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

Abstract

We consider a mass conserved Allen-Cahn equation $u_t=Δu+ \e^{-2} (f(u)-\eλ(t))$ in a bounded domain with no flux boundary condition, where $\eλ(t)$ 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 ${\gamma }_0$.

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