Convergence of the Allen-Cahn Equation to multi-phase mean-curvature flow

TL;DR

This paper proves that solutions of the vector-valued Allen-Cahn Equation converge to a multi-phase mean-curvature flow, extending previous results to include external forces and volume constraints.

Contribution

It establishes a convergence result for vector-valued Allen-Cahn solutions to multi-phase mean-curvature flow, including variants with external forces and volume constraints.

Findings

Convergence of Allen-Cahn solutions to multi-phase mean-curvature flow.

Extension to equations with external forces.

Extension to equations with volume constraints.

Abstract

We present a convergence result for solutions of the vector-valued Allen-Cahn Equation. In the spirit of the work of Luckhaus and Sturzenhecker we establish convergence towards a distributional formulation of multi-phase mean-curvature flow using sets of finite perimeter. Like their result, ours relies on the assumption that the time-integrated energies of the approximations converge to those of the limit. Furthermore, we apply our proof to two variants of the equation, incorporating external forces and a volume constraint.