On the low Mach number limit for 2D Navier–Stokes–Korteweg systems

Lars Eric Hientzsch; Lars Eric Hientzsch

doi:10.3934/mine.2023023

Mathematics in Engineering

2023, Volume 5, Issue 2: 1-26. doi: 10.3934/mine.2023023

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On the low Mach number limit for 2D Navier–Stokes–Korteweg systems

Lars Eric Hientzsch ^,

Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany

Received: 15 December 2021 Revised: 25 February 2022 Accepted: 25 February 2022 Published: 12 April 2022

This paper addresses the low Mach number limit for two-dimensional Navier–Stokes–Korteweg systems. The primary purpose is to investigate the relevance of the capillarity tensor for the analysis. For the sake of a concise exposition, our considerations focus on the case of the quantum Navier-Stokes (QNS) equations. An outline for a subsequent generalization to general viscosity and capillarity tensors is provided. Our main result proves the convergence of finite energy weak solutions of QNS to the unique Leray-Hopf weak solutions of the incompressible Navier-Stokes equations, for general initial data without additional smallness or regularity assumptions. We rely on the compactness properties stemming from energy and BD-entropy estimates. Strong convergence of acoustic waves is proven by means of refined Strichartz estimates that take into account the alteration of the dispersion relation due to the capillarity tensor. For both steps, the presence of a suitable capillarity tensor is pivotal.
- Navier–Stokes–Korteweg equation,
- incompressible Navier–Stokes equation,
- capillarity,
- quantum fluids,
- low Mach number limit,
- acoustic waves,
- Strichartz estimates,
- energy estimates,
- BD-entropy estimates
Citation: Lars Eric Hientzsch. On the low Mach number limit for 2D Navier–Stokes–Korteweg systems[J]. Mathematics in Engineering, 2023, 5(2): 1-26. doi: 10.3934/mine.2023023

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Abstract

This paper addresses the low Mach number limit for two-dimensional Navier–Stokes–Korteweg systems. The primary purpose is to investigate the relevance of the capillarity tensor for the analysis. For the sake of a concise exposition, our considerations focus on the case of the quantum Navier-Stokes (QNS) equations. An outline for a subsequent generalization to general viscosity and capillarity tensors is provided. Our main result proves the convergence of finite energy weak solutions of QNS to the unique Leray-Hopf weak solutions of the incompressible Navier-Stokes equations, for general initial data without additional smallness or regularity assumptions. We rely on the compactness properties stemming from energy and BD-entropy estimates. Strong convergence of acoustic waves is proven by means of refined Strichartz estimates that take into account the alteration of the dispersion relation due to the capillarity tensor. For both steps, the presence of a suitable capillarity tensor is pivotal.

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