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 \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.

References in corpus (1)

• The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system

Cited by in corpus (4)

• Unconditionally stable exponential time differencing schemes for the mass-conserving Allen-Cahn equation with nonlocal and local effects
• Global solutions to the volume-preserving mean-curvature flow
• Analysis of a time-discrete scheme for the Navier-Stokes/Allen-Cahn model
• Convergence of Landau-Lifshitz equation to multi-phase Brakke's mean curvature flow