Topics related to the dynamics of an ideal fluid.

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

The Euler equations, a set of partial differential equations that describe the motion of an inviscid fluid, are an exceptionally challenging system. Friedlander studies issues related to these equations that involve important mathematical problems and at the same time reflect basic properties of fluid behaviour. Friedlander and Pavlovic examine an infinite system of nonlinearly coupled ordinary differential equations that provide a simpler model of the Euler equations. They use a variety of tools to examine the "closeness" of the model, for which they prove the existence of finite time singularities, to the Euler equations. In a separate line of research Friedlander and her collaborators continue a project that examines the stability and instability of fluid configurations. They study the unstable spectrum of the Euler equation with the goal of a complete description of the structure of this spectrum over the energy norm for generic two- and three-dimensional flows. This spectrum detects not only instability in the linear sense but also is closely related to natural physical questions about the transition from stability to instability for the full nonlinear system. The investigator and her collaborators use a considerable range of mathematical techniques to carry out this research, including asymptotic methods, spectral theory, operator semi-group theory, dynamical systems, and harmonic analysis. The issue of stability or instability of a fluid flow is one of the central problems in fluid dynamics: stable flows are robust under inevitable disturbances in the environment, while unstable flows may break up, sometimes violently. Even though the topic has been the subject for intense study over more than a century because of its connection with many branches of science, such as engineering, physics, oceanography, and meteorology, many questions remain open. Friedlander uses mathematical techniques to answer some of these questions and her work shows that in some appropriate sense almost all fluid flows are unstable, although there are a number of different types of such instability. A student is involved in the project, and the investigator continues various mentoring activities to encourage women to enter careers in mathematics.