# Global existence of weak solutions to the compressible quantum Navier-Stokes equations with degenerate viscosity

## TL;DR

This paper proves the global existence of weak solutions to the three-dimensional compressible quantum Navier-Stokes equations with degenerate viscosity, including cases with damping and without damping, advancing mathematical understanding of quantum fluid models.

## Contribution

It introduces new methods to establish global weak solutions for compressible QNS equations with degenerate viscosity, improving previous results and removing certain assumptions.

## Key findings

- Existence of weak solutions with damping terms for large initial data.
- Global weak solutions without damping, removing lower bound assumptions.
- New a priori estimates avoiding velocity gradient assumptions.

## Abstract

We study the compressible quantum Navier-Stokes (QNS) equations with degenerate viscosity in the three dimensional periodic domains. On the one hand, we consider QNS with additional damping terms. Motivated by the recent works [Li-Xin, arXiv:1504.06826] and [Antonelli-Spirito, Arch. Ration. Mech. Anal., 203(2012), 499--527], we construct a suitable approximate system which has smooth solutions satisfying the energy inequality and the BD entropy estimate. Using this system, we obtain the global existence of weak solutions to the compressible QNS equations with damping terms for large initial data. Moreover, we obtain some new a priori estimates, which can avoid using the assumption that the gradient of the velocity is a well-defined function, which is indeed used directly in [Vasseur-Yu, SIAM J. Math. Anal., 48 (2016), 1489--1511; Invent. Math., 206 (2016), 935--974]. On the other hand, in the absence of damping terms, we also prove the global existence of weak solutions to the compressible QNS equations without the lower bound assumption on the dispersive coefficient, which improves the previous result due to [Antonelli-Spirito, Arch. Ration. Mech. Anal., 203(2012), 499--527].