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

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。扩散，即粒子、个人或能量从浓度较高的区域扩散到浓度较低的区域的过程，是许多重要物理系统的中心特征。理解这些系统的动力学导致了各种各样的数学挑战。本课题应用偏微分方程组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。这些模型具有重要的理论和实践意义，因为许多现代技术设备中都使用了等离子体。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不确定的几何区域和在该区域中定义的扩散方程之间的耦合。该项目的数学目标与教育部分相结合，包括对研究生研究人员的支持，以及围绕高级本科阶段引入的PDE研究问题组织的为期一天的研讨会。这些措施的目标是为这一主题领域的下一代研究人员增加机会，包括那些在科学、技术和经济研究领域传统上代表性不足的群体。本项目致力于发展两个经典动力学方程，Boltzmann方程和Landau方程的适定性和正则性理论，这两个方程的特点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，以及削弱解在给定时间之后规则化和持续所需的条件。在自由边界理论中，本项目包括对可能具有不规则扩散系数和无界耦合项的Alt-Caffarelli型泛函的极小项的定性研究。其目的是将众所周知的自由边界正则性结果推广到这些更一般泛函的极小化。这个项目还将使用自由边界技术来研究与一般椭圆算子的Dirichlet特征值相关的形状优化问题，包括目前尚不知道最优域的存在的问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.