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、液晶和水波中尖波和冲击波的形成是自然界和工程中出现的非线性现象的例子，可以用非线性偏微分方程来描述。该项目考虑波动模型的非线性偏微分方程，其解可能在有限时间内形成不同类型的奇点，例如浅水波动方程中的尖点奇点或非线性光学模型中的冲击波，总体目标是了解在何种条件下这些解决方案在任何时候都有效。该研究将为工程提供指导，例如设计和控制光学系统中的设备。该项目还将为研究生提供研究培训的机会。该项目的主要目标是描述非线性波速的力量如何影响解的规律性。研究人员将研究一类方程，其中包括非线性光学的短脉冲方程和水波的卡马萨-霍尔姆型方程，其解可能形成有限时间奇点。在研究向列液晶建模的波动方程系统的稳定性时，他们还将建立最佳传输度量。最后，他们将探索一种具有径向对称性的高维拟线性模型，即所谓的 O(3) sigma 模型，具有广义相对论、杨-米尔斯场和向列液晶的背景。该项目由 DMS Applied 共同资助数学计划和刺激竞争性研究的既定计划 (EPSCoR)。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和能力进行评估，被认为值得支持。更广泛的影响审查标准。

项目成果

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

拟线性波动方程的Finsler型Lipschitz最优输运度量

• DOI: 10.1016/j.jde.2023.01.035
• 发表时间: 2020-07-30
• 期刊: Journal of Differential Equations
• 影响因子: 2.4
• 作者: H. Cai;Geng Chen;Y. Shen
• 通讯作者: Y. Shen