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.

可压缩的Euler系统最早于16世纪引入，是最古老的部分微分方程模型之一，它被广泛用于物理和工程中，例如，以模拟气体动力学。但是，与其应用相关的基本理论问题尚未解决。该项目将在其中两个仍在开放的问题中带来见解：在长期跨越长期跨越物理性质的大幅变化的小区域，解决方案的存在和稳定性。研究者还将考虑一个波模型来研究液晶中形成的缺陷结构的时间进化，例如在显示器中使用的材料，其中理解和控制缺陷的结构具有技术后果。该项目还将为本科和研究生提供与研究相关的培训机会。在这个项目中，研究人员将针对可压缩的欧拉方程和列液晶的波浪模型解决一个长期而基本的问题：解决方案如何超越奇异性的形成，例如冲击波？分析技术和数值技术都将用于增强当前的理解。该项目的第一个目标是研究可压缩欧拉方程的溶液的稳定性，这些解决方案会产生冲击波，并且存在具有较大总变化的溶液。第二个目标是通过波型Ericksen-Leslie模型来研究管子中列液晶列晶晶体的Poiseuille流动溶液的奇异性形成和稳定性。为了克服由奇异性形成造成的挑战，所使用的分析技术将包括对坐标的新转换和最佳的运输指标。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的审查标准来评估的支持。