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

I. 粘性守恒定律中波的稳定性。

• 批准号: 9706842
• 负责人: Kevin Zumbrun
• 金额: $ 8.06万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-15 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706842&HistoricalAwards=false
• 关键词: I.; Stability; Waves; Viscous; Conservation

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.

9107990 尊布伦 Kevin Zumbrun提出了几个项目， 从流体动力学。 这些涉及结构和 奇异界面解的稳定性 极限问题，特别是冲击波和相位 边界 稳定性是休克的中心话题 波动理论，与物理等问题有关， 双曲波的容许性，收敛性， 差分格式和无粘极限问题。 Zumbrun建议研究各种相关问题， 从多维粘性冲击波的稳定性 在可压缩Navier-Stokes方程中， 振荡激波层在非局部， 分散沉降模型计划的方法 分析包括逐点分析、绿色分析、 功能技术，已被证明是有用的， 其他情况下的微妙稳定性，光谱分析 利用矩阵微扰理论和Evans函数 从反应扩散研究中借用的技术 战线 同样，相边界的结构是一个 相变研究的中心课题。 在 过渡层厚度为零的极限 相变模型简化到理想化最小 表面问题，和相边界，以最小 表面。 这种联系是直觉的丰富来源， 提出了几何学和PDE中的新问题 设置. Zumbrun提出了其中的几个供研究， 最值得注意的是“诺依曼”解的规律性 最小表面问题。 计划方法 的分析是通过结合偏微分方程和几何 测量理论技术 界面的行为和结构是一个基本的物理兴趣的主题，作为组织原则， 在自然界中看到的各种效应。 比如说， 众所周知， 能源结构的性质，他们确定， 具有最小的表面积。 这些也是稳定的， 根据系统运动的原理， 总是朝向低能态，在这种情况下，回到最小能量构型。 同时对 可见宇宙中物质的大尺度分布 似乎也有类似的结构， 被薄薄的星系层覆盖显然，我们生活在 接口--显然， 不同的能量可以通过一个 常见的数学机制 上面给出的例子只是PHASE接口的两个实例 边界 其他重要的接口是SHOCK WAVES， 或移动的边界， 属性（最常见的是，一个“前”分离空气非常 不同的温度或压力）。 就像相位边界， 这些在自然界中无处不在，从音爆到“波”， 在化学反应中的浓度。 再次是 稳定性或不稳定性，决定了一个特定的冲击结构将继续存在还是破裂。 数学 控制界面的稳定性相当微妙， 还没有完全被理解。 事实上，项目 Zumbrun提出的关于基本理论问题 在我们能够自信地 实际应用，如计算机模拟这些 在他们所处的各种环境中的现象。