*Slow Motion of Gradient Flows*

**Maria Westdickenberg**

Georgia Institute of Technology

School of Mathematics

mariaatmath.gatech.edu

**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.