Nonlocal PDE Models in Biology and Fluids

物理学和生物学中的许多现象都涉及到非局部效应，其中允许遥远的部分相互影响。这些现象通常由包含非局部和非线性项的偏微分方程（PDE）来建模。该项目的目的是研究各种非局部方程的解的定性性质，例如模拟细胞受自发化学物质吸引而集体运动的Keller-Segel方程;在研究大气湍流和海洋流动时产生的表面准地转方程;以及描述大群昆虫运动的聚集方程。本项目的目标是开发新的数学工具来分析解的基本性质，以便更好地理解物理和生物模型。在数学上，本项目将研究以下问题：对于流体模型，如Boussinesq方程的一维模拟，PDE的解是否具有全局适定性，或者解是否可以在有限时间内形成奇点？对于具有吸引核的聚集方程，其中奇点的形成总是发生在有限时间内，目标是找到一些解的概念，可以扩展到过去的爆破时间。我们还将研究珊瑚广播产卵，两个密度下扩散，反应和趋化性发展的非本地系统的动机，并严格分析反应速率的趋化性的影响。这些方程的非局部和非线性性质在分析中提出了许多挑战。为了克服这些困难，我们将应用和开发各种分析工具，包括能量方法，比较原理，最优传输理论和概率方法。

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.