# American Institute of Mathematical Sciences

July  2011, 16(1): 345-360. doi: 10.3934/dcdsb.2011.16.345

## Existence and some limit analysis of stationary solutions for a multi-dimensional bipolar Euler-Poisson system

 1 Department of Mathematics, Shanghai Normal University, Shanghai 200234

Received  February 2009 Revised  January 2011 Published  April 2011

In this paper, we study a two- and three-dimensional bipolar Euler-Poisson system (hydrodynamic model). The system arises in mathematical modeling for semiconductors and plasmas. We are interested in the steady state isentropic case supplemented by the proper boundary conditions. We first show the existence and uniqueness of irrotational subsonic stationary solutions for the two- and three-dimensional hydrodynamic model. Next, we investigate the zero-electron-mass limit, the zero-relaxation-time limit and the Debye-length (quasi-neutral) limit for above stationary solutions, respectively. For each limit, we show the strong convergence of the sequence of solutions and give the associated convergence rates.
Citation: 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
##### References:
 [1] P. Amster and M. P. Beccar Varela, Subsonic solutions to a one-dimensional nonisentropic hydrodynamic model for semiconductors,, J. Math. Anal. Appl., 258 (2001), 52. doi: 10.1006/jmaa.2000.7359. Google Scholar [2] A. Ambrose, F. Méhats and P. Raviart, On singular perturbation problems for the nonlinear Poisson equation,, Asymptot. Anal., 25 (2001), 39. Google Scholar [3] U. Ascher, P. A. Markowich and C. Schmeiser, A phase plane analysis of transonic solutions for the hydrodynamic semiconductor model,, Math. Models Meth. Appl. Sci., 11 (1991), 347. doi: 10.1142/S0218202591000174. Google Scholar [4] H. Brézis, F. Golse and R. Sentis, Analyse asymptotique de l'équation de Poisson couplée à la relation de Boltzmann. Quasi-neutralité des plasmas,, C. R. Acad. Sci. Paris, 321 (1995), 953. Google Scholar [5] S. Cordier and E. Grenier, Quasineutral limit of an Euler-Poisson system arising from plasma physics,, Comm. Partial Differential Equations, 25 (2000), 1099. doi: 10.1080/03605300008821542. Google Scholar [6] P. Degond and P. A. Markowich, On a one-dimensional steady-state hydrodynamic model for semiconductors,, Appl. Math. Letters, 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. Math. Pura Appl.(4), 165 (1993), 87. Google Scholar [8] I. M. Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductors,, Comm. Partial Differential Equations, 17 (1992), 553. Google Scholar [9] T. Goudon, A. Jüngel and Y.-J. Peng, Zero-mass-electrons limits in hydrodynamic models for plasmas,, Appl. Math. Letters, 12 (1999), 75. doi: 10.1016/S0893-9659(99)00038-5. Google Scholar [10] D. Gilbarg and N. S. Trüdinger, "Elliptic Partial Differential Equations of Second Order,", Springer-Verlag, (1998). Google Scholar [11] L. Hsiao and K. 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 [12] A. Jüngel and Y.-J. Peng, Zero-relaxation-time limits in hydrodynamic models for plasmas revisted,, Z. Angew. Math. Phys., 51 (2000), 385. doi: 10.1007/s000330050004. Google Scholar [13] A. Jüngel, "Quasi-Hydrodynamic Semiconductor Equations,", Progress in Nonlinear Differential Equations, (2001). Google Scholar [14] C. Lattanzio and P. Marcati, The relaxation to the drift-diffusion system for the 3-D isentropic Euler-Poisson model for semiconductors,, Discrete Contin. Dyn. Syst., 5 (1999), 449. doi: 10.3934/dcds.1999.5.449. Google Scholar [15] Y.-P. Li and J.-Z. Zhang, Stationary solutions for a multi-dimensional nonisentropic hydrodynamic model for semiconductors,, Math. Comput. Modeling, 49 (2009), 163. doi: 10.1016/j.mcm.2008.05.006. Google Scholar [16] Y.-P. Li, Stationary solutions for a one-dimensional nonisentropic hydrodynamic model for semiconductors,, Acta Math. Sci., B28 (2008), 479. Google Scholar [17] Y.-P. Li, Asymptotic profile in a one-dimensional nonisentropic hydrodynamic model for semiconductors,, J. Math. Anal. Appl., 325 (2007), 949. doi: 10.1016/j.jmaa.2006.02.018. Google Scholar [18] O. A. Ladyzhenskaya and N. U. Uraltsera, "Linear and Quasilinear Elliptic Equations,", Academic Press, (1968). Google Scholar [19] Y.-P. Li, Asymptotic profile in a multi-dimensional nonisentropic hydrodynamic model for semiconductors,, Nonlinear Anal. Real World Appl., 8 (2007), 1235. doi: 10.1016/j.nonrwa.2006.06.011. Google Scholar [20] P. A. Markowich, On steady state Euler-Poisson models for semiconductors,, Z. Angew Math. Phys., 42 (1991), 387. doi: 10.1007/BF00945711. Google Scholar [21] P. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation,, Arch. Rational Mech. Anal., 129 (1995), 129. doi: 10.1007/BF00379918. Google Scholar [22] P. A. Markowich, C. A. Ringhofev and C. Schmeiser, "Semiconductor Equations,", Springer-Verlag, (1990). Google Scholar [23] Y.-J. Peng, Some analysis in steady-state Euler-Poisson equations for potential flow,, Asymp. Anal., 36 (2003), 75. Google Scholar [24] Y.-J. Peng and Y.-G. Wang, Convergence of compressible Euler-Poisson equations to incompressible type Euler equations,, Asymptotic Anal., 41 (2005), 141. Google Scholar [25] Y.-J. Peng and Y.-G. Wang, Boundary layers and quasi-neutral limit in steady state Euler-Poisson equations for potential flows,, Nonlinearity, 17 (2004), 835. doi: 10.1088/0951-7715/17/3/006. Google Scholar [26] Y.-J. Peng and I. Violet, Asymptotic expansions in a steady state Euler-Poisson equations to incompressible Euler equations,, Math. Models Meth. Appl. Sci., 15 (2005), 717. doi: 10.1142/S0218202505000546. Google Scholar [27] Y.-J. Peng and Y.-F. Yang, Junction layer analysis in one-dimensional steady-state Euler-Poisson equations,, J. Math. Anal. Appl., 344 (2008), 440. doi: 10.1016/j.jmaa.2008.02.062. Google Scholar [28] M. D. Rosini, A phase analysis of transonic solutions for the hydrodynamic semiconductor model,, Quart. Appl. Math., (2003). Google Scholar [29] M. Slemrod and N. Sternberg, Quasi-neutral limit for the Euler-Poisson system,, J. Nonlinear Sci., 11 (2001), 193. doi: 10.1007/s00332-001-0004-9. Google Scholar [30] I. Violet, High-order expansions in the quasi-neutral limit of the Euler-Poisson system for a potential flow,, Proc. Roy. Soc. Edinburgh, 137 (2007), 1101. doi: 10.1017/S0308210505001216. Google Scholar [31] S. Wang, Quasineutral limit of Euler-Poinsson system with and without viscosity,, Comm. Partial Differential Equations, 29 (2004), 419. doi: 10.1081/PDE-120030403. Google Scholar [32] L.-M. Yeh, On a steady state Euler-Poisson model for semiconductors,, Comm. Partial Differential Equations, 21 (1996), 1007. doi: 10.1080/03605309608821216. Google Scholar [33] F. Zhou and Y.-P. Li, Existence and some limits of stationary solutions to a one-dimensional bipolar Euler-Poisson system,, J. Math. Anal. Appl., 351 (2009), 480. doi: 10.1016/j.jmaa.2008.10.032. Google Scholar

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##### References:
 [1] P. Amster and M. P. Beccar Varela, Subsonic solutions to a one-dimensional nonisentropic hydrodynamic model for semiconductors,, J. Math. Anal. Appl., 258 (2001), 52. doi: 10.1006/jmaa.2000.7359. Google Scholar [2] A. Ambrose, F. Méhats and P. Raviart, On singular perturbation problems for the nonlinear Poisson equation,, Asymptot. Anal., 25 (2001), 39. Google Scholar [3] U. Ascher, P. A. Markowich and C. Schmeiser, A phase plane analysis of transonic solutions for the hydrodynamic semiconductor model,, Math. Models Meth. Appl. Sci., 11 (1991), 347. doi: 10.1142/S0218202591000174. Google Scholar [4] H. Brézis, F. Golse and R. Sentis, Analyse asymptotique de l'équation de Poisson couplée à la relation de Boltzmann. Quasi-neutralité des plasmas,, C. R. Acad. Sci. Paris, 321 (1995), 953. Google Scholar [5] S. Cordier and E. Grenier, Quasineutral limit of an Euler-Poisson system arising from plasma physics,, Comm. Partial Differential Equations, 25 (2000), 1099. doi: 10.1080/03605300008821542. Google Scholar [6] P. Degond and P. A. Markowich, On a one-dimensional steady-state hydrodynamic model for semiconductors,, Appl. Math. Letters, 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. Math. Pura Appl.(4), 165 (1993), 87. Google Scholar [8] I. M. Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductors,, Comm. Partial Differential Equations, 17 (1992), 553. Google Scholar [9] T. Goudon, A. Jüngel and Y.-J. Peng, Zero-mass-electrons limits in hydrodynamic models for plasmas,, Appl. Math. Letters, 12 (1999), 75. doi: 10.1016/S0893-9659(99)00038-5. Google Scholar [10] D. Gilbarg and N. S. Trüdinger, "Elliptic Partial Differential Equations of Second Order,", Springer-Verlag, (1998). Google Scholar [11] L. Hsiao and K. 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 [12] A. Jüngel and Y.-J. Peng, Zero-relaxation-time limits in hydrodynamic models for plasmas revisted,, Z. Angew. Math. Phys., 51 (2000), 385. doi: 10.1007/s000330050004. Google Scholar [13] A. Jüngel, "Quasi-Hydrodynamic Semiconductor Equations,", Progress in Nonlinear Differential Equations, (2001). Google Scholar [14] C. Lattanzio and P. Marcati, The relaxation to the drift-diffusion system for the 3-D isentropic Euler-Poisson model for semiconductors,, Discrete Contin. Dyn. Syst., 5 (1999), 449. doi: 10.3934/dcds.1999.5.449. Google Scholar [15] Y.-P. Li and J.-Z. Zhang, Stationary solutions for a multi-dimensional nonisentropic hydrodynamic model for semiconductors,, Math. Comput. Modeling, 49 (2009), 163. doi: 10.1016/j.mcm.2008.05.006. Google Scholar [16] Y.-P. Li, Stationary solutions for a one-dimensional nonisentropic hydrodynamic model for semiconductors,, Acta Math. Sci., B28 (2008), 479. Google Scholar [17] Y.-P. Li, Asymptotic profile in a one-dimensional nonisentropic hydrodynamic model for semiconductors,, J. Math. Anal. Appl., 325 (2007), 949. doi: 10.1016/j.jmaa.2006.02.018. Google Scholar [18] O. A. Ladyzhenskaya and N. U. Uraltsera, "Linear and Quasilinear Elliptic Equations,", Academic Press, (1968). Google Scholar [19] Y.-P. Li, Asymptotic profile in a multi-dimensional nonisentropic hydrodynamic model for semiconductors,, Nonlinear Anal. Real World Appl., 8 (2007), 1235. doi: 10.1016/j.nonrwa.2006.06.011. Google Scholar [20] P. A. Markowich, On steady state Euler-Poisson models for semiconductors,, Z. Angew Math. Phys., 42 (1991), 387. doi: 10.1007/BF00945711. Google Scholar [21] P. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation,, Arch. Rational Mech. Anal., 129 (1995), 129. doi: 10.1007/BF00379918. Google Scholar [22] P. A. Markowich, C. A. Ringhofev and C. Schmeiser, "Semiconductor Equations,", Springer-Verlag, (1990). Google Scholar [23] Y.-J. Peng, Some analysis in steady-state Euler-Poisson equations for potential flow,, Asymp. Anal., 36 (2003), 75. Google Scholar [24] Y.-J. Peng and Y.-G. Wang, Convergence of compressible Euler-Poisson equations to incompressible type Euler equations,, Asymptotic Anal., 41 (2005), 141. Google Scholar [25] Y.-J. Peng and Y.-G. Wang, Boundary layers and quasi-neutral limit in steady state Euler-Poisson equations for potential flows,, Nonlinearity, 17 (2004), 835. doi: 10.1088/0951-7715/17/3/006. Google Scholar [26] Y.-J. Peng and I. Violet, Asymptotic expansions in a steady state Euler-Poisson equations to incompressible Euler equations,, Math. Models Meth. Appl. Sci., 15 (2005), 717. doi: 10.1142/S0218202505000546. Google Scholar [27] Y.-J. Peng and Y.-F. Yang, Junction layer analysis in one-dimensional steady-state Euler-Poisson equations,, J. Math. Anal. Appl., 344 (2008), 440. doi: 10.1016/j.jmaa.2008.02.062. Google Scholar [28] M. D. Rosini, A phase analysis of transonic solutions for the hydrodynamic semiconductor model,, Quart. Appl. Math., (2003). Google Scholar [29] M. Slemrod and N. Sternberg, Quasi-neutral limit for the Euler-Poisson system,, J. Nonlinear Sci., 11 (2001), 193. doi: 10.1007/s00332-001-0004-9. Google Scholar [30] I. Violet, High-order expansions in the quasi-neutral limit of the Euler-Poisson system for a potential flow,, Proc. Roy. Soc. Edinburgh, 137 (2007), 1101. doi: 10.1017/S0308210505001216. Google Scholar [31] S. Wang, Quasineutral limit of Euler-Poinsson system with and without viscosity,, Comm. Partial Differential Equations, 29 (2004), 419. doi: 10.1081/PDE-120030403. Google Scholar [32] L.-M. Yeh, On a steady state Euler-Poisson model for semiconductors,, Comm. Partial Differential Equations, 21 (1996), 1007. doi: 10.1080/03605309608821216. Google Scholar [33] F. Zhou and Y.-P. Li, Existence and some limits of stationary solutions to a one-dimensional bipolar Euler-Poisson system,, J. Math. Anal. Appl., 351 (2009), 480. doi: 10.1016/j.jmaa.2008.10.032. Google Scholar
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