2018/06/06 by Helmut Abels, Abels, Helmut, Maximilian Moser +1 · 1 citation
Computer Science · Materials Science · Mathematics · #35B25 #35B36 #35K57 #35R37 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Solidification and crystal growth phenomena
paper · pdf · doi:10.48550/arxiv.1806.02065
openalex publication_date 2018/06/06 · openalex created_date 2018/06/13 · openalex updated_date 2026/07/28
We consider the sharp interface limit of the Allen-Cahn equation with homogeneous Neumann boundary condition in a two-dimensional domain Ω, in the situation where an interface has developed and intersects ∂Ω. Here a parameter ε>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.