Hydrodynamic Stability in viscous, compressible flow

主要研究人员提出了几个关于可压缩、粘性和反应性介质中流动的多维稳定性的方案。这些理论既包括经典流体动力稳定性的剪切流动，也包括激波压缩流动和燃烧理论，前者表现出平行于流动的局部对称性，而后者表现为垂直于流动的局部对称性。这些问题的统一数学主题是出现与小尺度输送和大尺度对流效应相对应的多个长度尺度，并在线性化扰动问题中与之相关联的“刚性”。这导致了谱和半群理论中有趣的、非标准的问题。同时，从物理应用的角度来看，小尺度输运效应的包含是非常可取的，这通常发生在这些影响可能显著的尺度上。规则流型的稳定性是流体、气体和等离子体动力学中的一个古老而中心的话题，它决定了哪些(稳定的)模式通常会被观察到，而哪些(不稳定的)只是数学上的而不是物理上可观察到的解。从稳定到不稳定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)的更复杂的替代流型的出现--这是通过研究更简单和更好地理解的流型来理解复杂流动的一种方法。尽管早在1800年末S就有大量的、众所周知的关于这个问题的理论，但仍然有许多方面是鲜为人知的，特别是关于压缩、粘性或反应流动的。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，以及在燃烧中的应用，在这些应用中，这种通常被忽略的影响具有相当大的实际意义。我们的目标是，通过纳入这些在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.