Interfacial Phenomena and Pattern Formation

当一个连续体存在于至少两个不同的相（例如冰和水）中，并且存在某种机制产生或强制这两个相之间的空间分离（水的冻结）时，界面现象就会出现。分离边界称为接口或自由边界。该项目的目的是从以下角度研究边界的时间演变：(i)生成—界面如何从最初几乎均匀的非稳定平衡中出现；（ii）传播—界面如何移动；（iii）成核—相如何从一种转换到另一种；（iv）模式形成—如何使用数学模型来适应和预测常见的相模式，如旋转螺旋、振荡斑点、膨胀环等。该项目大部分将在反应-扩散方程系统上进行，如生物学中的活化剂-抑制剂模型。所使用的主要工具是偏微分方程理论和几何测度。通过计算机模拟的实验将是这个项目的一个组成部分，以刺激、预测和验证理论研究。界面现象在物理学、化学、材料科学、生物学甚至数学金融学中都很常见。在所有这些领域，预测并最终控制界面动力学是很重要的。该奖项的研究将研究控制这些看似复杂但普遍的自然现象的数学模型。这项工作还将导致新的和完善的数学理论和工具的发展，这些理论和工具可以处理不稳定和模糊的现象。在这些应用中经常观察到这种现象。

Interfacial phenomena occur whenever a continuum is present that can exist in at least two different phases (e.g. ice and water) and there is some mechanism that generates or enforces a spatial separation between these two phases (the freezing of the water). The separation boundaries are called interfaces or free boundaries. The objective of the project is to study the time evolution of the boundaries from the following points of view: (i) generation---how interfaces emerge from an initially almost uniform non-stable equilibrium, (ii) propagation---how interfaces move, (iii) nucleation--how phases switch from one to the other, and (iv) pattern formation--how mathematical models can be used to accommodate and predict commonly observed phase patterns such as rotating spirals, oscillating spots, expanding rings, etc. The project will for the most part be carried out on systems of reaction--diffusion equations such as activator-inhibitor models in biology. The main tools used are the theories of partial differential equations and geometric measure. Experimentation through computer simulation will be an integral part of this project in order to stimulate, predict, and validate theoretical studies.Interfacial phenomena are commonplace in physics, chemistry, material science, biology, and even in mathematical finance. In all these areas it is important to predict, and eventually to control, the interfacial dynamics. The research performed with this award will study mathematical model that govern these seemingly complicated yet universal natural phenomena. The work will also lead to the development of new and refined mathematical theories and tools that can handle phenomena of instability and ambiguity. Such phenomena are often observed inthese applications.