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

Relative energy for the Korteweg theory and related Hamiltonian flows in gas dynamics