# 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 *ut* = Δ_u_ + 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 γ0 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 γ0.

## 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