Analysis of Nonlinear Partial Differential Equations in Free Boundary Fluid Dynamics, Mathematical Biology, and Kinetic Theory

This project aims to develop new methods to advance the current level of scientific knowledge on a diverse collection of recognized questions in two different areas of the mathematical analysis of nonlinear partial differential equations (PDE) and to develop new mathematical methods to study these systems. The first part of the project concerns basic questions in the analysis of partial differential equations in Mathematical Biology.

The second part of this project concerns the study of partial differential equations from kinetic theory in the presence of the physical kinetic boundary conditions. This project will involve training in research and teaching of postdoctoral researchers, graduate students, and undergraduate students from the University of Pennsylvania and other Universities.

The project will also involve outreach to undergraduate students through the University of Pennsylvania Center for Undergraduate Research & Fellowships program. The principal investigator (PI) is fully committed to facilitating the training and education of these students through teaching courses, regular direct mentoring, and running regular research seminars.

The PI is actively working to develop new innovative mathematics courses at the University of Pennsylvania in order to further the goal of developing a diverse and globally competitive STEM workforce and to improve STEM education at the collegiate level. The PI is engaging in outreach activities to groups that are traditionally underrepresented in mathematics, and these activities will continue over the course of this project.

The PI is further consistently working to increase the scientific knowledge of the community by giving national and international research presentations. The results of this project will be further disseminated through publication in journal articles and they will be posted on the PI's website.

Problems in which an elastic structure interacts with the surrounding fluid, called Fluid-Structure Interaction (FSI) problems, are plentiful in science and engineering. These problems have many applications in Physics, Biology, and the Medical Sciences. Such FSI problems include the mathematical modeling of the flying of birds, the swimming of fish, and blood flow through the heart and blood vessels.

FSI models have been intensively studied using computational methods. Many numerical algorithms have been developed for such problems, and the scientific computing of FSI problems continues to be a very active area of research. Despite their importance, these computational methods are poorly understood from an analytical standpoint.

A major impediment for numerical analysis has been the lack of analytical understanding of the underlying nonlinear partial differential equations. A better theoretical understanding of the analytical aspects of these PDE should lead to improved computational algorithms for FSI problems. The PI seeks to address these issues by focusing on a set of canonical analytical FSI problems in the Stokes flow.

The second part of this project is in kinetic theory. The Landau equation with Coulomb potential and the non-cutoff Boltzmann equation for the long-range interaction potentials are two fundamental mathematical models in collisional kinetic theory which describe the dynamics of a non-equilibrium rarefied gas and a dilute hot plasma. Plasmas appear in fundamental physical problems from Astrophysics, Nuclear fusion, and Tokamaks.

The Boltzmann equation has been used as a mathematical model in a wide variety of places, for instance in high atmosphere aerodynamics, where the air is a very rarefied gas and fluid equations are probably not sufficient. Further, the study of boundary effects for these kinetic equations is physically very important because they describe the interaction, in the form of drag and heat transfer, between gas or plasma and a solid body.

The objective of this research in both parts of the project is to fully understand the local-in-time well-posedness for large initial data, and the global-in-time well-posedness in a close to equilibrium setting, for several different fundamental physical models in nonlinear PDE. We expect that the techniques developed as part of this project will be useful for future mathematical and physical developments.

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.