PDE Seminar Slow Motion of Gradient Flows

2006-03-22 16:30 - 2006-03-22 17:30
Faculty of Science Building #8 Room 309
Maria Reznikoff (Department of Mathematics, Princeton University)
Sometimes physical systems exhibit ``metastability,’’ in the sensethat states get drawn toward so--called metastable states and aretrapped near them for a very long time. A familiar example is theone--dimensional Allen Cahn equation: initial data is drawnquickly to a ``multi--kink’’ state and the subsequent evolution isexponentially slow. The slow coarsening has been analyzed by Carr\& Pego, Fusco \& Hale, Bronsard \& Kohn, and X. Chen.In general, what causes metastability? Our main idea is to convertinformation about the energy landscape (statics) into informationabout the coarsening rate (dynamics). We give sufficientconditions for a gradient flow system to exhibit metastability. Wethen apply this abstract framework to give a new analysis of the1--d Allen Cahn equation. The central ingredient is to establisha certain nonlinear energy--energy--dissipation relationship. Onebenefit of the method is that it gives a natural proof of the factthat exponential closeness to the multi--kink state is not onlypropagated, but also generated.This work is joint with Felix Otto, University of Bonn.