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

We show that, as epsilon --> 0, the solution of the Cahn-Hilliard equation partial derivative(t) phi - Delta u = 0, u = -epsilon Delta phi + 1/epsilon W'(phi) converges to a solution of the Mullins-Sekerka problem -Delta u=0 in each phase, V = -[del u] . nu; u = -c(w)K on the interface, where nu 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. (C) 1996 Academic Press, Inc.