Stability and dynamics of shock, detonation, and boundary layers

主要研究者建议研究各种问题的稳定性和行为的冲击和边界层类型的解决方案的方程的可压缩气体动力学，重点是全球（即，振幅无关）分析;新的功能分析工具，适用于治疗的时间周期性，离散，和/或动力波;和实际的数值测试的稳定性和分叉，特别是对于大型系统所产生的包括粘性热和磁或电磁感应，相变，和反应的影响，或通过傅立叶变换/截断在横向模式的真正多维的解决方案，如非平面（即，横向或非轴向坐标变化）在圆柱形管道中流动。后者预计将提供感兴趣的物理从业者在冲击和爆炸理论的定量信息。一个更大的目标是超越简单的稳定性分析，研究非平凡的动力学，包括分叉，相互作用和复杂流动的行为。该项目涉及奇异摄动理论，动力系统和分叉，线性算子谱理论和非线性偏微分方程的有趣和非标准问题，并应导致新的数学工具的发展的一般应用。攻击计划围绕埃文斯函数和相关的光谱技术最近开发的调查员和各种合作者。稳定性的定期流动模式是一个古老的和中心的话题，在流体，气体和等离子体动力学，决定哪些（稳定）模式通常会被观察到，哪些（不稳定）只是数学和物理上不可观察的解决方案。 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出现，更复杂的流动模式接近于原始（现在不稳定）的流动模式-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式。 我们的目标是将现有的理论从定性转向定量，在推进数学理论的同时获得对实践者有用的新信息。 计划开展的活动预计将加强和扩大现有的跨领域合作网络，并协助培训研究生和博士后学生。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。

The principal investigator proposes to study a variety of problems in stabilityand behavior of shock and boundary layer type solutions of the equations of compressible gas dynamics, with an emphasis on global (i.e., amplitude-independent) analysis; new functional-analytic tools suitable for treatment of time-periodic, discrete, and or kinetic waves; and practical numerical testing of stability and bifurcation, especially for the large systems resulting from inclusion of viscous heat- and magnetic- or electromagnetic-inductive, phase-transitional, and reactive effects, or by Fourier transform/truncation in transverse modes of genuinely multidimensional solutions such as nonplanar (i.e., varying in transverse, or nonaxial, coordinates) flow in a cylindrical duct. The latter is expected to provide quantitative information of interest to physical practitioners in shock and detonation theory. A larger goal is to move beyond simple stability analysis to the study of nontrivial dynamics including bifurcation, interaction, and behavior of complex flows. The project 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 plan of attack centers around Evans function and related spectral techniques developed recently by the investigator and various collaborators.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 appearance 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. Our goal is 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 expected to strengthen and extend existing networks of cooperation across field, 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.