Stability properties of the Euler-Korteweg system with nonmonotone pressures

TL;DR

This paper develops a relative energy framework for the Euler-Korteweg system with non-convex energy, enabling proofs of weak-strong uniqueness, convergence to Cahn-Hilliard, and limits to Euler system under certain conditions.

Contribution

It introduces a novel relative energy approach for non-convex Euler-Korteweg systems, establishing new convergence and uniqueness results.

Findings

Proves weak-strong uniqueness for the system.

Shows convergence to Cahn-Hilliard system in large friction limit.

Demonstrates convergence to Euler system in vanishing capillarity limit.

Abstract

We establish a relative energy framework for the Euler-Korteweg system with non-convex energy. This allows us to prove weak-strong uniqueness and to show convergence to a Cahn-Hilliard system in the large friction limit. We also use relative energy to show that solutions of Euler-Korteweg with convex energy converge to solutions of the Euler system in the vanishing capillarity limit, as long as the latter admits sufficiently regular strong solutions.