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

该奖项全部或部分根据2021年美国救援计划法案（公法117-2）资助。该奖项支持偏微分方程的综合研究计划和促进本科和研究生水平的科学计算和应用数学的教育计划。该项目的研究部分涉及物理学和工程学的数学模型，特别是用于描述长程力和质量输运/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创建实用且可靠的算法，用于复杂物理系统的计算机模拟（例如等离子体物理学、空气动力学、流体力学、材料科学中的算法），并创建实用且可靠的自主系统（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称为积分微分方程的方程来建模。这类方程的数学理论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中变得越来越普遍。从最广泛的意义上理解，质量输运的数学分析涉及大量“粒子”在背景力和所述粒子之间的碰撞的反应中移动的动力学。根据上下文，这些“粒子”可能代表实际的物理粒子或智能代理与他们的环境和彼此交互。因此，运输模型涵盖了许多情况：从统计力学中模型的动态公式到经济学中最优或稳定匹配的研究，该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），特别是将探索使用传输方法来分析涉及远程力的问题。在这个项目中的活动也提供了大量的支持，招聘，培训和辅导数学天才的本科生和研究生在得克萨斯州州立大学。这些活动包括与来自学术界和工业界的演讲嘉宾，在本科和研究生水平的专业高级课程，研究生和高级本科生的研究导师校园范围内的活动。该项目的教育工作的一个主要目标是增加毕业生进入需要最先进的数学和计算培训的工业和/或学术界的职业生涯的数量。该项目的研究活动，从主题上讲，在以下一个或多个领域：高维Hamilton-Jacobi方程，带非局部项的动力学方程，界面问题的积分-微分方法，雅可比方程/运输问题一个共同的线程连接这些不同的主题是搜索逐点估计和正则性估计的椭圆和抛物型偏微分方程的解决方案，并因此问题的估计，使相当一部分的这个项目的研究工作。虽然这个项目中考虑的一些问题完全属于上述类别之一，但它们的动机是在不同类别中的潜在应用-例如非局部雅可比方程，其动机是获得适用于积分微分方程的亚历山德罗夫型估计。各种调查的积极成果可能会导致Hamilton-Jacobi方程粘性解的无网格算法，非线性动力学方程解的最佳传输公式，两相自由边界问题的新规律性结果，这个奖项反映了NSF的法定使命，并已被认为是值得支持，通过评估使用基金会的知识价值和更广泛的影响审查标准。

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.