CAREER: Well-posedness and long-time behavior of reaction-diffusion and kinetic equations

职业：反应扩散和动力学方程的适定性和长期行为

• 批准号: 2337666
• 负责人: Christopher Henderson
• 金额: $ 45.39万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-06-01 至 2029-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2337666&HistoricalAwards=false
• 关键词: CAREER; Well; posedness; long; time

This project focuses on the behavior of physical, chemical, and biological systems that can be modelled by partial differential equations (PDEs). The forces that determine the time-evolution of these systems are complex, making their analysis subtle and technical. Two fundamental questions of interest are the qualitative behavior of these systems, e.g., whether solutions have large fluctuations, and their long-time behavior, e.g., by quantifying the speed with which an invasive species overruns a new environment. These questions are interdependent, with the latter relying on an understanding of the former. Our ability to understand the long-time behavior of PDE, including identifying the key quantities on which each long-time outcome depends, allows us to predict the behavior of real-world systems in a way that cannot be captured purely by numerical simulation, which, by necessity, is restricted to finite time scales. This project will develop novel methods for these goals. Graduate and undergraduate research will be integrated into the project, training the next generation of applied mathematicians and scientists. The project also involves a summer boot camp for entering applied mathematics PhD students transitioning from adjacent, but nonmathematical, fields that shore up their mathematical reasoning (logical thinking) and technical writing skills. Their training is impactful because these students have diverse interests (mathematical biology, machine learning, data science, PDE and numerical analysis, etc.) and go on to careers in industry, academia, and national labs.This project focuses on advances in reaction-diffusion equations and collisional kinetic equations. In the former, the project will develop a novel "Stein's method" approach to PDE that is based on the observation that monotonic steady states of a given PDE satisfy first order autonomous ordinary differential equations (ODE) and that, to show convergence of a generic solution of the PDE to such a steady state, it is enough to show that the generic solution converges to a solution of the ODE. The research will leverage new functional inequalities and ideas in the calculus of variations. In the latter, the project will import techniques from parabolic theory and stochastic analysis to characterize when blow-up occurs in generic domains (both with and without boundaries). This requires the precise and quantitative understanding of the regularity of solutions near the boundary in physical space and the decay of solutions at "large" velocities.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.

该项目的重点是物理，化学和生物系统的行为，可以通过偏微分方程（PDE）建模。 决定这些系统的时间演化的力量是复杂的，这使得它们的分析变得微妙和技术性。 感兴趣的两个基本问题是这些系统的定性行为，例如，解决方案是否具有大的波动，以及它们的长期行为，例如，通过量化入侵物种在新环境中的蔓延速度。 这些问题是相互依存的，后者依赖于对前者的理解。 我们理解PDE的长期行为的能力，包括确定每个长期结果所依赖的关键量，使我们能够以一种无法纯粹通过数值模拟捕获的方式预测现实世界系统的行为，而数值模拟必然限于有限的时间尺度。 该项目将为这些目标开发新的方法。 研究生和本科生的研究将被纳入该项目，培养下一代应用数学家和科学家。该项目还包括一个夏季靴子营进入应用数学博士生过渡从相邻的，但非数学，领域，海岸了他们的数学推理（逻辑思维）和技术写作技能。 他们的培训是有影响力的，因为这些学生有不同的兴趣（数学生物学，机器学习，数据科学，PDE和数值分析等）。并继续在工业界，学术界和国家实验室的职业生涯。该项目侧重于反应扩散方程和碰撞动力学方程的进展。 在前者中，该项目将开发一种新的“斯坦方法”的PDE方法，该方法基于以下观察：给定PDE的单调稳定状态满足一阶自治常微分方程（ODE），并且为了显示PDE的一般解收敛到这样的稳定状态，足以显示一般解收敛到ODE的解。 该研究将利用新的函数不等式和变分法中的思想。 在后者中，该项目将从抛物线理论和随机分析中引入技术，以表征在一般域（有边界和无边界）中何时发生爆破。 这需要精确和定量地了解物理空间边界附近解的规律性以及解在“大”速度下的衰减。该奖项反映了NSF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。