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

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 cross -diffusion 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.