Nonlinear and Nonlocal Partial Differential Equations

该项目涉及非线性和非局部偏微分方程及其应用。非局部方程被用来描述各种各样的物理和生物现象。非定域性意味着一个位置的小扰动可以影响整个系统。该项目将增强数学家可用的工具箱，并扩大可以严格研究的模型类别。有三个相互关联的主题。首先是肿瘤生长模型。目的是建立在细胞水平上描述肿瘤的模型和那些将肿瘤描述为具有控制其边界运动规律的区域的模型之间的联系。第二个主题是进化生态学。对有关的偏微分方程的严格分析将用于研究种群的动态，其中个体可以在代之间迁移以及发生突变。第三个主题涉及某些偏微分方程的数值方法。所涉及的数学思想也将用于开发算法，以使许多自主机器人执行合作任务（例如机器人授粉蜜蜂或自动监视）。组成该项目的三个主题有可能影响社会更广泛感兴趣的几个领域——即医学、生态和技术发展。该项目将通过增加我们对数学的理解和加强数学与其他学科之间的联系来促进科学进步。此外，首席研究员将教授和指导学生，并向更广泛的社区进行外展。该项目将阐明多种类型的pde。退化扩散方程和自由边界问题是肿瘤生长模型研究的基础。目的是研究这些方程并在它们之间建立严密的联系。进化生态学工作的目标是理解非局部反应扩散方程中的传播现象。一个重要的工具是这些偏微分方程和汉密尔顿-雅可比方程之间的联系。第三个主题涉及二阶偏微分方程的新数值方法，包括理解偏微分方程的结构及其正则化。目的是证明数值方法的收敛性，以及使用这些工具来开发机器人控制算法。连接这三个主题的一个重要主题是非定域性。非局部偏微分方程通常缺乏比较原理，而比较原理是研究经典偏微分方程的关键工具。此外，还推测了一些待研究的方程相对于初始条件是不稳定的。能够克服这一点，甚至研究不稳定的原因和影响，将是一个重大的发展，并可能导致在其他问题上取得进展。此外，这项工作将涉及理解和发展弱解的新概念。在许多现实世界的系统中，期望退化或不可微是很自然的；因此，要使pde在这些上下文中发挥作用，就需要一个新颖的、足够健壮的解决方案概念。该奖项反映了美国国家科学基金会的法定使命，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。

The project concerns nonlinear and nonlocal partial differential equations (PDEs) and their applications. Nonlocal equations are used to describe a wide variety of physical and biological phenomena. Nonlocality means that a small perturbation in one location can affect the entire system. The project will enhance the tool-box available to mathematicians and widen the class of models that can be studied rigorously. There are three interconnected topics. The first is tumor growth models. The aim is to establish connections between models that describe the tumor at a cellular level and those that characterize the tumor as a region with a law governing the movement of its boundary. The second topic is evolutionary ecology. Rigorous analysis of the relevant PDEs will be used to study the dynamics of populations in which individuals can migrate as well as undergo mutation between generations. The third topic concerns numerical methods for certain PDEs. The mathematical ideas involved will also be used to develop algorithms for getting many autonomous robots to perform a cooperative task (examples include robotic pollinating bees or automated surveillance). The three topics making up the project have the potential to impact several areas of broader interest to society - namely, medicine, ecology, and technological development. The project will promote scientific progress by increasing our understanding of mathematics and by strengthening the connections between it and other disciplines. In addition, the principal investigator will teach and mentor students, as well as conduct outreach to the broader community.The project will shed light on multiple classes of PDEs. Degenerate diffusion equations and free boundary problems underlie the work on tumor growth models. The aim is to study these equations and establish rigorous connections between them. The goal of the work on evolutionary ecology is to understand propagation phenomena in nonlocal reaction-diffusion equations. An important tool is the link between these PDEs and Hamilton-Jacobi equations. The third topic concerns novel numerical methods for second order PDEs and involves understanding the structure of the PDEs and their regularizations. The aim is to prove convergence of the numerical methods, as well as to use these tools to develop algorithms in robotic control. An important theme connecting the three topics is nonlocality. Nonlocal PDEs often lack a comparison principle, which is a key tool in the study of classical PDEs. Moreover, some equations to be studied are conjectured to be unstable with respect to initial condition. Being able to overcome this, and even studying the causes and effects of instability, will be a significant development, and may lead to progress on other problems. In addition, this work will involve understanding and developing new notions of weak solution. In many real-world systems, it is natural to expect degeneracy or non-differentiability to form; so, for PDEs to be useful in these contexts, a novel sufficiently robust notion of solution is needed.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.