Topics related to the dynamics of an ideal fluid.

欧拉方程是一组描述无粘流体运动的偏微分方程组，是一个极具挑战性的系统。弗里德兰德研究了与这些方程相关的问题，这些问题涉及重要的数学问题，同时也反映了流体行为的基本性质。Friedlander和Pavlovic研究了一个无限的非线性耦合常微分方程组，它提供了欧拉方程的一个更简单的模型。他们使用各种工具来检验该模型的“贴近性”，他们证明了该模型存在有限时间奇点，即欧拉方程。在另一项单独的研究中，弗里德兰德和她的合作者继续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，目的是在一般的二维和三维流的能量范数下完整地描述这种谱的结构。这一谱不仅检测到线性意义上的不稳定性，而且与完全非线性系统从稳定到不稳定的自然物理问题密切相关。研究人员和她的合作者使用了相当广泛的数学技术来进行这项研究，包括渐近方法、谱理论、算子半群理论、动力系统和调和分析。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：稳定的流动在不可避免的环境扰动下是稳健的，而不稳定的流动可能会破裂，有时甚至是猛烈的。尽管这个主题已经成为一个多世纪以来密集研究的主题，因为它与工程学、物理学、海洋学和气象学等许多科学分支有关，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其中一些问题，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。一名学生参与了这个项目，调查人员继续开展各种指导活动，鼓励女性进入数学职业。

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.