Stability of compressible flow in real media

主要研究者：Kevin Zumbrun，印第安纳州大学DMS-0300487摘要首席研究员提出研究“真实”介质中可压缩流动的稳定性，其特征是经常被忽视的影响，如粘度，热传导，电磁动力学，相变，非热平衡和化学反应，在物理上有趣的（通常是大振幅）区域，在该区域可能发生向不稳定性的转变：例如，强激波和爆震波的多维稳定性，或经典剪切流的多维稳定性。这涉及奇异摄动理论、动力系统和分叉、线性算子谱理论和非线性偏微分方程中有趣的非标准问题，并应导致新的数学工具的发展。物理学的最终目标是对稳定性现象的理解，这种理解比在简化模型中所能获得的更完整、更精确：一方面在数学基础的层面上解决哲学难题，另一方面在实际应用的层面上产生定量预测。攻击计划围绕埃文斯函数和最近在粘性激波阵面稳定性研究中发展起来的相关谱技术展开。规则流型的稳定性是流体、气体和等离子体动力学中一个古老而核心的课题，它决定了哪些（稳定的）流型会被典型地观察到，哪些（不稳定的）流型仅仅是数学上而不是物理上可观察到的解。 从稳定性到不稳定性的转变是特别重要的，因为它通常标志着接近原始（现在不稳定）流型的替代的、更复杂的流型的出现--这是通过研究更简单和更好理解的流型来理解复杂流动的一种方式。 尽管关于这个问题的大量和众所周知的理论可以追溯到1800年代后期，但仍然有许多方面知之甚少，特别是对于压缩流、粘性流或反应流。 在这里，我们建议研究这些问题中出现的可压缩气体和等离子体动力学，并在燃烧中，这种通常被忽视的影响是相当重要的实际应用。 我们的目标是，通过包括这些数学上有问题的条款，将现有的理论从定性到定量的制度，获得新的信息使用的从业者在同一时间，我们推进的数学理论。计划中的活动既有分析部分，也有数字部分，并涉及与国内外同事以及现任和前任研究生和博士后的合作。预计这将加强和扩大跨领域和跨机构的现有合作网络，并协助培训研究生和博士后。 这些研究的最终目的，即定量预测向不稳定性的转变，如果能够实现，将在工程、化学、制造和其他过程的水平上具有直接和实际的用途。

PI: Kevin Zumbrun, Indiana UniversityDMS-0300487ABSTRACTThe principal investigator proposes to study stability of compressible flows in ``real'' media featuring often-neglected effects such as viscosity, heat conduction, electromagneticdynamics, phase-transition, non-thermoequilibrium, and chemical reaction, in the physically interesting (usually large-amplitude) regime where transition to instability may be expected to occur: for example, multidimensional stability of strong shock and detonation waves, or of classical shear flows. This involves interesting and nonstandard issues in singular perturbation theory, dynamical systems and bifurcation, spectral theory of linear operators, and nonlinear partial differential equations, and should result in the development of new mathematical tools of general application. The ultimate physical goal is an understanding of stability phenomena that is both more complete and more precise than can be obtained within simplified models: on the one hand resolving philosophical puzzles at the level of mathematical foundations and on the other yielding quantitative predictions at the level of practical application. The plan of attack centers around Evans function and related spectral techniques developed recently in the study of stability of viscous shock fronts.The stability of regular flow patterns is an old and central topic in fluid, gas, and plasma dynamics, deciding which (stable) patterns will typically be observed, and which (unstable) are only mathematical and not physically observable solutions. The transition from stability to instability is of particular importance, since it usually signals the arising of alternative, more complicated flow patterns close to the original (now unstable) one- this is a way to understand complicated flows by the study of simpler and better-understood ones. Despite a large and well-known body of theory on this subject, dating back to the late 1800's, there are still many aspects that are poorly understood, particularly for compressive, viscous, or reacting flows. Here, we propose to study several of these issues arising in compressible gas and plasma dynamics, and in combustion, applications in which such usually neglected effects are of considerable practical importance. Our goal is, by including these mathematically problematic terms, to move existing theory from the qualitative to the quantitative regime, obtaining new information of use to practitioners at the same time that we advance the mathematical theory. The planned activities have both analytic and numerical components, and involve collaboration with domestic and foreign colleagues and with current and former graduate students and post doctorates. This may be expected to strengthen and extend existing networks of cooperation across field and institution, and to aid in training of graduate and postdoctoral students. The ultimate aim of these investigations, of quantitative predictions of transition to instability, would, if achieved, be of direct and practical use at the level of engineering, in chemical, manufacturing, and other processes.