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

## TL;DR

This paper studies the low Mach number limit of the 3-D quantum Navier-Stokes equations, proving strong convergence to incompressible flows using dispersive analysis and energy estimates, even for ill-prepared initial data.

## Contribution

It introduces a dispersive analysis approach for acoustic parts and establishes strong convergence results for weak solutions in the quantum Navier-Stokes system.

## Key findings

- Strong convergence of solutions to incompressible Navier-Stokes equations
- Dispersive analysis based on Bogoliubov dispersion relation
- Results hold for ill-prepared initial data

## Abstract

We investigate the low Mach number limit for the 3-D 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.