Topics related to the dynamics of an ideal fluid.

与理想流体动力学相关的主题。

• 批准号: 0503768
• 负责人: Susan Friedlander
• 金额: $ 12.53万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0503768&HistoricalAwards=false
• 关键词: Topics; related; dynamics; ideal; fluid.

The Euler equations, a set of partial differential equationsthat describe the motion of an inviscid fluid, are anexceptionally challenging system. Friedlander studies issuesrelated to these equations that involve important mathematicalproblems and at the same time reflect basic properties of fluidbehaviour. Friedlander and Pavlovic examine an infinite system ofnonlinearly coupled ordinary differential equations that provide asimpler model of the Euler equations. They use a variety of toolsto examine the "closeness" of the model, for which they prove theexistence of finite time singularities, to the Euler equations. In a separate line of research Friedlander and her collaboratorscontinue a project that examines the stability and instability offluid configurations. They study the unstable spectrum of theEuler equation with the goal of a complete description of thestructure of this spectrum over the energy norm for generic two-and three-dimensional flows. This spectrum detects not onlyinstability in the linear sense but also is closely related tonatural physical questions about the transition from stability toinstability for the full nonlinear system. The investigator andher collaborators use a considerable range of mathematicaltechniques to carry out this research, including asymptoticmethods, spectral theory, operator semi-group theory, dynamicalsystems, and harmonic analysis. The issue of stability or instability of a fluid flow is oneof the central problems in fluid dynamics: stable flows are robustunder inevitable disturbances in the environment, while unstableflows may break up, sometimes violently. Even though the topichas been the subject for intense study over more than a centurybecause of its connection with many branches of science, such asengineering, physics, oceanography, and meteorology, manyquestions remain open. Friedlander uses mathematical techniquesto answer some of these questions and her work shows that in someappropriate sense almost all fluid flows are unstable, althoughthere are a number of different types of such instability. Astudent is involved in the project, and the investigator continuesvarious mentoring activities to encourage women to enter careersin mathematics.

欧拉方程是一组描述无粘流体运动的偏微分方程，是一个非常具有挑战性的系统。 Friedlander研究的问题涉及到这些方程，涉及重要的物理问题，同时反映了流体行为的基本性质。 Friedlander和Pavlovic研究了一个无限系统的非线性耦合常微分方程，提供了一个更简单的模型的欧拉方程。 他们使用各种各样的工具来检查模型的“接近性”，为此他们证明了有限时间奇点的存在，欧拉方程。在另一项研究中，弗里德兰德和她的合作者继续进行一个项目，研究流体结构的稳定性和不稳定性。 他们研究欧拉方程的不稳定谱，目标是完整描述该谱在一般二维和三维流动能量规范上的结构。 该谱不仅检测线性意义上的不稳定性，而且与完全非线性系统从稳定性向不稳定性转变的自然物理问题密切相关。 研究者和她的合作者使用了相当广泛的分析技术来进行这项研究，包括渐近方法，谱理论，算子半群理论，动力系统和谐波分析。 流体流动的稳定性或不稳定性问题是流体动力学的核心问题之一：稳定的流动在环境中不可避免的扰动下是鲁棒的，而不稳定的流动可能会破裂，有时是剧烈的。 尽管这个主题由于它与许多科学分支，如工程学、物理学、海洋学和气象学的联系而成为一个多世纪以来深入研究的主题，但许多问题仍然悬而未决。 弗里德兰德使用数学技术来回答这些问题，她的工作表明，在某种适当的意义上，几乎所有的流体流动都是不稳定的，尽管有许多不同类型的不稳定性。 一名学生参与了该项目，调查员继续开展各种辅导活动，鼓励妇女进入数学职业。