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.

翻译：本文研究了多种类退化Ginzburg-Landau能量的若干性质及其与交叉扩散Cahn-Hilliard系统的关联。该模型受多组分混合物启发，其中考虑了不同物种间的交叉扩散效应，且仅有一种物种与其他物种分离。通过比较论证方法，我们获得了极小化子的严格界，并由此推导出一阶最优性条件，揭示其与单物种能量的联系，同时提供足够正则性以证明极小化子可作为演化系统的稳态解。我们还讨论了能量的凸性性质以及含时问题的长时间渐近行为。最后，针对含时问题引入了一种保结构有限体积格式，并在一维和二维空间维度上进行了若干数值实验。