Many phenomena in physics and biology involve nonlocal effects, where distant parts are allowed to influence each other. These phenomena are usually modeled by partial differential equations (PDE) that incorporate nonlocal and nonlinear terms. The purpose of this project is to study qualitative properties of solutions to a variety of nonlocal equations, such as the Keller-Segel equation that models the collective motion of cells which are attracted by a self-emitted chemical substance; the surface quasi-geostrophic equation arising in the study of atmospheric turbulence and oceanic flows; and the aggregation equation, which describes the movement of large swarms of insects. The goal is to develop new mathematical tools to analyze basic properties of solutions in order to gain a better understanding of the physical and biological models.
Mathematically, the following questions will be investigated in this project: For fluid models such as a 1D analog of Boussinesq equation, does the solutions to the PDE have global well-posedness, or can the solution form a singularity in finite time? For the aggregation equation with attractive kernels, where formation of singularities always happens in finite time, the goal is to find some notion of solutions that can be extended past the blow-up time. We will also study a nonlocal system motivated by coral broadcast spawning, where two densities evolve under diffusion, reaction and chemotaxis, and rigorously analyze the effect of chemotaxis on the reaction rate. The nonlocal and nonlinear nature of these equations poses many challenges in the analysis. To overcome these difficulties, we will apply and develop a variety of analysis tools including energy methods, comparison principle, optimal transport theory and probabilistic methods.
