Analysis of acoustic oscillations for a class of hydrodynamic systems describing quantum fluids

TL;DR

This paper analyzes how quantum effects modify acoustic oscillations in hydrodynamic systems for quantum fluids, providing refined estimates and applications to low Mach number limits, with implications for capillary fluid models.

Contribution

It introduces refined Strichartz estimates for quantum hydrodynamic systems and extends dispersive analysis to related Euler and Navier-Stokes-Korteweg models.

Findings

Quantum effects alter acoustic dispersion relations.

Refined estimates improve control of acoustic oscillations.

Applications to low Mach number limits in quantum fluids.

Abstract

Hydrodynamic systems for quantum fluids are systems for compressible fluid flows for which quantum effects are macroscopically relevant. We discuss how the presence of the dispersive tensor describing the quantum effects alters the acoustic dispersion at the example of the Quantum Hydrodynamic system (QHD). For the QHD system the dispersion relation is given by the Bogoliubov dispersion relation for weakly interacting Bose gases. We provide refined Strichartz estimates allowing for an accurate control of acoustic oscillations. Applications to the low Mach number limit for the quantum Navier-Stokes equations and the QHD system are discussed. The dispersive analysis generalizes to some Euler- and Navier-Stokes-Korteweg systems for capillary fluids.