From Differential Inclusions to Variational Problems: Theory and Applications

奇点在自然界中无处不在。例如流体动力学中的湍流、薄膜中的折叠和液晶中的缺陷。奇点周围的物理行为通常是高度复杂的，这给理解奇点的形成、结构和对邻近区域的影响带来了巨大的挑战。奇点性质的知识对于预测系统的行为或开发材料的有效应用是基本的。这个项目考虑了连续介质力学和材料科学中的系统，这些系统可以直接或间接地用一类偏微分方程(PDE)来描述。主要目标包括开发新的数学方法来分析这类偏微分方程组，并应用这些新方法来更好地了解它们的奇点性质。该项目将为研究生和本科生提供研究和培训机会。许多模拟物理系统和材料的非线性偏微分方程组都可以表示为微分包含。这些问题中奇点的性质与相关微分包含的刚性和柔性性质密切相关。这个项目包含两个主要主题。第一个主题旨在扩展微分包含的刚性和柔性的一般理论，并发展新的分析工具来研究被视为微分包含的标量守恒律和守恒律的刚性和柔性。具体地说，Eikonal方程和一个2乘2的等熵弹性守恒定律系统将作为模型问题进行研究。研究人员将结合微分包含和双曲守恒律的方法来推进对系统的熵解和标量方程的广义熵解的结构的理解。第二个主题寻求发展新的视角和分析方法，结合熵，研究与Eikonal方程密切相关的变分中的二阶标量和多值问题。这些问题出现在各种物理环境中，包括薄膜和层状弹性材料、液晶和自组织对流模式。研究人员计划将变分分析中的工具与该项目第一部分开发的工具结合起来，以表征低能量状态。这一分析将为不同物理环境下的复杂奇点结构提供信息，具有广泛的实际意义。该奖项反映了NSF的法定使命，并已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。

Singularities are ubiquitous in nature. Examples include turbulence in fluid dynamics, folds in thin films, and defects in liquid crystals. The physical behavior around singularities is generally highly complex, posing great challenges to the efforts of understanding their formation, structure, and influence on neighboring regions. The knowledge of the nature of singularities is fundamental to predict a system’s behavior or to develop effective applications for a material. This project considers systems in continuum mechanics and materials science that can be described directly or indirectly by a class of partial differential equations (PDE). The main goals include developing novel mathematical methods for analyzing this class of PDE and applying such new methods to better understand the nature of their singularities. This project will offer research and training opportunities to graduate and undergraduate students. Many nonlinear PDE modeling physical systems and materials can be formulated as differential inclusions. The nature of singularities in these problems is closely related to the rigidity and flexibility properties of the relevant differential inclusions. This project contains two main themes. The first theme aims to extend the general theory for rigidity and flexibility of differential inclusions, and to develop new analytical tools to study the rigidity and flexibility of scalar and systems of conservation laws viewed as differential inclusions. Specifically, the Eikonal equation and a two-by-two system of conservation laws for isentropic elasticity will be investigated as model problems. The investigator will combine methods from differential inclusions and hyperbolic conservation laws to advance the understanding of the structure of entropy solutions for systems and generalized entropy solutions for scalar equations. The second theme seeks to develop novel perspectives and analytical methods incorporating entropies, to study second order scalar and multi-valued problems in the calculus of variations, which are closely related to the Eikonal equation. Such problems arise in various physical settings, including thin films and layered elastic materials, liquid crystals, and self-organized convection patterns. The investigator plans to meld tools from variational analysis and those developed in the first part of this project to characterize the low-energy states. This analysis will inform the complex singularity structures in different physical settings of broad practical interest.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.