Relative Energy for the Korteweg Theory and Related Hamiltonian Flows in Gas Dynamics

We consider a Euler system with dynamics generated by a potential energy functional. We propose a form for the relative energy that exploits the variational structure and we derive a relative energy identity. When applied to specific energies, this 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, and prove stability for the NSK system via a modified relative energy approach. We prove the continuous dependence of solutions on initial data and the 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, compensated by higher-order gradients.