  In this paper, we construct a family of integral varifolds, which is a global
weak solution to the volume preserving mean curvature flow in the sense of
$L^2$-flow. This flow is also a distributional BV-solution for a short time,
when the perimeter of the initial data is sufficiently close to that of ball
with the same volume. To construct the flow, we use the Allen--Cahn equation
with non-local term motivated by studies of Mugnai, Seis, and Spadaro, and Kim
and Kwon. We prove the convergence of the solution for the Allen--Cahn equation
to the family of integral varifolds with only natural assumptions for the
initial data.
