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

TL;DR

This paper proves the global existence of weak solutions for a class of Quantum Hydrodynamics systems with large initial energy, using a wave function approach and compactness arguments, relevant for physics and semiconductor modeling.

Contribution

It introduces a new method combining wave function polar decomposition and fractional steps to establish existence of solutions in quantum fluid models.

Findings

Existence of weak solutions for large initial energy data.

Use of a fractional steps method for solution construction.

Establishment of a priori bounds ensuring compactness.

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, 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.