Nonlinear Partial Differential Equations

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The principal investigator will study mathematical models for superconductivity and liquid crystals. These materials are described by models posed at a variety of length scales.The intent of the research is to investigate the qualitative features of the solutions to these models, analyze the thresholds at which the models predict change in the materials' basic characteristics (changes of phase) and investigate the connections between the solutions of different models at different length scales. The methods to be employed are analysis of partial differential equations, asymptotic methods, homogenization, and numerical simulation.Superconductors are used in electronic sensing devices, magnets and power transmission.Applications for liquid crystals are visual displays and optical screens.These are examples of materials for which their characteristic features (their phases) can change dramatically if subjected to electro-magnetic fields or variations in temperature.This phenomenon is a great asset (essentially as if one has two materials in the same location) but it is also a difficult feature to control.The goal of the project is to add to the understanding of the behavior of the models and their interrelations so that these materials can be used more effectively. This will be done by means of mathematical analysis of these models in general settings.