Singularity and Asymptotics for Nonlocal Partial Differential Equations

从流体动力学到动物群，在广泛的自然现象中都可以观察到远距离相互作用。数学上，这些现象可以用非局部项和非线性项的偏微分方程（PDE）来建模。本项目致力于流体力学和生物学中出现的各种非局部方程的研究，如大气锋和海洋环流研究中出现的二维Boussinesq方程；聚集方程，模拟动物群体的集体行为；一个由珊瑚广播产卵驱动的两种系统。本课题研究的许多方程解的基本数学问题尚未得到解答。尽管如此，这些方程及其数值解被广泛用于理解和预测自然现象。本研究旨在增强这些方程对自然现象建模有效性的信心，并为应用提供坚实的理论基础。本课题的目标是严格研究非局部方程的解在有限时间内是否能形成奇点，以及如果解在时间上全局存在，其长期动力学是什么。本项目包含了关于非局部偏微分方程的奇异性和渐近性的三个不同但相关的研究方向。第一个方向涉及流体方程中有限时间奇点的形成。对于二维无粘Boussinesq方程的一些变化，目标是使用“双曲流场景”来构造在有限时间内在边界点爆炸的解。第二个子项目致力于具有梯度流结构的非局部偏微分方程，其中粒子/个体倾向于在短距离内相互排斥而在远距离上相互吸引。问题是，从长远来看，解决方案是收敛于某个能量泛函的全局最小值，还是耗散到零。第三个方向涉及两种系统，其中两种密度在扩散，反应和趋化性下演化，目标是研究趋化性项如何影响该系统的长时间动力学。在每个方向上，研究者计划应用和开发各种分析工具，如能量法、梯度流理论和比较原理方法来分析解的动力学特征。该项目将促进对非局部偏微分方程及其相关应用的数学理解，并将为这一活跃领域的学生提供教育和培训。

Long-range interactions can be observed in a broad class of natural phenomena, ranging from fluid dynamics to animal swarms. Mathematically, these phenomena can be modeled by partial differential equations (PDE) with both nonlocal and nonlinear terms. This project is devoted to the study of a variety of nonlocal equations arising in fluid mechanics and biology, such as the two-dimensional Boussinesq equation arising in the study of atmospheric fronts and oceanic circulation; the aggregation equation, which models the collective behavior of animal groups; and a two-species system motivated by coral broadcast spawning. Many fundamental mathematical questions about solutions of the equations under the investigation in this project are unanswered. Nonetheless, the equations and their numerical solutions are widely used for understanding and prediction of natural phenomena. This research aims to enhance confidence in the validity of these equation for modeling natural phenomena and to provide a firm theoretical foundation for applications. The goal of this project is to rigorously study whether solutions to nonlocal equations can form a singularity in finite time, and what is the long-time dynamics if the solutions do exist globally in time. This project contains three different but related directions of research on the singularity and asymptotics for nonlocal PDEs. The first direction concerns finite-time singularity formation in fluid equations. For some variations of the two-dimensional inviscid Boussinesq equation, the goal is to use the "hyperbolic flow scenario" to construct a solution that blows up in finite time at a boundary point. The second subproject is devoted to a nonlocal PDE with a gradient flow structure, where particles/individuals tend to repulse each other in short distance and attract each other in long distance. The question is whether in the long run solutions converge to a global minimizer of a certain energy functional, or dissipate to zero. The third direction deals with a two-species system, where two densities evolve under diffusion, reaction and chemotaxis, and the goal is to investigate how the chemotaxis term would affect the long time dynamics of this system. In each of these directions, the investigator plans to apply and develop various analysis tools such as energy methods, gradient flow theory, and comparison principle methods to analyze the dynamical features of the solutions. This project will advance the mathematical understanding of nonlocal PDEs and related applications, and will also provide education and training to students in this active field.