Back to Departmental Colloquium: Spring 2006
Departmental Colloquium
Date: Friday, Feb 17, 2006
Time: 4:15PM
Location: JWB 335
Maria Reznikoff
Princeton
Title |
Slow Motion of Gradient Flows |
Abstract |
Sometimes physical systems exhibit metastability,'' in the sense that states get drawn toward so--called metastable states and are trapped near them for a very long time. A familiar example is the one--dimensional Allen Cahn equation: initial data is drawn quickly to a multi–kink’’ state and the subsequent evolution is exponentially slow. The slow coarsening has been analyzed by Carr & Pego, Fusco & Hale, and Bronsard & Kohn. In general, what causes metastability? Our main idea is to convert information about the energy landscape (statics) into information about the coarsening rate (dynamics). We give sufficient conditions for a gradient flow system to exhibit metastability. We then apply this abstract framework to give a new analysis of the 1–d Allen Cahn equation. The central ingredient is to establish a certain nonlinear energy–energy–dissipation relationship. One benefit of the method is that it shows that exponential closeness to the multi–kink state is not only propagated, but also generated. This work is joint with Felix Otto, University of Bonn. |
Mathematical Biology
Data Science and Machine Learning
Applied Mathematics