Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Quantum Euler-Poisson systems: global existence and exponential decay

Authors: Ansgar Jüngel and Hailiang Li
Journal: Quart. Appl. Math. 62 (2004), 569-600
MSC: Primary 82C10; Secondary 35A07, 35B25, 35Q40, 76Y05, 82D37
DOI: https://doi.org/10.1090/qam/2086047
MathSciNet review: MR2086047
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Abstract | References | Similar Articles | Additional Information

Abstract: A one-dimensional transient quantum Euler-Poisson system for the electron density, the current density, and the electrostatic potential in bounded intervals is considered. The equations include the Bohm potential accounting for quantum mechanical effects and are of dispersive type. They are used, for instance, for the modelling of quantum semiconductor devices.

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  • [1] F. Brezzi, I. Gasser, P. Markowich, and C. Schmeiser, Thermal equilibrium state of the quantum hydrodynamic model for semiconductors in one dimension, Appl. Math. Lett. 8 (1995), 47-52. MR 1355150
  • [2] P. Degond and P. A. Markowich, On a one-dimensional steady-state hydrodynamic model, Appl. Math. Lett. 3 (1990), 25-29. MR 1077867
  • [3] K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math. 11 (1958), 333-418. MR 0100718
  • [4] I. Gamba and A. Jüngel, Positive solutions to singular second and third order differential equations for quantum fluids, Arch. Rational Mech. Anal. 156 (2001), 183-203. MR 1816474
  • [5] C. Gardner, Numerical simulation of a steady-state electron shock wave in a submicron semiconductor device, IEEE Trans. El. Dev. 38 (1991), 392-398.
  • [6] C. Gardner, The quantum hydrodynamic model for semiconductors devices, SIAM J. Appl. Math. 54 (1994), 409-427. MR 1265234
  • [7] I. Gasser and A. Jüngel, The quantum hydrodynamic model for semiconductors in thermal equilibrium, Z. Angew. Math. Phys. 48 (1997), 45-59. MR 1439735
  • [8] I. Gasser and P. Markowich, Quantum hydrodynamics, Wigner transforms and the classical limit, Asymptotic Anal. 14 (1997), 97-116. MR 1451208
  • [9] I. Gasser, P. A. Markowich, and C. Ringhofer, Closure conditions for classical and quantum moment hierarchies in the small temperature limit, Transp. Theory Stat. Phys. 25 (1996), 409-423. MR 1407543
  • [10] M. T. Gyi and A. Jüngel, A quantum regularization of the one-dimensional hydrodynamic model for semiconductors, Adv. Diff. Eqs. 5 (2000), 773-800.
  • [11] A. Jüngel, A steady-state potential flow Euler-Poisson system for charged quantum fluids, Comm. Math. Phys. 194 (1998), 463-479.
  • [12] A. Jüngel, Quasi-hydrodynamic semiconductor equations, Progress in Nonlinear Differential Equations, Birkhäuser, Basel (2001).
  • [13] A. Jüngel, M. C. Mariani, and D. Rial, Local existence of solutions to the transient quantum hydrodynamic equations, Math. Models Meth. Appl. Sci. 12 (2002), 485-495. MR 1899838
  • [14] A. Jüngel and H.-L. Li, Quantum Euler-Poisson systems: existence of stationary states, to appear in Arch. Math. (2003).
  • [15] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal. 58 (1975), 181-205. MR 0390516
  • [16] L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-relativistic Theory, New York, Pergamon Press (1977). MR 0400931
  • [17] H.-L. Li and C.-K. Lin, Semiclassical limit and well-posedness of Schrödinger-Poisson system, Electronic Journal of Differential Equations, Vol. 2003 (2003), No. 93, 1-17. MR 2000689
  • [18] H.-L. Li and P. Marcati, Existence and asymptotic behavior of multi-dimensional quantum hydrodynamic model for semiconductors, preprint (2002).
  • [19] H.-L. Li, M. Mei, and P. A. Markowich, Asymptotic behavior of solutions of the hydrodynamic model of semiconductors, Proc. Roy. Soc. Edinburgh, Sect. A 132 (2002), no. 2, 359-378. MR 1899826
  • [20] M. Loffredo and L. Morato, On the creation of quantum vortex lines in rotating HeII, Il nouvo cimento 108B (1993), 205-215.
  • [21] E. Madelung, Quantentheorie in hydrodynamischer Form, Z. Physik 40 (1927), 322.
  • [22] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Appl. Math. Sci. 53, Springer (1984). MR 748308
  • [23] T. Makino and S. Ukai, Sur l'existence des solutions locales de l'équation d'Euler-Poisson pour l'évolution d'étoiles gazeuses, J. Math. Kyoto Univ. 27 (1987), 387-399. MR 910225
  • [24] P. A. Markowich, C. Ringhofer, and C. Schmeiser, Semiconductor Equations, Springer, Wien (1990). MR 1063852
  • [25] C.-W. Shu, 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
  • [26] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci. 68, Springer (1988). MR 953967
  • [27] B. Zhang and J. Jerome, On a steady-state quantum hydrodynamic model for semiconductors, Nonlinear Anal., TMA 26 (1996), 845-856. MR 1362757
  • [28] E. Zeidler, Nonlinear functional analysis and its applications. Vol. II: Nonlinear monotone operators, Springer (1990).

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DOI: https://doi.org/10.1090/qam/2086047
Article copyright: © Copyright 2004 American Mathematical Society

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