Convergence of the Allen-Cahn Equation to the Mean Curvature Flow with $90^\circ$-Contact Angle in 2D

TL;DR

This paper proves that solutions of the Allen-Cahn equation in 2D converge to mean curvature flow with a 90-degree contact angle at the boundary as the interface thickness parameter approaches zero, under certain smoothness assumptions.

Contribution

It rigorously establishes the convergence of the Allen-Cahn equation to mean curvature flow with a contact angle condition in 2D, using asymptotic expansions and spectral estimates.

Findings

Convergence of Allen-Cahn to mean curvature flow with contact angle

Construction of approximate solutions via asymptotic expansions

Spectral estimates for the linearized operator

Abstract

We consider the sharp interface limit of the Allen-Cahn equation with homogeneous Neumann boundary condition in a two-dimensional domain $\Omega$, in the situation where an interface has developed and intersects $\partial\Omega$. Here a parameter $\varepsilon>0$ in the equation, which is related to the thickness of the diffuse interface, is sent to zero. The limit problem is given by mean curvature flow with a $90$\textdegree-contact angle condition and convergence using strong norms is shown for small times. Here we assume that a smooth solution to this limit problem exists on $[0,T]$ for some $T>0$ and that it can be parametrized suitably. With the aid of asymptotic expansions we construct an approximate solution for the Allen-Cahn equation and estimate the difference of the exact and approximate solution with the aid of a spectral estimate for the linearized Allen-Cahn operator.