A phase analysis of transonic solutions for the hydrodynamic semiconductor model
Author:
Massimiliano D. Rosini
Journal:
Quart. Appl. Math. 63 (2005), 251-268
MSC (2000):
Primary 82D37; Secondary 35B40, 35L67
DOI:
https://doi.org/10.1090/S0033-569X-05-00942-1
Published electronically:
February 23, 2005
MathSciNet review:
2150772
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Abstract: In the present paper we present a phase plane analysis of transonic solutions of the steady state one-dimensional unipolar hydrodynamic model for semiconductors, taking also into consider shocks.
AMN G. Alì, P. Marcati, R. Natalini, Hydrodynamical models for semiconductors, Z. Angew. Math. Mech., 76, Suppl. 2 1996, pp. 301-304.
AP A. M. Anile, S. Pennisi, Thermodynamic derivation of the hydrodynamical model for charge transport in semiconductors, Physical Review B Volume 46, Number 20 1992, pp. 186-193.
AMPS U. M. Asher, P. A. Markowich, P. Pietra, C. Schmeiser, A phase plane analysis of transonic solutions for the hydrodynamic semiconductor model, Math. Models Appl. Sci. 1 1991, pp. 347-376.
- P. Degond and P. A. Markowich, On a one-dimensional steady-state hydrodynamic model for semiconductors, Appl. Math. Lett. 3 (1990), no. 3, 25–29. MR 1077867, DOI https://doi.org/10.1016/0893-9659%2890%2990130-4
- Irene Martínez Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductors, Comm. Partial Differential Equations 17 (1992), no. 3-4, 553–577. MR 1163436, DOI https://doi.org/10.1080/03605309208820853
- Irene Martínez Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductors, Comm. Partial Differential Equations 17 (1992), no. 3-4, 553–577. MR 1163436, DOI https://doi.org/10.1080/03605309208820853
- Ingenuin Gasser and Roberto Natalini, The energy transport and the drift diffusion equations as relaxation limits of the hydrodynamic model for semiconductors, Quart. Appl. Math. 57 (1999), no. 2, 269–282. MR 1686190, DOI https://doi.org/10.1090/qam/1686190
- Ingenuin Gasser and Pierangelo Marcati, The combined relaxation and vanishing Debye length limit in the hydrodynamic model for semiconductors, Math. Methods Appl. Sci. 24 (2001), no. 2, 81–92. MR 1808684, DOI https://doi.org/10.1002/1099-1476%2820010125%2924%3A2%3C81%3A%3AAID-MMA198%3E3.3.CO%3B2-O
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Rosini2 M. D. Rosini, Stability of hydrodynamic model for semiconductor, to appear in Archivum Mathematicum, 2003.
- Massimiliano D. Rosini, Stability of transonic strong shock waves for the one-dimensional hydrodynamic model for semiconductors, J. Differential Equations 199 (2004), no. 2, 326–351. MR 2047913, DOI https://doi.org/10.1016/j.jde.2003.09.009
- Denis Serre, Systems of conservation laws. 1, Cambridge University Press, Cambridge, 1999. Hyperbolicity, entropies, shock waves; Translated from the 1996 French original by I. N. Sneddon. MR 1707279
Se2 D. Serre, System of conservation laws, Vol I, Cambridge University Press, Cambridge, 1999.
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AMN G. Alì, P. Marcati, R. Natalini, Hydrodynamical models for semiconductors, Z. Angew. Math. Mech., 76, Suppl. 2 1996, pp. 301-304.
AP A. M. Anile, S. Pennisi, Thermodynamic derivation of the hydrodynamical model for charge transport in semiconductors, Physical Review B Volume 46, Number 20 1992, pp. 186-193.
AMPS U. M. Asher, P. A. Markowich, P. Pietra, C. Schmeiser, A phase plane analysis of transonic solutions for the hydrodynamic semiconductor model, Math. Models Appl. Sci. 1 1991, pp. 347-376.
DegondMarkowich P. Degond, P. A. Markowich, On a one-dimensional steady-state hydrodynamic model for semiconductors, Appl. Math. Letters 3 (3) 1990, pp. 25-86.
Gamba1 I. M. Gamba, Stationary transonic solutions for a one-dimensional hydrodynamic model for semiconductors, Comm. Partial Differential Equations 17, no. 3-4 1992, pp. 553-577.
Gamba2 I. M. Gamba, Boundary-layer formation for viscosity approximations in transonic flow, Phys. Fluids A 4, no. 3 1992, pp. 486-490.
GN I. Gasser, R. Natalini, The energy transport and the drift diffusion equations as relaxation limits of the hydrodynamic model for semiconductors, Quart. Appl. Math., 57, no. 2 1999, pp. 269-282.
GM1 I. Gasser, P. Marcati, The combined relaxation and vanishing Debye length limit in the hydrodynamic model for semiconductor, Math. Methods Appl. Sci., 24, no. 2 2001, pp. 81-92.
GM2 I. Gasser, P. Marcati, A quasi-neutral limit in the hydrodynamic model for charged fluids, Monatsh. Math., 138, no. 3 2003, pp. 189-208.
M P. A. Markowich, Kinetic Models for Semiconductors, Nonequilibrium problems in many-particle systems (Montecatini, 1992), Lecture Notes in Math., 1551, Springer, Berlin 1993, pp. 87-111.
Rosini2 M. D. Rosini, Stability of hydrodynamic model for semiconductor, to appear in Archivum Mathematicum, 2003.
Rosini3 M. D. Rosini, Stability of transonic strong shock waves for the one-dimensional hydrodynamic model for semiconductors, Journal of Differential Equations, 199 (2004), pp. 326–351.
Rosini4 M. D. Rosini, Existence and Stability of Transonic Shock Waves in the Hydrodynamic Model For Semiconductors, Ph. Thesis Univerity of Naples (Italy), 2003.
Se2 D. Serre, System of conservation laws, Vol I, Cambridge University Press, Cambridge, 1999.
Smoller J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, Heidelberg, Berlin 1983.
Tatarskii V. I. Tatarskii, The Wigner representation of quantum mechanics, Sov. Phys. Usp. 26 1983, pp. 311-327.
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Additional Information
Massimiliano D. Rosini
Affiliation:
Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli “Federico II”, Complesso Monte S. Angelo, Via Cintia, 80126 Napoli-Italy
Address at time of publication:
Dipartimento di Matematica Pura ed Applicata, Università di L’Aquila, Via Vetoio, 67100 L’Aquila-Italy
Email:
mrosini@univaq.it
Keywords:
Transonic shock waves,
stability,
hydrodynamic models,
semiconductors
Received by editor(s):
April 1, 2003
Published electronically:
February 23, 2005
Article copyright:
© Copyright 2005
Brown University