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

TL;DR

This paper proves that the Allen-Cahn equation with a transport term converges to a mean curvature flow influenced by the transport, under bounded energy conditions, and demonstrates existence of such flows with gradient vector fields.

Contribution

It establishes the convergence of the Allen-Cahn equation with transport to mean curvature flow and shows existence results for flows with gradient vector fields as transport.

Findings

Limit interface is the mean curvature flow with transport

Existence of mean curvature flow with gradient vector field as transport

Energy remains uniformly bounded over time

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.