Nonlinear and Nonlocal Partial Differential Equations

该项目涉及非线性和非局部偏微分方程（PDE）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。其目的是在细胞水平描述肿瘤的模型与将肿瘤描述为具有控制其边界移动规律的区域的模型之间建立联系。第二个主题是进化生态学。对相关偏微分方程的严格分析将用于研究种群的动态，在种群中，个体可以迁移，也可以在两代之间发生突变。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，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，生态学和技术发展。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推广。该项目将阐明多类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，并建立它们之间的严格联系。进化生态学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哈密尔顿-雅可比方程之间的联系。第三个主题是关于二阶偏微分方程的新的数值方法，并涉及了解偏微分方程的结构和正则化。其目的是证明收敛的数值方法，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。非局部偏微分方程往往缺乏比较原理，这是研究经典偏微分方程的一个关键工具。此外，还证明了某些待研究的方程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，期望退化或不可微性的形成是很自然的;因此，对于在这些情况下有用的偏微分方程，需要一个新的足够强大的解决方案的概念。该奖项反映了NSF的法定使命，并已被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.