Topics related to the dynamics of an ideal fluid.

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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.