  For an Euler system, with dynamics generated by a potential energy
functional, we propose a functional format for the relative energy and derive a
relative energy identity. The latter, when applied to specific energies, yields
relative energy identities for the Euler-Korteweg, the Euler-Poisson, the
Quantum Hydrodynamics system, and low order approximations of the
Euler-Korteweg system. For the Euler-Korteweg system we prove a stability
theorem between a weak and a strong solution and an associated weak-strong
uniqueness theorem. In the second part we focus on the Navier-Stokes-Korteweg
system (NSK) with non-monotone pressure laws: we prove stability for the NSK
system via a modified relative energy approach. We prove continuous dependence
of solutions on initial data and convergence of solutions of a low order model
to solutions of the NSK system. The last two results provide physically
meaningful examples of how higher order regularization terms enable the use of
the relative energy framework for models with energies which are not poly- or
quasi-convex, but compensating via higher-order gradients.
