CAREER: Transport Equations in Fluids and Biology: Singularity, Dynamics, and Mixing

这个项目致力于对自然界中普遍存在的传输现象进行数学研究。它们涉及某种物质（例如污染物或颗粒材料）通过流动（流体或气体流动或细胞内流动）的运输。 在数学上，某个标量场或矢量场（例如密度、涡度、温度）由某个速度场携带。 这些现象可以用偏微分方程（PDE）来描述，偏微分方程通常涉及非局部（依赖于流动的历史或不一定接近的事件）和非线性项。例子包括模拟大规模大气和海洋流动的二维Boussinesq方程，以及模拟集体动物行为的扩散-聚集方程。由于这些方程的非局部和非线性性质，通常不知道解是否在时间上全局存在或发展为有限时间奇点。即使在已知解是全局的情况下，对于许多方程，它们的长期行为仍然不清楚。该项目旨在为流体动力学和生物学中产生的一系列输运方程开发新的分析工具，重点是解决方案的奇异性，渐近性和混合性。该项目的一个组成部分是教育部分，包括开发高级课程，监督本科生研究，并进行非线性偏微分方程的年轻研究人员的研讨会。研讨会的特色是由知名研究人员举办的小型课程和初级参与者的简短演讲，旨在向年轻研究人员介绍PDE研究的最前沿并促进合作。这个项目将推进非局部偏微分方程的数学理解及其在流体和生物学中的应用。 该项目还将为这一充满活力的领域的初级研究人员提供教育和培训机会。第一个方向是获得某些流体方程的有限时间奇异性。该计划是从一些一维模型方程开始，通过建立某种全局控制到爆破时间来证明有限时间奇异性。对这些模型方程的研究可能为更高维流体方程的全动力学研究提供新的思路。第二个方向是开发新的工具，用于理解聚集扩散方程的长期动力学，其中梯度流结构起着重要作用。第三个方向是不可压缩流的混合，目标是研究在给定速度场的定量约束下，密度混合的速度有多快。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project is devoted to a mathematical study of transport phenomena that are ubiquitous in nature. They relate to a transport of some substance, for example a pollutant or particulate material, by a flow, be it a fluid or gas flow or an intra-cellular flow. Mathematically, a certain scalar or vector field (e.g. density, vorticity, temperature) is carried by some velocity field. These phenomena can be described by partial differential equations (PDE), which often involve both nonlocal (dependence on the history of the flow or on the not necessarily close-by events) and nonlinear terms. Examples include the 2D Boussinesq equation that models large scale atmospheric and oceanic flows, and the diffusion-aggregation equation that models collective animal behavior. Due to the nonlocal and nonlinear nature of these equations, it is often unknown whether solutions exist globally in time or develop a finite-time singularity. Even in the cases where solutions are known to be global, their long-time behavior remains unclear for many equations. This project aims to develop novel analytical tools for a range of transport equations arising in fluid dynamics and biology, focusing on singularity, asymptotic, and mixing properties of the solutions. An integral part of the project is the educational component including developing advanced courses, supervising undergraduate research, and conducting a young researchers' workshop on nonlinear PDE. The workshop features mini-courses by established researchers and short talks by junior participants, aiming to introduce young researchers to the forefront of PDE research and facilitate collaborations. This project will advance the mathematical understanding of nonlocal PDE and their applications in fluids and biology. The project will also provide opportunities for education and training of junior researchers in this vibrant field.This project contains three different but related directions. The first direction is to obtain finite-time singularity formation for some fluid equations. The plan is to start with some one-dimensional model equations and prove finite-time singularity by establishing some kind of global control up to the blow-up time. The study of these model equations may shed new light on the full dynamics of fluid equations in higher dimensions. A second direction is the development of new tools for understanding the long-time dynamics of aggregation-diffusion equations, where the gradient flow structure plays an important role. A third direction is mixing by incompressible flows, and the goal is to study how fast the density can get mixed given some quantitative constraint of the velocity field.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.