Global Existence of Finite Energy Weak Solutions of Quantum Navier-Stokes Equations

TL;DR

This paper proves the global existence of finite energy weak solutions for the Quantum Navier-Stokes equations in two and three dimensions, including vacuum regions, without additional damping or pressure terms.

Contribution

It introduces a novel regular approximation system aligned with the effective velocity transformation, enabling the proof of global solutions for large initial data.

Findings

Global existence of weak solutions established

Vacuum regions included in the weak formulation

No extra damping or cold pressure terms used

Abstract

In this paper we consider the Quantum Navier-Stokes system both in two and in three space dimensions and prove global existence of finite energy weak solutions for large initial data. In particular, the notion of weak solutions is the standard one. This means that the vacuum region are included in the weak formulations. In particular, no extra term like damping or cold pressure are added to the system in order to define the velocity field in the vacuum region. The main contribution of this paper is the construction of a regular approximating system consistent with the effective velocity transformation needed to get necessary a priori estimates.