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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.
D.J. Korteweg. Sur la forme que prennent les équations du mouvement si l’on tient compte de forces capillaires causées par les variations de densité considérables mais continues et sur la théorie de la capillarité dans l’hypothèse d’une variation continue de la densité
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C. Bardos, T.-T. Nguyen. Remarks on the inviscid limit for the compressible flows
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D. Bresch, F. Couderc, P. Noble, J.-P. Vila. A generalization of the quantum Bohm identity: Hyperbolic CFL condition for Euler–Korteweg equations
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