Topics in Mathematical Fluid Dynamics

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

DMS Award Abstract Award #: 0202767 PI: Friedlander, Susan Institution: University of Illinois, Chicago Program: Applied Mathematics Program Manager: Catherine Mavriplis Title: Topics in Mathematical Fluid Dynamics This project uses a variety of techniques in nonlinear partial differential equations to study several different open questions connected with the equations that describe the motion of a fluid, namely the Euler and the Navier-Stokes equations. Stability of a fluid flow is one of the most basic problems in fluid dynamics: stable flows are robust under the inevitable disturbances in the environment, while unstable flows may break up, sometimes rapidly. The investigator will continue to explore the relations between different types of instability with the goal of defining scales or degrees of instability. Sufficient conditions are sought to demonstrate nonlinear instability for classes of inviscid fluids. Another topic studied in the project is the equation for a viscous fluid when there is a nonlinear relation between the stress and strain tensors (a so-called non-Newtonian fluid ). A fundamental mathematical question asked for any of the fluid equations is the possibility of the development of singularities in finite time. Motivated by results from a dyadic model for the non-Newtonian equations, the investigator and collaborators use techniques of wavelets and Littlewood -Paley theory to give an upper bound on the dimension of the singular set in the case of the Navier Stokes equations with nonlinear viscosity. The greater portion of our world is composed of fluids: e.g., the atmosphere, the oceans, even our own bodies. However fluids behave in very complex ways that are presently understood only to a very minor degree. This project uses rigorous mathematics to examine several questions that are fundamantal to the nature of fluid motion. These include the stability or instability of a fluid configuration. In the view of many scientists, waves and instabilities lie at the heart of long term weather prediction with all its practical implications for global change and the world economy. Another topic under investigation concerns fluids with a nonlinear viscous force relation. This happens, for example, in models for turbulent eddies and also in fluids with a special molecular structure such a blood or some polymers. The investigator seeks to bound the development of singularities (e.g. infinite energy spikes) in the motion of such fluids. Date: April 26, 2002