Applications of Qualitative Analysis of Nonlinear PDE's

使用非线性偏微分方程（PDE）的定性理论研究喷气发动机的喘振、失速和颤振动力学。模拟压气机流动的Moore-Greitzer方程将用于分析喘振和失速。首先确定动力学是高维还是低维，然后确定导致停滞失速的分岔，并开发控制失速和喘振的方法。研究结果将应用于实验室压缩机的研究和实验室数据的分析。建立了叶片动（静）环颤振的简单模型，用于研究叶片动（静）环颤振可能存在的模态，进而制定相应的控制策略。首先用数值方法，然后用分析方法。最后，这些方法将用于研究转子环的真实模型和实际颤振模式，分析其分岔并制定控制方法。非线性偏微分方程的定性分析将首先通过研究阻尼和驱动波动方程，在数值和解析上扩展到二维及更大的维度，并实现高维理论的应用。这项研究的目的是帮助设计出比当代喷气发动机更轻、更有效、更安全的喷气发动机。这些发动机将用于20世纪的飞机上，并将使空中旅行更快、更便宜、更安全。

9704874 Birnir The dynamics of surge, stall, and flutter in jet engines will be investigated using the qualitative theory of nonlinear partial differential equations ( PDE's). The Moore-Greitzer equations modelling the flow through a compressor will be used to analyze surge and stall. First it will be determined whether the dynamics are high or low-dimensional, then the bifurcation that causes stagnation stall identified and methods developed to control stall and surge. The results will be applied to the study of laboratory compressors and the analysis of laboratory data. A simple model for flutter in rotor (and stator) rings of blades is developed and used to study the possible flutter modes, then control strategies for these modes will be developed. This will first be done numerically and then analytically. Finally these methods will be used to study a realistic model of rotor rings and the real flutter modes and their bifurcations analyzed and methods developed to control them. The qualitative analysis of nonlinear PDE's will be extended to dimensions two and greater, numerically and analytically, first by studying damped and driven wave equations, and applications of the higher dimensional theory will be implemented. The purpose of the research is to help design jet engines that are lighter and can be operated more efficiently and safely than contemporary jet engines. These engines will be used in the airplanes of the 20th century and will make air travel faster, cheaper, and safer.