I. Stability of Waves in Viscous Conservation Laws. II. Phase Transitions and Minimal Surfaces

9107990 Zumbrun Kevin Zumbrun提出了几个源自流体动力学的项目。这些问题涉及奇异极限问题中出现的界面解的结构和稳定性，特别是激波和相边界。稳定性是激波理论中的一个中心问题，与双曲波的物理可容许性、差分格式的收敛性和无粘极限问题等问题有关。Zumbrun提出研究各种相关问题，从可压缩Navier-Stokes方程中多维粘性激波的稳定性到非局部分散沉积模型中振荡激波层的物理意义。计划的分析方法包括，除其他外，点式，格林函数技术已被证明在其他微妙稳定性的情况下是有用的，使用矩阵摄动理论的光谱分析，以及从反应扩散前沿研究中借鉴的埃文斯函数技术。同样，相边界的结构也是相变研究的中心问题。在过渡层厚度为零的极限下，Cahn-Hilliard相变模型简化为理想最小表面问题，相边界简化为最小表面问题。这种联系是直觉的丰富来源，在几何和pde设置中都提出了新的问题。Zumbrun提出了其中的几个用于研究，最著名的是最小曲面问题的“诺伊曼”解的规律性。计划的分析方法是通过pde和几何测量理论技术的结合。界面的行为和结构是一个基本的物理兴趣话题，作为自然界中各种效果的组织原则。例如，众所周知，肥皂泡形成最小的界面能结构，这是由它们具有最小的表面积这一特性确定的。它们在小的扰动下也是稳定的，根据系统总是向低能量状态移动的原理，在这种情况下，系统会回到最小能量构型。与此同时，可见宇宙中物质的大规模分布似乎也有类似的结构，巨大的空洞被薄层星系包围。很明显，我们生活在一个界面上——很明显，同样，通过一个共同的数学机制，不同能量的最小化可以导致相似的结构。上面给出的例子只是称为相位边界的接口的两个实例。其他重要的界面是冲击波，或者是分离性质非常不同的物质的移动边界（最常见的是分离温度或压力非常不同的空气的“前沿”）。就像相界一样，它们在自然界中无处不在，从音爆到化学反应中的浓度“波”。同样，是稳定还是不稳定决定了一个特定的冲击结构是持续存在还是破裂。控制界面稳定性的数学是相当微妙的，绝不是完全理解。的确，Zumbrun提出的项目涉及一些基本的理论问题，在我们能够自信地进行实际应用之前，这些问题必须得到回答，比如在各种环境中对这些现象进行计算机建模。

9107990 Zumbrun Kevin Zumbrun proposes several projects originating from fluid dynamics. These concern structure and stability of interfacial solutions arising in singular limit problems, specifically SHOCK WAVES and PHASE BOUNDARIES. Stability is a central topic in shock wave theory, connected with such issues as physical admissibility of hyperbolic waves, convergence of difference schemes, and the inviscid limit problem. Zumbrun proposes to study a variety of related questions, from stability of multidimensional viscous shock waves in compressible Navier-Stokes equations to physical significance of oscillatory shock layers in a nonlocal, dispersive sedimentation model. The planned methods of analysis include, among others, pointwise, Green's function techniques which have proved to be useful in other situations of delicate stability, spectral analysis using matrix perturbation theory, and Evans function techniques borrowed from the study of reaction diffusion fronts. Likewise, structure of phase boundaries is a central topic in the study of phase transitions. In the limit of zero transition layer thickness, Cahn-Hilliard models for phase transition reduce to idealized minimal surface problems, and the phase boundaries to minimal surfaces. This link is a rich source of intuition, suggesting new problems in both the geometry and pde setting. Zumbrun proposes several of these for study, most notably the regularity of "Neumann" solutions for the minimal surface problem. The planned method of analysis is by a combination of pde and geometric measure theory techniques. The behavior and structure of interfaces is a topic of basic physical interest, as the organizing principle for a variety of effects seen in nature. For example, soap bubbles are well known to form minimum interfacial energy structures identified by the property that they have minimal surface area. These are also STABLE under small perturbations , by the principle that systems move always toward lower-energy states, in this case back toward the minimal energy configuration. At the same time, the large-scale distribution of matter in the visible universe seems to have a similar structure, with vast voids surrounded by thin layers of galaxies. Apparently, we live on an interface--evidently, also, the minimization of quite different energies can lead to similar structure through a common mathematical mechanism. The examples given above are just two instances of the interfaces known as PHASE BOUNDARIES. Other important interfaces are SHOCK WAVES, or moving boundaries separating substances of very different properties (most usually, a "front" separating air of very different temperature or pressure). Like phase boundaries, these are ubiquitous in nature, from sonic booms to "waves" of concentration in a chemical reaction. Again, it is stability or instability that determines whether a particular shock structure will persist or break up. The mathematics governing stability of interfaces is rather subtle and is by no means completely understood. Indeed, the projects proposed by Zumbrun concern basic theoretical questions that must be answered before we can confidently pursue practical applications such as computer modeling of these phenomena in the variety of settings to which they pertain.