The quantum Navier-Stokes (QNS) equations describe a compressible fluid including a degenerate viscosity and a dispersive tensor accounting for capillarity effects. The QNS equations can be considered as a viscous correction of the Quantum Hydrodynamics (QHD) arising in the description of superfluid flow in Bose-Einstein condensation and superfluid helium. In this talk, we consider the QNS system with non-trivial farfield behaviour, namely non-vanishing density at infinity, providing the suitable framework to study coherent structures and the low Mach number limit.

In the first part of the talk, we show global existence of finite energy weak solutions with non-trivial farfield. In the second part, we discuss the low Mach number limit for the 3D QNS equations in the class of finite energy weak solutions. The main novelty is a precise analysis of the acoustic dispersion. The presence of the dispersive capillary tensor alters the dispersion relation, the linearized system is governed by the Bogoliubov dispersion relation.

Time permitting, we comment on work in progress regarding the quantum Hydrodynamic (QHD) system, namely the inviscid counterpart of the QNS eq. and quantum vortices.

These are joint works with P. Antonelli, S. Spirito and P. Marcati.