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

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

2008-02-06

arxiv.org/abs/0802.0846v1

Mathematics

Analysis of PDEs

Scientific paper

10.1007/s00220-008-0632-0

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, have been extensively used in Physics to investigate Supefluidity and Superconductivity phenomena and more recently in the modeling of semiconductor devices . 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 Schr\"odinger Madelung picture with a suitable wave function updating mechanism. Therefore several \emph{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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