Skip to content..
Carnegie Mellon Center for Nonlinear Analysis


Slow Motion of Gradient Flows

Maria Westdickenberg
Georgia Institute of Technology
School of Mathematics

Abstract: In metastable systems, states tend to be trapped in "valleys" of the energy landscape for a very long time even though they are not near any stable state. 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 and Pego; Fusco and Hale; Bronsard and Kohn; and X. Chen. The one-dimensional Cahn Hilliard equation is a conservative system that displays similar behavior.

In general, what causes metastability in a gradient flow system? 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 gives a natural proof of the fact that exponential closeness to the multi-kink state is not only propagated, but also generated. In other words, the initial data does not need to be close to the slow manifold. We will also discuss work in progress on the Cahn Hilliard equation.

This work is joint with Felix Otto, University of Bonn.