A Hilbert expansion method for the rigorous sharp interface limit of the generalized Cahn-Hilliard equation (Q2252223)

Summary: 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 \textit{E. A. Carlen} et al. [Arch. Ration. Mech. Anal. 178, No. 1, 1--55 (2005; Zbl 1076.76009)]. 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 [\textit{N. D. Alikakos} et al., Arch. Ration. Mech. Anal. 128, No. 2, 165--205 (1994; Zbl 0828.35105)].