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 bounds for the densities, which cannot be obtained from the energy or entropy inequalities alone.
Cite this article
Xiaokai Huo, Ansgar Jüngel, Athanasios E. Tzavaras, Existence and weak–strong uniqueness for Maxwell–Stefan–Cahn–Hilliard systems. Ann. Inst. H. Poincaré Anal. Non Linéaire (2023),DOI 10.4171/AIHPC/89