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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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., in press.

[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.

[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.

[17]

A. Jüngel, "Quasi-hydrodynamic Semiconductor Equations," Progress in Nonlinear Differential Equations and their Applications, Vol 41, Birkhäuser Verlag, Besel-Boston-Berlin, 2001.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations," Springer-Verlag, Vienna, 1990.

[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.

[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.

[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.

[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.

[33]

M. Mei, Best asymptotic profile for hyperbolic $p$-sytem with damping, SIAM J. Math. Anal., 42 (2010), 1-23. doi: 10.1137/090756594.

[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.

[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.

[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.

[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.

[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.

[39]

A. Sitenko and V. Malnev, "Plasma Physics Theory," Applied Mathematics and Mathematical Computation, 10. Chapman & Hall, London, 1995.

[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.

[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.

[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.

show all references

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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., in press.

[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.

[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.

[17]

A. Jüngel, "Quasi-hydrodynamic Semiconductor Equations," Progress in Nonlinear Differential Equations and their Applications, Vol 41, Birkhäuser Verlag, Besel-Boston-Berlin, 2001.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations," Springer-Verlag, Vienna, 1990.

[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.

[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.

[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.

[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.

[33]

M. Mei, Best asymptotic profile for hyperbolic $p$-sytem with damping, SIAM J. Math. Anal., 42 (2010), 1-23. doi: 10.1137/090756594.

[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.

[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.

[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.

[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.

[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.

[39]

A. Sitenko and V. Malnev, "Plasma Physics Theory," Applied Mathematics and Mathematical Computation, 10. Chapman & Hall, London, 1995.

[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.

[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.

[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.

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