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

该项目侧重于可以通过偏微分方程（PDEs）建模的物理，化学和生物系统的行为。决定这些系统时间演化的力量是复杂的，这使得它们的分析变得微妙而技术性强。我们感兴趣的两个基本问题是这些系统的定性行为，例如，解决方案是否有很大的波动，以及它们的长期行为，例如，通过量化入侵物种超越新环境的速度。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长期行为的能力，包括确定每个长期结果所依赖的关键数量，使我们能够以一种不能纯粹通过数值模拟捕获的方式预测现实世界系统的行为，因为数值模拟必须限于有限的时间尺度。这个项目将为实现这些目标开发新的方法。研究生和本科生的研究将整合到该项目中，培养下一代应用数学家和科学家。该项目还包括一个夏季训练营，面向从相邻但非数学领域过渡的应用数学博士生，以增强他们的数学推理（逻辑思维）和技术写作技能。他们的训练是有影响力的，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，学术界和国家实验室的职业生涯。本项目重点介绍了反应扩散方程和碰撞动力学方程的研究进展。在前者中，该项目将开发一种新颖的PDE“Stein方法”方法，该方法基于观察到给定PDE的单调稳态满足一阶自治常微分方程（ODE），并且为了证明PDE的一般解收敛于这种稳态，证明一般解收敛于ODE的一个解就足够了。这项研究将在变分学中利用新的函数不等式和思想。在后一种情况下，该项目将引进抛物线理论和随机分析的技术，以确定在一般领域（有边界和无边界）何时发生爆炸。这需要对物理空间边界附近解的规律性和“大”速度下解的衰减进行精确和定量的理解。该奖项反映了美国国家科学基金会的法定使命，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。

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.