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

Abstract

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 by I. Lacroix-Violet and A. Vasseur (2016)] 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. Our results are based on the fact that Euler-Korteweg systems and corresponding Navier–Stokes-Korteweg systems can be reformu-lated through an augmented system 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. 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)] and with less hypothesis required on the capillary coefficient: no concavity assumption needed in our result.