Quantum Euler-Poisson systems: Existence of stationary states

Archivum Mathematicum

Volumn 40, Issue 4, 2004, Pages 435-456

  Jüngel, Ansgar ^a   Hailiang, L I ^b  

^a JOHANNES GUTENBERG UNIVERSITY   (Germany)

^b CAPITAL NORMAL UNIVERSITY   (China)

Author keywords

Existence and uniqueness of solutions; Non monotone pressure; Quantum hydrodynamics; Semiconductors

Indexed keywords

EID: 12844282285     PISSN: 00448753     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (31)

1. Brezzi, F., Gasser, I., Markowich, P., Schmeiser, C., Thermal equilibrium state of the quantum hydrodynamic model for semiconductor in one dimension, Appl. Math. Lett. 8 (1995), 47-52.
2. Chen, G. and Wang, D., Convergence of shock schemes for the compressible Euler-Poisson equations, Comm. Math. Phys. 170 (1996), 333-364.
3. Courant, R. and Friedrichs, K. O., Supersonic flow and shock waves, Springer-Verlag, New York 1976.
4. Degond, P. and Markowich, P. A., On a one-dimensional steady-state hydrodynamic model, Appl. Math. Lett. 3 (1990), 25-29.
5. Degond, P. and Markowich, P. A., A steady state potential flow model for semiconductors, Ann. Mat. Pura Appl. 165 (1993), 87-98.
6. Gamba, I., Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductor, Comm. Partial Differential Equations 17 (1992), 553-577.
7. Gamba, I. and Jüngel, A., Asymptotic limits in quantum trajectory models, Comm. Partial Differential Equations 27 (2002), 669-691.
8. Gamba, I. and Jüngel, A., Positive solutions to singular second and third order differential equations for quantum fluids, Arch. Rational Mech. Anal. 156 (2001), 183-203.
9. Gamba, I. and Morawitz, C., A viscous approximation for a 2D steady semiconductor or transonic gas dynamics flow: existence theorem for potential flow, Comm. Pure Appl. Math. 49 (1996), 999-1049.
10. Gardner, C., Numerical simulation of a steady-state electron shock wave in a submicron semiconductor device, IEEE Trans. El. Dev. 38 (1991), 392-398.
11. Gardner, C., The quantum hydrodynamic model for semiconductors devices, SIAM J. Appl. Math. 54 (1994), 409-427.
12. Gasser, I. and Jüngel, A., The quantum hydrodynamic model for semiconductors in thermal equilibrium, Z. Angew. Math. Phys. 48 (1997), 45-59.
13. Gasser, I., Lin, C.-K. and Markowich, P., A review of dispersive limits of the (non)linear Schrödinger-type equation, Taiwanese J. of Math. 4, (2000), 501-529.
14. Gasser, I. and Markowich, P., Quantum hydrodynamics, Wigner transforms and the classical limit, Asymptot. Anal. 14 (1997), 97-116.
15. Gasser, I., Markowich, P. A. and Ringhofer, C., Closure conditions for classical and quantum moment hierarchies in the small temperature limit, Transport Theory Statistic Phys. 25 (1996), 409-423.
16. Gyi, M. T. and Jüngel, A., A quantum regularization of the one-dimensional hydrodynamic model for semiconductors, Adv. Differential Equations 5 (2000), 773-800.
17. Hsiao, L. and Yang, T., Asymptotic of initial boundary value problems for hydrodynamic and drift diffusion models for semiconductors, J. Differential Equations 170 (2001), 472-493.
18. Jerome, J., Analysis of charge transport: a mathematical study of semiconductor devices, Springer-Verlag, Heidelberg 1996.
19. Jügel, A., A steady-state potential flow Euler-Poisson system for charged quantum fluids, Comm. Math. Phys. 194 (1998), 463-479.
20. Jüngel, A., Quasi-hydrodynamic semiconductor equations, Progress in Nonlinear Differential Equations, Birkhäuser, Basel 2001.
21. Jüngel, A., Mariani, M. C. and Rial, D., Local existence of solutions to the transient quantum hydrodynamic equations, Math. Models Methods Appl. Sci. 12 (2002), 485-495.
22. Jüngel, A. and Li, H.-L., Quantum Euler-Poisson systems: global existence and exponential decay, to appear in Quart. Appl. Math. 2005.
23. Landau, L. D. and Lifshitz, E. M., Quantum mechanics: non-relativistic theory, New York, Pergamon Press 1977.
24. Li, H.-L. and Markowich, P. A., A review of hydrodynamical models for semiconductors: asymptotic behavior, Bol. Soc. Brasil. Mat. (N.S.) 32 (2001), 321-342.
25. Loffredo, M. and Morato, L., On the creation of quantum vortex lines in rotating Hell. II nouvo cimento 108B (1993), 205-215.
26. Madelung, E., Quantentheorie in hydrodynamischer Form, Z. Phys. 40 (1927), 322.
27. Marcati, P. and Natalini, R., Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation, Arch. Rational Mech. Anal. 129 (1995), 129-145.
28. Markowich, P., Ringhofer, C. and Schmeiser, C., Semiconductor Equations, Springer, Wien 1990.
29. Pacard, F. and Unterreiter, A., A variational analysis of the thermal equilibrium state of charged quantum fluids, Comm. Partial Differential Equations 20 (1995), 885-900.
30. Shu, C.-W., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, ICASE Report No. 97-65, NASA Langley Research Center, Hampton, USA 1997.
31. Zhang, B. and Jerome, W., On a steady-state quantum hydrodynamic model for semiconductors, Nonlinear Anal., Theory Methods Appl. 26 (1996), 845-856.