On the Low Mach Number Limit for Quantum Navier--Stokes Equations

We investigate the low Mach number limit for the three-dimensional quantum Navier--Stokes system. For general ill-prepared initial data, we prove strong convergence of finite energy weak solutions to weak solutions of the incompressible Navier--Stokes equations. Our approach relies on a quite accurate dispersive analysis for the acoustic part, governed by the well-known Bogoliubov dispersion relation for the elementary excitations of the weakly interacting Bose gas. Once we have a control of the acoustic dispersion, the a priori bounds provided by the energy and Bresch--Desjardins entropy type estimates lead to the strong convergence. Moreover, for well-prepared data we show that the limit is a Leray weak solution, namely, it satisfies the energy inequality. Solutions under consideration in this paper are not smooth enough to allow for the use of relative entropy techniques.