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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.

MSC:

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
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