Mathematical Sciences: Investigations of Phase Transitions Using Relaxation and Nonlocal Regularization

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9403844 Rogers This project will support an ongoing study of a family of problems in phase transitions. The problems come from three areas of application: mechanics (liquid-vapor, solid-liquid, and multiphase-solid transitions), superconductivity, and magnetism (rigid ferromagnets as well as magnetostrictive materials). The research primarily involves studying static problems of energy minimization and trying to characterize the set of metastable states (local minimizers) of the energy. The goal is to describe hysteresis (which occurs in all of these problems) in terms of these metastable states. The primary technique is to model the physical problems being studied using relaxed, nonlocal energy functionals. Stationary points and/or relative minimizers are found using techniques of integral equations, convex analysis, and methods from the calculus of variations. Numerical algorithms are developed to approximate solutions of the Euler-Lagrange equations. The most useful properties of many high-tech materials (e.g. shape memory alloys, magnetic and magnetostrictive materials, and superconductors) are caused by the material undergoing a change of "phase." Often, these phase changes do not occur smoothly, but instead, exhibit wild oscillations at a microscopic level. In the last century, there has been a great deal of empirical study of these phenomena by materials scientists. But it was only in the past decade that mathematicians began to turn these investigations into a more quantitative science. The current project aims to make quantitative predictions of the behavior of large-scale material structures based on fundamental models of the internal energy. The project combines elements of mathematical modeling, mathematical analysis, and the development and analysis of new numerical algorithms. The proposed applications are to liquid-vapor transitions, crystal growth, shape memory alloys, magnetic and magnetostrictive materials, and superc onductors.