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方面的重要工作。特别是，他将应用先进的数学工具来研究多相热力学的一般理论，因为它与界面动力学和涉及熔化和冻结的现象有关。他还将研究一些不同寻常的偏微分方程式，这些偏微分方程式存在于渗流介质和生物扩散中。首席研究人员的工作涉及数学模型的开发和研究，这些模型试图描述具有小面和拐角的晶体的生长，熔融冻结波的演化，以及更广泛地说，通过穿过介质的曲线锋面的作用在介质中创建有序结构等复杂的物理现象。举个例子，试着定量地描述一下，如果你拉一根电线穿过一块冰，会发生什么。这是一个涉及融化和重新冻结的复杂过程，很难研究，因为融化和未融化冰的边界对金属丝的温度和其他因素非常敏感。

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.