On the compactness of finite energy weak solutions to the quantum Navier-Stokes equations

TL;DR

This paper proves the compactness of finite energy weak solutions to the three-dimensional Quantum Navier-Stokes equations, including vacuum regions, without additional damping or pressure terms, using an effective velocity formulation.

Contribution

It introduces a novel approach using an effective velocity to handle third order terms and includes vacuum regions in the weak formulation of the quantum Navier-Stokes system.

Findings

Proves compactness of weak solutions for large initial data.

Includes vacuum regions in the weak formulation.

Eliminates the need for extra damping or pressure terms.

Abstract

We consider the Quantum Navier-Stokes system in three space dimensions. We prove compactness of finite energy weak solutions for large initial data. The main novelties are that vacuum regions are included in the weak formulation and no extra terms, like damping or cold pressure, are considered in the equations in order to define the velocity field. Our argument uses an equivalent formulation of the system in terms of an effective velocity, in order to eliminate the third order terms in the new system. It allows us to derive estimates similar to the ones available in the case of the compressible Navier-Stokes with degenerate viscosity.