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拟议的研究是在数学物理领域，具体涉及波图的正则性和稳定性理论。波映射是调和映射的双曲类比，其中区域是洛伦兹流形而不是黎曼流形。当考虑从洛伦兹流形到黎曼流形的映射的能量泛函时，所得到的欧拉-拉格朗日方程是双曲的--对于两个黎曼流形之间的映射，这些方程是椭圆型的。波图被物理学家称为西格玛模型，它们以各种形式出现在现代物理学中。例如，爱因斯坦的场方程可以简化为西格玛模型方程。到目前为止，对于波图还没有开发出太多的一般理论，因为双曲型偏微分方程组在数学上比椭圆型偏微分方程组更难处理。