A Micro-Local and Fourier-Analytical Approach to Some Non-Linear Problems in Fluid Mechanics and Elasticity

流体力学和弹性中一些非线性问题的微观局部和傅立叶分析方法

• 批准号: 0405803
• 负责人: Anna Mazzucato
• 金额: $ 11.13万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-01 至 2008-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405803&HistoricalAwards=false
• 关键词: Micro; Local; Fourier; Analytical; Approach

Project Abstract: 0405803 A Mazzucato, Pennsylvania State UniversityA Micro-Local and Fourier-Analytical Approach to Some Non-Linear Problems in Fluid Mechanics and Elasticity The investigator A. L. Mazzucato will address several questions in themathematical investigation of fluid flows and elasticity using methodsfrom Fourier and micro-local analysis. Micro-local analysis seeks toidentify points and directions along which a solution to partialdifferential equations looses regularity, by localizing it in both space and frequency. Modern techniques in Fourier analysis consist indecomposing a signal by testing it against a given set of waves orwave-forms at different length scales, so that relevant information can be extracted accurately and efficiently. Turbulent flows, for example, exhibits a complex behavior at both large and small scales. The coupling between different scales is often due to the non-linearity of the underlying equations. The investigator will concentrate on the following problems. She will study dissipation of enstrophy, the squared vorticity, for the two-dimensional Euler equations, which model inviscid fluid flow, by considering transport by irregular vector fields. Understanding howenstrophy is dissipated is important for two-dimensional turbulence. She will analyze certain weak solutions to the Navier-Stokesequations, which describe the motion of viscous fluids, with globallyinfinite energy by using generalized energy inequalities. Allowing for weaker control at infinity could in turn lead to refined estimates on the local behavior of solutions. She will investigate existence of mild solutions to the Navier-Stokes equations by semi-group methods in polyhedral domains, which are domains of particular interest in numerical simulations. Finally, the investigator will continue studying the inverse problem of unique identification of elastic properties by dynamicsurface measurements, exploiting the covariance of the elasticityequations under coordinate changes. The determination of elasticparameters has significant applications in medical imaging.The present project stresses the inter-disciplinary nature of the analysis of partial differential equations, which mathematically model physical phenomena. Theoretical tools developed to discern subtle properties of these equations have been successfully employed in real-life problems. Micro-local analysis studies how singularities are propagated by differential equations. Changes in material properties cause singularities to form in waves and can hence be determined when direct measurement is not possible, as in seismology, oil exploration, and medical diagnostics. Fourier analysis examines the content of a signal at a given frequency or length scale. Understanding crucial aspects of turbulent flows, forexample concentration and dissipation of energy and vorticity, atdifferent scales has an impact in disciplines ranging from aerodynamics, to meteorology, to human physiology. The need for numerical simulation and design has underlined the role of complex geometries, where a refined mathematical analysis is often necessary for a qualitative understanding. With this project the investigator also aims at strengthening hercollaborative effort with other female researchers both in the UnitedStates and abroad.

项目摘要：0405803宾夕法尼亚州立大学马祖卡托流体力学和弹性力学中一些非线性问题的微局部和傅立叶分析方法研究员A.L.马祖卡托将使用傅立叶和微局部分析方法解决流体流动和弹性数学研究中的几个问题。微局部分析试图通过在空间和频率上对偏微分方程解进行局部化，从而找出偏微分方程解失去正则性的点和方向。傅立叶分析中的现代技术在于通过对信号进行不同长度尺度上的一组给定的波或波形的测试来对信号进行分解，从而准确而有效地提取相关信息。例如，湍流在大尺度和小尺度上都表现出复杂的行为。不同尺度之间的耦合往往是由于基本方程的非线性造成的。调查员将集中精力解决以下问题。她将通过考虑不规则矢量场的输运，为模拟无粘流体流动的二维欧拉方程研究拟能耗散，即平方涡度。了解涡旋是如何消散的对于二维湍流很重要。她将利用广义能量不等式分析描述粘性流体运动的Navier-Stokes方程的某些弱解。允许在无穷远处进行较弱的控制，反过来可能会导致对解决方案的局部行为的精确估计。她将在多面体区域用半群方法研究Navier-Stokes方程温和解的存在性，多面体区域是数值模拟中特别感兴趣的区域。最后，研究人员将继续研究通过动态表面测量唯一识别弹性性质的反问题，利用坐标变化下弹性方程的协方差。弹性参数的确定在医学成像中有重要的应用。本项目强调偏微分方程分析的跨学科性质，它对物理现象进行数学模拟。为了辨别这些方程的微妙性质而开发的理论工具已经成功地应用于现实生活中的问题。微观局部分析研究奇点是如何通过微分方程传播的。材料性质的变化会在波中形成奇点，因此当不可能进行直接测量时，例如在地震学、石油勘探和医学诊断中，可以确定奇点。傅立叶分析在给定的频率或长度范围内检查信号的内容。在不同尺度上理解湍流的关键方面，例如能量和涡度的集中和耗散，在从空气动力学到气象学再到人类生理学的学科中都会产生影响。对数值模拟和设计的需要突显了复杂几何的作用，在这些几何中，精致的数学分析往往是定性理解所必需的。通过这个项目，研究人员还旨在加强她与美国和国外的其他女性研究人员的合作努力。