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September  2019, 24(9): 5149-5181. doi: 10.3934/dcdsb.2019055

Asymptotic behaviors for the full compressible quantum Navier-Stokes-Maxwell equations with general initial data

1. 

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

2. 

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

* Corresponding author: Min Li

Received  November 2016 Revised  January 2019 Published  April 2019

Fund Project: The first author is supported by NSFC grant 11871172.

In this paper, we study the asymptotic behaviors for the quantum Navier-Stokes-Maxwell equations with general initial data in a torus $\mathbb{T}^{3}$. Based on the local existence theory, we prove the convergence of strong solutions for the full compressible quantum Navier-Stokes-Maxwell equations towards those for the incompressible e-MHD equations plus the fast singular oscillating in time of the sequence of solutions as the Debye length goes to zero. We also mention that similar arguments can be applied to the Euler-Maxwell system. Remarkably, we eliminate the highly oscillating terms produced by the general initial data by using the formal two-timing method. Moreover, using the curl-div decomposition and elaborate energy estimates, we derive uniform (in the Debye length) estimates for the remainder system.

Citation: Xueke Pu, Min Li. Asymptotic behaviors for the full compressible quantum Navier-Stokes-Maxwell equations with general initial data. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5149-5181. doi: 10.3934/dcdsb.2019055
References:
[1]

A. Anile and S. Pennisi, Thermodynamic derivation of the hydrodynamical model for charge transport in semiconductors, Phys. Rev. B, 46 (1992), 13186-13193.  doi: 10.1103/PhysRevB.46.13186.  Google Scholar

[2]

D. Bohm, A suggested interpretation of the quantum theory in terms of "hidden" valuables, I, Phys. Rev., 85 (1952), 166-179.  doi: 10.1103/PhysRev.85.166.  Google Scholar

[3]

L. ChenD. Donatelli and P. Marcati, Incompressible type limit analysis of a hydrodynamic model for charge-carrier transport, SIAM J. Math. Anal., 45 (2013), 915-933.  doi: 10.1137/120876630.  Google Scholar

[4]

G. ChenJ. Jerome and D. Wang, Compressible Euler-Maxwell equations, Transport Theory and Statistical Physics, 29 (2000), 311-331.  doi: 10.1080/00411450008205877.  Google Scholar

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S. Cordier and E. Grenier, Quasineutral limit of an Euler-Poisson system arising from plasma physics, Comm. Partial Differential Equations, 25 (2000), 1099-1113.  doi: 10.1080/03605300008821542.  Google Scholar

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P. DegondF. Deluzet and D. Savelief, Numerical approximation of the Euler-Maxwell model in the quasineutral limit, J. Comput. Phys., 231 (2012), 1917-1946.  doi: 10.1016/j.jcp.2011.11.011.  Google Scholar

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D. Donatelli and P. Marcati, A quasineutral type limit for the Navier-Stokes-Poisson system with large data, Nonlinearity, 21 (2008), 135-148.  doi: 10.1088/0951-7715/21/1/008.  Google Scholar

[8]

D. Donatelli and P. Marcati, Analysis of oscillations and defect measures for the quasineutral limit in plasma physics, Arch. Ration. Mech. Anal., 206 (2012), 159-188.  doi: 10.1007/s00205-012-0531-6.  Google Scholar

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D. Ferry and J. Zhou, Form of the quantum potential for use in hydrodynamic equations for semiconductor device modeling, Phys. Rev. B, 48 (1993), 7944-7950.  doi: 10.1103/PhysRevB.48.7944.  Google Scholar

[10]

C. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math., 54 (1994), 409-427.  doi: 10.1137/S0036139992240425.  Google Scholar

[11]

I. Gasser and P. Marcati, The combined relaxation and vanishing Debye length limit in the hydrodynamic model for semiconductors, Math. Methods Appl. Sci., 24 (2001), 81-92.  doi: 10.1002/1099-1476(20010125)24:2<81::AID-MMA198>3.0.CO;2-X.  Google Scholar

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I. Gasser and P. Marcati, A vanishing Debye length limit in a hydrodynamic model for semiconductors, Hyperbolic Problems: Theory, Numerics, Applications, Vol. I, II(Magdeburg), (2000), 409-414. doi: 10.1007/978-3-0348-8370-2_43.  Google Scholar

[13]

I. Gasser and P. Markowich, Quantum hydrodynamics, wigner transforms and the classical limit, Asymptot. Anal., 14 (1997), 97-116.   Google Scholar

[14]

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

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F. Haas, Quantum Plasmas: An Hydrodynamic Approach, Springer, New York, 2011. doi: 10.1007/978-1-4419-8201-8.  Google Scholar

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S. JiangQ. JuH. Li and Y. Li, Quasi-neutral limit of the full bipolar Euler-Poisson system, Sci. China Math., 53 (2010), 3099-3114.  doi: 10.1007/s11425-010-4114-4.  Google Scholar

[17]

Q. JuF. Li and H. Li, The quasineutral limit of compressible Navier-Stokes-Poisson system with heat conductivity and general initial data, J. Differential Equations, 247 (2009), 203-224.  doi: 10.1016/j.jde.2009.02.019.  Google Scholar

[18]

T. Kato, Nonstationary flows of viscous and ideal fluids in ℝ3, J. Funct. Anal., 9 (1972), 296-305.  doi: 10.1016/0022-1236(72)90003-1.  Google Scholar

[19]

S. Klainerman and A. Maida, 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

[20]

C. LevermoreW. Sun and K. Trivisa, A low Mach number limit of a dispersive Navier-Stokes system, SIAM J. Math. Anal., 44 (2012), 1760-1807.  doi: 10.1137/100818765.  Google Scholar

[21]

H. Li and C. Lin, Zero Debye length asymptotic of the quantum hydrodynamic model for semiconductors, Comm. Math. Phys., 256 (2005), 195-212.  doi: 10.1007/s00220-005-1316-7.  Google Scholar

[22]

M. LiX. Pu and S. Wang, Quasineutral limit for the quantum Navier-Stokes-Poisson equations, Commun. Pure Appl. Anal., 16 (2017), 273-293.  doi: 10.3934/cpaa.2017013.  Google Scholar

[23]

M. LiX. 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

[24]

Y. Li and W. Yong, Quasi-neutral limit in a 3D compressible Navier-Stokes-Poisson-Korteweg model, IMA J. Appl. Math., 80 (2015), 712-727.  doi: 10.1093/imamat/hxu008.  Google Scholar

[25]

Y. Li and W. Yong, Zero Mach number limit of the compressible Navier-Stokes-Korteweg equations, Commun. Math. Sci., 14 (2016), 233-247.  doi: 10.4310/CMS.2016.v14.n1.a9.  Google Scholar

[26]

P. Lions, Mathematical Topics in Fluid Mechanics, vol. 1: Incompressible Models, Oxford Lecture Ser. Math. Appl., vol. 3, The Clarendon Press/Oxford University Press, New York, 1996.  Google Scholar

[27]

N. Masmoudi, From Vlasov-Poisson system to the incompressible Euler system, Comm. Partial Differential Equations, 26 (2001), 1913-1928.  doi: 10.1081/PDE-100107463.  Google Scholar

[28]

G. M$\acute{e}$tivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal., 158 (2001), 61-90.  doi: 10.1007/PL00004241.  Google Scholar

[29]

Y. Peng and Y. Wang, Convergence of compressible Euler-Poisson equations to incompressible type Euler equations, Asymptot. Anal., 41 (2005), 141-160.  doi: 10.1016/j.amc.2010.04.035.  Google Scholar

[30]

Y. 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

[31]

Y. 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

[32]

Y. 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

[33]

Y. 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

[34]

Y. PengS. Wang and G. Gu, Relaxation limit and global existence of smooth solutions of compressible Euler-Maxwell equations, SIAM J. Math. Anal., 43 (2011), 944-970.  doi: 10.1137/100786927.  Google Scholar

[35]

Y. PengY. Wang and W. Yong, Quasi-neutral limit of the non-isentropic Euler-Poisson system, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 1013-1026.  doi: 10.1017/S0308210500004856.  Google Scholar

[36]

X. Pu and B. Guo, Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction, Kinet. Relat. Models, 9 (2016), 165-191.  doi: 10.3934/krm.2016.9.165.  Google Scholar

[37]

X. Pu and B. Guo, Quasineutral limit of the pressureless Euler-Poisson equation for ions, Quart. Appl. Math., 74 (2016), 245-273.  doi: 10.1090/qam/1424.  Google Scholar

[38]

R. Racke, Lectures on nonlinear evolution equations, initial value problems, vol.19, Friedr. Vieweg & Sohn, Braunschweig, 1992. doi: 10.1007/978-3-319-21873-1.  Google Scholar

[39]

S. Schochet, Fast singular limits of hyperbolic PDEs, J. Differential Equations, 114 (1994), 476-512.  doi: 10.1006/jdeq.1994.1157.  Google Scholar

[40]

S. Schochet, The mathematical theory of low mach number flows, M2AN Math. Model. Numer. Anal., 39 (2005), 441-458.  doi: 10.1051/m2an:2005017.  Google Scholar

[41]

S. Wang and S. Jiang, The convergence of Navier-Stokes-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 31 (2006), 571-591.  doi: 10.1080/03605300500361487.  Google Scholar

[42]

E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749-759.  doi: 10.1103/PhysRev.40.749.  Google Scholar

[43]

J. Yang and S. Wang, Convergence of compressible Navier-Stokes-Maxwell equations to incompressible Navier-Stokes equations, Sci. China Math., 57 (2014), 2153-2162.  doi: 10.1007/s11425-014-4792-4.  Google Scholar

show all references

References:
[1]

A. Anile and S. Pennisi, Thermodynamic derivation of the hydrodynamical model for charge transport in semiconductors, Phys. Rev. B, 46 (1992), 13186-13193.  doi: 10.1103/PhysRevB.46.13186.  Google Scholar

[2]

D. Bohm, A suggested interpretation of the quantum theory in terms of "hidden" valuables, I, Phys. Rev., 85 (1952), 166-179.  doi: 10.1103/PhysRev.85.166.  Google Scholar

[3]

L. ChenD. Donatelli and P. Marcati, Incompressible type limit analysis of a hydrodynamic model for charge-carrier transport, SIAM J. Math. Anal., 45 (2013), 915-933.  doi: 10.1137/120876630.  Google Scholar

[4]

G. ChenJ. Jerome and D. Wang, Compressible Euler-Maxwell equations, Transport Theory and Statistical Physics, 29 (2000), 311-331.  doi: 10.1080/00411450008205877.  Google Scholar

[5]

S. Cordier and E. Grenier, Quasineutral limit of an Euler-Poisson system arising from plasma physics, Comm. Partial Differential Equations, 25 (2000), 1099-1113.  doi: 10.1080/03605300008821542.  Google Scholar

[6]

P. DegondF. Deluzet and D. Savelief, Numerical approximation of the Euler-Maxwell model in the quasineutral limit, J. Comput. Phys., 231 (2012), 1917-1946.  doi: 10.1016/j.jcp.2011.11.011.  Google Scholar

[7]

D. Donatelli and P. Marcati, A quasineutral type limit for the Navier-Stokes-Poisson system with large data, Nonlinearity, 21 (2008), 135-148.  doi: 10.1088/0951-7715/21/1/008.  Google Scholar

[8]

D. Donatelli and P. Marcati, Analysis of oscillations and defect measures for the quasineutral limit in plasma physics, Arch. Ration. Mech. Anal., 206 (2012), 159-188.  doi: 10.1007/s00205-012-0531-6.  Google Scholar

[9]

D. Ferry and J. Zhou, Form of the quantum potential for use in hydrodynamic equations for semiconductor device modeling, Phys. Rev. B, 48 (1993), 7944-7950.  doi: 10.1103/PhysRevB.48.7944.  Google Scholar

[10]

C. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math., 54 (1994), 409-427.  doi: 10.1137/S0036139992240425.  Google Scholar

[11]

I. Gasser and P. Marcati, The combined relaxation and vanishing Debye length limit in the hydrodynamic model for semiconductors, Math. Methods Appl. Sci., 24 (2001), 81-92.  doi: 10.1002/1099-1476(20010125)24:2<81::AID-MMA198>3.0.CO;2-X.  Google Scholar

[12]

I. Gasser and P. Marcati, A vanishing Debye length limit in a hydrodynamic model for semiconductors, Hyperbolic Problems: Theory, Numerics, Applications, Vol. I, II(Magdeburg), (2000), 409-414. doi: 10.1007/978-3-0348-8370-2_43.  Google Scholar

[13]

I. Gasser and P. Markowich, Quantum hydrodynamics, wigner transforms and the classical limit, Asymptot. Anal., 14 (1997), 97-116.   Google Scholar

[14]

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

[15]

F. Haas, Quantum Plasmas: An Hydrodynamic Approach, Springer, New York, 2011. doi: 10.1007/978-1-4419-8201-8.  Google Scholar

[16]

S. JiangQ. JuH. Li and Y. Li, Quasi-neutral limit of the full bipolar Euler-Poisson system, Sci. China Math., 53 (2010), 3099-3114.  doi: 10.1007/s11425-010-4114-4.  Google Scholar

[17]

Q. JuF. Li and H. Li, The quasineutral limit of compressible Navier-Stokes-Poisson system with heat conductivity and general initial data, J. Differential Equations, 247 (2009), 203-224.  doi: 10.1016/j.jde.2009.02.019.  Google Scholar

[18]

T. Kato, Nonstationary flows of viscous and ideal fluids in ℝ3, J. Funct. Anal., 9 (1972), 296-305.  doi: 10.1016/0022-1236(72)90003-1.  Google Scholar

[19]

S. Klainerman and A. Maida, 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

[20]

C. LevermoreW. Sun and K. Trivisa, A low Mach number limit of a dispersive Navier-Stokes system, SIAM J. Math. Anal., 44 (2012), 1760-1807.  doi: 10.1137/100818765.  Google Scholar

[21]

H. Li and C. Lin, Zero Debye length asymptotic of the quantum hydrodynamic model for semiconductors, Comm. Math. Phys., 256 (2005), 195-212.  doi: 10.1007/s00220-005-1316-7.  Google Scholar

[22]

M. LiX. Pu and S. Wang, Quasineutral limit for the quantum Navier-Stokes-Poisson equations, Commun. Pure Appl. Anal., 16 (2017), 273-293.  doi: 10.3934/cpaa.2017013.  Google Scholar

[23]

M. LiX. 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

[24]

Y. Li and W. Yong, Quasi-neutral limit in a 3D compressible Navier-Stokes-Poisson-Korteweg model, IMA J. Appl. Math., 80 (2015), 712-727.  doi: 10.1093/imamat/hxu008.  Google Scholar

[25]

Y. Li and W. Yong, Zero Mach number limit of the compressible Navier-Stokes-Korteweg equations, Commun. Math. Sci., 14 (2016), 233-247.  doi: 10.4310/CMS.2016.v14.n1.a9.  Google Scholar

[26]

P. Lions, Mathematical Topics in Fluid Mechanics, vol. 1: Incompressible Models, Oxford Lecture Ser. Math. Appl., vol. 3, The Clarendon Press/Oxford University Press, New York, 1996.  Google Scholar

[27]

N. Masmoudi, From Vlasov-Poisson system to the incompressible Euler system, Comm. Partial Differential Equations, 26 (2001), 1913-1928.  doi: 10.1081/PDE-100107463.  Google Scholar

[28]

G. M$\acute{e}$tivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal., 158 (2001), 61-90.  doi: 10.1007/PL00004241.  Google Scholar

[29]

Y. Peng and Y. Wang, Convergence of compressible Euler-Poisson equations to incompressible type Euler equations, Asymptot. Anal., 41 (2005), 141-160.  doi: 10.1016/j.amc.2010.04.035.  Google Scholar

[30]

Y. 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

[31]

Y. 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

[32]

Y. 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

[33]

Y. 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

[34]

Y. PengS. Wang and G. Gu, Relaxation limit and global existence of smooth solutions of compressible Euler-Maxwell equations, SIAM J. Math. Anal., 43 (2011), 944-970.  doi: 10.1137/100786927.  Google Scholar

[35]

Y. PengY. Wang and W. Yong, Quasi-neutral limit of the non-isentropic Euler-Poisson system, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 1013-1026.  doi: 10.1017/S0308210500004856.  Google Scholar

[36]

X. Pu and B. Guo, Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction, Kinet. Relat. Models, 9 (2016), 165-191.  doi: 10.3934/krm.2016.9.165.  Google Scholar

[37]

X. Pu and B. Guo, Quasineutral limit of the pressureless Euler-Poisson equation for ions, Quart. Appl. Math., 74 (2016), 245-273.  doi: 10.1090/qam/1424.  Google Scholar

[38]

R. Racke, Lectures on nonlinear evolution equations, initial value problems, vol.19, Friedr. Vieweg & Sohn, Braunschweig, 1992. doi: 10.1007/978-3-319-21873-1.  Google Scholar

[39]

S. Schochet, Fast singular limits of hyperbolic PDEs, J. Differential Equations, 114 (1994), 476-512.  doi: 10.1006/jdeq.1994.1157.  Google Scholar

[40]

S. Schochet, The mathematical theory of low mach number flows, M2AN Math. Model. Numer. Anal., 39 (2005), 441-458.  doi: 10.1051/m2an:2005017.  Google Scholar

[41]

S. Wang and S. Jiang, The convergence of Navier-Stokes-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 31 (2006), 571-591.  doi: 10.1080/03605300500361487.  Google Scholar

[42]

E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749-759.  doi: 10.1103/PhysRev.40.749.  Google Scholar

[43]

J. Yang and S. Wang, Convergence of compressible Navier-Stokes-Maxwell equations to incompressible Navier-Stokes equations, Sci. China Math., 57 (2014), 2153-2162.  doi: 10.1007/s11425-014-4792-4.  Google Scholar

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