LEAPS-MPS: Diffusive Partial Differential Equations in the Physical Sciences

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This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2).Diffusion, the process whereby particles, individuals, or energy spread from areas of higher concentration to areas of lower concentration, is a central feature of many important physical systems. Understanding the dynamics of these systems leads to a wide variety of mathematical challenges. This project applies the theory of partial differential equations (PDE) to study two classes of diffusive models. The first, kinetic models of gases and plasmas, feature diffusion generated by collisions between particles. These models are of both theoretical and practical importance, as plasmas are used in many modern technological devices. The second class, elliptic free boundary problems, arise in the study of phase transitions, insulation, and many other physical and industrial processes. These models are based on the coupling between an undetermined geometric region and a diffusive equation defined in that region. The mathematical goals of this project are combined with an educational component that includes support for graduate student researchers, as well as a one-day workshop organized around research questions in PDE introduced at the advanced-undergraduate level. The goal of these measures is to increase opportunities for the next generation of researchers in this subject area, including those from groups traditionally underrepresented in STEM. This project seeks to advance the well-posedness and regularity theory of two classical kinetic equations, the Boltzmann and Landau equations, which feature the interaction of nonlinear, nonlocal diffusion in the velocity variable, with transport in the time and space variables. Specific goals include constructing short-time solutions for general initial data by leveraging the smoothing properties of the diffusion term, as well as weakening the conditions required for solutions to regularize and be continued past a given time. In free boundary theory, this project includes the qualitative study of minimizers for functionals of Alt-Caffarelli type with possibly irregular diffusion coefficients and unbounded coupling terms. The goal is to extend well-known free boundary regularity results to minimizers of these more general functionals. This project will also use free boundary techniques to study shape optimization problems related to Dirichlet eigenvalues of general elliptic operators, including problems for which existence of optimal domains is currently unknown.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.