Extending relative entropy for Korteweg-type models with non-monotone pressure: large friction limit and weak-strong uniqueness

In this paper we study weak-strong uniqueness and singular relaxation limits for the Euler–Korteweg and Navier–Stokes–Korteweg systems with non-monotone pressure. Both weak-strong uniqueness and the relaxation limit are investigated using relative entropy technique. We make use of the enlarged formulation of the model in terms of the drift velocity introduced in [Bresch, D. et al., Arch. Ration. Mech. Anal., 233(3):975–1025, 2019], generalizing in this way results proved in [Lattanzio, C. et al. Commun. Part. Diff. Eqs., 42(2):261–290, 2017] for the Euler–Korteweg model, by allowing more general capillarity functions, and the result contained in [Cianfarani Carnevale, G. et al. J. Diff. Eqs., 269, 2020] for the monotone pressure case.