Mathematical Sciences: Some Mathematical Problems Associated with Phase Transitions

主要研究者将分析定性 几类偏微分方程解的性态 这是相变理论的一部分。 特别 感兴趣的是Cahn-Hilliard、Hele-Shaw和Phase场 方程 每个模型的均衡集的结构 也将讨论。 技术涉及的理论， 奇异扰动、不变流形和谱 分析微分算子以及经典 偏微分方程解的估计 这个项目涉及几个数学问题 与固体中的相变有关。 在这方面 区分非均质材料的不同相 由不同浓度的成分组成。 的 这个项目中涉及的数学方程也出现在 科学和几何学的许多其他领域，因此， 预计这里的成果将产生深远的影响。 之间 主要研究人员所处理的现象是 自发形成细粒结构，成核 由此广泛分离的颗粒经历快速生长，并且 材料微观结构的粗化。 所有这些 现象对于材料性能具有重要意义 例如合金的脆性和超导性， 陶瓷。

The principal investigator will analyze the qualitative behavior of solutions to several partial differential equations arising in the theory of phase transitions. Of particular interest are the Cahn-Hilliard, Hele-Shaw, and Phase field equations. The structure of the set of equilibria for each model will also be discussed. Techniques involve the theories of singular perturbations, invariant manifolds, and spectral analysis for differential operators as well as classical estimates for solutions of partial differential equations. This project is concerned with several mathematical problems associated with phase transitions in solids. In this context different phases of a nonhomogeneous material are distinguished by different concentrations of their components. The mathematical equations involved in this project also arise in many other areas of science and geometry and therefore it is expected that results here will have far-reaching effects. Among the phenomena addressed by the principal investigators are the spontaneous creation of a fine-grained structure, nucleation whereby widely separated particles undergo rapid growth, and coarsening of the micro structure of the material. All of these phenomena have significance with respect to material properties such as the brittleness and superconductivity of an alloy or ceramic.