June  2017, 37(6): 3183-3210. doi: 10.3934/dcds.2017136

Existence of the solution for the viscous bipolar quantum hydrodynamic model

1. 

Institute of Applied Physics and Computational Mathematics, China Academy of Engineering Physics, Beijing 100088, China

2. 

Graduate School of China Academy of Engineering Physics, Beijing 100088, China

* Corresponding author: Guangwu Wang

Received  August 2015 Revised  January 2017 Published  February 2017

Fund Project: The first author is supported by National Natural Science Foundation of China-NSAF No.11271052.

In this paper, we investigate the existence of classical solution of the viscous bipolar quantum hydrodynamic(QHD) models for ir-rotational fluid in a periodic domain. By applying the iteration method, we prove that the viscous bipolar QHD model has a local classical solution. Then we prove this solution is global with small initial data, based on a series of a priori estimates. Finally, we obtained the inviscid limit of this viscous quantum hydrodynamic model.

Citation: Boling Guo, Guangwu Wang. Existence of the solution for the viscous bipolar quantum hydrodynamic model. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3183-3210. doi: 10.3934/dcds.2017136
References:
[1]

M. G. Ancona and G. I. Iafrate, Quantum correction to the equation of state of an electron gas in a semiconductor, Phys. Revi. B, 39 (1989), 9536-9540.  doi: 10.1103/PhysRevB.39.9536.

[2]

M. G. Ancona and H. F. Tiersten, Microscopic physics of the sillicon inversion layer, Phys. Revi. B, 35 (1987), 7959-7965. 

[3]

P. Antonelli and P. Marcati, On the finite enegy weak solutions to a system in quantum fluid dynamics, Comm. Math. Phys., 287 (2009), 657-686.  doi: 10.1007/s00220-008-0632-0.

[4]

P. Antonelli and P. Marcati, The quantum hydrodynamics system in two space dimensions, Arch. Rat. Mech. Anal., 203 (2012), 499-527.  doi: 10.1007/s00205-011-0454-7.

[5]

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

[6]

L. Chen and M. Drether, The viscous model of quantum hydrodynamic in several dimensions, Math. Mod. Meth. in Appl. Sci, 17 (2007), 1065-1093.  doi: 10.1142/S0218202507002200.

[7]

P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy priciple, J. Statist. Phys., 112 (2003), 587-628.  doi: 10.1023/A:1023824008525.

[8]

I. Gamba and A. Jüngel, Positive solutions to singular second and third order differential equations for quantum fluids, Arch. Rational Mech. Anal., 156 (2001), 183-203.  doi: 10.1007/s002050000114.

[9]

C. Gardner, The quantum hydrodynamic model for semiconductors devices, SIAM J. Appl. Math., 54 (1990), 409-427.  doi: 10.1137/S0036139992240425.

[10]

M. P. GualdniA. Jüngel and G. Toscani, Exponential decay in time of solutions of the viscous quantum hydrodynamic equations, Appl. Math. Lett, 16 (2003), 1273-1278.  doi: 10.1016/S0893-9659(03)90128-5.

[11]

M. P. Gualdini and A. Jüngel, Analysis of the viscous quantum hydrodynamic equations for semiconductors, Euro. Jnl. of Applied Mathematics, 15 (2004), 577-595.  doi: 10.1017/S0956792504005686.

[12]

F. Haas, A magnetohydrodynamic model for quantum plasmas Physics of Plamas 12 (2005), 062117. doi: 10.1063/1.1939947.

[13]

C. C. HaoY. L. Jia and H. L. Li, Quantum Euler-Poisson system: Local existence, J. Partial Diff. Eqns., 16 (2003), 306-320. 

[14]

L. Hsiao and H. L. Li, The well-posedness and asymptotics of multi-dimensional quantum hydrodynamics, Acta Math. Sci., 29 (2009), 552-568.  doi: 10.1016/S0252-9602(09)60053-9.

[15]

F. HuangH. L. Li and A. Matsumura, Existence and stability of steady-state of one-dimensional quantum Euler-Poisson system for seimiconductors, J. Diff. Eqns., 225 (2006), 1-25.  doi: 10.1016/j.jde.2006.02.002.

[16]

F. Huang, H. L. Li, A. Matsumura and S. Odanaka, Well-posedness and stability of multi-dimensional quantum hydrodynamics for semiconductors in R3, Series in Contemporary Applied Mathematics CAM 15 High Education Press, Beijing, 2010.

[17]

Y. L. Jia and H. L. Li, Large time behavior of solutions of quantum hydrodynamical model for semiconductors, Acat. Math. Sci., 26 (2006), 163-178.  doi: 10.1016/S0252-9602(06)60038-6.

[18]

A. Jüngel, A steady-state potential flow Euler-Poisson system for charged quantum fluids, Comm. Math., 194 (1998), 463-479. 

[19]

A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for qunatum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045.  doi: 10.1137/090776068.

[20]

A. Jüngel and H. L. Li, Quantum Euler-Poisson systems: Existence and of stationary state, Arch. Math.(Brno), 40 (2004), 435-456. 

[21]

A. Jüngel and H. L. Li, Quantum Euler-Poisson systems: Global existence and exponential decay, Quart. Appl. Math., 62 (2004), 569-600.  doi: 10.1090/qam/2086047.

[22]

A. JüngelH. L. Li and A. Matsumura, The relaxation-time limit in the quantum hydrodynamic equations for semiconductors, J. Diff. Eqns., 225 (2006), 440-464.  doi: 10.1016/j.jde.2005.11.007.

[23]

A. JüngelM. C. Mariani and D. Rial, Local exitence of solutins to the transient quantum hydrodynamic equations, Math. Mod. Meth. Appl. Sci., 12 (2002), 485-495.  doi: 10.1142/S0218202502001751.

[24]

A. Jüngel and S. Q. Tang, Numerical approximation of the viscous quantum hydrodynamic model for semiconductors, Applied Numerical Mathematics, 56 (2006), 899-915.  doi: 10.1016/j.apnum.2005.07.003.

[25]

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

[26]

H. L. Li and P. Marcati, Exitence and asymptotic behavior of multi-dimensional quantum hydrodynami cmodel for semiconductors, Comm. Math. Phys., 245 (2004), 215-247.  doi: 10.1007/s00220-003-1001-7.

[27]

H. L. LiG. J. Zhang and K. J. Zhang, Algebraic time decay for the bipolar quantum hydrodynamic model, Math. Models Methods Appl. Sci., 18 (2008), 859-881.  doi: 10.1142/S0218202508002887.

[28]

B. Liang and K. J. Zhang, Steady-state solutions and asymptotic limits on the multi-dimensional semiconductor quantum hydrodynamic model, Math. Models Methods Appl. Sci., 17 (2007), 253-275.  doi: 10.1142/S0218202507001905.

[29]

B. Liang and S. M. Zheng, Exponential decay to a quantum hydrodynamic models for semiconductors, Nonlinear Analysis: Real World Application, 9 (2008), 326-337.  doi: 10.1016/j.nonrwa.2006.11.001.

[30]

E. Madelunge, Quantentheorie in hydrodynamischer form, Z. Phys., 40 (1927), 322-326.  doi: 10.1007/BF01400372.

[31]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations Springer, New York, 1990. doi: 10.1007/978-3-7091-6961-2.

[32]

A. Unterreiter, The thermal equilibrium solution of a generic bipolar quantum hydrodinamic model, Comm. Math. Phys., 188 (1997), 69-88.  doi: 10.1007/s002200050157.

[33]

G. W. Wang, Exponential decay for the viscous bipolar Quantum hydrodynamic model, Ann.of Appl. Math., 31 (2015), 329-335. 

[34]

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

[35]

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

[36]

X. H. Yang, Quasineutral limit of bipolar quantum hydrodynamic model for semiconductors, Frot. Math. China, 6 (2011), 349-362.  doi: 10.1007/s11464-011-0102-4.

[37]

B. Zhang and J. Jerome, On a steady state quantum hydrodynamic model for semiconductors, Nonlinear Anal. TMA, 26 (1996), 845-856.  doi: 10.1016/0362-546X(94)00326-D.

[38]

G. J. ZhangH. L. Li and K. J. Zhang, Semiclassical and relaxation limits of bipolar quantum hydrodynamic model for semiconductors, J. Diff. Eqns., 245 (2008), 1433-1453.  doi: 10.1016/j.jde.2008.06.019.

[39]

G. J. Zhang and K. J. Zhang, On the bipolar multidimensional quantum Euler-Poisson system: The thermal equilibrium solution and semiclassical limit, Nonlinear Anal. TMA, 66 (2007), 2218-2229.  doi: 10.1016/j.na.2006.03.010.

show all references

References:
[1]

M. G. Ancona and G. I. Iafrate, Quantum correction to the equation of state of an electron gas in a semiconductor, Phys. Revi. B, 39 (1989), 9536-9540.  doi: 10.1103/PhysRevB.39.9536.

[2]

M. G. Ancona and H. F. Tiersten, Microscopic physics of the sillicon inversion layer, Phys. Revi. B, 35 (1987), 7959-7965. 

[3]

P. Antonelli and P. Marcati, On the finite enegy weak solutions to a system in quantum fluid dynamics, Comm. Math. Phys., 287 (2009), 657-686.  doi: 10.1007/s00220-008-0632-0.

[4]

P. Antonelli and P. Marcati, The quantum hydrodynamics system in two space dimensions, Arch. Rat. Mech. Anal., 203 (2012), 499-527.  doi: 10.1007/s00205-011-0454-7.

[5]

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

[6]

L. Chen and M. Drether, The viscous model of quantum hydrodynamic in several dimensions, Math. Mod. Meth. in Appl. Sci, 17 (2007), 1065-1093.  doi: 10.1142/S0218202507002200.

[7]

P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy priciple, J. Statist. Phys., 112 (2003), 587-628.  doi: 10.1023/A:1023824008525.

[8]

I. Gamba and A. Jüngel, Positive solutions to singular second and third order differential equations for quantum fluids, Arch. Rational Mech. Anal., 156 (2001), 183-203.  doi: 10.1007/s002050000114.

[9]

C. Gardner, The quantum hydrodynamic model for semiconductors devices, SIAM J. Appl. Math., 54 (1990), 409-427.  doi: 10.1137/S0036139992240425.

[10]

M. P. GualdniA. Jüngel and G. Toscani, Exponential decay in time of solutions of the viscous quantum hydrodynamic equations, Appl. Math. Lett, 16 (2003), 1273-1278.  doi: 10.1016/S0893-9659(03)90128-5.

[11]

M. P. Gualdini and A. Jüngel, Analysis of the viscous quantum hydrodynamic equations for semiconductors, Euro. Jnl. of Applied Mathematics, 15 (2004), 577-595.  doi: 10.1017/S0956792504005686.

[12]

F. Haas, A magnetohydrodynamic model for quantum plasmas Physics of Plamas 12 (2005), 062117. doi: 10.1063/1.1939947.

[13]

C. C. HaoY. L. Jia and H. L. Li, Quantum Euler-Poisson system: Local existence, J. Partial Diff. Eqns., 16 (2003), 306-320. 

[14]

L. Hsiao and H. L. Li, The well-posedness and asymptotics of multi-dimensional quantum hydrodynamics, Acta Math. Sci., 29 (2009), 552-568.  doi: 10.1016/S0252-9602(09)60053-9.

[15]

F. HuangH. L. Li and A. Matsumura, Existence and stability of steady-state of one-dimensional quantum Euler-Poisson system for seimiconductors, J. Diff. Eqns., 225 (2006), 1-25.  doi: 10.1016/j.jde.2006.02.002.

[16]

F. Huang, H. L. Li, A. Matsumura and S. Odanaka, Well-posedness and stability of multi-dimensional quantum hydrodynamics for semiconductors in R3, Series in Contemporary Applied Mathematics CAM 15 High Education Press, Beijing, 2010.

[17]

Y. L. Jia and H. L. Li, Large time behavior of solutions of quantum hydrodynamical model for semiconductors, Acat. Math. Sci., 26 (2006), 163-178.  doi: 10.1016/S0252-9602(06)60038-6.

[18]

A. Jüngel, A steady-state potential flow Euler-Poisson system for charged quantum fluids, Comm. Math., 194 (1998), 463-479. 

[19]

A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for qunatum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045.  doi: 10.1137/090776068.

[20]

A. Jüngel and H. L. Li, Quantum Euler-Poisson systems: Existence and of stationary state, Arch. Math.(Brno), 40 (2004), 435-456. 

[21]

A. Jüngel and H. L. Li, Quantum Euler-Poisson systems: Global existence and exponential decay, Quart. Appl. Math., 62 (2004), 569-600.  doi: 10.1090/qam/2086047.

[22]

A. JüngelH. L. Li and A. Matsumura, The relaxation-time limit in the quantum hydrodynamic equations for semiconductors, J. Diff. Eqns., 225 (2006), 440-464.  doi: 10.1016/j.jde.2005.11.007.

[23]

A. JüngelM. C. Mariani and D. Rial, Local exitence of solutins to the transient quantum hydrodynamic equations, Math. Mod. Meth. Appl. Sci., 12 (2002), 485-495.  doi: 10.1142/S0218202502001751.

[24]

A. Jüngel and S. Q. Tang, Numerical approximation of the viscous quantum hydrodynamic model for semiconductors, Applied Numerical Mathematics, 56 (2006), 899-915.  doi: 10.1016/j.apnum.2005.07.003.

[25]

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

[26]

H. L. Li and P. Marcati, Exitence and asymptotic behavior of multi-dimensional quantum hydrodynami cmodel for semiconductors, Comm. Math. Phys., 245 (2004), 215-247.  doi: 10.1007/s00220-003-1001-7.

[27]

H. L. LiG. J. Zhang and K. J. Zhang, Algebraic time decay for the bipolar quantum hydrodynamic model, Math. Models Methods Appl. Sci., 18 (2008), 859-881.  doi: 10.1142/S0218202508002887.

[28]

B. Liang and K. J. Zhang, Steady-state solutions and asymptotic limits on the multi-dimensional semiconductor quantum hydrodynamic model, Math. Models Methods Appl. Sci., 17 (2007), 253-275.  doi: 10.1142/S0218202507001905.

[29]

B. Liang and S. M. Zheng, Exponential decay to a quantum hydrodynamic models for semiconductors, Nonlinear Analysis: Real World Application, 9 (2008), 326-337.  doi: 10.1016/j.nonrwa.2006.11.001.

[30]

E. Madelunge, Quantentheorie in hydrodynamischer form, Z. Phys., 40 (1927), 322-326.  doi: 10.1007/BF01400372.

[31]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations Springer, New York, 1990. doi: 10.1007/978-3-7091-6961-2.

[32]

A. Unterreiter, The thermal equilibrium solution of a generic bipolar quantum hydrodinamic model, Comm. Math. Phys., 188 (1997), 69-88.  doi: 10.1007/s002200050157.

[33]

G. W. Wang, Exponential decay for the viscous bipolar Quantum hydrodynamic model, Ann.of Appl. Math., 31 (2015), 329-335. 

[34]

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

[35]

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

[36]

X. H. Yang, Quasineutral limit of bipolar quantum hydrodynamic model for semiconductors, Frot. Math. China, 6 (2011), 349-362.  doi: 10.1007/s11464-011-0102-4.

[37]

B. Zhang and J. Jerome, On a steady state quantum hydrodynamic model for semiconductors, Nonlinear Anal. TMA, 26 (1996), 845-856.  doi: 10.1016/0362-546X(94)00326-D.

[38]

G. J. ZhangH. L. Li and K. J. Zhang, Semiclassical and relaxation limits of bipolar quantum hydrodynamic model for semiconductors, J. Diff. Eqns., 245 (2008), 1433-1453.  doi: 10.1016/j.jde.2008.06.019.

[39]

G. J. Zhang and K. J. Zhang, On the bipolar multidimensional quantum Euler-Poisson system: The thermal equilibrium solution and semiclassical limit, Nonlinear Anal. TMA, 66 (2007), 2218-2229.  doi: 10.1016/j.na.2006.03.010.

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