Large solutions for systems of hyperbolic conservation laws and wave equations in one and multiple space dimensions

本研究项目旨在研究气体动力学和液晶物理中广泛使用的两种描述波状运动的数学模型。本项目考虑的第一个模型是欧拉微分方程组。作为流体力学最基本的系统之一，欧拉系统在物理和工程中得到了广泛的应用，是最古老的偏微分方程组之一，于16世纪首次提出。然而，与欧拉方程有关的许多基本理论问题仍未解决。最重要的悬而未决的问题之一是，产生强烈冲击波的解决方案在长时间跨度内如何表现。本项目旨在根据首席调查员和合作者的最新进展，对这一问题有一个基本的了解。这个项目中考虑的另一个数学模型是描述液晶的波动模型。首席调查员(PI)和他的合作者最近发现了一种新的尖点状奇异解。这为这一领域开辟了新的方向。本研究旨在加深对奇点形成后解的结构的理解。该项目将包括几个与研究相关的本科生和研究生培养项目。在这个项目中，PI的目标是捕捉一些非线性波动方程包含奇性的解的特征，例如激波和尖点奇性。这将回答这些方程的一个非常重要和基本的问题：解在奇点形成之外是如何表现的？将使用分析和数值技术来加强目前的理解。该项目的第一个目标是更好地理解可压缩欧拉方程的大数据解，这会产生冲击波。PI将研究具有大总变分的解的存在性、解的一般性质以及密度的最优下界。第二个目标是通过Ericksen-Leslie模型研究向列相液晶全Poiseuille流解的整体存在性、唯一性和稳定性。在最近的一项发现中，PI和合作者发现了尖点奇点的形成。为了克服新奇点带来的挑战，将使用许多分析技术，包括新的坐标变换和最优传输度量。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This research project is aimed at the investigation of two types of mathematical models, which are widely used to describe the wave-like motion in gas dynamics and physics of liquid crystals. The first model considered in this project is the Euler system of differential equations. As one of the most fundamental systems for fluid dynamics, the Euler system has been extensively used in physics and engineering and is among the oldest partial differential equations, first introduced in the 16th century. However, many fundamental theoretical issues pertaining to the Euler equations have remained unresolved. One of the most important open problems is how the solutions that develop strong shock waves behave over long time spans. This project aims at the fundamental understanding of this problem, following the related Principal Investigator's and collaborators’ recent progress. Another mathematical model considered in this project is a wave model describing liquid crystals. The Principal Investigator (PI) and collaborators recently found a new type of cusp-like singular solutions. This opens a new direction in this field. This research project seeks to deepen understanding of the structure of solutions after the singularity formation. This project will include several research-related undergraduate and graduate student training projects. In this project the PI aims to capture features of solutions that include singularities, such as shock wave and cusp singularities, for some nonlinear wave equations. This will answer a very important and fundamental question for these equations: How do solutions behave beyond the formation of a singularity? Both analytical and numerical techniques will be used to enhance the current understanding. The first goal of the project is to better understand the large-data solutions for the compressible Euler equations, which develop shock waves. The PI will study the existence of solutions with large total variation, the generic properties of solutions, and the optimal lower bounds on the density. The second goal is to study the global existence, uniqueness, and stability for solutions of full Poiseuille flow of nematic liquid crystals via the Ericksen-Leslie model. In a recent discovery, the formation of cusp singularities was found by the PI and collaborators. To overcome the challenge caused by the new singularity, many analytical techniques, including a new transformation of coordinates and an optimal transport metric, will be used.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.