Mathematical Sciences: Smoothing of Dispersive Waves

数学科学：色散波的平滑

• 批准号: 9204510
• 负责人: Thomas Kappeler
• 金额: --
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-08-15 至 1994-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9204510&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Smoothing; Dispersive; Waves

The primary area of research supported by this award is partial differential equations, especially those used to model wave phenomena. Work includes the investigation of smoothing properties for a large class of linear and nonlinear dispersive evolution equations. Dispersive waves are characterized by the property that the group velocity depends in a nontrivial way on the wave number. An example would be Korteweg deVries equation. The smoothing phenomenon refers to the observed behavior of solutions of certain equations - they are smoother than their initial conditions. The approach to this work is the use of the commutator method together with scattering theory. One needs microlocal versions of Sobolev weighted norms to replace the standard norms which do not persist with time. Other research directions include regularized determinants: the study of variational formulas for the regularized determinants of elliptic operators on Riemannian manifolds of dimension two; global aspects of completely integrable Hamiltonian systems and the Schrodinger operator on tori and Heisenberg manifolds. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them and validate methods for expressing solutions.

该奖项支持的主要研究领域是 偏微分方程，特别是那些用来模拟 波动现象 工作包括调查平滑 性质的一大类线性和非线性色散 演化方程 色散波的特征在于 群速度以非平凡的方式依赖于 波数。 一个例子是Korteweg DeVries方程。 平滑现象是指观察到的 某些方程的解-它们比它们的 初始条件。 这项工作的方法是使用 交换子方法和散射理论。 人需要 微局部版本的Sobolev加权范数，以取代 标准规范不会随着时间而持续存在。 其他研究 方向包括正规化的决定因素：研究 椭圆型方程正则化行列式的变分公式 二维黎曼流形上的算子;整体 完全可积的哈密顿系统和 环面和海森堡流形上的薛定谔算子。 偏微分方程是 物理科学中的数学建模。 的现象 包括连续变化，如运动、材料 和能量都服从某些普遍规律， 可以用相互作用和关系来表达 偏导数之间的关系 数学的关键作用不是 来陈述这些关系，而是为了提取定性的 并从它们的定量意义和验证方法 表达解决方案。