CAREER: Integro-differential and Transport Problems in Partial Differential Equations

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。该奖项支持一个关于偏微分方程的综合研究项目，以及一个促进本科生和研究生水平的科学计算和应用数学的教育项目。该项目的研究部分涉及物理学和工程学的数学模型，特别是用于描述远程力和质量传递/动力学效应的模型，并试图回答一些基本问题。这些问题的答案将有助于为复杂物理系统（如等离子体物理学、空气动力学、流体力学、材料科学）的计算机模拟创建实用可靠的算法，并有助于创建实用可靠的自主系统（包括但不限于机器人和控制理论）。在数学中，长程力和重尾分布通常通过一类被称为积分-微分方程的方程来建模。这类方程的数学理论并不像常微分方程或偏微分方程那样先进，但积分微分方程在理论和应用上正变得越来越普遍。从最广泛的意义上讲，质量传递的数学分析处理的是“粒子”质量的动力学，它们在背景力和粒子之间的碰撞的作用下运动。根据上下文，这些“粒子”可以代表实际的物理粒子或与环境相互作用的智能代理。因此，运输模型涵盖了许多情况：从统计力学模型的动态公式到经济学中最优或稳定匹配的研究，这个项目的数学结果可能对这些学科有用。该项目探讨了这两个领域中相互关联的问题（其中许多与等离子体建模有关），特别是将探索使用输运方法来分析涉及远程力的问题。该项目的活动也为德州州立大学数学人才的招聘、培训和指导提供了大量支持。这些活动包括来自学术界和工业界的演讲嘉宾，本科和研究生水平的专业高级课程，以及研究生和高级本科生的研究指导。该项目教育工作的主要目标是增加毕业生进入需要最先进的数学和计算训练的工业和/或学术界工作的人数。从主题上讲，本项目的研究活动分为以下一个或多个领域：高维Hamilton-Jacobi方程，非局域项动力学方程，界面问题的积分-微分方法，以及Jacobian方程/输运问题。连接这些不同主题的一个共同线索是对椭圆型和抛物型偏微分方程解的点估计和正则估计的搜索，并且关于估计的问题是这个项目研究工作的相当一部分。虽然本项目中考虑的一些问题完全属于上述类别之一，但它们是由不同类别的潜在应用驱动的-例如非局部雅可比方程，它是由获得适用于积分-微分方程的aleksandrov型估计的问题驱动的。在各种研究领域的积极成果可能会导致Hamilton-Jacobi方程黏性解的无网格算法，非线性动力学方程解的最优输运公式，两相自由边界问题的新规则结果，以及非局部方程的新Harnack不等式。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 award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). This award supports an integrated research program on partial differential equations and an educational program promoting scientific computing and applied mathematics at the undergraduate and graduate level. The research component of the project is concerned with mathematical models from physics and engineering, specifically models used to describe long-range forces and mass transport/kinetic effects, and seeks to answer a number of fundamental questions. Answers to these questions will aid in the creation of practical and reliable algorithms for computer simulations of complex physical systems (such as those found in plasma physics, aerodynamics, fluid mechanics, material science) and in the creation of practical and reliable autonomous systems (including but not limited to robotics and control theory). In mathematics, long-range forces and heavy-tailed distribution are often modeled via a class of equations known as integro-differential equations. The mathematical theory for such equations is not as advanced as that of ordinary or partial differential equations, but integro-differential equations are becoming increasingly common in theory and applications. The mathematical analysis of mass-transport, understood in the broadest sense, deals with dynamics of masses of "particles" moving in reaction to both background forces and collisions between said particles. Depending on the context, these "particles" may represent actual physical particles or intelligent agents interacting with their environment and with one another. As such, transport models cover many situations: from dynamical formulations of models in statistical mechanics to the study of optimal or stable matchings in economics, and the mathematical results of this project may be of use to such disciplines. The project explores interconnected questions in these two fields (many of which are relevant to the modeling of plasmas) and in particular will explore the use of transport methods to analyze problems involving long-range forces. The activities in this project also provide substantial support for the recruitment, training, and mentorship of mathematically talented undergraduate and graduate students at Texas State University. These activities include campus-wide events with guest speakers from academia and industry, specialized advanced courses at the undergraduate and graduate levels, and research mentorship of graduate students and advanced undergraduate students. A prime objective of this project's educational efforts is increasing the numbers of graduates entering careers in industry and/or academia that require state of the art mathematical and computational training.The research activities in this project fall, thematically speaking, in one or more of the following areas: high dimensional Hamilton-Jacobi equations, kinetic equations with nonlocal terms, integro-differential methods for interface problems, and Jacobian equations/transport problems. A common thread connecting these diverse topics is the search for pointwise estimates and regularity estimates for solutions of elliptic and parabolic partial differential equations, and as such questions about estimates makes for a considerable portion of this project's research efforts. While a few of the problems considered in this project lie squarely within one of the categories above, they are motivated by a potential application in a different category - such as the Nonlocal Jacobian equation, which is motivated by the question of obtaining Aleksandrov-type estimates adapted to integro-differential equations. Positive outcomes in the various lines of inquiry could potentially lead to a mesh-free algorithm for viscosity solutions of Hamilton-Jacobi equations, an optimal transport formulation of solutions for nonlinear kinetic equations, new regularity results for two-phase free boundary problems, and new Harnack inequalities for nonlocal equations.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.