972), no. 4, 517–544.
- Linjie Xiong, Incompressible limit of isentropic Navier-Stokes equations with Navier-slip boundary, Kinet. Relat. Models 11 (2018), no. 3, 469–490. MR 3810835, DOI 10.3934/krm.2018021
- JianWei Yang and Shu Wang, Convergence of compressible Navier-Stokes-Maxwell equations to incompressible Navier-Stokes equations, Sci. China Math. 57 (2014), no. 10, 2153–2162. MR 3247540, DOI 10.1007/s11425-014-4792-4
- Jianwei Yang and Shu Wang, Convergence of the Euler-Maxwell two-fluid system to compressible Euler equations, J. Math. Anal. Appl. 417 (2014), no. 2, 889–903. MR 3194520, DOI 10.1016/j.jmaa.2014.02.035
- Jianwei Yang and Shu Wang, Non-relativistic limit of two-fluid Euler-Maxwell equations arising from plasma physics, ZAMM Z. Angew. Math. Mech. 89 (2009), no. 12, 981–994. MR 2590892, DOI 10.1002/zamm.200900267
- S. Zheng, Nonlinear parabolic equations and hyperbolic-parabolic coupled systems, CRC Press, 1995.
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Additional Information
Yi Peng
Affiliation:
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, People’s Republic of China
Email:
20170602018t@cqu.edu.cn
Huaqiao Wang
Affiliation:
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, People’s Republic of China
Email:
wanghuaqiao@cqu.edu.cn
Keywords:
Two fluid Navier–Stokes–Maxwell equations,
compressible Navier–Stokes equations,
singular limit,
energy estimates
Received by editor(s):
December 18, 2022
Received by editor(s) in revised form:
January 30, 2023
Published electronically:
June 21, 2023
Additional Notes:
The research was supported by the National Natural Science Foundation of China (No. 11901066), the Natural Science Foundation of Chongqing (No. cstc2019jcyj-msxmX0167) and projects Nos. 2022CDJXY-001, 2020CDJQY-A040 supported by the Fundamental Research Funds for the Central Universities. The second author is the corresponding author.
Article copyright:
© Copyright 2023
Brown University
