On the low Mach number limit for Quantum Navier-Stokes equations

Abstract

In this paper, we investigate the low Mach number limit for the 3D 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.

Publication
preprint on the arXiv