Analysis of acoustic oscillations for a class of hydrodynamic systems describing 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.

Long-time stability of the quantum hydrodynamic system on irrational tori

“…Here the dispersive character of the equation is taken into account. For recent developments in this direction see [1] and reference therein. It is worth mentioning also the scattering result for the Gross-Pitaevsii equation [16].…”