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.

小行星9403871 非线性波在科学和工程中的偏微分方程系统中起着重要的作用。拟议的工作的主要目标是开发有效的方法来了解许多这样的波的稳定性。提出者和合作者最近介绍了一些有前途的新方法来研究孤立水波简化模型的稳定性。我们计划将这些方法扩展到研究各种重要系统中的波，包括完整的水波方程和粒子物理和相变的简化模型。第二个主要的研究课题将是在零重力和1重力条件下，研究液-气临界点附近流体的低速流动。高压缩性和异常的物理性质是这种状态的特征，在航天器和地球上的实验中观察到了一些无法解释或只能部分解释的现象。 拟议的稳定性分析涉及使用和进一步发展的几个数学工具，包括埃文斯函数，和克莱因的扰动行列式。两者都是解析函数，其零点对应于本征值或谐振极点。利用线性稳定性分析来证明色散方程孤立波的非线性稳定性将得到进一步发展。此外，一种新的方式重新制定的哈密顿系统，使约束成为守恒量，将调查：水波，弯曲的电缆在零G，和各向异性材料是“刚性”在一个方向。应用数学中关于燃烧中的“零马赫数流动”的最新发展将适用于近临界纯流体的研究。