Interplay Between Interfacial Patterns, Singular Perturbations and Heterogeneous Medium

YipDMS-1009102 研究者和他的同事们研究材料中的缺陷结构（如相和晶界）与背景非均匀介质之间的相互作用。 异质性可以来自内在的原子结构，外部强加的强迫，甚至是数值近似。重点是理解界面的结构和传播。 由于材料界面和背景不均匀性之间的高度非线性相互作用，出现了许多新的特征，最显著的是图案之间的边界而不是纯相，钉扎和脱钉扎转变。 在数学学科方面，该项目与多尺度共存的奇摄动问题的均匀化概念有关。 为了解决当前的非线性问题，需要超越传统方法的新技术和新解释。 该项目研究模式之间的边界的结构和传播，以及脉动波和奇异扰动之间的联系。 对离散和连续两种状态进行了分析。 该项目是高度跨学科的，并受到来自应用科学，特别是材料科学的问题的启发。 它汇集了严格的应用分析，数值模拟和各种物理现象的建模。 它强调多尺度之间的联系，因为实际材料通常由具有不同长度尺度的结构组成。 结果可以导致更好地理解如何施加强迫和外部环境的变化可以影响材料的响应行为。 这反过来又可以导致制造更强和更好的材料。

YipDMS-1009102 The investigator and his colleagues study the interactionbetween the defect structures in materials (such as phase andgrain boundaries) and the background heterogeneous medium. Theheterogeneity can come from intrinsic atomic structures,externally imposed forcing, or even numerical approximations. The emphasis is on understanding the structure and propagation ofinterfaces. Due to the highly nonlinear interaction between thematerials interface and background inhomogeneity, many newfeatures arise, most notably boundaries between patterns insteadof pure phases, and pinning and de-pinning transitions. In termsof mathematical disciplines, the project is related to theconcept of homogenization of singularly perturbed problems inwhich multiple scales coexist. To tackle nonlinear problems asin the current setting, new techniques and interpretations beyondtraditional approaches are required. The project investigatesthe structure and propagation of boundaries between patterns, andthe connection between pulsating waves and singularperturbations. Both the discrete and continuum regimes areanalyzed. The project is highly interdisciplinary and iswell-motivated by problems coming from applied science, inparticular materials science. It brings together rigorousapplied analysis, numerical simulations and modeling in a varietyof physical phenomena. It emphasizes the connections betweenmultiple scales, as practical materials often are composed ofstructures that have greatly different length scales. Resultscan lead to better understanding of how applied forcing andchanges of external environments can affect the behavior ofmaterials responses. This in turn can lead to the manufacturingof stronger and better materials.