Mathematical Sciences: "The Wave Map Program: Toward a Theory of Regularity and Break-down in Classical Nonlinear Fields"

数学科学：“波图程序：走向经典非线性场中的规律性和分解理论”

• 批准号: 9504919
• 负责人: Abdolreza Tahvildar-Zadeh
• 金额: $ 4万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9504919&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Wave; Map; Program

9504919 Tahvildar-Zadeh The proposed research is in the area of mathematical physics and deals specifically with regularity and stability theory of wave maps. Wave maps are a hyperbolic analogue of harmonic maps, where the domain is a Lorentzian manifold rather than a Riemannain manifold. When one considers the energy functional for maps from a Lorentzian manifold to a Riemannian manifold, the resulting Euler-Lagrange equations become hyperbolic - for maps between two Riemannian manifolds these equations are elliptic. Wave maps are called sigma models by physicists, and they arise in modern physics in various guises. For example, Einstein's field equations can be reduced to sigma model equations. Not much general theory has been developed for wave maps thus far as hyperbolic partial differential equations are mathematically harder to deal with than the elliptic ones.

9504919 Tahvildar-Zadeh提出的研究是在数学物理学领域，专门涉及波图的规律性和稳定理论。波图是谐波图的双曲线类似物，其中域是洛伦兹的流形，而不是riemannain歧管。当人们认为从洛伦兹歧管到riemannian歧管的地图的能量函数时，所得的欧拉 - 拉格朗日方程变为双曲线 - 对于两个riemannian歧管之间的地图，这些方程是椭圆形的。 波浪图被物理学家称为Sigma模型，它们以各种阵利的现代物理形式出现。例如，爱因斯坦的场方程可以简化为Sigma模型方程。由于双曲线部分微分方程在数学上比椭圆形更难处理，因此对于波图的一般理论并不多。