Mathematical Sciences: Waves and Stability in Nonlinear Partial Differential Equations

数学科学：非线性偏微分方程中的波和稳定性

• 批准号: 9403871
• 负责人: Robert Pego
• 金额: $ 9.24万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-06-01 至 1998-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9403871&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Waves; Stability; Nonlinear

9403871 Pego Nonlinear waves play a significant role in systems of partial differential equations of importance in science and engineering. A primary goal of the proposed work is to develop effective methods for understanding the stability properties of many such waves. The proposer and collaborators have recently introduced some promising new methods for studying the stability of simplified models of solitary water waves. We plan to extend these methods to study waves in various systems of importance, including the full water wave equations and simplified models of particle physics and phase transitions. A second major topic of investigation will be the study of low velocity flow in fluids near the liquid-gas critical point, both in zero-G and 1-G conditions. High compressibility and anomalous physical properties characterize this regime, and several phenomena have been observed in experiments in spacecraft and on earth which are unexplained or only partly explained. The proposed stability analyses involve the use and further development of several mathematical tools, including the Evans function, and Krein's perturbation determinant. Both are analytic functions whose zeros correspond to eigenvalues or resonance poles. The use of linear stability analysis to prove nonlinear stability will be further developed for solitary waves of dispersive equations. Also, a new way of reformulating Hamiltonian systems, so that constraints become conserved quantities, will be investigated for: water waves, flexure of cables in zero-G, and anisotropic materials that are `stiff' in one direction. Recent developments in applied mathematics regarding `zero-Mach number flows' in combustion will be adapted and extended for studying near-critical pure fluids.

9403871 PEGO非线性波在科学和工程中重要性偏微分方程的系统中起重要作用。拟议工作的主要目标是开发有效的方法来了解许多此类波浪的稳定性。提议者和合作者最近引入了一些有希望的新方法，用于研究简化的单型水浪模型的稳定性。我们计划将这些方法扩展到研究各种重要性系统中的波，包括完整的水波方程以及简化粒子物理和相变的模型。研究的第二个主要主题将是研究液体加速度临界点附近流体中低速流的研究，无论是在零G和1-G条件下。高压和异常的物理特性表征了这种制度，并且在航天器和地球上的实验中观察到了几种现象，这些现象无法解释或仅部分解释。 提出的稳定性分析涉及几种数学工具的使用和进一步开发，包括Evans函数和Kerin的扰动决定因素。两者都是分析功能，其零对应于特征值或共振电线。使用线性稳定性分析来证明非线性稳定性将进一步开发用于分散方程的孤立波。此外，将研究一种新的重新设计汉密尔顿系统的方法，以使限制变为保守的数量，将研究：水波，零G中的电缆弯曲以及各向方向上“僵硬”的各向异性材料。关于燃烧中有关“零摩托车数”流的应用数学的最​​新发展将被调整并扩展，以研究近乎临界的纯流体。