Hydrodynamic Stability in viscous, compressible flow

摘要主要研究人员提出了可压缩介质、粘性介质和反应介质中流动的多维稳定性问题。这既包括经典水动力稳定性的剪切流，也包括激波和燃烧理论的压缩流，前者表现出与流动平行的局部对称性，后者表现出与流动垂直的局部对称性。这些问题中统一的数学主题是对应于小尺度输运和大尺度对流效应的多重长度尺度的出现，在线性化扰动问题中具有相关的“刚度”。这导致了谱和半群理论中有趣的、非标准的问题。与此同时，从物理应用的角度来看，包括小规模运输影响是非常可取的，因为这些影响往往发生在这些影响可能会很大的尺度上。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话题，它决定了哪些（稳定）模式将被典型地观察到，哪些（不稳定）模式只是数学上的而不是物理上可观察到的解决方案。从稳定到不稳定的转变是特别重要的，因为它通常标志着替代的出现，更复杂的流动模式接近原始的（现在不稳定的）——这是通过研究更简单和更好理解的流动来理解复杂流动的一种方法。尽管在这个问题上有大量著名的理论，可以追溯到19世纪后期，但仍然有许多方面没有得到很好的理解，特别是压缩流、粘性流或反应流。在这里，我们建议研究在可压缩气体和等离子体动力学中出现的几个问题，以及在燃烧应用中，这些通常被忽视的效应具有相当大的实际重要性。我们的目标是，通过包括这些数学上有问题的术语，将现有的理论从定性转移到定量制度，在我们推进数学理论的同时获得对实践者有用的新信息。

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.