On the entropy of parabolic Allen–Cahn equation

Abstract

We define a local (mean curvature flow) entropy for Radon measures in ${\mathbb{R}}^n$ or in a compact manifold. Moreover, we prove a monotonicity formula of the entropy of the measures associated with the parabolic Allen–Cahn equations. If the ambient manifold is a compact manifold with non-negative sectional curvature and parallel Ricci curvature, this is a consequence of a new monotonicity formula for the parabolic Allen–Cahn equation. As an application, we show that when the entropy of the initial data is small enough (less than twice of the energy of the one-dimensional standing wave), the limit measure of the parabolic Allen–Cahn equation has unit density for all future time.