On the Entropy of Parabolic Allen-Cahn Equation

TL;DR

This paper introduces a new entropy concept for Radon measures related to the parabolic Allen-Cahn equation, proving its monotonicity and applying it to show unit density of limit measures under certain initial conditions.

Contribution

It defines a mean curvature flow entropy for Radon measures and establishes a new monotonicity formula for the parabolic Allen-Cahn equation in specific manifolds.

Findings

Monotonicity of the entropy for measures associated with the parabolic Allen-Cahn equation.

Small initial entropy ensures the limit measure has unit density over time.

New monotonicity formula applicable in manifolds with non-negative sectional curvature.

Abstract

We define a (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.