Free Boundary Problems and Interfacial Dynamics

University Of Pittsburgh, Pittsburgh PA

Investigators

Abstract

Interfacial phenomena are common in nature. They occur whenever there is a substance that can exist in at least two different phases and there is some mechanism that generates a spatial separation of these phases. For most substances there are at least three phases (vapor, liquid, and ice for water, for example). The interfaces that separate different phases may evolve with time. A typical two-phase transition is the 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. To date, a significant amount of knowledge has been accumulated on two-phase transitions. For multi-phase transitions, however, rigorous mathematical studies are far from complete. 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 polycrystal. The main tools used here will be theories of partial differential equations, singular perturbations, asymptotic expansions, complex analysis, bifurcation, center manifolds, global analysis, dynamical systems, and geometric measure. Computer assistance will be needed to first simulate, then validate, and further extend theoretical conclusions. The project includes training of undergraduate and graduate students through their participation in research.