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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 and Continuous Dynamical Systems, 2017, 37 (9) : 4677-4696. doi: 10.3934/dcds.2017201
##### References:
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Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductor, Comm. Partial Differential Equations, 17 (1992), 553-577.  doi: 10.1080/03605309208820853. [7] 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. [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. [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. [10] L. J. Han, J. 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. [11] 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. [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. [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. [14] 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. [15] 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. [16] 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. [17] 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. [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. [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. [20] H. L. Li, P. Markowich and M. Mei, Asymptotic behavior of solutions of the hydrodynamic model of semiconductors, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 359-378. [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. [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. [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. [24] 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. [25] P. A. Markowich, On steady state Euler-Poisson models for semiconductors, Z. Angew. Math. Phys., 42 (1991), 389-407.  doi: 10.1007/BF00945711. [26] M. Mei, B. 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. [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. [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. [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. [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. [31] N. Tsuge, Uniqueness of the stationary solutions for a fluid dynamical model of semiconductors, Osaka J. Math., 46 (2009), 931-937. [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. [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. [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. [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. [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.

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. [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. [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. [4] 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. [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. [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. [7] 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. [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. [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. [10] L. J. Han, J. 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. [11] 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. [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. [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. [14] 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. [15] 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. [16] 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. [17] 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. [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. [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. [20] H. L. Li, P. Markowich and M. Mei, Asymptotic behavior of solutions of the hydrodynamic model of semiconductors, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 359-378. [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. [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. [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. [24] 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. [25] P. A. Markowich, On steady state Euler-Poisson models for semiconductors, Z. Angew. Math. Phys., 42 (1991), 389-407.  doi: 10.1007/BF00945711. [26] M. Mei, B. 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. [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. [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. [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. [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. [31] N. Tsuge, Uniqueness of the stationary solutions for a fluid dynamical model of semiconductors, Osaka J. Math., 46 (2009), 931-937. [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. [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. [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. [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. [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.
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