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

一维和多维空间双曲守恒定律和波动方程组的大解

• 批准号: 2008504
• 负责人: Geng Chen
• 金额: $ 26.5万
• 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: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2008504&HistoricalAwards=false
• 关键词: Large; solutions; systems; hyperbolic; conservation

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.

本研究项目旨在研究两类数学模型，这两类模型在气体动力学和液晶物理中广泛用于描述波状运动。在这个项目中考虑的第一个模型是微分方程的欧拉系统。欧拉系统是流体力学最基本的系统之一，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之一，于16世纪首次引入。然而，有关欧拉方程的许多基本理论问题仍未得到解决。最重要的开放问题之一是，产生强冲击波的溶液如何在长时间跨度内表现。本项目旨在根据相关首席研究员和合作者的最新进展，从根本上理解这个问题。本项目考虑的另一个数学模型是描述液晶的波动模型。首席研究员（PI）和合作者最近发现了一种新型的尖状奇异解。这为该领域开辟了一个新的方向。本研究项目旨在加深对奇点形成后解结构的理解。该项目将包括几个与研究相关的本科生和研究生培训项目。在这个项目中，PI的目标是捕捉一些非线性波动方程的解的特征，包括奇点，如激波奇点和尖点奇点。这将回答这些方程的一个非常重要和基本的问题：在奇点形成之外，解是如何表现的？分析和数值技术将被用来加强目前的理解。该项目的第一个目标是更好地理解产生激波的可压缩欧拉方程的大数据解。PI将研究大总变分解的存在性、解的一般性质以及密度的最优下界。第二个目标是利用Ericksen-Leslie模型研究向列型液晶全泊泽维尔流解的全局存在性、唯一性和稳定性。在最近的一项发现中，PI及其合作者发现了尖端奇点的形成。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。