A Hilbert expansion method for the rigorous sharp interface limit of the generalized Cahn–Hilliard equation

Abstract

We consider Cahn–Hilliard equations with external forcing terms. Energy decreasing and mass conservation might not hold. We show that level surfaces of the solutions of such generalized Cahn–Hilliard equations tend to the solutions of a moving boundary problem under the assumption that classical solution of the latter exists. Our strategy is to construct approximate solutions of the generalized Cahn–Hilliard equation by the Hilbert expansion method used in kinetic theory and proposed for the standard Cahn–Hilliard equation, by Carlen, Carvalho and Orlandi, [14]. The constructed approximate solutions allow to derive rigorously the sharp interface limit of the generalized Cahn–Hilliard equations and higher order corrections to the limiting motion. We then estimate the difference between the true solutions and the approximate solutions by spectral analysis, as in [1].