On Navier-Stokes-Korteweg and Euler-Korteweg Systems: Application to Quantum Fluids Models - Archive ouverte HAL
Article Dans Une Revue Archive for Rational Mechanics and Analysis Année : 2019

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

Résumé

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 D. Bresch, P. Noble and J.–P. Vila, (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 I. Lacroix-Violet and A. Vasseur (2017)] 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 get weak-strong uniqueness for the Quantum-Euler and 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 D. Bresch, F. Couderc, P. Noble and J.–P. Vila, (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 D. Donatelli, E. Feireisl, P. Marcati (2015) and J. Giesselmann, C. Lattanzio, A.-E. Tzavaras (2017)] with less hypothesis required on the capillary coefficient.
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Dates et versions

hal-01496960 , version 1 (28-03-2017)
hal-01496960 , version 2 (29-01-2018)
hal-01496960 , version 3 (01-02-2018)
hal-01496960 , version 4 (21-06-2018)

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Citer

Didier Bresch, Marguerite Gisclon, Ingrid Lacroix-Violet. On Navier-Stokes-Korteweg and Euler-Korteweg Systems: Application to Quantum Fluids Models. Archive for Rational Mechanics and Analysis, 2019, 233 (3), pp.975-1025. ⟨10.1007/s00205-019-01373-w⟩. ⟨hal-01496960v4⟩
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