  We study some properties of a multi-species degenerate Ginzburg-Landau energy
and its relation to a cross-diffusion Cahn-Hilliard system. The model is
motivated by multicomponent mixtures where crossdiffusion effects between the
different species are taken into account, and where only one species does
separate from the others. Using a comparison argument, we obtain strict bounds
on the minimizers from which we can derive first-order optimality conditions,
revealing a link with the single-species energy, and providing enough
regularity to qualify the minimizers as stationary solutions of the evolution
system. We also discuss convexity properties of the energy as well as long time
asymptotics of the time-dependent problem. Lastly, we introduce a
structure-preserving finite volume scheme for the time-dependent problem and
present several numerical experiments in one and two spatial dimensions.
