Use of Harmonic Analysis Methods for the Equations of Fluid Motion

该项目应用调和分析技术来研究流体动力学中涉及重要数学问题并同时反映流体行为内在特性的问题。第一个问题是证明具有超耗散的Navier-Stokes方程的部分正则性结果。 这项工作将提供一个上界的Hausdorff维数的一组奇点。 数学技巧涉及Littlewood-Paley算子和伪微分算子。第二个问题考虑了流体运动方程的精细二元模型，该方程描述了沿二元树沿着的可能的能量级联。该项目采用小波来探索这些模型的爆炸场景。 第三个领域的工作建立了部分规律性的结果，涉及粘性应力张量和应变率之间的非线性关系的Navier-Stokes方程的修改，有用的非牛顿流体的建模行为。 数学技术涉及广义能量不等式和微观局部分析。拟议的活动包含对流体动力学问题的跨学科方法。所使用的方法包括谐波分析的复杂技术，最近取得了重要的类似问题的进展。 首席研究员将传播结果的项目之间的纯和应用数学家，她将把结果的研究活动纳入研究生课程。

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.