• Previous Article
    Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach
  • DCDS Home
  • This Issue
  • Next Article
    Spreading speeds and transition fronts of lattice KPP equations in time heterogeneous media
September  2017, 37(9): 4677-4696. doi: 10.3934/dcds.2017201

Stability of stationary solutions to the compressible bipolar Euler-Poisson equations

1. 

School of Mathematical Sciences, Xiamen University, Fujian, Xiamen 361005, China,

2. 

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

* Corresponding author: Zhong Tan, tan85@xmu.edu.cn

Received  April 2016 Revised  April 2017 Published  June 2017

Fund Project: The authors are supported by National Natural Science Foundation of China-NSAF (No.11271305, 11531010)

In this paper, we study the compressible bipolar Euler-Poisson equations with a non-flat doping profile in three-dimensional space. The existence and uniqueness of the non-constant stationary solutions are established under the smallness assumption on the gradient of the doping profile. Then we show the global existence of smooth solutions to the Cauchy problem near the stationary state provided the $H^3$ norms of the initial density and velocity are small, but the higher derivatives can be arbitrarily large.

Citation: Hong Cai, Zhong Tan. Stability of stationary solutions to the compressible bipolar Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4677-4696. doi: 10.3934/dcds.2017201
References:
[1]

G. Alí 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. Google Scholar

[2]

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. Google Scholar

[3]

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. Google Scholar

[4]

D. DonatelliM. MeiB. 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. Google Scholar

[5]

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. Google Scholar

[6]

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. Google Scholar

[7]

I. GasserL. 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. Google Scholar

[8]

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. Google Scholar

[9]

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. Google Scholar

[10]

L. J. HanJ. J. Zhang and B. L. Guo, Global smooth solution for a kind of two-fluid system in plasmas, J. Differential Equations, 252 (2012), 3453-3481. doi: 10.1016/j.jde.2011.12.004. Google Scholar

[11]

L. HsiaoP. 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. Google Scholar

[12]

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. Google Scholar

[13]

F. M. Huang and Y. P. Li, Large time behavior and quasineutral limit of solutions to a bipolar hydrodynamic model with large data and vacuum, Discrete Contin. Dyn. Syst., 24 (2009), 455-470. doi: 10.3934/dcds.2009.24.455. Google Scholar

[14]

F. M. HuangM. 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. Google Scholar

[15]

F. M. HuangM. MeiY. 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. Google Scholar

[16]

F. M. HuangM. MeiY. 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. Google Scholar

[17]

F. M. HuangM. MeiY. 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. Google Scholar

[18]

Q. C. Ju, Asymptotic behavior of global smooth solutions to the Euler-Poisson system in semiconductors, J. Partial Differential Equations, 15 (2002), 89-96. Google Scholar

[19]

Q. C. Ju, Global smooth solutions to the multidimensional hydrodynamic model for plasmas with insulating boundary conditions, J. Math. Anal. Appl., 336 (2007), 888-904. doi: 10.1016/j.jmaa.2007.03.038. Google Scholar

[20]

H. L. LiP. Markowich and M. Mei, Asymptotic behavior of solutions of the hydrodynamic model of semiconductors, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 359-378. Google Scholar

[21]

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. Google Scholar

[22]

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. Google Scholar

[23]

Q. Q. Liu and C. J. Zhu, Asymptotic stability of stationary solutions to the compressible Euler-Maxwell equations, Indiana Univ. Math. J., 62 (2013), 1203-1235. doi: 10.1512/iumj.2013.62.5047. Google Scholar

[24]

T. LuoR. 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. Google Scholar

[25]

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

[26]

M. MeiB. Rubino and R. Sampalmieri, Asymptotic behavior of solutions to the bipolar hydrodynamic model of semiconductors in bounded domain, Kinet. Relat. Models, 5 (2012), 537-550. doi: 10.3934/krm.2012.5.537. Google Scholar

[27]

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. Google Scholar

[28]

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

[29]

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. Google Scholar

[30]

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. Google Scholar

[31]

N. Tsuge, Uniqueness of the stationary solutions for a fluid dynamical model of semiconductors, Osaka J. Math., 46 (2009), 931-937. Google Scholar

[32]

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. Google Scholar

[33]

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. Google Scholar

[34]

Y. Wang and Z. Tan, Stability of steady states of the compressible Euler-Poisson system in $\mathbb{R}^3$, J. Math. Anal. Appl., 422 (2015), 1058-1071. doi: 10.1016/j.jmaa.2014.09.047. Google Scholar

[35]

Z. Y. Zhao and Y. P. Li, Global existence and optimal decay rate of the compressible bipolar Navier-Stokes-Poisson equations with external force, Nonlinear Anal. Real World Appl., 16 (2014), 146-162. doi: 10.1016/j.nonrwa.2013.09.014. Google Scholar

[36]

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. Google Scholar

show all references

References:
[1]

G. Alí 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. Google Scholar

[2]

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. Google Scholar

[3]

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. Google Scholar

[4]

D. DonatelliM. MeiB. 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. Google Scholar

[5]

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. Google Scholar

[6]

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. Google Scholar

[7]

I. GasserL. 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. Google Scholar

[8]

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. Google Scholar

[9]

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. Google Scholar

[10]

L. J. HanJ. J. Zhang and B. L. Guo, Global smooth solution for a kind of two-fluid system in plasmas, J. Differential Equations, 252 (2012), 3453-3481. doi: 10.1016/j.jde.2011.12.004. Google Scholar

[11]

L. HsiaoP. 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. Google Scholar

[12]

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. Google Scholar

[13]

F. M. Huang and Y. P. Li, Large time behavior and quasineutral limit of solutions to a bipolar hydrodynamic model with large data and vacuum, Discrete Contin. Dyn. Syst., 24 (2009), 455-470. doi: 10.3934/dcds.2009.24.455. Google Scholar

[14]

F. M. HuangM. 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. Google Scholar

[15]

F. M. HuangM. MeiY. 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. Google Scholar

[16]

F. M. HuangM. MeiY. 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. Google Scholar

[17]

F. M. HuangM. MeiY. 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. Google Scholar

[18]

Q. C. Ju, Asymptotic behavior of global smooth solutions to the Euler-Poisson system in semiconductors, J. Partial Differential Equations, 15 (2002), 89-96. Google Scholar

[19]

Q. C. Ju, Global smooth solutions to the multidimensional hydrodynamic model for plasmas with insulating boundary conditions, J. Math. Anal. Appl., 336 (2007), 888-904. doi: 10.1016/j.jmaa.2007.03.038. Google Scholar

[20]

H. L. LiP. Markowich and M. Mei, Asymptotic behavior of solutions of the hydrodynamic model of semiconductors, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 359-378. Google Scholar

[21]

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. Google Scholar

[22]

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. Google Scholar

[23]

Q. Q. Liu and C. J. Zhu, Asymptotic stability of stationary solutions to the compressible Euler-Maxwell equations, Indiana Univ. Math. J., 62 (2013), 1203-1235. doi: 10.1512/iumj.2013.62.5047. Google Scholar

[24]

T. LuoR. 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. Google Scholar

[25]

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

[26]

M. MeiB. Rubino and R. Sampalmieri, Asymptotic behavior of solutions to the bipolar hydrodynamic model of semiconductors in bounded domain, Kinet. Relat. Models, 5 (2012), 537-550. doi: 10.3934/krm.2012.5.537. Google Scholar

[27]

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. Google Scholar

[28]

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

[29]

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. Google Scholar

[30]

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. Google Scholar

[31]

N. Tsuge, Uniqueness of the stationary solutions for a fluid dynamical model of semiconductors, Osaka J. Math., 46 (2009), 931-937. Google Scholar

[32]

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. Google Scholar

[33]

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. Google Scholar

[34]

Y. Wang and Z. Tan, Stability of steady states of the compressible Euler-Poisson system in $\mathbb{R}^3$, J. Math. Anal. Appl., 422 (2015), 1058-1071. doi: 10.1016/j.jmaa.2014.09.047. Google Scholar

[35]

Z. Y. Zhao and Y. P. Li, Global existence and optimal decay rate of the compressible bipolar Navier-Stokes-Poisson equations with external force, Nonlinear Anal. Real World Appl., 16 (2014), 146-162. doi: 10.1016/j.nonrwa.2013.09.014. Google Scholar

[36]

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. Google Scholar

[1]

La-Su Mai, Kaijun Zhang. Asymptotic stability of steady state solutions for the relativistic Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 981-1004. doi: 10.3934/dcds.2016.36.981

[2]

Yeping Li, Jie Liao. Stability and $ L^{p}$ convergence rates of planar diffusion waves for three-dimensional bipolar Euler-Poisson systems. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1281-1302. doi: 10.3934/cpaa.2019062

[3]

A. Alexandrou Himonas, Gerard Misiołek, Feride Tiǧlay. On unique continuation for the modified Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 515-529. doi: 10.3934/dcds.2007.19.515

[4]

Yeping Li. Existence and some limit analysis of stationary solutions for a multi-dimensional bipolar Euler-Poisson system. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 345-360. doi: 10.3934/dcdsb.2011.16.345

[5]

Masahiro Suzuki. Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics. Kinetic & Related Models, 2011, 4 (2) : 569-588. doi: 10.3934/krm.2011.4.569

[6]

Yongcai Geng. Singularity formation for relativistic Euler and Euler-Poisson equations with repulsive force. Communications on Pure & Applied Analysis, 2015, 14 (2) : 549-564. doi: 10.3934/cpaa.2015.14.549

[7]

Manwai Yuen. Cylindrical blowup solutions to the isothermal Euler-Poisson equations. Conference Publications, 2011, 2011 (Special) : 1448-1456. doi: 10.3934/proc.2011.2011.1448

[8]

Jiang Xu, Ting Zhang. Zero-electron-mass limit of Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4743-4768. doi: 10.3934/dcds.2013.33.4743

[9]

Haigang Li, Jiguang Bao. Euler-Poisson equations related to general compressible rotating fluids. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1085-1096. doi: 10.3934/dcds.2011.29.1085

[10]

Sasho Popov, Jean-Marie Strelcyn. The Euler-Poisson equations: An elementary approach to integrability conditions. Journal of Geometric Mechanics, 2018, 10 (3) : 293-329. doi: 10.3934/jgm.2018011

[11]

Ming Mei, Yong Wang. Stability of stationary waves for full Euler-Poisson system in multi-dimensional space. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1775-1807. doi: 10.3934/cpaa.2012.11.1775

[12]

Zhigang Wu, Weike Wang. Pointwise estimates of solutions for the Euler-Poisson equations with damping in multi-dimensions. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 1101-1117. doi: 10.3934/dcds.2010.26.1101

[13]

Xueke Pu. Quasineutral limit of the Euler-Poisson system under strong magnetic fields. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2095-2111. doi: 10.3934/dcdss.2016086

[14]

Shu Wang, Chundi Liu. Boundary Layer Problem and Quasineutral Limit of Compressible Euler-Poisson System. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2177-2199. doi: 10.3934/cpaa.2017108

[15]

Myoungjean Bae, Yong Park. Radial transonic shock solutions of Euler-Poisson system in convergent nozzles. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 773-791. doi: 10.3934/dcdss.2018049

[16]

Masahiro Suzuki. Asymptotic stability of a boundary layer to the Euler--Poisson equations for a multicomponent plasma. Kinetic & Related Models, 2016, 9 (3) : 587-603. doi: 10.3934/krm.2016008

[17]

Qiangchang Ju, Hailiang Li, Yong Li, Song Jiang. Quasi-neutral limit of the two-fluid Euler-Poisson system. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1577-1590. doi: 10.3934/cpaa.2010.9.1577

[18]

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

[19]

Corrado Lattanzio, Pierangelo Marcati. The relaxation to the drift-diffusion system for the 3-$D$ isentropic Euler-Poisson model for semiconductors. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 449-455. doi: 10.3934/dcds.1999.5.449

[20]

Laiqing Meng, Jia Yuan, Xiaoxin Zheng. Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3413-3441. doi: 10.3934/dcds.2019141

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (18)
  • HTML views (26)
  • Cited by (0)

Other articles
by authors

[Back to Top]