Variational studies of nonlocal models with fine properties

Models for material damage or fracture, cellular biological processes, and social network dynamics all involve interactions over long distances. Such nonlocal mathematical models have seen major use by scientific communities in recent years. This project aims to develop rigorous mathematical foundations for this class of models, thereby increasing their reliability and expanding their applicability. In each of the application areas, the modeled system contains quantities operating at different scales, yet each quantity has a significant effect on the whole system. These different scales are challenging to analyze mathematically. As a result, the project outcomes are expected to inform the implementation of models in said applications, including solid mechanics, damage prediction, and network science (especially at large scales). The project provides training for graduate students and postdoctoral researchers, supports existing outreach programs that promote scientific literacy, and provides opportunities for collaborations with researchers from other universities and national laboratories. Nonlocal models have emerged as powerful tools for describing phenomena that classical partial differential equations cannot easily capture, such as discontinuities, long-range interactions, and fine-scale structures. The project addresses fundamental open problems for nonlocal models in three application areas: (1) Dynamics on graphons, (2) Nonlocal continuum mechanics, and (3) brittle fracture. The work combines variational analysis, (non)linear functional analysis, and asymptotic methods to establish qualitative properties and convergence results, enabling rigorous justification and accurate modeling of fine-scale behaviors across these applications. The project results will also develop new analytical tools for use in function space theory and operator theory. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.