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

项目摘要：A. Mazzucato，宾夕法尼亚州立大学流体力学和弹性中的一些非线性问题的微局部和傅立叶解析方法研究员A. L. Mazzucato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的解进行局部化，从而确定其解失去规律性的点和方向。傅里叶分析的现代技术是通过对给定的一组不同长度尺度的波或波形进行测试来分解信号，以便准确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底层方程的非线性。调查员将集中研究以下问题。她将通过考虑不规则矢量场的输运来研究二维欧拉方程的熵耗散，即涡度的平方。了解涡旋是如何耗散的对于二维湍流是很重要的。她将用广义能量不等式分析具有全局无限能量的navier - stokes方程组的弱解，该方程组描述粘性流体的运动。允许在无穷远处较弱的控制反过来可以导致对解的局部行为的精确估计。她将用半群方法研究多面体域中Navier-Stokes方程温和解的存在性，多面体域是数值模拟中特别感兴趣的域。最后，利用坐标变化下弹性方程的协方差，继续研究动力面测量弹性特性唯一识别的反问题。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，偏微分方程是物理现象的数学模型。为识别这些方程的微妙性质而开发的理论工具已成功地应用于实际问题。微局部分析研究奇异点如何在微分方程中传播。材料性质的变化会导致奇点在波中形成，因此在无法直接测量时可以确定奇点，例如在地震学、石油勘探和医学诊断中。傅里叶分析在给定的频率或长度范围内检查信号的内容。了解湍流的关键方面，例如能量和涡度的集中和耗散，在不同的尺度上对从空气动力学、气象学到人体生理学等学科都有影响。对数值模拟和设计的需求强调了复杂几何的作用，其中精细的数学分析通常是定性理解所必需的。通过这个项目，研究者还旨在加强她与美国和国外其他女性研究人员的合作努力。

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.