The principal investigator and his colleague study the mathematical properties of nonlocal models, particularly the peridynamic model of continuum mechanics. The project focuses on establishing the well-posedness of both linear and nonlinear models for dynamic and equilibrium problems involving nonlocal boundary (volume-constrained) value conditions; classifying emerging discontinuities and developing a regularity theory for solutions of nonlocal models as functions of given data; and studying conditions that enable nucleation and propagation of singularities in peridynamic equations and other nonlocal models. The investigators use analytical tools ranging from perturbation methods to calculus of variations. The project addresses technical challenges that are absent from the traditional local approach, leading to extensions of classical mathematical concepts and techniques to the nonlocal setting.
The project offers a mathematical framework for studying nonlocal operators and equations that may be useful to the study of many common nonlocal phenomena. It helps establish the necessary theoretical footing for nonlocal models. The project findings are integrated by the investigators into classroom teaching and other educational endeavors. The investigators' long-term goal is to work with mechanical engineers, physicists, biologists and materials scientists to make nonlocal models useful tools for studying complex systems in various scientific and engineering applications such as anomalous diffusion, nonlocal heat conduction, phase transition, and biological aggregation.
