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

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 nonconvex part of the energy is assumed to have a bounded Hessian. 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.