JournalsifbVol. 12, No. 4pp. 527–549

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