Applications of Qualitative Analysis of Nonlinear PDE's

9704874 比尔尼尔 将研究喷气发动机的喘振、失速和颤振的动力学 使用非线性偏微分方程（PDE）的定性理论。模拟压缩机气流的Moore-Greitzer方程将用于分析喘振和失速。首先将确定是否 动力学是高维或低维的，然后是导致 确定了滞止失速并开发了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。转子（和定子）环颤振的一个简单模型 的方法，并用于研究可能的颤振模态，然后针对这些模态提出控制策略。这将首先进行数值计算，然后进行分析。最后将这些方法用于研究一个真实的转子环模型和真实的颤振模态及其分叉 分析并制定控制方法。 非线性偏微分方程的定性分析将扩展到二维和更大的，数值和分析，首先通过研究阻尼 和驱动波方程，以及高维理论的应用 将予以实施。 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。 这些发动机将用于世纪的飞机，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.