Global existence and monotonicity formula for volume preserving mean curvature flow (Regularity, singularity and long time behavior for partial differential equations with conservation law)

Abstract

In this note, the global existence of the weak solution for the volume preserving mean curvature flow is considered. We compare two phase field methods for the volume preserving mean curvature flow from the viewpoint of L2-estimate of the mean curvature, which is the key estimate of the existence theorem. We also study the monotonicity formula for the volume preserving mean curvature flow.