Nonlinear and Nonlocal Partial Differential Equations

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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.