Mass conserving Allen–Cahn equation and volume preserving mean curvature flow

Abstract

We consider a mass conserving Allen–Cahn equation $u_t=\Delta u+{\epsilon }^{–2}(f(u)–\epsilon \lambda (t))$ in a bounded domain with no flux boundary condition, where $\epsilon \lambda (t)$ 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 $u^{\epsilon }$ with appropriate initial data tends, as $\epsilon \searrow 0$, 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 ${\gamma }_0$.