Mathematical Sciences: Studies in Continuum Mechanics and Partial Differential Equations

首席研究员将继续他在连续介质力学各个领域中出现的偏微分方程的重要工作。特别是，他将运用先进的数学工具来研究多相热力学的一般理论，因为它涉及到界面动力学和涉及融化和冻结的现象。他还将研究在多孔介质流动和生物扩散中发现的一些不寻常的偏微分方程。首席研究员的工作涉及数学模型的发展和研究，这些模型试图描述复杂的物理现象，如带面和角的晶体的生长，融冻波的演变，以及更一般地说，通过在介质中移动的弯曲锋的作用在介质中产生的有序结构。举个例子，试想一下，如果你用一根热电线穿过一块冰，会发生什么。这是一个涉及融化和再冻结的复杂过程，由于融化和未融化的冰之间的边界对电线的温度和其他因素非常敏感，因此很难研究。

The principal investigator will continue his important work on the partial differential equations that arise in various areas of continuum mechanics. In particular, he will apply advanced mathematical tools to study the general theory of multiphase thermomechanics as it relates to interfacial dynamics and phenomena involving melting and freezing. He will also investigate some unusual partial differential equations that are found in flows through porous media and in biological dispersal. The work of the principal investigator concerns the development and study of mathematical models that attempt to describe such complicated physical phenomena as the growth of crystals with facets and corners, the evolution of a melting-freezing wave, and more generally the creation of ordered structures in media through the action of curved fronts that move through the media. As an example, think of trying to describe quantitatively what happens if you draw a hot wire through a block of ice. It is a complicated process involving melting and re-freezing, which is difficult to study because the boundary between the melted and unmelted ice is so sensitive to the temperature of the wire and other factors.