Non-Uniqueness of Weak Solutions of the Quantum-Hydrodynamic System

TL;DR

This paper explores the non-uniqueness of weak solutions in the Quantum-Hydrodynamic system, linking it to changes in the support's connected components and introducing the concept of trajectory-uniqueness.

Contribution

It demonstrates non-uniqueness conditions related to support component changes and introduces trajectory-uniqueness in quantum hydrodynamics.

Findings

Non-uniqueness occurs when the number of support components decreases.

Brouwer invariance of domain theorem is used to establish non-uniqueness.

Introduction of trajectory-uniqueness concept for full solution trajectories.

Abstract

We investigate the non-uniqueness of weak solutions of the Quantum-Hydrodynamic system. This form of ill-posedness is related to the change of the number of connected components of the support of the position density (called nodal domains) of the weak solution throughout its time evolution. We start by considering a scenario consisting of initial and final time, showing that if there is a decrease in the number of connected components, then we have non-uniqueness. This result relies on the Brouwer invariance of domain theorem. Then we consider the case in which the results involve a time interval and a full trajectory (position-current densities). We introduce the concept of trajectory-uniqueness and its characterization.