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.

该项目涉及非局部和非局部部分微分方程（PDE）及其应用。非局部方程用于描述各种物理和生物学现象。非局部性意味着一个位置的小扰动会影响整个系统。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。有三个相互联系的主题。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与将肿瘤描述为具有法律的区域的模型之间的连接，该区域统治了其边界的运动。第二个主题是进化生态学。对相关PDE的严格分析将用于研究人群的动态，其中个人可以迁移以及世代之间的突变。第三个主题涉及某些PDE的数值方法。涉及的数学思想还将用于开发算法，以获取许多自主机器人来执行合作任务（例如，包括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兴趣的几个领域，即医学，生态和技术发展。该项目将通过增进我们对数学的理解并加强IT与其他学科之间的联系来促进科学进步。此外，首席研究员将教和指导学生，并向更广泛的社区进行宣传。该项目将阐明多种类别的PDE。退化扩散方程和自由边界问题是肿瘤生长模型的工作。目的是研究这些方程并在它们之间建立严格的联系。进化生态学的工作的目的是了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PDE和汉密尔顿 - 雅各比方程之间的联系。第三个主题涉及二阶PDE的新型数值方法，涉及了解PDE及其正常化的结构。目的是证明数值方法的收敛性，并使用这些工具在机器人控制中开发算法。连接这三个主题的一个重要主题是非局部性。非局部PDE通常缺乏比较原理，这是对经典PDE的研究的关键工具。此外，对初始条件的某些方程式被认为是不稳定的。能够克服这一点，甚至研究不稳定性的原因和影响，将是一个重大发展，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决方案的新概念。在许多现实世界中，自然可以期望形成变性或非差异性。因此，要使PDE在这些情况下有用，需要一个充分强大的解决方案概念。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评估标准来通过评估来获得支持的。

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