CAREER: Mathematics of Vorticity in Ginzburg-Landau Theory and Fluids

This project aims to establish rigorous qualitative behavior of problems arising in nonlinear partial differential equations and comprises three parts. The first part consists of a family of problems arising in the study of phase transition equations, including those that model superconductivity and superfluidity. When a certain parameter becomes asymptotically large, these materials form vortices, which are regions of high energy and spin. This part of the project studies the dynamical behavior of asymptotically large numbers of vortices, each of which contain an asymptotically large amount of energy. The second part of the project analyzes the nucleation and dynamical behavior of vortices in models for thin micromagnetic materials. Such problems arise in magnetic information storage devices. The analytical methods used in the first and second part of the project come from geometric measure theory and the calculus of variations. The third part of the research project entails an analysis of water waves over variable topographies. The analytical methods employed will include bifurcation theory, harmonic analysis, and spectral theory, along with numerical experiments. Tied to the three research areas are several educational projects that aim to train undergraduate and graduate students in analysis, along with the goal of raising interest in applied mathematics.

In fluid dynamics and materials science, vorticity roughly measures the local rotation of either a fluid or a phase function. In many important physical contexts, vorticity is a crucial component, and in many cases understanding the vorticity leads to understanding of the entire physical problem. The first two parts of the project study the qualitative behavior of vorticity in two physics problems of importance to materials science, superconductors and thin micromagnetic films. These materials are of increasing use in applications, such as in powerful magnets and information storage devices. The third part of the proposal centers on understanding the behavior of ideal fluids with free boundaries, which arise in the study of waves on the surface of the ocean or fast-moving streams. The equations that model these problems are complicated; however, since these problems arise with great frequency, their analysis is important. Educational and training objectives are clearly and closely tied to the research projects.