Convergence of the Cahn-Hilliard equation to the Mullins-Sekerka problem in spherical symmetry

Abstract

We show that, as ε → 0, the solution of the Cahn-Hilliard equation ∂tφ - Δu = 0, u = - ε Δφ + 1/ε W′(φ) converges to a solution of the Mullins-Sekerka problem - Δu = 0 in each phase, V = -[∇u] · v; u = -cwK on the interface, where v denotes a normal, V the normal velocity and K the sum of principal curvatures of the interface, provided the solutions are radially symmetric. We use energy type estimates to show that the solution of the Cahn-Hilliard equation can be approximated by the well-known stationary wave solution that corresponds to the potential W. © 1996 Academic Press, Inc.