复杂流体中的两相流及其界面的动力学近年来备受应用数学领域的关注。这些模型被广泛用于描述自然界和现代工业中的各种现象，如生物膜的输运，金属侵蚀过程，液晶的缺陷以及石油开采中油水混合物的分离过程。研究相应的偏微分方程模型可以加深对这些现象的发生机制的理解并为数值计算提供依据。本项目将重点关注复杂流体中与界面相关的三个课题。首先是分析Allen-Cahn能量的二阶变分流，它是生物膜形变理论的重要模型。第二个是相场/流体耦合系统，它是两相流的基本模型。最后是液晶中的向列相-无序相相变及其界面生成问题。尽管这些模型源于复杂流体的不同方面，相应的偏微分方程也具有不同的结构，但它们都涉及小参数渐近极限及界面生成。本项目申请团队将利用内外层展开法和伽马收敛法严格论证它们的界面极限，将它们各自的研究方法融会贯通从而开发新路径，提出新问题，最终实现对该主题的全面认识。

The coexistence of two phases in complex fluid and the dynamics of their interfaces are gaining extensive attentions among engineers and applied mathematicians in the last two decades. Such models are widely employed to describe various phenomenon in nature and modern industry, such as bio-membranes deformation and transportation, erosion process of metal, phase-transition in liquid crystals material, and oil-water mixture segregation in oil exploitation. Studying the corresponding partial differential equations serves as a theoretical way of achieving an overall understanding of the mechanisms. In this project we shall focus on three models arising from two phases physics. The first one is the second order variational flow of Allen-Cahn energy, which is used as a simplified model for the deformation theory of bio-membranes. The second model is the Allen-Cahn/Stokes system which characterizes the evolution of a viscous fluid carrying bubbles. The last model is about the isotropic-nematic phase transition in liquid crystals, especially under the framework of the Onsager’s molecular theory. Despite that these models origin from different aspects of complex fluid and the corresponding partial differential equations retain individual properties and analytical difficulties, all of them can be considered as diffusive interface models that involves phase transitions in the small parameter asymptotics. We shall rigorously study their sharp interface limit by means of the formally matched asymptotic method or/and the Gamma-convergence theory. This will involve many other branches of mathematics such as geometric measure theory, geometric analysis, operator theory, and so on. Through these investigations, we wish to borrow existing technics from each of these problems and develop new methods, and eventually achieve an overall understanding of the subject.