The Quantum Hydrodynamics system in two space dimensions

TL;DR

This paper proves the global existence of weak solutions for the 2D Quantum Hydrodynamics system without requiring regularity or smallness assumptions, using a novel polar decomposition approach and advanced mathematical techniques.

Contribution

It introduces a polar decomposition method that handles vacuum regions, enabling the analysis of weak solutions without regularity or smallness constraints.

Findings

Established global existence of weak solutions in 2D quantum hydrodynamics

Developed a polar decomposition approach avoiding velocity field definitions in nodal regions

Utilized uniform Strichartz estimates and local smoothing properties in the analysis

Abstract

In this paper we study global existence of weak solutions for the Quantum Hydrodynamics System in 2-D in the space of energy. We do not require any additional regularity and/or smallness assumptions on the initial data. Our approach replaces the WKB formalism with a polar decomposition theory which is not limited by the presence of vacuum regions. In this way we set up a self consistent theory, based only on particle density and current density, which does not need to define velocity fields in the nodal regions. The mathematical techniques we use in this paper are based on uniform (with respect to the approximating parameter) Strichartz estimates and the local smoothing property.