Stability, Uniqueness, and Existence for Solutions of Hyperbolic Conservation Laws and Nonlinear Wave Equations

双曲守恒定律和非线性波动方程解的稳定性、唯一性和存在性

• 批准号: 2306258
• 负责人: Geng Chen
• 金额: $ 23.5万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2306258&HistoricalAwards=false
• 关键词: Stability; Uniqueness; Existence; Solutions; Hyperbolic

The compressible Euler system, first introduced in the sixteenth century, is amongst the oldest partial differential equation models, and it is extensively used in physics and engineering, for instance, to model gas dynamics. However, fundamental theoretical issues, relevant to its applications, remain unresolved. This project will bring insights in two of these still open issues: the existence and stability of solutions that develop strong shock waves, that is small regions where large changes in physical properties occur, over long-time spans. The investigator will also consider a wave model to study the time-evolution of the structure of defects that form in liquid crystals, which are materials used for example in display devices, where understanding and controlling the structure of defects has technological ramifications. This project will also offer research-related training opportunities for undergraduate and graduate students. In this project, the investigator will address a long-standing and fundamental question for compressible Euler equations and for a wave model for nematic liquid crystals: How do solutions behave beyond the formation of a singularity, such as shock waves? Both analytical and numerical techniques will be used to enhance the current understanding. The first goal of the project is to study the stability of solutions of the compressible Euler equations, which develop shock waves, and the existence of solutions with large total variation. A second goal is to study the singularity formation, global existence, and stability for solutions of Poiseuille flow of a nematic liquid crystal in a tube, via the wave-type Ericksen-Leslie model. To overcome the challenge caused by the singularity formation, the analytical techniques used will include a new transformation of coordinates and an optimal transport metric.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.

可压缩欧拉系统于十六世纪首次引入，是最古老的偏微分方程模型之一，广泛应用于物理和工程领域，例如模拟气体动力学。然而，与其应用相关的基本理论问题仍未解决。该项目将为其中两个仍然悬而未决的问题带来见解：产生强烈冲击波的解决方案的存在和稳定性，即在很长一段时间内物理特性发生巨大变化的小区域。研究人员还将考虑使用波模型来研究液晶中形成的缺陷结构的时间演化，液晶是用于显示设备等的材料，其中理解和控制缺陷结构具有技术影响。该项目还将为本科生和研究生提供与研究相关的培训机会。在这个项目中，研究人员将解决可压缩欧拉方程和向列液晶波模型的一个长期存在的基本问题：除了形成奇点（例如冲击波）之外，解决方案如何表现？分析和数值技术将用于增强当前的理解。该项目的首要目标是研究产生冲击波的可压缩欧拉方程解的稳定性，以及总变差较大的解的存在性。第二个目标是通过波型 Ericksen-Leslie 模型研究管中向列液晶的泊肃叶流解的奇点形成、整体存在性和稳定性。为了克服奇点形成带来的挑战，所使用的分析技术将包括新的坐标变换和最佳传输度量。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响进行评估，被认为值得支持审查标准。