A convergence rate of the discrepancy for Allen-Cahn type equation

TL;DR

This paper introduces a modified Allen-Cahn equation (MAC) with a time-dependent parameter and Z-transform term, providing improved energy discrepancy estimates and establishing a connection to Brakke's motion as a weak mean curvature flow.

Contribution

The paper proposes a novel MAC equation with enhanced energy discrepancy estimates and demonstrates its relation to Brakke's motion, advancing the understanding of mean curvature flow modeling.

Findings

Improved estimate on the discrepant energy for the MAC.

Establishment of a connection between MAC and Brakke's motion.

Introduction of a time-dependent parameter and Z-transform in Allen-Cahn equation.

Abstract

This paper presents a new partial differential equation to build Brakke's motion, which is a weak notion of mean curvature flow. We call the equation a modified Allen-Cahn equation abbreviated to MAC. After that we introduce its benefit: An improved estimate on the discrepant energy that means the equipartition of the first energy. This equation is obtained by rendering the parameter in Allen-Cahn equation time-dependent and adding a term like Z-transform. Furthermore, we mention the existence of a Brakke's motion through our MAC.