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We study some properties of a multi-species degenerate Ginzburg-Landau energyand its relation to a cross-diffusion Cahn-Hilliard system. The model ismotivated by multicomponent mixtures where crossdiffusion effects between thedifferent species are taken into account, and where only one species doesseparate from the others. Using a comparison argument, we obtain strict boundson the minimizers from which we can derive first-order optimality conditions,revealing a link with the single-species energy, and providing enoughregularity to qualify the minimizers as stationary solutions of the evolutionsystem. We also discuss convexity properties of the energy as well as long timeasymptotics of the time-dependent problem. Lastly, we introduce astructure-preserving finite volume scheme for the time-dependent problem andpresent several numerical experiments in one and two spatial dimensions.

Stationary solutions and large time asymptotics to a cross-diffusion-Cahn-Hilliard system