COLLABORATIVE: Mathematical Sciences: Dynamics of Interfaces and Phase Transitions

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9622785 Bates The main goal of our research program is the development and analysis of mathematical models of continua which admit phase transitions or structural defects. The mathematical problems address the general questions: How do nonlinear systems relax to equilibrium? How do interfaces and defects form and how do they propagate? These questions arise both for diffuse and sharp interface models. We also encounter interesting questions in phase interface dynamics which are closely tied to interesting questions in geometry. We intend to obtain information on the qualitative behavior of phase states predicted by several new models as well as classical modelsfor phase transition. We will consider isothermal solid-solid transitions which occur without a change in the total amount of each species and we will also study liquid-solid transformations where a heat equation is coupled to that for the order parameter. There are natural situations where the perimeter or the geometry of the interface, does not change significantly during the evolution. Then a reduction to a finite dimensional dynamical system is often possible. A good deal of our effort in these investigations is spent towards identifying and understanding the underlying finite dimensional dynamics. %%% We investigate certain physical phenomena where surface tension plays a role. Surface tension is responsible for the shape of planets, as well as for the shape of rain drops. It is also the reason why skating on ice is possible. Generally it is a second order effect. However when the other forces balance each other, surface tension can become the determining factor. Here is anexample: In the first stages of the space program scientists were puzzled by the fact that as soon as the rocket was exiting the earth's gravitiational attraction the engines would shut off as if they were running out of fuel. Eventually they realized that in space the fuel in the tank forms into sphe rical shapes due to surface tension effects with many blobs staying away from the walls of the container. We are interested in phase change phenomena , for example those involving ice and water (different phases of the same substance which can coexist near the critical temperature of 32F). Understanding the way certain phases mix is important for determinig the properties of materials. Superconductivity is a good example to which our work applies. For understanding all this, one, in principle, could start with the basic equations of physics. The problem with this approach is the enormous complexity that one encounters and the difficulty in the numerical simulation. For dealing with this scientists are proposing adhoc, phenomenologically convincing models, which have the advantage of great simplicity. Our work is of qualitative nature. It aims to establish that some of these simple mathematical models exhibit the wealth of behavior making them capable of describing and predicting the phenomena that are observed . ***