Mathematical Study of Ginzburg-Landau Asymptotics and the Stability of Fluids

This proposal aims to rigorously study several problems arising in nonlinear partial differential equations and is split into two parts. The first part consists of a family of problems coming from the study of phase transition equations. Such equations model several types of phenomena, including superconductors, superfluids, Bose-Einstein condensates, and liquid crystals. These models form interesting defects in singular asymptotic regimes, in which the equations simplify substantially. This simplification is useful for improved numerical techniques and qualitative understanding. The first part of the project studies the behavior of these defects in both static and time dependent situations with techniques developed from the calculus of variations and harmonic analysis. The second part of the proposals centers on examining the stability of free boundaries in fluid equations with nontrivial vorticity in the fluid. The equations that model such fluids have both geometrical and physical components. The primary aim of this part of the project is to study the stability of such fluids under perturbations. The techniques used to analyze these problems come from energy methods, spectral theory, and harmonic analysis.

Helium, when cooled to temperatures close to absolute zero, becomes a liquid with unusual characteristics. The liquid loses all internal friction, and if stirred, will rotate without end. Such liquids are called superfluids. If the stirring is too strong, little eddies (or vortices) form with quantized momentum, each of which are no longer in the superfluid state. Knowledge of the behavior and location of the vortices is the fundamental piece of information needed to describe the superfluid. The first part of this proposal studies properties of such vortices in superfluids, superconductors, and other related physics problems where quantum mechanical effects appear in large scales. Such physics problems are of increasing use in applications. The second part of the proposal centers on understanding the behavior of regular fluids with free boundaries, such as the surface of the ocean and the surface of stars. The equations that model these problems are complicated; however, since these problems arise with great frequency that their analysis is important