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June  2020, 28(2): 879-895. doi: 10.3934/era.2020046

Quasineutral limit for the compressible two-fluid Euler–Maxwell equations for well-prepared initial data

1. 

Faculty of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, China

2. 

School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

3. 

College of Applied Sciences, Beijing University of Technology, Beijing 100124, China

* Corresponding author: Xueke Pu

Received  April 2020 Revised  April 2020 Published  June 2020

Fund Project: This work is partially supported by NSFC grant (11871172, 11831003, 11771031, 11531010) of China, the Natural Science Foundation of Guangdong Province of China (2019A1515012000), the Youth Research Fund for the Shanxi University of Finance and Economics (QN-2019022) and starting Fund for the Shanxi University of Finance and Economics doctoral graduates research (Z18204)

In this paper, we study the quasi-neutral limit for the compressible two-fluid Euler–Maxwell equations for well-prepared initial data. Precisely, we proved the solution of the three-dimensional compressible two-fluid Euler–Maxwell equations converges locally in time to that of the compressible Euler equation as $ \varepsilon $ tends to zero. This proof is based on the formal asymptotic expansions, the iteration techniques, the vector analysis formulas and the Sobolev energy estimates.

Citation: Min Li, Xueke Pu, Shu Wang. Quasineutral limit for the compressible two-fluid Euler–Maxwell equations for well-prepared initial data. Electronic Research Archive, 2020, 28 (2) : 879-895. doi: 10.3934/era.2020046
References:
[1]

D. Gérard-VaretD. Han-Kwan and F. Rousset, Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries, Indiana Univ. Math. J., 62 (2013), 359-402.  doi: 10.1512/iumj.2013.62.4900.  Google Scholar

[2]

D. Gérard-Varet, D. Han-Kwan and F. Rousset, Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries Ⅱ, J. Éc. Polytech. Math., 1 (2014), 343–386. doi: 10.5802/jep.13.  Google Scholar

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Y. GuoA. D. Ionescu and B. Pausader, Global solutions of the Euler–Maxwell two-fluid system in 3D, Ann. of Math. (2), 183 (2016), 377-498.  doi: 10.4007/annals.2016.183.2.1.  Google Scholar

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Q. C. Ju and Y. Li, Quasineutral limit of the two-fluid Euler-Poisson system in a bounded domain of $\mathbb{R}^3$, J. Math. Anal. Appl., 469 (2019), 169-187.  doi: 10.1016/j.jmaa.2018.09.010.  Google Scholar

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T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205.  doi: 10.1007/BF00280740.  Google Scholar

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S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524.  doi: 10.1002/cpa.3160340405.  Google Scholar

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Y. P. Li, The asymptotic behavior and the quasineutral limit for the bipolar Euler-Poisson system with boundary effects and a vacuum, Chin. Ann. Math. Ser. B, 34 (2013), 529-540.  doi: 10.1007/s11401-013-0782-z.  Google Scholar

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M. LiX. K. Pu and S. Wang, Quasineutral limit for the compressible quantum Navier-Stokes-Maxwell equations, Commun. Math. Sci., 16 (2018), 363-391.  doi: 10.4310/CMS.2018.v16.n2.a3.  Google Scholar

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A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, 53. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[10]

Y. J. Peng, Global existence and long-time behavior of smooth solutions of two-fluid Euler-Maxwell equations, Ann. Inst. H. Poincaré Anal. Non Lińeaire, 29 (2012), 737–759. doi: 10.1016/j.anihpc.2012.04.002.  Google Scholar

[11]

Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to compressible Euler-Poisson equations, Chin. Ann. Math. Ser. B, 28 (2007), 583-602.  doi: 10.1007/s11401-005-0556-3.  Google Scholar

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Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to incompressible Euler equations, Comm. Partial Differential Equations, 33 (2008), 349-476.  doi: 10.1080/03605300701318989.  Google Scholar

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Y. J. Peng and S. Wang, Rigorous derivation of incompressible e-MHD equations from compressible Euler-Maxwell equations, SIAM J. Math. Anal., 40 (2008), 540-565.  doi: 10.1137/070686056.  Google Scholar

[14]

Y. J. Peng and S. Wang, Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters, Discrete Contin. Dyn. Syst., 23 (2009), 415-433.  doi: 10.3934/dcds.2009.23.415.  Google Scholar

[15]

Y. J. Peng and Y. G. Wang, Boundary layers and quasi-neutral limit in steady state Euler-Poisson equations for potential flows, Nonlinearity, 17 (2004), 835-849.  doi: 10.1088/0951-7715/17/3/006.  Google Scholar

[16]

X. K. Pu and M. Li, Asymptotic behaviors for the full compressible quantum Navier-Stokes-Maxwell equations with general initial data, Discrete Contin. Dyn. Syst. B, 24 (2019), 5149-5181.  doi: 10.3934/dcdsb.2019055.  Google Scholar

[17]

M. H. 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), 1761-1787.  doi: 10.1137/070703272.  Google Scholar

[18]

S. Wang, Quasineutral limit of Euler-Poisson system with and without viscosity, Comm. Partial Differential Equations, 29 (2004), 419-456.  doi: 10.1081/PDE-120030403.  Google Scholar

[19]

L. J. Xiong, Incompressible limit of isentropic Navier-Stokes equations with Navier-slip boundary, Kinet. Relat. Models, 11 (2018), 469-490.  doi: 10.3934/krm.2018021.  Google Scholar

[20]

J. Xu, Global classical solutions to the compressible Euler-Maxwell equations, SIAM J. Math. Anal., 43 (2011), 2688-2718.  doi: 10.1137/100812768.  Google Scholar

[21]

J. W. Yang and S. Wang, The non-relativistic limit of Euler-Maxwell equations for two-fluid plasma, Nonlinear Anal., 72 (2010), 1829-1840.  doi: 10.1016/j.na.2009.09.024.  Google Scholar

[22]

J. W. Yang and S. Wang, Convergence of the Euler-Maxwell two-fluid system to compressible Euler equations, J. Math. Anal. Appl., 417 (2014), 889-903.  doi: 10.1016/j.jmaa.2014.02.035.  Google Scholar

show all references

References:
[1]

D. Gérard-VaretD. Han-Kwan and F. Rousset, Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries, Indiana Univ. Math. J., 62 (2013), 359-402.  doi: 10.1512/iumj.2013.62.4900.  Google Scholar

[2]

D. Gérard-Varet, D. Han-Kwan and F. Rousset, Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries Ⅱ, J. Éc. Polytech. Math., 1 (2014), 343–386. doi: 10.5802/jep.13.  Google Scholar

[3]

Y. GuoA. D. Ionescu and B. Pausader, Global solutions of the Euler–Maxwell two-fluid system in 3D, Ann. of Math. (2), 183 (2016), 377-498.  doi: 10.4007/annals.2016.183.2.1.  Google Scholar

[4]

Q. C. Ju and Y. Li, Quasineutral limit of the two-fluid Euler-Poisson system in a bounded domain of $\mathbb{R}^3$, J. Math. Anal. Appl., 469 (2019), 169-187.  doi: 10.1016/j.jmaa.2018.09.010.  Google Scholar

[5]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205.  doi: 10.1007/BF00280740.  Google Scholar

[6]

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524.  doi: 10.1002/cpa.3160340405.  Google Scholar

[7]

Y. P. Li, The asymptotic behavior and the quasineutral limit for the bipolar Euler-Poisson system with boundary effects and a vacuum, Chin. Ann. Math. Ser. B, 34 (2013), 529-540.  doi: 10.1007/s11401-013-0782-z.  Google Scholar

[8]

M. LiX. K. Pu and S. Wang, Quasineutral limit for the compressible quantum Navier-Stokes-Maxwell equations, Commun. Math. Sci., 16 (2018), 363-391.  doi: 10.4310/CMS.2018.v16.n2.a3.  Google Scholar

[9]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, 53. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[10]

Y. J. Peng, Global existence and long-time behavior of smooth solutions of two-fluid Euler-Maxwell equations, Ann. Inst. H. Poincaré Anal. Non Lińeaire, 29 (2012), 737–759. doi: 10.1016/j.anihpc.2012.04.002.  Google Scholar

[11]

Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to compressible Euler-Poisson equations, Chin. Ann. Math. Ser. B, 28 (2007), 583-602.  doi: 10.1007/s11401-005-0556-3.  Google Scholar

[12]

Y. J. Peng and S. Wang, Convergence of compressible Euler-Maxwell equations to incompressible Euler equations, Comm. Partial Differential Equations, 33 (2008), 349-476.  doi: 10.1080/03605300701318989.  Google Scholar

[13]

Y. J. Peng and S. Wang, Rigorous derivation of incompressible e-MHD equations from compressible Euler-Maxwell equations, SIAM J. Math. Anal., 40 (2008), 540-565.  doi: 10.1137/070686056.  Google Scholar

[14]

Y. J. Peng and S. Wang, Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters, Discrete Contin. Dyn. Syst., 23 (2009), 415-433.  doi: 10.3934/dcds.2009.23.415.  Google Scholar

[15]

Y. J. Peng and Y. G. Wang, Boundary layers and quasi-neutral limit in steady state Euler-Poisson equations for potential flows, Nonlinearity, 17 (2004), 835-849.  doi: 10.1088/0951-7715/17/3/006.  Google Scholar

[16]

X. K. Pu and M. Li, Asymptotic behaviors for the full compressible quantum Navier-Stokes-Maxwell equations with general initial data, Discrete Contin. Dyn. Syst. B, 24 (2019), 5149-5181.  doi: 10.3934/dcdsb.2019055.  Google Scholar

[17]

M. H. 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), 1761-1787.  doi: 10.1137/070703272.  Google Scholar

[18]

S. Wang, Quasineutral limit of Euler-Poisson system with and without viscosity, Comm. Partial Differential Equations, 29 (2004), 419-456.  doi: 10.1081/PDE-120030403.  Google Scholar

[19]

L. J. Xiong, Incompressible limit of isentropic Navier-Stokes equations with Navier-slip boundary, Kinet. Relat. Models, 11 (2018), 469-490.  doi: 10.3934/krm.2018021.  Google Scholar

[20]

J. Xu, Global classical solutions to the compressible Euler-Maxwell equations, SIAM J. Math. Anal., 43 (2011), 2688-2718.  doi: 10.1137/100812768.  Google Scholar

[21]

J. W. Yang and S. Wang, The non-relativistic limit of Euler-Maxwell equations for two-fluid plasma, Nonlinear Anal., 72 (2010), 1829-1840.  doi: 10.1016/j.na.2009.09.024.  Google Scholar

[22]

J. W. Yang and S. Wang, Convergence of the Euler-Maxwell two-fluid system to compressible Euler equations, J. Math. Anal. Appl., 417 (2014), 889-903.  doi: 10.1016/j.jmaa.2014.02.035.  Google Scholar

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