Mathematical Sciences: Studies in Continuum Mechanics and Partial Differential Equations

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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.