This project involves the study of mathematics underlying systems of equations that model fluid flow. Applications of such equations include the motion of plasma in blood flow, oil moving through pipelines, and the formation of gaseous planets such as Jupiter and Saturn. The aim of the research is to develop methods that can be used to establish the global existence of solutions to systems of nonlinear partial differential equations that assert the conservation of mass and momentum in a variety of physical phenomena. In addition to involving students in the research, the investigator will pilot a "bridge to PhD" program intended to increase the number of women and minorities earning PhD degrees in mathematics at the University of Pennsylvania. The investigator will also work with educators at Philadelphia's Workshop School to introduce mathematical modeling into their curriculum and to train teams of students to compete in the annual Moody's Mega Math Challenge contest for high school juniors and seniors organized by the Society for Industrial and Applied Mathematics.
The focus of the proposed research is based on the generalized Euler-Poisson equations, a system of partial differential equations used to study the behavior of materials modeled as continuous masses. These equations arise as critical points of an action in the space of probability measures; the project will investigate sufficient and necessary conditions for minimizers of this action to exist. Moreover, the investigator aims is to lay a foundation for a theory of Lagrangian mechanics, calculus of variations, and control theory in the space of probability measures. To this end, he will employ methods from fluid mechanics, optimal transport, functional inequalities, and evolution equations. In collaboration with the University of Pennsylvania's Center for Teaching and Learning, the investigator will organize an interactive seminar based on these topics as a way recruit and train undergraduates to do research. The educational and outreach efforts proposed in this project will enable the investigator to motivate students at all levels from diverse backgrounds to use mathematics in tackling some of our most pressing scientific challenges.
