LEAPS-MPS: Diffusive Partial Differential Equations in the Physical Sciences

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。扩散，即粒子、个体或能量从浓度较高的区域向浓度较低的区域扩散的过程，是许多重要物理系统的中心特征。理解这些系统的动力学会带来各种各样的数学挑战。本课题应用偏微分方程理论研究了两类扩散模型。第一种是气体和等离子体的动力学模型，以粒子间碰撞产生的扩散为特征。这些模型在理论和实践上都很重要，因为等离子体被用于许多现代技术设备中。第二类，椭圆自由边界问题，出现在相变、绝缘和许多其他物理和工业过程的研究中。这些模型是基于一个未确定的几何区域和该区域中定义的扩散方程之间的耦合。这个项目的数学目标与教育组成部分相结合，包括对研究生研究人员的支持，以及围绕PDE在高级本科阶段引入的研究问题组织的为期一天的研讨会。这些措施的目标是增加该学科领域下一代研究人员的机会，包括那些传统上在STEM中代表性不足的群体。本项目旨在推进两个经典动力学方程的适定性和规律性理论，Boltzmann和Landau方程，这两个方程的特征是速度变量中的非线性非局部扩散与时间和空间变量中的输运的相互作用。具体目标包括通过利用扩散项的平滑特性来构建一般初始数据的短期解，以及削弱解规范化和持续超过给定时间所需的条件。在自由边界理论中，对可能具有不规则扩散系数和无界耦合项的Alt-Caffarelli型泛函的极小值进行了定性研究。目标是将众所周知的自由边界正则性结果扩展到这些更一般的函数的最小化。本项目还将利用自由边界技术研究与一般椭圆算子狄利克雷特征值相关的形状优化问题，包括目前未知最优域存在的问题。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).Diffusion, the process whereby particles, individuals, or energy spread from areas of higher concentration to areas of lower concentration, is a central feature of many important physical systems. Understanding the dynamics of these systems leads to a wide variety of mathematical challenges. This project applies the theory of partial differential equations (PDE) to study two classes of diffusive models. The first, kinetic models of gases and plasmas, feature diffusion generated by collisions between particles. These models are of both theoretical and practical importance, as plasmas are used in many modern technological devices. The second class, elliptic free boundary problems, arise in the study of phase transitions, insulation, and many other physical and industrial processes. These models are based on the coupling between an undetermined geometric region and a diffusive equation defined in that region. The mathematical goals of this project are combined with an educational component that includes support for graduate student researchers, as well as a one-day workshop organized around research questions in PDE introduced at the advanced-undergraduate level. The goal of these measures is to increase opportunities for the next generation of researchers in this subject area, including those from groups traditionally underrepresented in STEM. This project seeks to advance the well-posedness and regularity theory of two classical kinetic equations, the Boltzmann and Landau equations, which feature the interaction of nonlinear, nonlocal diffusion in the velocity variable, with transport in the time and space variables. Specific goals include constructing short-time solutions for general initial data by leveraging the smoothing properties of the diffusion term, as well as weakening the conditions required for solutions to regularize and be continued past a given time. In free boundary theory, this project includes the qualitative study of minimizers for functionals of Alt-Caffarelli type with possibly irregular diffusion coefficients and unbounded coupling terms. The goal is to extend well-known free boundary regularity results to minimizers of these more general functionals. This project will also use free boundary techniques to study shape optimization problems related to Dirichlet eigenvalues of general elliptic operators, including problems for which existence of optimal domains is currently unknown.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.