Mathematical Analysis of Interfacial Dynamics

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ChenDMS-1008905 The objective of this project is to provide a systematicstudy of multi-phase transitions. There are macroscopic freeboundary models, which are based on observable quantities and areneeded to deal with singularities due to topological changes offree boundaries, and microscopic continuum models, derived at themolecular level and (always) well-posed. One goal of the projectis to derive critical information from continuum models for freeboundary models when the latter become indeterminate. Another isto develop new microscopic continuum models for phase transitionsamong phases such as solid/liquid and different grains in apolycrystalline material. The main tools used here are theoriesof partial differential equations, singular perturbations,asymptotic expansions, complex analysis, bifurcation, centermanifolds, global analysis, dynamical systems, and geometricmeasure. From time to time, computer assistance is used to firststimulate, then validate, and further extend theoreticalconclusions. The investigator studies interfacial phenomena, which arecommonplace in nature and occur whenever there is a continuumthat can exist in at least two different phases and there is somemechanism that generates a spatial separation of these phases. The spatial boundaries that separate these phases, referred to asinterfaces or free boundaries, evolve with time due to someenforcement of transitions of phases from one to another. Atypical two-phase transition is a solidification process of aliquid. A classical multi-phase transition arises fromevolutions of grains in a polycrystal where alignments of atomsin different grains do not match at grain boundaries. The growthof a solid polycrystal from a liquid involves a two-phasetransition between liquid and solid and a multi-phase transitionamong grains. These phenomena motivate new theoreticaldevelopment in free boundary problems, systems of parabolic orelliptic partial or ordinary differential equations, anddynamical systems. Insights gained from the project increaseconnections between mathematicians and materials scientists. Newmodels developed here provide scientists with tools forunderstanding and designing new materials.