Developing Robust Techniques for the Analysis of Multiple-Scale Behaviors

University Of Arizona, Tucson AZ

Investigators

Abstract

The long term goal of this research project is to develop the mathematical tools needed for a robust and statistical description of multiple scale phenomena. Synthesizing ideas first developed for dynamical systems and methods for variational problems (Gamma-convergence), this project studies (1) elasticity of thin sheets, (2) stripe patterns in Rayleigh-Benard convection, (3) two-dimensional electrostatics and connections to Hele-Shaw flow and diffusion-limited aggregation, and (4) pattern formation in extended, periodically-forced systems. The insights gleaned from studying these particular problems will be used to address the general questions motivating this research. A significant problem in nonlinear partial differential equations and in the physics of extended systems is to understand and analyze phenomena on multiple spatial and temporal scales. This project is part of a long term research program to understand multiple scale behaviors with particular emphasis on non-convex variational problems (energy driven pattern formation). A simple example is the behavior of a crumpled sheet. All crumpled sheets "look" the same "statistically", but it is impossible to crumple two sheets of paper into identical configurations. This suggests many natural questions, including "What are the statistics that describe the global behavior of the system?" and "Is there a theory for these statistics?" Important questions of this type arise repeatedly in the study of physical, geophysical, and biological systems. This project aims to develop tools needed to answer such questions.