Applications of Qualitative Analysis of Nonlinear PDE's

9704874 Birnir将利用非线性偏微分方程组的定性理论研究喷气发动机的喘振、失速和颤振动力学。模拟压气机流动的摩尔-格雷泽方程将被用来分析喘振和失速。首先将确定动力学是高维的还是低维的，然后确定导致停滞失速的分叉，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)子环颤振的简化模型，并利用该模型研究了叶片可能出现的颤振模式，进而提出了颤振模式的控制策略。这将首先用数字来完成，然后再用解析来完成。最后，用这些方法研究了一个真实的转子环模型，分析了真实的颤振模式及其分叉，并提出了控制方法。非线性偏微分方程组的定性分析将从数值和解析两个方面扩展到二维或更高维，首先通过研究阻尼波方程和驱动波方程，并应用高维理论。这项研究的目的是帮助设计比当代喷气发动机更轻、更高效和更安全的喷气发动机。这些发动机将用于20世纪的飞机，将使航空旅行更快、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.