Hydrodynamic Stability in viscous, compressible flow

本文的主要研究者提出了几个关于可压缩、粘性和反应介质中流动的多维稳定性的研究项目。 这些包括剪切流的经典流体动力学稳定性和压缩流的冲击波和燃烧理论，前者表现出局部对称平行于和后者垂直于流动。 在这些问题中统一的数学主题是多个长度尺度对应于小规模的运输和大规模的对流效应，与相关的“刚度”的线性化扰动问题的外观。 这导致了有趣的，非标准的问题，在谱和半群理论。 同时，从物理应用的角度来看，包含小尺度输运效应是非常可取的，这通常发生在这些效应可能被预期为显著的尺度上。规则流模式的稳定性是流体、气体和等离子体动力学中的一个古老而核心的主题，决定了哪些（稳定）模式通常会被观察到，而这些（不稳定的）只是数学上的，而不是物理上可观察到的解决方案。 从稳定性到不稳定性的转变是特别重要的，因为它通常标志着替代的，更复杂的流动模式接近原始（现在不稳定）的模式-这是一种通过研究更简单和更好理解的流动来理解复杂流动的方法。 尽管在这个问题上有一个大的和众所周知的理论体系，可以追溯到19世纪后期，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，我们建议研究可压缩气体和等离子体动力学中出现的这些问题中的几个，在燃烧中，这种通常被忽视的effectsare相当实际的重要性的应用。 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，我们推进的理论。

ABSTRACTThe principal investigator proposes several projects concerning multi-dimensional stability of flows in compressible, viscous, and reacting media. These include both shear flows of classical hydrodynamic stability and compressive flows of shock wave and combustion theory, the former exhibiting local symmetry parallel to and the latter normal to the flow. The unifying mathematical theme in these problems is the appearance of multiple length scales corresponding to small-scale transport and large-scale convective effects, with associated ``stiffness'' in the linearized perturbation problem. This leads to interesting, nonstandard issues in spectral and semigroup theory. At the same time, the inclusion of small-scale transport effects is highly desirable from the point of view of physical applications, which often occur at scales where these effects might be expected to be significant.The stability of regular flow patterns is an old and central topic in fluid, gas, and plasma dynamics, deciding which (stable) patterns will typicallybe observed, and which (unstable) are only mathematical and not physicallyobservable solutions. The transition from stability to instabilityis of particular importance, since it usually signals the arisal ofalternative, more complicated flow patterns close to the original(now unstable) one- this is a way to understand complicated flowsby the study of simpler and better-understood ones. Despite a largeand well-known body of theory on this subject, dating back to the late 1800's, there are still many aspects that are poorly understood, particularlyfor compressive, viscous, or reacting flows. Here, we propose to studyseveral of these issues arising in compressible gas and plasma dynamics,and in combustion, applications in which such usually-neglected effectsare of considerable practical importance. Our goal is, by includingthese mathematically problematic terms, to move existing theory fromthe qualitative to the quantitative regime, obtaining new informationof use to practitioners at the same time that we advance the mathematicaltheory.