Interfacial Phenomena and Pattern Formation

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Interfacial phenomena occur whenever a continuum is present that can exist in at least two different phases (e.g. ice and water) and there is some mechanism that generates or enforces a spatial separation between these two phases (the freezing of the water). The separation boundaries are called interfaces or free boundaries. The objective of the project is to study the time evolution of the boundaries from the following points of view: (i) generation---how interfaces emerge from an initially almost uniform non-stable equilibrium, (ii) propagation---how interfaces move, (iii) nucleation--how phases switch from one to the other, and (iv) pattern formation--how mathematical models can be used to accommodate and predict commonly observed phase patterns such as rotating spirals, oscillating spots, expanding rings, etc. The project will for the most part be carried out on systems of reaction--diffusion equations such as activator-inhibitor models in biology. The main tools used are the theories of partial differential equations and geometric measure. Experimentation through computer simulation will be an integral part of this project in order to stimulate, predict, and validate theoretical studies.Interfacial phenomena are commonplace in physics, chemistry, material science, biology, and even in mathematical finance. In all these areas it is important to predict, and eventually to control, the interfacial dynamics. The research performed with this award will study mathematical model that govern these seemingly complicated yet universal natural phenomena. The work will also lead to the development of new and refined mathematical theories and tools that can handle phenomena of instability and ambiguity. Such phenomena are often observed inthese applications.