Mathematical and computational problems in energy driven pattern formation

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The configuration of any complex physical systems can be described by an associated energy. The dynamics then follow the gradient flow of this energy as the system dissipates energy. The first natural step in studying these systems is to analyze the critical points of the energy. For the systems of interest in this proposal these critical states and their energies depend on multiple parameters. There is generally some noise in the original physical systems being modeled thus systems generically evolve to global minimizers of the energy.? Most energy based models in the applied sciences involve parameters which often cannot be determined with absolute certainty. It is therefore important to study such models for whole parameter ranges, and in particular, to detect parameter values at which the system behaviour changes qualitatively. A first step towards accomplishing this goal is the understanding of the set of equilibria of the model and their respective energies. The long-term goal of my research program is the development of computational tools which can simulate complex mathematical models with accurate computed error bounds where every computation is an existence proof. Towards this goal I will work with HQP over the next five years on developing methods to address questions related to the Ohta-Kawasaki energy and related models from material science. The project will tie together various research objectives over three interconnected themes: Rigorous computation of phase diagrams; Extensions of the Ohta-Kawasaki energy and Confinement and geometric effects.