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

Abstract

Abstract. We consider an Euler system with dynamics generated by a potential energy functional. We propose a form for the relative energy that exploits the variational structure and 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: 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, compensated by higher-order gradients.

On Navier–Stokes–Korteweg and Euler–Korteweg Systems: Application to Quantum Fluids Models

“…This also helps to get rid the concavity assumption on 1/K(ρ) which is strongly used in [27]. For the interested readers, we provide a comparison of the quantities appearing in our relative entropy to the ones introduced in [27] and remark that they are equivalent under the assumptions made in [27].…”

“…Secondly, we develop relative entropy estimates for general cases of the Euler-Korteweg and the Navier-Stokes-Korteweg systems extending the augmented formulations introduced recently in [13] and [14]: more general viscosities and third order dispersive terms. This gives a more simple procedure to perform relative entropy than the one developped in [27,21] for the Euler-Korteweg system but asks to start with an augmented version of the Euler-Korteweg system. This allows us to provide a weak-strong uniqueness result for the Euler-Korteweg and Navier-Stokes-Korteweg systems.…”

Remarks on the derivation of finite energy weak solutions to the QHD system

“…see also [32]. Unfortunately the chemical potential µ cannot be used to carry out a satisfactory mathematical analysis in the framework of weak solutions to (1.1).…”

High-friction limits of Euler flows for multicomponent systems

“…While such relaxation and high-friction limits are widely explored in mono-species situations, there are no results for multicomponent Euler-Korteweg flows. The aim of this paper is to compute the Chapman-Enskog expansion and to justify the expansion via a relative entropy approach, extending results for the mono-species case to fluid mixtures [10,15,16].…”

“…Model (1)- (2) belongs to the general realm of multicomponent fluid mixtures whose thermodynamical structure has been extensively analyzed; see, e.g., [3,18,19] and references therein. On the other hand, we adopt the mathematical structure espoused in [10], in that the dynamics of the flow is determined by the functional E(ρ) of potential energy, with δE/δρ i standing for the variational derivatives with respect to the partial densities ρ i . Several isothermal models fit into this framework.…”