Spectral analysis and stability for wave patterns and multidimensional waves

波型和多维波的频谱分析和稳定性

• 批准号: 0906370
• 负责人: Peter Howard
• 金额: $ 21.35万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906370&HistoricalAwards=false
• 关键词: Spectral; analysis; stability; wave; patterns

This project concerns stability of solutions to systems of nonlinear conservation laws. Though the methods under development will be broadly applicable to general conservation laws, the project focuses on three particular models: (1) the reacting Navier-Stokes model of combustion theory; (2) the Cahn-Hilliard equation of phase separation; and (3) a recent model of thin-film dynamics. For (1) (combustion), the project goal is to identify distinguished solutions corresponding to combustive behavior (such as an explosion), and to categorize each as stable or unstable. Each such solution corresponds with a particular speed of combustion, and so the rate of combustion can be found by determining which of these solutions is stable. For (2) (phase separation), the project goal is to determine rates of phase separation. In practice, the properties (flexibility, hardness, etc.) of a particular compound will depend on the amount of phase separation that has occurred in its development, and so the rate of phase separation is particularly important to industrial manufacturing. For (3) (thin films), the project goal is to determine when a "fingering" instability will occur. While the practical applications considered are on the scale of silicon chips, the basic idea can be understood on the level of painting a wall. If a thin layer of paint is brushed across the top of a wall, we expect it to drip down in long vertical lines, or fingers. The goal of this project is to identify conditions, in certain applications, under which such fingers do not form. This project focuses on certain partial differential equations (PDE) that describe conserved quantities such as mass, biomass, energy, or charge. Such equations are often quite complicated, and generally speaking cannot be solved explicitly for general initial conditions. Moreover, numerical evaluation of such equations can be extremely time-consuming, and results can be highly dependent on parameter values, which may not be accurately known. In light of these difficulties, PDE of this type are often studied through consideration of certain distinguished solutions that represent specialized modes of behavior. Once such a distinguished solution has been identified, a natural and important question regards its stability: roughly speaking, do general conditions exist under which the solution occurs/persists in nature? The primary goal of this project is to categorize distinguished solutions of certain standard PDE models as stable or unstable, and to use this information to understand the dynamics modeled by these equations. The work will be of use in modeling a variety of important physical processes, including combustion dynamics and thin-film flow.

本课题主要研究非线性守恒律方程组解的稳定性问题。尽管正在开发的方法将广泛适用于一般守恒定律，但该项目专注于三个特殊的模型：(1)燃烧理论的反应Navier-Stokes模型；(2)相分离的Cahn-Hilliard方程；以及(3)最近的薄膜动力学模型。对于(1)(燃烧)，项目目标是确定与燃烧行为(如爆炸)相对应的不同解决方案，并将每种解决方案归类为稳定或不稳定。每一种这样的溶液都对应着特定的燃烧速度，因此可以通过确定这些溶液中哪一种是稳定的来确定燃烧速度。对于(2)(相分离)，项目目标是确定相分离的速率。在实践中，性能(柔韧性、硬度等)一种特定化合物的成熟度将取决于其发展过程中所发生的相分离量，因此相分离的速度对工业制造特别重要。对于(3)(薄膜)，项目的目标是确定何时会发生“指法”不稳定。虽然所考虑的实际应用是在硅芯片的规模上，但其基本思想可以在油漆墙壁的水平上理解。如果在墙上刷上一层薄薄的油漆，我们预计它会沿着长长的垂直线或手指滴落下来。这个项目的目标是确定在某些应用中，在哪些条件下不会形成这样的手指。本项目主要研究描述守恒量的偏微分方程(PDE)，如质量、生物量、能量或电荷。这样的方程往往相当复杂，一般情况下，对于一般的初始条件，不能显式求解。此外，这种方程的数值计算可能非常耗时，并且结果可能高度依赖于参数值，而参数值可能不是准确知道的。鉴于这些困难，通常通过考虑代表特定行为模式的某些不同的解决方案来研究这种类型的PDE。一旦确定了这样一个杰出的解决方案，关于它的稳定性就有一个自然而重要的问题：粗略地说，在自然界中，解决方案是否存在发生/持续的一般条件？这个项目的主要目标是将某些标准PDE模型的不同解归类为稳定或不稳定，并使用这些信息来理解由这些方程模拟的动力学。这项工作将用于对各种重要的物理过程进行建模，包括燃烧动力学和薄膜流动。