Mathematical Sciences: Thermistor Problems and Interfacial Dynamics

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This project contains two research areas: (1) thermistor modeling, which involves a nonlinear system of two degenerate elliptic partial differential equations, and (2) interfacial dynamics which involves the motion of hypersurfaces, the motion of level sets of solutions of parabolic partial differential equations, and the relations between these two motions. For the first problem, the investigator plans to use the classical analysis of elliptic partial differential equations, variational inequalities, free boundary analysis, and some complex analysis to establish the existence, uniqueness, and regularity of the solutions and to provide numerical schemes for the calculation of the solutions and the fundamental relations between the voltage (or current) applied to the thermistor and its total resistivity, which is essential for designing and testing thermistors. For the interfacial dynamics problem, the investigator will use singular perturbation methods, viscosity methods, functional analysis, etc., to establish some physically based criteria for the motion of the interfaces, and to provide some mathematical justification on the uniformity between the modern models and the classical models in solidification theory and related areas. A thermistor is a heat sensitive electrical resistor which has been widely used in current regulation, switching, gas detection, control and alarms, etc. The investigator will use mathematical tools to find relations among the characteristics of thermistors, thereby providing guidelines for practical design and testing. Interface, in the solidification case, is the boundary between solid and liquid. Motion of the interface is essential to the solidification theory. Many good models concerning the motion of interfaces have been proposed. The investigator plans to justify rigorously these relations, especially the common aspects of various models, and find physically based criteria for the motion of the interfaces where the current models give multiple choices.