Convergence of a mass conserving Allen–Cahn equation whose Lagrange multiplier is nonlocal and local

Abstract

We consider the mass conserving Allen–Cahn equation proposed in [8]: the Lagrange multiplier which ensures the conservation of the mass contains not only nonlocal but also local effects (in contrast with [14]). As a parameter related to the thickness of a diffuse internal layer tends to zero, we perform formal asymptotic expansions of the solution. Then, equipped with this approximate solution, we rigorously prove the convergence to the volume preserving mean curvature flow, under the assumption that a classical solution of the latter exists. This requires a precise analysis of the error between the actual and the approximate Lagrange multipliers.