In this talk, we discuss the low Mach number limit for the system of Quantum Navier-Stokes equations in three dimensions. The QNS equations, arising as dissipative model for quantum fluids, are barotropic compressible Navier-Stokes equations with a degenerate viscosity tensor and augmented by a third-order dispersive quantum correction term. For general ill-prepared initial data of finite energy, we show strong convergence of finite energy weak solutions towards weak solutions to the incompressible Navier-Stokes equations.
The main novelty is that our analysis only requires the initial data to be of finite energy, in particular no smallness, regularity or well-preparedness of the data is assumed. Our method is based on a precise analysis of acoustic waves based on some new Strichartz estimates for the linearised system being governed by the Bogoliubov dispersion relation. Further, we exploit a priori bounds derived from Bresch-Desjardins type inequalities.
Time permitting, we comment on work in progress regarding the quantum Hydrodynamic (QHD) system, namely QNS without viscous term, that has more physical relevance, for instance in superfluidity and Bose-Einstein condensation. We discuss some similar singular limits with applications to quantum vortex dynamics.
This is joint work with P. Antonelli and P. Marcati.