Asymptotic behavior of subsonic entropy solutions of the isentropic Euler-Poisson equations

Authors:
Hailiang Li, Peter Markowich and Ming Mei

Journal:
Quart. Appl. Math. **60** (2002), 773-796

MSC:
Primary 35L60; Secondary 35B40, 35L45, 35L67, 76X05

DOI:
https://doi.org/10.1090/qam/1939010

MathSciNet review:
MR1939010

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The hydrodynamic model for semiconductors in one dimension is considered. For perturbated Riemann data, global subsonic (weak) entropy solutions, piecewise continuous and piecewise smooth solutions with shock discontinuities are constructed and their asymptotic behavior is analyzed. In subsonic domains, the solution is smooth and, exponentially as , tends to the corresponding stationary solution due to the influence of Poisson coupling. Along the shock discontinuity, the shock strength and the difference of derivatives of solutions decay exponentially affected by the relaxation mechanism.

**[1]**R. Courant and K. O. Friedrichs,*Supersonic flow and shock waves*, Applied Mathematical Sciences Vol. 21, Springer-Verlag, New York, 1948 MR**0029615****[2]**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.**1**, 347-376 (1991)**[3]**K. Bløtekjaer,*Transport equations for electrons in two-valley semiconductors*, IEEE Trans. Electron Devices ED-17, 38-47 (1970)**[4]**G. Chen, J. Jerome, and B. Zhang,*Particle hydrodynamic moment models in biology and microelectronics: Singular relaxation limits*, preprint**[5]**G. Chen and D. Wang,*Convergence of shock schemes for the compressible Euler-Poisson equations*, Comm. Math. Phys.**179**, 333-364 (1996) MR**1400743****[6]**P. Degond and P. A. Markowich,*On a one-dimensional steady-state hydrodynamic model*, Appl. Math. Lett.**3**, 25-29 (1990) MR**1077867****[7]**P. Degond and P. A. Markowich,*A steady-state potential flow model for semiconductors*, Ann. Math. Pure Appl.**IV**, 87-98 (1993)**[8]**W. Fang and K. Ito,*Steady-state solutions of a one-dimensional hydrodynamic model for semiconductors*, J. Differential Equations**133**, 224-244 (1997) MR**1427851****[9]**I. Gamba,*Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductor*Comm. Partial Differential Equations**17**(3 & 4), 553-577 (1992) MR**1163436****[10]**I. Gamba and C. S. Morawetz,*A viscous approximation for a**steady semiconductor or transonic gas dynamic flow: Existence theorem for potential flow*, Comm. Pure Appl. Math.**49**, 999-1049 (1996) MR**1404324****[11]**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**, 269-282 (1999) MR**1686190****[12]**H. Hattori,*Stability and instability of steady-state solutions for a hydrodynamic model of semiconductors*, Proc. Roy. Soc. Edinburgh A**127**, 781-796 (1997) MR**1465420****[13]**H. Hattori and C. Zhu,*Asymptotic behavior of the solutions to a non-isentropic hydrodynamic model of semiconductors*, J. Differential Equations**144**, 353-389 (1998) MR**1616897****[14]**L. Hsiao,*Quasilinear hyperbolic systems and dissipative mechanisms*, World Scientific, 1998 MR**1640089****[15]**L. Hsiao and Hailiang Li,*Shock reflection for the damped p-system*, Quart. Appl. Math.**60**, 437-460 (2002) MR**1914435****[16]**L. Hsiao and T. Luo,*Nonlinear diffusive phenomena of entropy weak solutions for a system of quasilinear hyperbolic conservation laws with damping*, Quart. Appl. Math.**56**, 173-198 (1998) MR**1604829****[17]**L. Hsiao and S. Q. Tang,*Construction and qualitative behavior of solutions for a system of nonlinear hyperbolic conservation laws with damping*, Quart. Appl. Math.**53**, 487-505 (1995) MR**1343463****[18]**L. Hsiao and S. Q. Tang,*Construction and qualitative behavior of solutions of perturbed Riemann problem for the system of one-dimensional isentropic flow with damping*, J. Differential Equations**123**, 480-503 (1995) MR**1362883****[19]**L. Hsiao and T. Yang,*Asymptotic of initial boundary value problems for hydrodynamic and drift diffusion models for semiconductors*, J. Differential Equations, 170, 472-493 (2001) MR**1815191****[20]**L. Hsiao and K. Zhang,*The relaxation of the hydrodynamic model for semiconductors to drift diffusion equations***[21]**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**, 1333-1361 (2000) MR**1796567****[22]**J. Jerome,*Analysis of charge transport: A mathematical study of semiconductor devices*, Springer-Verlag, Heidelberg, 1996 MR**1437143****[23]**J. Jerome and C. Shu,*Energy models for one-carrier transport in semiconductor devices*, preprint**[24]**H. Li, P. Markowich, and M. Mei,*Asymptotic behavior of solutions of the hydrodynamic model of semiconductors*, Proc. Royal Soc. Edinburgh, A:132, 359-378 (2002) MR**1899826****[25]**T. Li and W. C. Yu,*Boundary value problem for quasilinear hyperbolic systems*, Duke Univ. Math. Ser. V, 1985**[26]**T. Luo, R. Natalini, and Z. Xin,*Large time behavior of the solutions to a hydrodynamic model for semiconductors*, SIAM J. Math. Anal.**59**, 810-830 (1998) MR**1661255****[27]**P. Marcati and M. Mei,*Asymptotic convergence to steady-state solutions of the initial boundary value problem to a hydrodynamic model for semiconductors*, preprint**[28]**P. Marcati and R. Natalini,*Weak solutions to a hydrodynamic model for semiconductors: The Cauchy problem*, Proc. Royal Soc. Edinburgh**A:125**, 115-131 (1995) MR**1318626****[29]**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**, 129-145 (1995) MR**1328473****[30]**P. A. Markowich,*The Stationary Semiconductor Device Equations*, Springer, Vienna, New York, 1986 MR**821965****[31]**P. A. Markowich,*On steady-state Euler-Poisson model for semiconductors*, Z. Angew. Math. Phys.**62**, 389-407 (1991) MR**1115198****[32]**P. A. Markowich and C. Schmeiser,*The drift-diffusion limit for electron-phonon interaction in semiconductors*, Math. Models Methods Appl. Sci.**7**, 707-729 (1997) MR**1460701****[33]**P. A. Markowich and P. Pietra,*A non-isentropie Euler-Poisson model for a collisionless plasma*, Math. Methods Appl. Sci.**16**, 409-442 (1993) MR**1221036****[34]**P. A. Markowich, C. Ringhofer, and C. Schmeiser,*Semiconductor Equations*, Springer, Vienna, New York, 1989 MR**1063852****[35]**F. Poupaud,*On a system of nonlinear Boltzmann equations of semiconductor physics*, SIAM J. Appl. Math.**50**, 1593-1606 (1990) MR**1080510****[36]**F. Poupaud,*Diffusion approximation of the linear semiconductor Boltzmann equation: Analysis of boundary layer*, Asymptotic analysis**4**, 293-317 (1991) MR**1127004****[37]**F. Poupaud, M. Rascle, and J.-P. Vila,*Global solutions to the isothermal Euler-Poisson system with arbitrarily large data*, J. Differential Equations**123**, 93-121 (1995) MR**1359913****[38]**S. Selberherr,*Analysis and Simulation of Semiconductor Device Equations*, Springer, Vienna, New York, 1984**[39]**L. Yeh,*Subsonic solutions of hydrodynamic model for semiconductors*, Math. Methods Appl. Sci.**20**, 1389-1410 (1997) MR**1475152****[40]**B. Zhang,*Convergence of the Godunov scheme for a simplified one-dimensional hydrodynamic model for semiconductor devices*, Comm. Math. Phys.**157**, 1-22 (1993) MR**1244856****[41]**B. Zhang,*On a local existence theorem for a simplified one-dimensional hydrodynamic model for semiconductor devices*, SIAM J. Math. Anal.**25**, 941-947 (1994) MR**1271318****[42]**K. Zhang,*Global weak solutions of the Cauchy problem to a hydrodynamic model for semiconductors*, J. Partial Differential Equations**12**, 369-383 (1999) MR**1745406****[43]**K. Zhang,*Zero relaxation limit of global weak solutions of the Cauchy problem to a hydrodynamic model for semiconductors*, preprint

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
35L60,
35B40,
35L45,
35L67,
76X05

Retrieve articles in all journals with MSC: 35L60, 35B40, 35L45, 35L67, 76X05

Additional Information

DOI:
https://doi.org/10.1090/qam/1939010

Article copyright:
© Copyright 2002
American Mathematical Society