Strong Shock Waves in Cosmology, General Relativity, and Classical Fluids

宇宙学、广义相对论和经典流体中的强冲击波

• 批准号: 0406096
• 负责人: John Temple
• 金额: $ 18.33万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-09-01 至 2007-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0406096&HistoricalAwards=false
• 关键词: Strong; Shock; Waves; Cosmology; General

This project investigates fundamental open problems in the theory of nonlinear partial differential equations concerning large amplitude or highly oscillatory solutions that generate shock waves. The first part of the project addresses both mathematical and cosmological issues related to shock wave propagation in the Einstein equations of general relativity. The second part of the project addresses unsolved problems regarding oscillatory and resonant interactions of strong waves in classical fluids, particularly the Euler equations for a compressible fluid.Shock waves are steep fronts that develop and propagate in fluids. Mathematical models of such fluid flow involve highly nonlinear differential equations that have solutions with very singular structure, which is difficult to analyze. For large amplitude or highly oscillatory solutions that generate shock waves, there is no general mathematical theory that applies, and many fundamental problems remain unsolved. Such problems are the topics of this research project, which develops the mathematical theory of gravitational shock waves and of shock waves in compressible fluids. The work has potential application to cosmological models of the universe and to modeling of transonic flow for aircraft design.

本计画研究非线性偏微分方程式理论中的基本开放问题，这些问题涉及产生冲击波的大振幅或高振荡解。 该项目的第一部分涉及与广义相对论爱因斯坦方程中的冲击波传播有关的数学和宇宙学问题。 该项目的第二部分解决了经典流体中强波的振荡和共振相互作用的未解决问题，特别是可压缩流体的欧拉方程。冲击波是在流体中发展和传播的陡峭前沿。 这种流体流动的数学模型涉及高度非线性微分方程，其解具有非常奇异的结构，这是难以分析的。 对于产生冲击波的大振幅或高振荡解，没有适用的通用数学理论，许多基本问题仍未解决。 这些问题是本研究项目的主题，该项目发展了引力冲击波和可压缩流体中冲击波的数学理论。 这项工作有潜在的应用宇宙学模型的宇宙和跨音速流动的飞机设计建模。