Global weak solutions to the Quantum Navier-Stokes system in the whole space with a confining external potential

We consider a dissipative quantum fluid on the whole space $\mathbb{R}^d$ ($d\geq 1$) confined by an external harmonic potential. The dynamics of the quantum fluid is described by the Quantum Navier-Stokes (QNS) system which is a particular case of the Navier-Stokes-Korteweg systems. The goal of this paper is to prove the existence of global weak solutions to the QNS system. To this end, we write the evolution equations with respect to a Gaussian reference measure and follow the general strategy of previous works. Nevertheless, several substantial modifications have to be done due to our choice of the reference measure.