Spectral analysis and stability for wave patterns and multidimensional waves

这个项目关注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模型：（1）燃烧理论的反应Navier-Stokes模型;（2）相分离的Cahn-Hilliard方程;和（3）最近的薄膜动力学模型。 对于（1）（燃烧），项目的目标是识别与燃烧行为（如爆炸）相对应的不同解决方案，并将每个解决方案归类为稳定或不稳定。 每个这样的解对应于特定的燃烧速度，因此可以通过确定这些解中的哪一个是稳定的来找到燃烧速率。 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（柔性、硬度等）特定化合物的产率将取决于在其发展过程中发生的相分离的量，因此相分离的速率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“指进”不稳定性。 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粉刷墙壁的水平上理解。 如果在墙上刷上一层薄薄的油漆，我们希望它以长长的垂直线或手指状滴落。 该项目的目标是确定在某些应用中不会形成这种手指的条件。 这个项目的重点是某些偏微分方程（PDE），描述守恒量，如质量，生物量，能量，或电荷。 这样的方程往往是相当复杂的，一般来说，不能明确解决一般的初始条件。 此外，这些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，而参数值可能并不准确。 鉴于这些困难，这种类型的PDE通常通过考虑代表特定行为模式的某些特殊解决方案来研究。 一旦确定了这样一个独特的解决方案，一个自然而重要的问题就是它的稳定性：粗略地说，在自然界中存在解决方案发生/持续的一般条件吗？这个项目的主要目标是将某些标准PDE模型的杰出解分类为稳定或不稳定，并使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。

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.