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：本文研究了多物种退化Ginzburg-Landau能量的一些性质及其与交叉扩散Cahn-Hilliard系统的关系。该模型源于多组分混合物，其中考虑了不同物种之间的交叉扩散效应，且仅有一个物种与其他物种分离。通过比较论证，我们得到了极小元的严格界，并由此推导出一阶最优性条件，揭示了与单物种能量的联系，同时提供了足够的正则性以证明极小元是演化系统的稳态解。我们还讨论了能量的凸性性质以及时间依赖问题的长期渐近行为。最后，针对时间依赖问题，我们引入了一种保持结构特征的有限体积格式，并展示了在一维和二维空间中的若干数值实验。