Existence and weak-strong uniqueness for Maxwell-Stefan-Cahn-Hilliard systems

TL;DR

This paper proves the existence and uniqueness of solutions for a complex Maxwell-Stefan-Cahn-Hilliard system modeling fluid mixtures, overcoming mathematical challenges posed by degeneracy and high-order derivatives.

Contribution

It introduces a novel analysis method for Maxwell-Stefan-Cahn-Hilliard systems, establishing global weak solutions and weak-strong uniqueness despite degeneracy issues.

Findings

Proved global existence of weak solutions.

Established weak-strong uniqueness property.

Developed energy and entropy estimates for high-order PDEs.

Abstract

A Maxwell-Stefan system for fluid mixtures with driving forces depending on Cahn-Hilliard-type chemical potentials is analyzed. The corresponding parabolic cross-diffusion equations contain fourth-order derivatives and are considered in a bounded domain with no-flux boundary conditions. The main difficulty of the analysis is the degeneracy of the diffusion matrix, which is overcome by proving the positive definiteness of the matrix on a subspace and using the Bott--Duffin matrix inverse. The global existence of weak solutions and a weak-strong uniqueness property are shown by a careful combination of (relative) energy and entropy estimates, yielding $H^2(\Omega)$ bounds for the densities, which cannot be obtained from the energy or entropy inequalities alone.