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

Matthieu Alfaro (1) , Pierre Alifrangis (1)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier

We consider the mass conserving Allen-Cahn equation proposed in \cite{Bra-Bre}: the Lagrange multiplier which ensures the conservation of the mass contains not only nonlocal but also local effects (in contrast with \cite{Che-Hil-Log}). As a parameter related to the thickness of a diffuse internal layer tends to zero, we perform formal asymptotic expansions of the solutions. Then, equipped with these approximate solutions, we rigorously prove the convergence to the volume preserving mean curvature flow, under the assumption that classical solutions of the latter exist. This requires a precise analysis of the error between the actual and the approximate Lagrange multipliers.

Domaines: Equations aux dérivées partielles [math.AP]

Matthieu Alfaro, Pierre Alifrangis. Convergence of a mass conserving Allen-Cahn equation whose Lagrange multiplier is nonlocal and local. Interfaces and Free Boundaries : Mathematical Analysis, Computation and Applications, 2014, 16 (2), pp.243-268. ⟨10.4171/IFB/319⟩. ⟨hal-00800909⟩