Atlanta, GA | Posted: January 1, 1997
Halil Mete Soner got his Ph.D. in applied mathematics from Brown University in 1986. He was a research associate at the Institute for Mathematics and Applied Sciences in Minneapolis, MN and is currently at Carnegie Mellon University. His interests are in nonlinear partial differential equations, viscosity solutions, stochastic optimal control and mathematical finance.
Mathematical models for phase transitions postulate that there are several regions corresponding to different phases, and they are separated by interfaces. In sharp interface models, interfaces are assumed to be hypersurfaces, while the diffused interface theories use an additional field variable called the order parameter to model the interface. Typically, in the second approach, the order parameter is modelled as a solution to a reaction-diffusion equation with a small parameter.
In this talk, I will discuss both the sharp and the diffused interface models for the supercooled solidification. These models are natural extensions of the classical Stefan problem, and the sharp interface model is the asymptotic limit of the one with diffused interfaces. I will outline the methods used in this asymptotic result and, then, I will show the connection between these models and some geometric evolution problems, such as the mean curvature flow.
The Ginzburg and Landau theory for superconductivity is a phenomological model that was proposed in the late '50s. Mathematically, it is a variational problem for the magnetic vector potential and the order parameter which takes values in the unit ball of the complex plane. The length of the order parameter is proportional to the density of superconducting electrons. The Euler-Lagrange equations are the Maxwell's system coupled to a reaction diffusion equation, and it is very similar to the phase field model for supercooled solidification, which will be discussed in the first talk.
After a historical introduction, I will discuss recent results of Bethuel, Brezis, Helein, Jerrard, Lin and Struwe on a related variational problem. I will then outline some asymptotic results for the evolutionary Ginzburg-Landau system that I have obtained with R. Jerrard of the University of Illinois. In one particular limit, normal regions reduce to several points called vortices, and the dynamics of these vortices is governed by a system of ordinary differential equations.