Finite Energy Weak Solutions to the Quantum Hydrodynamics System

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 [24], have been extensively used in Physics to investigate Supefluidity and Superconductivity phenomena [10], [19] and more recently in the modeling of semiconductor devices [11]. 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 SchrÖodinger 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. A more detailed exposition of the results is given in [2].Key wordsAnalysis of PDEsmathematical physics