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March  2016, 36(3): 1583-1601. doi: 10.3934/dcds.2016.36.1583

Large-time behavior of the full compressible Euler-Poisson system without the temperature damping

1. 

School of Mathematical Sciences and Fujian Provincial Key Laboratory, on Mathematical Modeling and Scientific Computing, Xiamen University, Xiamen, 361005, China

2. 

Department of Mathematics, University of Southern California, Los Angeles, CA 90089, United States

Received  January 2015 Revised  April 2015 Published  August 2015

We study the three-dimensional full compressible Euler-Poisson system without the temperature damping. Using a general energy method, we prove the optimal decay rates of the solutions and their higher order derivatives. We show that the optimal decay rates is algebraic but not exponential since the absence of temperature damping.
Citation: Zhong Tan, Yong Wang, Fanhui Xu. Large-time behavior of the full compressible Euler-Poisson system without the temperature damping. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1583-1601. doi: 10.3934/dcds.2016.36.1583
References:
[1]

G. Alì, Global existence of smooth solutions of the $N$-dimensional Euler-Poisson model, SIAM J. Math. Anal., 35 (2003), 389-422. doi: 10.1137/S0036141001393225.

[2]

G. Alì, D. Bini and S. Rionero, Global existence and relaxation limit for smooth solutions to the Euler-Poisson model for semiconductors, SIAM J. Math. Anal., 32 (2000), 572-587. doi: 10.1137/S0036141099355174.

[3]

G. Alì and A. Jüngel, Global smooth solutions to the multi-dimensional hydrodynamic model for two-carrier plasmas, J. Differential Equations, 190 (2003), 663-685. doi: 10.1016/S0022-0396(02)00157-2.

[4]

F. Chen, Introduction to Plasma Physics and Controlled Fusion, Vol. 1, Plenum Press, New York, 1984. doi: 10.1007/978-1-4757-5595-4.

[5]

G. Q. Chen and D. H. Wang, Convergence of shock capturing schemes for the compressible Euler-Poisson equations, Comm. Math. Phys., 179 (1996), 333-364. doi: 10.1007/BF02102592.

[6]

P. Degond and P. A. Markowich, On a one-dimensional steady-state hydrodynamic model, Appl. Math. Lett., 3 (1990), 25-29. doi: 10.1016/0893-9659(90)90130-4.

[7]

P. Degond and P. A. Markowich, A steady-state potential flow model for semiconductors, Ann. Mat. Pura Appl., 165 (1993), 87-98. doi: 10.1007/BF01765842.

[8]

D. Donatelli, M. Mei, B. Rubino and R. Sampalmieri, Asymptotic behavior of solutions to Euler-Poisson equations for bipolar hydrodynamic model of semiconductors, J. Differential Equations, 255 (2013), 3150-3184. doi: 10.1016/j.jde.2013.07.027.

[9]

W. F. Fang and K. Ito, Steady-state solutions of a one-dimensional hydrodynamic model for semiconductors, J. Differential Equations, 133 (1997), 224-244. doi: 10.1006/jdeq.1996.3203.

[10]

I. Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductor, Comm. Partial Differential Equations, 17 (1992), 553-577. doi: 10.1080/03605309208820853.

[11]

I. Gasser, L. Hsiao and H. L. Li, Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors, J. Differential Equations, 192 (2003), 326-359. doi: 10.1016/S0022-0396(03)00122-0.

[12]

I. Gasser and R. Natalini, The energy transport and the drift diffusion equations as relaxation limits of the hydrodynamic model for semiconductors, Quart. Appl. Math., 57 (1999), 269-282.

[13]

L. Grafakos, Classical and Modern Fourier Analysis, Pearson/Prentice Hall, Englewood Cliffs, NJ, 2004.

[14]

Y. Guo and W. Strauss, Stability of semiconductor states with insulating and contact boundary conditions, Arch. Ration. Mech. Anal., 179 (2006), 1-30. doi: 10.1007/s00205-005-0369-2.

[15]

Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations, 37 (2012), 2165-2208. doi: 10.1080/03605302.2012.696296.

[16]

L. Hsiao, Q. C. Ju and S. Wang, The asymptotic behaviour of global smooth solutions to the multi-dimensional hydrodynamic model for semiconductors, Math. Meth. Appl. Sci., 26 (2003), 1187-1210. doi: 10.1002/mma.410.

[17]

L. Hsiao, P. A. Markowich and S. Wang, The asymptotic behavior of globally smooth solutions of the multidimensional isentropic hydrodynamic model for semiconductors, J. Differential Equations, 192 (2003), 111-133. doi: 10.1016/S0022-0396(03)00063-9.

[18]

L. Hsiao and T. Yang, Asymptotics of initial boundary value problems for hydrodynamic and drift diffusion models for semiconductors, J. Differential Equations, 170 (2001), 472-493. doi: 10.1006/jdeq.2000.3825.

[19]

L. Hsiao and K. J. Zhang, The global weak solution and relaxation limits of the initial boundary value problem to the bipolar hydrodynamic model for semiconductors, Math. Models Methods Appl. Sci., 10 (2000), 1333-1361. doi: 10.1142/S0218202500000653.

[20]

L. Hsiao and K. J. Zhang, The relaxation of the hydrodynamic model for semiconductors to the drift-diffusion equations, J. Differential Equations, 165 (2000), 315-354. doi: 10.1006/jdeq.2000.3780.

[21]

F. M. Huang, T. H. Li and H. M. Yu, Weak solutions to isothermal hydrodynamic model for semiconductor devices, J. Differential Equations, 247 (2009), 3070-3099. doi: 10.1016/j.jde.2009.07.032.

[22]

F. M. Huang, M. Mei and Y. Wang, Large time behavior of solutions to $n$-dimensional bipolar hydrodynamic model for semiconductors, SIAM J. Math. Anal., 43 (2011), 1595-1630. doi: 10.1137/100810228.

[23]

F. M. Huang, M. Mei, Y. Wang and T. Yang, Long-time behavior of solutions to the bipolar hydrodynamic model of semiconductors with boundary effect, SIAM J. Math. Anal., 44 (2012), 1134-1164. doi: 10.1137/110831647.

[24]

F. M. Huang, M. Mei, Y. Wang and H. M. Yu, Asymptotic convergence to stationary waves for unipolar hydrodynamic model of semiconductors, SIAM J. Math. Anal., 43 (2011), 411-429. doi: 10.1137/100793025.

[25]

F. M. Huang, M. Mei, Y. Wang and H. M. Yu, Asymptotic convergence to planar stationary waves for multi-dimensional unipolar hydrodynamic model of semiconductors, J. Differential Equations, 251 (2011), 1305-1331. doi: 10.1016/j.jde.2011.04.007.

[26]

N. Ju, Existence and uniqueness of the solution to the dissipative $2D$ Quasi-Geostrophic equations in the Sobolev space, Commun. Math. Phys., 251 (2004), 365-376. doi: 10.1007/s00220-004-1062-2.

[27]

A. Jüngel, Quasi-hydrodynamic Semiconductor Equations, Progr. Nonlinear Differential Equations Appl., Vol. 41, Birkhäuser Verlag, Basel, Boston, Berlin, 2001. doi: 10.1007/978-3-0348-8334-4.

[28]

A. Jüngel and Y. J. Peng, A hierarchy of hydrodynamic models for plasmas: Zero-relaxation-time limits, Comm. Partial Differential Equations, 24 (1999), 1007-1033. doi: 10.1080/03605309908821456.

[29]

H. L. Li, P. Markowich and M. Mei, Asymptotic behaviour of solutions of the hydrodynamic model of semiconductors, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 359-378. doi: 10.1017/S0308210500001670.

[30]

Y. P. Li, Global existence and asymptotic behavior for a multidimensional nonisentropic hydrodynamic semiconductor model with the heat source, J. Differential Equations, 225 (2006), 134-167. doi: 10.1016/j.jde.2006.01.001.

[31]

Y. P. Li, Diffusion relaxation limit of a nonisentropic hydrodynamic model for semiconductors, Math. Methods Appl. Sci., 30 (2007), 2247-2261. doi: 10.1002/mma.890.

[32]

Y. P. Li, Global existence and asymptotic behavior of solutions to the nonisentropic bipolar hydrodynamic models, J. Differential Equations, 250 (2011), 1285-1309. doi: 10.1016/j.jde.2010.08.018.

[33]

Y. P. Li and X. F. Yang, Global existence and asymptotic behavior of the solutions to the three-dimensional bipolar Euler-Poisson systems, J. Differential Equations, 252 (2012), 768-791. doi: 10.1016/j.jde.2011.08.008.

[34]

T. Luo, R. Natalini and Z. P. Xin, Large time behavior of the solutions to a hydrodynamic model for semiconductors, SIAM J. Appl. Math., 59 (1999), 810-830. doi: 10.1137/S0036139996312168.

[35]

P. A. Markowich, On steady state Euler-Poisson models for semiconductors, Z. Angew. Math. Phys., 42 (1991), 389-407. doi: 10.1007/BF00945711.

[36]

P. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation, Arch. Ration. Mech. Anal., 129 (1995), 129-145. doi: 10.1007/BF00379918.

[37]

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

[38]

M. Mei and Y. Wang, Stability of stationary waves for full Euler-Poisson system in multi-dimensional space, Commun. Pure Appl. Anal., 11 (2012), 1775-1807. doi: 10.3934/cpaa.2012.11.1775.

[39]

R. Natalini, The bipolar hydrodynamic model for semiconductors and the drift-diffusion equations, J. Math. Anal. Appl., 198 (1996), 262-281. doi: 10.1006/jmaa.1996.0081.

[40]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.

[41]

S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors, Osaka J. Math., 44 (2007), 639-665.

[42]

S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a thermal hydrodynamic model for semiconductors, Arch. Ration. Mech. Anal., 192 (2009), 187-215. doi: 10.1007/s00205-008-0129-1.

[43]

Y. J. Peng and J. Xu, Global well-posedness of the hydrodynamic model for two-carrier plasmas, J. Differential Equations, 255 (2013), 3447-3471. doi: 10.1016/j.jde.2013.07.045.

[44]

F. Poupaud, M. Rascle and J. P. Vila, Global solutions to the isothermal Euler-Poisson system with arbitrarily large data, J. Differential Equations, 123 (1995), 93-121. doi: 10.1006/jdeq.1995.1158.

[45]

A. Sitenko and V. Malnev, Plasma Physics Theory, Appl. Math. Math. Comput., Vol. 10, Chapman & Hall, London, 1995.

[46]

V. Sohinger and R. M. Strain, The Boltzmann equation, Besov spaces, and optimal time decay rates in $\mathbbR_x^n$, Adv. Math., 261 (2014), 274-332. doi: 10.1016/j.aim.2014.04.012.

[47]

D. H. Wang, Global solutions to the Euler-Poisson equations of two-carrier types in one dimension, Z. Angew. Math. Phys., 48 (1997), 680-693. doi: 10.1007/s000330050056.

[48]

D. H. Wang and G. Q. Chen, Formation of singularities in compressible Euler-Poisson fluids with heat diffusion and damping relaxation, J. Differential Equations, 144 (1998), 44-65. doi: 10.1006/jdeq.1997.3377.

[49]

D. H. Wang and Z. J. Wang, Large BV solutions to the compressible isothermal Euler-Poisson equations with spherical symmetry, Nonlinearity, 19 (2006), 1985-2004. doi: 10.1088/0951-7715/19/8/012.

[50]

Y. J. Wang, Decay of the Navier-Stokes-Poisson equations, J. Differential Equations, 253 (2012), 273-297. doi: 10.1016/j.jde.2012.03.006.

[51]

J. Xu, Energy-transport and drift-diffusion limits of nonisentropic Euler-Poisson equations, J. Differential Equations, 252 (2012), 915-940. doi: 10.1016/j.jde.2011.09.040.

[52]

B. Zhang, Convergence of the Godunov scheme for a simplified one-dimensional hydrodynamic model for semiconductor devices, Comm. Math. Phys., 157 (1993), 1-22. doi: 10.1007/BF02098016.

[53]

C. Zhu and H. Hattori, Stability of steady state solutions for an isentropic hydrodynamic model of semiconductors of two species, J. Differential Equations, 166 (2000), 1-32. doi: 10.1006/jdeq.2000.3799.

show all references

References:
[1]

G. Alì, Global existence of smooth solutions of the $N$-dimensional Euler-Poisson model, SIAM J. Math. Anal., 35 (2003), 389-422. doi: 10.1137/S0036141001393225.

[2]

G. Alì, D. Bini and S. Rionero, Global existence and relaxation limit for smooth solutions to the Euler-Poisson model for semiconductors, SIAM J. Math. Anal., 32 (2000), 572-587. doi: 10.1137/S0036141099355174.

[3]

G. Alì and A. Jüngel, Global smooth solutions to the multi-dimensional hydrodynamic model for two-carrier plasmas, J. Differential Equations, 190 (2003), 663-685. doi: 10.1016/S0022-0396(02)00157-2.

[4]

F. Chen, Introduction to Plasma Physics and Controlled Fusion, Vol. 1, Plenum Press, New York, 1984. doi: 10.1007/978-1-4757-5595-4.

[5]

G. Q. Chen and D. H. Wang, Convergence of shock capturing schemes for the compressible Euler-Poisson equations, Comm. Math. Phys., 179 (1996), 333-364. doi: 10.1007/BF02102592.

[6]

P. Degond and P. A. Markowich, On a one-dimensional steady-state hydrodynamic model, Appl. Math. Lett., 3 (1990), 25-29. doi: 10.1016/0893-9659(90)90130-4.

[7]

P. Degond and P. A. Markowich, A steady-state potential flow model for semiconductors, Ann. Mat. Pura Appl., 165 (1993), 87-98. doi: 10.1007/BF01765842.

[8]

D. Donatelli, M. Mei, B. Rubino and R. Sampalmieri, Asymptotic behavior of solutions to Euler-Poisson equations for bipolar hydrodynamic model of semiconductors, J. Differential Equations, 255 (2013), 3150-3184. doi: 10.1016/j.jde.2013.07.027.

[9]

W. F. Fang and K. Ito, Steady-state solutions of a one-dimensional hydrodynamic model for semiconductors, J. Differential Equations, 133 (1997), 224-244. doi: 10.1006/jdeq.1996.3203.

[10]

I. Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductor, Comm. Partial Differential Equations, 17 (1992), 553-577. doi: 10.1080/03605309208820853.

[11]

I. Gasser, L. Hsiao and H. L. Li, Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors, J. Differential Equations, 192 (2003), 326-359. doi: 10.1016/S0022-0396(03)00122-0.

[12]

I. Gasser and R. Natalini, The energy transport and the drift diffusion equations as relaxation limits of the hydrodynamic model for semiconductors, Quart. Appl. Math., 57 (1999), 269-282.

[13]

L. Grafakos, Classical and Modern Fourier Analysis, Pearson/Prentice Hall, Englewood Cliffs, NJ, 2004.

[14]

Y. Guo and W. Strauss, Stability of semiconductor states with insulating and contact boundary conditions, Arch. Ration. Mech. Anal., 179 (2006), 1-30. doi: 10.1007/s00205-005-0369-2.

[15]

Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations, 37 (2012), 2165-2208. doi: 10.1080/03605302.2012.696296.

[16]

L. Hsiao, Q. C. Ju and S. Wang, The asymptotic behaviour of global smooth solutions to the multi-dimensional hydrodynamic model for semiconductors, Math. Meth. Appl. Sci., 26 (2003), 1187-1210. doi: 10.1002/mma.410.

[17]

L. Hsiao, P. A. Markowich and S. Wang, The asymptotic behavior of globally smooth solutions of the multidimensional isentropic hydrodynamic model for semiconductors, J. Differential Equations, 192 (2003), 111-133. doi: 10.1016/S0022-0396(03)00063-9.

[18]

L. Hsiao and T. Yang, Asymptotics of initial boundary value problems for hydrodynamic and drift diffusion models for semiconductors, J. Differential Equations, 170 (2001), 472-493. doi: 10.1006/jdeq.2000.3825.

[19]

L. Hsiao and K. J. Zhang, The global weak solution and relaxation limits of the initial boundary value problem to the bipolar hydrodynamic model for semiconductors, Math. Models Methods Appl. Sci., 10 (2000), 1333-1361. doi: 10.1142/S0218202500000653.

[20]

L. Hsiao and K. J. Zhang, The relaxation of the hydrodynamic model for semiconductors to the drift-diffusion equations, J. Differential Equations, 165 (2000), 315-354. doi: 10.1006/jdeq.2000.3780.

[21]

F. M. Huang, T. H. Li and H. M. Yu, Weak solutions to isothermal hydrodynamic model for semiconductor devices, J. Differential Equations, 247 (2009), 3070-3099. doi: 10.1016/j.jde.2009.07.032.

[22]

F. M. Huang, M. Mei and Y. Wang, Large time behavior of solutions to $n$-dimensional bipolar hydrodynamic model for semiconductors, SIAM J. Math. Anal., 43 (2011), 1595-1630. doi: 10.1137/100810228.

[23]

F. M. Huang, M. Mei, Y. Wang and T. Yang, Long-time behavior of solutions to the bipolar hydrodynamic model of semiconductors with boundary effect, SIAM J. Math. Anal., 44 (2012), 1134-1164. doi: 10.1137/110831647.

[24]

F. M. Huang, M. Mei, Y. Wang and H. M. Yu, Asymptotic convergence to stationary waves for unipolar hydrodynamic model of semiconductors, SIAM J. Math. Anal., 43 (2011), 411-429. doi: 10.1137/100793025.

[25]

F. M. Huang, M. Mei, Y. Wang and H. M. Yu, Asymptotic convergence to planar stationary waves for multi-dimensional unipolar hydrodynamic model of semiconductors, J. Differential Equations, 251 (2011), 1305-1331. doi: 10.1016/j.jde.2011.04.007.

[26]

N. Ju, Existence and uniqueness of the solution to the dissipative $2D$ Quasi-Geostrophic equations in the Sobolev space, Commun. Math. Phys., 251 (2004), 365-376. doi: 10.1007/s00220-004-1062-2.

[27]

A. Jüngel, Quasi-hydrodynamic Semiconductor Equations, Progr. Nonlinear Differential Equations Appl., Vol. 41, Birkhäuser Verlag, Basel, Boston, Berlin, 2001. doi: 10.1007/978-3-0348-8334-4.

[28]

A. Jüngel and Y. J. Peng, A hierarchy of hydrodynamic models for plasmas: Zero-relaxation-time limits, Comm. Partial Differential Equations, 24 (1999), 1007-1033. doi: 10.1080/03605309908821456.

[29]

H. L. Li, P. Markowich and M. Mei, Asymptotic behaviour of solutions of the hydrodynamic model of semiconductors, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 359-378. doi: 10.1017/S0308210500001670.

[30]

Y. P. Li, Global existence and asymptotic behavior for a multidimensional nonisentropic hydrodynamic semiconductor model with the heat source, J. Differential Equations, 225 (2006), 134-167. doi: 10.1016/j.jde.2006.01.001.

[31]

Y. P. Li, Diffusion relaxation limit of a nonisentropic hydrodynamic model for semiconductors, Math. Methods Appl. Sci., 30 (2007), 2247-2261. doi: 10.1002/mma.890.

[32]

Y. P. Li, Global existence and asymptotic behavior of solutions to the nonisentropic bipolar hydrodynamic models, J. Differential Equations, 250 (2011), 1285-1309. doi: 10.1016/j.jde.2010.08.018.

[33]

Y. P. Li and X. F. Yang, Global existence and asymptotic behavior of the solutions to the three-dimensional bipolar Euler-Poisson systems, J. Differential Equations, 252 (2012), 768-791. doi: 10.1016/j.jde.2011.08.008.

[34]

T. Luo, R. Natalini and Z. P. Xin, Large time behavior of the solutions to a hydrodynamic model for semiconductors, SIAM J. Appl. Math., 59 (1999), 810-830. doi: 10.1137/S0036139996312168.

[35]

P. A. Markowich, On steady state Euler-Poisson models for semiconductors, Z. Angew. Math. Phys., 42 (1991), 389-407. doi: 10.1007/BF00945711.

[36]

P. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation, Arch. Ration. Mech. Anal., 129 (1995), 129-145. doi: 10.1007/BF00379918.

[37]

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

[38]

M. Mei and Y. Wang, Stability of stationary waves for full Euler-Poisson system in multi-dimensional space, Commun. Pure Appl. Anal., 11 (2012), 1775-1807. doi: 10.3934/cpaa.2012.11.1775.

[39]

R. Natalini, The bipolar hydrodynamic model for semiconductors and the drift-diffusion equations, J. Math. Anal. Appl., 198 (1996), 262-281. doi: 10.1006/jmaa.1996.0081.

[40]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.

[41]

S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors, Osaka J. Math., 44 (2007), 639-665.

[42]

S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a thermal hydrodynamic model for semiconductors, Arch. Ration. Mech. Anal., 192 (2009), 187-215. doi: 10.1007/s00205-008-0129-1.

[43]

Y. J. Peng and J. Xu, Global well-posedness of the hydrodynamic model for two-carrier plasmas, J. Differential Equations, 255 (2013), 3447-3471. doi: 10.1016/j.jde.2013.07.045.

[44]

F. Poupaud, M. Rascle and J. P. Vila, Global solutions to the isothermal Euler-Poisson system with arbitrarily large data, J. Differential Equations, 123 (1995), 93-121. doi: 10.1006/jdeq.1995.1158.

[45]

A. Sitenko and V. Malnev, Plasma Physics Theory, Appl. Math. Math. Comput., Vol. 10, Chapman & Hall, London, 1995.

[46]

V. Sohinger and R. M. Strain, The Boltzmann equation, Besov spaces, and optimal time decay rates in $\mathbbR_x^n$, Adv. Math., 261 (2014), 274-332. doi: 10.1016/j.aim.2014.04.012.

[47]

D. H. Wang, Global solutions to the Euler-Poisson equations of two-carrier types in one dimension, Z. Angew. Math. Phys., 48 (1997), 680-693. doi: 10.1007/s000330050056.

[48]

D. H. Wang and G. Q. Chen, Formation of singularities in compressible Euler-Poisson fluids with heat diffusion and damping relaxation, J. Differential Equations, 144 (1998), 44-65. doi: 10.1006/jdeq.1997.3377.

[49]

D. H. Wang and Z. J. Wang, Large BV solutions to the compressible isothermal Euler-Poisson equations with spherical symmetry, Nonlinearity, 19 (2006), 1985-2004. doi: 10.1088/0951-7715/19/8/012.

[50]

Y. J. Wang, Decay of the Navier-Stokes-Poisson equations, J. Differential Equations, 253 (2012), 273-297. doi: 10.1016/j.jde.2012.03.006.

[51]

J. Xu, Energy-transport and drift-diffusion limits of nonisentropic Euler-Poisson equations, J. Differential Equations, 252 (2012), 915-940. doi: 10.1016/j.jde.2011.09.040.

[52]

B. Zhang, Convergence of the Godunov scheme for a simplified one-dimensional hydrodynamic model for semiconductor devices, Comm. Math. Phys., 157 (1993), 1-22. doi: 10.1007/BF02098016.

[53]

C. Zhu and H. Hattori, Stability of steady state solutions for an isentropic hydrodynamic model of semiconductors of two species, J. Differential Equations, 166 (2000), 1-32. doi: 10.1006/jdeq.2000.3799.

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