  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.

Analysis of PDEs