Archive for Rational Mechanics and Analysis

, Volume 233, Issue 3, pp 975–1025 | Cite as

On Navier–Stokes–Korteweg and Euler–Korteweg Systems: Application to Quantum Fluids Models

  • Didier BreschEmail author
  • Marguerite Gisclon
  • Ingrid Lacroix-Violet


In this paper, the main objective is to generalize to the Navier–Stokes–Korteweg (with density dependent viscosities satisfying the BD relation) and Euler–Korteweg systems a recent relative entropy (proposed by Bresch et al. in C R Math Acad Sci Paris 354(1):45–49, 2016) introduced for the compressible Navier–Stokes equations with a linear density dependent shear viscosity and a zero bulk viscosity. As a concrete application, this helps to justify mathematically the convergence between global weak solutions of the quantum Navier–Stokes system (recently obtained simultaneously by Lacroix-Violet and Vasseur in J Math Pures Appl 114(9):191–210, 2018) and dissipative solutions of the quantum Euler system when the viscosity coefficient tends to zero; this selects a dissipative solution as the limit of a viscous system. We also recover the weak–strong uniqueness for the Quantum-Euler as in Giesselmann et al. (Arch Ration Mech Anal 223:1427–1484, 2017) and extend the result for the Quantum-Navier–Stokes equations. Our results are based on the fact that Euler–Korteweg systems and corresponding Navier–Stokes–Korteweg systems can be reformulated through an augmented system such as the compressible Navier–Stokes system with density dependent viscosities satisfying the BD algebraic relation. This was also observed recently by Bresch et al. (2016) for the Euler–Korteweg system for numerical purposes. As a by-product of our analysis, we show that this augmented formulation helps to define relative entropy estimates for the Euler–Korteweg systems in a simplest way compared to recent works (see Donatelli et al. in Commun Partial Differ Equ 40:1314–1335, 2015; Giesselmann et al. 2017) with less hypothesis required on the capillary coefficient.


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The third author acknowledges support from the teamINRIA/RAPSODI and the Labex CEMPI (ANR-11-LABX-0007-01). The first author acknowledges the project TELLUS INSU-INSMI “Approche croisée pour fluides visco-élasto-plastiques: vers une meilleure compréhension des zones solides/fluides”. The first and the second authors acknowledge the ANR Project FRAISE managed by C. Ruyer-Quil.

Conflict of interest

The authors have no conflicts of interest to declare.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Didier Bresch
    • 1
    Email author
  • Marguerite Gisclon
    • 1
  • Ingrid Lacroix-Violet
    • 2
  1. 1.Laboratoire de Mathématiques, CNRS UMR 5127Université Savoie Mont-BlancLe Bourget-du-LacFrance
  2. 2.Laboratoire de Mathématiques, CNRS UMR 8524Université de Lille 1Villeneuve d’AscqFrance

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