Mathematical Sciences: Some Mathematical Problems Associated with Phase Transitions

首席研究员将分析相变理论中出现的几个偏微分方程解的定性行为。特别令人感兴趣的是卡恩-希利亚德方程、海尔-肖方程和相场方程。每个模型的均衡集的结构也将被讨论。技术包括奇异摄动理论、不变流形、微分算子的谱分析以及偏微分方程解的经典估计。这个项目涉及几个与固体相变有关的数学问题。在这种情况下，非均质材料的不同相通过其组分的不同浓度来区分。本项目涉及的数学方程也出现在许多其他科学和几何领域，因此，预计这里的结果将产生深远的影响。主要研究人员处理的现象包括自发产生细粒结构，广泛分离的颗粒迅速生长的成核，以及材料微观结构的粗化。所有这些现象对于合金或陶瓷的脆性和超导性等材料特性都具有重要意义。

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.