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

### Xinfu Chen

University of Pittsburgh, United States### Danielle Hilhorst

Université Paris-Sud, Orsay, France### Elisabeth Logak

Université de Cergy-Pontoise, France

## Abstract

We consider a mass conserving Allen–Cahn equation $u_{t}=Δu+ε_{–2}(f(u)–ελ(t))$ in a bounded domain with no flux boundary condition, where $ελ(t)$ is the average of $f(u(⋅,t))$ and $–f$ is the derivative of a double equal well potential. Given a smooth hypersurface $γ_{0}$ contained in the domain, we show that the solution $u_{ε}$ with appropriate initial data tends, as $ε↘0$, 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 $γ_{0}$.

## Cite this article

Xinfu Chen, Danielle Hilhorst, Elisabeth Logak, Mass conserving Allen–Cahn equation and volume preserving mean curvature flow. Interfaces Free Bound. 12 (2010), no. 4, pp. 527–549

DOI 10.4171/IFB/244