Existence of Weak Solution for Volume-Preserving Mean Curvature Flow via Phase Field Method

We study the phase field method for the volume-preserving mean curvature flow. Given an initial data which is a measure-theoretic boundary of a Caccioppoli set with a suitable density bound, we prove 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 for the reaction diffusion equation.