Mass conserved Allen-Cahn equation and volume preserving mean curvature flow

TL;DR

This paper analyzes a mass-conserved Allen-Cahn equation and shows that, as the parameter approaches zero, solutions converge to a sharp interface evolving according to volume-preserving mean curvature flow.

Contribution

It establishes the connection between the mass-conserved Allen-Cahn equation and volume-preserving mean curvature flow in a rigorous asymptotic limit.

Findings

Solutions approach a sharp interface as epsilon tends to zero.

The interface evolves according to volume-preserving mean curvature flow.

The limit takes only two values separated by the evolving hypersurface.

Abstract

We consider a mass conserved Allen-Cahn equation $u_t=\Delta u+ \e^{-2} (f(u)-\e\lambda(t))$ in a bounded domain with no flux boundary condition, where $\e\lambda(t)$ is the average of $f(u(\cdot,t))$ and $-f$ is the derivative of a double equal well potential. Given a smooth hypersurface $\gamma_0$ contained in the domain, we show that the solution $u^\e$ with appropriate initial data approaches, as $\e\searrow0$, to a limit which takes only two values, with the jump occurring at the hypersurface obtained from the volume preserving mean curvature flow starting from $\gamma_0$.