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February  2016, 36(2): 981-1004. doi: 10.3934/dcds.2016.36.981

Asymptotic stability of steady state solutions for the relativistic Euler-Poisson equations

La-Su Mai 1, and  Kaijun Zhang 2,

1. 

Department of Mathematics, Capital Normal University, Beijing 100048, China
2. 

School of Mathematics and Statistics, Northeast Normal University, Changchun, MO 130024

Received  June 2014 Revised  June 2014 Published  August 2015

Under the conditions of both the initial data being the small perturbation of given steady state solution and the boundary strength being small, the global existence of smooth solution to the initial boundary value problem of the relativistic Euler-Poisson equations is proved. The convergence of the global smooth solution to smooth steady state solution in time exponentially is also obtained when time goes to infinity.
Keywords: steady state solution, asymptotic stability, global smooth solution., Relativistic Euler-Poisson equations.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: La-Su Mai, Kaijun Zhang. Asymptotic stability of steady state solutions for the relativistic Euler-Poisson equations. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 981-1004. doi: 10.3934/dcds.2016.36.981
References:
[1]

P. Degond and P. A. Markowich, On a one-dimensional steady-state hydrodynamic model for semiconductors， Appl. Math. Lett., 3 (1990), 25-29. doi: 10.1016/0893-9659(90)90130-4.  Google Scholar

[2]

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

[3]

J. P. Goedbloed, R. Keppens and S. Poedts, Advanced Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas, Cambridge University Press, 2010. doi: 10.1017/CBO9781139195560.  Google Scholar

[4]

Y. Guo and W. Strauss, Stability of semiconductor states with insulating and contact boundary, Arch. Ration. Mech. Anal., 179 (2006), 1-30. doi: 10.1007/s00205-005-0369-2.  Google Scholar

[5]

L. Hsiao and K. J. Zhang, The relaxation of the hydrodynamic model for semiconductors to the drift-diffusion equations, J. Differential Equations, 165 (2000), 315-354. doi: 10.1006/jdeq.2000.3780.  Google Scholar

[6]

F. M. Huang, M. Mei, Y. Wang and H. 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

[7]

H. L. Li, P. A. Markowich and M. Mei, Asymptotic behaviour of solutions of the hydrodynamic model of semiconductors, Proc. Roy. Soc. Edinburgh Sect. A., 132 (2002), 359-378. doi: 10.1017/S0308210500001670.  Google Scholar

[8]

T. Luo, R. Natalini and Z. Xin, Large time behavior of the solutions to a hydrodynamic model for semiconductors, SIAM J. Math. Anal., 59 (1999), 810-830. doi: 10.1137/S0036139996312168.  Google Scholar

[9]

La-Su. Mai, J. Y. Li and K. J. Zhang, On the steady state relativistic Euler-Poisson equations, Acta. Appl. Math., 125 (2013), 135-157. doi: 10.1007/s10440-012-9784-1.  Google Scholar

[10]

A. Majda, Compressible Fluid Flow and System of Conservation Laws in Several Space Variables, (Appl. Math. Sci. 53), Springer, 1984. Google Scholar

[11]

P. A. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation, Arch. Ration. Mech. Analysis., 129 (1995), 129-145. doi: 10.1007/BF00379918.  Google Scholar

[12]

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

[13]

P. A. Markowich, The Stationary Semiconductor Device Equations, Computational Microelectronics, Springer-Verlag, Vienna, 1986. Google Scholar

[14]

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

[15]

S. Nishibata and M. Suzuki, Relaxation limit and initial layer to hydrodynamic models for semiconductors, J. Differential Equations, 249 (2010), 1385-1409. doi: 10.1016/j.jde.2010.06.008.  Google Scholar

[16]

V. Pant, On I. Symmetry Breaking Under Perturbations and II. Relativistic Fluid Dynamics, Ph. D. Thesis, University of Michigan, 1996.  Google Scholar

[17]

Y. J. Peng, Asymptotic limits of one-dimensional hydrodynamic models for plasmas and semiconductors, Chinese Ann. Math. Ser. B, 23 (2002), 25-36. doi: 10.1142/S0252959902000043.  Google Scholar

[18]

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

[19]

S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit, Comm. Math. Phys., 104 (1986), 49-75. doi: 10.1007/BF01210792.  Google Scholar

[20]

B, Zhang, On a local existence theorem for a simplified one-dimensional hydrodynamic model for semiconductor devices, SIAM J. Math. Anal., 25 (1994), 941-947. doi: 10.1137/S0036141092224595.  Google Scholar

show all references

References:
[1]

P. Degond and P. A. Markowich, On a one-dimensional steady-state hydrodynamic model for semiconductors， Appl. Math. Lett., 3 (1990), 25-29. doi: 10.1016/0893-9659(90)90130-4.  Google Scholar

[2]

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

[3]

J. P. Goedbloed, R. Keppens and S. Poedts, Advanced Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas, Cambridge University Press, 2010. doi: 10.1017/CBO9781139195560.  Google Scholar

[4]

Y. Guo and W. Strauss, Stability of semiconductor states with insulating and contact boundary, Arch. Ration. Mech. Anal., 179 (2006), 1-30. doi: 10.1007/s00205-005-0369-2.  Google Scholar

[5]

L. Hsiao and K. J. Zhang, The relaxation of the hydrodynamic model for semiconductors to the drift-diffusion equations, J. Differential Equations, 165 (2000), 315-354. doi: 10.1006/jdeq.2000.3780.  Google Scholar

[6]

F. M. Huang, M. Mei, Y. Wang and H. 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

[7]

H. L. Li, P. A. Markowich and M. Mei, Asymptotic behaviour of solutions of the hydrodynamic model of semiconductors, Proc. Roy. Soc. Edinburgh Sect. A., 132 (2002), 359-378. doi: 10.1017/S0308210500001670.  Google Scholar

[8]

T. Luo, R. Natalini and Z. Xin, Large time behavior of the solutions to a hydrodynamic model for semiconductors, SIAM J. Math. Anal., 59 (1999), 810-830. doi: 10.1137/S0036139996312168.  Google Scholar

[9]

La-Su. Mai, J. Y. Li and K. J. Zhang, On the steady state relativistic Euler-Poisson equations, Acta. Appl. Math., 125 (2013), 135-157. doi: 10.1007/s10440-012-9784-1.  Google Scholar

[10]

A. Majda, Compressible Fluid Flow and System of Conservation Laws in Several Space Variables, (Appl. Math. Sci. 53), Springer, 1984. Google Scholar

[11]

P. A. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation, Arch. Ration. Mech. Analysis., 129 (1995), 129-145. doi: 10.1007/BF00379918.  Google Scholar

[12]

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

[13]

P. A. Markowich, The Stationary Semiconductor Device Equations, Computational Microelectronics, Springer-Verlag, Vienna, 1986. Google Scholar

[14]

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

[15]

S. Nishibata and M. Suzuki, Relaxation limit and initial layer to hydrodynamic models for semiconductors, J. Differential Equations, 249 (2010), 1385-1409. doi: 10.1016/j.jde.2010.06.008.  Google Scholar

[16]

V. Pant, On I. Symmetry Breaking Under Perturbations and II. Relativistic Fluid Dynamics, Ph. D. Thesis, University of Michigan, 1996.  Google Scholar

[17]

Y. J. Peng, Asymptotic limits of one-dimensional hydrodynamic models for plasmas and semiconductors, Chinese Ann. Math. Ser. B, 23 (2002), 25-36. doi: 10.1142/S0252959902000043.  Google Scholar

[18]

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

[19]

S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit, Comm. Math. Phys., 104 (1986), 49-75. doi: 10.1007/BF01210792.  Google Scholar

[20]

B, Zhang, On a local existence theorem for a simplified one-dimensional hydrodynamic model for semiconductor devices, SIAM J. Math. Anal., 25 (1994), 941-947. doi: 10.1137/S0036141092224595.  Google Scholar

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