Abstract

In this talk we consider the global existence of weak solutions to the Quantum Hydrodynamics System (QHD) with initial data in the energy space, without any smallness conditions. These types of models, initially proposed by Madelung, have been extensively used in Physics to investigate superfluidity phenomena and, more recently, in the modeling of semiconductor devices. Our approach is based on the formal equivalence between the QHD system and a nonlinear Schroedinger equation, and on the fractional step method to construct a sequence of approximate solutions. A key ingredient will be a wave function polar decomposition in order to develop a suitable updating mechanism and to give the quadratic terms in the system a stable structure depending on the wave function. Therefore several a priori bounds of energy, dispersive and local smoothing type allow us to prove the compactness of the approximate sequences. No uniqueness result is provided.