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.

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

Antonelli, Paolo
Supervision
;
Hientzsch, Lars Eric
Writing – Original Draft Preparation
;
Marcati, Pierangelo
Supervision
2020

Abstract

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.
compressible and incompressible Navier--Stokes equation, quantum fluids, low Mach number limit, acoustic waves, Strichartz estimates, energy estimates Read More: https://epubs.siam.org/doi/abs/10.1137/19M1252958
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.12571/15362
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