Nonlinear and Nonlocal Partial Differential Equations

非线性和非局部偏微分方程

• 批准号: 1907221
• 负责人: Olga Turanova
• 金额: $ 8.49万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-08-15 至 2022-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1907221&HistoricalAwards=false
• 关键词: Nonlinear; Nonlocal; Partial; Differential; Equations

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.

该项目涉及非线性和非局部偏微分方程（PDE）及其应用。非局部方程用于描述各种物理和生物现象。非定域性意味着一个位置的微小扰动可能会影响整个系统。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。共有三个相互关联的主题。第一个是肿瘤生长模型。目的是在细胞水平上描述肿瘤的模型和将肿瘤表征为具有控制其边界运动的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个体在代际间迁移和突变的种群动态。第三个主题涉及某些偏微分方程的数值方法。所涉及的数学思想还将用于开发算法，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）。该项目的三个主题有可能影响社会更广泛关注的几个领域，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并加强数学与其他学科之间的联系来促进科学进步。此外，首席研究员还将教授和指导学生，并向更广泛的社区进行推广。该项目将阐明多个类别的偏微分方程。简并扩散方程和自由边界问题是肿瘤生长模型工作的基础。目的是研究这些方程并在它们之间建立严格的联系。进化生态学工作的目标是理解非局域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 Hamilton-Jacobi 方程之间的联系。第三个主题涉及二阶偏微分方程的新颖数值方法，并涉及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局域性。非局部偏微分方程通常缺乏比较原理，而比较原理是经典偏微分方程研究的关键工具。此外，一些待研究的方程被推测相对于初始条件是不稳定的。能够克服这个问题，甚至研究不稳定的原因和影响，将是一个重大的进步，并可能导致其他问题的进展。此外，这项工作将涉及理解和开发弱解的新概念。在许多现实世界的系统中，很自然地会出现简并性或不可微性。因此，为了使偏微分方程在这些情况下发挥作用，需要一种新颖的、足够强大的解决方案概念。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Non-local competition slows down front acceleration during dispersal evolution

非局部竞争减缓了扩散演化过程中的前沿加速

• DOI: 10.5802/ahl.117
• 发表时间: 2022
• 期刊: Annales Henri Lebesgue
• 作者: Calvez, Vincent;Henderson, Christopher;Mirrahimi, Sepideh;Turanova, Olga;Dumont, Thierry
• 通讯作者: Dumont, Thierry