Local existence of solutions to the transient quantum hydrodynamic equations. (English) Zbl 1215.81031

Summary: The existence of weak solutions locally in time to the quantum hydrodynamic equations in bounded domains is shown. These Madelung-type equations consist of the Euler equations, including the quantum Bohm potential term, for the particle density and the particle current density and are coupled to the Poisson equation for the electrostatic potential. This model has been used in the modeling of quantum semiconductors and superfluids. The proof of the existence result is based on a formulation of the problem as a nonlinear Schrödinger–Poisson system and uses semigroup theory and fixed-point techniques.


81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
82D37 Statistical mechanics of semiconductors
82D50 Statistical mechanics of superfluids
Full Text: DOI


[1] DOI: 10.1007/s002050000114 · Zbl 1013.76009 · doi:10.1007/s002050000114
[2] DOI: 10.1109/16.69922 · doi:10.1109/16.69922
[3] DOI: 10.1137/S0036139992240425 · Zbl 0815.35111 · doi:10.1137/S0036139992240425
[4] DOI: 10.1109/43.24878 · Zbl 05448063 · doi:10.1109/43.24878
[5] DOI: 10.1016/S0893-9659(00)00149-X · Zbl 0978.82103 · doi:10.1016/S0893-9659(00)00149-X
[6] Gasser I., Asymptotic Anal. 14 pp 97– (1997)
[7] DOI: 10.1080/00411459608220710 · Zbl 0871.76078 · doi:10.1080/00411459608220710
[8] Gyi M. T., Adv. Differential Equations 5 pp 773– (2000)
[9] DOI: 10.1007/s002200050364 · Zbl 0916.76099 · doi:10.1007/s002200050364
[10] DOI: 10.1016/S0362-546X(01)00704-0 · Zbl 1042.82630 · doi:10.1016/S0362-546X(01)00704-0
[11] DOI: 10.1007/BF02874411 · doi:10.1007/BF02874411
[12] DOI: 10.1007/BF01400372 · doi:10.1007/BF01400372
[13] Pietra P., Proc. of the Workshop on Quantum Kinetic Theory 9 pp 427– (1999)
[14] DOI: 10.1016/0362-546X(94)00326-D · Zbl 0882.76105 · doi:10.1016/0362-546X(94)00326-D
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.