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

Abstract

We consider the quantum Navier–Stokes (QNS) 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. This will allow to obtain the same compactness properties as for the Navier–Stokes equations with degenerate viscosity.