Regularity and Stability for Solutions of Quasilinear Wave Equations with Singularities

具有奇异性的拟线性波动方程解的正则性和稳定性

• 批准号: 2206218
• 负责人: Yannan Shen
• 金额: $ 24.38万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-15 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2206218&HistoricalAwards=false
• 关键词: Regularity; Stability; Solutions; Quasilinear; Wave

The formation of cusp and shock waves in optical systems, liquid crystals, and water waves are examples of nonlinear phenomena arising in nature and engineering that can be described by nonlinear partial differential equations. This project considers nonlinear partial differential equations for wave models whose solutions might form different types of singularities in finite time, such as cusp singularities in the shallow water wave equation or shock waves in models of nonlinear optics, with the overall aim of understanding under which condition the solutions are valid for all times. The research will give guidance in engineering, for example for designing and controlling devices in optical systems. The project will also provide opportunities for research training of graduate students. A main goal of the project is to describe how the power of nonlinear wave speed impacts the regularity of solutions. The investigator will study a class of equations, which include the short pulse equation from nonlinear optics and Camassa-Holm type equations from water waves, whose solutions might form finite-time singularities. They will also establish an optimal transport metric when studying the stability of a system of wave equations modelling nematic liquid crystals. Finally, they will explore a higher dimensional quasilinear model with radial symmetry, the so-called O(3) sigma-model, with background in general relativity, Yang-Mills field and nematic liquid crystals.This project is jointly funded by the DMS Applied Mathematics Program and the Established Program to Stimulate Competitive Research (EPSCoR).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.

光学系统，液晶和水波在自然界和工程中产生的非线性现象的例子是，非线性偏微分方程可以描述。该项目考虑了波浪模型的非线性偏微分方程，其解决方案可能会在有限的时间内形成不同类型的奇异性，例如浅水波方程中的尖尖象奇异性或非线性光学模型中的冲击波，其总体目的是在哪个条件下了解哪种情况。该解决方案始终有效。该研究将为工程提供指导，例如用于设计和控制光学系统中的设备。该项目还将为研究生的研究培训提供机会。该项目的一个主要目标是描述非线性波速度的力量如何影响解决方案的规律性。研究者将研究一类方程，其中包括来自非线性光学和Camassa-Holm型方程的短脉冲方程，其溶液可能形成有限的时间奇异性。在研究建模列神经液晶体系统的稳定性时，他们还将建立一个最佳的运输度量。最后，他们将探索具有径向对称性，所谓的O（3）Sigma-Model的较高维度的准线性模型，具有一般相对论，Yang-Mills Field和Nematic Liquid Crystals。数学计划和启发竞争研究的既定计划（EPSCOR）。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响评估标准，被认为值得通过评估来获得支持。

项目成果

A Finsler type Lipschitz optimal transport metric for a quasilinear wave equation

• DOI: 10.1016/j.jde.2023.01.035
• 发表时间: 2020-07
• 期刊: Journal of Differential Equations
• 影响因子: 2.4
• 作者: H. Cai;Geng Chen;Y. Shen