Existence of weak solution for volume preserving mean curvature flow via phase field method

TL;DR

This paper proves the existence of weak solutions for volume-preserving mean curvature flow using a phase field approach, establishing key properties like monotonicity and density bounds.

Contribution

It introduces a novel phase field method to demonstrate the existence of weak solutions for volume-preserving mean curvature flow with nonlocal reaction diffusion equations.

Findings

Existence of weak solutions for volume-preserving mean curvature flow.

Monotonicity formula established for the reaction diffusion equation.

Density upper bounds proved for the phase field model.

Abstract

We study the phase field method for the volume preserving mean curvature flow. Given an initial $C^1$ hypersurface we proved the existence of the weak solution for the volume preserving mean curvature flow via the reaction diffusion equation with a nonlocal term. We also show the monotonicity formula and the density upper bound for the reaction diffusion equation.