Free Boundary Problems and Other Partial Differential Equations

University Of Maryland, College Park, College Park MD

Investigators

Abstract

One aspect of this research project concerns the mathematical analysis of so-called self-organizing phenomena, that is, the process by which a large group of agents (which could be fish, birds, molecules, etc.) behaves as a coherent structure without the direction of a leader but through only local interactions of individual agents with their nearest neighbors. The goal of the project is to investigate how the one-to-one interactions between the agents determine the global behavior of the group. A second aspect of this project concerns the mathematical study of the motion of small droplets of liquid on a solid support. Various models have been developed to describe this phenomenon; this project investigates the detailed mathematical properties of several of these models, as a step toward better understanding which models can be used to accurately describe specific behaviors of the droplets. Of particular interest are the geometric properties of the contact line (the interface where air, liquid and solid meet). Finally, this project also investigates mathematical problems related to the modeling of heat propagation through certain materials (mostly insulating crystals). One of the challenges is to better understand how the microscopic laws, which describe the motion of the atoms of the crystals, yield the classical macroscopic models for heat diffusion (such as the classical Fourier law). Most of the mathematical problems under study in this project are free boundary problems (partial differential equations to be solved on domains that are not known a priori, but must be found as part of the solution). This is the case in particular of the moving drop problem, where the free boundary is the contact line mentioned above. The project investigates how various models (corresponding to different choices of free boundary conditions) affect the regularity of the contact line. The project also studies the relationship of self-organizing phenomena under simple attraction/repulsion mechanisms to fine regularity properties for some obstacle problems of order higher than two (such as the biharmonic obstacle problem). While the harmonic obstacle problem is a classical free boundary problem, there are still many unanswered questions concerning its higher order counterpart. Many classical tools (such as the maximum principle) seem no longer relevant, while other tools, in particular tools from geometric measure theory, should play an important role. The project also investigates related evolution problems that model the phenomenon of phase separation (leading to patches with positive concentration separated by empty regions). The equations under study are reminiscent of Cahn-Hilliard equations, but involve degenerate and non-local diffusion phenomena. Finally, the project explores the rigorous derivation of non-local integro-differential equations from kinetic models for heat propagation (and other transport phenomena). One such kinetic model is the Boltzmann phonon equation. The study of this equation is still in its early stage compared to the considerable amount of work devoted to the Boltzmann equation for dilute gases, and the project will contribute to the development of this theory.