Use of Harmonic Analysis Methods for the Equations of Fluid Motion

调和分析方法在流体运动方程中的应用

• 批准号: 0304594
• 负责人: Natasa Pavlovic
• 金额: $ 10.68万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0304594&HistoricalAwards=false
• 关键词: Use; Harmonic; Analysis; Methods; Equations

This project applies techniques of harmonic analysis to investigate problems in fluid dynamics that involve important mathematical questions and at the same time reflect intrinsic properties of fluid behavior. The first problem is related to proving partial regularity results for the Navier-Stokes equations with hyper-dissipation. The work will provide an upper bound on the Hausdorff dimension of the set of singular points. Mathematical techniques involve Littlewood-Paley operators and pseudodifferential operators. The second problem considers refined dyadic models for the equations of fluid motion that describe a possible cascade of energy along the dyadic tree. The project employs wavelets to explore blow-up scenarios for these models. The third area of work establishes partial regularity results for a modification of the Navier-Stokes equations involving a nonlinear relation between the viscous stress tensor and the rate of strain, useful in modeling behavior of non-Newtonian fluids. The mathematical techniques involve generalized energy inequalities and microlocal analysis.The proposed activity contains an interdisciplinary approach to questions arising from fluid dynamics. Methods used include sophisticated techniques of harmonic analysis that recently yielded progress on important similar questions. The principal investigator will disseminate results of the project among both pure and applied mathematicians, and she will incorporate results of the research activity into a graduate course.

该项目应用调和分析技术来研究流体动力学问题，这些问题涉及重要的数学问题，同时反映流体行为的内在特性。第一个问题与证明具有超耗散的纳维-斯托克斯方程的部分正则性结果有关。 这项工作将提供奇异点集的豪斯多夫维数的上限。 数学技术涉及 Littlewood-Paley 算子和伪微分算子。第二个问题考虑流体运动方程的精细二元模型，该模型描述了沿二元树可能的能量级联。该项目使用小波来探索这些模型的爆炸场景。 第三个工作领域为纳维-斯托克斯方程的修正建立了部分正则性结果，涉及粘性应力张量和应变率之间的非线性关系，可用于非牛顿流体的行为建模。 数学技术涉及广义能量不等式和微局域分析。拟议的活动包含解决流体动力学问题的跨学科方法。使用的方法包括复杂的谐波分析技术，这些技术最近在重要的类似问题上取得了进展。 首席研究员将向纯粹数学家和应用数学家传播该项目的结果，并将研究活动的结果纳入研究生课程。