# American Institute of Mathematical Sciences

September  2012, 11(5): 1775-1807. doi: 10.3934/cpaa.2012.11.1775

## Stability of stationary waves for full Euler-Poisson system in multi-dimensional space

 1 Department of Mathematics, Champlain College Saint-Lambert, Quebec, J4P 3P2, Canada 2 Institute of Applied Mathematics, Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing, 100190, China

Received  February 2011 Revised  November 2011 Published  March 2012

This paper is concerned with the nonisentropic unipolar hydrodynamic model of semiconductors in the form of multi-dimensional full Euler-Poisson system. By heuristically analyzing the exact gaps between the original solutions and the stationary waves at far fields, we ingeniously construct some correction functions to delete these gaps, and then prove the $L^\infty$-stability of stationary waves with an exponential decay rate in 1-D case. Furthermore, based on the 1-D convergence result, we show the stability of planar stationary waves with also some exponential decay rate in $m$-D case.
Citation: 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
##### References:
 [1] G. Ali, Global existence of smooth solutions of the N-dimensional Euler-Possion model, SIAM J. Math. Anal., 35 (2003), 389-422. doi: 10.1137/S0036141001393225.  Google Scholar [2] G. Ali, D. Bini and D. Rionero, Global existence and relaxation limit for smooth solutions to the Euler-Possion model for semiconductors, SIAM J. Math. Anal., 32 (2000), 572-587. doi: 10.1137/S0036141099355174.  Google Scholar [3] G. Ali 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 [4] 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 [5] P. Degond and P. A. Markowich, A steady-state potential flow model for semiconductors, Ann. Math. Pure Appl., 4 (1993), 87-98. doi: 10.1007/BF01765842.  Google Scholar [6] W. 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 [7] I. Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductor, Comm. Partial Diff. Eqns, 17 (1992), 553-577. doi: 10.1080/03605309208820853.  Google Scholar [8] 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.  Google Scholar [9] 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 (1996), 269-282. doi: 10.1.1.53.9991.  Google Scholar [10] Y. Guo and W. Strauss, Stability of semiconductor states with insulating and contact boundary conditions, Arch. Rational Mech. Anal., 179 (2005), 1-30. doi: 10.1007/s00205-005-0369-2.  Google Scholar [11] L. Hsiao and T.-P. Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Comm. Math. Phys., 143 (1992), 599-605. doi: 10.1007/BF02099268.  Google Scholar [12] 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.  Google Scholar [13] 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.  Google Scholar [14] F.-M. Huang, M. Mei, Y. Wang and T. Yang, Long-time behavior of solutions for bipolar hydrodynamic model of semiconductors with boundary effects,, SIAM J. Math. Anal., ().   Google Scholar [15] F.-M. Huang, M. Mei, Y. Wang and H. Yu, Asymptotic convergence to stationary waves for unipolar hydrodynami model of semiconductors, SIAM J. Math. Anal., 43 (2011), 411-429. doi: 10.1137/100793025.  Google Scholar [16] F.-M. Huang, M. Mei, Y. Wang and H. 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 [17] A. Jüngel, "Quasi-hydrodynamic Semiconductor Equations," Progress in Nonlinear Differential Equations and their Applications, Vol 41, Birkhäuser Verlag, Besel-Boston-Berlin, 2001.  Google Scholar [18] L. Hsiao and S. Wang, Asymptotic behavior of global smooth solutions to the full 1D hydrodynamic model for semiconductors, Math. Models Methods Appl. Sci., 12 (2002), 777-796. doi: 10.1142/S0218202502001891.  Google Scholar [19] L. Hsiao and K. 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: 10.1016/j.mcm.2009.04.013.  Google Scholar [20] L. Hsiao and K. Zhang, The relaxation of the hydrodynamic model for semiconductors to drift-diffusion equations, J. Differential Equations, 165 (2000), 315-354. doi: 10.1006/jdeq.2000.3780.  Google Scholar [21] 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 [22] H.-L. Li, P. Markowich and M. Mei, Asymptotic behavior of solutions of the hydrodynamic model of semiconductors, Proc. Royal Soc. Edinburgh, Sect. A, 132 (2002), 359-378. doi: 10.1017/S0308210500001670.  Google Scholar [23] H.-L. Li, P. Markowich and M. Mei, Asymptotic behavior of subsonic entropy solutions of the isentropic Euler-Poisson equations, Quart. Appl. Math., 60 (2002), 773-796.  Google Scholar [24] 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.  Google Scholar [25] 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.  Google Scholar [26] C.-K. Lin, C.-T. Lin and M. Mei, Asymptotic behavior of solution to nonlinear damped p-system with boundary effect, Int. J. Numer. Anal. Model. Ser. B, 1 (2010), 70-92.  Google Scholar [27] T. Luo, R. Natalini and Z. Xin, Large time behavior of the solutions to a hydrodynamic model for semiconductors, SIAM J. Appl. Math., 59 (1998), 810-830. doi: 10.1.1.55.4600.  Google Scholar [28] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations," Springer-Verlag, Vienna, 1990.  Google Scholar [29] P. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors: the Cauchy problem, Proc. Roy. Soc. Edinburgh, Sect. A, 125 (1995), 115-131 doi: 10.1017/S030821050003078X.  Google Scholar [30] 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), 129-145. doi: 10.1007/BF00379918.  Google Scholar [31] A. Matsumura, Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equation with the first-order dissipation, Publ. Res. Inst. Math. Sci., 13 (1977), 349-379. doi: 10.2977/prims/1195189813.  Google Scholar [32] A. Matsumura and M. Mei, Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary, Arch. Rational Mech. Anal., 146 (1999), 1-22. doi: 10.1007/s002050050134.  Google Scholar [33] M. Mei, Best asymptotic profile for hyperbolic $p$-sytem with damping, SIAM J. Math. Anal., 42 (2010), 1-23. doi: 10.1137/090756594.  Google Scholar [34] M. Mei, Nonlinear diffusion waves for hyperbolic $p$-system with nonlinear damping, J. Differentoial Equations, 247 (2009), 1275-1269. doi: 10.1016/j.jde.2009.04.004.  Google Scholar [35] 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 [36] S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors, Osaka J. Math., 44 (2007), 639-665. doi: 10.1007/BF01210792.  Google Scholar [37] S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a thermal hydrodynamic model for semiconductors, Arch. Rational Mech. Anal., 192 (2009), 187-215. doi: 10.1007/s00205-008-0129-1.  Google Scholar [38] 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.  Google Scholar [39] A. Sitenko and V. Malnev, "Plasma Physics Theory," Applied Mathematics and Mathematical Computation, 10. Chapman $&$ Hall, London, 1995.  Google Scholar [40] 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.  Google Scholar [41] 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 [42] C. Zhu and H. Hattori, Asymptotic behavior of the solution to a nonisentropic hydrodynamic model of semiconductors, J. Differential Equations, 144 (1998), 353-389. doi: 10.1006/jdeq.1997.3381.  Google Scholar

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##### References:
 [1] G. Ali, Global existence of smooth solutions of the N-dimensional Euler-Possion model, SIAM J. Math. Anal., 35 (2003), 389-422. doi: 10.1137/S0036141001393225.  Google Scholar [2] G. Ali, D. Bini and D. Rionero, Global existence and relaxation limit for smooth solutions to the Euler-Possion model for semiconductors, SIAM J. Math. Anal., 32 (2000), 572-587. doi: 10.1137/S0036141099355174.  Google Scholar [3] G. Ali 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 [4] 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 [5] P. Degond and P. A. Markowich, A steady-state potential flow model for semiconductors, Ann. Math. Pure Appl., 4 (1993), 87-98. doi: 10.1007/BF01765842.  Google Scholar [6] W. 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 [7] I. Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductor, Comm. Partial Diff. Eqns, 17 (1992), 553-577. doi: 10.1080/03605309208820853.  Google Scholar [8] 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.  Google Scholar [9] 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 (1996), 269-282. doi: 10.1.1.53.9991.  Google Scholar [10] Y. Guo and W. Strauss, Stability of semiconductor states with insulating and contact boundary conditions, Arch. Rational Mech. Anal., 179 (2005), 1-30. doi: 10.1007/s00205-005-0369-2.  Google Scholar [11] L. Hsiao and T.-P. Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Comm. Math. Phys., 143 (1992), 599-605. doi: 10.1007/BF02099268.  Google Scholar [12] 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.  Google Scholar [13] 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.  Google Scholar [14] F.-M. Huang, M. Mei, Y. Wang and T. Yang, Long-time behavior of solutions for bipolar hydrodynamic model of semiconductors with boundary effects,, SIAM J. Math. Anal., ().   Google Scholar [15] F.-M. Huang, M. Mei, Y. Wang and H. Yu, Asymptotic convergence to stationary waves for unipolar hydrodynami model of semiconductors, SIAM J. Math. Anal., 43 (2011), 411-429. doi: 10.1137/100793025.  Google Scholar [16] F.-M. Huang, M. Mei, Y. Wang and H. 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 [17] A. Jüngel, "Quasi-hydrodynamic Semiconductor Equations," Progress in Nonlinear Differential Equations and their Applications, Vol 41, Birkhäuser Verlag, Besel-Boston-Berlin, 2001.  Google Scholar [18] L. Hsiao and S. Wang, Asymptotic behavior of global smooth solutions to the full 1D hydrodynamic model for semiconductors, Math. Models Methods Appl. Sci., 12 (2002), 777-796. doi: 10.1142/S0218202502001891.  Google Scholar [19] L. Hsiao and K. 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: 10.1016/j.mcm.2009.04.013.  Google Scholar [20] L. Hsiao and K. Zhang, The relaxation of the hydrodynamic model for semiconductors to drift-diffusion equations, J. Differential Equations, 165 (2000), 315-354. doi: 10.1006/jdeq.2000.3780.  Google Scholar [21] 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 [22] H.-L. Li, P. Markowich and M. Mei, Asymptotic behavior of solutions of the hydrodynamic model of semiconductors, Proc. Royal Soc. Edinburgh, Sect. A, 132 (2002), 359-378. doi: 10.1017/S0308210500001670.  Google Scholar [23] H.-L. Li, P. Markowich and M. Mei, Asymptotic behavior of subsonic entropy solutions of the isentropic Euler-Poisson equations, Quart. Appl. Math., 60 (2002), 773-796.  Google Scholar [24] 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.  Google Scholar [25] 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.  Google Scholar [26] C.-K. Lin, C.-T. Lin and M. Mei, Asymptotic behavior of solution to nonlinear damped p-system with boundary effect, Int. J. Numer. Anal. Model. Ser. B, 1 (2010), 70-92.  Google Scholar [27] T. Luo, R. Natalini and Z. Xin, Large time behavior of the solutions to a hydrodynamic model for semiconductors, SIAM J. Appl. Math., 59 (1998), 810-830. doi: 10.1.1.55.4600.  Google Scholar [28] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations," Springer-Verlag, Vienna, 1990.  Google Scholar [29] P. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors: the Cauchy problem, Proc. Roy. Soc. Edinburgh, Sect. A, 125 (1995), 115-131 doi: 10.1017/S030821050003078X.  Google Scholar [30] 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), 129-145. doi: 10.1007/BF00379918.  Google Scholar [31] A. Matsumura, Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equation with the first-order dissipation, Publ. Res. Inst. Math. Sci., 13 (1977), 349-379. doi: 10.2977/prims/1195189813.  Google Scholar [32] A. Matsumura and M. Mei, Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary, Arch. Rational Mech. Anal., 146 (1999), 1-22. doi: 10.1007/s002050050134.  Google Scholar [33] M. Mei, Best asymptotic profile for hyperbolic $p$-sytem with damping, SIAM J. Math. Anal., 42 (2010), 1-23. doi: 10.1137/090756594.  Google Scholar [34] M. Mei, Nonlinear diffusion waves for hyperbolic $p$-system with nonlinear damping, J. Differentoial Equations, 247 (2009), 1275-1269. doi: 10.1016/j.jde.2009.04.004.  Google Scholar [35] 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 [36] S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors, Osaka J. Math., 44 (2007), 639-665. doi: 10.1007/BF01210792.  Google Scholar [37] S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a thermal hydrodynamic model for semiconductors, Arch. Rational Mech. Anal., 192 (2009), 187-215. doi: 10.1007/s00205-008-0129-1.  Google Scholar [38] 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.  Google Scholar [39] A. Sitenko and V. Malnev, "Plasma Physics Theory," Applied Mathematics and Mathematical Computation, 10. Chapman $&$ Hall, London, 1995.  Google Scholar [40] 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.  Google Scholar [41] 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 [42] C. Zhu and H. Hattori, Asymptotic behavior of the solution to a nonisentropic hydrodynamic model of semiconductors, J. Differential Equations, 144 (1998), 353-389. doi: 10.1006/jdeq.1997.3381.  Google Scholar
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