Convergence of the Allen-Cahn equation with transport term in a bounded domain

Abstract

We study the Allen-Cahn equation with respect to a transport term in a bounded domain. We prove that the limit interface is the mean curvature flow with the transport term, given the condition that the energy is uniformly bounded with respect to time. Using this result, we show the existence of the mean curvature flow with a gradient vector field as the transport term.