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

TL;DR

This paper develops a relative energy framework for gas dynamics models like Euler-Korteweg and Navier-Stokes-Korteweg, establishing stability, uniqueness, and convergence results, especially for systems with complex, non-monotone pressure laws.

Contribution

It introduces a novel functional format for relative energy applicable to various Hamiltonian flow models, enabling stability and convergence analysis for complex Korteweg-type systems.

Findings

Proves stability and weak-strong uniqueness for Euler-Korteweg system.

Establishes stability and continuous dependence for Navier-Stokes-Korteweg system.

Demonstrates how higher-order gradients facilitate analysis of non-convex energies.

Abstract

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.