Slow motion of gradient flows

We present sufficient conditions on an energy landscape in order for the associated gradient flow to exhibit slow motion or “dynamic metastability.” The first condition is a weak form of convexity transverse to the so-called slow manifold, N. The second condition is that the energy restricted to N is Lipschitz with a constant δ≪1. One feature of the abstract result that makes it of broader interest is that it does not rely on maximum principles.As an application, we give a new proof of the exponentially slow motion of transition layers in the one-dimensional Allen–Cahn equation. The analysis is more nonlinear than previous work: It relies on the nonlinear convexity condition or “energy–energy-dissipation inequality.” (Although we do use the maximum principle for convenience in the application, we believe it may be removed with additional work.) Our result demonstrates that a broad class of initial data relaxes with an exponential rate into a δ-neighborhood of the slow manifold, where it is then trapped for an exponentially long time.