Mathematical Sciences: Some Mathematical Problems Associatedwith Phase Transitions

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The investigator studies various models of phase transitions in either a pure substance or a binary alloy. The fundamental models are singularly perturbed partial differential equations whose solutions are evolving continuous fields that represent either an order parameter, measuring degree of solidification, or local concentration of one of the species in a binary alloy. The three basic models are known as the Cahn-Allen, the Cahn-Hilliard, and the Phase Field equations. Taking certain distinguished limits as the singular parameter approaches zero leads to equations for the motion of sharp interfaces, which are also studied. Materials science has grown more mathematical even as it has become a more critical technology. This work can contribute to a better understanding of behavior of materials and the development of new materials. Similar mathematical problems involving phase transitions arise in other areas of physics and engineering also.