Stability and dynamics of shock, detonation, and boundary layers

首席研究员建议研究可压缩气体动力学方程的激波和边界层解的稳定性和行为的各种问题，重点是全局(即，与幅度无关)分析；适用于处理时间周期波、离散波和/或动态波的新的泛函分析工具；稳定性和分叉的实用数值测试，特别是对于包含粘性热感、磁感、相变和反应效应的大系统，或者由于真正的多维解的横向模式的傅立叶变换/截断，例如圆柱形管道中的非平面(即，横向或非轴坐标变化)流动。后者有望为冲击波和爆震理论的物理从业者提供感兴趣的定量信息。一个更大的目标是超越简单的稳定性分析，研究包括分叉、相互作用和复杂流动行为在内的非平凡动力学。该项目涉及奇异摄动理论、动力系统和分叉、线性算子的谱理论和非线性偏微分方程的有趣和非标准问题，并应导致新的普遍应用的数学工具的发展。攻击计划围绕着埃文斯函数和最近由研究人员和各种合作者开发的相关光谱技术。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的中心主题，决定了哪些(稳定的)模式通常会被观察到，哪些(不稳定的)只是数学上的，而不是物理上可以观察到的。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)的替代的、更复杂的流型的出现--这是一种通过研究更简单和更好地理解的流型来理解复杂流动的方法。我们的目标是将现有的理论从定性系统转移到定量系统，在我们推进数学理论的同时，获得对实践者有用的新信息。计划开展的活动将加强和扩大现有的跨领域合作网络，并帮助培训研究生和博士后。这些研究的最终目的，即对过渡到不稳定的定量预测，如果实现，将在工程、化学、制造和其他过程中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.