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From Differential Inclusions to Variational Problems: Theory and Applications

从微分包含到变分问题：理论与应用

基本信息

批准号：
2206291
负责人：
Guanying Peng
金额：
$ 19.8万
依托单位：
Worcester Polytechnic Institute
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2206291&HistoricalAwards=false
关键词：
Differential Inclusions Variational Problems Theory

项目摘要

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.

奇点在自然界中无处不在。例子包括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，对理解它们的形成，结构和对邻近区域的影响的努力构成了巨大的挑战。奇异性的性质的知识是基本的预测系统的行为或开发有效的应用程序的材料。该项目考虑了连续介质力学和材料科学中可以直接或间接由一类偏微分方程（PDE）描述的系统。主要目标包括开发新的数学方法来分析这类偏微分方程，并应用这些新方法来更好地理解其奇点的性质。该项目将为研究生和本科生提供研究和培训机会。许多非线性偏微分方程模型的物理系统和材料可以表示为微分包含。这些问题中奇异性的性质与相关微分包含的刚性和柔性性质密切相关。该项目包含两个主题。第一个主题旨在扩展微分包含的刚性和柔性的一般理论，并开发新的分析工具来研究作为微分包含的标量和守恒律系统的刚性和柔性。具体地说，Eikonal方程和二乘二系统的守恒定律等熵弹性将作为模型问题进行研究。调查员将联合收割机方法从微分包含和双曲守恒律推进系统和广义熵解标量方程的熵解的结构的理解。第二个主题旨在发展新的视角和分析方法，结合熵，研究变分法中的二阶标量和多值问题，这是密切相关的Eikonal方程。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本项目第一部分开发的工具相结合，以表征低能态。这项分析将为不同物理环境中的复杂奇点结构提供广泛的实际利益。该奖项反映了NSF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。