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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  September 2019 Early access  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 and 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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[13]

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

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[39]

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

[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.

[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.

[42]

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

[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.

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[13]

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

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[39]

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

[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.

[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.

[42]

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

[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.

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