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

作者：Antonopoulou Dimitra C^*; Karali Georgia; Orlandi Enza

来源：INTERFACES AND FREE BOUNDARIES, 2014, 16(1): 65-104.

DOI：10.4171/IFB/314

• Cahn-Hilliard equation
• forcing
• sharp interface limit
• Hilbert expansion

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].

• 出版日期2014