Randomness in Fluids and Waves

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

NSF Award Abstract - DMS-0203938 Mathematical Sciences: Randomness in Fluids and Waves Abstract 0203938 Bronski Fluid mechanics is an important subject that plays a central role in modern applied mathematics. Many phenomena in fluids, such as turbulence and intermittency, are still understood poorly even at the physical level, and as such provide a rich source of mathematical problems. In this project we will study mathematically the transport properties of a randomly stirred fluid. In earlier work, the principal investigator and collaborators were able to deduce a great deal of information about the transport properties and induced intermittency in a particular model of fluid flow, using ideas of asymptotic analysis, semiclassical eigenvalue problems, and probability. In current work we extend these ideas to more complex flow fields, where exact solution formulae are not available. In addition to the above techniques, we intend to make use of the Donsker-Varadhan theory, giving large deviations principles for quantities defined via Feynman-Kac integrals. The goal of this project is to bridge the gap between physical understanding and mathematical proof in the area of fluid transport. At present only a few very simple models for the transport of a passive quantity, such as a dye, by a turbulent fluid are understood in a rigorous mathematical sense. By extending the mathematical understanding to more complex models, we will clarify the question of what properties these simple models do and do not share with the more realistic flows of interest to scientists and engineers. We also hope to expand the physical understanding of transport properties, which is still far from complete.