| Title:
|
Quantum Euler-Poisson systems: Existence of stationary states (English) |
| Author:
|
Jüngel, Ansgar |
| Author:
|
Li, Hailiang |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
40 |
| Issue:
|
4 |
| Year:
|
2004 |
| Pages:
|
435-456 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A one-dimensional quantum Euler-Poisson system for semiconductors for the electron density and the electrostatic potential in bounded intervals is considered. The existence and uniqueness of strong solutions with positive electron density is shown for quite general (possibly non-convex or non-monotone) pressure-density functions under a “subsonic” condition, i.e. assuming sufficiently small current densities. The proof is based on a reformulation of the dispersive third-order equation for the electron density as a nonlinear elliptic fourth-order equation using an exponential transformation of variables. (English) |
| Keyword:
|
quantum hydrodynamics |
| Keyword:
|
existence and uniqueness of solutions |
| Keyword:
|
non-monotone pressure |
| Keyword:
|
semiconductors |
| MSC:
|
35Q55 |
| MSC:
|
35Q60 |
| MSC:
|
76Y05 |
| MSC:
|
82C10 |
| MSC:
|
82D37 |
| idZBL:
|
Zbl 1122.35140 |
| idMR:
|
MR2129964 |
| . |
| Date available:
|
2008-06-06T22:44:48Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107926 |
| . |
| Reference:
|
[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. MR 1355150 |
| Reference:
|
[2] Chen G., Wang D.: Convergence of shock schemes for the compressible Euler-Poisson equations.Comm. Math. Phys. 179 (1996), 333–364. MR 1400743 |
| Reference:
|
[3] Courant R., Friedrichs K. O.: Supersonic flow and shock waves.Springer-Verlag, New York 1976. MR 0421279 |
| Reference:
|
[4] Degond P., Markowich P. A.: On a one-dimensional steady-state hydrodynamic model.Appl. Math. Lett. 3 (1990), 25–29. MR 1077867 |
| Reference:
|
[5] Degond P., Markowich P. A.: A steady state potential flow model for semiconductors.Ann. Mat. Pura Appl. 165 (1993), 87–98. MR 1271412 |
| Reference:
|
[6] Gamba I.: Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductor.Comm. Partial Differential Equations 17 (1992), 553–577. MR 1163436 |
| Reference:
|
[7] Gamba I., Jüngel A.: Asymptotic limits in quantum trajectory models.Comm. Partial Differential Equations 27 (2002), 669–691. MR 1900558 |
| Reference:
|
[8] Gamba I., Jüngel A.: Positive solutions to singular second and third order differential equations for quantum fluids.Arch. Rational Mech. Anal. 156 (2001), 183–203. MR 1816474 |
| Reference:
|
[9] Gamba I., 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. MR 1404324 |
| Reference:
|
[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. |
| Reference:
|
[11] Gardner C.: The quantum hydrodynamic model for semiconductors devices.SIAM J. Appl. Math. 54 (1994), 409–427. MR 1265234 |
| Reference:
|
[12] Gasser I., Jüngel A.: The quantum hydrodynamic model for semiconductors in thermal equilibrium.Z. Angew. Math. Phys. 48 (1997), 45–59. MR 1439735 |
| Reference:
|
[13] Gasser I., Lin C.-K., Markowich P.: A review of dispersive limits of the (non)linear Schrödinger-type equation.Taiwanese J. of Math. 4, (2000), 501–529. MR 1799752 |
| Reference:
|
[14] Gasser I., Markowich P.: Quantum hydrodynamics, Wigner transforms and the classical limit.Asymptot. Anal. 14 (1997), 97–116. Zbl 0877.76087, MR 1451208 |
| Reference:
|
[15] Gasser I., Markowich P. A., Ringhofer C.: Closure conditions for classical and quantum moment hierarchies in the small temperature limit.Transport Theory Statistic Phys. 25 (1996), 409–423. Zbl 0871.76078, MR 1407543 |
| Reference:
|
[16] Gyi M. T., Jüngel A.: A quantum regularization of the one-dimensional hydrodynamic model for semiconductors.Adv. Differential Equations 5 (2000), 773–800. Zbl 1174.82348, MR 1750118 |
| Reference:
|
[17] Hsiao L., Yang T.: Asymptotic of initial boundary value problems for hydrodynamic and drift diffusion models for semiconductors.J. Differential Equations 170 (2001), 472–493. MR 1815191 |
| Reference:
|
[18] Jerome J.: Analysis of charge transport: a mathematical study of semiconductor devices.Springer-Verlag, Heidelberg 1996. MR 1437143 |
| Reference:
|
[19] Jüngel A.: A steady-state potential flow Euler-Poisson system for charged quantum fluids.Comm. Math. Phys. 194 (1998), 463–479. MR 1627673 |
| Reference:
|
[20] Jüngel A.: Quasi-hydrodynamic semiconductor equations.Progress in Nonlinear Differential Equations, Birkhäuser, Basel 2001. Zbl 0969.35001, MR 1818867 |
| Reference:
|
[21] Jüngel A., Mariani M. C., Rial D.: Local existence of solutions to the transient quantum hydrodynamic equations.Math. Models Methods Appl. Sci. 12 (2002), 485–495. Zbl 1215.81031, MR 1899838 |
| Reference:
|
[22] Jüngel A., Li H.-L.: Quantum Euler-Poisson systems: global existence and exponential decay.to appear in Quart. Appl. Math. 2005. Zbl 1069.35012, MR 2086047 |
| Reference:
|
[23] Landau L. D., Lifshitz E. M.: Quantum mechanics: non-relativistic theory.New York, Pergamon Press 1977. MR 0400931 |
| Reference:
|
[24] Li H.-L., Markowich P. A.: A review of hydrodynamical models for semiconductors: asymptotic behavior.Bol. Soc. Brasil. Mat. (N.S.) 32 (2001), 321-342. Zbl 0996.82064, MR 1894562 |
| Reference:
|
[25] Loffredo M., Morato L.: On the creation of quantum vortex lines in rotating HeII.Il nouvo cimento 108B (1993), 205–215. |
| Reference:
|
[26] Madelung E.: Quantentheorie in hydrodynamischer Form.Z. Phys. 40 (1927), 322. |
| Reference:
|
[27] Marcati P., 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. Zbl 0829.35128, MR 1328473 |
| Reference:
|
[28] Markowich P., Ringhofer C., Schmeiser C.: Semiconductor Equations.Springer, Wien 1990. Zbl 0765.35001, MR 1063852 |
| Reference:
|
[29] Pacard F., Unterreiter A.: A variational analysis of the thermal equilibrium state of charged quantum fluids.Comm. Partial Differential Equations 20 (1995), 885–900. Zbl 0820.35112, MR 1326910 |
| Reference:
|
[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. MR 1728856 |
| Reference:
|
[31] Zhang B., Jerome W.: On a steady-state quantum hydrodynamic model for semiconductors.Nonlinear Anal., Theory Methods Appl. 26 (1996), 845–856. Zbl 0882.76105, MR 1362757 |
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