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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

 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.
Citation: La-Su Mai, Kaijun Zhang. Asymptotic stability of steady state solutions for the relativistic Euler-Poisson equations. Discrete and 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. [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. [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. [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. [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. [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. [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. [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. [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. [10] A. Majda, Compressible Fluid Flow and System of Conservation Laws in Several Space Variables, (Appl. Math. Sci. 53), Springer, 1984. [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. [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. [13] P. A. Markowich, The Stationary Semiconductor Device Equations, Computational Microelectronics, Springer-Verlag, Vienna, 1986. [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. [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. [16] V. Pant, On I. Symmetry Breaking Under Perturbations and II. Relativistic Fluid Dynamics, Ph. D. Thesis, University of Michigan, 1996. [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. [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. [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. [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.

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. [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. [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. [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. [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. [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. [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. [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. [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. [10] A. Majda, Compressible Fluid Flow and System of Conservation Laws in Several Space Variables, (Appl. Math. Sci. 53), Springer, 1984. [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. [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. [13] P. A. Markowich, The Stationary Semiconductor Device Equations, Computational Microelectronics, Springer-Verlag, Vienna, 1986. [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. [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. [16] V. Pant, On I. Symmetry Breaking Under Perturbations and II. Relativistic Fluid Dynamics, Ph. D. Thesis, University of Michigan, 1996. [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. [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. [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. [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.
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