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

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.