PDE Seminar Slow Motion of Gradient Flows
 Date

20060322 16:30 
20060322 17:30
 Place
 Faculty of Science Building #8 Room 309

Speaker/Organizer
 Maria Reznikoff （Department of Mathematics, Princeton University）

 Sometimes physical systems exhibit ``metastability,’’ in the sensethat states get drawn toward socalled metastable states and aretrapped near them for a very long time. A familiar example is theonedimensional Allen Cahn equation: initial data is drawnquickly to a ``multikink’’ 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 the1d Allen Cahn equation. The central ingredient is to establisha certain nonlinear energyenergydissipation relationship. Onebenefit of the method is that it gives a natural proof of the factthat exponential closeness to the multikink state is not onlypropagated, but also generated.This work is joint with Felix Otto, University of Bonn.
http://coe.math.sci.hokudai.ac.jp/sympo/pde/