Mathematical Sciences: Non-convex Energies and Dynamical Metastability

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9501060 Grant The principal investigator analyzes mathematical models involving nonconvex energy functionals that incorporate both bulk and gradient effects, in an effort to better understand the nature of dynamical metastability, which typically appears in the form of slow migration of transition structures, such as steep fronts or vortices. Energy methods of the type developed by Bronsard and Kohn have proven successful in establishing the existence of dynamically metastable solutions to the Allen-Cahn equation, the Cahn-Hilliard equation, Cahn-Morral systems, and equations for modeling phase transitions on lattices. The investigator will apply similar techniques to equations in higher-dimensional domains, with multiple state variables, with multiple (or a continuum of) low-energy phases, with gradient energies involving convolutions, or with complex dynamical behavior. %%% In several mathematical models of physical systems (in particular, models from materials science) certain structures have been discovered that eventually undergo significant changes but take an extraordinarily long time to do so. This phenomenon, sometimes called dynamical metastability, is important to identify and understand because macroscopic properties of a material (e.g., brittleness) may depend sensitively on the underlying structure. This project will contribute to this understanding and, therefore, to the development of new materials and the improved use of existing ones. The models exhibiting dynamical metastability are general enough that the results of this project should have implications in other contexts, as well. ***