The objective of this project is to provide a systematic study of multi-phase transitions. There are macroscopic free boundary models, which are based on observable quantities and are needed to deal with singularities due to topological changes of free boundaries, and microscopic continuum models, derived at the molecular level and (always) well-posed. One goal of the project is to derive critical information from continuum models for free boundary models when the latter become indeterminate. Another is to develop new microscopic continuum models for phase transitions among phases such as solid/liquid and different grains in a polycrystalline material. The main tools used here are theories of partial differential equations, singular perturbations, asymptotic expansions, complex analysis, bifurcation, center manifolds, global analysis, dynamical systems, and geometric measure. From time to time, computer assistance is used to first stimulate, then validate, and further extend theoretical conclusions.
The investigator studies interfacial phenomena, which are commonplace in nature and occur whenever there is a continuum that can exist in at least two different phases and there is some mechanism that generates a spatial separation of these phases. The spatial boundaries that separate these phases, referred to as interfaces or free boundaries, evolve with time due to some enforcement of transitions of phases from one to another. A typical two-phase transition is a solidification process of a liquid. A classical multi-phase transition arises from evolutions of grains in a polycrystal where alignments of atoms in different grains do not match at grain boundaries. The growth of a solid polycrystal from a liquid involves a two-phase transition between liquid and solid and a multi-phase transition among grains. These phenomena motivate new theoretical development in free boundary problems, systems of parabolic or elliptic partial or ordinary differential equations, and dynamical systems. Insights gained from the project increase connections between mathematicians and materials scientists. New models developed here provide scientists with tools for understanding and designing new materials.
