From nonlocal Euler-Korteweg to local Cahn-Hilliard via the high-friction limit

TL;DR

This paper rigorously derives the nonlocal and local Cahn-Hilliard equations from the Euler-Korteweg system in the high-friction limit using the relative entropy method, providing a new approach for equations lacking classical solutions.

Contribution

It introduces a novel method to derive complex equations from fluid models via the high-friction limit and relative entropy, especially for measure-valued solutions.

Findings

Derivation of nonlocal Cahn-Hilliard as a high-friction limit of Euler-Korteweg

Rigorous derivation of local degenerate Cahn-Hilliard equation

Application of relative entropy method to measure-valued solutions

Abstract

Several recent papers considered the high-friction limit for systems arising in fluid mechanics. Following this approach, we rigorously derive the nonlocal Cahn-Hilliard equation as a limit of the nonlocal Euler-Korteweg equation using the relative entropy method. Applying the recent result by the first and third author, we also derive rigorously the local degenerate Cahn-Hilliard equation. The proof is formulated for dissipative measure-valued solutions of the nonlocal Euler-Korteweg equation which are known to exist on arbitrary intervals of time. Our work provides a new method to derive equations not enjoying classical solutions via relative entropy method by introducing the nonlocal effect in the fluid equation.