Title: Slow Motion of Gradient Flows
Speaker: Professor Maria G. Reznikof
Speaker Info: Princeton University
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.Date: Thursday, January 12, 2006
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.