Spectral analysis and stability for wave patterns and multidimensional waves

本项目涉及非线性守恒律系统解的稳定性。虽然正在开发的方法将广泛适用于一般守恒定律，但该项目侧重于三个特定的模型：(1)燃烧理论的反应Navier-Stokes模型；(2)相分离的Cahn-Hilliard方程；(3)薄膜动力学的最新模型。对于(1)（燃烧），项目目标是识别对应于燃烧行为（如爆炸）的不同解决方案，并将每种解决方案分类为稳定或不稳定。每一个这样的溶液对应一个特定的燃烧速度，因此，通过确定这些溶液中哪一个是稳定的，就可以找到燃烧速度。对于(2)（阶段分离），项目目标是确定阶段分离的速率。在实践中，特定化合物的性能（柔韧性、硬度等）将取决于在其开发过程中发生的相分离的量，因此相分离的速度对工业制造尤为重要。对于(3)（薄膜），项目目标是确定何时会发生“指指”不稳定性。虽然考虑的实际应用是在硅芯片的规模上，但基本思想可以理解为在墙上涂漆的水平。如果在墙的顶部刷上一层薄薄的油漆，我们预计它会以长长的垂直线或手指滴下。这个项目的目标是确定在某些应用中，这种手指不会形成的条件。该项目侧重于描述守恒量（如质量、生物量、能量或电荷）的某些偏微分方程（PDE）。这类方程往往相当复杂，一般来说，不能在一般初始条件下显式求解。此外，这类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，而这些参数值可能无法准确知道。鉴于这些困难，通常通过考虑代表特定行为模式的特定解决方案来研究这种类型的PDE。一旦确定了这样一个特殊的解决方案，一个自然而重要的问题就会涉及到它的稳定性：粗略地说，在自然界中是否存在解决方案发生/持续存在的一般条件？本项目的主要目标是将某些标准PDE模型的不同解分类为稳定或不稳定，并利用这些信息来理解这些方程所模拟的动力学。这项工作将用于模拟各种重要的物理过程，包括燃烧动力学和薄膜流动。

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.