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 Mazzucato，宾夕法尼亚州立大学的微本地和傅立叶分析方法，用于流体力学和弹性的某些非线性问题。微区分析寻求识别点和方向，通过将其定位在空间和频率中，对partialDifferention方程的解决方案却散发了规律性。傅立叶分析中的现代技术通过在不同的长度尺度上针对给定的波浪形式进行测试，从而使信号的现代技术不合理，从而可以准确有效地提取相关信息。例如，湍流在大小尺度上表现出复杂的行为。不同尺度之间的耦合通常是由于基础方程的非线性引起的。 研究人员将集中精力于以下问题。她将研究二维Euler方程的肠爆散，平方的涡度，该方程模拟了刻薄的流体流，通过考虑不规则矢量场的运输。 了解HOSTROPHY是消散的，对于二维湍流很重要。她将通过使用普遍的能量不平等的全球能量来分析Navier-Stokesequations的某些弱解决方案，这些解决方案描述了粘性流体的运动。允许在无穷大的控制下进行较弱的控制又可能导致对解决方案的局部行为的精致估计。她将通过多面体结构域中的半组方法调查对Navier-Stokes方程的温和溶液的存在，这些域是数值模拟中特别感兴趣的域。 最后，研究者将通过DynamicSurface测量值继续研究弹性性质唯一识别的反问题，从而利用了在坐标变化下的弹性方程的协方差。弹性参数的确定在医学成像中具有重要的应用。目前的项目强调了部分微分方程分析的跨学科性质，而偏微分方程的分析是数学上对物理现象进行模拟的。为辨别这些方程的微妙特性而开发的理论工具已成功地用于现实生活中。微本地分析研究了如何通过微分方程传播奇异性。材料特性的变化会导致在波浪中形成奇点，因此可以在无法直接测量的情况下确定，例如地震学，石油探索和医学诊断。 傅立叶分析以给定频率或长度尺度检查信号的内容。了解湍流的关键方面，大型能量和涡度的耗散量，量表对从空气动力学到气象学到人类生理学等学科具有影响。对数值模拟和设计的需求强调了复杂几何形状的作用，在定性理解中通常需要进行精致的数学分析。通过这个项目，研究人员还旨在加强与美国和国外其他女性研究人员的强力努力。