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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 & Continuous Dynamical Systems - A, 2016, 36 (3) : 1583-1601. doi: 10.3934/dcds.2016.36.1583
References:
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G. Alì, Global existence of smooth solutions of the $N$-dimensional Euler-Poisson model,, SIAM J. Math. Anal., 35 (2003), 389. doi: 10.1137/S0036141001393225. Google Scholar

[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. doi: 10.1137/S0036141099355174. Google Scholar

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

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F. Chen, Introduction to Plasma Physics and Controlled Fusion,, Vol. 1, (1984). doi: 10.1007/978-1-4757-5595-4. Google Scholar

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

[6]

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

[7]

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

[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. doi: 10.1016/j.jde.2013.07.027. Google Scholar

[9]

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

[10]

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

[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. doi: 10.1016/S0022-0396(03)00122-0. Google Scholar

[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. Google Scholar

[13]

L. Grafakos, Classical and Modern Fourier Analysis,, Pearson/Prentice Hall, (2004). Google Scholar

[14]

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

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

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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. doi: 10.1002/mma.410. Google Scholar

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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. doi: 10.1142/S0218202500000653. Google Scholar

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[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. doi: 10.1137/100810228. Google Scholar

[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. doi: 10.1137/110831647. Google Scholar

[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. doi: 10.1137/100793025. Google Scholar

[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. doi: 10.1016/j.jde.2011.04.007. Google Scholar

[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. doi: 10.1007/s00220-004-1062-2. Google Scholar

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A. Jüngel, Quasi-hydrodynamic Semiconductor Equations,, Progr. Nonlinear Differential Equations Appl., (2001). doi: 10.1007/978-3-0348-8334-4. Google Scholar

[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. doi: 10.1080/03605309908821456. Google Scholar

[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. doi: 10.1017/S0308210500001670. Google Scholar

[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. doi: 10.1016/j.jde.2006.01.001. Google Scholar

[31]

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

[32]

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

[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. doi: 10.1016/j.jde.2011.08.008. Google Scholar

[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. doi: 10.1137/S0036139996312168. Google Scholar

[35]

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

[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. doi: 10.1007/BF00379918. Google Scholar

[37]

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

[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. doi: 10.3934/cpaa.2012.11.1775. Google Scholar

[39]

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

[40]

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

[41]

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

[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. doi: 10.1007/s00205-008-0129-1. Google Scholar

[43]

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

[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. doi: 10.1006/jdeq.1995.1158. Google Scholar

[45]

A. Sitenko and V. Malnev, Plasma Physics Theory,, Appl. Math. Math. Comput., (1995). Google Scholar

[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. doi: 10.1016/j.aim.2014.04.012. Google Scholar

[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. doi: 10.1007/s000330050056. Google Scholar

[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. doi: 10.1006/jdeq.1997.3377. Google Scholar

[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. doi: 10.1088/0951-7715/19/8/012. Google Scholar

[50]

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

[51]

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

[52]

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

[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. doi: 10.1006/jdeq.2000.3799. Google Scholar

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. doi: 10.1137/S0036141001393225. Google Scholar

[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. doi: 10.1137/S0036141099355174. Google Scholar

[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. doi: 10.1016/S0022-0396(02)00157-2. Google Scholar

[4]

F. Chen, Introduction to Plasma Physics and Controlled Fusion,, Vol. 1, (1984). doi: 10.1007/978-1-4757-5595-4. Google Scholar

[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. doi: 10.1007/BF02102592. Google Scholar

[6]

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

[7]

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

[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. doi: 10.1016/j.jde.2013.07.027. Google Scholar

[9]

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

[10]

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

[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. doi: 10.1016/S0022-0396(03)00122-0. Google Scholar

[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. Google Scholar

[13]

L. Grafakos, Classical and Modern Fourier Analysis,, Pearson/Prentice Hall, (2004). Google Scholar

[14]

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

[15]

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

[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. doi: 10.1002/mma.410. Google Scholar

[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. doi: 10.1016/S0022-0396(03)00063-9. Google Scholar

[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. doi: 10.1006/jdeq.2000.3825. Google Scholar

[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. doi: 10.1142/S0218202500000653. Google Scholar

[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. doi: 10.1006/jdeq.2000.3780. Google Scholar

[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. doi: 10.1016/j.jde.2009.07.032. Google Scholar

[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. doi: 10.1137/100810228. Google Scholar

[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. doi: 10.1137/110831647. Google Scholar

[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. doi: 10.1137/100793025. Google Scholar

[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. doi: 10.1016/j.jde.2011.04.007. Google Scholar

[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. doi: 10.1007/s00220-004-1062-2. Google Scholar

[27]

A. Jüngel, Quasi-hydrodynamic Semiconductor Equations,, Progr. Nonlinear Differential Equations Appl., (2001). doi: 10.1007/978-3-0348-8334-4. Google Scholar

[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. doi: 10.1080/03605309908821456. Google Scholar

[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. doi: 10.1017/S0308210500001670. Google Scholar

[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. doi: 10.1016/j.jde.2006.01.001. Google Scholar

[31]

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

[32]

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

[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. doi: 10.1016/j.jde.2011.08.008. Google Scholar

[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. doi: 10.1137/S0036139996312168. Google Scholar

[35]

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

[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. doi: 10.1007/BF00379918. Google Scholar

[37]

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

[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. doi: 10.3934/cpaa.2012.11.1775. Google Scholar

[39]

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

[40]

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

[41]

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

[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. doi: 10.1007/s00205-008-0129-1. Google Scholar

[43]

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

[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. doi: 10.1006/jdeq.1995.1158. Google Scholar

[45]

A. Sitenko and V. Malnev, Plasma Physics Theory,, Appl. Math. Math. Comput., (1995). Google Scholar

[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. doi: 10.1016/j.aim.2014.04.012. Google Scholar

[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. doi: 10.1007/s000330050056. Google Scholar

[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. doi: 10.1006/jdeq.1997.3377. Google Scholar

[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. doi: 10.1088/0951-7715/19/8/012. Google Scholar

[50]

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

[51]

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

[52]

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

[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. doi: 10.1006/jdeq.2000.3799. Google Scholar

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