  We establish a relative energy framework for the Euler-Korteweg system with
non-convex energy. This allows us to prove weak-strong uniqueness and to show
convergence to a Cahn-Hilliard system in the large friction limit. We also use
relative energy to show that solutions of Euler-Korteweg with convex energy
converge to solutions of the Euler system in the vanishing capillarity limit,
as long as the latter admits sufficiently regular strong solutions.
