The principal investigator will analyze the qualitative behavior of solutions to several partial differential equations arising in the theory of phase transitions. Of particular interest are the Cahn-Hilliard, Hele-Shaw, and Phase field equations. The structure of the set of equilibria for each model will also be discussed. Techniques involve the theories of singular perturbations, invariant manifolds, and spectral analysis for differential operators as well as classical estimates for solutions of partial differential equations. This project is concerned with several mathematical problems associated with phase transitions in solids. In this context different phases of a nonhomogeneous material are distinguished by different concentrations of their components. The mathematical equations involved in this project also arise in many other areas of science and geometry and therefore it is expected that results here will have far-reaching effects. Among the phenomena addressed by the principal investigators are the spontaneous creation of a fine-grained structure, nucleation whereby widely separated particles undergo rapid growth, and coarsening of the micro structure of the material. All of these phenomena have significance with respect to material properties such as the brittleness and superconductivity of an alloy or ceramic.
