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On the Finite Energy Weak Solutions to a System in Quantum Fluid Dynamics

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Abstract

In this paper we consider the global existence of weak solutions to a class of Quantum Hydrodynamics (QHD) systems with initial data, arbitrarily large in the energy norm. These type of models, initially proposed by Madelung [44], have been extensively used in Physics to investigate Superfluidity and Superconductivity phenomena [19,38] and more recently in the modeling of semiconductor devices [20] . Our approach is based on various tools, namely the wave functions polar decomposition, the construction of approximate solution via a fractional steps method which iterates a Schrödinger Madelung picture with a suitable wave function updating mechanism. Therefore several a priori bounds of energy, dispersive and local smoothing type, allow us to prove the compactness of the approximating sequences. No uniqueness result is provided.

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  1. Dipartimento di Matematica Pura ed Applicata, Università degli Studi dell’Aquila, Via Vetoio, 67010, Coppito (AQ), Italy

    Paolo Antonelli & Pierangelo Marcati

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  1. Paolo Antonelli
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  2. Pierangelo Marcati
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Communicated by P. Constantin

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Antonelli, P., Marcati, P. On the Finite Energy Weak Solutions to a System in Quantum Fluid Dynamics. Commun. Math. Phys. 287, 657–686 (2009). https://doi.org/10.1007/s00220-008-0632-0

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