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

可压缩欧拉系统于16世纪首次提出，是最古老的偏微分方程模型之一，它广泛应用于物理和工程中，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得到解决。该项目将带来两个仍然悬而未决的问题的见解：发展强冲击波的解决方案的存在和稳定性，即物理性质发生大变化的小区域，在很长一段时间内。研究者还将考虑一个波动模型来研究液晶中形成的缺陷结构的时间演变，液晶是显示设备中使用的材料，其中理解和控制缺陷结构具有技术分支。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，研究者将解决可压缩欧拉方程和向列液晶的波动模型中一个长期存在的基本问题：在奇点（如冲击波）形成之外，溶液的行为如何？分析和数值技术将被用来加强目前的理解。项目的第一个目标是研究产生激波的可压缩欧拉方程解的稳定性和大总变分解的存在性。第二个目标是通过波型Ericksen-Leslie模型研究管中向列液晶泊泽维尔流解的奇点形成、全局存在性和稳定性。为了克服奇点形成带来的挑战，使用的分析技术将包括新的坐标变换和最佳传输度量。该奖项反映了美国国家科学基金会的法定使命，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。

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.