  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.
